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The field of thermal ,system design and analysis continues to develop. The number of workers is gro\\"ing, technical papers appear in greater numbers, and new textbooks are _being written. . . The major objective of this third ,edition is tQ organize so.me of the new approaches that are now ,avaHabJe and to provide more flexibibty to - instructors who use Design of Thermal Systems as a texL The changes to the twelve chapters of the second edition are modest and mainly constitute the inclusion of some additional end-of-chapter problems. Chapters 13 tPJ"Dugh \ 19, ho\v~yer, are all new_ One possible use of th~ text is to cover the first twelve chapters in an advanced-level undergraduate course and the remaining seven chapters as a graduate COjJfse. In some engineering schools students already have some J.01d of optimiiatlOn course prior to ' taking the' thermal design course. For those classes certai)1 chapters of the fIrst hvelve (usually the ones on search methods, dynamic prograrrul:ring, and linear proirarruning) can be omi.tted and material can be supplemented from the, new seven chapters. Several of the new chapters are exten~ions of the introductions offered in the lust twelve chapters, especially mathematical moqeling, steady~state system simulation', and search methods. Chapter 14 addresses $.Orne of the challenges that arise when simulating Jarge, thennal systems. New material appears in Chapter 15 on dynamic be'h~yior, in Chapter 18 which introduces calculus of variations as a companion to dynamic programming, and in Chapter 19 on probabilistic approaches to design, which is exploratory. '., The author thanks colleagues both at the University of nlinois at Urbana-Champaign and at other engineering schools for continued input and suggestions during [he past several years on how to keep the ,book fresh. ~qraw2Hjl1 would rus,9. like to thank the foHowing ' reviewers for their useful comments: John R. Biddle, California State Polytechnic.
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PREFACE TO TI-IE THIRD EDmON
X
University, Po'mona; Theodore F. ' Smith,.:1be University 0 '- 0 a; Edward Q~ Stoffel" CalifonUa State Polyt~chnic .Uriivers~ty; San Luis Obispo; John A. Tichy, ,Rensselaer Polytechnic Institute;, Daniel T. Va1~I! . e, ~larkson ,College; ?lld 'William' J. 'Wepfet" qe'
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The title, Design of Thermal Systems, refie,cts the three concepts embodied in this book: design, thennal~ and "systems. '
DESIGN A frequ~nt product of the erig?neer's efforts is a drawing, a set' of calcu~a t!ons, or a report that is an abstraction an? .descriptio~ of hardware.' Within ' . ,epgineering education, thl? cookbook approach to design, often practiced during the 1940s, discredited tpe ' design effort so that many engineering schools dropped design courses 'from their curricula in the 1950s. But now design has returned. This reemergence is Dot a re~apse to the earlier procedures; design is reappearing as a cre"ative and highly technica1 , ~ctivity.
THERMAL Within many mechanical engineering curricula the tenn design J,s limited to machine design. In order to compensate for this frequent lack of recognition of thermal design some special emphasis on this subject for the next few years is warranted. The designation thermal implies calculations and activities 'based on principles of thermodynamics, heat transfer, and fluid mechanics. , ,The, hard,ware associated '1'ith therm~. systems iqcludes fans) pumps, compressors, engines, expanders, ,'turbines, heat and mass exchangers, and reactors .. all interconnected with some form of conduits. ' generally, the .,working substances are fluids. These types 9f systems appear in such industries as power generatio'n, electric'and gas utilities, refrigeration, air conditioning and heating. and in the food, chemical, and process industries . I
. '
xiii
xiv
PREFACE TO nIE FIRST EDITION
SYSTEMS Engineering e¢lucation is predominantly pro,cess )orielZted, while engineering practice is predornioantly system 'orien(eil. . cO,urses of study in e;ngi- " " _' neering .proyide the student wi~ effec~ve exposure to such processes as the flow of a compressible fluid through' a nozzle and the .bel;l3rytor of " ,~hydrodynamic and theI"ID:al bou,ndary l~y~rs ,at solid,$q.rfaGes,._lhe ptacti~ing " . en'g ineer, ,however) is lik~Iy be confronted with, a task ~ucp' as. desigmng , : ~an: _~eG,o,n.~c:-:.sy.st~,Itl __!p~t ,}"~ceiv~~ p,~t~~ g~ _from a. pipeline- aijc;l .,stores , it underground '.fo~ )~fef~'~sag~'~ ·- 'Th~r~,~~~~:' a~- big~/iap.:l>.e~,een:Jmoyw~~~qg.~:,;:,~(,.j -,::,:;;" .~'S' ~': , inqi~id~hl proces~es 4tld rl1e ,~,~tegraticm- ~f these prQc~s.s~s in ~n.·;engiriee~g. , -: '~:~. : " ,. ~~', enterprise. Clos~g the gap should ~ot b~ accomplished by diminishing the empha- . si~ ' on pro~esses. A faulty .~no\vledge,' of fi?1dainentals may result in subs~:' quent failure of the ~ystem. But -within a university enviroJ!illent, it is beneficial for future engineers to begin thinking in te~s of. systems, Another re~sonfor more emphasis on systems in the llnjversity envirQnment, in additio,q to, influencing the (hought patterns of students, is that there are sqme technjques-such as simulation and optinrizatjo~-which only re~ently have been applied to thennal systems '. 'These are usefui t601s and the graduate should have s6me facility with them. ' , While the availability of procedures of simulation and optimization js not a new situation, the practical application of these procedures has only recently become widespread because . of ,the- availability of. the: djgital computer. Heretofore, the lirriltation of time, did not permit hand calcula'tiqns: for example, of an pptirnization of a function that was dependent upon dO,zens or hundreds of independent variables. This meant that in designing systems consisting of dozens or hundreds of comp
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OD,T LINE OF THIS BOOK' The goal of this book ' is the design of optimum thennal systems. Chapters 6 through 11 cover topics and specific procedures in optimization. After Chap. 6 explains the typical statement of the optinlization problem and illustrates how this staten1enl derives from the physical situation, the chap-
ters chat follow explore optimization procedure,s such as calculus m~thods, search methods, geometric programming, dynamic programming, and linear programming. All these merhods have applicabiH ty' 10 nlany other types' of problems besides thennal ones and, ·in this sense, are general. On 'the other hand, the applications are chosen fr~m the therma1 field to emphasize the opportunity for optimization in this cJass of problems .
..
PREFACE TO THE F1RST EDITION
If the .engineer inLmediately sets out to try to optimize a rnoderately , c.omplex thermal system he 'is soon struckl lDy the need for predicting the performance of that sy~tem. gi veil CeI1aiIl inpu t conditions 3J."1d performance charac'telistics of' components . This is the process of systein sirn.ulatiolL System simulation' fl9t. only rp.ay be a, step the optirmzation' -but, ·may hav'e a usefuJne~s iI! its' O'1}JI;1 'right.' ,A-,sysren1 may.J~l~, ..gesigrted on. the ' . --,' ··bq.sis of some ..IDaXlrllUm;:-1oad",conditiori\'-but'-maybper~t~' g5·'·: perd~rit'dttl1e··~f~ tiille ' 9-t 1es.s'::t)i~in-maxjmiiiQ. load. System' simulitt-ioll.periuirs an exar.nination of the. ope~a"tini' condi'ti~'n's that m~y pinpoint pos_s5hle operatiI;tg and c~ntrol problems at non-design c.onditions. ' . . Since. system simulation and pptimjzation on .any but· the simplest problems .4?Ie complex. operations, the executio~~ of the ptob~en1 must be performed on ·a computer. When using .~. computer, the equation forin. of representation. of the performance of corpp~nents and expression of .prop- . erties of ,su-bstances is much more' convenie.nt tabular or graphical representations. Chapter 4 on mathe~atical mo~eling: presents sOTI?-e techniques for equation development for the case where there is and also where '. the;re is not some insight into the relationships based in thennallaws. Chapter J~ .on econom.ics~ is appropriate because engineering design , and economics are inseparable. and because ,a frequent cnterion for optimization is [he economic one. Chapter 2, on workable systems, attempts to convey one simple but imponant dis(inc~ion =-the difference be~een the . design process that resui ts. in a workable system in contrast to an optimum. system. The first chapter on engineering design emphasizes ·the importance . of design in an engineering under1aking. The appendix includes problem statements of sevemJ. comprehensive projects which may run as part-time assignments during an entire tenn. These tenn projects are industrially oriented b~[ require application of some of the topics explained in the text. . . Th~ ~udience for which this book \vas written includes senior or fiisryear graduate ·students in mechanical or chemical engineering, or practicing engineers in the thermal field. The background assumed is a knowledge of thermodynamics heat transfer, fluid mechanics. and an awareness of the performance charactens[ics of such thermal 'equipment as heat exchangers, pumps, and compressors. The now generally accepted facility of engineers' to do basic digi tal computer programming is also a requiremen..t. 9
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ACKNOWLEDGI\1ENTS (, Thennal system design is gradually emerging as an identifiable discipljne. ·.. Special recognition should' be given to the program coordinated by the University of Michigan on Computers in Engineering Design Education, which in 1966 .clearly delineated topics and defined directions that have .since..P'oYed to be productive. Acknowledgment should be given to activities .. ~.
"A.-vi"
PREFACE TO THE FIRST EDITION -
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within the chemical engineering field for developments .that are clos.ely related, and -in some cases identical, to those in thy theIDlal stem of ll1echanical . I ..engmeenng. _ . . M~y faculty members during the past five years have arrived, often ' .ind~pendently, at the same 'conclusion as the author: the rime .is -oppprtune' . '. for .dever~pments in th~rmal-. q.~.s.ign. :Mjap.y ,of th.ese. fa<:;]Jlty memb~rs have . .-:).,. . -. ':share<;i some of.th~ir ~xperiences in the theilllal design·section'
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This ,.manus'cn p~ ,is'- the'" '::. third~: id{tio~·~· (jt.~· t~:it~ ~in~ftte~a)·:: :~t:~ed~~;:iD:rtht:·:. ~'.~~ :.'~'.',. ': : ' .Design of Thermal Systems c611rse " at,th~ ,University 'ofTIlinois"at 'Urbana--- '-' .-Champaigll7.-I thank the students-Vl·ho have .worked with in this :course for their suggestions. for improvement of the manuscript. The. second edition · , . . I was an at!:factively printed ~ook1et prepared by my D~partment Publication Office, George Morris, Pirector; June J(empka and Dianne Merridith, typists; and Don Anderson, Bruce Br~ckenfeld, 'and Paul Stoecker, draftsmen. Special thanks are due to the Engine~Png Department of Alpoca Chemicals .Corporation, Chicago, for their intert:st jn ,engineering·,.education and for . their concrete evidence of this interest shown by printing the second edition. Competent colleagues are invaluable as sounding boards for ideas and a~ contributors ~f ideas of their own. Professor L. E. Doyle offered suggestions on the econoDljcs chapter and Prof C. O. Pedersen, a coworker in ·the development of the ·thermal systems prqgrarn "at the Unive'rsity of illiriois at ·Urbana-Champaign, prov]d.ed .a dvice at many stages . Mr. Donald -R. \'\tin and a class of architectural engineering students at Pennsylvania State University class-tested '~he manuscript and pf9vided valuable sugges. tions from the point of view of a user of the book. Beneficial comment~ and criticisms aJso came from the Newark College of Engineering, where Prof. 'Eugene Stamper and a group of students tested the manuscript in one of their classes. Professor Jack P. Holman of Southern Methodist University, -consulting editor of ~cGraw-Hil1 Book Company, supplied perceptive .con1ments both in. terms of pedagogy as well as in the technical feattlfeS of thennal systems. The "illustrations in this book 'Were prepared by George Morris of .Champaign~ IlJinojs. By being the people that they are\ my wife Pat and children Paul, Janet. and Anita have made the work on this book, as well as anyt..hing else 1 dO'1 seem worthwhile.
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W. F. Stoecker
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Typical ,professional activities of engineers include sales, construction, . research, development, and design. Design will be our special concern in this book ..The immedia,te 1>roduct of the design process ·is a report, a set of calculations, andJor a drawing that are abstractions 'of hardware. -T he subject of the design may be a process,' an elem,e nt or component of a larger assembly, or an entire system. OUf emphasis will be system design, .w here a system is defined as a collection of components with interrelated performance. Even this definition often needs interpretation, becau-se a large system sometimes includes s·ubsystems. Furthe nn ore , we shall progressively focus on thermal systems, where fluids and energy in the form of heat anel work are conveyed and converted. Before adjusting this focus, howe'ver, this chapter will examine the larger picture fn~o which the technical engineering activ-
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··implying that engineering plays a aecisive role but also dovetails with other considerations. Engineering undertakings include a wide variety of commercial and industrial enterprises as well as municipally state-. and federally sponsored projects. I
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The advantage of analyzing the de<;:ision process, especjal~y in com~ plex undertakings, is that it ,l eads ,to a more logi~a1 coordinati0D of the ~aJ;ly individual efforts consti~ting the entire venture. The flow diagram in Fig. 1-1 shows ·typical steps followed in the conception, evaluation, .and execu- , tion of ,the plan. The rectangular boxes, which indicate actions, may'represent consid,erable effort and e,xpenditures on large projects. The diamond boxe~ represent decisions, e.g., whether to c;oIitinue the project or to mop , J it. " The technical engineerlng occurs mostly in activities 5 and 7, product or system design and research an~ development. Ljttle will be said in this chapter about product or system design because it will be 'studied in the chapters to follow. The'tlow diagra!ll shows only how this design procedure fits into the larger pattern of the '·undertaking. The individual nondesign acri virie.s will be discussed next.
1.3 NEED OR OPPORTUNITY (STEP 1) Step 1 in the flow diagram of Fig. 1-1 is to define the n~ed or, .opportunity. It may seem easy to state the need or opportunity, but it is Dot always a simple task. For example, the officials of a city.may suppose that their need is to enlarge the reservoir so that it' can store a larger quantity of water for municipal purposes. The officials may not have specified the actual need but instead may have leaped to one possible solution. Perhaps the need would better have been stated as a low water reserve d'uring certain times of the year. Enlargem'ent of the reservoir might be one possible solution, but other sol utions might b~ to restrict the consumption of water and to se6-k other soutCes such as wells. Sometime's· possible-solutions are precluded by not stating the need properly at the begmning. ,The word "opportunity" has po"sitjve connotations, whereas uneed" suggests a defensi ve action. Sometimos the two cannot be distinguished. For -example, an industrial firm may recognize a new product as an opportunity, but if the company does not then expand its line of prpducts, business is likely to decline. Thus the introct'uction of a new product is also a need.
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within the organization. A new product may be developed intentionally Qf . ,a ccjperitally. Sometimes. a new use .of all eXlsting proQ.uct· can be found by making a slight modification of it. An orgahization .maY know how to" manufacture ,a g~JJ?my, sticky sup.stance and a.,ssign ' to the research ,and developmefl;t department the task of fiIid,ing som~. use Jar it. . . ''':'~J .' Of interest to us· a~ the ni~IPent is ~e nee~ or ()PP9rWni..ty th.a~ r~q~~es :.. " ' engineering design at a subsequent stage:" ' . , . . ' .
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mterion of sllcc,ess is' 'showillg a ,profit~" providing a certairi" rate of telurtr .on the -investment. In public ' projects . the criterion of success is the d~gree tq which Q-le need' is satisfied in relatipn . to tile cos~~ monetary or otherwise: ' . . In a ,profit-and-loss ecopbmy, 0e expected earning power of a ' proposed conimercial project is a .dqmjn~tiDg influence on the ·decision pro- . ceed with the project. Strict monetary concerns are always tempered, bow~ ever: by. hunlan~ social, and political considerations to a .greater or ~esser degree. In other words, a price tag is placed, on the' nonmonetary factors . ..A. factoD' may be located' at- a more remote site at a penalty in !be form of , transportation costs so that'its atmospheric. pollution or noise affects fewer people. As an alternative, the plant may spend a lot on superior pollut~on . c0!ltIo] in order to be a good neighbor to the surroundIng community . . ' So:metlmes' a finn will desjgn and ' manufacture a product· that offers ·Jj'a le opportunity for profit sin1ply to round out a line of products. The a\'aiI~biliry of this product, product A, peMits the sales force to say to a prospective customer, "Yes, we can sell you product A, but'we recommend product 'B 7~' which is a more profitable item in the company's line an~ may acrually be superior to product A. ," . Often a decjsion, particularly in an emergency, appears outside the realm of economics. If a boiler providing steam for beating a rental office building fails, the decjsion \vhether to repair or replace the boiler inay seem, to be outside the realm of econQmics. The question can still be considered an economic one. however, the penalty for nor executing the project being nn overpov';'ering loss. In· c'ommercial ente'r prises the' usual i.e.~
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1.5' PROBABILITY QF , SUCC~~S (STEP 3) Plans and designs are a]ways dirdcted roward the future, for which only probabiI ity not cenajnty, is applicable: There is no absolute assurance Chl;H the pJanr will n)eet the success. criteria discussed in Sec .. 1.4, 'Only a - li.k;]lihood or probability thai it wilJ po so. The mention of probability suggests [he nonna} distribution cUrY"e (Fig. 1·2), an excel~en[ starting point for expressing uncertainty in the decisionmaking l?J:Oces . s The significance of rhe distribution curve lies particolarly I
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24
ENGINEERING DESIGN
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been analyzed more carefuUy C).nd mar.keting studies have been conducted_ more thoroughly. _ . I L The. probab il ity distribution curves at Hvo other stages~ after construction and after 1 year.of operation, show progressively greater q~grees of . confidence. in·.the rate of retuill;after a 5-ye~r life. A.fier:.5 . ye~rs; th~ rate of return- is kllQWn exactlY1 and th.e . :prob)abi1ifY;,distiibut~dn·"c~rve - degeD.e~tes- .>-,"-' into' 'a: c urve-.. that:- is' infinites tmally' ··ttli·riw·and'::illfini-tely. .:;hi gh~,~:;' -;,',;\ .:~:o ;'>:->.:~ ,-.. :~.,:-:~:.~;;. :'\.~.':. -~h¢ re¢9ghition th.~~· pr:edj.¢.ti9~ :pf:fjJ.Jilte ?-eli.~vi6r is "rioi :deterrmriisii.c!t·" so that only one ~Set of events or cOD:ditions \vill prevail, has spawned a ne\v probabilistic approach ' to design (see the addirional readings p-~ the e~d of the chapter and Chapter 19). One of the activities of this new study is .that" of quantifying the curves sho\vn in Fig_ 1-4. It is valuable for the .decision maker to know not .only the most likely v3Jue of the rerum on investrTIent . but also whether there i$' a high- or low.probability of achieving this mO${ _ likely value~ "
1.6. I\tIARKET .ANALYSIS (STEP 4) If the undertaking is one in which a product or service' must eventually be sold or leased to customers, there mus! be some indication of favorable reaction by the potential consumer. An ideal fonn of the information provided by a' market ana-lysis would- be a set of cu[\(es like tl:ose in Fig. l~? With an increase in .price, the potential volume of sales decreases until - such a high price is reached that no sales can be made. The sales-volume _ to price relationship affects the· size of the plant or process because the unit -price is often lo\ver in a large plant. For this reason, (h.e marketing and.plant capabilities must be evaluated in conjunction with each other.
/
High sales and advertising effort
~ ~------------------------------~------~----~~
Prier: f1GURE;M~_
End resulTof-a market analysis.
8
DESIGN OF THERMAL SYSTEMS
Because the sales and advertising effort influences the!"Volume of sales " for a given price, a family of curves -is'expected. Since a cost is associated' :'~,:, with the sales and advertising effort,sib~e a continuous increase of ,', this effort resuits ~n dirninishffig improvement in sales there eXls'ts an " " optimum level of sales and' adv~rtising effort. A marketing plan, should ,:' ' emerge 'simultaneously \vith the ,tec~cal plans Jor the undertaking~' ,
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1.8 RESEARCH AND DEVELOPl\1ENT (STEP'. 7) If the product or process is one new to the organization, the results from research and qevelopment (R&D) may be an important input' to the decision 'process. Research efforts may provIde the origin or improvement of the " basic idea, and development ,work may supply . \vorking mo_dels or ,a pilot pJant, dep~nding upon the natl.l!e of the undertaking. Placing R&D in a late stage of decision' ~aking, as was done in Fig. 1-1, suggests that an idea originates somewhere else in the organization or in the field and eventually is placed at the doorstep of R&D for transfonnation into a workable idea. The possibility of the idea's originating in the research group should also be exploited .and is indicated by the dashed line in Fig. 1-},. Rese:arch people often learn of new ideas in other'fields which might be applied to their--Q\vn activity.
1.9 ITERATIONS The loop in Fig. 1-1 emphasizes that the decision-making process involves many iterations. Each pass through the loop improves the amount and tbe quality of infQtmation and data., Eventual]y la point lS reac~ed" ,where final decisions are made regardi~g the de~ignt prod\lction, and marketing of the product. The substance that circulates through ~s flow diagram
is infonnatio,n which ' may be in. the form of reports and ~onyersatjons a'nd f!1ay be both verbal and pictorial. The iterations are ' accompl ished by c6mmunicatron between people, an'd this communication is interspersed by t
go-~r-no-go
decisions.
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ENGINEERL"IG DESIGN
1.10 OPTI1VIIZA.TIOI\T
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9
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The 1low diagr:ln1 of Fig. 1-1 termipates with1the constnJction or begin~ ing of lnanufacture of a product. service. Actually lli.~other stage takes over . at thi:s point, which seeks to optimize rh~ ope~ation of- a given facil.ity. The facility was designed on the ba~is , of.qc~ain,9-~sig.r:t p~ram~Fer5. which almost jnevit~bly cha.11ge by the tirne the facility )s iri-'optration~"rr(e ,ilext ,challeng'e, :
or
. ,~thell ~ is": to··'op~.rate ihe-1~cility'· iri':'rJ1e-'bes't-~p'6s,s'i5ret'fri'a.n.:ne';:' ip-- 'tii:e~ Jigl11:~6..r" 'such facto(s a$-acnt~al costs arid ' pric~s. A' painful' activity occurs when ,th~" project is not profitable and the ob.jective becomes ,that of minimizing the
loss.
1011 ' TECHNICAL DESIGN (STEP 5) I
5 in Fig. 1-1, the product or system design, ,has no~,. been discussed~ 'The reason for this omission is that the system design is 'the subject of this book from this point on. This step 'js where the largest portion of eng'i neering , time, is spent. System design as an activity lies some\vhere between the study and analysis of individual processes or cOillponents and the larger decisions, \vhich are heavily economic. Usually one person coordi~ates the planning of the undertaking. This manager normally ~merges with a background gained from experience in one of the subactivities. The manager's experience might be in finane'e , engineering, or marketing, for example. Whatever the original - discipline the manag~r must become conversant with -all the fields that play role in the .decision-making process. .. The word "desjgn~' encompasses a wide range of activities,. Design may be applied to (he act of selecting a single member or part, e.g., the size of a tube in a heat exchanger; to a ]arger co'mponent, e.g., the entire shell-and-tube heat exchanger; or to the design of the system in which the heat'ex:chang'er is only one component. Design activities can · be, directed toward mechanical devices which incorporate linkages, gears'~ and orher moving solid members, electrical or electronic systems, thermal systems, and a multitude of others. -OUf concenrration win be on thennal systems such as those in power generation, heating and refrigeration plants, the food-processing jndustry, and in [he chemjcal and process industries. .
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1~12
7
SUMMARY
The flow diagram and description
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10
DES1GN OF THERMAL SYSTEMS
ADDITIONAL READINGS Introductory 'bQoks on engineering design
I )
",
Alger, 1. R. M .. and C. V. Hays: Creative Synthesis in Design, Prentice-Hall. Englewo,od Cliffs, N. J.~ 1964. Asimow, M.: Introduction to Design, 'Prentice-Hall, Englewood Cliffs, N. J.,c .1962.,_. " ,_) Beakley, Q. ~., and H. W. Leach: El1gi~1eering, ,f-n Introduction to a Creative Profession, , , Macm,!l1an, New York, 1967. " , ' "B'i ml; .H .', ,R.:. c..r.e~riv,e. ~ngi.neering Design, Iowa State Un,iversity Press, 'Ames, 1960-.·- .' ,~ '. " DJxori.-J:;R::·.bes"gn-'Erfifrreeri1'):g:~il:tVe!llil~eness, Analysis~' ,a nd Decision Ma~ng, :M:c(h:ay.;- ' , , Ifill, 'New' Y~~k,_ ~966. , "-, ~'~c/:'~' :~~::~.'i:,:,,::'..~:~,-::.,~'~i;-i,_,/.,:,<,~ , , ~ ~/ .~::,; _ « ,(,,: ~,"' ' :~'< ~ :~ ';'~_: " ',;-,;,~:.~ ':'J ~1:';:,!.,_:v ~ , , Harrisberger, L.: Engin{!ersmD;lzshipJ .. APhilosopny· 'oj Des;in~ Brb6kS1C6re~· Berriiont~ '·Caiif.'~ '-., ., -, ', '
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' : .',
, :::,;
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' ' , KriGk. E. V.,: 'An Introduction to Engineen·ng and Engineering Design, Wjley, New York,_ I
1966.
1965. Middendorf, W. H.: Engineering Design, Al~yn ·1'4.isc;hke~
and Bacon, Boston, 1968.
c. R.: ~ lnrroduciion [0 Computer-Aid,ed,Des(gn
I!entice-Hall, Englewood Cliffs, N. l., 1968. , 11orris, G. E.: Engineenng. 'A pecision-Making Process, Houghton -Mifflin Company, Boston~ 1977. Woodson, T. T.:-Introduction to Engineering Des,ign, McGra~-Hill, New York, l ,~§6 . 7
. Probabilistic approaches to design Ang, A. H-S., and \V, H. Tang: Probal;iliT), Concepts in Engineering ,Planning and Design. Wiley, New York. 1975. Haugen, E. B.: Probabilistic Approaches to Design, Wiley. New Yqrk, 1968. Rudd, D. F., and C. C. \Vatson: Strategy of Process Engilleen'ng, Wiley, New York, .1968.
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DESIGNING .A WORKABJ-.JE SYSTEIVl· I
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2.1 WORKABLE' AND OPT1l\1UM SYSTEMS The simple but important point of this chapter' is the distinction ·betwee:p.' designing a workable system and an optimum syst~m. This chapter also continues the progression from the broad concerns of an undertaking~ as d~scribed in Chapter 1, to a concentration on engineering systems and, even ,more specificull y, on thermal systems. It is so often said that ~4there are many possible ans'wers to a design prpblem'" tba.t the idea is sometimes conveyed that all solutions are equally desira151e~ Actu~lly only·one solution is the optimum, where the optimum is base~ Or:l some defined criterion, e.g., cost, size, or weight. The distinction then win be made between a workable and an optimum system. It should not be suggested that a workable system is' being sc·omed. Obviously, a workable syster:IJ is infinitely preferable to a nonworkable system. Fu rthenn ore , extensive effort in progressing from a workable toward an optimum system may not be justified because of limitations in calendar time, cost of engineering time, or even ' the r~liability ,of the fundamental data on which the design is· based. One point to be explored in this chapter is how superior solutions may be ruled out in the design process by prematurely eliminating so~e system concepts. Superior soiutions may also be precluded by fixing
interconnecting parameters betweerl components and selecting the components based on these parameters instead of letting the parameters float unti1 the optimum total system emerges. ~
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11
12
DESIGN OF TIfERMAL SYS1EMS
2~2
A WORKTCAJ3LE SYSTEM
.
A .workable system is one that .
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2.3 STEPS IN ARRIVING AT A WORKABLE . . ... . . :SYSTEM . :~
The
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rriaj~r' steps in .achieving a workable system are (1) to select the
concept to be 'used an~ (2) 't 6 fix whatever parameters ar~ necessary to select
the components of the system. These parameters must be chosen so that 'the design requirements .and constraints are satisfied.
2.4 . CREATIVITY' IN CONCEPT' SELECTION 'Engineering~ especially engineering design. is a potentially creative activity.
In practice creativity may not 'be exercised because of lack of ~ime for adequate exploration, discouragen1ent by supervision or environment, or the laziness and timjdity of the engineer. It is particularly in selecting the concept that creativity · can be exercised. Too often only' one concept is ever considered, the concept that was used on the last similar job. As a standard practice, engineers should discipline themselves to review all the alternative concepts in some manner appropriate to.· the scope of the project. . Old ideas that were once discarded as impractical or uneconomical should be constant1y reviewed. Costs change; new devices or materials on the market D1ay make an app'roach successful today that was not .attractive 1a. years
. ago.
2.5 WOEKABLE VS. OPTIMUM 'SYSTEM .
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The distinction between the approaches used in aniving at a workable system and an optimum system· can be inustrated by a simple example. Suppose that the pump and piping are to be selected to convey 3 kgls from one location to another 250 m away from the original position and 8 m higher... If the design is approached with the limited objective of achieving a workable system, the foJlowing procedure might be followe'd:
13
DESIGNlNG A WORKABLE SYSTEM
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Arbitrarily choose. an additional 100 kPa to COlnpensate for, fg~tion in the 250 ill of pipe. , ' . I ... .'.~' :'. ,~.".. '. ' . ' , '~2o' itccording to' the foregoi]J,g decision select a pur.o.p ,;vh:kI~l'. delivers 3 legIs , '. '. -,--Y:.agalnst a -pres~llre-~-d.i[ference...of~\L1,8 . 5); k:f)~.<~jnqllY-'~;iselect~,a~~:g~H~,}si?;~:~~:) r:,:f.~'Y\'" frqni- a h,!IldqQok such "thaf the pressure' drop:in 250 m··'of.Jength is-)UO··::···· ., . " kPa or less. A pipe size of - 5~ rlli-n' (2 'hi) satisfies the requirement . ~ . 1
. 'A~poaching the' same problem with the 9bjective of achieving an optimum system presupposes agreement on a c~terion to optimize .. A frequeptly chosen criterion .is cOS.t (sometimes fn-st .~ost only in speculative projects, .and sometimes the lifetime cost, GOI?-sisting of first plus lifetime pumping and maintenance costs). ~ designing th~ optimum pump piping system for minimum 'lifetime cost~ ,the pressure rise to be developed by the -pump is not fixed immediately but left free to float. If [he three major contributors to cost are (1) the rust cost of the .pump, (2) the fIrst cost' of the pipe, and (3) the .lifetime pumping costs, these costs will vary as a function. of pump pressure, as shown in Fig. 2-1. As the pump-pressure rise jnc.reases, the cost of the Pl!mp probably Increases for the required flow rate of 3 kg/s because of the - need for higher speed ~dJor larger impeller piameter. W~th the increase in pressure rise) the power required. by' the' pump increases and is reflected in
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DESIGN OF THE.KMAL SYSffi.\1S
a higb~~ lifetime 'pumping cost.' The first cost of the pipe, the third contrib- . uto.r to, i:p.e t~t~ co.st,. "i;>ec~m~s 'e nornollsly hi~h as th~ presSUre available -to' overcome fricTIon ill the pIpe .r educes to ~ero. The avaIlable press~e for. the' .pipe is the pump-pr~ssuIe rise minus 78.5 .kPa. need~d for the difference in , '.elevation. An appropriate' optimization technique ~ ~~ used to deten;n1ne the optim~ pump-pressure rise,.which in Fig. 2-1 is approximately-ISO kPa~ '-, _' ,
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-towax:d designing optinium systems.-· To·, temper thi.s bias, several additional c~nsiderations should ,be_mentiqried. ,I f the job is a small one" the' cost 'of, the increased engineering tim~ required Jor optimization ~a'y devour the , -savings, if any. Not only the engineer's time but pressure of c!iIendar time, -nlay not permit. the desigil to proceed beY0!1d a workable design. t
2 .. 6 ' DESIGN OF A FOOD-FREEZING _PLANT' Large-s'cale engineering projects are extremely complex, ~'d decisions are often intricately interrelated; not onJy do they influence each other in· the purely technical area but als9 c~oss over into the technoeconomic ) social, and human fields. To illustrate a few of the decisions.involved in a.;-ealistic , ~ commerclal undertaking and- to provide a further example "B(the contrast between a workable ~ystem and an optimum system~ consider the following project.
.(\. [ooq company can buy sv/eet com and peas from farmers during the ' season and'sell the vegetables as frozen food throughout the year in a city 300 Ian away. What are the decisions and procedures involved in designing the plant to process and freeze the crops? . The ,statement of the task · actually starts at an advanced stage jn the decision process, because it is already assumed that a plant will be constructed. This decision cannot realistically be made until some cost data are available to evaluate the attractjveness of the project. Let us aSSUDle, therefore, that 'arbitrarily selected soJuriq,p has been priced out and found to be potentially piofita ble. We are likely then, to ve at a sol utian that is an improvement over the arbitrary selection. Son1e major decisions tqat _ rpy..~,~ , .pe , ,m'lc;le . are (1) thew location, (2) size, ~nd (3) type of freezinp pl~t. The plant could be located near (he producing area, in the market. city, or somewhere between. The size \vilI be strongly influenced by the market expectati'on. The third decision, the type of freezing plant, embraces ,the engineering desjgn ~ These three major ·decisions are interrelated. For example, 'the location an d' size of plant might reasonably influence the type of system selected. The seJec£ion of the type of freezing plant includes choosing the concept on wh'cb the freezing-
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DESfGNING A WORKABLE SYSTE:v{
15
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plant des ign \vv. li be based. A.fter the cO.ncept has been decided:. the inter:nal design of the plant can proceed. I ) . An outline of the sequence of tasks and decisions by vvhich a wQrkable , design could be arrived at ' i~ as follows: , ...:.- :
' . ...' _- . --l. · I?~Gjde-.t.o locate the .plant i~ the ·.tharket ~ity adjacenf::t~ a , ." ... :--'·:"~::' w~reflouse~~perated"by :.the -co_inpany.· . "~'" ' .0.·,,,,. -. ,; . . - . ' " : 2~ Select the , freezing · capac'i ty of the plaiil
-availability of fini!l~.iQg.
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.the· : basis o.f the :. curre.nt .'~. the crop, the pqtential sale in the city, ~nd il vaiIahIe ' . .
-'34 Decid~ upon' the co~cept to be used in the freezing plant, e.g. the one 7
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shown in Fig.' 2-2. In this system the food 'particles are frozen in a fluidized' bed4 in' \vhich low-temperature air blows up through a conveyor chain, suspending the product being ·frozen. This air returns from the ' fluidized-bed conveyor to a. heat exchanger that 'is the evaporator of a : re~rigerating unit. The refrigerating unit uses a reciprocating cOlnpressor and water-cooled condenser. A cooling to\ver. in tum, cools the condenser water. rejecting heat to the atmosphere. 4. The design can be quantified by establishing certain valueS. Since the throughput of the plant has already been detelmined, [he freezing capac~ ity in kilograms per second can be .c omputed by d.eciding upon the nUffi.' ber of shIfts to ·be operated. Assume that one shift is s.elected, so that DQ":' the .r efrigeration load can be' calculated at, say, 220 kW. To proceed with the design, the parameters shown in Table 2.1. can be pinned down.
Air
Water
Pump
f:m FlG~
Schemaric flow di:lgram of freezinE
Condtnser
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16
DESIGN OF ~ SYSTE."1S
TABLE 2.1 ,
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5 . :After, thes~ valu~; : DaVe·been-·fi,,,ed,..·th·e-.. :indivrdual'·comp6n'ents can be selected. The flow of chilled be. C'alculated to remove 220 kW with a temperature" rise" of 7°C. The c'onveyor length' and speed mllst now be chosen to achieve the required rate heat transfer. Th~. air~cooling evaporator can be selected from catalog because the airflow rate, air temperatures, 'and r~frigerant evaporating temp~rature fix the choice. The compress'o r must provide 220 kW of.r~frigeration with an evaporating temperature of -38°C and a condensing ·temperature 'of 45°C, which ' is adequate information for selecting the compressor or perhaps a two-stage . compression system. The heat-rejection rate at the condenser exceeds the .220-kW refrigeration capacity by the amouD:t of work added in the compresso~ and may be in the neighborhoo.d 'of 300 kW_ The condenser and cooling to\ver can be sized on the basis of the ra~e of he'at flow and' the water temperature of 30 ' and 35°C. Thus.,' a workable' system can be -designed.' . ,-
-
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Unlike the above procedure, an attempt to achieve on optimum system returns to the pojnt \vhere the first decisjons are made. Such decisions, as the locat~on, size, and freezing concept should be considered in connection with ea'ch other instead of independently. The choice of fluidized-bed freezing \vith a conventi.onaI refrigeration plant is only one of the commercially available concepts, to say nothing of the possibility (admittedly remote) of devising an entirely new concept. Other concepts are a freezing tunnel, \\'here the air blows over the toP. of the product; packaging the product first and immersing the p'ackage in cold brine until frozen; or freezing the product with liquid nitrogen purchased in Jiquid fann in bulk. An example of the interconnection of decisions is that the location of the plant that is best for one concept may not be best for another concept. A compression ,refrigeration pI ant may be best located ,in .the, cjry .as ~n, ~xtension of e):.isting freezing facilities, and it may be ' un~jse to locate" it close to the produdng area because' of Jack of t'rained operators. The liquid-nitrogen freezing p]ant, . on the . other hand, is simple in 'operation 'and could be located close to we field; furthermore, it could be shut down for the idle off-season more conveniently than the compression plant. If the possibiJiry of two or even three shifts were considered. · the processing , rate of the plant could be reduced py a factor of·2 or 3 respectively. for the same daily throughput.
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DESIGNJrjG A \VORKAELE SYSTElvI
17
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rh,e .internal design of the compression re£-:-igerfl.tioD pla.nt? the procedure was' to select r~asonable 'tempenatures and then design each
.'
component around lhose temperatures and resw.tlng flo'w rates.· 'Vhen one approaches the design \vith the 'objective of optimization, aU those interconnecting parameters _are left .free to Jfloat. and one 'finds . th~ comhination
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.'
'-:'., :.
:
~ •
-'.',.
~
. '
TO .THE STrFDY
Any attempt to apply optimization theory to .thennal systems at .this st.age 'is destined for frustration. There are a variety of optimizarion techniq'ues available, some of which are introduced ill Chapters 8 to 12. The first . attempts to optimize a thermal system \virh a· dozen· components, however~ \vill be detoured by the need to predict the performance :6f the system with given input: conditions. This assignment. called system si}nuiatiolZ, must'be studied first' and will be considered in Chapters 6 and '14 .. For complex systemS, system simulation must be performed with a computer. For this purpose the performance characteristics.~aking up the system could possibly ,be stored as tables~ but a far more efficient and llsefu1 form is equ'ation-t;ipe . formulation;', . ·Translating . . . . , catalog tables into· equatioos, called: cOJnponent . . silnulation or mathernatical In ode lillg , is a routine preliminary step to system · simulation and \vill ' be treated in Chapters 4, 5, and 13. Finally. since optimization presupposes a cliterion, 'which in 'engineering practice is often an economic one. a review of investment econ'omics in · Chapter 3 will be appropriate. The sequence to be followed in the ensuing srudies; then, will be . (1) economics, (2) mathematical modeling, (3) system simulation, and (4) optimization.
PROBLEMS 2.1. Location S in Fig .. 2-3 is an -adequate source of \l./at~r. and location A) B ,
and C are points at which water' must be provided at . flow:
Us
. 2.5
3.5
the following rates of
1.5
Points S .A, B. and C are all at the same elevation. The demands for water at A and C occur inrerminen.tly and only during the working day, nnd they may ·coinCide. The demand for wUler o.t B occurs only during nonworking hours nnd is also intermittent. .GrQund-level ·access exists in II 3-m border ~g the building. ~.ccess is not permined over, through, or under the bUilding.
'18
pESIGN OF THERMAL SYSTEMS
' '':;''J
. , ... . :~ .
.
~.
:
... .
_;!_l /
\
.
\
./
/~~~--------~---~m---------------.~~
fiGURE 2-3 SuppJy and consumption points in water-distribution system.
1.0 .
0.8
0.6 . 0.4
c..
-c
L-
0.2
"C
2 ;; v.
:...0
!1: O. J 0 0.08
0.06
0. 04
0.1
0.2
2 0.4 0.6 FJow of wat~r. LIs .
10
FIGURE 2-4 Pressure drop in pipe.
.-. - ......
DESIGNlNG A WORKABLE SYSTEM -
120
--~~~"'-"Q~-
~IOO
~g-
80
.
;
I I
"1 ~T
--
,,'
Z~ ~ 0
~~~~
~
19
,
,
,S'
,-:- J
-~
~-~
,
I, '
3
2
,
,~
"4
6
Flow rare, LIs ..
FIGURE 2-5 Pump perfonnance curves.
(a) Describe all the concepts ot" workable methods you can devise to fulfill the assignment . .(b) The influence of such factors as the expected life of the system has resulted in the decision to use a' system in which a pump delivers water into an elevated storag~ tank, which supplies me piping system. A water, level switch starts and stops the pump. Design .. the system; this includes sketchjng the pipe network chosen, ,listing all the pipe sizes,"-select-ing the pump, and speci~ying the elevation of the 'storage tank. Use pr~ssure drop data from' Fig. 2-4 and pump performance from F~g_ 2-5. ,ENeglect the pressure drop in, the pipe fittings and pressure conversions due to, kinetic energy.) Fill out Table 2.2. TABLE 2.2
Design data for Prob. 2.1 Pipe
Design flow, Us
size, Pipe section
Day
rom
Apt kPa
NIght
Day
Night
(S to A. for
example)
..
-
'-
(
,
"
I
"
----+'Pl~l1mp
..........
selected _'_ ->' Elevation of water in stonge tank __
20
DESIGN OF THERMAL SYSTEMS
(c) Review the design and list tb~ decisio~s that preclude~possible optimiza. ..' tion later in the design. 202 A heating ventilating syst~m for ~'publ~a indoor swimmjng ,designed. . '
and
l.
i. i
I i
I
,I'
.~ ..
l.- 'f
J' I ~
I!
:
,.-
, Specifications ,
.
0
, , po~l is
to be'
'.::.- )
.
I ·
•
' .
Pool water tempe~tute, 26. ,C ) Indoo~ air diy-bulb' ~'emperature;,·Q'1
I I ',
I( -'- ,':;:: ,.' ,~ "~';:'''~.'' ' ''~
II r' . '. .. . . ,<':..":::Oqt~oordesign humidity:~o'f'a:iiai(j5:il¢i~:;'M'.'~:;i}.i,~, .?,,,,~{,: "'~.'{'{!:,' •....,/:-'" I III~' ..';~; ~t:r~::~:!;::::;: ~!=~~:~,a:/:~~~atjpn, S:lHgls- . , ..., I
· IJr J
.
peonstr~ctio~ .r:eatuTses
i,/) ,
l'
.
45
dl.Q1enSlozu;,.l x.. m Essentially the structure has rnaseIiI}' walls and a glass roof:
ij;ji;i
,00
I ~ pn;~i
~~~:::~6~~:;~: W/(m -K)
,ilIII!: , ;
<_.'
2
:1 !t~
..
SupplementaT)' information , Rate of evaporation of water from pool. g/s
~: /' '
= (0.04)
(ar~, ' m 2 )(Pw - Pa.)
where Pw = water-vapocpressure at pool temperature', kPa Pa = p~-tia~ 12.ressure of water vapor in surrounding air, kPa
A
~hojce
of the type of window must be made tsing1e s double; or triple pane), They have the folloWing heat-transfer coefficients: Heat-transfer coefficent, W/(m 1 . K)
Single Outside air film Glass (between external and internal surfaces)
'Inside film
34 118 11.4
Double ,
Triple
34
34
.8 11.4
3.4 11.4 '
The basic purpose of the system is as follows: 1. Te maintain the indoor temperature when design conditions prevail out-
doors 2. To abide by th~ cons~ints ','.. '- . . . 3. To prevent condensat)on of. w~ter vapor on the lnsJde of the glass (a) Describe at least two different concepts for accomplishing the objectives
,
of the design. Use schematjc diagrams if useful. (b) Assume that the concept chosen is one where ourdoor rur i-s drawn in, heated, and introduced to the space and an equal quantity of moist room air is exhausted., Should additi'onal 'heating be required, it is proyjded
.'
DESIGN!NG A \NORKABLE SYSTE;\,1 '.!-
f::ither by heating recirculated air 'or by using convectors around the perimeter of the building. PerfOffil the desigI.l ca~culations in order to specify ~he
following:
)
F1o-w rate of ventilation air_'_kg/s Temperarure, of air e~teting space_ ,, _~C_, . ' Typ~ , .of.gla~s sete~ted~ ,-- "~;""",' " 1"~~ ~ "';,:!- . "-"
. , ~
•
'
. ....-
• •<.....:..
~
-
. {,
...
.. ' .
.~
Temperature.).of.-jiis-idel·surrace::.of' gla'ss",.,·:~-.!. .~G·:.:-.,:.;>:· .~··,~""·l'~' "~:~" "::':"- "~" '''~' ~ .. :': '-<.i':" " COJ:ldition ,at-indoor '-air:-:-:dev.i. 'point_ '- _ . °c" '. :. '.' .- ::.:.. -. --:' .'
: \
, Rate ~ef water evaporafea .from pODl_'._ 'kg/s . , : Heat 10's5 by conduction through walls and ,glass_:_ kW
,Heat supplied to raise temperature of ventilation ,a~r fronl room temperature to supply temperature__ kW . Heat to recirculated air or perimeter convectors_ . kW
(c) Review' the 'design
and list the decisi<;>ns that precluded possible opti-
. mization laterin the design. 2.3. You have just purchased a remote uninhab~ted island where you plan to give parties lasting till late at night. You need to insWI an electricpower system that will,provide at least 8 kW of lighting. (a) List and describe, in severai sentences, three merhods of generating power in this remote location. assuming that , equipment and supplies can be . transported to the is1.and. ' (b) Assume that the decision has been made to produce. the electric power by means of an engine:"driven generator. .Specjfically, .an engine will be (Frect-connected to a generator that delivers power to bulbs. Available choices are as follows:
Bulbs. IOO-W bulbs at 115 V. The bulbs ,may be 'c onnected in parallel. ip 'series, or in combination, and th~ current flowing through a , bulb must be above 70 percent of the rated current to obtain satisfactory lighting efficiency but below 110 percent of the rated current to achie~ . long life. E'!8 ines . A choice can be made between two engines whose power. deliveries at wide-open throttle are as follows:
,Power delivery, kW, at given speed, rls
,
Engine
10
1 2
3.4
2.2
20
' 30
,5.5 - " .. ·..·,,9.0 . 7.1 ILl
40 ·~· , lL2
13.8
50 12.7 15.2
Generators. Single ph~se, alternating current. No adjuStment of output voltage is p
following rwq.generators:
22
,
' .J
DESIGN OF THERMAL SYSTE.1v1S
,
" I
J
I
Ii
Maxim~)
!, I,
allowable current, A
, ~D.erntor
PoleS
2
I. '2
Efficiency
', 0.72 ." ,
.~
OA83
.4
100 50 J' ,
)
Output. vGRlage, V, at given speed1 rls
30
50
10
20
38
7r ·-106·-:-140·. ..:~:174 II7 , 170 -224 , - 270
~9
40
Engine Dumber_ ·. _ Generator' number__ Engine-generator speed__ rls'
, Frequency_ Hz Power delivered ' by . engine~ kW Power d~livered by, generator__ kW . Voltage_ '_ V ·Current_·_.· A
Number of bulbs aI;ld
,
circuitin~
(c) Review the' design and list the 'decisions that preclude possible optimiza-
.0\:"
lion larer in the design. . 2.4. rude oil is be transported overland in Alaska In suc,h a way that the , environnlent is not adversely affected. I ' . • (0) Describe two workable methods of transporting this oil. (b) Of· the wOTkabfe methods, a pipeline is- the method chosen for ' further examination. The inside diamete.f of the pipe is 600 mm,and the pjpe will carry a crude oil flow rate of8:ii0 The distance between pumping stations is 32 km. To facilitate pumping, a heater will be instaJled of each pumping station, shown in Fig. 2-6. The pipeline is to be buried in pennafrost whose temperature at design conditions is -4°C. The penna fro s1 is not to be mdted. Insulation may be- used on the pjpe~ and the external surface temperature of the pipe or insulation in contacl with the permafrost must pe ·maintained ~t O°C or be]ow. Heat-transfer ',data appljcable to the pipe and insulation are shown in Fig. 2-7. The
to
as
1 Numbered
references appear at the end of the chapter.
~---1
Heater ~--------~--------32km------------------~
FlGURE 2-6 'Oil pipe]i~.
----
•.
23
DESIGNING A W9RKABLE SYSTEM
-.
:;.
.. .
~
FIGURE 2-7 Heat-transfer
da~.
available thicknesses 'of insulation are 25~ 50, 75, and 100 mill. The overall heat-transfer coefficient U between the oil and permafrost in watts per ,square meter per kelvin 'bas¢d on the .inside pipe ~ea A i is ,1
I
.
UA,.
I13A;
k[(Ai
--=--+
x
+ Ao)!2]
1
+--7.2 Ao
wh~re Ao is .the outside area of the insulation. The 1emperarure 'change dt
a differential length of pipe (see Fig. 2-8) is expressed by lVcp(-dt) = 'UlID[t - (-4°C)]dL
where w = mass rate of flow = 44 kg! s cp = specific heat of oil. = 1930 J!(kg. • K) t = oil temperature, °C 'dr = change in oil te!Dperature in length dL U = overall heat-transfer coefficient between oil and pennafrost, W/(m'l. . K)
D
= pipe diameter = 0.6 m
L = length of pipe, m
._AJ _4 ,
,
' I
I
I
r
J
t
J
dl
J
I
f
I~
F1GURE2 ..8 Differential length of pipe.
.
J
~I
I
Insulation
ill
1800
-5
'-
0
.
5
15
10
20
25
, 30
35
Outler oil temperature,oC
flGURE 2-9 Oil pressure drop in 32-km length of pipe.
The pressure drop in the 32-km sect jon of pipe is a'function of the inlet' and outlet temperatures of the oil because of the influence of these temperatures on viscosity. Figure 2-9 shov.'s pressure drops for 32 krn. The maximum pressure the pipe can withsra,nd is 2350 kPa ga'uge: Specify the follo\ving:
/i 'p )
Insulation mm Inlet oil temperature Il_oC Outlet oil temperature f2_oC Pressure drop__ kPa Temperature of surface in contact 'with permafrost (highest in 32-km run)_OC (c) Review the design and list the decisions that preclude possible optimization later in the design.
2.5. The rube spacing and fin height are to be selected for th~ steam-generating section of a furnace, ,shown in Fig. 2-10. The furnace section is 1.8 m wide; [he tubes are 2.5 m long and arranged in a square array, six rows high. The tubes have an OD of 75 min, and they ,can be either bare 'Or equipped
are
with fins. The fins are 2 m~ thick and 'are spaced 6 mm apart along the length of the rube. ' 'The stearn temperature in the boiler is 175°C, and the entenng ten'lperature of the stack gases is 560°C. The minimum spacing between the center· ' Jines of the rubes, whiCh is dictated by the smallest U bend available, i'5
t
Prohlem based on
~omp;my.
8
suggestion of O.B. TaJiaferro. Exxon Res.earch and ~.'
Engin~rjng
DESIGNING A \yORKABLE SYSTEM
r-I
r£) +
+. +-
+
+
+ + +
.+ I !
· ' l@
I
T
:.. J.@. ,:,~,"' ."- ~~. ' ,:~
,.. . . .
·0
--- .
'.
• =....... • •
~.'.':O
I. '.~
.
.
( .~
. .
..
.
-. '~
J •
''' ~
.. ::: .~.
'~
.." ' ..' \
,.".,.
'
.
~,'" ' '
.'
,":.,j."-'. . n, ,· :..r· ·.t::...~J ' .. • •
":
.... ..~ ..
",
25
1
I
. ..:.- )
.::.:.+. ".. .
l.,·.! ... :>.
.
I
My
.@) .
Sqmre array
l-l@ · @ . @ @ ..@ @ @ @ H ~--------~--------~1.8m ------------------~~
;,~
FIGURE 2-10 Steam-generaring secrion in a fumace_
·125 mm. ·There -are ~wo restrictions on the design: (1) th.::re must be 3 rom clearance between the fins of adjacent tubes and betwee·n the fins ~~·d the walls and (2) the maximum meta( temperature t which occurs at the tip of the fins, must not exceed 41SoC in order to limit oxidation. . The essential requirement of the furnace is that the produc[ .of V A for t.ne boiler section .·be 18,500 WJK or higher. The U value, based on steam-
side area, is I
U
=
Rto'tJI
= Rncam + Rtubc +
Rgm
where the resistances are
where
h, Aprlme
Ast.~:lm
.-
A fin 'TJ
1
.,
Rst.e.lm
= ~-O = O.0001754(m-· K)/~V 570 .
Rrubc
= o.oZ~
!iCk =
O,OOO0909(m 2 • K)/W •
= g~s-side heot·eransfer coefficient 1
= 70 W/(m"!.· K)
= 0.157 m per meter of tube length ni 2 ·per meter of tube length == fin area per meter of tube length
= 0.2 J0
= fin effectiveness
26
DESIGN OF THERMAl.. SYS1EMS
TABLE 2.3
Fin data
)' ,
)
.~
Fin height, Imn, meas~ed from 'outside'of tube ,
560 - t'tip , 560 - !root "
.",;(,.
,
.'
"
~ ".
. ' ..:-,
:~c:7.~ ,'~If':" ';.:~.'~';.' '.;:',:.";j;~ ;"'~:~:g~!!"":';',:'. !~li~:.~"~~(i~~·~ c~..:; .• . :~.C"~;~,:,~..,'.~•.~:"" >1,::.
31 :37 ',
.' 3.591
,0.64 0..55 0.46
4~503
44
5.6,65
0.58 0.49 0.41
Table 2.3 gives fro 'areas per meter of tube length, the fin effec'tiveness, and values 'o f the expressIon for the fin-tip, temperature as' a :fu.riction of th~ , root temp~rature. In' tum the root temper~ture can be computed from
175 560 - 175 troot -
, Specify the numb~r of tubes in each horizontal row and the'fill height so that the assembly fits into the space available (abiding by the necessary clearances), provides ~ UA of 18,500 WIK or more, and maintains a, fin-tip tenlperature of 415°(: or less.
REfERENCES 1. D. F. O[hmer and J. W. E'. Griern-?mann, "Moving the Arctic OiJ: Pipelines and the Pour Point:'·Mech. Eng .• vol. 93, no. II, pp. 27-32, November 1971. describes further approaches to moving all under Arctic condjtions.
#)'
I I
'
C~IAPTEP
>·3
. . . _;
, r'
"
J'.
.
";:.'
.
'
~ '.0;,
!.
..
" ~'_. , ~.:. :,~l-:.·
. - ". '.
'ECONOMICS
3,,1 INTRODUCTION The -basis of most engineering decisions is economic. Designing and building a device or system that functions properly is only part of the engineer's , task. The device or system must, in addition~ be economic, which means
that the investment must show an adequate return. In the study of thermal systems. one of the key ingredients is optimization t and the function that is most frequently optimized is the, potential profit. Sometimes the designer seeks the solution having minimum fIrst cost or, more frequently, the minimum total lifetime cost of the facility. Har~ly ever are decisions made solely on , the basis - of - monetary considerations. Many ,noneconorrllc factors affect the decisions of industrial organizations. Deci'sions are often influenced by legal concerns, such as zoning regulations, or by so~ial 'concerris~' such as the displacement of workers, or by air or stream pollutjon~ Aesthetics also have their influence, e.g., when extra money is spent to make a neW factory building attractive. Since these social or aesthetic con'cerns almost always require the outlay of extra money t they revert to such economic questions as how much a' fum is 'willing able to spend for locating a plant wh~re ·the employees will live in a district with good schools . . , ,
or
~.
27
28
,
. .,-., .
DESIGN OF TIlERMAL SYSTEMS
This' chapt~~ fIrst explai~s tbe practice of charging interest and the~ ' ,p:roceeds to the ~il)p]ication of interest .in eva}.uating the worth of lump sums, .:. of series of uniform paym.ents" .and of p~ymeJi1ts that vary linearly witp. time . . Numerous applications of these fac~or~ win be explored, including such " staridard ' and important .ones as' ·c
'
"
~- )
.
~;"' ~,-"~,,;;r,:~:' will :?e~~*p!f~e~~ i;~" r,;" '~'C:
d'
"i :.. -
.. -:'':~:~~).'~',~'J..
. W.',',
....
~ :, .... •
,.. ;
i'
• .;
-~ "
-' #
....
~.,
...
....
._~> ~
:· ·I~·
":.~., ~
,
..'
3.2 INTEREST .
,
Interest is the rental chcu-ge for the use of. money ~ When renting a house, a tenant pays rent but also returns possession of the house to the owner after ' the stip:ulated period. In a simple .la,an, the borrower of money pays the , interest at stated periods throughout the duration of the, loan, e.g., every' 6 .' ' months or 'e very , y~ar, and then i,etums the o~igin~l sum ~o the lender. The existence'of inter~st gives money ,a time value. Because of jnterest , it is not, adequate simply ,to. total 'aU expected lifetime receipts and ,in 'another coluum total 'all the expected lifetime expenditures of a facility and subtract the latter from the former to determine the profit. A dollar at year 4 does not have the same value ,as a dollar at year 8 (even neglecting possible inflation) due to. the existence of interest. A thought process that rrn.i.st become ingrained in anyone 'm aking economic calculations is that the \vorth of money has [wo dimensions ~ the dollar' alllount and the time. Because of this extra dimension of time equations structured forso]ution of economic problen1s must equate amounts that are all referred to a common time base. The most fundamental type of inter~st 'is silnple inlerest, which will be qui.ckly dismissed because it is hardly ever applied . .
the
I
Example 3.'1. Simple interest of 8 perce!lt per year is charged on a 5-year 'Joan of $5.00. How much does the borrower pay to the. lender? ,The annual interest )$ ($500)(0.08) = $40. so at the end of 5 years the b'o'rrower pays back to the lender $500 + 5($40) == $700.
Sol/ltio~!;.
3.3 LUMP SUM, C01V1POUNDED, ANNUALLY Annual con1pounding means that [he interest on the principal becomes avai1able nt the end of each year, this interest ,is added to the principal, and the con1qination oraws jnte~est during the next year. ~n Example 3. J jf [he interest were compollnded annually, the value at the end of the first , I year wou] d be $500
+
..
($500)(0.08)
= $540
ECONOMICS
TABLE 30J. . Year
; mo unt S ail: end, Df year P + Pi = P(l + i) 'j~, (l + il) + P(l + i)i
Pi .-
I 2
..
I I
Interest dur ing year
.PO +.
i)i .
J
P(l
;=:
+
£)2.<
,j;I!~':;.-; ;··:" :·;·/t<·~·/-I~/ i,e" "0""; ("I: ~: ~)~~;.:~:; (;~:' j;:'--;~',,-=; ;tr:~~":;r~'o .'.'":' .•. '. ..
:
, During the second year the interest is computed on $540; o~ the second year t.he value is .$540
At the end
, [0
+ ($540)(0.08)
===
""".
so that at the end
$583.20
of the third year the value is $583.20
,
• ...;:-.'
+
($5~3.20){O.08)
= $629 ..86
The pattern for computing the value of an original amount P subjected ' interest compounded annually at a .r:-ate i is shown in Table 3.1.
,
Exampl~ 3.2. \Vhat amount must be repaid on the $500 loan.in Ex.ample 3~ 1
if the interest of 8 percent is compounded annually? Solution. Amount to be ~epaid after 5 years $734.66.
=
(5500)(1
+
0.08)5
LlJlVIP SUM COMPOUNDED MORE OFTEN THAN ANNUALLY 3D4
In most business situations compounding is often semiannually, quarterly, or even daily. At compounding periods other than annual a special interpretation is placed on ~pe designation 'of the interest rate in that_i refers to . norninal annual interest rate. If interest is compounded semiannually, an interest of il2 is assessed every ha!f year. Example 3.3. What amount must be repaid on a 5-year $500 loan at 8 percent annual interest compounded quarterly?
Solution t
Amount
[0
.
. .= (S5P<) (0.08) 1 + -4-
be repaId.
20
= $742.97
, The interpretation of the 8 percent interest in Example 3.3 is that one .. fourth of that rate is assessed each quarter and there are 20 quarters, or compOltD}:ing periods. in the ~-year span of the loan.
30
DESIGN OF THERMAL SY STElviS
3.5
CO,l YIPOUNO-AMOUNT FACTOR (f/p)
~
AND' PRESENT~WORTH FACTOR (pit) .
)
"
The factors that' translate the value of lump sums bet\veen present-and futu~e worctis are ': ....;:-,
,
,
The two factors are the' re¢iprocal of each other, and the' exp,ressions ' , for the factors are' ' , f/p '= ,
(1 + lJ; and
p/f
mn
(1
171
+
1 ' i/nz)mn
where' i '= nOll11nal annual interest rate i2 = nun1ber of ye'ars 1]] = number of cOJ!lpounding periods per year
r',
u' r·I" J
I,
Exa~ple 3.4. You inves~ $5000 in a credit unjon which compounds 5 percent interest quanerly. \Vhm 1S the value of the investment after 5 years?
SolutiOJi
Future amount
,
(0.05 -4- •. 20 periOds)
= (present amount)l f/p,
where the meaning of the convention is (factor, rate, period). ' Future nm.oun! = ($5000).[ J
0.05)120
+ -4-'
= ($5000)( 1.2820)
= $6410
Example 3.5. A family wishes to invest a sum of n10ney when a child begins ,elen)enwry school so [hat the accumulated amount will be $10,000 when the child begins college 12 years ]~teL The m0!ley can be invested at 8 percent, compounded semiannually. What amount must be invested 7..
Solutiol1 ')( p"If'"-2-' 0.08 24 periods J Present an10unt = ~ f,uture 'amount I
= (J +$ JO.ooo O.08/2)~~ ...
'
:c:
,
10.000 2.5633
:1
$3901.20
ECONOMICS
3.,1'
FiUTVJR~E
-VVORTI-f (f /a) OF A. Ur{[FORIYI
. SER IES OF AJVI0UI\Tl' S
i :)
The next factor to be d~veloped . trarlslates the value of a unifonn series' of amounts into the value some tir:p.e in the ,future. The equiv?len~~_Jis sho\vp ~ymbolically in Fig. 3-1, where R' is rhe uniform amount 'a~ e~c.h time period. : Tbe miovv','JndicatiD.g)(q~~~e:.' V,.o!1h i~ ' thG,.:eqgiy.a.lent ~l~l1?e ·.ip ' the future of '.
that senes of'ieg'Ji~ ;anl9unis :~iifi';ilie; apR~opn:ate int~r,~s'(~pplt~tr:>: ..-:..... , ' :.>,,, .. , . Suppose that in Fig. 3-1 the magnitude ·of R is .$1.00 a~d that this ·. amount llppe.ars~ on p.n annual b~$is . Further assume that the interest rate is' 6' .percent, compounded annuaH y. At· the end o~ the first year the accumulat.ed sun1 S is the $1-00 that has just become ·available: At the. end of 2 years the .accunlulated sum is the $100 from the first year with its interest pl~s the new $100 amount: .. .
S
=
($100)(1 +.0.06) + .$100
~
($100)[(1 +0.06)
+ 1]
At the end· of 7, years
S
= ($100) (( 1 + O.06}6 = ($100)( 8.3938)
-
+( 1
.+ 0.06)5 + ... '. +
(1
+ 0.06)
+·1]
== $839.38
In general. the future Worth S of a unifonn series of amounts, each of. . . which is R, wi.rh an interest rate i that i.s coinpoun~ed at the .same frequency '8.S the R amounts is . .
S = 'R[(l
+
+
i)"-1
+ i)n-2 +' ... ~ (1 + i) + 1]
(1
The term in the brackets is called the series-compound-alnount factor (SCAF or f/a) . . f/a
=
(1
+
i) JI -
1
+
(1
+
i) Il - 2
+ ... +
(1
'+
i)
+
1
(3 . 1)
A closed form for the series can be develop~d by first multiplying both sides of Eg. (3.1) by 1 + i (fJ a) ( (+ i) = (1
+
i) n
+ (1 +
i) n - 1
+ ... +
Subrrac[ing Eq. (3.1) from Eq. (3.2) yields (f/a) [ (1 + i) -- 1] = (1
R
o
+
i) n
-
1
Time
/ 2
3
4
5
6
7 Furure worth
(1
+
i)
(3 .2)
32
DESIGN OF THERMAL SYSTEMS
and so .ffa
.
=
alf ..=
(1
'
(1
+ i)n
.'. i I I
- 1 .
l
+ i)n
_
(3.4)
- 1 .
A.
" .
'." !
1- . I ·
pqint to ~Ipphasize js ,the' convention for .when the first :regular amount becomes available. Figure 3-1 'shows that the 'first amount R is available at the end of the first period and :r:t0t at. time O. Sometimes when a series is to be evaluated the first amount may be available at .time 0; such conversions can' be made and will be explained later in the chapter. The important point is that whenev~r Eq. (3.3) or (3.4) is used, .it applies to the' . conyention by \vruch the rust of the regular amounts occurs at the end ,of . the frrst time period. Example 3.6. A ne\\; machine has just been installed in a factory, and the management \\~ants to set aside and invest equal .!lmotints .each year starting 1 "year from now so that $16,000 will be available in' 10 years for the replacement of the . machine.' The interest received on the money invested is 8 percent, con1pounded annually. How much must be provided each year?
Solution. The annual amount R is
R
= ($ ~ 6,000) (alf, 8%, 10) .= ( $ 16,000) (1
= ($16,000) (0.06903)
+
0.08 0.08) 10
-
1
= $1104.50
3.7 PRESENT WORTH (pIa) OF A " UNIFORM SERIES OF AMOUNTS The series-present-l1:orrh jactor (SPWF or pIa) trans] ates the yaJ ue of a series of uniform amounts R into the present worth, as shown symbo}jcalJy in Fig. ,3'-2". As in the previous section, the convention is that the ftrst payment occu~ at the end of the first time period and not at tjme O. The present worth of the series can be found by applying the p/f factor to .each of the R amounts: ·' " ; " -.;.""
Present worth =
.-
"
"
R (1
+
i)1
+
R (1
+
;)1
R ( 1 + ;)n
+ . . . + --.......--
ECONOMICS
A
A
A
R~~~!_ " ~I----,r
{I
A
33
'1\
.]A
I_I .?i:
------,----I-.-------1-1-----'---------,--"
]
.
2
3
6
5
4·
7
~---- Present worth ,( . ..... '. :
,
(' " • • j •.
'~
:i -.'
. '. ". '.::,FIGURE .3-2 ....'- .'. ,. c;:'-'." "''''~,;,\, ,-:,:,- .....)~~ ..-:-._ .•;.~...~ . Present worth of a"seri,es .of ~ourts",~ ~." .
.','·:1
' j.
-
-' .
. ' .. ,~ ' . =' ~::: ,.. - (,
, •..'. , :; .'
~:
.<",
or
Pre~eDt wor.-t.b. .-: R[
1
+
1
+
1
+ .. . + 1 . . J
(3.5)
+ i)2 ... .( 1 + i)n I~. both sides of Eq. (3.5) ar~ mUltiplied by (i -+- i)n, the expression in 'the b~ackets after mUltiplying by (1 + i)n is .the same as ffa in Eq. (3.1). Thus . (1 + i)11 - 1 (Present worth)( r + i)n R(f/a) = R . . . . '1 (1
i)
(1
:=
Ine series-present-worth factor pia is' the one which when multiplied by R yields the present worth. Thus (1 + i)n - 1 pia = i( 1 + i)n ' .. 4
.
.
'
(3.6)
.
. It is not surprising to find from a comparison of Eqs. (3.3) and (3.6) that pia is (f/a) [1/(1 7- i)n]. Thus , pia , (f/a)(p/f)
The reciprocal of the series-present-worth factor alp capital-recovery factor.
a/ _
i( 1
p - (1
+
+
i)n i)n - 1
IS
called the
(3·7)
Example 3.7. You borrow $1000 from a loan company chat charges IS percent norn.jnal annual interest compounded monthly. You can afford to pay off $38 per month ·on the Joan. How many months will it take to repay" the loan?
Solution $1000 = ($38)(pJa, 1.250/0. n months) (1.0125)ft - 1
1000 = (?8) (0.0125)( 1.0125)11 (L0125)ft = 1.490
.-
n 6'=
32.1 months
34
DESIGN OF THERMAL SYSTEMS
3e8 GRADIENT-PRESENT ~ WORTH FACTOR The factors pIa,. fla, and their reciproFaIs ,apply 'where the amounts in the'. series are u7Jifonn. Instead the .pr~sent Worth of a series of increasing" ~ounts may be sought. Typical c~ses that- tll~ ~eries of ¢creasing amounts . 'approximates are maintenan.ce costs 'ar:td energy costs. The c:os1;·-.of Ulalnte- · _': 'nance equipment is .expected to . incr~'.lse progressively as. the equipment . ~:J..;:~,;,.:ag~s,:.·apd;. e~e~gyj;.~:os~~~W.~Y~~~f:.P'~oj_ec~ed .to mcr~ase as fossil ~els b~co~~ - "'moreexpensjve:~ ' " '. .. :'. .:...... '.. ";;';;"':;':';';''';''~''.,. · i.. ,. , ".:.::... ... . . ." . . - ·The. gradient-pr~se.In~worth . factor' (GPwF) .:appli~~"~~hen:··th~~~~;i~?n~·:':}:·i; . cost during the first ye'ar, a costG 'at "the. e'n d of-.the .seco.nd.".year, 2.G at .~ .. . ·t he .end 'o f the third year, and so on, as' shown graphically in Fig_ 3~3 . .'The straight line in fig. 3-3 'starts 'at the end of ·ye·a rl. The present worth of this .. s~ries of incr~asing ' amounts is the SUfi' of·the individual present worths
.of
. . . ' Present worth
=
G
(1
+
i)
2
+
· 2G (1 + i) 3
(n-l)G (1 + i)ll
+ ... + - - - -
. This series can be expressed ip. closed form as Present worth .
= G (GPWF). = G { -:-1 [ (1 . _+ .
l
i) n - 1 ' n ' ]} . .. 1 (1 + 1) n (1 .+ 1) n _
ExampJe 3.8. The annual cost of energy for a (acility is $3000 for the fIrst the end of the year) and increases by ]0 percent, year (assume payable or $300, each year thereafter. What is the present worth this 12-year serjes of energy costs. as shown in Fig. 3-4, if the interest rate is 9 percent compounded annually?
at
of
(n .,.. 1)G
4G 3G
o
. ).
3
2
4 y~
, FIGURE 3·3 Gradient
~erit::s ,
~---
..
5
n
35
!ECONOll;L!CS
I I
6300
~
i"
3~OO
"3300
,. 6r
II
l
,.j:~J ~,,' .":O~O_:",_,; ,. '. "'", . 1.,., -, A';: _,", -"",:< :',." ~.,: :, V
o
2
• ,..: ... ;
'" ' "
~
.
..
.
''' .~
-
3
I"J"
12
Year flGURE3~
"
Progressively increasing energy" cosrsfi1"ExampIe 3.8.
" SoZutf.on~ "The series "of amounts shown in Fig. " 3-4 could be reproduced by a combinatiori of "two series, a unifoffil 'series of $3000 and a. gradient series "' in which G == ' 300. Thus " ,Present worth = (3000)(p/a, 990,12)
+
(300)(GPWF7 ' 9o/c, 12)
PW=($3000 (1.09)12-1 + ' -;oo'{"_l_[ (l.09)1:!-1 , ) (0.09)( 1.09) 12," (S- ) 0.09 (0.09) (L09) 12 PW
~
+ $9648
$21,482
12' (
~l}
1.09) 12
= $31,130
3.9 SUMMARY OF INTEREST F.A CTORS The factors p/f, fla, pia, their re.ciprocals, and the GPWF are tools that can be judiciously applied and combined to solve a variety of economic problems. These factors are summarized in Table 3.2. Sections 3.10 to ·3 .. 13 and a number of problems at the end of the chapter illustrate ,how these factors can b~ combined to solve more compliTABLE 3.2 Summ~ry
of intere~t factors
Factor
Formula
f/p fia
+ i)" 'i( 1 + 1)"
(-l
pia
GPWF
I
1[( 1+ olt - I n ] j
/(
1
+ i)1t
(1
+
;)~
36
DESIGN OF THERMAL SYSTEMS
.
,.."T" ..
.
.
cated sitUations. In most of these~problems the solution depends on setting " -.',' :up an.,equation that expres$es the equivale:pce of amo-qnts existing at, differ-,~' : ' ent times. Many of these problems ~e' thus -solved by qanslating amourits t~' ...' '. the same basis, such 'as the futuie ' worth orall arrlOl1!1ts, tJ?e present \Vorth~' orthe annual ~orth. One point to emph3$.iz:e.is that the infJuep~Y-Jof interest '. me?Us tha.t agiv~:q.,.sum.1:;t~s.differi~g v~ues at diffe!~nt times; As a con~- . ,. quence .the .equatiC?ns just. mentioned sho~ld never. add-or s'u btract ,amounts· . ':<":~':' ;'appl~caRJe.~tQ:Aiff~r§~?r ~~me~~' The amounts ~~s(~ ~.~a'ys ~'e J:~:arlsl~ted to ~. first- . " :",-<,,,,~,::,~ :';"~' J,j,("i~::" -t;; '.' .>:: : ':. ' .. '~.~ ! "': ~ ~. , . '. , ~,. . One of ~ the i~~~;~t 'fi~~n~ia.i '· ~~~~~t"i6~s. ·is ...ili~.· .iss·~~~~ .·sei~g:,·. and 'buying of bond$. Bonds will be the (list applic-ation of the combination .of factors shovv'n in Table 3.2.
. eamon- base
J-:"•••• ,
••
....
oJ'. . . :...•• _
.:"
'.: '
'....
.:: ., ••
:,
".
3.10 BONDS ' 'C ommercial finTIS have· V'ariOllS' methods for raj~i.ng capital to conduct ·or .. expand .their business. They may borrow money ~rom b~nks, sell common stock, sell preferred stock, or issue .bonds. A bond is usu,a lly issued \vith a face va]ue of $1000, wnich means that at maturity the bond', s owner surrenders it to tpe fmrt that issued it and receives $1000 in return. In . addition, interest is paid by the finn to the owner at a rate specified on ' the bond, usually semiannually _ The owner of a bond therefore receives (i/2)($1000) every 6 months plus $1000 at maturity. The original purchaser ·does not have to keep the bond for the life of the bond but may 'sell it at a price agreed on with a purchaser. The price may be higher or lo\ver than $1000. Interest rates fluctuate, depending upon the availability of investment money_ A fum may .have to pay 10 percent il!terest in order to sell-the bond . at the time it' is issued, but 2 years later, for example, the going rate of interest may be above or below 10 percenr. The owner of a 10 percent bond can demand more than $1000 for the. bond if the going interest .rate 1S say, 8 percent, but would haye to sell the bond for less than $1000 if the going rate of interest were 12 percen 1. In other words, the current price of the bond is· such that the total investment earns the current rate of interest. " The principle of setting ,up an equation that ref]ects alJ vaJues to a common basis will now be applied by equating what should be paid for the bond to what i's acquired from the bond. Arbitrarily choose the future worth as the basis of all amounts 1
Pb [flp,
~. 211) (f/a, ~, =1
2n
where Pb ·= price to be paid fori bond now i c = current rate of interest ib = rate of jnterest on bond 11 = years to maturity ~
r;
oJ
(l 000)
+ 1000
(3.8)
31
ECONOMICS
.c 11 n th e ,I e_t f·d Eq. ('J~J" 8' - j are as ro_ O\VS.:"";j} SI .e Of- tne equatIon is the future vrorth of the investment, \vhich1i)S the price to be paid for the bond translated to .the future. The interest applicable is L"1e current rate of interest, -v:.'b.ich is assumed to continue from now until fficlttllitv of the bond. , . . The [list tetm, o'n the right,of'Eq. · (3 ..'8) is the futu~e worth of ~tUnifolm' .series 9'f 'the',s'emiiLrinli:a1.inte paYm;~l).ts 'on
, ,'. '; . ' :the"'mves:toF "immediately. .remVests·"'the interest i:paYin-ent·s·~·.at;· tb,e:'::goingyrate;.<·,.:::,' 11, . . of iriterest ·ic·. . ·The other ' term .cur.the right' 'o tthe' eqLi~ti6n is" $TOOa~ ' whIch '..... ,"IS the 'am~~t ili~ fmn'wii{ b~ckto th~ owner 'of the bond .at ~aturity. This $1000 is 'alre~dy at the .lruh~e time, so no correction need' be made.on it.
Ihe
rT1-
.'It
ten.T}S JIl
....--.
'
•
.rest
J
'
pay'
Example ' 389. A $1000: bond that has '10 years to maturity pays interest . semiafi~llally at a nominal · aimual rate of 8 percent. .,An investor wishes to earn 9 percent on her investment. What price could she ·pay for the bond in order to achieve this'9 percent interest rate? .
Solution. Application of Eq. (J.8) gives . Pb(f/p,
O~045,
20) = (f/a, 0.045. 20) (0.04)( 1000)
Pb(2.4117) = (31.371)(0.04)( 1000)
+
+
1000
1000
. Ph = $9.34.96'
3.il
SHiFT 1N TIME OF A SERIES
In the series shown in Fig. 3-1 for future worth and Fig. 3-2 for present worth, the cO.Qvention is that the fust regular amount appears. at the end and not the b~gjnning of the rJIst period. In an actual situation the rust amount . may appear at time 0, and ,no amount appears at the end, as illustrated in ' Fig. 3-5. If the future worth of this series is sought, for example~ Eg. (3.1) would be, modified by multiplying each of the terms by 1 + i. The closed '. form in Eq. (3.3) could be multiplied by 1 + i, with the result that the f/a for the series in Fig. 3-5 is (f/a)shi~ ' ~
t
!1t
0"
2
3
(1
8
i)
(l+i)n-l
i
1 .t .1 .. 1 I
4
5
Year
FlGURE 3-S First amourif' in
+
series appearing at time O.
7
8
.
(3.9)
38
DESIGN OF THERMAL SYSTEMS ' .
~
3012 DIFFERENT FR.EQDENCY OF SERIES
. AMOUNTS AND COMPOUNDING . ..
.
• ..
• <"
"
'j " " "
.'
•
I
The fonnulas for f/a and pia were developed on the basis 'of ident;ical c~m.; pOWliling and payment frequencies, e.g., annual Of semiannuaL A .situation often "arises where' We .peri.oq.s . are .different, e·.g ~ , quarterly compounding . wIth annual. seri~s amOUI1ts::Mor~ freque~t compounding than payme.nts can · be 'a ccommodated by 'deterinining 'an 'equivale'n t ~ate of .appIi<;able·
>. =~: . -:;~ 'J?~~een ~·061?~~ynient;pe.li?.d..:s.: ~ :(~:~. :.~~,."".:. .~.".>'~; ~.t.;. ' :' ,.:':. :"-.. : :·.,.: -... 7~::
mteres{
.•"'·' :-.""_fi,.". ..:.
._
'~_.,-. __.:.: -: t:: 5}'":''' ".1'._.. -:: .. ( ~. _:,~.~ '::/' :7
a
Example. '3 .10 Annual in:ve~tmenl:S of-$.120.0. a,re.~to ~ ~_e mad~. ?-.t. sayings ,and loan iqstitution for· 10 years. beginning at the end of the firs.t year. The. institution compounds interest quarterly at a nominal annual rate of 5 percent. What js the expected 'value of the investment at the' en(i of 10 years? .' . . .
.
(
a
SOlUtiOll . .If an' amount X starts drawing interest at. the beginning of year is compounded .q~arterly at an. annual rate of 5 percent, the value Y at the end of the year is
and
Y =X 1 (
0.05)4.
+' 4
=X(L0509) =X(l
.'
+ 0.0509)
The equivalent annual rate of interest due to the quarterly compounding js therefore 5.09 percent. This rate can be used jn the f/a factor for the annual
amounts Future worth
=
=
(f/a, 5.09%,10 years) ($1200)
( 1.0509) 10
. 0.0509
-
I
. (1200)
..
= (12:631)(1200) = $15,157
3.13 CHANGES IN MIDSTREAM A plan involving a series of payments (or withdrawals) js set up on the basis of certain regular amounts and expected interest rates. During the Hfe of the plan the accepted interest rate may change and/or the ability to make the regular payments may change, and an alteration in the plan n1ay be desired. This class of problems can generally ,be sol ved by establ ishing the worth at the rin1e of change ' and making new calculations for the rem~ining term. Ex~mpJe 3.11. A sinking fund is established such that $12.000 w~ll be available to replace a facility aJ the end of ] 0 years. At the end of.4 years following the fourth uniform payment, management decides to retire the facility at the end of 9 years of life. A~ an inteft'st rate of 6 percent. compounded annually, what are the puyments during the tirst 4 and last 5 years? t
Solution', According to the. original plan. years were
.-
th~
payments to be made for 10
ECONOM~CS
($1 2, 000) (L1/f, 60/0> [0)
39'
= (12, OOO}(O,,075868) = $9
I I
-
The worth -of the _ser~es of payments at the end of 4 years is ($910.42)(fla. 6%9 4) =,(910:42)(4.3746) -.::. $3982072 '
,
.. .:,-,
This acc~mulated vai~e will drr,iw! interest for 'the next 5 years.. bu[ [he •• " ,'.--..-'1 .... - '_t _ . • ·i~ri}ai.nder of- [he $12,000 must be -provided by-·th~ five additi6iia~ p~y'!nJ~n-~s '"_ 0""
. :.-..-"" '.' 1.
:~'plus
interest
.~:."'.·i,,;·',"
."
.'
.' .-
~.,:
'.
'.. '
.'.
.. - :, .
.I
..
R(f/a, 6%~ 5) = $12,'000 -f3982.71)(f/p~ 6%? 5)
and so [he
paymex:~
R
=
during the last 5' years must be ,
12,000 -
(3~
982Q 71) (1.3382)
-. 5. 6 J7 ( .
.
'.
= .$1183.30
'
3Ql4. EVALUATING POTENTiAL INVESTMENTS . .
'.
An importa~t func~ion of economic analyses in ,engineering"-ente:prises i~ to evaluate proposed investments. A commerci~iI firm must develop a rate of return on its. investnlent that is sufficient to pay corporation taXes and still leave enpugh to pay in~erest on the ,bonds or dividends on the stock ~haf provide the investment capital. The evaluations can become very intri~ate,. and only the basic investment situations will be explained. This fundamentai approach is. however, the starting point from \Yhich modifications and' refinements can be ro'ade in more complicated sitUations. Four elements \viIl be considered in investment analyses: (1):first' cost, (2) income~ (3) 9perating expense, and (4) salvage value. The rate of return is treated as though it were interest. Example 3.12. An owner-manager firm of re~tal offic,e buildings has a choice of buying building A or buiJding B with the intent of operating 'the building for 5 years and then selling it. Building A is in an improving !ocation, so that ., the expected value is to be 20 percent higher in 5 years, while building B is in a declining neighborhood, with an expected drop in value of 10 percent in 5 years. Other data applicable to these buildings are shown in Table 3.3. -Whar. yvill be the rate of return on each buikiing?
-
Solution. The various quantities must be translated to a cOrnfl:1on baSis, future worth, annual worth, or present forth. If all amounts are placed on a presentworth basis, the following equatlo~S apply: Building A:
,
= (160,000 -
73, OOO}(p/a, ,. %, 5) .:,.. (960, 000) (p/f, i%, 5)
.... (155.000 -
5b~300)(p/a.
BOO,ooo ~uiIding
B:
6Of!~OOO
i%, 5) +
(54D,OOO)(p/f~ i% ,5)
40
DESIGN OF THERMAL SYSTEMS
TABLE 3.3
Economic
d~~a
_
for Examp]e 3.12 : ' :-'.
Building A
.' Fmt cost $800,000 Annual income from rent .. ' . ..1.60,000 ' ., _ Annual operating ·.a nd' ,;-:",) .:_.,:_" . ""~'''' I .~ ..... :,: . -,,; maintenance cbs~ . . , 73,000
,, '
.,,- .' . AntS9pated .sdilng,
$600,090 155,000 .' "_50,300
rile" ";': :.:,•_ ..' ;.:~,~o"';?'<"""" ~~~;Qrii:> ':; ~u ,~,:,'j:';"'::: _. •.. :,;;. ' :.
, The unknown
iri' each of the two pres~nt-wortb 'equations' is the rate 'o f return
. i. Since the expressions ·are not ·linear in i, . .an jt~ative techniq~e must be used to flnd th~ value of i: ' i' = {13.9% buildin~ A ' 16.0% ' building B Building B provides a greater "tate of return, aI?d the flrm must ·now decide whether even ~e 16 percent is adequate to justify the investment.
3.15 TAXES' The money for operating the government and for financing services provided . by the government COllles primarily from taxes. The inc'lusion of taxes in an econorrUc analysis is- often important because in some cases taxes may be the factor deciding whether to' undertake the project or not. In certain other cases the introduction of c~Dsiderations may influence willch of two alternatives will be more attractive economically .. In most sections of the United States, . property taxes are levied by a substate taxing district in order to pay for schools, city government and sen'ices, and perhaps park and sewage systems. Theoretically, the real estate tax should decrease as the facility depreciates, resulting in lower real es't ate taxes as the facility ages. Often on investments such as buildings, the tax, as a dollar figure, n.ever decreases; It is therefore a common practice to plan for a constant real estate tax when making the investment analysis. The eff~ct of the tax is to penalize a· facility which has a high taxable value. ~ederal corporation income. taxes on any but 0e smallest enterprises
tax
- amount to approximately 50 percent of the profit.' tn Example 3.12 the rate of return on the investm~n.[ in building B was 16 percent, which may seem very favorable cOlnpared with the usua] ranges of interest rates of 5 to ] 0 percent. The rate of return of 16 percent is before taxes, however;· ,
after the corporation income lax has been extractM the rate of return available for stock or bond holders in the company is of the order of 8 percent. Since income tax _is usuaUy a much more significant factor in the I
.-
.'
ECONOMICS
41
ecpllomic analysis than p.foperty tax; income tax \Ivill be discussed furthe~. f\n ingredient of income tax calculatio I1 s is) ?epreciarion, ex.plored in the next .section. ' ..;:- J
'.':':,.
."
..
;~. Depreciation. . is' can -·~amo unt-· thai..; is'.;-.} isted'.- -:as .. ·an~ -annhaI,' expense~' -~"" the':, lax'!"" ':':t,~"
.c~cu'l9-tion,
to a~lqw- for replaceme:lIi ··of·rhe.:./?c'ility''"ar the: end" 'o f its -HfeD '
Numerous rrtethdds of computing depreciation ire p~m1itted 'by th~ Int~mal Revenue 'Service-, e.g., straight line, sum-'Of-the-yea{~s digits? and doublef?te declining balance. The first' two will ·be explained. Straight-line depreciation consists .- simply 9f dividing the 4iffen~nce' between fIrst cost and salvage value of the facility by the number of years' of life. The result is the annual depreciation. The tax- life to be used is prescribed by the Internal Revenue Service and mayor may not be rl?-e "same as the economic life used in the economic ana1ysis~ . In the sum-of-the-year's~digits (SYD) method., the depreciation for a given year is represented by the fOffi1ula
the
tax
.. DepreclatiOn., dollars . . ".
=
2
N-t+l . (P - s) N (-N + 1)
, (3.10)
where 'N == tax life, years t = year in qu~stion P = first cost, dollars S -.-: sal va.ge value, dollars If the tax life is 10 years, for example, the depreciation is First year Second year
= 2( 10/110) (P
= 2(9/110) (P
~
S)
- s)
Tenth year = 2(1I110)(P - S)
A comparison of the depreciation rates calculated by the straight=-line method with those calculated by the SYD method shows that the SYD method permits greater depreciation in the early portion of the life. With the SYD method the income tax that must b~~-opaid earJy in the .life of the jacility is less than with the straig!lc-line ~etqo,d, a)tbough near the end of the life the SYD tax is greater. The total tax paid over the tax life of the .facility i~ the same by eIther method, but the advantage of using the SYD method is that more of [he tax is paid in later years, which is advantageous in view of the ti'me value of money. I The straighr.l1ne method has an advantage, however, if it is likely ,the tax rate will increase. If [he rate jumps, it is bener to have paid the low tax on a lar~r fr(lction of the investment. I
;-
6'
'-..
42
.
'
DESIGN OF THERMAL SYSTEMS
3 .. 17 INFLUENCE OF INCOME TAX ON E;CON01VUC Al~ALYSIS )
...
'
~.
'
)
To see the -effect of depreciation and' feq.eral income tax, consider' t.he, , following simpl,e exampl~ of choosing betw,e en ,ltemative investments A and B~ 'f or ,whjch ,the .Qa.ta.~f:l. Table, 3,-4 ,apply. \ .. ' ~~.1 A calculation "of the .annual c'ost of both aI'tematives without inclusion of' the""' income taX is ' ~s follows: ' "
,
.
-
'.:,
, r:
.
-.
. .•.
' . ,, ' 1. - . '
• •.
~"
~ ,'
.
'.
.
~
.'
•
, . .i
r
o,
,'~
I
.
'. -
. .
Alternative A: ' FIrSt c'ost on annual basis (200,o.OO)(a/p, 9%, 20) , $21,910 .Annual opera~ng expense 14.000 10,000 Real esta'te tax and insurance Total $45,910 ,Alternative B: 'Frrst cost on annual basis (270,OOO)(aJp, 9%, 30) $26,280 . 6,200 Annual operating expense 13,500 Real estate tax and insurance $45,980 Toial"
,T he economic analysis of alternatives' A and B shows approximately the same annual costs and incomes' (in fact~ the example was rigged to accomplish this). In the computation of profit on which to pay in<;ome tax, the aG.t:u,al interest p_aid is listed as an expense: an'd jf straight-line depreciation is applied, the expenses for the first year for the two alternatives are as shown in Table 3.5. ' A higher income tax must be paid on ,A than on B during the early years. In the later years of the project, a higher tax will be' paid on B. The example shows that' even though the investment analysis indicated equal profit on the two alternatives, the inclusion of inco~e tax shifts the preference to B. The advantage of 'B is that the present worth of the tax payments is less than for A. TABLE 3.4 '
Alternative investments.
_ First cost
~
Life SaJ\'age vaJue Annual income A nnual operating exPense Re~1 e~lale tax and insurance (5% of first cost) Interest
Alternath'e A
Alternative B
$200,000
$270.000 30 years
,20 years
o
o
$60.000 $14,000.
160,000 $6,200
I
'
5JO,000 9%
$J3,500 9%
, • ••
;~-"
. -
L~.3
ECONOMICS
TABI...E 3.5 mC0111 e
First-year
tax n
LV{O
Cl.lteJrn a i uves
I
J
:;;:~::enses
Depreciarion . Interest. ,.9%.o.f..iJripnid::'baladce· ""~ .. Operating expet!se,
... .
~.:aQA::~;~l!surance .. ~:~.:.: •...:: ..
:";··· · T~t~I~?cpenses .. :::--<·· ··.
SIO,OOO.$·. 9~OO'O . 118.000' 24.300
: -='.1
.
,_
'.' '.11" -)~_:.·OOOOOO';·:": -i':;'- " ·:;';' · ' :'.-:.; ..:.-', .,'. '$'}39~010600~'' ' ~~'~'''' i'''~';:-~'.,- .,..~/~::~:'?::.:.:".::.. I-';~ .
.. . ,~Profit . = . inc01pe .~ expenses' . Income Lax (50% of profits) . :.
~
J
I
, 8.000 . 4,000
•
7 ~OoO · . : 3,500
'. 3~18 CONTI~{UOUS CO·M'POUNDKNG 'Frequencies of compounding such as annuaJ, semiannu'al, and quarterly have been discussed. Even shorter c~:nl).pounding periods are coinmon~. e~g._-: the daily compounding offered investors by many savings and loan ins-ritutioI1s. High frequency of compounding is quite re~llistic in busipess operation . . be~c'a'llse the 'notion ·of accumulating money 'for a quarterly or semiannual payment is not a typk:al practice. Businesses controi their money more on a flo'w bas'fs than on a batch basis. The limit of compounding frequency is c0l1ri~7.1l011S c07npoli.nding with an infinite number of compounding periods per year. This section dIscusses three factors applicable to continuous compounding: (1) the continuous compounding factor correspo'n ding to flp, _ (2) unifo~ lump sums continuously compounded, and (3) continuous flow continuously compounded. The tIp term with a nominal annual interest rate of i compounded In times per year for a period of n years is
= (1 + ~
f/p
rn .
The flp factor for continuous compounding (f/P)cont is found by letting approach -infinity .
(f/p)coo,
= (1 +
~ fl-+a>
(3.11 )
To evaluate the limit, take the logarithm of both sides
In
(r/p)con,
=
~n [In (I + ~ )L~
. Express , . In (1 + ilm) as an infmi[e ~eries of ilm )~ (f/p)cOII\ == mn 10 [
.;2 i +. m + (consc) m 2 + . ~ .]
J11:"
. m~
44
DESIGN OF TI1ER..lI..1AL SYSTEMS
Cancel m and let 'm
co
-7
, Therefore , ," . ', (f/p\orit ' . e~n •
' ~ :.-J '
·... ...
~.--:l
'-.'
.~ ~-:~''''~I'' ..... ~-
::1 ~ . ~.
' ! -'
6
~
~ample 3.i3 .. · Compare the VallleS of (flp, 8%, 1.0) '-.
Solutio~ ,~"
" "
" . [(fJP)~on" 8%, lQJ
and [(f/p)
cant ,
8%~ 10].
.
. . ....
(fJp~' 8"%, In)
(3.12)
• ..:. .. ,
, "1" ' . '
= . (1.:+
:O~08)'~O,, ~, 2~i5&9~
= "eO' S ' = 2.2255
The next f~ct~r presented is that of the future worth of a uniform series of lumped amounts. compounde~ continuously. Th~ continuous f/a ' factor for annual amounts R can be derived by modifying Eg. (3.1) to cOJ;1tinuous compounding by replacing 1 + 'j with e i . . '= (ei)n (f/a) cont.lump
.1
+ ' (e i )n.-2 + + i can
. In the closed-form expression of Eq. (3.3) l i by e i - I ~ yielding (f/a)cont.
+
ei
+
1
be replaced by e i and
e in - 1 lump
(3.13),
, -.--
el
1
-
The final continuous-compounding' factor to be presented is the COJ1linuous-flol1' !ztfure-'I"vorth faclor (f/a)llow, which expresses the future worth of a continuous flow compounded continuously. If $1 per year is' divided into ]]z .equal amounts and spread unifonnly over the entire year', and if each of these JJ1 amounts begi~s drawing interest immediately,. the future worth of this series after 11 years is
~(l + iJ·mn-l + In
111
!!.(l + ~)mn-2 + ... + g(l + ill + £ In
n!
172
m
m
J
(3.14)
If we use' the analogy between Eqs. (3; 1) and (3.3), Eq. (3.14) can be translated into the closed form
1 (1 + i I nz ) mil 111· ilm
"
(f/a)tlo\\, for $ J per year = ~ '
As
In
approaches infin"ity, t~e term (1
, (f/a)~(JW
+
1
(1 + jim) m J1 - 1 - ------
i
i/nl)mn , approache~ e in
for $1 per year -
..
-
,
s,o that
(3.15)
ECONOM1CS
45
Example 3.144 I/JciJ-tion A :For a. facrory is expected to prcctnce ~m ;;'lT1RV:<.I profit that is $10,000 per year greater than ifllpcation B is chosen. Thi~ additiODZ.J profiT is spread evenly over the entire year and constitutes a contjnuous , fIo .v of $10,000 per year. The prQfi~ is continuously rein~'est'ed at the Kate of return, of 14 percent per yeaL .At the end of 8 years, the expected iEconomic 'life?" 9(.,~:e Ja~Jor:i.~s, .. what'is the difference ' between the worths of fue ~wo , , "':':"""'-~ :- , iu'V'estrri'ents-?, :-' .. ,:,:,'"""~';',1,"'--~' ,
,
""
:'
,' "
_,
,,'
",' ,'., "
.. .. '
-
Solution.; Since, the' $10,000 per ye'~ diff~renc'e' is a conti~uous''flow 'couti~~. uously compounded, the difference between future ,~/orths is :;e fO .i4)(S)
($10,000)
0.14
-
1
='$147:490
There are several , levels of economic analysis higher than that approached by this chapter., The complications in accounting, financing" and tax computations involve sophistications beyond, those presented here. The ,stage achieved by this chapter might be describ~d as ~he second level of economic analysis. The first leve'l \vould be a triv'ial one of simply totaling costs with no consideration of the' tirpe value of money. ' The second Ievei introdu~es the influences of interest, which imposes the dimension of time as well as .amount in assessing the value of money. The methods of investment analyses explajned in this chapter are used repeatedly in engineering practice, and in most cases engineers are not . required [0 go beyond these princip~es. These meth.ods also are the base for extensions int~ more complex economic analyses.
PROBLEMS 3.1. Using a computer, calculate your personal set of tables for the factors f/p. fJa, pIa, arid GPWF. Devote a separate page to each of the factors, label adequately, and c()1culate at the interest rates of4, 5,6,7,8,9,10,11. 12. 14, 16, 18, 20, 25 percent. Calculate for the following interest periods: 1 to 20 by' ones, 22 to 30 by two~, and 30 to 60 by fives. Print out the factors to" four places after the decimal point.
3.2. Annual investments are being made so that $20,000 will be accumulated at ' the end of 10 years. The jnteres~rate on these investments is initially expected to be 4 percent compounded annually. After 4 years, the rate of interest is unexpectedly increased to 5 percent, so that payments for the, r~maining 6 years can be reduced. What amounts should be invested annually for the first 4 years nnd what sums' for the I~t 6? Ans.: Final payment, $1547. 3.3., A fum wishes to set aside equal amounts at the end of ea'Ch of ]0 years, beginniRB at the end of the flist year, in order to have $8000 in maintenance
46
DESIGN OF THERMAL . SYSTEMS ,
,
#'-
funds available at the end of the seventh, eighth, ni nth, and tenth years.' What is the required annually,payment if the1money is invested and d,raws 6 per,cen't compounded annually? " I , ' Ans~: ,
$2655.
'
3.4. ,A ,home inortgage ' extends for 20 years at 8 pe.rcent interest compounded monthly. The' payments are ,also ni~de monOlly. After hoW 'many months 15· half of the principal ' pkid offlJ ' " , ,. , , " , " : ', , ' Ans_: 164 months., :. ' , ,- ',: ::' 3.5.,"·A lender 'offe~,a,-, l~yeat. I9?P ,a,t ~:~hat;:,he!, ,c;alI,S;:8~R~rc,~~)n t,~~e~t J:~~~:::requ~re~:,~: , the inJ.erest ,to be paid at, th~.,beg~n~iI:lg ' rather: than- at' "the', ie~'~ . 'pf'~tij¥:ye~~ ~:~" as the usual practice is. To what interest rate ,qomputed ·in .. the. :conventional manner does this interest ~h~ge c~r:t?sp?nd? ." ' Ans .. : 8.7%. 3.6. A loan of $50,000 at 8 percent compounded ann u all y is to' be p~id off in 25 years by upifoffil annual payments beginning at the end of (he first year. These ann'u al payments proc~ed on schedule until the end Df the 'eighth ,y~ar. ,when the bo!!ower is unable to pay and misses the payment. He negotiate$ with the 'lender to increase the remaining 17 payments in such a way that the lender continues to rec,eive 8 percent., What is the amount of the original ~d' the :(inal payments in the series? Ans.: , Final payments, $5197.44. 3.7. An $18,000 mortgage, on which 8 percent interest is paid, compounded monthly, is to be paid off in 15 years jn equal monthly installments. What is the total .amount of interest paid du,ring [he life of this mortgage? Ans.: $12,964. 3.8. What wiJl be the future worth 'of a series of 15 annuaJ $ I 000' payments if the nominal annual interest rare is 8 percenI and the interest is compounded J
"
quarterJy? Ans.: $27.671. 3.9. A sum of sufficient magnitude is to be invested now so that starting 10 years from now an amount of $2000 per year can be paid' in each of 8 succeeding years. The unexpended money remajns invested at 8 percent c,o mpounded annually. How much must be allocated now? Ans.: $5749.50. ' 3.10. A mortgage that was originally $20,.000 is being paid off in regular gunner])' , payments of $500. The interest is 8 percent compounded quarterly. How much of the principal remains after 9 years, or 36 payments? , Ans.: $14.800.60. 3.11. A' 20-year n)ortg~ge se[ up for' unifonn monthly payments with 6 percent interest compounded monthly is taken over by a new owner after 8 years. At that lime $12.000 is stil1 owed on the principal. What was the amount of the original loan~ , Ans.: $16.345. 3.12. An investor buys common stock in a firm for $)000. At'the end of the first' ,year and every year thereof/er, she receives a divide'nd of $100. which she immediately in\'ests in a savings and Joan institution that pays 5percenl intet· es" compounded annua)]y. At the ,end of the' tenth 'year. just after rece'iving . ... .
. '
ECO;VOMlCS
<~7
. her di vide nd) she sells the stock fOf $ 1. 200. \Vh:J.t is the f3te of interest (on. aD arlnu~l compounding basis) yiel.ded by Itf1is investment program?
Ans:: 9.41%. A. sun~ of $20,000 . is' borrovved a.t an interest rate of 8 pefcern on tpe. unpaid ~a1ance compounded semiannuaUy. The loan is to be paid back · \v'ith~-10 equal . paymemsjn,., 20 .y.ears .,.,The.·.paymenlts 'a~e , to be 'made every 2 ye;:rrs; sl~rting ; . ':.,:". '
att he endoLthe·.~ec.o.nd.~y.ew.,.,. 'V~at".is, _ the..am6unt:. ,oJ..~.;!ph . bi. ~np.ial p~Ym~Il~?"!..."~ " ... ': · ··.-'Ati.~L: $429-r~r . .. -.. . '.~ ~ . , '. . . . . . ....: .".. : . . . . . . . . .
.-
'. .
The packing in' a ,cooling tower that cools condensing water for :1. power ' plant progressively deterio!'!tes and results .in gradually.rising costs due to ·re.d uced plant efficieJ)cy. These costs are treated as lump sum.s at the end of the ' year, as shown in Fig. 3-6. The cost "is zero for the first year a.nd men increases $1000 per year unril me packing'is 16 years oJd. when replace~ent is mandatory. At a point 8 years into the life of the pilck,:ing (ju~t after 'the $7000 ann.ual cost has been assessed), a de~ision is to be made on ~e plan for the next 8 years, i.e., whether r~pltlce the packing or to continue with the existing packing. Money can b~ borrowed at 9 pen;::el~t interest. compounded annually_ What is the maximum amount that could be paid for the packing in or~er to justify its replacement?
to
Ans.: $44,279. 3.15. The anticipated ta'\es on a facility for its IO-year tax life decline in a stp3.ight·line fashion a~ follows: ' . At end of year 1 At end of year 2
At end
~
of year
S10,OOO
.
10
9~OO&
'.1,000 . o o
.
o~
'"
~
./
./ ./ ./
Vi
o
u
d
=:J C
New packing
c::
<-
. 0
2
4
6
B
10
12
Year
FIGURE 3-6 Increasing !¥F':t., dn~ to cooling#[ower d~Jeriora[ion.
14
16
48
DES1GN OF TIIERMAL SYSTEMS
..
In an economic analysis of the facili ty the present worth of this series must be .computed on the basis of 6 -percent interest compounded- annually. - _ (a) Using a combination of ~va11abl~'!factors, determine a formula for ,'the ' present ,worth of a, declining series l~e this 'o ne. . (b) Using the formula from part (a), compute the pre~ent worth of the above -' series. . '. . ''''ADS~:-(~) $4$~999': :".,-" . )... •
. '
.
.
'f',
"
. " ;..,,,rf,,,:. ':';!{",." 3.16..' .Calculate the uniform .aillluaJ. profit ,.
c····
'.~. ':"
on
:dollowing·:4ata•.al>p~r;::; ';":":';:';;~~Te r >:; ."
a processi-!lg plant for which the
,.<".c.:' ;;.:! · ~.~i·
.;:;;.
: .
. Life . Frrst cost Annual real estate tax. and insurance Salvage value at end of 12 yeats Annual cpst of raw materials, labor, and other supplies Annual income ' Maintenance costs, during first year At' end of second year : At end of third year At end of twelth year
.~-.J
1,2
"~"'f' ~:: . :i;;. ''0'
".
ye~
$280,000 4% of first cost
$50,000 £60,000 -$140,000
o ,$1000 S2000
S11,000
.
The interest rate applicable is 6 percent compou!l.d~d annnally.
Ans.:
$33~560.
3.17. A $ 1000 bond was issued 5 years ago and wiD mature 5 years from now. The bond yields an interest rate of 5 percent, or $50 per year. The o~ner of the bond wishes to sell the bond. bur since interest rates have· incre'a sed, a prospective buyer wishes to earn a rate of 6 percent on his inve~tment. What should the selling price be? Remember that the purchaser receives $50 , per year, which is reinvested, and receives the $1000 face value at maturity. Interest is compounded an'nualIy. Ans.: $957 ~88. 3.18. Equation (3.8) relat.esthe value of a bond Pb to the bond intere'sl and current rate of interest by refiecring all values to a future worth. Develop an equation' that reflects all val ues. to a unifonn semi?nnual worth and so} ve Example 3.9 '. with this equation. 3.19. Using'a computer program. caJculale tables of the price of a $1000 bond that will yield 5.0,5.5,6.0,6.5,7.0.7.5,8.0,8.5,9.0,9.5.1 10.0, ]0.5, and
,
, 11.0 percent interest when the jnterest rates on· the bond are 5'.Ot 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5,' 9.0. 9.5, and JO.O percent. C,?mpute the foregoing .table for each of the following number of years to maturity: 2, 3, 4,,5, 6, 7, 8, 9, and ] O. Interes[ is ~omp'ounded ,semiannually. must buildI a new electric generating plant and can choose 3.20. A municipality ' , between a' steam or a hydro fllci1ity. The anticipated cost of the steam plant '$10 million., Compararive d~la for the IWO plants are
is
i?
Generating cost~ including main~ ter'ancf' ? per kvVh
ECONOfl.'lICS -
Salvage vnbe, percentage of firs'! ·cost .
Equip;m nj: ]ife~ y~~;xs
. ...:. ~
Steam ~y~o'
,.'
c"
50.0040.002"-' :"-
.,' ..~O
"
10 10
, 30'" . •-
49
I'
rr
Tpe ex.pect~d annual consumption of power . is'_.300,OOO,lvrWh .. ~oriey is borrowed a.t S' percent interest compounde(J. annually, what first cost 'or' the : hydro plant would make the two alternatives equolly attractive investrnen-ts? . AllJls~: '$21 ?593~OOO.' ,
3011~ A proposed investment consists of constructing a b~ilding, purchasing pr~
duction m~chinery, and operating for 20.years~ The expected life of the building is 20 years; its first cost' is $250,000 with 'a' saly.~ge value of $50,000. , Since ,the maximum life of the machinery is 12 years, it wili be necessary ~o · renew the machinery once ,during ¢e 20 years. The first cost of the machinery is $132,000, and its salvage value is $132 OOO/(age, years)~ The annual income less the operating expense is expected to be $50,000. Annual interest is 6 percent compounde~ annually. (a) When is the Inost favorable time to replace the machinery? , (b) Compute the present worrh of the profir if the machinery is replaced at the time iiJdicated by part (a). f
Ans.: (b) $152,100. 3.22: Owners of a plant tha.t manufacrures edible oil are co~sidering constructing a tank to store unrefined oil; this will permit buying raw oil at more favorable prices. The cost of [he tank is $150,000, it has an expected l,i fe of 10 years and a salvage value at the end of its life of $20,000. The anticipated annual saving in oil cost is $25,000; What is the rate of return on the investment? , Ans.: 11.650/0. 3.23. A new facility- is expected to show zero profit during each of the first 5 years. For the 10 yea~ then~_after the ' expected profit is to be $80,000 pe:r- year. If 12 percent interest (compounded annually) is desired as the re~rn on the investment, what amount of investment is justified?
Ans.: $256,500. 3.24. The anticipated income from an investment is $40,000 per year for the first 5 years and $30,000 per YC;~ f~r the re'maining 5 years of J ife. The desired rat.e of rerum on the investment is 12 percent. The salvage val ue at the end of 10 years is expected to be 20 percent of the first cost. Determine .. the flISt _ cost that will result in the 12 percent retuql. . Ails.: $219,700. , 3.25. How many years will be required to double an investment if it draws interest-, at a rute of 8 percent, compoundeq semiannuaIJy? 3.26. You just received a notice of an insurance premium of $45 per month starting • op[ion of making one annual payment October I but were, also offered' the on October I. If you want 10 percent nnnual rate of interest what annual J
paymen-r would you be wining to make? -- AM. ~ $ 5 ~ 6. 12. .'
50
DESIGN OF THERMAL SYSTElv.lS
(' ~
3 .. 27. A sum .of $10,000 is in-yested and draws<. interest at a rate of 8· percecompounded annually. ' Starting at the end of the first year ,and each y __ thereafter $lOOO is withdrawn.' For how many years can this plan contiL , until the mopey is exhausted? .
Ans.: 21 years. 3.28. A frrin borrows $200,000 at 9 percent nominal -interest, compounded month:_ and is- to ' repay th~' loatf ill "12 years-'with regular -monthly payments -= .:.:__ _ $2276,Q,6,. 'The. firm has the option 6f paying off in ,advance, 'an,d aft~r -. .: "sixth year. makes an, additiq~,~l.~~,~,~O.O, ~axmeDt_.)! if-c~n~ipue'~"the::$g~l~~ ~: pay_rpents" h0}V inany' additiona!,-·moiitl:;ls,. are. ,re'qu'ired: to 'pay 'off·the.-16an? . Ans.: 39
,months .
,
,'
,
,~t29. A car rental agency which leases cars to another firm buys cars for $9,00-:
anq. sells them for $6,000 two years later. It, charges a monthly rate th~ second year Df rental of 80 percent of that of the rust year. The agency ,seeks "1 percent per month return. What are the monthly , rates each year? ," . Ans.: 2nd year, $177.74 3 .. 30. A 20-year loan is to be paid off by monthly payments of M. The nomina: 'annual interest rate is i. Develop' a, closed-form expression for the unpaic 'balance at year n. 3~31.· A finn has capital on hand and is consjdering an investment in a plant 'that is expected to show a net annual return (income less expen,s.e) of $80,000 per year. The life of the facility is 10 years~ and the salvage value at the eDd o · rhat time js 20 percent of the first cost.. If the firm wishes a ]2 percent ,retum on its investment, how much can it justify a.s the first cost? Ans.: $483,131. 3.32. The first cost of an lnvestment js $600,000 borrowed at 1] percent interest compounded semiannuaJly_ The expected income (less operating expense) for 'every 6-month period is $60~OOO. If there is no salvage value, how jong . must the plant operate in order to payoff, the investment? Ans.: 7~ years 3 .. 33. A IS-year mo{tgage of $40,000 at 10 percent interest 'c ompounded monthly is to be paid off with 'monthly payments. How much tota] interest will be paid during the fust two years? Ans.: $7,763.57 3.34. A sum of $1,000 is invested and draws interest at the rate of 8 percent compounded annually. At the end of the first year and each year thereafter $50 is withdfi3wn from the invested amount. How much money is still availabJe in the investment after {he 20th annual withdrawal? Ans.: $2372.96 3.35. In a certain financing arrangement the sum of $iOO,OOO is loaned at 12 percent compounded monthly as though it were 'to' be paid off in 25 years. At th~ end of 5 years the agreeme'nt calls for the borrower to pay back the unp3id balance at that time. What is the unpail balance after 5 years? Ans.~
,
$95,.653. 3.36. The expected annual inGome from a new fndlitY_,.that is under con!-iideration L<; $120.000, and the anticipated annuaJ operating exp~nditures are $60,000. The salvage value ar the end of the - expected life of 12 years wjJl be 20
ECONOMICS
51.
perceni: of the first cost. \Nhat first cost would result in a rate of retum Df L5 percent? )) A nsD: $337,868. 30370 .A. .processing plant has a first .cost of $600.000 and an expected life of 15 years with rio salvage value. Money i~ borrowed ' at 8 percent comp-ounded annuaUy, and the ,f'ust cdsLi"s .paid· cj!ff with 15 equal annual paymejjrso The , ;;(.,:~~h·;"" expected" , ~nnua1,~.iu.~pp}~e. ~s . $~OQ,OO.o·,: and' ann: i.lqI;)?pe~-~~~g exp'enses.. ,r:lfe . .".... ,.:,:. , ," ,$40,090 .. ' C6rpq'ratiiji( income ":tax' ,"is '5a:-p'ei~ehr of' the·' profits''5ef5i-e' :;iliXe's: =:. , .< -.. ' . and the'.' SYD, method of de'predation ' 'I s :applicable on the ~taXJife of .'¢~: .. ' . fac.ility I whic'h is 12 years 'with no salvage value. Compute th~ income .fa.'\: for (a) the first year and (b) the se60nd year. " , . , . Ans.:' (a) '$ 9846; (b)$14~576. ;f.. 3.38a A dieDt who is constructing a warehouse instructs the contractor. to omit insulation. The client .explains that he will operate the building for severa) months and . then install the in~ulation as a repair, so that he can deduct the expense from income tax at the end of tbe first yeat rather than spread it as straight-line depreciation over the 8-year tax , life of the warehouse. The . contractor points out that a' later installation will cost more than the $20~OOO cost of installing the insulation with the original construction.' How much . could the cli~nt afford to pay for the .later installation ,for equal profit 'i f he ,plans on a 15 p~rcent return on his jn\~~~tment p.nd corporarion income taxes are 50 perceiii?' " ' . - - .. .... ---. I
.'
Ans.: $25,461. 3.39. A $200,000 facility has an 8-year tax life, and the fiim exp~cts a 12 percent return on its investment and pays 50 percent corporation income ta."t on profits. The firm is con1paring the relative advantage of the SYD and straightline methods of depreciation. If the taxes computed by the two methods are expressed as 'uniform annual amounts~ what . is the advantage of the SYD
t
meiliod? . Ans.: $1630. 3640.. A fum borrows $250,000 for a facility that it will payoff in 10 equal installments at 12 percent interest, compounded annually. In computing income tax the firm can deduct the aCTUaL interest paid during the year. What is the actual interest paid the second year? Ans.: $28,290. 3.41. An investor pays $80,000 for a building nnd expects to sell it for twice that amount at the end of eight years. He can depreciate the building on a straightline basis during the eight years, or he can charge off no dep~eciation at all. On the capital gains of the sale at the end of eight years, which is $160,000 ($80,000 - depreciation) he pays half the income tax that he does 'On regular income. State which is the mo~t profitable depreciation pIon and ,give all the Ie3,$ons why it is most profitab1e. 3.42. Regular payments of $1400 ~e to be made annuaIJy, starti~g at the end ' of the first year. These nmounts' wil1 be invested at 6 percent compo~nded , continuously. How many yenrs will be needed for the payments plus interest to accumulnte to" $24,OOO? Ans.: 12 years.
»
52
J
-DES1GN OF TIIERMAL SYSTEMS
3.43. An investment of $300,009 yields an annual profit~of $86,000 that is ~pr~~d unnormly over the year and is reinvested w. mediately (thus continuously , compounded). The life is 6 years, arl.q th~re is no salvage value. What is th~ rate of return on the investment? AnSo: 20%. .,
." ,
'~ - J
_._. ," _ AnDITIONAt. ~ADrn-GS::" ;;
, 7'
•.. . .
:" - , '~ -,'", ,,-',
t"
::~;;,:;~; ~. ·N~~·. ffc~n~;"i~::;naiys;s f~~ i;;guie~i/ng"'~'i>1ailitg!!riiJ1.i>~r5;.iiq1t~ Mqking,·,. ..
-McGraw-Hill. New ,York, ' 1962" " " ' ', " DeGarmo, E:P.: Engineering Economy, Macmillan, New'Xork, 1967. Grant, E. L. t anq. W. G. Ires,on:, Prindples of Engineering Economy. _4th ed. Ronaid Press, ,New York. 1960. ._ _ , Smith, G. W.: Engineering Economy, Jowa State University Press, Ames, 1968. Taylor, G, 'A.: Managerial and Engineering Economy, Van Nostrand, Princeton, 'N. J., .1964. J
,
', 4
. : . :.'
J•
• ,
,J
. ,
~.
__ ...,,:':
,.
\:
•
•"'
'~'I
•
•
.
'\
:"'"','
.
"EQUATIO'N' FITTING
ttl . MATHEMATICAL MODELIN"C This chapter 'and the next present procedures for developing equations that represent the performance Gharacteristics of equipment, the ,behavior of processes, and thermodynamic prope~es of substances. Engineers may have a variety of reasons for wanting to develop equations} but the crucial ones, in the design of _thermal systems are (1) to facilitate the process of system simuiation and (2) to develop a mathematical statement for optimization. ' Most large,. realistic simulation 'a nd optiniization problems must be executed on the computer, and it is usuilly more expedient.to operate with equations than with tabular data. An em~rging need for expressing equations is in equipment'selection some designers are automating ,equipment selection, storing' performance data in the computer, and then automatically retrieving them when ~,component is being selected. _ . Equation oevelopment will div~ded into two different categories; this chapter ~eats equation fitting and Chapter 5 concentrates on mo~eling thennal equipment. The distinction between the two' is that this chapter approaches the development of eq uations as purely a number-processing operation, while Chapter 5 uses some physjcal··.Jaws to help equation development. Both approaches are' appropria~e. In modeling a reciprocating comp~ssor, for example, obviously th~re physical explanations for the _ perfonnance, but by the time the complicated flow processes, c'ompression, reexpansion, B?d valve mechanics are incorporated. the model is so complex that it is simpler to use experimental or catalog data and treat the problem as a Dumb~r-processing exercise. On the other hand heat exchangers follow . .53 ' J
be
are
I
,~
54
DESIGN OF THERMAL SYSTEMS
certain laws that suggest a fo~ for the equation, and this insight c~ b: used to advantage, as shown in Chapter 5. . .. ' >,.;....:. Where do data come from on \Vhich equations are based1··Us~an· . the da~a used by a designer come from tables.. or ~aphs~ . Expe_riment~ .' data from the laboratory might provide the basis , and the techniques' E " .
the
, "
.
-:-)
"
. this 4lld the· next ·ch~pter: ~e ap'plicable ·to.processing laboratory . data ..~ Bu: . . system designerS,'are usually onb."step removed from the laboratory and ~e : '.,.: .',:>."... ' . selecting CQgID1~rclally available components for :which the manufacture. r.·· . ." . . .... , ., /!~:: 'has_. p~o vid~~ ~'peifoim~ce;.:datif>In/.aiJew:~ei~iMt.31?:C;:~,§.'JP:?~ ~X~ctnrers~ma) .. reserve several im~s a ··pag.~ of..tabl+lai. d_ata to provide~ the·. 'equ·a'tion:~ti1at represents -the table'. If and wpen -that··practice b~comes. wi.d~sp[ead/.lhe system 'deslgner~s t~sk will'be made easier. That .stage",' ,h owever, has p'o t yet been re'ache~l ' : . Much of thls chapter presents systen1atic' techniques for detennining . the constants and coefficients jn equations~ a process of following ,rules . Th~ other facet' of equation fitting is that of proposing the fonn of the equation, and' th,is operatiqn is. art. Some suggestions will be offer~d for .the execution of this art. Methods will be presented for determining equations that ·fit .a limited numbe'[ of .data .points perfectly. Also explained is the Dlethod of least squares, which provides 'an equation of best fit to a .large number of poin~s. . .,," ', . ~ "':"_"".
on
an
4.2 MATRICES All the operations in this chapter can be perf9ITned without using matrix tenninoJogy, but the use of matrices provides .several conveniences and insights, In particular, the application of matrix tenninology is applicable ' to the solution of sets of simultaneous equations. A manix ·is a rectangul8! array of numbers, for exa~ple,
[~
-r] [-~~] [; ~l
-2 1
[
all
'012
021
022
a I~l;
.
.. .
'Q'm"2' • " .. .
OIn] 12n a mn
The numbers that nlake up the array are caJled eleJJ7enfs. The orders of these matrices, from left to right, are 3 X 3,3 x 2,2 x 2, and nl X n . A transpose of a matrix (A], ~esjgnated [A]T, is formed by interchang~ ing rows and columns, Thus,. if '" (AJ
,
=
3 -1] ~ [ 2 0 4 '-2
then
[A]! =' [
3
-1
2
0
·41
-2
. To multiply two matric'es,' multiply elements of t.he fjlrst ·ro1w of ,the left matrix by the corresponding elements of the first column of [he righr
EQT6-ATION FITIlN.G
J .:?
Inarrix~
b1en sum the products to give t~~ .ele,ryent of the first row and fir~t . column of the product matrix. For ex.ample~ , the multiplication of the t'i,:vo
m_atnces
-,
....
, gIves ,
,
(1)(-2~ +, (~1)(3), + (0)(1).(1)(1) '+ (-1)(0) + (0)(4)] == [-5 Ij[ (2)(-2) + (0)(3) + (l)(J) (2)(1) + (0)(0) . +. (1)(4) 6 ' ~3
the convention for the multiplicarion of two matrices offers a slighriy shorter form for writing a system of simultaneous linear equations. The three eq~ations
2t I
-
, Xl
+ -3x 2,
4Jt- I
-
Xl
2x 2.
'+
3x)
+
x-)
=6 =1 0
,
ca!1. ,be vlrritten in matrix form as 2
3][XI] .[6] X2 == 1
'.-1
1 3 [ 4 -2
0 1
0
x)
'· -The determinant is a scalar (which is simply a number) and is written between vertical lines. For a 1 x 1 matrix it is the element itself; thus lalll = all
A technique for evaluating the detenninan-t of a 2 x 2 matrix is to sum the products of diagonal elements, assigning a positive sign to the diagonaJ moving downward to the right and a negative sign to the pr:oduct moving upward to the right: '
An extension of this method applies to
compu~ing
the detenninant 'Of a 3 X 3
matrix: al2
aJ3
a22
a23
032
a33
------;;-
- +\+ \ + \ I
".= a II a22 G33 + a l2 a n G31 + a 13 -aJt"£j22 G 13 -
J
-) -) -
a32 Q 23 Q 11 -
Q
21 a32
a33 Q 21 0 12
56
,DESIGN OF THERMAL SYSTEMS
Evaluation of d etennin ants' 4 x,4 and larger requires a more general . procedure~ , which applies als,o to ,2 >(2 an!q 3 X 3 ,matrices. This proc.edure ',is row, ·expansion or column ' expansion~ The determinG$t of a 3 x ~ matrix' ., ,found.,b y expanding abo~t the ['list column is .•
', .,~et =':~l)Al}j,
'+ a21 A ,21 + . a31~31
, .
is f~und as' .' .<:, ...... :-~::~~;.> ·t:~-:~'. I:~~.; ,:': .:,~(,~:,,,~.;.~/.:?~-»~::·i~j"i~i~:~~,./,,::.~1·.~,>. '.~,; ,:' ..->: .:.:~:,.:~ ' '/,:;?~;::~,'~<:>_.~' '" '< ::':, ,
a
, , submatrix, fomied '/
A .. , ,
:."
i; :~e cofa.~tor
,::,,,,~,:,,:,.,? }'Ihere-AU, ,iB}be".c:tJja;tor ~f tlie~l~~en~ .' .' ....:,:·f911ows;'
'''::-1,-
= [( -l)i~j J
?y strikingo~t " " zth row and lID
fl, '
,'
','" "
column of [A]
~ '/
For example, the cofactor, o~ a21, which.is A 21 , is
Example 4.1. Eyaluate' ._, 1
-1 0/· 2 0 1 2 1 5
2 1
o 3
-1
4
·2
, Solution.' Two elements of the' second ,row are zero,
so that row would be a
convenient one about which to ,·e xpand.
del =
Q:!l
A 2J
+ 022 A 22 + G'2JA 23 + 1
= (O)A21 + (1)( _1)2+2 +
1 (2)( -1):!-:-3 3
4
= 0,+ ,
10
+
2
3 4
-1
024A24
0
1- ·2
1 5
0
-1 . 2 + (O)A 24
2 5
46 + 0
= 56
4.3 SOLUTION OF'SThfULTANE'O US EQUATIO'NS . There are many ways of solving sets of simultaneous equations, two of which wjJl be descrlbed jn' 'this section, Cramer's rule and gaussian eJimipa!ton. For a set of llnear simultaneous equations
EQrUA1l0N FITTING
+ Q12 .X 2 ·:+ il13:f3 = b l . ,anAl .+ a2~x2 + a23 X 3 = b2 . ·Q31X 1 +- a32 x 2 + a33 x 3 == b 3 J .. ' . . . . . v/l).ich' co.:o,· oe·'\,;ritten- in .matro(IoiID .". ".. . . -': . .: ., .,. . .. ail x
,
..
--
.,.-~
.
.. .
.
.
.
:.
. ..
.
;
...
!
.•
.. ;"
.: '
Xi=
(4. 1) . ' ..: ":
.:
. .
,-
::r~~][;:]H:r ··[BJ· ·
.·.• IAJfXJ. [;:: Cramer's rule
c
,-
'.: '
57
(4.2)
states that
j [AJ rp~trix with fBJ matrix substituted in ith colurim· I
\AI
Example 4.2. Using Cramer's l1,lIe. solve for . linear equations:
:
.
. -".
[i -}
1
-2· 0
X2.
( 4.3)'
in this set of simultaneous
-lfl] -fJ 2 3
X2
9
-
0
X3
Solution
~l2
2 3 1 9 -1 0
=
.2 1
-1
.....,...1
2
3
-11
1 -2 . 2
=-
3
0
30
= , -2
-]5
.
Equation (4.3) suggests that none of the x's can be detennlned if IA I is zero. The equations are dependent ·in this case, and there ·is no unique solution to the set. . Another method of solving simultaneous linear equations is gaussian elimination, which will be ilJustrated by solving XI
3.x 1
- 4x 2 + 3x 3
+. X2
2x J +
"2
-
+
7
(4.4)
2x3 =
14
(4.5)
=
5
(4.6)
X3
::= -
The two major steps in gaussian eli~ination are conversion of the coefficient matrix into a triangular matrix and solution for Xn to Xl- by back substitution. In the example set of equations, the first part of step 1 is to eHminate
the coefficients of Xl in Eq. (4.5) by ·multiplying Eq. (4.4) by a suitab1e constant and adding the product to Eq. (4.5). Specifically, multiply Eq. (4.4) by: ~ aud add to Eq. (4~). Similarly. multiply Eq. (4.4) by -2 and add
[0
Eq. (4.6):
58
DESIGN OF THERMAL SYSTEMS
Xl -
4X'2
t
== -7
3X3 ,
!
(4.7)
~3X2 ~:' llx3l = 35
:9X 2 -
<"
. , ,. . .
(4.8) ,
5 X 3 == 19 ,
, ,(4~9},
· Th~ last part ofstep lis .to m~l tipI Y F1 ,!4J~) 1?y.~nlWdadd to ~Eq;
~:-.,.'. ",~, .;;";,,,.biSl:tS~p1p,Je ~~~~th e, tri~gul ;ui~~tkm'c0,-,;~ ,~::<::.,:;•.,:" . :" , , ': ', ."':'~-. ' .. : . ' x}'.:..- 4x 2 +. ,3X-3-' ==. - ,,7 " " .' "-, .
.'
(.4.9), ...
..",,"'.' :.; .;
....;".
,(4.JD). ... :: '-: _
~~
,, 34 ' ,
,(4.11)
35
13x2 - -llX3 , .
68 '
. (4.12)
. -X3 ,= -,--:-' ,13
'
' 13.
In 'step 2 the value' of XJ Cp.D be detennined directly from ~q. (4~I'2) as, X3 = -2.-,.Substituting the 'value .of X3 into' eq. (4.11) and solving' gives ' .x2 == 1.' Finally, substitute the values ,of -X2 and X3 into Eq. (4.10) t?, find that x 1== 3. If a different set of equations were. being solved, and in the equation corr~sponding to Eq.' (4.8) if the coefficient of X2 bad been zero insteaq of 13, it Wquld have been necessary to exchange ,the posjtions of Eqs. (4.8) and (4-.9). If both the X2 coefficients in' Eqs,. (4.8) and (4.9) had been zero, this would indicate that the set' of equations is dependent. ' Ivlost computer departn1ents have in their library a routine for solving set of simultaneous linear equations which can be called as needed. It may be convenient to \vrite one ~ s own subprogram using a method like gaussian elImination. 1 It \vil1 be useful for future work in this text to have access to
a
, an equation-solving routine on a digital computer.
4.4 ' POLYNOlVIIAL REPRESENTATIONS .
.
Probably the most obvious and most useful form of equation representation is a polynomiaJ. If y is ,to be represented as a function of x, the polynomial form is )' = a 0
_
+
alx + a"x':J. _ + ...
+ a II xn
(4.13)
\vhere a o ,to Gil are constants. The degree of the equation is the hi,ghest _ exponent of x, which in Eq. (4.13) is 11. Equation (4.13) is an expression giYlng the function of one variable in tenns of another. In other common situations one variable is a function of [wo.or more variables, e.g., ~n an axial-flow compress~.r
, Flow rate .=
f (inlet pressure, inlet temperature
I
compressor speed. outlet This ......£ann Qf equarion will be presented"jn Sec. 4.8 . . .,. ~
c'
pres~ure)
EQUATION FlITJNG
59
b
,\Vhert the :lumber of data points aV :?l..~Iable is precisely the saIne as the degre.e of .the equation plus I, J2 + 1, 3! polynomial can be devised that exactly expresses dIose data. points- \Vh~n d1e number of available data points exceeds n -Iit may, be advisable to seek a polynomial ~hat gives . ".:;-; ,- . - the ~I.best fit" to the data points (see Sec. 4.10)0' , . ' .' .- J' .' ' . .' . .1JJe· fi~sL and -.'simples.t caseJo "be considered :is where one variable is. ?
"
. -Y·a1u:nc.ti6~ of anot~er"van~b~e.·;and·, .the·{ nl1mbef..·:bf~data~':p·oints!".equarSr:-n-'~.:T~,'h'· :t::,:·<.;:::.'.·~' "
. "
4,,5 POLYNOIVIIAL, ONE VARrABLE.A
~JCTION OF p}NOTfIER VARIABLE n + 1 DATA POINTS
AND
Two available data poi.nts are adequate to describe a ,[ ust-degree, or lineaL . equation (Fig. ,4-1). The fOmi of this first-degree' e'qu.ation is (4_14) ~.
The ,x y
.pairs for ~e two known points (x 0' Yo) and .(x 1 , }' t) can be substituted into Eq. (4.14), providing two linear equations with two unknowns,
.'
Qoandal
Yl = a o
+ alx I
For a second-degree, or q uadratic, equation~ three data points are needed; for example, points 0, '1, and 2 in Fig. 4-2. ' The x y pairs for the three known points ca~ be substiruted into the general fonn. for the quadratic equation
(4.15)
y
, F1GURE 4-1 Two points describing
8
liDeru equation.
60
DESIGN OF THERMAL SYSTEMS
y . l .)
-2 , .... .. ,
,
:
... ','
"
'
-,: ,:~'," , ::,',~ ?,~,",,;' ~ . ,;. , ,.....:.:
"':
"
:...:.-
.,
,.'
. ..:. . '.
-'
":"
.., '
.
" ,:
'~' .
.J
'
. .,
" . j
";-"
:". j, _
~
_
•
A•• , ; • • :
~•
• • ,J
;-:,: ' : ' .
"
~
_.
• _
.
J
_ .
•
4
.;:._,
~ . ~
.
. .
.
,
,. . . . .~ \'
' ,-
"
,
flGURE 4-2 ' x
~ p~in1S describing a 'q,uadratic eq.u~tion.
which ,gives three equations ~
.. : ~ -,-
"
"
.-.. :
[i ;~;m~~] [~~] =
2
2,
The solution of these three linear simultaneous equations provides the v~lues of 00' a J, -and 02 . . The coefficients of the high-degree terms in a polynomial may be quite small~ particularly if the independent variaqle is large. For example, if the , 'enthalpy of saturated water vapor h is a function of temperature t in the equation h =
Go
+ all + ... + 0515 +
Q6t6
where the range of f extends into hundreds of,degrees, the value of as and
may be so small that precision problems result. Sometimes thjs difficulty can be sunnounred by defining a new independent variable, for example, f /100. a6
-
h =
Do
+
f 0]-
100
- 1
.-~
-
+ ... + a5(--~r + \ J 00 .
f)6 06(-'100
4.6 SIMPLIFICATIONS WHEN - THE INDEPENDENT VARIABLE IS UNIFORMLY SPACED ' ,
Sometimes a polynomial is used to represenr a function, say y = I(x), where the values of yare known ,at eRually spaced values of x. ThlS situation t exists, for ins[ance when the data points are read off a graph and the pojnrs can be chosen at equal intervals of x. The solution of simultaneous t
\'
"
EQUATION FITTING
~~r"
6JL
equations to detennine the coefficients in the polynomial cc~n be performed symbolically Dl advance,2 and the execution pf the calculations Iequir~s a relati vel y s_maJl effort thereafter. Suppose that the curve in Fig. 4-3 is to be rep~_:oduced by _~ _ ~ourili degree- polynomial. The n + 1 data P9llts (five in this -caSe)estahlish a -P9l ynomiaI . degr~.e ' (J., ,.:(fo.llr.,~,:irr.·. .this_~·cq,se).·. , Tn.e. spacing qf' ~_e points .is
of
.
;"0 ' ~- 'x 2 -- X -·il':-.. " .: Xj'>./-.r.-;;'[ 1"- == ~ x ~t,- "x 3'~·"··1?fle,',rang&·},qf-:.i'~~~r~' ': ~;~~i'~ .(\ is,;;:!;~··: ;.i~ ~; '. --. ,' : ':,-designaied.1?·~ 'and :the' symhols' are 'ily I -~ Yl "- Yo~" L\Yi ' " Y2 '~ Yo; etc. - '
.. -
'~~~:&:·;-~~;.:·-i ~:: ~,
, Instead of the polynomial fonn ofEq. (4.13), a~ alternate form is 'us,ed .
Y --Yo
=
'1
' .
-3
QI[;(X -Xo)1 + G2[;C1: -Xo)1 -+ aJ[~(X ~xo)J
+~4[;(X -xo)f , To find a 1 to /1V '1
.,
=
04,
at
·
(4.16)
first s~,bs'titute the (Xl, ),1) pair into' E'q. (4.16)
2 [4(X1 -'X O)J , [4(Xl -~Xo)13. + G'I + a3 ' , -. R ' R
4(xJ - ,x o)
+a4 [
R 4(X 1
-: xo)l~'
R
(4.17)
'
Because of the unifonn spacing of the points -along the x xo)! R = .1 and so Eq. (4.17) can be rewritten as
axlS, n (x ,! -
t
(4.18) 4
y
I
4
Range
~
R
g
,
I
,
'
FIGURE 4..3 PoIynomjal....repr~ntation when
....
:-
point! are equalJy spaced along the x ..,
axi~.
62
DESIGN OF THERMAL SYSTEMS
TABLE 4.1
~ons,tants .41. Eq . (4 .. 16) ' -I .
:.
:
I1:Y2
I "
2
· ·Fourth
.
.
-fly]
." . -3~~'''-'7a~ •
~ :'."'~~'~."~;
• .)\'
•
~.""', ~I-
: _ :.,
1"1" : '
••
: ••
----------..,............,.....-.--.,~-~:___--.:...~~--_o__-____:_--'---~ .
. Cubic .
.kC;~yr:.+:·A)'J . :
. .'
: ~C~'2. .~.··i.AY"I) . ". ' ~ A);I '-: q2 '~: a~'
-:- 3~y.2)
"
. - 3a3
Quadratic
' . .6.)' I
Linear
U sing the' (X2;: X2)
parr 'in4 th.~ 'fa~t that ;1(X2 Ay~
::;:
2al
+ 40 2 +
xo)/R
.
== 2 gives ', (4.~9)
8P3 '+ 1604
Similarly, for (X3 ~.: )'3) and (x 4'- Y4) . L\)'J'
==
30 1
.
AY4 = ·4aj
+ 90 2 + 27aJ'+ ' 81a 4 . .. . +. 1601 + 6403 + 256a4
(4.20)
( 4.21)
The expressions for Q! to a4 found by solvjng Eqs. (4. 18) to (4.21) sitnul~ taneously are shown in Table 4.1, along \yith the constants for the cubic, qua.dratic, and linear e9uations.
,·4.7 LAGRANGE INTERPOLATION 'Another form' of polynomial results when using Lagrange interpolatjon~ , This n1ethod is applicable, unlike the method described in Sec. 4.6, to arbi,rrary spacing along the x axis. Ir has the advantage of not requiring the sih1ultaneous solution of equations but is cumber$ome to wrire our. This disadvantage is nor applicable if the calculation is perfomled a digital computet. in 'v/hich case the programming is qui~e compact. With a quadratic equation as an example, the usual form fPf a function - of one variable is
on
(4.22)
For Lagrange interpolation, a revised form is
.v :::: C d x
- x:?) (x - X 3)
+
I
c 2 (x -
X J)
,u~ed
(x - X3)
+ c 3.( x
- Xl) (x -).' 2 ) (~.23)
"';'" ," :"
,~
.
EQUAT10N F1ITING
63
&-
Tbg tJ:rc:c a:vail~b e d~t 8.. points are (Xl: ~~JI)! r'~21 Y2), an~ (X37 Y3). E.quation (4.23) could be mul tiplIed out and teI!TIs coUected to show the c9rrespoDdence to "he form in Eq', (4:22). . By se tting ~'( = . Xl, Xl, and X3 in tum in Eg. (4.23) th~ .cqn~t?-nts .can be found quite ' sID:plX:.. ..< .~ ~.~~ . ,'.J.,:... .... . ;- ...... . . '. . _~ .. ...:1-'.:.:';_ . .
"'<7':"'; .,-.:" '. .
..' ,'.
=
.
.,CI
I.
.
~
......
I...
. , ' .... . ~
(..t· l·:~.:~ti).Cx ,
~.
"
~.'
v 1 .. ",. '.;.',., :.:.... ,:'..: :::".:
. ·__
...
.
i
"
.
~"x"j)
......
....:.~ . " ..... ;", '
,'- ', .....
I
-,'
. ..d.
,The general form oJ the equation''for fllldmg the value of J; 'for'a give:q w~
\vhen
11 . data
points · are' known is ' ~
V' "'
== LY'
nn. -ex. -
i= 1 .1 j
Xj)
.
omitting
(x -.x i)
= 1 (X; - XJ) omitting (Xi - Xi)
(4.24)
where the pi~ or product sign, indicates mUltiplication. The equation represented by .Eq. (4.24) is a _polynomial of .degree Jl - 1. '.
4.8 FUNCTION OF TWO VA1QABLES A performance variable of a component is often a function of tWo other variables,3 not just one. For eX,ample, the pressure .rise developed by the centrifugal pump shown in Fig. 4-4 is a function of both the speed S and tp.e flow rate Q. . , If a polynomial expression for the pressure rise /1p is sought in tenns of a second-degree equation in S and Q, separate equations can be written for each of the three curves in Fig. 44. 1bree .points on the 30 rls curve' would provide the constants in the equation
D.Pl ==
al
+
b1Q
+ cIQ2
(4.25)
Similar equations for the curve~ for the 24 and 16 rls speeds an~ . .
L'lP2
~.p3
= 112 + = a3' +
b2 Q +
b3 Q
2
C2Q
+ C3Q 2
(4.26)
(4.27)
'Next the a constants can be expressed as a second-degree equation in tenns S. using the three data points (~'1 30), (a2. 24), and (a3. 16). Sucp an equation would have the form
'of
.-
a
= Ao
+ A JS + A 2 S2
(4.28)
I
!• I
64
'
DESIGN OF
TIIERMAL
SYSTEMS
I
", ~
,
.~...,::-
J
-
.. ~ .~~,~ :,. :,~,~ >'; ~-~,; "","
',\,. :'-, ';,
.
'
.
rI . "
' .'
••••••
'.
. ... . ,. • • • • • ,'
.,.: .. :::~". :
FIGURE 4-4 performance of a centrifugal pump ..
Similarly for b and c
+, B1S + #').S ,2 Co + C1S + C2 S2
b' = Bo " C
==
, (4.29) (4.30)
Finally, the constants 'of Eqs. (4.28) to (4.30) ~e ,put into " the general .
equatio.n
.11p == Ao + A1S + A 2 S2
+ (Bo +
B1S
+ B 2 S2)Q
+(Co + CIS + C2 S 2)Q2 (4.31)
, The A9 B, and C constants can be computed.if nine dpta points from Fig. 4-4 are available. Example 4.3. Manufacturers of coolmg towers often present catalog data showing the outlet-water temperature as a function of the wet-bulb temperature of the ambient rur and the range. The range is the difference between the in1et and outlet temperatures of the water. In Table 4.2, for example, when the wet0 bulb temperature is 20°C and the range is 10 e, the temperature of leaving w~ter is 25. 9°C and so the temperature of the entering water is 25,'9 + 10 = 35.9°C. Express the outlet-water temperature t in Table 4.2 as a function of the wet-bulb temperature (WET) and the range R. I
Solution. A second-degr~ polynomial equation in both independent variables will be chosen as the form 'of the equation and three dlffere-nt methods for developing the equation will be illustrated. , Method J. The three pairs of points for WBT = 20°C, (10, 25.9), (16. 27.0). and (22. 28.4), can be represented by a parabola I
,
t = 24.733
.-
.'
+ 0.075006R + O.004146R2
/"
'5J~
EQUATION FITTING
TABLE lL2 ~O ~t.lu. ~'~'aDl ;' - - ;;ia " \:: 0. "(" "'e ,., P .? ~<1: k. ~ ' ~ o ... .... t it .5. l il _J _ .!I. al '; U'.' "-: ?
°iL r-· , (' t 'V_
I )
cooHn g tov1er in Example 4,,2 ,
ForW'BT =23°C I
t = 26.667
'
+ O.041659R +. O.0041469R 2
and for WBT =26°C
+ O.024999R + O.00414&1R 2
r:= 28.733
Next. the const~nt terms 24. 733. 26~~67 .. and 28.733 can be expressed by a second-degree equation of WBT. 15.247
+
O.32637\VBT
+
2
O.007380WBT
The coefficients of Rand R2' can also be expressed .by equations in terms the WBT. which then provide the complete equation t
=
(15.247
+
+ (0.72375
O.32637WBT
+ O.007380\VBT2)
.:. . O.050978\VBT
+ (0.004147 +
OWBT
of
+ O.000927WBT2)R
+ OWBT 2)R'2
(4.32)
!vi~thod 2. An' alternate polynomial form using second-degree expression's for Rand WBT is
(4.33)
The nine sets of t-'R- WBT comb'ina'tions expressed in Table 4.2 can be substiruted into Eq. (4.33) CO develop nine simultaneous equations, which can be solved for the unknowns c1 to C9' The c values thus obtained are C4
= ~5., 247 = 0.723753
ci =
C7
= 0.00092704
Cs
Cl
,
C2
= 0.32631
0.0041474
= 0.0
, C3 = 0.0073991 C6
C9
=-
0.05097'82
= 0.0
It is possible' to mU'ltiply and collect the terms in Eq. (4.32) to develop the equation of the form of Eq. (4.c3). Method 3. Section 4.7 described a polynomial representation of a dependent variable as a function of one independent vanable by use of ~nterpolatjon. La.$.range interpolation can be extended to a function
66
DESIGN OF TIIERMAL SYSTI:MS ~
-of two independent variables,' for- example, if z chosen 1
= I(x ,Y), the form
-
can t:
'
',I ,.
'z = co(x - X2)(X -'X3)(Y - Y2)(Y - Y3) , ' +C12(X' -X'2)(i:.-',X3)(Y -Yl-)(Y ,
-:
-
.,-,'·,,=;Y I ....
~•
• "
••••
-':')'3).
. -- . " +C13(~'~ X2)(~ ~'X3).()' -~ Yl?(~ ~~~2)' ,
-
_ ~..
'
':
••• •" : •••> , ' .
.. -..
~
:>-', , "
, :'. :
,
~
'
. .::-:
-
,
-.: . .
~
,'+.~..... ~."C31(~ ~~ :X.t)(x. ~.X2)(y·-' YO~(y -- Y.2}-::i ..,: ·:..·.(1·34
~6re~~s~ni~e 4a~ ta~le 4~2; ~ couldref~
outl~t-\\1ater
in to !he teolPerature, x the WBT, arid Y the· range. In Eq. ·(4.34.) Xl .- 20, X2 == 23, and X3 ' ,= 26, while YI ' = 10,}'2 = '16, and')'3 =' 22. " .. ,'. ' "'- To determine the magnitUde of for example', values. applicabJe when x = ' x. land y =)'2 can be substituted into Eq'. (4.34). ' ' . . .. .. -
cn,
27.0 C12
= "(20 - 23)(20 - 26)(16 '- 10)(] 6 - 22) = -0.04167
.
,
. 4.9 ExPONENTIAL FORMS The dependence of one variable on
~
second variable raised to an exponent
is a physical relation occurring' frequently jp engineering practice. The graphical method of determining the ~onstants b and m in the equation y'
== bxm -
(4.35)
is a simple example of mathematical modeling of an exponential faun. On a graph of the known values of x and), on a log-log plot (Fig. 4-5) the slope of the straight line through the points equals m, and the intercept at x = 1 defines b . ' .
." JOOO
I
-
,
,-
JO
I I I I I
O.t
FIGURE 4·5
10
lob
1000
Graphic~ determination of the constant b and exponent m. ,.
~
EQUAT10N FInING 0-
y
V
67
iI
)')
..:. ' . .,.
.': ~.
~----~----------~~
x (h)
(a) FIGURE 4-6 Curve of the equation y
=b +
ax nJ •
The simple exponential form ofEq., (4.35) can be extended. to include a constant
(4.36) The equation permits representations of curves"similar to those ShOVvD in .. Fig. 4-6'. The curve shown in Fig. 4-6b is especially c"ommon i!1 engineering ·practic.e. The function yapproaches some valu~ b asymptotically as x
mcreases. One possible graphical method of determining G, b, and m in Eq. (4.36) when pairs of xy values are known is as follows: '
JL. Estimate the value of b. 2 .. Use the steep portion ' of the curve to evaluate m by a log-log plot of y - b vs. x in a manner similar to that show~ in Fig. 4-5. 3,. With the value of m from step 2, plot a graph of y vs. x nJ• The resulting curve should be a str;iight line with a slope of a and an intercept that indicates a more correct value of h. Iterate starting at step 2 if desired.
4 . 10 BEST FIT: METHOD OF LEAST SQUARES
fir
, This chapter has concentrated so pn finding equations that give a perfect fir to a limited number of points. If nl coefficients are to be detennined in an equation, ,m data points are required. If more than m points are available,
it is possible to determine the nz coefficients that result in the best fit of the equation to the data. One definition ,of a best fit is the one where the sum , of the absolute values of the deviations from the data points is a minimum. In anothei ."type of best fit sligRtly''different from 'the one just mentioned the sum of the squares of the deviation is a minimum.' The procedure in
.'
~.
.,.
68
DESIGN OF THERMAL SYSTEMS
is
establis.hing the 'coefficients in such ~ eqUf.tion called the method of least :'. . I ." ' . squares. ' . . Some people proudly announce their use of the method of least .squares in order to emphasize the care they have lavished on ·their dacii · ~aIysis. . : Misuses of." ~e .method ·as. ill:ustratedj~;·B.ji~~A.~J,ct-:and·1i;~~e~·n~f.ti~cQnnnf)lfA~.:~ I . .., ,. . ._ . . ..... . . ~.Q.;.,In/ . ~ig~:·: .4-74;. . ¢hile)a:: straigp.t" line can, he"fourid thaf~resllltS iIi the least":" , . :'squares' .deviation," the 'cprrelation betWeen the .,. x ·~d y variables seems , questionable ~d perhaps no such device can i,m prove the corr~l~iob.. The scatt~r J!lay be 'due to the 6~ssion of SOIl1e significant variable(s). In Fig. " - :-~"4-7b , it w'o uld 'have' been p~eferable to- eyeball.in the, curve, rather than to .fit. a straight line to the data by the least -squares method. The error was not in llSing least squares but in applying a curve of too low a degree. Th,e pro.cedure for using the least-squares method fo~ frrst- and seconcldegree polynomials will be explained here. Consider -frrst the linear equation of the fonn 1.
-.
" " .< .
'
.,;.
'
,
.
y =
a+, bx
(4.37)
where m pairs of data points are ayailable:' (xJ' YI), (x2,Yi), .'~., (x m ', Ym). The deviation of the data point from that calculated from the equation is· a + bx i - Yi' We, wish to choose an a arid a b such that the sumrriation m
L (a + bx
j -
Yi)2
,---7
rilln~um
(4.38) .
i=l
The minimum occurs when the 'partial derivativ'es ofEq. (4.38) with respect to a and b equal zero. ' m
~
.
'
2 a"'-' (a+bx·-y·) I= I I ' - - - - - - - - = L2(a +
aa
y
bXi -
Yi) = 0
'Y
o
o o
o
o. ·0
,
I
X
Co) FJGUR.E 4:2 Mjsus~ of th~ method of least squires.
(b)
77 -
EQUATION fffTING i-
=
An eqLation of the fOITH 'v I
-
-
[l'X
+b/x has been chosen to fit the folIowlDcr
:::7
pairs of points: O. 10.5), (3, 2), 211°)(8,18). Choose a and b to give the best fit to the points' 'in the sense of least sUf0. of the deviatioris squared. ' Auso!, b = 3.14. 'L.t16a The propos'ed form of the equation to represent z as a function of-x-and}' js z = .qx + b[ln(xy)], wh.~le" a and b are co~stants. Some" -dart;!: rela,ting these (x~ y)
, ,vanna es "are '" "
'
0
1,'
,
.:
x
i
..
y
2
2 :5
:2
4
2
2 ;
.
Determine the values of' a and b that give the best fit oJ the equation to the data in the sense of least square deviation. Ans.: b = ,-:-0.35. 4.17 .. With the method of least squares, ' fit the enthalpy ,of saturat~d ' liquid Iy by ' means of a cubic equation to the temperatur,~ t in degrees Celsius using the 11 points .on Table 4.3. Then compute th~ values of hi at the 11 po~nts with , the .equation just developed. A~.: hi = -0.0037 '+ 4.2000t - 0.000505r 2 + 0.000003935-t 3. A frequ~ntly used form of equation to relate saturatjon pressures t9 temperatures is B Inp=A+-
,
where p T
=
T
saturation pressure, kPa absolute temperature K
=
7
With the method of least squares and the 11 points for Table 4.3, determine the values of A and B that give the best fit. Then compute the values of p at the 11 points using the equation just developed. Ans.: In p = 18.60 - 5206.9/T. tt19: The variable z is to be expressed in an equation of the fonn
z
=
ax
+
by
+ ex y
The foJlowing data points are available, and a least-squares fit is desired:
,
Z
:c
0.1 -0.9
J
2.0 -1.8
2 3
y
~
1 2 2 1
Detennine the v.alues of a, b ~ 'an'd c. ""-.- Ans.: -2.0467, -0.9"167, and 1~8833.
78
DESIGN OF THERMAL SYSTEMS
)'
Approaches y axis
Approaches ~ straight line
.-
,,- -
-
' . . _. ~..~. ~ .
...
'
•
... '
•
J
\
aSymptoticallY
.~ : -~' J'
I
, .. ...::-~
.'
I'
,~:!
'
-.~ -.'-" ~t.
"
.
'.~. ,;
. -.
' ~'.
.
---:f· :.
'-. -- .
... - .L.:..:..:.-...:........-£.._ _ _ _ _
-_--'--_~----'-----1f1-
x
FIGuRE 4·13 Function in Frob. 4.19. 4.20. Tbree points, (XJ,Yl), (X2,Y2), and (X3,)'3), lie precisely on the straight line y = a + bx. If a least-:-squares best fit were applied to these three points to detenninc the values of A and ·B in th~ equation y = A + Bx, show that the process wo.UJd indeed give A = Q, aT1d B ~ b. 4.21. An eguatjon is to be found that represents the. function shown in Fig. 4-13. Since one single simple expression seems .inadequate, propose that .)' = 11 ex) + fz(x). Suggest appropriate fonns for II and f2 and -sketch these
functions . ./' 4~22. The enthalpy of a solution is a function of-the temperature t and the concentration x and consists of straight lines at a constant temperature, as shown in Fi.g. 4-14. Develop an equation that accurately repre~ents h as a function of .x and t. 4~
40 35
,
30
2
:!5
c
20
r: .r:. !J.)
_. 10
5
0
,.
0
,
p.S
1.0 '
I
Concentration. :r
FIGURE 4·14 Enlha~clion
of
lempera!~rc and
concentration in Prob. 4.20.
BtUATION FrITI;-..IG
-I- x 2 - 4x 3 + "OX..} tI- I3.t• 5 - 2r 6 = 16 + 2r 2 + 3x 3 + 5x..j. - 2.r 5 = 7 x I ~ 2'( 2 - ,SX 3 + 3x 4 -I- 2x 5 + X 6 = i 4x 1+ 3X2 - 2'(3 + ::~,:r4 + X6 = -
75
2r, 1 - X I
, . 3Xj'
*
,
"..,;":
-I- X2 -', X3,+ 4.r~ +3X,5 6X6 = """:'11,, ' 3x.~ +" ~t 5 ~-~',:( 6 = 5'
, 5x,j' '+,2x,2 -,;- '2r3 . .\ . . . .
+'
' ,
.. ' " ,', ~~~ ..:\ ' ~ 1.;:' t no.~.3:~ ~A. , 4.4~ A second-degree eq'uatiQn of. the form' , y
I. '
, = a + ox + cx")
has been prqposed, to pass through the -thr~e (x ,y) points (C3). (2.4)·~ aod (2, 6). Proceed with the solutio l1 for a. b? and c. ' (a) Describe any uriusual problems encountered. . ' (b) Propose an alternate second-degree relation between x arid y ' [hat .. will successfully' ~epresent these three points. 0 :f4.So Us~ data from Table 4.3 ?t I ' ~ o~ 50, and 100 to establish seconddegree p.olynomial that fits hg to t. Using the equation. compute hg a[ 80°C. 'Ans.: :2643.3 kJ/kg. ' ' 4.6. Using th,e data froD1 Table 4.3 for Vg at I = 40, 60, 80 and 100°C. develop a third-degree eguation 'similar in form to Eq. (4.16). Compute Vg at 70°C
a
e
using this equation.
Ans.: 4.91 m 3 Jkg. ,t 7 .., Lagrange interpolation is to be llsed to represent the epthalpy of sary.rated air~ hs kJlkg, as a function of the temperature tOC. The pairs of (h s , t) values to be used as the basis are (9.470, 0), ,(29.34. 10), (57.53, 20), and -(99.96, 30). (a) Determine the values of [he coefficients Cl [Q C4 in' the equatiori for' lIs. (btCaIculare hs at 15°C. ' A.ns.: (b). From tables 42.09 klfkg. TABLE 4.3
Prope .... ties of saturated water Enthalpy
Temperature
Pressure
Specific volume "r' m) /kg
l,oC
,T,K
0 - 10 20 30
, 273.15 283.15
0.6108
206.3
J.227
293,15 303.15
2.337 4.24'}
J06.4 57.84 32.93
40 50
313.15 323.15
7.375 12.335
.. 60
333.15 343.15 353.15 363 .J5 ·-i13. J5
70
SO 9()
100
p, kPa
19.92 3 1. 16 47.36
7.0.11 101.33-
19.55 12.05 7.679 5.046 .
3.409 2.361 1.673
. .
hb kJ/kg
hi' kJ/kg
-{)'04
2501.6
41.99 83.86 125.66 167.45 209.26 251.09
2519.9 2538.2 2556.4
292.97 ' 334.92 376.94
419.06
2.574.4 2592.2 2609.7 2626.9 2643.8 2660.1 2676.0
76
DES1GN OF THERMAL SYSTEMS
4Jt ,An equation of the form y - y~
=
,+ a2(x
al(x:'- 1)'
- 1)2
js to fit the following three . (x~y ) point,s: (1., 4), (2, 8), al1d (3, 10). What · , are the vallle~ · of Yo? al ~ and Q2? . '~.'} . , ~'.' , . . . . , . Ans.: a 1 . = 5 ~ .. . J ,. . , ~. ' .. .,h,>-',' . ,. :.~~:: The pumping capacity of a refrigerating 'compressor (and thuS the 'capability ..'. ;' .' .~~-<.·,.f9! :~eve~~p'ing~. r~frigerating· ·~Q~P,.~·S.rU.').::~:i,~ '~~\~~~:~9~\"~f .~~ :, evaporating, and· " '. ~ condensing pre~.s.~s:·. The)'efrigerating~ capaCi.'ti~:s..,irr' kilovhftti:·.o4~?;.q~~t~·i~{.::,:;; ,. . reciprocatirig compressor 'at combiiiations·of . thr.ee·,o.jff~rent 'eyaporating"and:~:~ ...,-":.". c9ndensjn~ temperattires ~e'shown in Ta~le 4.4. Develop'an equ'atiori ,simi1ar. ~ .: . to the fonn of Eq. · (4.33), D.runeJy, ' ~
lir~ok4:"
'.
........
-•• "
-
, , ' ""
"
.
•
qe . = Cl
- ........I
+ C2 t
t:
_
+ .C3 f ;
•
- . '
•
+ C4 t c . +
...•
+ C9 t ;t;
A~.: Cl to c~ are 239.51, '10.073, -0.10901, -3.4100, .~O.0025000, '-0.20300, 0.0082004, 0.0013000, -:-0.000080005. 4.10. The data in Table 4.4 are to be fit to an eqllati,on llsing Lagrange interpolation with a 'torm similar to Eq.. (4.34). The vari~ble X .c orresponds to t~, y corresponds to fe, and z to' qe' Comput~ the coefficjent C2~' Ans.: -tL02026 . . 4.11. The values of C J and C2 are to be determined so that the curve represented by the equation y = Cl)(C~ + X)2 passes through the (x,y) pomts (2, 4) and (3, 1). Find the ty.,,'o CJ - c:! combinations. Ans.:· One value of Cl is ~. ' . ~4.12. Using the graphical method for the fonn y '=' b + QX m described in Sec. 4.9, determine the equation that represents the following pairs of ,(x, y) .pojnts: (0.2.26), (0.5, 7)~ (I, 2.8), (2, } .3), (4~ 0.79), (6, 0.65)~ (10,0.58), (15,
0.54).
' Ans.: y = 0.5 + 2.3.1"-1.~. 4.13. A function y is expected to be of the form), = CJ: m and the xy data develop a straight 1ine on log-log paper. The line passes through the (x, y) points (100, 50) and (1000, 10). What are the values of c and nz? Ans.: c = . 1250. 4.]4. Compute the constants in the equation." = ao + a I x + 02x2 to 'provide a best fil jn the sense of least squares for the following (x, y) points: 0,. 9.8), (3. 13.0), (6,9.1), and (8,0.6).
Ans.: 6.424,3.953, -0.585. TABLE 4.4
Refrigerating capacity' q, kW ;
Condensing temperature, It,
Evaporating temperature /"
, 0 .5 , 10 .;;:=. *
°c
°c
.25
35
1.52.7 182.9
J 17. J
81.0
14J .9
~15.4
J70.7
JOl.3 126 ..5
.'
4S
•
73
EQUATION FlTTiNG
y /.
j
','
-'
'..::.-:
,-
- .
( .
·X
FIGURE 4':12 Combination of two forms.': .
, sales ~oIume vs ~ years for many products w~~ch J;1a¥~~ low sales when first introduced, experic;nc'e a period of rapid increase, then reach saturation. The personnel required in many projects also often follows the 'curve. The form . th~t re presen ts Fig. 4-11 is ..
Y == ab~ where
Q
7
I
(4.44)
b, and c are constants and band c have magnitudes .less than
unity.~
Combination of Forms It may be possible to fit a curve by com.bining·: two or more fOnTIs. For example, in Fig. 4-12, suppose that the value of y approaches asymptotically ?l straight line as :i increases. A reasonable y!ay to attack this modeling task would be to propose that
y = YI
+ Y2
= (a
+ bx) +
(c
+
dx
m
)
where m is -·a negative exponent.
4613 AN OVERVIEW OF EQUATION· FIlTING The task of finding suitable equations to represent the perfonnance of com~ po'I)ents or thennodynarnic propertie~ is a co~on preliminary step to sim-
ulating and optimizing complex systems. Data may be available in tabular or graphic fonn, and w'e seek to represent the data with an equation that is boch simWe;~ aod faithfuL A .reqHirement for keeping the equation simple
74
DESIGN OF THERMAL SYSTEMS •
.
-
•
~
•
&
"
~
is to choose the proper terms (exponential, polynomial, etc.) to include .in the equation. It'is possible, of : cour~e, to include all the terms that could possibly be imagined~' ev.aluate the) coefficients by .the method of least squar~s, and then eliminate tenns that proviqe little contribution . . _process -is essentially .one of regression. analysis,5 whicl;1 aJS.9_,lS us~d ,.to- . " < '. ,:assess .w hich varia~les are ipiport?Jilt in representing the. depeJ;1dent vanable, _ : " .... The field of statistical ,analysis of data is an exrensive one, and this .: , ':-',: ' ", ,~~' ". ~h~~ptei ti~·~~~i'y ~,c~~qh~~.:;.~~:.s.pifa~~·. ~Q9. 9?e other hand, much of the ~ffort' in th~: statisticii "ana:lys'ls, oT·:·data' is:.di:tec:t~10Waffu;fi~;~g~;~~p~~~nt~da~;;t:::;~:. ~ to ' equations wher~ 'rartdom .experirrie;ntal. e~9i' .occtirs-:··· ' IIr· eqil'arion,\£1'tiltrg"':-~:~ for of thermal systems, since .c atalog tables- and 'charts --me . most frequent ,source of data, there usually has already been a proc~ss of " smoothing of the experimenta.l data coming from the laboratory. Because ' of the growing 'need for -fitting catalog d~ta to '- equatiOll'S1 many designers -h ope that Dlanufacturers will present the equation th~t represents the table' or graph to save each engineer the effort of developing the equation again ' when needed. .. . This chapter presented one approach to math~matica1 modeling where' '. the relationship of dependent apd independent variables was'- developeq. \vithout the help of physicaJ laws. Chapter 5 explores some special irriportant cases where physical insight into some thermal equipment can be used to advantage in fitt1ng equations to performance data. Chapter- 13 'extends the experience on mathematical modeling and also con'centrates on the impOJ;1:ant . topic of thennoqynanllc properties,
This
are
the -design
4.2. Test the coefficient matrix in the set of Unear equations
~.[Xx~Jj _
1 .-2. ,_2 ) 1: -2 3 . -2 [- 1 1 ,
3 - 3
.I 5
- 4 .~ 3 ~\.,t
-
[5J 18 [
-
6,' I
13
·and detemline whether the 'S/~t of equalions is dependent or independent. 4.3. Using n computer program (gaussian elimination or any other that is .av'ailable for solving a set of linear simultoneous' equations), solve for the )."5:
EQUATION FITnNG
:: .
)~.
I I
70
?o
71
R
-~~ ~; '-'~."
.
~
, 15
-
,
'•
•
I
'
__ , ' · r
,"
':
.
-
~~.
· 4,~~;"i~·
"I;':':; :.:. -: .
:,~~.
... :fo··
5~Y~3 ~~Y=4
5 ,
;-,.. .. .,
o
-0
o·
2
5
4
3
y
4
2
0 '
-, (b)
(a)
FIGURE 4-8 Cross plots to aid in developing the :form of the equation.
The iRS~ght provided by Fig. 4-8a is that'Z bears a linear relation to y, and the 'fact that the straight lines, parallel shows no influence of x on the slope. Figure 4-8b suggests at least a second-degree representation of z as a. function of x. A reaspnable form to propose, then, js
are
'
.
-
Z
=
aD
,
+ a IX + D2Y + a3 x
".•.J .
2
Several frequently used fonns merit fur¢er discussion.
P9lynomiaJs If there is a lack of special indicators that other forms are more applicable, a polynomial would probably be explored. When the curve has a reverse curvature (inflection p'o int), as shown in 'Pig. 4-9, at least a third-degree polynomial must be chosen. Extrapolation of a polynomial beyond the
y
x
.~'
FlGURE 4 ..9 At lease a third-degree poJynomial Deeded.
x
72
DESIGN OF THERMAL SY$1EvfS •
y
I
'. .I I
I
.
. , . ...
."
"
.
,::
~'
:'~
.
"
•
}
.
.
r- ;-
.. .. .. ... !; ~~'I ~· ;:-."·1, · . ',,~.
;
, • . ; .. .'
,
'.
, ·FIGURE 4il0' .'
" '- . -', ' .: ., . . _ . Ne~ative exponents.'-.of~R91yp.-omia1s for _ a curve that ...'-.-....;........,,...;.....:..---,----,-..,.....,. ... . .......,~=-:-::;~ : .•-.-..-,'--:'-. x-'~ .. ' .!~~::::rfl~~~;:otf.,';·~ '.~. I ; .. ' : ...~.: ~.{.~.;.: .~.~.. ~:.. .}:.:...:' . ?;~ ~;'~~-:".;\ ~.· <::,f :"':: ."i l l
. -borders of 'the data used ' ~o develop the ~qllation often results in serious error. .
. Polynomials with
N~gative
Exponents' .
When curv'es approach a constant value at large magnitudes .of the independent variable! .polynomi.als with negative exponents . y = qo +
Ql X - I . + a2 x - z -
may provide a good representation; see Fig. E~onential
4-10~
Equations
. Section 4.9 has described sever~ examples of exp-onential forms. The shape of the curve in Fig. 4-10 might also include a c-~ term. Plots on log-log paper wO\lld be a routine procedure, although a simple plot of log y vs . log x yields ·a straight line only with equations in the form of~. (4.35),
Gompertz Equation . The G.ompertz equation,4 or S curve (Fig. 4-11), appears frequently in engIneenng practice. The Gompertz curve. for example, represents the
----"----"-'------
,
----------F1GURE 4-11
..
:x '
Gompertz. or S curve .
EQu~_nON FTITING
and ,
69
I I
m
.. a i 2: (o -+ . hr_y.)2 = 1\ -' I l .
!
- - - -.a-b---- == 2:2(a .'
bXj -
y;)Xj
I
_0
,I
''';:1' ;
.",_ .
.'. , -;'-." DiYi4in.g.. -by.2: and seti~Ja.tiD.g the :at~ove-t':'vvo ·'eqhat?',ons" inltd'.~qividlla1 tf;rrn.-s· . : g1yes,,' '" ::. .: .,'" ~:.: .:.-.:,~.~~~.:.,~ ';' j"i') " :'~:~ ~'~':.';:':/'.-'.' t_ .. ,~ ,:;,~;::,:: .. _.: ~.": .~ ,~--:.?":~: .'. ::"':>' ":-,.:\. ..:~.:-:;.~-:>.-: ' --:': ,>":;", _.. j.,.
+.bIx i
.: "'111(1 .
+
a"x4 I
;
.
.
'2:Yi
."
"C4.39)
')
b>x:-J .-I
=
"x-y· £..i I I
'(4.40)
Example 4.. 4& Detennine 00 and a I in me equation y = Qo + a IX to prov·ide · a best fit in the sense of least-squares ' deviation to me data P?ints (1, 4.~). (3; 11.2), (4,. 13.7)J and (6,20.1). ,
Solution. The summatioris
substitute into Eqs. (4.39) aI:ld (4.40) are
[0
,
,X I'
~ ..t...
and m
4.9
1
4.9
9 16 36
33.6
13.7
6 14
20.1 49.9
The
X;}'j
11.2
l' 3 4
=
x:-I
)'i
54.8 120.6 213.9'
62
4 ~imul~eous
equations to be solved are
4ao + 14a, = 49.9 14ao
+ 62a, = 213.9
yielding ao = 1. 968 and a 1 = 3.019. Thus y = 1.908
+ 3.019.x
A similar procedure can be followed when fitting a parabola of the fonn y
= a + bx + cx 2
( 4.41)
. <
to m data points. The summation ~o be minimized ~s m
, ' 2:(a'
+ bXr.t c.--II -
y/)2 ~ minimum
I ... 1
~
Differentiating partially with respect' to at hi and c, in-tum, results in three linear simultaneous equations expressed in matrix form m
.-
[
!x; Ixf
( 4.42)
...
:'
.
70
DES1GN OF THERMAL SYSTEMS
A comparison of Lhe ,matrix equation (4.42) with Eqs. (4.39) .(4 ..40) shows a pattern evolv~g whiqh by analogy permits developing :::.'equations ,for higher degree polynomials .without even differentiating - _ s,u~ation of the squared deviation.', " -, '
....::-,
' ;" " ,~, 4.11 METHOD ,OF LEAST s,QuAREs~,~ " '"\- ,' '" ~,. " " " .',' ~ ,::' ; :~," -" ' ~""-?~:APPLIED-.:T, O ',:NONPOLYNOl\lIIAi, FORMs ," ,- " ,
', ,'
J:'-::" :'-r :-~~!~ ,,-i,;'; ~::;.":~'.' :',:, ~~);:,':, :;:~:o:i,· ,:-i . - : 0':- ,~,~'..... "" ',~, :._ ~ , ,;:;,:' , :' .::' The- e7Cplana.tion of the ,p1~Ui:6d bf 'Ie'ast 'squares ,was~, 'appl-ied-~:to" jJolyno ' ' forms in Sec. '4.10, but' it ,should not be suggested' that ' ,the--:metho :~ limite4 to those faIms. The method is applic,a ble ,to any form which con ... ·- constant -coefficients. For ~xaIDp~eJ- if th~· form otthe ,e quati,o n is ": -~ ' , , - '"
"
,, ' ,-"
' ,' ,- ,: '~" ",''<',
'I, :
.-=
y = a sin 2x
the
su~mation comparab~e
+.b
In x 2
to Eq .. (4.38) is
m
L(Yi '-asin2xi~ 'blnxl)2 . i=1 .
Partial differenti4ttion with respect to ,'Q.and ,h yields . a I(sin 2x i )2
+ b 2: (sin
, a 2: (sin 2x )(ln xl) j
+
2x i )(ln x~)
b 2:( 1~ x1)2
=
2: Yi sip 2x i
= 2: Yi
In
x7
which can be solved for a and h . . A crucial ch~racteristic of ~he equation form that makes it tractable to the metho~ of least squares is that the equarjon 'have constant coefficients . . In an equation of th~ form
y -= sin 2ax
+
bx c
' the terms a ,~nd c·do not appear as coefficients and this equation cannot be handled in a straightforward manne~ by least squares. I
4.12
THE ART OF 'EQUATION FITTING
While there are methodica1 prpcedures for fitting equations to . data, the process is also an art. The art of intuition is partic\llarly needed in deciding - upon ,the form of the equation, namely, the choice of independent variables to be included and the form in' which these variables should appear .. There
are no fixed _rules for knowing what variables to inc}ude or what their form , should be in the equation, but· making at least a rough -'plot of the data , wjJl often provide some insight.t If the dependent variable is a function of two independent variabJes... _Jls in l == j(x ,y) two plots might be made, as . ilJ ustrated in Fig. 4-8. . . I
;
-.
.
, . ...,,r,:. . '
-,J , .
'
,
'
. '!',
:
.. .'
,;.
'. ~
;"\
'.
... ~
,
~.
".
~', .
';:;
,. I: ' ,
.. .. .
.::~;
. _
".
, ...
.,..
' 1
';-
.
~. ::",' ~.' .
,
.'
~
!.:,~. ';r ~ . . '
, '. , '
1' - '
" .,
:
.
.
.:
' ~, loII! ' - "
•• •
':...;'
"
.
'
. ; .•
• .• •
.
.'
501 ' 'USING PaYSICAL INSIG'H T ,
TDis chapter continues the objective of the previol;ls chapter, of fitting equa, tibns to the perfonnance of cornpone~~s. Chapter 4, ' however, assumed that ' physical insight into the equation either did not exist- ~r offered:'no parti.cular , advantage. In contrast, this chapter concentratt~s on three classe~ 'e{ 'co~po: - nents "that a'ppe~ almost universallY , in thermal systems, heat exchangers, distillatio.n' separators, and turbomachinery, . where' the knowledge of the physical relationships helps struc;ture .the equations. ' The reasons for singling. out these three compon~nts are different for each component.' The preface of this book indicates that· 'a bac}t;,g'rouhd 'in heat trans~er is assumed. In the study' of thennal systems a segment' of that knowledge is p~icularly important, namely, predicting th~ performance of an existing heat exc~anger. Not only is tJ:le sele~tion of ,a heat €xcpanger important, but it is also crucial to be able to calculate how a certain heat exchanger will .- perform wpen ,9perating at off~design conditions. A useful tool to ,be stressed in this chapter is the effectiveness of hea~ exchangers. The techniques this book explains are particularly applicable to the thermal processirig industry, which includes petroleum refining, and other process industries where the separation of mixtures of several substances by , means of distillation is an integra] process. , Even an - unde'rstanding of the separation of binary m~Xtures which is explained in, thi~ chapter t e~pands t
80
.
, " J'
. .-.... . .
: -
f
" .:
..
MODELING THER.."vt-\L EQUIPMENT
81
the .h.orizons of applications 'of tht? future ~opics of simult;ltion and optimii:a~ , .
tIOD.
•
.
.
J
.
,
. '.
. ,T he third 'class of thennal equipment tr¢ated in th,is chapter. i~' Ujrb~-' ". ~' ,'::--,..'.'<;', machi~erY' the; performanc.e of which can often be .~x:piessed in ' .· :of -.: " .
terms
'>/:: , «~. ": 'diWensionl~ss. gro~ps. This chapter wlvi sho~v hpw " ~§_ :~$e' o{dim~.nsi~.Q.~e~~ · " :~,: -. "
•• -
._~. -
':
~; ,~:;' ,
';
-
\.. ! : : :
II
,
'-'
" ,
•
[ ;. '
-'.
•
-
.
,
:...
•
.:
'::
, ~.> ~', ~:"·'\'5~2' · '. 8ELE CTlNG·· VS·.- .SIl\tIULATING · .... , ' . " . . ' . ' .. " . '. ',"_ ..... 1,":" . -A',HEAT EXCHANGER . " -,." ,
•
,
'
•
.,
' .' , .
J
A cbriunon engineering task'is'
..
to .selec~.~desigri; or speCifY'aheat ~xchaDge,r· . , ,~ - .
to perform a certam . heat-~ansfer duty~ .Th~ ·. ellgitleerthen -decides on .th~· " type heat ' exchanger and its deta.iis.- Three of the 's~vernl: dozen types' . '. of he'at· exchangt:~ '~va,i'~able' ate:' S~o'~Iljn Fig. 5-1. Figure 'S-la 'shows , .~. sfiel I"" and-tUbe' heat ~xchanger, com,monly .p sed transfer.lieat between two ", ~.iqu.ids. On~ ~9f ,the fllfids 'ffow~ j!1side the 'tubes and i~ called the tube-side .... '.~ 'fluid, while · the .'other flo~s over' rh,e
ot
a
to
t
t +
W!uer (b)
FIGURE 5-1 Several types of heat exchangers.
Finned-coil
(c) Compact
EQUllJ10N FITTING
Y
79
A
.~- ;
.; -
:' !'"
,. -1'
,. ' ,
x
FIGURE 4-15 Gompertz equation in Prob. 4~23.
4.23. In a certain Gornpertz equation: which. is y
,=
ab
cx
and represented by Fig.
4-15, 'c = O.'5,yo = 2 and the asymptote has a value of 6. Determine the
values .of a and b. Ans.:a- = 6.
REFERENCES' .1. Procedures for Simulating the PeifomzQn.ce of Components and Systems for Energy Cal-~:: culations, American Society of Heating, Refrigerating, arid Air-Conditioning Engineers, New York, 1975. 2. M. W. Wambsganss. Jr .• "Curve Fitting with Polynomials," Mach. Des., vol. 35, uo. 10, p. 167, Apr. 25, 1963. 3. C. Daniel. and F. S. Wood, Fitting Equations to Data. Wiley-Interscience, New York, 1971. 4. D. S. Davis, Nomography and Empirical Equations, Reinhold, New York. 1955. 5. N. R. Draper and H. Smith. Applied Regression Arullysis; Wiley, 'New York, 1966 .
.-
. , '--~ "-'
82
:
~..
cor:rugated an~ arranged so that th~ t\vo.f1uids tJow through alternate spaces between the plates. " ') ' .. ' , We now return to ,the ' distinction ,between selectiIlg ,anq simulating a ,heat exchangeI:' To select a sheIl-and-tub~ heat exchanger; for example, the ,flov/ rates, entering temperatures, and'lea~ing t~mp'efatuies 'm both fluids ',; 'would be ,known~ The task 'o f the ,designer is to select the combinaticin":of
-.,
-
"
". i !"
DESIGN bF.11-I.ERM4 SYSTEMS
,. . .... .. ~ .,'
~. :."
, :>·. ~h~ll_Plame.ter.,.,;.tub~·'" le~gth~ ,f:tri.mPe~ pt::~~~j~~,,>,'nllpJber-"of tUbe,passes~
and
,: ~the," baffle , spatmg. that. will::accomplish,: tli¢' ~p~pi:fi~d':'he.a,t.~tpt,n,~~8~" duty =,. Ihe __ de~ign 'must also en,s ure :'that certain 'p ressure-drop Iiini,tatiopso~',of-ttie.JJbiclS~flowmg through the heat exchanger Dot ~xceeded. ' ' ' . In ~imulation" on the other hand, the heat exchanger already exists, either in actual hardware or as a sp;c1flc , design. ' FllrthenDOre the pedoi": mancechantctenstics of the heat exch~g~r are- available, such 'as Ithe ,area and overall heat-transfer coefficients,_ Simulation of a heat exchanger COllsists of predicting outlet ~ondjtions,. slIch as temperatures, for'variolls inlet ' terp.peratures and flow rates. The emphasis of the next several sectionS wi,11 be on' predicting outlet condit~ons of a -given heat exchanger when the inlet conditions are known.
are
o
5.3 , COUNTERFLOW HEAT EXCHANGER For a heat exchanger bet\\'een two fluids with' given inlet and olltlet tempera. tures, the nlQst favorable difference in temperature between these two flujds is achieved wi~h a counterflow arrangement. A counterflow heat exchanger is therefore a good choice for a standard of comparison. Figure 5-2 shows a ,
'
12
./C
}1'2
kgls
cp'J.1/(kg ",K) II,jOe
l(
U',
cp
.
kg/s k,....g_-K_)..L-L--"'--_ _
~
, FIGURE 5-2 A counterflow hear exchanger.
MODELL.'\fG lliER¥.AL EQU1PMENT
n-;
OJ
coun terFo'vv heaL txctanger 'Vt/ith the sYrn?ols Ithat vlill be used in developi,ng equ~tions. , ) Th·.-ee equations for the rate of heat transfer q. in w'arts are ;:r .
q .q
= WI ( t I ' ; - t~;o)
,v-I
...;.-:
~J
.
-
* v-1!...-:-.,;.. ..
(5.1)
'~':W2(t2:;,· .. ,'./2;iJ,,'~:~·:"·'·<"" .;.:.:.;;;;::>.: ;.:'f-('J,' ' ., .(S,.2} .~;: .
(11
q = UA
' ~, ~ 'f?_.0 }' ~!;l ~"t,,:) qJ ,0, ' _.1
.f
(5.3)
In[(tl,i -:- t2~o)/(tI.o -,.t2.i)] , \vhere
q
=
rate of heat transfer, \V of respe,ctive fluids? kg/s Cpl, cp 2 :::: specific heats of respective fluids, J/(kg,- K) t = temperature, ~C . , U = ov~r:all heat-:transfer coefficIent, 'W/(m- . Ie) wI,
W2
= flow rates
,
')
A = heat,-transfer
[~ = TV Y ? -
Wi
~ea,
)lciiJ ~ Q
= H:..,C p 2 -
I
,,., "
-' l)(
m26":cti\ ..~----,\./ 7~,_~/,~~
. . . .,--. '
\
j.... "'\
..t--
~~/'
,
,/ ....,"--- / "
_,,,'--
.
Equations (5.2) to (5.3) have made no ' assumption about which is the hot fl uid, and which ,the col d. I( q is· pas i ti v e', fluid 1 is ho tter than-iJ.8-icl--2, , ~ut if q is negative, ' t!Ie opposi~~~_~S(..L-_ILtM._Ws, UA, and the entering , temper~tures are known, the ~ee equations contain three unkno.w~s~ ~ and .t(!5 The number of equatIons can be reduce,~, to two by eliminkfin~ to gIve (5.4) ./ ,/ '
~v (t . I
Solving for
12.0
1,/
1
) 1,0
==
UA (fl.; - {2,o) - (fLo - t2';) In[(tl.i - 12',0)/(/1,0 - 12';)]
,
in Eq. (5.4) and substituting into Eg. (5.5) gives
Define D as
~ (1 -1) D=UA WJ
.
W2
TIJen II./ -:-/2/ - (W J IW2 )(tl.l -/1.0)
t 1.0....
-
12.i
= e~'
-
(5.5)
. 84
DESIGN OF THERMAL SYSTEMS
Sqlving for
t1,0
gives t10
·
•.
':-;'".i ·o':'f.;"
". :;-' ... .. ~
=
"
. tl,i(W llW2
.
I
-
1)
-
W 1 /W 2
+ (2.i( 1 -
eD)
eD
-
I
:- ,':_-':_.", .oi~ -.m ~temate fo~_.,_ .. ~- .. ~---:--~·--'"-.-~ r<
". -;": . ........ . -, ; .
, '.: .~~-•.: .~ ' . ' " ,
\ ."..."~ tl
_. '
" .
0
)
.
. '~-l
. ', :
~ "e- Dn :.~ . . , · : -...- -:.~,o , '. 'D , (5:6 '1 .... ..••
= 11 i - (t1 i ...:- 12 i)
~ .~': ",', " ,,-~; .." ? >:c;-:'.~~':,::, ~" _"-,.._"'_~_~ /
"
w'-
2 -
I
e
:.--.'
,
• •'
'
,"
"
. ., _ . '. . Equation .(5.6}·cp~rinit{:t.O~P¥t~~o:n.,';o~
.
in
5.4 SPEC)AL CASE OF COUNTERFLOW HEAT EXCHANGER WHERE PRODUCTS OF FLOW RATES AND SPECIFIC HEATS ARE EQUAL
The
direct application of Eg. (5.6) is unsuccessful when ,
, WlcpI
=W2 cp2
thUS
,
~.
, .
The value of D is zero~ and Eq. (5.6) is indetemlinate. There are two ways to develop an a1ternate expression for the outlet temperature, one mathematicaI and lhe other by physical reasoning. , The mathematical solution uses the expression for eX as a series.
x2
l+x+-+'" . 2 The indetenninate part of Eq. (5.6) can be written
1 '-
1eD
----- -
W1/W2 -e D
J + UA WJ
(1 - ~Vl + ~ [ UA (1 _.WI
J' ]
J'
W2
2 WI
UA(
WI [ '. 'W))' W2 - 1 + WI 1 - W + 2
2
+ ...
W2
I[VA( W1J·]2 } WI 1-'W + ...
2'
2
Canceling where possible aAd dividing both the numerator and denominator by 1 - l¥]/ W2 re.suIts in ,
~'
MODELING ~"\!AL EQUIPMENT
Finally let ~Vl ~,
Wi
1 ~ eD 'WlVf - e D -.'
;
: 0_.'"
"", .
Stlbs,t~~~itin,~,' back into , Ego ··;.i··.' · .. ·.
COD1.t'lJpn \f~llie
and call this
+
1
(5J5i gives
• ~-
Cr& ~ ,: - 5\ t2ft,:..-~\:~" , l't-,,' Jl \'
I ,
'. t1 j ' ~· t2 i = t '. - ' " . - '
&
t,
1.1
.1,0
.'
ii
. .~-;
UAjW - "W"JUA + 1
' •
H' . Then
,U fil Vi
1
85
"'i_--·
.
' .,
WI UA + l
'
'.'
.
..J'-\ 'L -'
.
' t2,i}
Since the sldpes through dA are identical, the temperature d,ifference after dA is stil~ tl,o - 12,i' and qy similar reasoning the temperature difference remains constant through iPe entire heat exchanger. ~us the temperature
/ r.J
1 I. ;
12.0 • • ~ , -.-
-
_ • • ,'..... . : • ..:.;._
•
f"
I I I ' I
, 1,0
I
~ J
I~j
I I I
.-
I
dA
~
I I I J
A -
Arc,a. m2'
FIGURE S~3 Counterflow heat exchanger where WI W1 . Q
'
"
1".
Starting at the left edge of the graph in Fig. 5-3, the inlet temperature t2.f of fluid is ~pecified, and there is some o'u tlet temperature of fluid I, designated t1.01 yet to be determined., Regardless ~f the value of {I,o' the slopes..,of the temperature lines are fixed for the first rncrement of area dA (fl,o -
..
C) :, t~7)- 'j' .', ':. <:';. ," --
__ . "
vV'mch is . the equation for computing one _Qutlet temperature . . The' pnYsica!· analysis that leads to Eq . (5.7) is to recognize that the change of tempera.ture of one "fluid while flowing past a differential area flA ' in Fig. 5-3 equals the change in temperature of the other fluid ~' The slopes of the tWo : temperature lines are then -.the same at all positions -along the ' ;ar~a' . ' (This condition does not' yet'establish that the lines are straight.) A ,further stipu~ation demanded 9Y the heat~transfer rate equation is -that
dtl dt2 U dA ::::: dA = W
l~
'
,
:
86
DESIGN OF THERMAL SYSTEMS
line$ are straight" parallel lines. The choice of 1'1.0 \-vas arbitrary, but now , w'e see that it must be chosen so th~: the) tIl 'line ~errninates' at the specifi,ed inlet'temperature of t I,;. Since t.he, mean temperature difference in the he,at e~changer is t I ,0 ' - t2,i, the rate' equation can: be written
.
'..;.-,
I'
-
.
.-
.. 'r: ",,~ :: :"f:! :~ ,;;.
,' . I ""
.~
I
r -:
••"
. ' i :·
and t 1.0 ' -~
WIl ..;
,+
UAt2.;
,UA +W
Add zero in the form of (UAtlj - UAtLi)/(UA then.
"
,'"
.' .. '
. : . 11,0
''''-''iiJ''= '~
+
side;
W) to the right
- .
= f U - W / UA + .1/)
which checks the rna thernatical deriv ati on, Eq. (5. 7) .
· 1 '»/-:-
y-~ b:'-~7.' . I<~ 1: "-
"'
> 0_
'
, Example 5.1. In the counterflow heat exchanger shown in Fjg. ·5~, '2 flow rate of 0.5 kg/s of water enters one circuit of the heat exchanger ata temperature of 30°C, and the same flow' rate of water enters the other circuit at a temperat.ure of 65°C. The VA of the heat exchanger is 4 kWIK. What is the Dlean temperature difference b~twe~n th,etwo streams?
Solution. The pr,odl!cts of the flow rates and specific heats of the two streams are the same, (0.5 kgJs) [4.19 kJ/(kg' K)] =2.095 kWIK. For this speciaJ case whe!e the wC p products are equal, Eq. (5 .. 7) gives the outlet.temperature for the hot stream 1l.0 = 65 -
1 1.0
65 - 30= 420C 2.095/4 +
The terripenltu~e difference at either end of the 'heat exchanger prevails, throughout and so the mean temperature di,fference is 42 - 30 = 12°C. I
,
I
O.S kg/s
VA
s:
4 kW/l<
0.5 ksls
I I." ~
.......
....
1....,.~
J
FlG URE 54 Heal exchanger in Example 5. J.
f
MODELING
THERf/~ EQUIPMENT
87
A specjaI set '-"f equ.ations is possible-and indeed necessary-:-':tvb.el1 one. of the fluids Ho wing through a heat excp.anger Chfu"lges phaseQ In an evaporator
. orcondenser,.as shown in Fig. 5-5, as~t1me that there)~ no_supetheatihgor . .. '.subcoolin,g of llie fluid that changes pl),,aseD .'1)l.at fluid 'will then remaiIi .at..a .'. . ·./:r·-·::constailf Lemperp.-t!lre;."provided~.thae'its):pk¢§.stiY~':.d$·6:~(rib:t":'¢,liahg~·~·:'·"·:,:"' ('·" ,·<:-' ~;.''-''· :,' ',,'.' ,,. " TP.e· tEtrijperature' difference' 's till 'appli'es and in conlbirtation , . , . , with a heat b'a lance, gives .
log-mean'
c q = UA (t - Ii) - (Ie - to~ == In [( t e - t i) / ( t c - to) ]
' (
wCp
to -
ti
)
Equation (5.9) can be converted into the form
, UA ' In t c
-
ti =
Ie ---: to
WC p
-In t f;
to
-
te --. Ii
Taking the antilo$ gives tc - to
+
te -
ti - ti tj
(5.10) For a heat exchanger of known characteristics Eq. (5.10) can be used to compute the outlet te~perature of the fluid that does not change phase when its entering temperature and the temperature of the boiling or condensing fluid tc are knOWD. The characteristic shape of the temperatUre curVes, of the two fluids is shown in .Fig. 5-6, applicable to a condenser.
t (
.
tD
tI
w
Cp
" Ie
FIGURE 5-~.. - - An evaporator oi- condenser where one (lwd remains !It B. constant temperaO,lCc.
J. .
, :...::-,
.....
~
, ' .t·'
.
'.'
..I.
~.
I.
•
..'
. 1'. ...
#
..
"
-FIGuRE 5-6 Temperature distribution ids in a condenser.
At:ea
Ex~ple 5.i~ Water is continuQusiy heated from 25
W,
flu-
to 50°C by steam con-
densing at 110°C. If the water flow rate remaUis constant but its inlet perature drops to 15°(:, what will its new outlet temperature be?
Soluiion. The terms VJ A,
~il
tem-
cP ' and te all remain constant, so
Original
New
to - Ii . 1 to - Ii - = -e-UA/we'P = --
te - ti
50 -
.
is
Ie - ti
-
110 - 25
15 110 - 15 To -
New to = 42.9°C. 5.6 HEAT--EXCHANGER EFFECTIVENESS The effectiveness
£
of a
~eat
exchanger is defmed as
c= where
qactJlJl1
~- -
qmax
(5.11)
= actual rate ~f heat transfer, kW qrrul. = maximum possible rate of heat transfer, kW, with same i:nlet .temperatures, flow rates, an~ ~pecific heats as ~ctua1
qBCruaJ
case
(
Another approach to ' defining qrnu is to designate it as the rate of heat . . transfer that a heat exchanger of infinite area would transfer with given ,inlet temperatures, flow rates t anp specific heats. Example ,5.3. What is the maximum rate of heat transfer possible in a
counterflow heat exchanger shown in Fig. 5·7 if waLer enters at 30°C and ~0is OJI entering at 6Q°C9
.
I..
.,7 • .1.-' •
:
~
MODELING THERMAL EQUIPMENT I
I
'
! I
89
'Oil 2,6 kg/s
,V 60°C 2.2 klJ(kg " K)
1..5 !~g/s 30°C .,.-,,: -3!' ''~
,
', / '.'
.
- -: .;
, 4.19 r-J/(kg ~ K) .. '-:- ... ..
,_" _":.:.~-::. .. .!
. ~- :
',-
.
, ',
'1
, .'
"
--i'»-
~=="-====<
Infinite area
FIGURE 5-7 -Oil cooler in Example 5.3.
,
' Fl ow rate
=
".
,
{ 2.6 kg/s 1.5 kgls
oil water
2.2 kJ/(kg' K) Specific heat = { 4. 19 kJI(kg . ,K)
oil water
Solu.tion. The break in the heat exchanger indicates that to achieve the maximum rute of heat transfer the area must be made infinite. The ,next question, then, is: What are th~ outlet ,temperatures? Does the oil leave at 30°C, or does the water leave at 60°C? . From energy balances those two options give the folIowi,ng consequences:
Oil
1.
.
leaves at 30°C
'
---,
q = (2.6 kgls)[2.2 kJ/(kg . K)](60 - 30°C) and water leaves at
+
300C
,
= r71~
'
.
=
, 171.6 kW
57.30C
(1.5 kg/s)[4.19 kJ/(kg· K)J 2.
Water leaves at 60° C q
=
(1.5)(4.19)(60 -- 30) = 188.6 kW
and oil leaves at 600C _<
188.6 "
(2.6)(2.2)
~" /
--The second case is clearly impossible because the oil temperature would drop ,
below th~r of the entering water,· which would violate the second law of thermodynamics. Thus. t}rnu. := 171.6 kW.
Th~--G,?:c,ept [hat Example' .s.3 has exposed is that the maximum rate of heat tran·sf~r occurs when the fluid with the minimum produ'ct of flow
. 90
DES1GN OF THERMAL SYSTEMS
rate and specific heat changes .temper~ture to the entering temp~rature " of tl:J.e other flu.id. Equation '(5.11) can 'be re\yritten .. ': ,J , ". ', .'
E=
"".; .
.
,
qacrmJ.
. '
( wep).min ( ~ hot.in
-
,,' .... ,;,~.. : , . :.:. ... ' ..where.' (W' Cp)~n"! is' tPe: sma11erjvtp .of th~" tw()
--',;..~ : .?-~- .::.;... . . .. -~ _"
..
. '
:•
.. '
~
•
'
:
.
'':''
•
'
... .
,- ..
I
..,.
_.
t col!:f.in)
.
:l
":, .(5.)2) ,
.
. ~
.. :
riu,ids ..
. . ....
;5.:; ~ .'fE~ERATcirE~R6~ES·~ · . . ,,:,.. ••. .•": • " ". "
..' """
_
. .
'"
- ,'
-
~~ ~p.?:pes of the t~~pe;ratuf(~ curves in . the coUnterflow heat exchanger · . are 'shown 'by Fig. ·S-Sa o(.l!. The minimtim produc~ .o f wC p' is possessed by fluid 1 in Fig. 5~8a by flUid 2 in' Fig. 5-8b~ \Yhich is indicated by' ·.
and
. the fluid whose temper~ture changes mO$t. The c~rves . are ste.epes( in the' portion of the heat exc.h~ger where the rate.of heat transfer is 'highest~ anq . this region occurs ' where the temperature differences are larges~.
5.8 EFFECTIVENESS OF A COUNTERFLOW HEAT EXCHANGER The form of Eg. (5.12) can be applied to the equati6rl' for a counterflow heat exchanger, Eq. (5.6}, to 'd evelop an expression for the effectiveriess of the counterflow heat exchang~r. Denote fluid 1 as the fluid with the lesser value of wC p . Note that
,
(a)
FlGURE 5·8 Te!p-pcra.lUCc profiles ;n
-
~
2
(b)
counterflow heal exchanger. ~
MODELING THOOAAL EQUIPMENT
Combinin.g \vith Eq . (5.6) gi'~res
~'!
91
....... :
.. . - •••••'
~...
~- '
J
-."
where
(5.14) (
Examination of Eqs. (5.13) 'a nd (5.14) ,.shows that toe effectiv'ene~s of the counterflow heat exchanger can be, expressed as a function of two" dim~nsionless groups, UAIWmin and Wmi~/W2.
509 NUlVIBER OF TR~SFER UNITS, NTU'S The ability to ' express the effectiveness as a function of UAJW1J1in and
, W:ill/~~ leads to gra~hic presentations .l§ fecti,-:~ness of heat exchangers ~~s .co~~a.lIons. Th~ group ' AJ~ ill IS €alled the,- . J2Lll~ber , of transfer lmzts ~lJ~. c presentatlons are ,shown m FIgS.. 5...:9 'and 5-1.0 for 'two m1fer~~t confi,g~rations of heat exchangers. Figure 5-9 is . the graphlc presentation of Eqs ,. (5.13) and (5.14). , '
~
Z?/!//I/J/}/R//I//////1///1/4
L He:t-transfer surf~ce
'
lOQ 80
w
~ 60
c
1.)_
.~
t;
~
40
"'"""'-----0.75
W
20,
o
'-----LOO
I
o
.-
4 2 3 NTU = UA/WrrM •
F1GURE 5..9 Effectiveness of 8 counterflow heat exchanger. I (From \Y. M. Kays and A. L. London. Compact Heal &chansers. 2d ed;, McGraw·Hill Book Company. New York, 1964, p.SO, used by perntission.)
92
DESIGN OF TI-IERMAL· SYSTEMS
100
. '_ .
. -. •. -:..t.'r••
".= ....
J . ...
~,~.,
'.J
..She~ fluid . ' .'
", .
,
'0 '
f . ....
.
'"
,' >,
J
'~60
w'.
,0.75 '
One ~hell ,pass . 2.4. 6,... tube passes
o '0
2
3
4
FJGURE 5·10 Effectiveness of a parallel counterflow heal exch.anger. (From. W. ,M. Kays and A. L. London, Compact Heat Exchangers, 2d ed., McGraw-Hill 'E ook Company, New. York. 1964, p.54. ' used by pem1ission . . ) · ' .
Example 5.4~ Compute the effectiveness of a counterflow heat exchanger having a U value of 1.1 kW/(m/ K) and an area of 16 m:! when one fluid . has a flow rate of 6 kg/s and ~·'specific heat of 4.1 kJ(kg . K) and the otherfluid a flow rate of 3.8 kg/s and a specific heat of 3.3 kJ/(kg : K).
So lu ti.o J1 l1'a = (6 kgls)(4.1 kl/(kg' K)J=24.6 kW/K ~Vh
= (3.8)(3.3)
NYU == , VA Wrni~
=
=
designate Wmin
12.54 kW/K
(1.1)(16) = 1 40 ] 2.54 .
Wmin 12.54 1V = i4".6 = ,0.51 2
From Fig. 5-9 E
= 0.67
or from Eq. (5.14) D~
= 1.40( I - O.S 1)
= 0.686
. and from Eq. (S.13)
,
E
T
J - eO.t.8f, 0.51 _ e O. 6S6
,
= 0.668,
It is now appropriate to put the previous n.ine ' sections on modeling of-?reat exchangers into perspective. The direction from which the mod-
5
MODELING THERtiAL EQUIPlv®f[
ro
~.'"
93
'
eUng was approached was to predict the pe~foI)il1ahce C?f an existing heat exchan,ger, which is a common assignment in systems simulatiolJ. The outlet remperatlu-es can be C01TIpllf.ed if the inlet temperatures and effectiveness ~e known. The effectiveness can be , compu[~d or detennined froni '.X/hen ~e product of u.A:, ~~ .,tf?:e ,pro,~¥ct~ wa:$:~ . Ia'~~ -of ,flow and;"specific he2tt ":- ':,) ,', a:re, ·l~~o-v/n_ 'EqJl,atlons ,fOF i4~ ,~~ff.~~ti,y..eness_qf.a;
9t
agraph
5 . 10 ' BINARY SOLUTIONS, Figure 5~11 illustrat~s a binary solution where equilibrium prevails between liquid and vapor. The liquid is a soludon 'of substances A and B, both , exertip.g a vapor pressure, such that both substances A :and B 'exist in the vapor. The concentration of substance A in,tpe:;,v apor phase is not necessarily the sam~ ,as its concentration in the liquid pha,s e. A frequent.assignment in petroleum', cryogenic, petrochemical, food, and other thermal processing industries is separate substance A from substance 13. System simulation computer programs (Chapter 6) are heavily us~d in thes~ industries, and the sep'aration process (usually by distillation) of binary solutions is often an integral function of these programs. We wish to be able to incorporate dis:.tillation equipment into systems along with heat exchangers, fans, pumps, compressors, expanders, and other thennal components.
to
5011 TEMPERATURE-CONCENTRATIONPRESSURE CHARACTERISTICS The temperature-concentration, t-vs.-x, graph of a binary mixture for a given pressure is shown in Fig. 5-12. The abscissa is the fraction of material A and a complementary scale is the fraction' of material B. TPe fractions can be expressed either as mass fractions, for example, ~
,
Mass fraction of A
mass of A. kg
:= ............------~--......---
mass of A. kg, + mass of B, kg (
FIGlLRE 5-11 A binary 50lution.
94
..
DES1GN OFTIiERMAL SYs1EM.?
- ':'9' : ....:. ...
Boiling
Vapor
.' ,' I :. . .·
1.:
'#
.-
• ..::"/
,:
...... :,-
. : • • ' . ......" _ . ':. : : ... LJ.. • : : . •
:, "
. .:
~~;'::;,":.,.; '.~
=..
•
• J-
BoiHng temperature .. . of pure A
o
LO . F~.ct~on of A,
Xa '. ' - '
LO
.0
Fraction of B,
or·
FIGURE 5..12 Temperatnre-concen~tion
Q.iagram of Ii
. binary soiut~bD ~t a cdnstant preSsure.
Xh
as mole fractions
. mol A mole fractIon of A = . . 1. A + mol B . rno (mass of A, kg)/CMWa) -
----------------~--------~------------
(mass .of A, kg)/(1v.1W a) + (mass of B, ·kg)/(MW b)
where MWa . and MWb are the molecular w~ights of substances A and B, respectively. The curves on Fig. 5-12 mark · off three regions on the graph. The upper region is vapor (actua]]y superheated vapor), If the temperature of vapor is · reduced at a constant concentration, the state reaches the dewpoint. line, where some vapor begins to ·condense. The region between the . two curves is the liquid-vapor region, where both vapor and liquid exist in equilibrium, as represented by the vessel in Fig. 5-11. The lowest region in Fig. 5-12 is the liquid region (actually subcooled ljquid). If the temperature of subcooled Jiquid is increased at a constant conc·entration, the ~tate reaches - [he bubble:-point line, where sO(11e liquid begins to yaporize. The temperature: where the curves join at the left axis is the boiling temperature of substance B at the pressure in question. and the temperature . intercepted at the right axis is the boiling temperature of substance A. , The pair of curves shown in'Fig. 5 .. 12 apP1.ies ,to a given pressure t and there will be a pair of curves for each different pressure. as shown in Fig. 5-13.
' ,'
•
.. .
.
MODELING THEP.Jv{;AL EQU1PMENT
" .::.:
. ~-
'l....
. . ;. ....: ~ '. . - :.. ( .... :J. ~.- ."--"" ...-'"
Low pressure..
o
LO'
95
~
:' : ~. ./(
FlGURE 5-13 T vs. x diagr~ for two different pressures. '
An alternate fonn of ~xp'ress1ng the 'properties of a binary solution IS to present.a p-vs. -x 'diagi-arnp as in Fig. 5-14, where the pair of curves· apply to a given temper?ture.
5. 12
DEVE~OPING
A T-VSo"X DIAGRAM
Several ideal rela~i9nships can be assembled to compute the data for ~ graph .like Fig. 5-12. In some cases.the results win be very close to th¢'. actual binary properti~s, ,while with othe.f .combinations of substances the real properties deviate from the ideal .. :In most cases the idealization ,:viII give a fair approximation of the properties 9f the ·real mix~. .
Liquid
p
Vapor
o
1.0
,.-
FIGURE ,5.14 Pressure--coDcen~ioQ
diagram.
·96 . DESIGN OF THERMAL ~YSTEMS ~
.
, - .
The' three tools used to deve~op the binat>j properties are (1) ¢e' :s~hl ration pressure-temperature rel~~~$hip~ the tv/o substances, C2) RaoillCs la~~ and (3) Dalton's lawo ' . . . . . ':.. ' -
?f
, 1, '
'. Saturation pressUi-e-temperature reI~ti.on. A sllnp).e .equatiqn fo~ that , . : relates .pressure ·m.d .temp~ratur~!at sa~ted 'c onditions (see ~ob .. 4.-l8) .45 _ .' . D -.. ' ' . .:. :.::;~;.::::; " f..~:~,tI.:7-" :'~ ":' .':. " . ~. .... In P .. .. .. _ 9~~ .~. T':~::':':~:': l" - :._. : .:.:.>; .'. "'''..' ·-:·1.-_. .. .,.'J.:...... ... . :
.... ..... .....
I
.-
"., '
0( ,
J'
~
,J
~
I
' :.
,-
~
•
J •
:'~ ~ ·, .':i · \-, ::.;........ _
••
_
where- : '- p ;··~··satllr4.pQp :·pressure>kPa · 7 .' , -. _ T = , ~bsoliite temperature, K " . C, D = constantS . .
.
The values of C imd' D are 'u ruque to each substance and mus~ be developed from 'experimenf?3l data. It is presnrp.ed that Co and Da fo~ substance A and Cb and Db' for subs~ce B arp kno~ in the equations ". . . D (5.15) .In Pa . Ca + ;.'
In Ph = Cb
Db
+y '
(5.16)
Raoult's .law .. Raoult's law states that·the vapor·pressure of one component in a mixtme is . (5.17) -where ' Pa = vapor pressure of substance A in miXture, kPa x a,l == mole fraction of substance A in liqu~d, dimensionless Pa = .saturation 'p ressure of pure A at existing temperature, kPa . Thus, as for tile binary mixture of A and B in Fig. 5-15, if at the temperature T the saturation pressure of pure A is 500 kPa and the mole fraction i'n the liquid Xa,1 = 0.3, the partial pr~ssure 'of substance A in the vapor is (0.3)(500) = 150 kPa . . . Dalton's Jaw. Dalton's law states that the total pressure of the.v·apor mixture is the sum of the partial pressures of the constiruents .
PI
. FUI1hennore
, pa
I
and
Pb
= Pa + Ph = Xa.vPI
(5.19)
= X b,,,Pr
(5.20)
where P, = tota] pressur~ k,Pa X == mole fraction of A in vapor ~ b.\' = mole fraction of B in vapor I
Q ."
(5.18)
MODELING
p~
_~.-'. ':.~ '"
===
e··..··. . ·· . ·.•..
THER~AL
,
97 1
EQUIPMENT
I )
t 50 kPa ~..........-.~.:
I ";:''','
.. ...
".
"
. ....i;.
~
. ..'
.
~ ., . ....
' . .... _.
,.
.... ...1
,/,',.'
.
.;
p
Exam ie .5.S. A binaDr ~oLution of n-butane .and n~heptane exists i,n 'Iiquid-,:,upor equilibriu.m at a pressure of 700 ~a. The "saturation pressuretemperature reiationsh.i:ps are .
,
.
~utane:
. " ' ·2795 In P =!= 21.77 -
T
"
3949
Heptane: In P = 22.16 - - .. T .
.
where P is in pascals. Compute the n101e fraction of butane in the liquid and in the vapor at a temperature of 120°C. .
Solution. At 120P C the saturation pressures of pure s~bstances are Butane: Pbo, = exp (21,77 Heplane: Phep
=
~7:: J ~ 2322 !cPa
(22016- ~9~) = 182 !cPa
eXP
The combination of Eqs. '(5.17) and (5.18) along with the recognition that xhep. I
and
+ Xbut. I = 1.0 yields Phep + PbUl = (I -
Xbu!.
J)( 182)
+
(Xbut, J)(23~2)
= 700
so Xbut. J
= 0.242
To find the ·concentration of butane in the yapor use Eqs. (5.18) to '(5.20) t
Xbu(, Xbl.ll.v
=
'Pbut = (0.242)(2322) = 0.803
P,
700
The results may be compared with the experimentally detennined properties .s hown in Fig. 5-16. ..
5.13 . CONDENSATION OF A BINARY MlXTURE'
tat
When a pure substance condenses cons,rant pressure. ·the 't emperature remains constant but in a binary mixture the tem.perature progressively changes even though the pressure remains constant.· The condensation process of ai»nary system taking p~ce in a rube (Fig. 5-17) ,c an be represented t
98
DESIGN OF THERMAL SYSTEMS
300 280 '1-
. z~o, .'
• -
. -"
'1
. 240'- ·
I"
. ."···,:;:,<··<.l.·:t~~::::,:·, ','
\
'.
.
.'
-:;
'
220
"
~~~~~.~.~ '_'_::?:>.:', ':': ~.'! :-.·~~.20i /1 -
"
1
. '
"
.
•
'0
' 0
•
. ...¥
. •
~
. . ••
.
-,'
-.': ;,. "
,.1
•
.. .
'
'
" "
100 80·
60 p = 700kPa ' ..
40
20
o
o
0.2
b.4
0.6
~ 0.8
FlGURE 5~16 1.0
Mole fraction of n":butaie
Binary system of n-butane and
n-heptane.
. for a specjfic binary mixture as in Fig. 5-16. Assume that th.e condensation takes place at a constant pressure 0[--2800 kPa and vapor enters wjth a mo1e fraction of 0.6 n-butane. Point. 1 is superheated vapor at a temperature of 223°C, which is fIrst cooled until point 2, where the temperature is 218 C ~d condensation begins. Removal of heat from the mixt~re results in continued condensation. At point 3 the temperature has dropped to 190°C, and here the liquid has a concentration of 0.33 and the vapor 0.80. The system is a mixture of liquid and vapor at point 3, and a mass balance of the butane can indicate how much liquid and how much vapor exists at 3. For 1 mol of combination C
0.33 mol in liquid state
+ (0.80)(1
- mol in liquid state) = (I mol)(O.60)
Therefore at poi,nt 3 in 1 mol of combination there is 0.427 mol of liqu.id and 0.573 mol .of vapor. . ..
,
2
3
....-,::::!5. ~~ . .' . "
AU liquid
,
SubcooJed
liquid
FlGURE,5-17 Coc:do.ll&aIoiQA of
8
binary
mixtu~ ,.
MODELING THER~AL EQUIPMENT
99
CO"1de s ation continues to poirt:4~ , \vhere the tempe~ature is 158°C; here .011 the vap'"'r has 'condensed. :,?urther ren10val of heat results -in' sub~ooling the liquid. During the cpl1densa~ion process the temperature drops from 2 18 to- 158.°C., ' . , ., ', . ' ~- : . . ' '",-
I '.-
,; , , ' " J', .,.
" '.',:.'i •.• ,'
4
••
~~-
A s~~ple di'stillation unit,is the single-stage 'still shown ,in' Fig. 5-18. If the ' still operates at pressure for v/~ch the bubble-point and dev/-point ,curves are sll1ywn in' Fig. , 5-19, and jf the entenng Jig uid to partial vaporizer is at point 1, various outlet -conditions o,f the -vapor~and , liquid are possible. ' Th~ limiting cases are combination 2-3 and combination 2 "_ 21 If,the iiquid is heated to point 3, only. liquid wouJd leave the still and no vapor. If the vaporizer carries on the process to 2' I, only vapor leaves the still and no
a
the
I.
liquid. The desired operating condition will be 2 /-3', for example, where · the sense 'that the vapor leav~s with a high ' there IS some separation
in
r'------'-------;!>-'
Vapor
2
Separator
Liquid feed Partial vaporizer
3 '--------!l- Liquid
F1GURE 5-18 Single-stage sriH.
,
- .o
.. 1.0
FlGURE 5.. 19 Some possibte outlel conditions from the sriH in Fig. ,Sa 18. .
10,0
DESIGN OF THERMAL SYSTEMS ::--
concentration of A and, the ,liquid .Iea~es \;vith a lower concentration ,thar: .. entered at. 1. . . . . . " .1 . ', ~ample 5.6. A' single-stage ,disti!I~on tow~r receives 3'.2 molls"of.:butane,heptane (Fig. 5-16) ~ Liq'ni: ~,'!. ::,~'~:...:~.;.~ :;" J<.<'::. .': , ." , ..:<~ .~ < ~.
but:ane,.
i
of
' -J·-.1
. ":'.:'; .'.'" .:. .
. " SO[l!ti.on. At 120°C· aI?-d. ,2 . pr:es~tiIie .of ?,oq kPa; "the 'mol'if"fr,i'ction: of'bir~e:- .-:-- ~in the liquid is 0.242 and in the vapoi:. is··O.80?:. A .~aterial hruance OD ·the:..
butane . states that .
.
'(3.2 mol/s)(O.4) = (w'/ molls)(Q.242) .
.
WJ'=
2.30 IJ?olls
+ .
WI'
(3.2 - w/)(O.8q3) .
I
= 3.2 - 2.30. = ' 0 . 90 molls
5 . 15 RECTIFICATION . . The single-st~ge still described in Sec. 5.14 perfOrn1S a separation but has rather poor per.formanc~. The other end of the performance spectrum IS'
'-'::, . :' ::- -fII-
r.
. Cooling
j
Liquid feed
.
-....
" FIGURE 5-20 A rectific8rion column',
.'
Vapor
f>
101
MODELIT--JG THER.i'v1AL EQUIPME."iT
FIGURE 5-21
o
,Stat,es' of binary system in rectificaMole fraction
1.0
Xa
tion column.
an, id~al r~ctificatio~ ' tower~ ' shown in ,FIg. 5-20. The ' i-vs_-~t' properties, , ,cor:responding to the key locations in the rectification celumn are shown in , 'Pig. 5-21. 'Liquid feed, assumed saturated, enters at point i., passes- through , a partial vaporizer, and is heated to temperatu~e 2. In the 'Col,uffiIl; vapor, flo\vs upward and liquiq downward. , The liquid is heated at the bottom to drive off some vapor, and a heat exchanger at' top of the to\y.er ~ondense~ some of the vapor, providing a s.ource of liquid to drain down the ' ' column. ' Figure 5-21 shows that the separation of the two componeJ?ts is quite effective, since the liquid leaves the bottom of the tower at condition 3L and vapor leaves tl,le top at condition 1V. There is a continuous trapsfer of , heat an'd mass between the rising vapor and descending liquid. Furthermore in the ideal rectification tower there is equilibrium of temperature and vapor pressure between the liquid and vapor along the column. For accurate simulation of towerS there must be a' temperatur.e difference between the vapor and liquid to transfer heat and vapor-pressure difference between the vapor and liquid in order to transfer mass. 3--6 -
,
me
5.16 ENtHALPY Enthalpy values of binary solutions and mixtures of vapor are necessary when' making energy calculations. < For sys tern simulation (Chapter 6) -the enthalpy data would be most convenient in equation form. More frequently the enthalpy data appear in graphic fonn, as shown in the skeleton diagram - of fig. 5-22. Figure 5-22 is an entha\py-concentration, h-vs.-x, diagram for solutions and vapor. Since pressure has a negligible effect on the enthalpy of the liquid, the chart is applicable to subcooled as well as satuIated liquid but becalJ§C_'be enthalpy of the v.?por is somewhat sensitive to the pressure, the ,enthalpy curves for vapor apply only to saturated conditions. I
·102
DESIGN, OF THERMAr-- SYSTEMS
':
_. ~-: ......
.....
!
·1
.
L
.' .. :,..•:0 , .
.~", .,
• ;.;'
., . ~: ,~<'
\
.-
-".:.")
.J
,;
..
"
'
' j'
.
-
.
,",
.'
t. '
I
i.
o
1.0
Mole fraction .of A .
FIGURE 5-22 Form of an ~ntha1py:- . concentration diagram.
5 .. 17 PRESSURE..DROP AND.PUl\1PlNG
POWER A ' cost that appears in most economic analyses 'Of thennal systems is the pumping cost. The. size of a heat exchanger tral?sfemng heat to.a.liquid can be reduced, for example, if the flow rate of liquid or the' velocity for a given flow rate is increased. The cost whose increase eventually overtakes the reduction in the cos.t of the heat exchanger as the velocity or flow lI)crease . is the pumping cost. Another example of the ,e mergence of pumping cost is in the. se1ection of optimum pipe size. The smaller the pipe the less the first cost but (for a gi ven flow) the higher the pumping cost for the life of the . system. . Since the pumping-cost term appears so frequently, it is appropriate to review the expression for pumping power. The pressufe drop of an incompressible fluid flowing'turbulently through pipes,. fittings, heat exchangers, -and almost any confining conduit varies as w n
t1p
= C(wn)
whereC is a constant, w is the mas.s rate of flow, and the exponent n varies - between about 1.8 and 2.0. Generally the .value of Ii is close to 2.0, except for flow in straight pipes at R~yno)ds numbers in the low turbulent range', The idea] work per unit mass required for pumping fluid in steady flow .is J v dp. and for an incompressible fluid tbe power required is , w Power == -Ap
p
wheJt-p is the densi;ty.
..
MODELING TIiERJyfq\L EQUIPMENT
Th
~S
Ihe equz(ion for the pumping
pO'\Vrf
103
is
. .t.-:
'vvhieb is· . further··modified bY;:di~'idirlg . by. ~he · pUl)ip:~· .f9P~ -9'r .. cQr,4p~:es.sor, . :., .. , -...;- -ef-flt?iency . '<"' :!""/'<.:~,. '.:~' --....--.;;:· -.'7~,_.;.:.·.:··-· :>.':_ :I:J";"':~-"/;-'- "~ . : ,·' ,: " ,.' .' . ' : . ~ . -'
The rn.ethods of ffi2.thematical fi):odeling explained. in. this .c hapter- have geperally been limited to expressi~g one variable as a functio-n of one or two qther' . varial?les. In prinoiple ' it is possible to extend these m~thods functions of tlire.e variabl~s, but the execution might· be formidable. Turbomachines,' such ~ f~ns, pumps? compre~sor~j and turbines, are used in 'pnictic:,!lly all themial systems, an~' in these components the dependent variable m~y. be a fun~tion 'of three of more 'i:ndependent yariables'. Fortunately, the tool of dimensional analysis, frequently pennits reducing the number of indepen- ' dent variab l.es -to a, smaller numb.e r by treating groups of tenus as i.n dividual variables~' The perfonnance of a centrifugal compressor, for example, 'w ill typicaUy appear as in 'Fig. 5-23. Instead 'of attempting to express P2 as a . function of six v'ariables, an equ-ation could be developed 'to express P2/P( . as' a function -of the two other dimensjonless groups. Then in calculFtting"
to
roD
- - ::: constant
""c T I p
ww.
,
2
D PI"
FlGURE 5-23 of s, centrifJgaJ compres.sor expressed in dimensionless groups to reduce the
Perfonn;lll~
l1umber
oT iril1ependel"i [ 't1ar1abJes"
..
.
104
DESIGN OF 1HERMAL SYSTEMS
.,. . ·P2 ,the .two iIidependent variables wDI -JcpTj and :,v )Cp TI/P2pl would.·be "c~culC1ted frrst ~d · ~ext p21p~; ·fiDai~y') P1.} would l?e c~mput~d from /!2iPl and Pl. •
,/:. -
_'""_,
I
•
," ' .
t'
'. '
' :'
-
•
: \
.
',' -
•
~
•• :
.,
-
' .•
I
.
•
_
' .
_
-
. , ;>'.:'" ',":-i:::'>'.. ~', ~.~l .. · ·~h~t..is the.. '~ffectiveness' of. a ~Qunterllo~. heat _exch~ger
~
•
-
. ,
that. ha~ a
UA :
- ~ ,': - ', . .<~~'~:: .value' pf 24- -kW!K~if:th~;:-r~$.p,~cti¥~,;;m~:s:.1:"at~ ,of flow and sp~d.pc . h~ats of J. . . . ·thetw~~dt__te~.IO kg~S; 2·kJjcki·:kr~41~!Vs;:'~kJf(k~: ~)~,._~:f;;~;·· : ~ .
a
,5520. · W~ter fl.ows tJ:rrough one ~ipe of a heat~xchanger with -flow -rate- of,'O:2 , kgls rising in temperatUre f~om · 20 to 50~C~ The specific heat of w~ter is 4.19 lcJ/(kg . K). The fluid on the 'other side of the Q~at exchanger enters. at 80°C ~nd leav'e s' at 40°C. What is the effectiveness' of the heat exchanger? ' , Ans.:. ,O.667. , . '5.3. A flow 'rate of 2 kg/s of ,water, ,cp = 4~19 kJ/(kg . K), enters one end of. a ~ounteiflow heat exchanger at a temperanrre of 20°C and leaves at 40~C. Oil enters the ' o~er side of-the heat exchang~r at 6Q'?C and leaves -at 30°C. If the hept exchanger were mao.e infmitely.large while the entering t~mper£).~es and flow rates of the water and oil rem·?ined constant, what would the rate of heat transfer ~ the exchanger be? , Ails.': 223.S kW. 5.4. Tht? evaporati,n g fluid in an evaporator has a temperamre of SoC and the heat . exchanger transfers 50 kW. The U , A of the exchanger js 12 kW/K, and the u'cp of the fluid being c;:ooled is 8 l-wlK. What are the' entering and leavIng temperatures of the fluid being chilled? ' 5.5. Stream 1 enters a mUltjpass heat exchanger at a temperature of 82°C V!jth a, , flow rate of 4.1 kg/s; the flujd has a specific heat of 4.19 kJ/(kg . K). Stre·am 2 enter~ at a temperature of lSoC, wjth ~ flow rate of 4.5 kg/s; the fluid has a specific heat of 3.2 kJ/(kg . K). The effectiveness of ·the heat exchanger is 0.46. ,What is the rate of heat transfer in kW in the heat exch~ger? 5.6. A counterflo.w heat e.xchanger having a VA value of 2 kW/K receives stream 1 at a temperature of 60°C and stream 2 at a temperature of 20°C: The flow rates are 0.3 and 0.4 kg/s, and the cp in kJ/(kg' K) are 3.2 and 4.19 for · streams 1 and 2, respectively. What is the outlet ,temperature of stream 2'1' 5.7. Fluid 1 enters a counterflow heat exchanger at a temperature of 60°C having a f1o~ rate of 1.0 kg/s 'and a cp of 3.2 kJ/(~g . K). F1uid' 2 el1rers with a temperature of 20°C havjng a ~ow rate of O.S kg/s and a cp of 4.'0 kJ/(kg . K). The , VA value of. the heat ex~hanger is 3.0 kWIK. What is the outlet temperature .q'f Fluid l? . . , 5.8. A./1ow rate of 0.8 kg/s of water' is heated in a heat exchanger by conqensing steam at lOOcC. When wa-ter enters at 15'cC. it leaves the he~~ cxch'anger at 62°C. If the inlet water temperature were changed to 20°C while its flow
rnteand the condensing temperature remained constant, what would its outlet temperature be? :Ans.: 64.2°C;
7 ' ,
[f:" ~n: '.
~~i:,
MODELING THERM;}L EQUIPMENT
I-
105
5.9. A CDU_,[cr:[ , -: -h eal exchanger coo:s5 kg/, of oil, cp = 2.4 kJ/(kg - K), wirh vlater that has a flow rate of 7.5 legis. 1-be specific heat of water is 4. 19
~g~~: '
r.,~":t;_-'.,~:·~;_.',r: .
kJf(kg . K). Under the original oper;,ting conditions the oil is cooled from' 75 ·to 40°C ~wben. water enters at, -25.~C. To what tenlperature w~l1 ..£the oil rbe· . cooled if it enters at '65°C ,and if there is no change'in the entering water ~".;. :.,,;.:,.,. ,...... ,. " : . remper~.i~re? the: ~9f e.~me~· fluid, or ,the .he a t-transle< ~gefflci~nts? , ." .' .~ :; '.' A\ns 37°C' :.,,=.,:-.:,- > .......:' " .: . . ' .... . ' 5~<:Ul,~ ~n a pr~ce~Osing pI;~;': ~~~~a~erial ' m~~ti be .~·b~~!edl~orri ..~O · ·~o~·'8Q~C~. i~ · oi~ei~· , : for· the desired reaction to proceed, yvhel!eupon. the material. is'·c.ooled in a' , . " 'r egenerative heat exchanger" as' shown in Fig. 5-24 ..The specific .heat of .the , ma~eriaI before and . after' reaction ' is 3.0k1/(kg· K).' If the ·UA o{this' counterflow regenerative hea'£ exchanger is 2.1 kVIfK and the flow rate' is' . 1.2 kg/s, ·what is the .temperature t.leaving the 'heat exch~ngei? ' /~.. Am.: 57·.9°~. . ',. '~. , . -", ' . . 5 .. 11 .. A condenser having a UA value 'o f 480 kW/K condense.s steam· at a temp~r,a ture of 40°C. The cooling \var.er enters at 20°C with a flow [(lte of 160, kg/so ·What is the outlet temperature of the coolirig water? The specific heat of ,water is 4.19 .k:J/(kg . K). . .. . , Ans.: 30.2°C A heat exchanger with one shell pass and t\~tube ~asses (Fig. 5-10) uses seawater at 15°C, cp .3.8 kI/(kg . K), to' cool a flow rate of fresh water , of 1.6 kg/s enteri.pg at. 40.°C. The specific heat of the fresh water is 4.19 kJJ(kg . K). If the VA of the heat exchanger is 10 k;WIK, ~hat must the flow •.
flow ,ra'tes
4
!'
.
the
'=
'rate of seawater be in 'order to cool the fresh water to 22.S0C? Ans.: 7kg/s. 5.13 .. A double-pipe heat exch\anger serves as an oil cooler with oil flowing in .one direction through ,the inner tube and cooljng water in the opposite direction 'through' the annulus. The oil flow rale is 0.63 kg/s, the oil has a specific heat of 1.68 kl/(kg . K), the water flow rate is 0.5 kg/s. and its specific heat is 4.19 kl/(kg . IC). In a test of a prototype, -oil entering at 78°C was cooled to 54°C 'when the entering wat~r temperature was.30°C. The possibility of increasing the area of the heat exchanger by increasing the length of the double pipe is' to be considered. If the flow rates, fluid properties, and entering temperarures: remain unchanged, what will the expected outlet temperature of the oil be if the area is increased by 20 percent? Ans.: 51.3°C . .
~/'
~
.P/./ . l~
kg/s
, F1GURE 5-24 Regenerative heat
exd: ~ nger in
Prob. 5.10.
.106
DESIGN OF THERMAL SYSTEMS
~4~O ventilate a factory ~uild~g, 5kgls offuctfu-y air; ~lemperature Of. 27°C ' .~s exhausted,
and an IdentIcal f19w rate of outdoor rur at .a temperature of i -12°C is introduced ~o take i~ ,plaoe. To r....cover so.me of the ' h~t 9f the . 'exhaus't ai,r, heat excI-;tangerS 'are :,p laced 'in exhaust .at-1rd ventilation:·air
the
.. , "::' .'. ".:, }.' " ~ (,,'" ~Y:'~', .
' ,
ducts, as shown ~ Fig. 5 ..25, ~d 1 'kg/s of water is pumpeq b~~~en tli~two " ~eai ~~chaJ?ge~., :!.?~" .~~,.~~~f., ~~_2?~ ' ~f.~~s~,:cou~nterfl~w .he~t ,exchangers ", is 6.33 kWIK. ~hat :Is' the tempera~~ of au- entenng ,t:p.t?' fact9~?: : ' ' .....
"
she.et, ~ sho,:,n' in Fig. ·. $.-2? ~e .absorbing.. s~eet..absorbs 500: ·WIi!?-'2' delivers all thIS ,heat to ¢e aIr, belDg h,ea~, which loses son;te to the atmo.. sphere through the. transpru:erit sheet. convection lieat-tninsfer coefficIent
,: ,,) ":" ~::f1S:1 A,' SOI;'~~~ ~c'con~~.;~ZOf;'1i~t}a~>~,,?uct,~mp~~~cl,:<)a:~e,:~S!,~9; ~~7 an" 6,' , ,\7, aoS'orblng sheet- backecl:,by msulatIon- and.:.~ii, tl,ie'. ntfiet:.slde',;by.· a·;t:ran:sparent :~ and
. ',,,.
f' /_
yV
,
rJ\')
'j~
The
~
.
\l\A,Cp "-5
Air. -12C
.
-
I
';-(5 1- i'l,)~ ' , ,~ - . "';v ... . 1vv ~ y4l~~£
.
____
.
5 kg/s Cp =
I k1f(kg.K)
2 kg/s water
VA == 6.33 kVf/K '
Cp
(b"
= 4.19 kJ/(kg·K) FIGURE 5-25 Heat-recovery system in Prob, 5,'14 .
.500 W/m2 absorbed
Ambient 15°C
'~'\-\-\ '-'~ \
\
\
\
\
~
'
~
Trruispannt sheet )-m
\
~
Airflow 0.02 kp)s I = J5°C cr == 1000 JI(kg· K)
sheer
, FIGURE 5·26 S'oJar air heater ·in Prob, 5.) 5,
'i"\'
;;:::J'. ~:~~
I
h m ilie
".t;i~~r',!":;.~:_,;'~,: ,;:l:
[ansp~ent
sheet to me
a~bientl ::~~L::G :/::A~ ::U~::~:m ::: Vv/(m~
air being heated to the transparent sheet is 45 . K). The air enters wi th a temperature' that is the san1e .as the aD;1~i~n.t, namely lSoC, ana t~t;_ flow ' r~te ' of 'a~r is 0.02 kg/s' per meter width. Develop the equ~tjon fQfth~ t~rhper:al'Jre ' -. of h~4t~d, air ,t " ~s .. a .func'ti:o~ :,length " aI~ng the ': collector' ~1(, aSsuming. no
'. , 4 .
[i~:
'of
)?
k:'
,< -<:o~!l,~:s~; i~, ~(!f~~~j~~~~r;:~~ij~- airflow. - s-:':" '.-'< . . ' '.> ~ir @~r The chain of·heai exchangers sho;"'nin Fig. 5-27 has the purpose of elevating ' c. .
':0'
'.
of
the t~mpera.ture a fluid t? 3~O K~ at \vhic~. temperature_~q,~~sired chemical. reactIOn ta~es, place. The flUid has a speclfic heat 9(3:2~PI(kg . K) ~oth,
be~ore -and after the r~3ctlon., and its flow r~te is~ '.~-kg;s. , .The en[eri~_g ·temperature of the flUId to heat exchanger I IS 290 K and the UA of tins heat exchanger is 2.88 kW/K. Steam is suppljed {O heat exchanger ~I at 3)5 "K" and condensate leaves at the same temperature. The UA values of heat , exchangers II and III are 4.7 and 9.6 k\VIK. respectively._What are the values ,of temperatures t I to 1>l? '' , l
. Ans.: to{ = 365 K. 5.17. A binary solption of liquid and vapQr exists at equilibrium. In the liquid the molal fraction 'of material A ' is 0.4 and of material B is 0.6. At the· exis'tinE!: . te~peratu[e the saruration pressure of pure A is 530 kPa' and of. pute 13 i~ 225 kPa. (a) What is the pressure in [he vessel, and (b) what is the fraction of A in the vapor? ' Ans.: (b) 0.6.15~1~. A single-stage still, as in Fig. -5-18~ is supplied with a, feed of 0.6 mole fraction of n-butane and 0.4 mole fraction of IZ-heptane with properties as shown in Fig. 5-16. The still operates at a pressure o'f 700 kPa. How many moles of vapor are derived from 1 mol of feed if the vapor is to leave the still with a mole fraction of butane of O. 8? Ans.: 0.5l. 5.19. 'A vapor mixture of ,n-butan~ and n-heptane, with properties as shQwn -in' Fig. 5-16, at a pressure of 700 kPa and a tempe~ature of 170°C, and a mole fraction of butane of 0.4 enters a condense!,_
Steam 375 0 K
II
Heal
e~changer
II
Heat exchntnger I
Rextor
flGURE 5·27 Chain of hea~t.-e""'xc""h'--a-ngers in Prob. 5. J 6~'
«
108
DESIGN OF THERMAl.. SYSTEMS
.(a) At what temperature does cOI?-densation begin? . (b) At what temperature is condenS·~tion c?mplete? (c) When the temperature is .120°C, what ~s · the fraction in liquid fonn? .
. . 5~20 • .A mixtur~ . of. butane· and propane i~ often sold as a fu~~. ,Wei-are in.tt;r.. : .: ested in detenninfug .the"T-Vs. ..-x-lrelanonship.,of a binary mixture of bit~e . . : 004 prop,Ule .at statIdaid atmosp~·eric · p.ressure of 10 l . 3 ~ .. pr~sure- . .. ·
.Ansg: (c) 0.78. ·
.
.
.
The
. .. . . -:.:'!~<~.~. .:'" ~ =, :-:~~.. .' J :', ,.. te~pera.~e re!~~p~.~p.ip'~..~.~ .~·aWIJl~~fP~~i.ti~p:s ~?r .,~~~ p~e ~~~~tances ~e . .
: ., .. ."'.-
. .
'~. '
:
. . ,. . ·:i· ·· ...t . /..
f:
~ ... ::.';.~" '~
..•;-: .
':: • .'.
1
. . _. _
where P =
t
ill P -
::
:....:.
f;~·86 ~ ~. 5·~'<:~:. ·
"2~~
::::e-·-. '· '··
pres$ure Fa . ~ absoh,lte temperature; K t
· Presentthe T -vs.-x curves for vapor and liquid neatly on a full-size sheet of grapb paper, where x represents the mole fraction of propane. Ans~: One pOlnt on vapor.c~rve, x = 0.5 when T = ,260 K. 5.21. A distillation tower (Fig . 5-28) receives a two-component solution in liquid · form. The two components are designated as A and B, and x indicates the mass fnKtion of material A. The ceflcentration of the feed .x 1 = 0.46 and · the entbalpies entering and leaving the still ru-e hI = 80 kIJkg, h3 . = 360 kJlkg, ·and 115 = 97 kJ/kg.The condenser operptes at .1 = 30 D e, at which state X2 = 0.92, h2 = 320 kJlkg, X4 = 0.82, h4 = 23 kJlkg, and the condenser rejects·S50 kW to the cooHng water. The reboiler operates at t ·= 210 ClC, at which temperature X6 = O.081h6 = 108 kJ/kg, X7 = 0.13, h7 = 41S ·kJ/,kg, and the reboiler receives 820 kW from high-.pressure steam. Complete Table 5.1. Ans.: Flow rate at f = 2.18. kg/so
4
- Feed
Sri H
7
, 6
.
': '. ' .'.:.~:: :~.:. ; .:~..~:~. .';~):/./.:. :~): : :;. :.
nGURE 5..2.8 DistilJation tower in Prob . 5.21
Iv10DELfNG
THERMAL EQUlPl'lfENT ~ 109
'IABLE5~1
Opera6.ng
C,
flu-if s of distiHation tnvver ' ..&.-}
. li, !{j"lkg
Position
'1-·' ' . .~
>,11--' ,..,.,
.-.;.• . . ' ".,....,'.- . -,.,
'.l.',
': ""0.46' ,"
, ". ,- 80
~~ ~~;~ ; " -~~"~~~~~~'~-~~~~~~~~~~~~~~~~M~~~~~J~~~;~'~~~-~~. : ' . :' ~d"~~ ~; .:.:, ...... .. . ~," ~ ~~~~ .
. D..92
320,
.3
360 23
0.82
97
5
6
0.08
108
7
0.13
415
5022a Dunensional analysis s~ggests that the performance of a centrifugal fan can be express~d as a function of hvo dimensionless gro~ps: SP
and
D 2 w 2p
where S P = static pressure Fa D = diameter of wheel,r'm ,w = rotative speed, radls p = density, kglm 3 Q = volume rate of airflow, O1 3/s . For a given fan operating with air at a constant densitY, it should be possible to plot one curve, ~s in Fig. 5-29, that represents the perfonnance at all speeds. The performanc~ of a certain O.3-m-diameter f~' of Lnu Blower COIl1pany js presented in- Table· 5.2. (a) Plot neatly on graph paper the, above performance data in the form of Fig. 5-29. 7
,
.-
Qlhl
FIGURE 5-29 Performsnce of 8 centrifugal fan .
110
DESIGN OF THERMAL SYSTEMS
TAJtHJ~ 5 .. 2 ·Perfor~ance of
•
.
fan in ·Prob. 5.-22
rad/s
Q,m'3/'s ·
SP, Pa
157 .
1.42 J.89 2}6 ···.
'861 '.
. '.
<
..
861. ;
...'72.~·~·.· \-~ .':,
<"-,'
126
3.30
114
0.94
304
1.
~ ~ ';:: .~:
-
SP, Pa '
.: ~. 94 ' .. ' ,: ..
.
..
radis I'
~ ··r·~'~> ~ "
>. ,.':.;._ '.,] :27. '.
.; " 299
. '., >.. , ,.1..89" · ... ~. .'" 21?:t_'~'" \ '~';~-;~"_"~' "':'; . ' .,.. ... .2..22 -....... ,134 . ~~ .. ·- '';'1' . . .. . .. 52?' -, ,:-: ~ . . :'"-'~:'.:' .-·~·~:·:~';::~·,2·~'3.6_·.·~·~':~~ · ·' ::'· '100" '-~~ " . '~'::""'" _. '.. .. .
.;:/:>'(:.Y_'. -.,.::'~ ,,:~,'O::,; , ._,. '~ " . 3~·02;~:>" 2.83 694,' . :·. ·-,:·:' :~,. 635.;.~: ....:-:--., .· " .. 3:;30 .
,.
1"26
L42: 1.79 2.17 2.36 2.60
548: 530 ·473 428 351
. ' 63-. -.'
. O~80....... ~':.
(34 L94.l W22 . ·1.42 . 70 L51.55 .
(b) If the SP is ~o be computed as·a·function.of Q ~d w,'propose a Gorivenien~
form of the equation Gust use symbols for the coefficjents; do not evaluate them numerically).
REFERENCES L. London~ CompaCl Heat ExclzQngers; 2d ed., McGraw-Hill, New . York, 1964.' ' . 2. P. Wo[s~e-Schrnidt and H. J. H¢gaard Knudsen, "The~aJ Modelljng o(Heat Exchangers for Simulation Purposes!·' 25th Hear Transfer Fluid Mech. lnst., Davis, Calif.: June 19'76. 3. Vi. Badger and J. T. B-anchero, introduction loChemica/ E11gineering, McGraw-Hill, New York~ 1955. -1. W. L. McCabe· and 'f. C. Smith, UniT Operatiolls of Chemical Engineering, 3d ed., McGraw':Hill, New York, 1 9 7 6 . ' . 5. C. M. Thatcher, Fundamentals of Chemical Engineering. Merrill, Columbus, Ohio, 1962. 6. C. G. Kirkbr!de,. Chemical Engineering FlIlzdc:menlals, McGraw-Hill, New York, 1947. 1.' ·\V. M. Kays and A.
~.
.-
. .....
) I
..... . ..l :•. . :.,
. , ... . J ' ...: .. - -, ..: . .. '- ...... " . •
.:~
. ....
. .... J.
'_,r: _. :"~.l~ : ~ '.
'0
' , '"
'~J :- _, ="
..
. ,...: ..... "
.
• ' -~
. :"'_' ~ . . .~ . . '?::;'.;, ;-;,,~::.~ ,.-"., '.'
.f
'6' " ..'.,-. .
.l
. ~.~
-: ,' : \
.
.
.
-
.
-
=--
I.... .
'
,.,
J ' . ""', - . -,' : .~.
,
~.
',
'I '':'
.-',
' ,"
,,'.
. . .SYS TElVI · '~ .. ~SIJ\Ij[ULp~TIOl\J ...,
.
601 DESCRIPTION OF SYSTElVI SIlVfULArION . .
.
.
'
System simulation, as practiced in this chapter, .is the calculation of operat-. j'n g variables (such as pressures, temperatures,' and flo'vv rates of energy and fluids) in a thermal system '.operating in a steady state. System simulation pre~unles lqlowledge of the perform'arlee character'istics of aU ~omponents as well as equarions fOf thennodynamic properties of the 'w orking su.bstances . The equ~tions for performance characteristics of the' components and thermodynamic prope'r des, alongw'ith energy and mass balances. form a, set of ·simultaneous equations relating £~e operating variables. The mathemati- ' cal description of system simulation is that of solYing these simultaneous equations. mnny of which may be nonlinear. A sysleJn is a collection of components whose performance parameters are interrelated. System" ii/ntllarion means observing a synthetic syslem that imir.ates t~e. pe~6rmance of a rea~ system. The type of simulation studied in this chapter can accomplished by calculation procedur~s, in. contr.ust
be
of
to simulating one physical system by observing the perfonnance another pfiysicaJ system. An' example . o,~ two corre·sponding physical systems is \vhen an electrical system of resistors and capacitors represents the heatflow system in a solid wall. .
.
6.2 SOME USES OF SI1\WLATION System simulation may be used in the de,sign stage to help achieve an irnproved-'Qrsigl.l. or it may 00. applied to an existing sys·£.e m to explore III
112
DESIGN OF mERMAL SYSTEMS .
.
'"
"~
.
prospective modifications. S!IDulation is not needed at the design coJidi~ioi:: beca~se in the design process 'rp.e engineer probably chooses reasonab.: values of-the operating variables (pressures~ temperatures, flow ,rates, etc. and selects the components (pumps, .c ompressors,· heat exch~gers~ et~ .. :thai· corr~spond to the ope!ating yar1ables~ ,It :would .he .for;tbe D'o ndesiir . 'condition~ ·that. ·system. ·simulation would pe applied, e.g..,- as at, part.:.loac ~ or oyedoa~ .conditions. The designer. may ~ish to' investigate 'pff-design
.">':: .'>. , r . '<~ :.'
,.::~., :~ ~;:::op.~i3.~on. to~ -,}).e
;>.
too high or: tQo.low·" -". '. , .. ~ '''.,<'~ . .f':,<'·~~'-~"r;;-. '.": ..-- .... ,. ~.-: ~ , '.';;'-:'." "<." . . . '" The steep in the' cost'of eneFgy~.has·"PtobablY'-been...resp()~si~ . ble fo'r. the .blos·soming of system simulation d~g recent years. Thermal sy's tems (power'. generation, thermal processing, heatipg;' arid "re~gera~ori) operate ·m ost of the ~e at off-design cond,itions_ To perform energy studies in the design stage the operation of the,system must be simulated throughout th~ ' range of operation th~ system will experience. ' . System simulation ,is sorp.etimes applied to existing systems when th~re .; is an operating problem 'o r ?l possible improvement is being considered. The effect on the system of changing a component can be examined before the actual change to ensure.. that the operating probl~m will be corrected and to find the cheapest means· of achieving the desired improvement. After listing some of '[he classes of system simuJatjon, this. 'chapter . concentrates on just one class for the remainder of the chapter. ~Next .the use of infonnation-flov" diagrams and . the application to sequentia1 and simul taneous ~a1culations are disc~ssed. The process of simulating thennal systems operating at steady stare usually sinm1ers ".down to the solution of simultaneous nonlinear a]gebr~ic equations, and procedures for their solution ar--e examined.
increase
r-··' . ··
6.3 CLASSES OF SIMULATION System simulation is a popular term and is used in different senses by various workers. \Ve shall first list ' son1e of the classes of system simulation and then designate [he type to which our attention \vil] be ,confined. Systems may be classified as continuous or discreTe. In a continuous
. system, the flow through -the systen1 is that of a continuum e.g., a fluid or I
even -'solid particles, flowing at such rates relative to .particle sizes that the
stream can be considered as a conrinuum. In discrete ~ystemSt the flow is treated a certain number of integers. The · analysis of the flow of people · through a supl:lmarket invoiving the ,time spent at various shopping areas and the checkout counter is a discrete system. Another example of a djscretesystem analysis is that perfonned in traffj~ control expressways and cit;y streets. OUf concern, since ir is primarily directed toward fliuid and energy systems, is continuous sys[e~s. . Another classific~tion js delerlnini.sfic v. slochastic. In the detennin .. isti,e ~Jlal;'6 Iis the input vadables are precisely specified. In s[ochas[ic .anal),-
as
on
SYSTs:M SIMULAT.ION
-'i_l.l .'l:.. .
sis the . input con trIaDS are uncertain., .eithrr being cornplet~ly random o r .' (more commonly) following some probability distributionc In simul:J.ting the pelfoHl1ance of a s[eam-electric generating· plan t tIl.at supplies both. process steam and electric povler to '3. facility 'J for exam:ple, it dererm~rtistic . analy~,~s s~?Its \Vi~.h ·'Qn~. ~~p~c.i.fi~~t Y-~lJ.~ .of ~be st~arp demand cliong with one . . .: sp~ci4e~ y~~Iue _.Qf th~ pqwe~ ~d.em_~nd. ~ ..f." 'st6cQas:tic ,:~ui~y_~is .migJJ:t_beg.~,., : ...
......-'.: . :.".~ ~":with.
some
probal#titY ··\ge~C'~ptiq~. ·of·~pe:.~$teti'rii' a~n(r p'Qw.er :Jqeiriands~.:·~· We""·' '." ~:~~.
. -: .: . ;:shall cop.centrate i'n :Lhis 'chapter on deterministic analysis, 'and reserve,Jor.Chapter 19' the stUdy of probabilis.tic influences. ." .' ... . ' Fin~lly. system 'simula~ion ~ay be classed as st~ady,~s~ate·or. d}·nar[lic, . \vhere in a dynamic simulation the~e ·are cha~ges of operating. \:arjables w.iiJ~ respect to time. py:o.amic analyses are .used for such 'purposes as the 'study of a control system in order to- achieve greater precision of coritrpl and to avoid uns.table operating cqhditions (Chapter 15). Th,e djnamiG .sirp.u!a.:tioP.1 of a given system is more difficult than the' steady-state simulation, .since the steady. state fall$ out as one special case of the transient analysis. On ~he other hand~ sreaO:y'-state simulations are req~ired much more often than dynamic simul-arjons and are normally app~ied to much larger systems. The simulation to be prac~iced here will be that of continuouS'~ deterministic steady-state systems. .
(
"
Ruid- and' energy-flow diagrams aTe standard engjneering tools. In system , sjrnulatiori, an equ-ally useful tool i$ the inforrnation-flow ~i~gram. A block diagram of a control system ·is an information-flow diagram in- which a block signifies that an output can be calculated when the input is known. in the block diagram used in automatic-control work the blocks represent transfer junctions, which ·could be considered differential equations. In steady-state system si.mularion'the block represents an algebraic eguation. A cenuifug~ pump might appear in a fluid-flow diagram like that shown in Fig. 6-1 a, while in the infonnation-flow diagram the blocks (Fig. 6-1 h) represent
PI--~
'-8
W
- ~~>tl ~
-~-.fI_0-t;_,",_.t_e
or P 'I ----a..-t
P2
(0)
Pump 't - - - - t - w I(P 1, Pr w) co '
(b)
FIGURE 6·1 (0)
Ct:nlrifogtlf pt:rT.p -. fluid-flow di,ng.ram (b) possible infonnation-f}ow Mocks, representing
pump,
114
DESIGN OF THERMAL SYSTEMS
fil:nctions or expressions that pennit calcnlati !1 of the outlet pressure ~- _ the one block and the flow rate fC?r the oilier_A block, as in Fig. 6: ~ is .usually an equation, here designated f(y I ,P2, w) = 0, .or' jt m.ay _.: ,tabular 'data to .\vhic:Q in.tetpolati6n v/ould be applic-able. -. . : ": ' . ' . ,'. Figure 6-1 onlY'one component.',To illustr~t~ how~these indivi":. ' ~ b~ocks can b'uild tne irifoiInatJion-Jlow'diagram'for a--sy-stem~ rcons~der' tt:
hs
_~' . .
shows
;' :: .:":::' . '.; -':' ,.'.:".( ", ~' ..,-. ',fire-w~ter ~~cility shown -in Fig., 6-2. A pump .having 'ijres.sure~fl-6w ,charac-,:::' ". '~ ." ".'',. ;"';~te:ns.tics·~ sli6wD.··.jn:Fig;,,',6r-2 \~~WS,..W.~t~r frQ~ . ~. op~~i re.servoir ~d :delivers ,'it through a lengJil o{pipe'to':'hyOrarit A>:·:~jl(w~~qme··.wate£".con-tint.tjJJg-'*9..ti~ . additional pipe to hyd:dint B. -The" w~ter flo.w -rate~,.,in ~e"pip'e" sectiori~' ..~ ., desjgnated w] a~d W2, and .the .f1ovl fates pas-sing out the'~':fiy:drants -are" 1.v. and .l-VB. The 'equations for -the water flow rate througb open hydrants are l-VA = CA ~P3 ~ Pat an'd WE ,~ CB ~P4 ~ Pai, whe~e. CA and CE . are. constants ,and Pat is. the atmospheric p.ressure~ The "equation for .the pipe section 0..;.1 is Par -: PI, = C1 W + hpg, wher,~ el w_ aCCO,unts ~or frictIon an4 hpg is the pressure drop due to the change elevation h. In .pipe sections 2-3 ane 1-4 . . . .!_
i
in
i
and
,
.
P3 - P4
=
C
2 311'2
These fjve eguatj'o ns can be wrin~n in functional fonn fl(WA, P3) ' Q
(6.1)
0
(6.2)
· !3(Wt,Pl) =0
(6.3)
f4(Wl,P2,P3) =0
(6.4)
=0
(6.5)
. f2(WB, P4)
f
S(w2, 'P3, P4)
:=
The atmospheric pressure Par is not listed as a variable 'since it will haye a
.t
,-....:~2~·
t
t
WA
WB
t
_ _ _ _ _ _ _t.'' H_Y_d_ran_'_1_ A __ 'kll~
3
, hm 0 J ~~. ----1
P.,
FIGURE 6-2 Fire~w3ller ~y51em and pump characteristics.
w:! -...:.
1,
Hydrant B
.. Pump characteristics J( P" P2' w);:: 0
SYSTEM SIMULATION /;~
115
Hydrmlt B
. ,h. (\~B' P4,)
FIGU~ 6-3 Infonnation-flow diagram for fire-water system.
known value. An additional function is provided 'by the pump characteristics (6.6) ,
The preceding ,six equations can all be designated ' as con~ponent performance characteristics. There are eight unknown variables~ WI, 'Wl, \OvA, WB, p'r, P2, P3, and P4, .but only six equations :s o far. Mass balan.-c es , .pro~ide the other tv(o equations ' •
or
(6.7)
3.l1d 'Or
(6.8)
Several correct flow diagrams can be developed to express this system, )ne 'Of which is shown in, Fig. 6-3. ,'Each block is arra,n ged so that there is )nly one output, which jndicates that the equation represented by that plock 5 solved for the ou't put variable. ).~
SEQUENTIAL AND SIMULTANEOUS =ALCULATIONS >ometimes it ,is possible to start with the input information and immediately alculate the ou tput of a component. The output infonnation from this rust "omponent is all that is needed to calc~late the output infonnation of the next ' omponent, and so on to the final component of the system. whose output
.; the output infonna.tion of the sys'tem". Such a system simulation consists -·f se~uenfia/ calculations. An example' of a sequenti~ calculation might ·ccur in an on--site power-generating plant using heat recovery to generate ,earn for heating or refrigeration. as shown schematically in Fig. 6-4. The xhaust gas !.-m.F_Dthe engine flows through the boiler, which generates steam
116
DESIGN OF THERMAL SYSTEMS
Steam :...
J Absorpt ion .re.frigenui~n
unit . -,' . ',=.
~
•
,, -
. ...: .. ;
• I
. . '. _.~ondensale
.:· Y;~-~' f"::;~~-:~':' :.: .:,~~ :><.~,... .;,." .
-
q. ~frigeration
.. .
Engin .-e- '-,- 1----11· Ge~erafor·
I
I,
FIGURE' 6-4 On-site power generarion with b eat recovery to develop
ste~m ~or
refrigeration.
to operate an absorption refrigeration unit. , If the output information is tt:: refrigeration capacity that wo~ld ' be . available when tile unit gerierates ~ given electric-power requirement, a possible' information-flow diagram fo:: this simu~ation is shown jn Fig. 6-5. " " " Starting with the knowledge of the engine-generator speed an~ electricpower demand, we can solve the eq~atjons representing performance characteris~~s of the components in sequence t6 ariive at the output information, the refrigeration capacity , . Th'e sequential sir:nulation shown by the information-flo\v diagram 0""
EXhaust-gas fi o-
Speed
Input
I
-
Engine
....... 'Electricpower demand
Elec[ric generator
"
~
"
Engine
..
,
,capaci.y
tcmper.lwrc
RcfrJgenJlion
'.....-_fl_o_w_--I
un.i t. 10-_ _-
_ _....
flGURE 6-5 information-flow diagram for an on-site, power-generating plan', of Fig, 6-4,
-
.-
Exhaust-gas
Steam '
Re£riget:Hion
,
r""1
Power
BoBer
Sy~ SIMULATION
117
Fig. 6-5 is I r:s:- ~-:.:.~ :::; :-':'- c S ~·i'!1 u.ltaneous s/r..f1ulation requ.ired for the illfor. mation-flo\¥ di2~gram or Fig. 6-3 . Sequential s1ffiulations' are straightfo0~ard~ but simultaneo us s imulations are the ~baH:enges on iNhich 0-e remainder of th.e cha.i?ter c n ~ eri.trates . ' .,. : .. " ~.. . ;.'. !., . ;
':'-:"".: '::>-":~J;-..
.: .· . 1\ , ."," ' . •
~ , ;.1'",
r
••
~'.:. . .
.... •. ,.:,
..f~ .'-'--"-::~:
,...... ;.
. . ,'"
:
'T'W' ~-~~HODS::·,()F"~SjrMUtliTIOI~:.'·~':~ :'·:<;;· ~-- · "":'' -:';' ' ·
::,~r _ .~ . . ·.SUCCESSIVE
§ UJ]STITUTJrON ' AIW"
' . . ' .,
.,'
NE1tYTON~RAPI-IS'O N ·The ta'sk 9f simulating a system~ after -the functional relatiopships and interconnections have been established, is one' of solving a set of sirnultan~ous . . algebraic equations, some or all of which may b~ n·onlin~. Two of ;the methods available for t't~.is simultaneo.us sp1ution are sLlccessive substitution. and Newton-Raphson.Each method has advantages and pis advantages which will be pdipted out. '.
607 . SUCCESSIVE· SUBSTITUTION ..'The method of suc~essive substitution is closely 'a ssociated with the'information-flow qiagram of the system (Fig. 6-3). There "seems to be no way .to find a t6e-hol~ to begin the calculations. The problem is circumvented by assufning a value of one or more variabl~s, beginning the calcuHttion, and 'p roceeding through the system until the originally-assumed variables have been recalc~lated. The recalculated values pre substiruted successively (which is the basis Jor the naJjJe of the method), and the calculation loop' is .repeated until s~tisfactory convergence is ac·h ieved. ' Example 6.1 A water-pumping system consists of two parallel pumps drawing . water from a lower ,reservoir and delivering it to .a nother that is 40 m higher, as illustrated in Fig. 6-6. In addition to overcoming the pressure difference due .to the elevation, the friction in the pipe js 7.2w2 kPa, where w is the combined flow nite in kilograms per se~ond. The pressure-flow-rate chara'cteristics of the pumps are .
Pump 1:
/J.p. kPa' :::: 810 - 25wl - 3.75wi
Pump 2:
flp. kPa
= 900 ~ 65w2
,
- 30w~
WI and W2 are the flow rates through pump 1 and pump 2. respectively. Use sLl!ccessive substitution ,t o simulate ,this system and determlne the va!uesofl1p, WI. W1, and w. .
where
~oJution.
The system can be represented by four simultaneous The pressure difference due to elevation and friction is
.'
equatioDs~
~18
DESIGN OF TIIERMAL $YSTEMS
T
.
J.
.
..,. •
. : . . ..
~.: •
'. ,; .........
40m ' .
: , ,:
.
'
.. '" .
.W2 . .
.
"
)
..
)1' "
~~Pl F1GURE 6-6 \Vater:"pumping systeTI?- in Example 6.1 . ., ' (40'm)(1000 kg/~3)(9.807 mls /1p =. 7 .2w- + . 1000 PaIkPa
2
)
(6.9)
Pump 1:
IIp = 810 - 2SwJ - .3 .75H.·· i
(6.10)
Pump 2:
/1p = 906 - 65w2 - 30H"~
(6.11)
Mass balance:
(6.12)
One possible 'information-flow diagram that represents this system 'js shown in Fig. 6-7. If .a triai value of 4.2 is .chosen for WJ, the value of Ap can be computed from Eq. (6.10). and so 'on about the ·l oop. The values of . the variables resulting from these i·te~tj.9ns are shown in Tabje 6.1. The calculation appears to be converging slowly to the values 1.99, w = 5.988, and !:ip = 650.5.
111'
Elevation and fricrion
WJ
= 3.991" W2
=
U'
Mass ~1
Balance
,
Pump 2 I
,
I
FIGURE 6·7 Informat'ion-flow diagram J for Example 6.). 't.
wJ
Ap Pump 1 ...
~
SYSTE!'v1 SL\1ULATlON
',:
1: j
:J.p
119
\)
focmation-flovY diagr axn
·w~ .-.
,
.
,
2.0.60... .. , , ~.<. . 5...852. : _ ,' " _.: 3:.,:}9.2: .. <: .... ~-<.• •,,';:-. ,,;,~ ,.~;., .:•• : .~~.:... . ~;:. -<.~.~:.~.:;, :.. ;" ~: '.~ ' 9~J.16 ":....~ .... '; 1:939" ., ..... . . '6 ..112 ,-:'.' " 4 . 174-·., -, ... ...... . 640.34 2~052 .: ~5.-870 . . 3-,818 659.90 . . 1.946 6.097 4.151 .
. _-.638J15" - ,..
•
.. -
.....
-
Q
•
47 48 49 50
-
..
_
......
"
, "
. . . . . . . . . . . . . . . . . . . . . . . . . . . .j,
......
t
........
5.~83
1'.000 1.995 2.000
649.98 650.96 . 650 ..04 ' 650.90 .
.......
..
3.983 3.999 3.984 3.998
5.994 5.983 5.993
1.995
v
6 .. 8 ·PITFALLS IN THE METHOD OF SUCCESSIVE SUBSJ1TuTION Figure 6~7 is only one .of the possible information-flo:w diagrams that can be generated from the set of equation$ (6.9) to (6.12). Two additional flow diagrams are shown in Figs. 6-8 and 6~9. Tpe trial value of W2 = 2.0 w~s chosen for the successive s~b~titution method on. information-flQw diagram 2, and the, results oJ the ite~ations are ' shoyv'n in Table 6.2. A trial value of lV :::: 6.0 was chosen for the solution of information-flow diagram 3, and the results .cu·e sho\vn in Table 6.3 . . Information-flow diagram 1 converged to the solution, while diagram~ :2 and 3 diverged. This e'xperience is typical of ·successive substitution. It should be obseryed that tpe divergence' in diagrams 2 and 3 is a~ributable . to the calcl:1iation sequence and not a faUlty chqi¢e .of the trial value. I~ both cases rhe trial value was essentially the correct solution: W2 :::: 2.0 and \V == 6~O. .
Elevacion and ." friction
l:lp ~
w
I
1
-
I
..
Mass
w,
,
Balance ,I
Pump I I
I
,
I
.:IGURE 6-8
:)formation-f)o\t'dtagram 2 fcr Ex~mple 6.
r
I
W'2
Pump 2
IIp
120
DESIGN OF THER.,\1AL SYSTEMS
.'
,". .:
. ~. '-~" .
I
Pump 1
'.
.
l
~1
Mass Balance
\i:'2
w
-
,
.
.-
-
Elevaticm ' : '" ,", li.~ . .---and "
friction ', ,'- , '
j I "
', : ' .,
, " . ' < 'i" · . · ',
FIGURE. 6~9 ' Infom:iation~f1ow diagr~ 3 for Exampl~ 6.! .
JrAJBIJ8 6.2 .
,
(
.
It.erations of information..flpw diagram 2 ' Iteratio~
.
Ap
w
650.0' 653.2 , 635.5 726.5 42.8
..,1
.L
3 4
5
WJ
5.983 6.019 5.812 6.814
4.000 3.942 '4.258 2.443
1.983 2.077
1.554 4.371
.L
I
t Value of w became imaginary. TABLE 6.3
Iterations of informatio~-flow diagram 3 Iteration
/lp
651.5 648.5 654.6 642.'1 667.6 616.1 722.3 514.5 951.2
2 3 4
5 6 7 8 9
)1'1
W,2
3.973
1.992
4.028 3.916 4. J42
2.008
~.671
4.593 2.539, 6.149
' 1.975 ' 2.042 1.903 2. ]78 1.580 2.662
w
5.965 6.036 5.891 6.184 5.575 6.771 4.120 8.811
t ..
tValue 0("·1 became imaginary.
~
Are there means of checking a ~ow diaEram in advance to detennine
whelher the, calculations will Qonverge or diverge? Yes, ~nd a [technique , will be explained in Chapter 14~ but die effort of such 8 check is probabJy gre2:ter than simply experimenting with various diagrams until one is found
-
that ~onyerges.
..
..
1..t, 1
Sy',)YC,Jvf SiMULATION ' I
.
I
lri the m.E;[hod of succes.s ive substitution each .equation is solved for one var i1ble? and the equation may be nonlinear in that varjable, as was trLle, for example"1 for the calculations of w ~d W2 in diagram 1 f',To 'p'articular problem resulted in compil~i.I}g. ,y~~.,\~.nd ~W2' h~re ·because the. eqnat~ ons 'wer.e G
'- H~.aq.ra1=ico· An lier~ti~vi iechntq~.e·,._·.\vhich.m.ay J~e_ requited....in~~9me...cases~js:-:;:·.i':J·,:..,··:·:. . .'.':' .~' des·crj.be.~ .iIl" S~~. 6 ~]'O:' . "~.:. " I .. " ".
.
;."
.
. .. " :
" : ....
- .. . '
.'
.. '.'
. .
..:'
.
•..
.•
.609 " T;.lYLOR-SERJIES E){PplNSJ[Ol\J·
anq
The second technique of system simulation, presented in Secs~ :6~10 6.1 l', the Newton-Raphson method, is based on a Taylor-series expansion. Tt is therefore appropriate to review the' Taylor-series expansion. If a fu,nctio~
·w hich 'is dependent upon' .t wo variabl~s x and j,., is to' be exp'arided about .. . . the point (x = a ~ y . b), the fonn' of the series expansio~ is.
. z.
: " const
+
first-degr.ee terms . + second-degree terms
+
higher-degree terms
. or, more specifically ~
z
=: Co
+
[Cj(x - a)
+
C2(Y - b)]
+c~(x ~ a)(y -6)
+
+ [C3(-::-
- a)2
es(y - b)2]
+ ...
(6.13)
Now determine the values of the constants in ·Eq. (6.13). If x is set equal . . ro a and -y is set equal to b,' all the terms on the right side of'lhe equation: reduce to zero except Co, so that the value of the function at (a,b).is Co
(6.14)
= z(a, b)
To find Cr , partially differentiate Eq. (6.13) with respect to x; then set x ~ · a and.y = h . .The only term remaining on theright side of Eq. (6.13) is c},
so
alta'. b) Cl -
rn
(6.15)
ax
a similar manner, rJz(a, b)
(6.16)
( oy
The constants C31 c4, _and Cs are found by partial differentiation twice followed by substitution of x = a and y x: b to yield
a2 z(a, b) 2i1 x 2
a2 z (a, b)
I
C3 =
C4
=
ax a y es
1 a2 z(a b) I
="2 ay 2
( 6.17)
For the spe~.oe:9·e where y is a fl3nction of one independent variable .x, .he Taylor-series ·expansion about the point x == a is
. :..
,' '12~
DESIGN OF THERMAL SYSTEMS
,
y = y (a)
. .
+ .dy.(a) " (x
: - a}
'9 x.
,
[ I-d 2y
+ . +-
(a).] , .' . -' (x '- a).2 + ' ~,~ ~
2 dx 2 , : .' , ,The general expression' for-the Taylor-series expansion if y'is a f
,
I
x~' .. aroun~ ,the '-point- ~(X.1 ,='.a;.,. x: ": ;;'i.',~~, ' ~l:~1.:;.~.: ~~;./!).:~,.~P~!;;,~;.t.i:. <~: .,/.' , ; / /:,c:.~ i,' "' . (':;::' ,c i;".' ':'~' , :;;,;:-:,,-,~,:< ::" -~ . " , Y.{Xl,":X2,' .. ~ xnJ -:- y(al', 't;li, ~ .. ,' an) . . .. . '
i .., '
'j '•.•.,
,, -.
'., +.'L' n
.
. J ~1
_
,
"
8y(al;' ~' .. ' , an) '.
Example 6.2. Express the function In (x 2Iy) as ' a about the point (x = 2,y, = 1). ' .
i}x· J
._' ..
=
x2
In
y =
Co
' Cl(X
+
~ 2)
+
1)
C1(Y -
.+
C3(X -
,
2)2
+ 'C4(X - 2)(y -'1) + cs(y - '1)2+ . " ", Evaluating the constants, Wf. get C.O
i2 ' = In. T ==
CJ
= a z (2, 1)
ax
az(2~ I)
C"2
=
-..:..-~
C3
=
~ a~ ~.~;
, C4
ay
1n 4
=
1.39
= 2-.; Jy
= ~
.:'('1/)'
x
1
x:'/v2 " 1 =.--.,"- = - - =.-1 x-I),
1)
=
y
M-x2~) -l =
_ o2 z (2, 1) _ -:-~",' , _ '.'ox'a" . - 0 'J a2z(2,t 1) ] 1 - -- ay"J - 2.\'2 -
c - ~ - '1
,
=
"2
The first sev,eraJ terms'elf 'the expansion are
_.__._,_.., _z =
1.39
+ (x
:
2) -(y
-
(x" - c.
J.
Tay~or-;,s.erie.s expans:_ =
. Solution' Z
•
I~ - (~)(X - 2)~ + (~)(y - 1)1 +
lr_
SYSTEM SL'vYULATION
123
6010 I'~EVvT N~RAJPI-ISOrJ V~/ITtl Or'iJl E QIJjs).,_~ ION A IV 0 f\lE UI"~K roV~lr~In the' Taylor-series expall:si.on, of Eq (6.18) ,whe~ x is close to a, thebigher' order ternlS become n~gligible . The". eqtI:ation then n~duces ,ap:proxiI11:ately ,to ,: ", .' 0
. ·-."i ,"
:
.•
, ' ."';" ",:,;:" ?"':',;t~)::: ~~,:[.y' l(alJ(x'~~·';:~" 'dY~'~~~-:"·"- :' , -:. ,- ", '~:' (6:: 26Y: ~:~':'~\-;
_ . I. '
..:
.
.....
~
..
. #
-
'
.
.
, "
.
•
•
•
_
, Equation (6.20) is the basis of the Newton-Ritphson iterative tecl1I1:iqtie for' solving a nqnIinear algebraic equati01"l. Suppo,se,.that the value of x is sought, that s,atisfjes the equation ' ,
+ 2
x
Define y
= eX'
,
as y(x)=x+2-e i ,
(6.22)
and denote x c - as the corr~ct value ofx that solves Eq. (6.2'1) and make,s ' y
.
0
(6..23) ,The Newton-Raphson process requires an initial assumption of the , value of x~ Denote as x { this temporary value of x. Substitudng x t into Eq., (6.22) gives a value of y whjch almost certainly does not provide the desired value of}' , . O. Specifically, if x t = 2" Y(x I )
:= Xl
+
2-
=2 +2-
eX,
7.39 ='-3'.39
Our trial value of x is incorrect, but now the question is how the value of , x should, be' changed in order to bring y closer to zer09 Returning to the Taylor expansion of Eq. (6.20), express y in tenns of x by expanding about ,x c
y (x) = y (x c)
+
[y' (x c) ] (x - xc)
(6.24)
+
[y'(Xt)](x, -xc)
(6.25)
For x' = x I, Eq. (6.24) becomes Y(x r ) ~ y(x,c}
Equation (6.2:5) contains the further approximation of evaluating the derivative at XI . rather than at Xc, becau~e the value of Xc is still unknown". From Eq. (6.21) Y (x ,) := 0, and so Eq. (6.25) can be solved approximately for
the unkn,Qwn value of x, '
<
_,
Xc ~ X, -
,
,y(x,) '(
• y x,
)
In ¢e numerical example·
,-
Xc
-3.39 1 - e2
= 2 - ..
= 1.469
(6.26)
,
124
DESIGN OF THERMAL SYSTEMS ..
-
4 1
S~,ond
I
. ", ~
'
tri'P' x = 1.469
,
'
.,.:
\.
dyJdx , ~ ..
.
:.
. ..
2
~
I
...... .-
-
. ... . ;"
'-.'
-2
--4 '
,
F1GURE 6~ 10 , Newron-Raphson iteration..
,
.
The' value of,x == 1.469 is a more 'correct value a,nd sbould be used for the next iteration. The results of the next 's everal iterations are
x J (x)
1.469
1.208
1.152
-0.876
-0.132
-0.018
The graphic visualization- of the iteration is shown' in Fig. 6-] 0, where we seek the roOt of the equation y = x ~ 2 - eX. The first trial is at x = 2, and the deviation of y from zero divided by the slope of the curve there suggests a new trial of 1.-469. The ,Newton-Raphson n1ethod\ while it is a powerful iteration technjque, should be used carefully because jf the initial trial is too far off from the correct result, the solution, may not converge. Some insight into the nature of the function being so] ved is therefore a] ways helpful. ~.11
NEWTON ..RAPHSON METHOD WiTH IVrULTIPLE EQUATIONS AND UNKNOWNS The solution of a nonlinear equation for the ,unknown vatitll>le discussed in Sec:" 6.10 is only a special cas'e of the solution.of a set o.f mUltiple nonlinear , equatjons. Suppose that three nonlinear e,quations are to be solved for the ,three unkno\-vn variables x J X:!. and x) t
=0
(6.27)
0
(6.28)
~(xJ ,X:2,'>:3) := 0
(6.29)
fl(X).X2.X;4)
,
I
f2(X, ,X2,X3) =
f .-
125
SYSTEM ZI..\liULATION
,The' procedure for solving the equa~iolls \is an iterative one' in v;hich the follo,vin,g step~ ar~ follo\.\led: ,.t;.-
,
1. Rewrite the equations so ' that "all .terms are on one side of L'1e: e,qualiry , ' , sign [Eqs. ,(6.,27) to (6.29} alre~ulY ,.eki,stir. 'tJJis' fom~i.r· ', :",'. ,' ,'
,,' ",;
2':.:.~j~.ssfime~' ieill:pofaDL~
"3~ Caic4Iate·the' y(lln~'s off" ~'-'i' f2".,~df3 :.cit'the:'t,emporary :vaiu~ of ~l~ :~'2~ . ,.and ,x 3· 4. Compute the partial derivatives of all functio.ns with respect to ~l1., vari- . ables', 5. Use the Taylor-series expansion of the form of Eq. (6.19) to establish a set of simultaneous, equations. The' Taylor-series expansion for: Eq'. (6.27), for ex~mple, is ' ' . ,
'
, fl(xi.r-. Xl. r , X3.r-) , fl(.Xl,c, X2',~, X3.e) ~ Xl. X3.1) ( _" +,- afl(Xl.l' a ,x 1,1 . Xl"
~)
I :
+
ail (.x1.r
I
X2.I'
.1:3.1)
aX2
(X?
,-.
r -
X I.e '
') x? c " -~ .
(6.30) The set of equations is '
6. Solve the
iJ!l
ail -ail
ax}
r3x2
af1
af?
ax) aff
ax}
BX2
aX3
af3
af3
_at3
ax)
rJX2
aX3
x I.e
~ X
i1
I.e
X2.1 -
x2.c
X3.t -
xJ,c
-
12
(6.31)
i3
set of linear sim,u ltaneou,s equatioI!s (6.31) to determine ~l.r
-
X i,e'
7. Correct the x s t
X:L new
= X3. old :-'" (X3,t
~ Xl. c)
8. Test for convergence. If the absolute magnitudes of all thef's or all the /).)( '5 are satisfactorily small, -tenninate; otherwise rerum to step 3.
;,' ',',
126,
DESJGN OF THERMAL SYSTEf.1S
, Example 6.3. Solve Example 6.1 by the Newton-Raphson ,method.' , "<~ '~ I
'Solution STep 1.
.:, •
• 1. -
'.:>"\ ". .
\,'
..~. . :' j7' \'
'.
~.<.-
-,';':::-;r.'~:~~·':~':~,~,~.";:?!:- : l :,': , . - .:'
\' .
.~
,
'-.,: '
Step 2. Choose ¢aI values of the variables, whj~h are here selected as /lp = 750, ' 11.'1 = 3. l"'2, ::::; 1.5, ,and w = 5. I Step 3. 'Calculate the magnitudes of the /'s at the te:r;nporary values 'of . the variables, 11 = '177.7,12 = 48.75. '1 3',= .15.0, andf4 = O.50~, ' , Step 4. The p~rtial derivatjv~s are shown in Table 6.4~ " ' Step 5.' SUbsti.tuting 'the temporary vaJue~ of the variables into the equations for the p~ial derivatives fonns a set of linear simultaneous equations to be sol ved for the co;rrections to x: LO " 0.0 1.0 47.5
' _[ 1.0
0.0 -...: 1.0
,0.0
where
j,'Yi '
= .r;'~ I
-72.0][01:J.-l] [177.7_] 0_0 .6. x 48.7)
0.0 0.0 155:0 -1 '.0
2
0.0 ~x 3 1.0 ~x -f
_
,-
15.0 0.50
Xj. c
-
Slep 6. Solution of the simultaneous equations
.6.x J = 98.84
L\.r~
= -1.055
.1.:\'3 = -0.541 ,
L\X4
= -1.096
STep 7. The c?rrected values of the variables are
Jlp
= 750.0 -
98.84
= 651.1?
H'I
= 4.055,
w;?
= 2.041
11'
= 6.096
These values of the variables are returned to step 3 for the next iteration. The values of [he I's and the variables resulting from continued iterations are shown in Table 6.5.
The calcufations converged sut·isfnctorily ~fter three iterations.
TABLE 6.4 ,
a/aA" ' al dW,
'.
af,1 j} i)f ~Jii /If ,fa
J
af~/D
0
~.
J J
ala\\'
a/PH~
0 25 + 7.5\\',
0 0
- J4.4w
.0
65 + 601"2 -J
0 -J
,- ,
...
0
J
127
SYSTEIIJr s1MULATION
After iteration.
f
.,
:W2
I
.
w ·
.:O.oqO ... ·__.-;:.?~l..~~..,. 4-~055: ·· . . 2:04"'1: "":;.6·.09~~: .. ' 0.000, .. <650.48 '" '3.992" . ·' L998.- .. .5 ~989··. ' ' . ' 'O~O??,: ~.. ~0:000.' .:",:'..\_. ~~O.qo~~,-::;.O_OOO, ~~ ~ ~~0-:49:': :':~ ~ .- 3':·9.9-1~)."~:,..,:ji997J~..;,,,,;.5~988-.:~:':~,,~!~.,,~,~.:, -"".
'- 8.68.) . ~O:.0'81 ,
"4.170
. 8.778:
O~OI48"·'· . O.fJ~~6·
.
~. , ..
0"
• • ..:
~
.'
6u12 SIIVIULATION' OF A GAS TUR'BJITiE SYSTEl%,1 .' . A ··simulatioll: of a more extensive thermal system' will be given for a rio.nre~ generative gas-turbine cyc'le, This cycle? -shown ' in fig _ 6-11, cons] sts of a 'compressor< combustor, and turbine Whose perronna.p.ce characteristics are known", The .tl).rbine-compressor combi~ation .op~riltes at 120 rls. The objective 'of the simulation is· to det~nnine the' p~YJei output at the "shaft~ .Es kW, if 8000 J(\V of energy added at the combustor by burning fueL ' The. turbine draws. air' ~nd -rejects the rurhine exha.ust · tQ 'atmospheric ,p ressure' of 101 kPa. The entering 'air temperature is 2S9.C. Certain simplifications will be introduced in the solut~on, but it is understood , [hat the simulation 'method to more refined calculations', . . can be , extended .. . The simplifications are .
is
I)
Combustor .
2
P3
'2 p..,
3
"
Turbine
qkW
Shafl
Compressor
Ec kW . £ JeW
P
"
..
I
w kg/s
P,
:=
4-
10 J 'kPa
I
FIGURE 6.. 11 Gas-rurbine cycle.
-.-- -
P4
= 101 kPa
128 ,
DESlGN 9F'rJ:IERMAL ?YSTEMS .
.
1" Assume perfect-gas propert.ies·~o~ghput the cyc~~ arid a cp constant at . 1.03 kJ/(kg· K ) ' .: , ! . '. '.: ,
~o .Neglect the r,nass added 'm the form of fuei in tlJe combustor ~o'~that : the , mass fate 'of flow w is constant throughout the cycle .-
3". ,Neglect ~e pre~sure 'drop ,in ~eJ c?~bust.o~ so Lh~t = p~ :'~d ~~ ~gli . pre~sure ill ,tl}e system. c~ ,be desIgnated, slI~ply as p. ' : :',," ' .:' ". ' :' ' ...," "-:~:'~ :~Negl~ct :hc:a,~ ~~.t~~, .t? ~e env~~nw~~t~: . ' . ' ., .: ~ ' :. ',, ~_ . - ,.' '... .. ; ,',' ~ ; '.'.-~::t ;" ,~:' ~-:,:,:: ~:r.:.,.( . ;.'. .,~,,~. .,.~ . i~;· · " -::'.':! <.~ -.:,,~:' _' ,~ '>" '. ','., ::;~::'-;''''', .:.._'"~..~<-.;!::~-
P2
,I
'
'
,"
.
.
. The .perfO:nriance. ch~acteristics.,of tb~- axia.l~fIow. ~o.tnp'ressor,, ~d ,the' . gas tUrbine l operating at 120 rls with an atmospheric pressure,of lO~.kPa that ,wjll be used,in the simulatiqn are sho~nin Figs. 6':"12,and 6-13, respectiyeiy . . With the techniques preseI!ted in Chap. 4 equations can ·be developed for ,d?-e curves ih Fig. 6-12, '. . " .1
'p
== 331 + 45.6w ,- 4.03w 2 ,
•
•
(6.32)
a,nd Ec \\,lhere p =
= 1020 -
O.383p
+ O.00513p~
_
(6.33)
discharge pressure of compressor, kPa
w = ' mass rate of flow, ·kg/s E c = power required by compressor, kW
When operating at a given speed and discharge pressure, the characteristics of the turbine take the form shown in Fig. 6.. 13. V{ith the techniques 600 500
:E oX c.i
2000 ~
oX
400
..:
3
v.
8.
Ir.
e c-
~
300
§.
o
f!l
r. ..r::: (.)
v.
] 800
IJ
L..
::l
] 6QO-
~
Co.
200
E
8
0
100
1400
1200 l~
6
8
)0
12
Flow rale.• kgls
Co)
,
14
~
100
__
~
______
200
300
~
__
400
~
.500
Dischruge pressure, kPa (b)
FIGURE 6 J2 . Performance ofaxial~f1ow compressor operating at 120 rls with 101- kPa inJet pressure. e
-
.
.
'
SYSTEM SJitIULATION
-woo
... ': _.:-.. ,. ..;~::. 1;4.' .,...
:
.
i
<. '.'
~~: 12'"
.:L. .
.I
129
'-:..l
e3 10
. =:J.
u: . 8 6 1000
200 Inl~[
~----&-'_ _ _....L.....-_ _- - - - - t _ - . l I
400
300
.300
200 .
400
'. Inlet pressure~ kPa
pressure. kPa
(b)
(0)
.; FIGURE 6-13· . Pertormance of gas turbine operating at 120 rls and 101 kP'.:i discharge pressure.
of .Sec. 4.8 eqyations can be developed for the curves in Fig . .6-13
= 8.5019 + O.02332p + . +0.1849
X
.
.
'--
.'
w
IO-.:l,2.
0..48 x IO-4p 2
+ O~000121pt
-
.
0.02644t .
- 0.2736
X
lO~6p2t
(6.34)
anQ E, = 1727.5 - IO.06p
+ O.033033p2
- 7.4709t
+O.050921pt - 0.8525 X lO-4 p 2f
+ 0.4473 x
-
+ O.OO39~9t2
0.2356 x lO-4pt 2
10-7p~t'2-
(6.35)
where" t . entering temperatUre:::::: 13, °C E, = power delivered by turbine, kW
To achieve [he simuiation the values of the following unknowp variables.must be determin~1 w, p. Eel /21 E,H t3t and E,. Seven independent equations must be found to solve fo~ this se,t of unknowns. Four equations are available from the performance characteristics of the compressor and ,turb~ne. The three other equations come from energy baJances: Compressor: Combustor: Turbine power:
Ec = WC;
(6.36)
= WCp (13 - ' 12) '
(6.37)
8000
E., ="-E c
+ E$
(6.38) .
130 . DES1GN OF THERMA'L SYSTEMS TABLE 6.6
.
'.'
.' t, .;
. ·Newton-Raphson. solution o~ gas~tllrbin~ . simulation Ec
111 .
'. Trial . ' ., .': . . · value." · ·· , .... : HJ.OO
;300.0
Es.
l000~'n
250.0
. ·.1509;~.9 :. -
.. '
. -t3 .
.4
" .'. . . .. ..
. . 354.8' '10.77 . 354.9 10.7'7 _ .... 35.4..9
'1~30~O '. ' ~ 'I62.8'"
1530.1 . 1530.1
'173.0 173.,0
.
_
". ",,:-
. ::·80,O.~, J:~, . ~ ,?~9.9· '.
K
:~;.: '10.77
3
',
E/ ' ". '. '
'.....
:. ' ~i '_:;C"~f;~on~~' c" I~: ~ -:':~:1~~~ ';:· ;;o~ ·:::":';~O.8: .: J569;:h~;:';&7;5 :';" i .'. ., >. i
.
3(1'11;0::
1~97:.7 . ... ; :·884:2·..· ·3.127.7 . ,. 1598.5 . .884.5 ,.. ---3'12.8.6 1598.5 . :884.5 3128.6
. ·The· exe'cuti~n of the solution follows the steps outl~ned in Sec. '6 ~ 11. A summary of the trial valueS and results aft'e r the Newton-Raphson ite~tions ·. " . is presented in Table 6.6. ' . ' .. , shaft power delivered by this system is 1598.5 kW. , . The . . . '.
,
6.13 OVERVIE\V OF SYSTEM SIMULATION Steady-statesimularion of therrnaJ systems js fast in.creasing in applicability. uses of sjm,ulation include evalll:ation of part-load operation di!ected lo'\.vard identifying potential operating problems and also predicting' annual e.nergy requirements ~'systems.2 System sjmulation can also be one.'st,ep in an optimization, process. For e~amp]e, the effect ,on the output of the system of D1aking a sn1alJ change in one component, e. g., the size of heat exchanger, js essentiaUy a partial derivative of the type that 'will be needed certain of the optimization techniques to be explained in later chapters. If the exposure [0 system simulation in this chapter was the reader~s first experience with it, wrestling \-vith the technjques may be the major pre·o ccupation. After the methods hav~ been mastered, setting up the equario'ns becomes the major challenge. In large systems it 'may not be simple to choose the proper combination of equations that precjsely specifies the . system while avolding con1binations of dependent equation's. Unfortunately'. no methodical procedure has yet been developed for choosing the equations; a thorough grounding in thermal principles anda bit of intuition are still the
The
a
in
_necessary tool s.
·
.
The mathematical descdpfion of steady-state system. simulation is that
of solving a sjmultaneous set of algebrai~ equa~ionst, some of which are . nonlinear. One impulse might be first to e.Jiminate equations and variables by . suostitution. This, strategy is no,t normally recommended when using a COffi'puter [0 perfonn the successive substitution or Newton-Raphson solution. \Vorking with the fu N set of equations provides the solution to 2. l~ger .-
..
SYSTEM SDAULATION
131
number oivaJ.iables ,directly; some of \~hich) may be of int~r~st. Performing the ,substi[ution alv!2~ys prese~ts the hazard Clf rnaking an' ~lgeb[aic errOL and the equatiQDs in ·coll.bined fonn ate more difficult to check ..than their , simpler bas:ic an4anger,OJ~~L . I . " " : ,, "J' "'' . ',. ' ' -7'"
"
.,.~.. ' :,.,.
.
,T.y/Omethogs ..o~ ~ysten1 s i~tllatton J1ave ,~¢en,pp::,sented)n.this'::.ch~prt:c,~.:.... . :. : :.
" , . :, .sllc;cessi\re ·~.u~~ti~~j:Qn"' 'and; Ne'w~oD:-~Rai?hsop.. -?u~ces'~jv~"·~ubstrflido:if . 'is: }i"'~::'''<':: straigbt~forward teGhnique ~d j~ 'usually easy,to program. It uses',compute'r memory sparingly. The disadva~tages are that sometimes the seque.nce may either converge' very slowly or.diverge. As. Jar as can be detennined through " the \iveb of commercial secrecy'~ many of the large siuluIation programs' u~ed in the petrole~m, chemical, and thermal ,processing industries -rely hea;vily on the successive-substitution method. 3 ,4 The experienced programmer will , enhance his chances of a ,convergent sequence ,by c,hoosing the blocks in the inforrriatioQ.':'f1o,v 'diagram in such a way that the output js only moderately ~ected by large changes . in the input. The Ne\Vl:Oll-:-Raphson technique, while 'a bit more complex,' is powerful. ' This chapter se,rves as an i~troduction to' system simulation and provides the tools for ~olvit;lg llsefcl engineering problems. Chapter ' 14 continues the study of steady-state system simulation.in greater depth. In that chapter .the successive-substitution method is explor~d further by identifying the nature of calculation sequences that result in convergence. Methods are peveloped to aGcelerate con'vergence or damp divergeIlc'e. The NewtonRaphson t~que as explained in the foregoing chapter required extractipg partial denvatives by hand~a process that is tedious. chapter 1,4 explaiils the strucWre of a generalized progra~ 5 [hat extracts the partiaJ derivatives numerically_ A challenge also addressed by Chapter 14 is .t he simulation of large systems where the number of equations and unknowns, n, becomes very large. The Newton-Raphson technique requif>es :the solution of an )1 X n matrix, so acceleration techniques become' valuable.
PROBLEMS
~~.; ...-/
The operating point of a fan-and-duct system equations for the ,two components are
Duct
Fan: where
s.p
IS
to be detennined. The
+ 10. 73QJ.8..
= 80
Q == 1'5- (73.5 x lO-6)~P2
SP
=
Q
=
st~tic, pressure Fa airflow rate , nJ) Is t'
t
Use successive substitution to solve for the operating point, choosing as trial values SP = 200 Pa or Q = 10 m 3/s . ...4 D.S • 6 m31s and ,350.P'J.
.
132
~:J ---
DES1GN OF THERMAL SYSTEMS
)
-
1
' I I S? ' ~
,-7>'"
\
'
o o
'"==-_~~'O""""--_----'
IL...-_ _ __
'
20
40 Ol' ,
ris'
'/
,
, '(2.
60
FIGURE 6-14 : Ton}ue-rotativ.e:speed curves of engine drive and load on a truck.
The, torque-rotativ~-speed
of the' -engine-and-drive . train of' a 'operating at a certain transmission, ~etting is S.hOYlll in Fig. 6-14. The T ' . UJ curve for the load :on the truck is also shown and is appropriat - r the truck ~oving slowly uphill. The ·equations for the two curves·-1ife Engine drive:
CUIve
T = -:-17,0 + 29.4w - O.284w:oIr. T = , IO.Sw
where T
=
w =
,.
torque~
N .m rotative spe.e d, rls
(0) Detenninatian of the operating condition of the truck js a simula-tion of a two-component system.' Perfonn this simulation with both· flow diagrams' . shown in'Fig. 6-15. Use an initial value for both simulations qf ~ == 40 11~1 rls and show the results in the form of Table 6 .7. t7 (b) Fro.m a physical standpoint, explain' the behavior of the system when operating in , the' immediate vicinity (on either side) of A. t"L) :::..
Engine .
Engine T
•
.Load ,
t'c. r)
(J)
Load (b)
(O)I .rI!bt1
FIGURE 6·15 FJow diagrams in Prob. 6.2.
...
SYSTEvr
Tfo.J3LE C. f'·
P~suHs
of iteratlLJDS ~r.
.. 2
. IteratioA1. .
T
133
1.1
PrG~) ~
6-15~ FJOW ~Ha~~~ F.igo"-;~
Flow diagrarn :'ig.
- , ,:y I
I
':f~
tV
.'
""J
,": ~_ .'
.' eO'" ~~ r:(.S'_~,." '~.",'.:' "':~'i;''''''~i-'c'~;;';; ,"C,_, " " · ··· 1 ·
1.
.J
2
2
3
3
o Converging to point 1
,.;; ' "
::<::c"
•
•
1
'[
ID' .Co~verging 'to point ~_ 0 Diverging ,.
, 0 Diverglng } " 6
SIi~i1JUl.TION
, "
.
.;L seawater desalination plant operates on the cycle shown in Fig. 6-16,
(f..;eawater is pressurized.~ flows through a heat exchanger .. where its [emperai
W'r e is elevated by the condensation of what b'ecomes the desalted water, and , flows next through a ste~m he,a t e=<:ctIanger, where itis heated but is still in , a li~d. state at poin.t 3. In passing '¢rough the flo'at valve the pressure drops and soipe of the liquid flashes into vapoI:, which is the vapor that condenses as fresh Water. The portion at point 4 that remains liquid flows out as waste . at point 6. The following conditions and relationships ?Ie kno\.vn: ~ Temperature and flow rate
~nd
of entering seawater.
.
,
V'v'-Q ~i...
='V'\I'>..~
heat exchangers. L.-.:Enfi('alpies of saturated liquid and sa~a[ed vapor of seawater and
/
@of the
:
~ ,~
. '
the
fresh water as functions of temperature: " V\
hI
== fl(t) ~and
hg
= 12(1) ~
For heat exchangers with one fluid condensing, use Eq. ·(5.10).
The system operates so that esse~tially · t4 = 15 ~ ' 16: \' ,
V~ t
',... Y(\L!-
. - ~-'~ , I....J.---...J
•
= 15 0 C
~
r "oJ.. \
~~
Seawarer
w
5
= 10 kg/s
Fresh w8't er,
6
,
3
FIGURE 6·16 Desalinatioo..ptW[
-
,-
if! Prob. 6.3.
Condcnsare 150°C
134
.
_
ESIGN OF THERMAL SYSTEMS
Set up an information-flow' di~gram that, would be used for a SllC- , :_ cessive-subs~tutjon syst~m simul~tio~; indiea~ing which equations .apply· . " .~ .--to 'eac4 block. For cpnvenience in checking, use these variables.: t.,. fl. h 3 , . . ' .' _ h4;,hf.4, -hg . 4, 14.' WS',X4, and q ~ere@is the 'fraction of vapor at "'_ ·~.,.PQint 4...a.ncl·-zJt~1Ile ra~e ofh~at transfer a'tthe fresh-water condens.e.r:.. '.
" _: _~ '-:,-f"" 'J?; ,-. ,.•-, ~ri·.,~OIP~
,. -,-- .
high-sp~e~ aircrm.~an · ~ir-crY~]e r~fri~e~a~jon , unit,is ~s~~ .for ~~bin -:';;. .. ' _=. . ' '- , coobng, . one concept of:whic~ IS shown In FIg. 6-:17. Eq~~t~tms -available '.' ' . ' for th~ :compress6r' are: powet= fl(tJ,'Pl, 'P2,' in) ~lJ;ll= 12(11, Pl', €I), pi).: .,': .~., .~::. ,"'. ·.For-t11.e: turbine . .the ..equ.ations .are:" po\Y.ef .~ /3.(t3 ,t ,P3~:"i:L~, P4) ~~d Tjt .= <:.. '- >-~ .~, ,:
.
-' . ' . "-:f4·(t3· ~ 'P3," -w''- P4):~ .w.heie,~jt) )rt~j~at~~~'a:::'fubftt~o~T' o'r'·:,eq4~tiq~:::fi1- ~teb:ii~~~of ~<~~ -,~,"."~ ~.~-.-~e variables- in .the. p£!.r~n.~~s~s, .. A~s,um~·.no. i!~ssure,dIop Urrough-..the.. heat.::. -..... ·...:·~
, exchanger. The compressor-turbine combination operate.s when p1 is greater ' ': '", '"~4' The following data ~. impose{fand.lmown: PI, II,' P4, the VA of'" the beat exchanger, and: the temperature and flow rate of ambjent.air through, !pe heat exchanger. 'Construct" an inform~tion-f]ow diagram- to simulate the system llsiIig the equations and variabJe~ previously listed: as well as others ~ fthat are necessary. . . 6.5'. Steam !Jailers s~etimes use a continuous b)owdown of water control the n " am~t.. or: impurities in ¢.e w~t~r. ~is high-temper?-tu~e -water 'is "c apable , ,- n.), ofheatlng feedwater, as shown lfl:1Flg.. 6-18.'A flow rate of '0.2 kg/s at a . ( l\ temperature of 340°(: is. blown down ~rom th~ boiler. The. fI0:-V rat: of the V feed water to the heater IS 3 kg/s and Jts entenng temperature IS 80 C. The . UA~alue ~f ~. he feed-water hea. teT ~O kWfK. Equations-for the enthalpy of s~turated lJqUld and vapor are, respectively, hi = 4.191' and hg = 2530 + 0.41, where t .is the iemperarure jn °C. The system is to be _simulated and the following variables computed: 11: 1CJ~ WA and WB. (a) .Construct.. an information-flow diagram. (b) Using successive substitution.• comp\lte the value.s of the variables. Ans.·: 12 = IOB.2°C. '/Jy Gfa... Co~ {\.IyJ. 6.6. I!l a synthetic-~mmonia planrFig. 6-19) a 1:3 mixture, on a ,ml ~lar basis,___ ./ .___ -.~f N2 and H2 along wit~ ~f!lpuritYI argon, passes through a reactor
to
"j.
'>
01'" ..
js
. .~)~.
:&~. P ~ "';c.r' C> "Ie
~;~.~
~
'\,,"'"I>
~)
Amb~ent .if IM.~~
2
I
fu
-
shaft
, Compressor (J)
rev/s
F r~r:n cQmpress()r
Cool, sir to cabin
of engine
4
,
FlGURE 6-17 Air-cycle refrigeration unit in Prob. 6.4:
-
135
SYS~1 SThHJl.ATION
\ .'
~;~ .~ ::
.,.'
l
.
,
' .i
.~ . :;
", :, '
I Feedwater hea~er ··
I .
~===-=''''-===:l
Ft!ed water"!
laooe
-<:i;---~~f.----I"1V\.1rv V\r---fl~~_ ~ ~.......
B[o~own
3 kgls'
Co~densate Sep~tor 1-
r-f I I Level____ . ,tIl_...:...._. ~
control
FIGURE 6-18 Blowdown from a boiler in Prob. 6.5. !., .
where . some of the nitrogen and hydrogen combine-'to form ammonjuo The ammonia product formed leaves the system at the condenser and the remaining 'H 2t N'2 • .and AI recycle to the reactor. 'The presence of .the inert gas argon is detrimental to the rea~tion. If no argon is present? the reactor converts 60 percent of the incoming N2 and Hl into ammonia, but as the flow rate of argon through the re~c[or increases. the
300 moJ/s H.., 1 moVs Ar 100 molls N2
-
I NH J • N 2 ' H 2 • AT Reactor I
J
j ~AAA .."..,.."
I,
,
Condenser
I 3 I----
'2
4
• Bleed,
F1GURE 6-19 A syntht:rlt:.!lllf1iiiO.llia plant.
~ moVs 0(- mixture
..
- ', ' (
136
.
DESIGN OF TIIERMAL SYSTEMS .
percent
.
co~version
decreases. The conversion effi,ciency follows the equation
,' ,
Conversio~,. % ~ 60e-
.where w is the flow fate of argon through the reac'tor i.n moles per ,second:. . ::. . To pre\~ent reaction from coIning to standstill, a"cqntipnolls bleed " .' . of i5 ·. inolls·of mixture of N2 , Ib, :.a,nd Ar)s provided. Ifthe.in-;;omirig fe~d ' ." ": .. . :.,., . -' . consists of :l'OO:mol/s oi .N2 , 30b m6l!s:of H 2,, :and 1" molls oCA.f;·~·s'imii14~~ e'~ ~ .~.':/::·i~·.:> --:' , .' , this' syst,e;m b)(successive sti1?stitUti,9n' to d~terniine . the "flo\l!'. rn.:tff· 9f..~xi1:rre ' . - .: . :' .: :~'. -~:'::<..:~: througfi :th,ei·re~ctor::~c?:Jhe:,:raie-.::of;~jq~i¢:. @.UIlb:n:~:a::I~iQdiidtiBii::J1ifi-b:ol~~1>~ri.':;-.:;
a
the
'.second~ "
.-.
Q . :.
/
·r . . •
.. :
·Ans.: ·S93.2.and·.188··mo]Js.· " . .' '. .. . . , . ~ For -:r(tan 'x) = .2.0, ' wl:tere x is in radians , use the Newton-Rapbson method' to determine the 'value of x. . . ~.: 1.0769. . '. f 6.8. The heat·. exchanger in Fig. 6-20 heats water entering at 30°C wi.th .steam' entering as sa~ated vapor at 50~C and leaving as condens'at~ .at 50°C. The ' ,f low of water is to be chosen so that the heat exchanger transfers 50 kV!. The area of th'eheat exchanger is 1.4 m 2 , and the .a value of·the heat exchang~r based on this area is given by . .
'
1.,
-(m- . K)/kW
V
.
'
=
0.0445 wO. 8
where Ap:;: pressure ri's e in the station, kPa HI = fJ ow rate of oil, kg/s,
( Water. 30° C, W = ?
A~ 1.4 m 2
) .
~ Condensate. 50°C FIGURE 6-20 Heal exchanger in Prob. 6.8.
+ 0.185
-In normal operation the flovv rate is 25 kg/so The pressure drop in the pipe' is proportional to the sq uare of the fLay! [a~e. If one pumping stalion falls and Lh9:t station ~ypassed? v~l o,t ~ill be the flotv rate. provided by ~e .re~aining n_~ ne
stations?
6~]Lt In a 'clos~d·loop a eentrifiigal and 1-/
.
. through
a long pipeo The
gear pump operate in series to d~1ivef ,fluid
~qllati~:mr reI~ti~g ~P and the :q,?~,~~,~.fo~ the th~,:~'
' . .: . ;:"~- :':','- .. <.C?InP"Qrie~ts~,~e::,.,w::~,.,,:.~,.~.,.,:,..:::r,,:'. ',":' ".-,:: .", "":""""',>, . '.:;,.,." ,.,", . ,,' ~ centrifugal pump-:---: .. , .. flp :=,'!i.~t 2Q '-_.oo,5Q_~ ... ': ' ' ','
' .. :"",",""":' ~ -~
,
gearpurnp:
Lip
.
~\40
-
5Q
'
/.
oJ
.' .'
pipe: .where I1p '~ pressur~'J;is~' (or drop in the pipe), kPa
__ ,.Q, ,= flow ra~e~ .~Js . ,
,
(a) Plot on a /1p -:- Q graph the performance of all cortJ.pdnentS_ (b) If 'a system ~imula~ion were performed (no need to perforrrf this simulation)" what wouid be the .appr.oximate solution? Discuss the physical implications of the solution. . 6 .. 12 .. In so'me cryogenic liquefaction systems the temperature 'of a stream of liquid is reduced by flashing off'some of the liquid ·into vapor through a ,throttling valye and heat exch~ger a,s shown in Fig. 6~21. With the va]ues shown In 'Fig. 6-21, use a N-ewton-Raphson simulation t6 detennine the flow rate of liquid leaving the hear exchanger and its temperature. Ans.: outlet t~mperature of liquid is 157.3 K. 6 ..13 .. A centrifugal pump, operates with a bypass as sbown in Fig. 6-22. The pressure drop through the bypass line is given by the equation
;
I?p
= 1.2(Wbp)~
the characteristics of the pump are"expressed by the equation
I1p
= 50 +
5wp - O.lw~
\rv\Vapor
h
120 K
= 1460 kJ/kg
\e\-Mv" T = 200 K. h = 230 kJlkg
,VA ;:.. ··1.57 kW/K .:
~
: "i':. 4.. •.
liquid. flow r.lte = 1. J kg/s-
,
cp s::
T == ?
2 kJ/(kg • K)
FIGURE 6--21 Cryogenic liquid cooler in Prob. 6. J2 .
.-
flow rate = ?
..
138
DESIGN OF THERMAL SYSTEMS fr'
,
.~ .
I
I
w
." P2 '- P.1 ~ O.018w 2 where the flow rates ,are in kg/~ and' the, pressures -in ,kPa,. ,U se the N ewton- . Ra.e.hson.techniqll_~ ~? _d:et~ne w ~ w p , Wbp and P2 6.l4. Air'at 28°C with a flow rate of 4 kgls flows through a cooling coil counterllow to ~oId wate~ that eJ;'iters at 6°C, as shown ,in Fig. 6-23 . .Air has a spe,ti:6'c " heat, c~ ' ~ 1.0 ,kJ/(kg . K). No. dehumidffjcati~n .of the air occurs as it passe~ through the coil. The . product of the area and heat-rransfer coefficient for the heat exchanger is 7 kWIK. The pump just overcomes the pressure drop ,through the control valve and coil, such that Pl = P4' The pre~sUre-flow . c~aracteristics of the pump are
P'2 - Pl, Pa = 120,000 ,- 15, 400w~, where ·W lt • is the f1o~ ~ate of \\'atei in kilograms per second. The specific hear of the water is c p = 4. I 9 kJ/(kg . K). The pressure 'drop tlu;"ough ,the coil is P3 - p~ = 92QOw·~;. The outlet-~ir temperature regulates the control
Air
{:llr QuI
~--~--~------~~~~~-------'28°C
H~
4 M·.,
Water
, FIGURE 6-23 CooHng coil in Prob. 6.14.
...
kgls
c
4 kg/s
.
.:; ~
;':" . :"~: ...
139
SYSTE:VI SIMuLATION j-
v;::,IVe tv In aintain an outlet-air temperature somewhere between ·.n and. 12°C. The flovv-pressure-drop relation for the Jalve is ll:l\' = ,C \' .jp'"!. - P3. \vhere C \' is a function of the degree of varve ope'n ing, a linear rebtion', as shown in Fig, 6-24.'Tbe fully open value of C,. 'is O~Ol·2. , " " Use the Newl0f.l-Rap.hson method to sirnuI:J.[e this syste~: d-:;t~rininlng at least , t~e foll,o~~.i,?,&..~:~"9.~~~:!~~; }J w ~...r~., (~\r)~l,H. P,~, ~"P3;' aBd.: C \<~ lise', as :the . : -' .' . ,-' "t~st' [Qr cp.~.vergence.. rpal}~~ _absol~,~i:yalue~.:qf;.~~~~:?~~~','~?-rig~ J~.s~ , V!~fL:~::'., .,:.:. . t' . 1.0, absol~~te values. of tempera.tures, l~ss. ·than_O',OOJ~ ..·aIid·.'abso]u,te· \ialue~Qf.' , . ~: C I' Iess ~,than 0.000001 ~ Li~t 'the n~mber 'of iterations-,to 10. " " . , Afis.~P2:= 64~355 (ba~edonpl = O) .. [~ == 14~14 •. Cl' ...:...·0.0108-. 60 15~ A two-stage tiir' compressor with Intercooler shown in ,Fig .. 6-25 compres'ses . a.ir (which is assumed ..dfy) from 100 to 1200 kPa absolute. The folloV'jing . data apply to the compo~ents: . ,
..
Displac~men[
rate =
0.2 rn 3 /s lovj-stage compressor ' { 0.05 'm3/s,high-srage compressor
"
Valu,m etric efficiency 1]~ % =
,
flow rate rne~su~ed at cornpresso: suction. m /s ('100) d,.splacen1ent rate, _ny~ Is ' 3
~nd
for both
compr~.s.s0rs
TJ. % = 104 - 4.0[
P~i~h )
1A
\Psuc[lon
The
.
,.
The polytropic exponent n in the equation PI v7 = P2V~ is 1.2. intercoole.r is a counterflow heat exchanger receiving' 0.09 kg/s of water at 22°C. The
Full opening. C,. ~ 0.012
/
-----------
I I r I
I
.I
II
I I I
I I J
I .
,
" 10
11
12
Our Jet air temperntun:.oC
---.-
..-
FIGURE ~24 Ch'aractenslics of valve in Prob. 6.14.
140
DESJGN OF THERMAL SY~TEMS
1200 kPa Intercooler
1 1 .,;".
, A.i!. ,·,100 kP3. .
.':•... \
.
'~. '.
'
.
. .... '., . ~ .~ ,.:&" -.:.
.
,~~:...':.:' ~' ..
.'-
.'
.,'
:--:"':: ~':.-:
. Cooling water' 22°C, 0:09 kgls
flGURE .6-25 Two-stage air compressio~ in Prob. 6.15.
product of the oyer~ll heat-transfer coefficient and, the area of tills h~t exchanger is VA = 0.3 kWIK. Assume that the air is a perfect gas. , Use 'the NeM,on-Rapbson meth.o d to sim1:l-late'this syste.m, ~eterm.injng. , at least the values of W ,Pi, 1'), and r3.'Use as a test for convergence that all 'variables change less than 0.001 during an ite~atio~. Limit the number of iterations to 10. Ans~: w = 0.18 kg/s. I]. = lOl.Soc, Pi = 387~7 kPa, 13 = 43.84°C . . 6.16. A helium liquefier operating 'according to the flov,' diagram shown in Fig. , 6-26 receives .high-pressure helium vapor, liguefi~s '? fmc[ion of the 'vapor, and returns the remainder t.o recycled. The for"lowingoper;:tdng conditions
be
prevail:
h
Point (~/apor entering warrp side of heat 'exchanger), ! == 15 K, kJlkg, W = 5 gls. p' = 2000 kPa Point 5 (vapor leaving lUrbjne), T = 8 K, h =:== 53 kJlkg, w = 4 g/s
= 78.3
8 w
= 5 g/s
7 VA
=:;
6
Separator 100 kPa
lOO\V/K
4.2 K 3
'.
.
4
-- w
= '4 g/s
T=~K
h =
'FIGURE 6..26 Helium liquefier in Prob. 6.16.
53 k1/Kg
ThrouJing valve
:
141 h-:+ !
=
Separator, p = IOO~ kPa. ~a tofation temperature at 100 ldJ.,} 10 .kJ{'/ru Jz g .-:. . .. ;:J':> .•.'i ,'''1~J/J(a I I:t.....~. -;:... ~
= 4,2 K~.,
Heat .exchanger, tJ.A. J~ lOa W/K 'Specific heat of heE ~m \rapo~: - .;:-
.
.'
~:, ':"' - " i
. -.... .
= {,:6.4 kJl~~g .".K) .~t.299~ k~a
.r
' • . "
'
.'-
p
:
:' . .<._.':.,. . ··~)t k IiQrg _~,;l(.,L:~:al~ l.OU.kEi.. ,.. -' ~:.:'.' ~', d
.
"
• . ,_ • •, . . .
~
. Usir:g the l'Tewton-:Raphson method, s.ifnuIate this sist6:~: deteITflining . the values of W4. T'2" T7 • and Ts- Use ' ;;ls the test for convergence that,' all ' variabl~s . chang~ . less than 0 . DO i ,during an i terati'on.. Lirni t the Dumber of . ' iterations' to 10. Ans.: W4 = .0.447 glS9 .T1 = -7.3i6, T7 = 5.97, and Tit ~ 1,0.93 rKOo . .' . . A refriger~tion plant that ~perates on the cycle shown in fig. 6-27 se~es as a water chiller. Data on t.h~ individual components are as follows~ . '
.VA -.: { 30~600 W/K evaporator 26,500 WIK condenser Rate of water flow
-
,
. {6.~' kg/s
evaporator
7.6 kg/s
condenser
.
.
The ref,rigeratio.n capacitY of the compressor as a ftInction of the evaporaqng and condensing temper?rUres '!e and tel respectively, is giv,e n by the equation developed in Prpb. 4.9. qe .. kW = 239_5
+
IO.073! e =- .0 .109t; - 3.41tc - .O.00250t~ 'J
-O.2030r ~tt" +. :O. O~820t;tc
I C"
'")
+ O.0013tet~
Condenser
Evnporator I
Expansion valve
I,
. FIGURE fr21 Refrigeration plnn~ in Prob. 6. 17.
.-
qt·
2 .,
- O.OOO080005t e't~
142
DESIGN OF THER..Iv1AL SYSTEMS
)Jle compression power is expressed bjr'the equation ,
P.,
.
kW == -2.634 - ,O.·3081t e
I
O.00301t~
-
+
.
l.066t c =-- 0.00528t; .. ".-:, ,
. ~
...
..~:~_... {':'" ~':' .: ~e .condens~r '.~ust ~reje~~ 'rh~' ,e nfrgy a~ded in b~th t~e evapor~tor ·~d .·~~ '" , ", ': " ': '~-). , . -, .::'\ ',' .compressor. DeterID}ne the vaIues ~f ..Ie, t c~_ , 'Lj~·, 'i .an~' .P Jor th~ . foIl~wlng · .; ; :
...
~
.~
~:~_~:~ ~::-...:-- ~ ~
..
'.
' .1 '"
'_;.',':.
"'I' ';{;~ ;~_~:: ,~. ~\ :'~,. ,~_.' .,~: " '~ : ...
. .:. .
-
"
'.
~~~PQ'tp1or ihI~t water .fa:oC·
Condenser inlet water
f b.
'
'I I
.-
I
10
"'~
"
0,25
.1'
i5
10
25
-30
15 1
30
I
. Continu.~ iterations unta all variables change by ari absolute value less than 0.1 percent'duiing an iteration. Ans.: For ia ~ 10°C ai;ld tb = - 25°C. t~ = 2.84°C, Ie = 34.0SoC • . q~ ....: 134.39 kW and p. = 28.34 kW.
REFERENCES 1. G. ·M. Dusinberre and J. C. Lester, Gas Turbine Power, International Textbook, Scranton. Pa.;,1958. 2. Proceilures for SiJ1lularing the Peifonna,l1ce of Componems C/Jld 5.,,·srems for Energy Cal.. clilan'oJls~ American Society of 'Heating, Refrigerating, and ' Air-Cond'i rioning Engineeri'. New York, 1975. . 3. H. A. MosIer, ··PACER-·A Digital Compu~er E~ecutjve Routine for ,Process Simulation and Design," ?\1.S. thesis, PurdlJe University,- Lafayetre, lnd.~ January 1964. 4. C. M. Crowe. A. E. Hamie1ec, T. W. Hoffman, A. 1. Johnson, D. R. Woods, and P. T. Shannon; Chcmi,ca! Plant SimulaTion; an 1l1!rodllcTtoll 10 ComplJler-Aid~d Sready· . Slale Process Allalysis, Prentice-Hall, Englewood Cliffs, N. J., 1971. 5. \V. F. Stoecker, "A Generalized Program for Sready~Srate System Simulation:' ASHRAE TrOllS., vo1. 77, pt. 1. pp. 140-148, 1971.
ADPITIONAL READINGS Chen. C-C .• and L. B. Evans: "More Computer Programs for Chemical Engineers .. ' ChcJ1I. Eng-.£.-\,oL 86. no. IJ, pp. 167-173. May 21. 1979. Henley. E. J., and E. M. Rosen: MOlerial alld Ellergy Balance ComplIIa!ions. Wiley, New York. 1969. . Naphtali. L. M .• "Process Heat and Material Balances," Chelll. Eng. Prog .• vol. 60. no. 9. ' pp. 70-74, September 1964. , Peterson. J. N .. C. Chen, and L. B. Evans: "Computer Programs (or Chemica} Engineers: 1978:' pl. L Ch~m. ~ng .. yoL 85" no. ·13. pp. 145-154. June 5, }978.
,
.-
· .~~/· ---~- · ·. ..
., - :'1,' , . • - ' , • •••
.-.,
.
:.:.-,1 ..··. ;'
• J . . .. .
, ,,,,:- ,, ';
.'
"
. ,
,
~/ -
,... ...:~.~ .;...:..,':,"()PTIlVnZATI Ol ~'. ' . '. -
.'
"
.
"
.
:~
7.1 INTRODUCTION Optimization is the' process of finding the conditions that giye rriaximurr; or minimum, values of a function. Optinl1zation has always been an expected . role of engineers, although sometimes on s,malI projects the' cost of engineering tjme may not justify' an optinlization effort. 'Often a design is difficult to optimize because of its complexity. In such cases, it may be possible to ·optimize· subsystems and ,then choose the optimum combination of them. There is no assurance, how'ever, [hat this procedure will lead to the true optimurp. Chapter 1 pointed out that in desigriing a workable sys,terp ,the process often COn~lS!S of arbitrarily . ~.§uming certai~!:parame.ters and selecting individual f:omponents around these assumptions. In contrast, when optimizaJion. is an integral part of the design. ,the parameters are f~e float until the combination of p'a rameters i$ reached. which cipthnizes the design. Basic to any optimization process is tije decision regarding which crherion is to be optimized. In an ~ircraft Or space vehicle, minimum weight 'may be [he criterion. In an aUltomobHe t the size of a system'may 'be the
to
eriterion. Minimum Cost is probahl.y the most common criterion. On the the minimum owning and ,operating cost, even ·including such factors _as those srudi,ed in Chapt,er 3 on economics, may not n1w~ys be ot.h~r . hand, ~
y
143
.
144
DESIGN OF THERMAL SYSTEMS 6-
followed strictJy. A manufacturer .of domestic refrigerators, for example, does not try to de~ign his system to pr~vide minimum total cost to the consumer,duIjng the life 'of the equipmen~. The la'chievemerr~ of minimum first . ' co~t; w1;ricb enhances sales, is more important .than. operating c9st, although ·: . ~' . :; . : ,. the' openitip.g cost cannot · be completely out :of. b~uI)ds. Industrjal ,organi.- :'.: :-, ....:.. .: :,~.:, . zations often turn - .as~de i!om.:the m.qst economi~aI solution. by jntrod':lcin~ :-' " -:' ~:, - . . .. .' human, social, and aesthetic. concerns. What is h~ppening ·is· that their c~~e- ' _. " . ,:,'-: ~. ".;,.<'..:,:,'ri'OIl, :functiop inchides not only monetary factors but also some other factors ':,:, .. .::',,:: tb~i'jn~y' ~.dmitfedly ·:be';!q~lY·'Y;l:lgQ~iy.~ ~ett~~~,_.; :~~ ~ " ';"'~'~' ,',." :~.,,;, . \. ,': '~,".;_"':-:' .. ~'~:;::~'. ,~..;.,~.:.J:' , ..' , Optimii~ti·o:~'. a.cil\'iti~~i" ,are·roften·".·practi'c~d.~··undei;· .1the~natrie:: Cif. :opera"-" ' ': - .
. tions research. Many ' ctev~IopI?erits '·in ·operations-··tesear~h-·emerge(j'·'from -: ," . .. attemptS to optimize mathematical models of. econOnllC s·ystems . 'It is only recently that rnec~anical and chemical engineers have used certain the . ' discipliries to ' optiplize fluid- lWd .energy-flqw systems.' . , Component simulation. and system simulation ?ie oft.en prelimin¥)'
or
ste;ps,.to 'optimizing .thermal systems, since ifmay be necessary to simulate' , ,the performance over a wide range of operating conditions. A system that may be optimum for d.e~ign loads Il).ay not be optimum over the entire rapge . of 'its expected operation. . .
7.2 LEVELS OF.OPTIMIZATION Sometimes a design engineer \\lill s'ay: I have op"tin1ized the design by examining four alternate concepts to do the job, ~hjch probably means that the engineer has compared' workable systems of four different concepts . . The statement does en)phasjze .the two levels of optjmization, co~parison of alternate concepts and optimization within a concept. All the optimization 'methods presented in the' following chapters are optimiz~tions within a concept. 'T he flow diagram and m.atl:1ematical representation of the system must be available at the beginning, and the optimization process consi sts of a g~ye and take of sizes of inciividual components. All this optimization is , done within a given concept. There is notning in the upcomin,g procedures that"-will jump from one mode] to a betrer one. N.o optimi~ation procedure wj'll automatically shifr the system under consideration from a steam-electric generating plan t to a fuel-ceH concept, fur examp.le. A complete optimizatioh procedure, then, consists of proposjng al,] reasonable alternate concept~, optimizjng ,the design of ·each. concept, and then choosing the best of the op'rimized designs.
,
7.3 MATHEMATICAL REPRESENTaTION OF OPTIMIZATION PROBLEMS I
The ele,ments of the mathematical $'tatement of o.ptjmi~a'ljon include s-pec. 'jfication of the fu!nction .and ,the constraints~ Let )' represent the 'function (hat is ....to ..4.be opti·mized. ca]]ed.,.. the objeclh't junction: )' is a function of
145
o PTl MIZATION ~
-.....
...~ -
.X. I, vt..
2., . ....
... _
1 . \..
ii'
: ,-,..1-
,-: '( , "1.
",",,'"
function" then? is -
t.
-,
0
._
.....
0
cis well as -inequality Ico.nstraints' . "
f'
,. ' / 1ft 1 ~ •
•
T~
......
.o.f.
- _.
•
01
" ":<:1
...
. ,
1[-
.- -~/
lfzaepL1I£kl1l ~ all r_ lD
A
•
I I
Q
ill
•
e.). . .
~-"'~., 0
J
...,
l:,\;",;,
\~ : ,..
~ ·, -7 '!1
OIUjCC U \. e
"
.......
'
Q
. ,¢m(XI,X2, . ","' ,J'~I1) ' ~ 0
'
J
lLe
":'·;-1
.
...
"'
1 1'J-"''-~
•
,ale Cad"c u,
v\ [l! L-!.!.
•
0,
"
'V I Ct: I ,-~ 2, • . •
II!;
-
' •••
-
-
..
•
..
..
•
•
•
•
'l!j '= 11'j (x l,X1~ ,' . . . . ~ Xii) •
'
-<, 'L 1
X;l)
• ,
•
,.
(7.5) /
-< 'Lj
.
j
'The physical conditions , dicta~e' the sense of the inequaliti~s in Eqs.' (7.4) to (7.5). ' An a~ditive constant appearing i,n the objective, function do~s not af~ect the value~ Clf the independent variables -a~ which the optinluin occurs. Thus~ ,
,
~.
'
y = a
,+ Y (x 1' '' ....
,.X n)
wl1ere a is a constant, the. minimum of y caD be written min [a
+ r(x!,
... , xn)] = a
+ min
[Y(Xl,··· ~ x n, )]
(7.. ~)
A further property of the optimuqI is that the maximum of a function ~ occurs at the same state point at which the minin1um of the negative of the function occurs, thus ' . (7.7) .
'
7.4 A WATER .. CIDLLING SYSTEM A water-chilling system, shown schematically in Fig. 7-1. will be used to iJIustrate the mathematical statement. The requirement of the system is that it coo] 20 kg/s of wate,r from 13 to 8°C, ,rejecting the heat to the atmosphere through a cooling tower ... We seek a system with a minimum flIStco~t to perform this duty. , Designate the s,izes of the components in the system by x CP t X E Vt .x CD. X Pi and.x CT. which repre'sent rthe sizes of the compressor, evaporator" c~ndenser, pump, and cooling row~r, respectively. The total cost y is the
sum of It he individual first plus instaUation costs, and this is the quantity that we wish to minimize. y (x CP ,x E V
• :t'eD,
X'F:. x CT) ~ minimize I
(7:8)
146
DESIGN OF THERMAL SYSTEMS
Cooling Cooling wa.,~er
tow~r (7
I I
..
'
• ;' r
.
.
,-
. ' _', r !
' .,:
'1
'. -
.I~ j. .-'~
" 1- .
\
"; '
.. -'.
.' .:-
~ ,
' .
--~ ~'vaporator EV
Water 20 kg/s, l30e
Com pressor CP .
FIGURE '7-1
\Vater-chiHing 'unit b~ing optimized for ?riIDmurri first) cost.
With only the statement of Eq. (7.8), the minimum could be achieved by shrinking the sizes of all ~omponents . to·zero_ Overlooked is the requirement that the' combination of sizes be such that the water-chiDing assignment of ·providing . . . '(20kg/s)(I3 - goC) [4. 19 kJ/(kg .- K)J == .419 kW c;>f refrigeration is accomplished. Equation (7.9) expresses this constraint
(7.9) where ¢ is understood to mean the cooling capacity as a function of component sizes when 20 kg/s of water' enters at 13°C. Actually, Eg. (7.9) could be inequality constraint, because probably no one would object t
an
,
-
.,
lr.v(XCP,XEV,XCD,XPtXcr)
f:: O°C
(7.10)
An extremely high dis~harge temperature t d of the refrigerant leaving the , compressor may 'impair the lubncation .
(7.11)
There: '!lay be other inequality constraints, such as. limiting
th~
condenser
OPTIIvHZATION
,"
CO(}Hng--water
F1'
LO~i:!
-
- " l l' . to (tne SIze Of.I: [ he COGung to-;,Ner to DreYC:tlt It
-,
In reJ.atlOII
, from splashing out. ,
3_4 7
'
'
I I
'
,'-"
JJ,-
,The elements qf the ,optimiza~ion problem are all iJlesent h~[e,. the , objective ftihction" equality constral,nts~ and inequalIty. constr.a~Qts~ _, all in , ,teTITIS of-the indepenqent ,v~, ?-?~es-, 'yhich ate the sizes of ,c~mp~~ehts~, ,- _,_,' , .
.'
.'
...
..
.- '·'~-7v5. ,OPTTh1IZATI N~PR6cEDUREs" ..
~.I.
•• ' .
•
In the' ne~t fe\v s' ecti~~s " ~~y~r~i, .'~ptiilli~'~tio.n ·metllodS ~/ill be'listed. Although this .list hlcludes most, of the, frequently used methods engmeer:ing pra~lice, .it is novvhere- nem:' 'exhaustive. Tn the optimization of systems., --.it is almost axiomatic that.'th'e objective function is.. dependent upon ,more , " . ' I than one ,variable. 'In 'fact, some theffilal systems may have dozens or even hundreds of variaples w~ich demand sophisticated optirriization techniques. Whil~ considerable effort may be required 'in the op1timization process, ' , developing mathematical relationships for the function .to be op,timized the,' 'constraint~ m~y ~so require consiqerable, effort.
in
and
706 CALCULUS METHODS: LAGRANGE MVLTIPLIERS The basis of optimization by calculus, presented in CGhapter 8 and continued in Chapter 16) is to use derivatives'to indicate the opti1!1um. The method of Lagrange multipljers perfonns an optimization where e ualit cOQ.strainrs exist but-t' 'method cannot dlrec y acco_mmodate ine uali constraints-~' A neees requrremen or ' USIng ealeu us methods is the ability -to extract derivatives of the .objective functi~n ,and constralnts~
707 ,S EARCH METHODS These methods, covered in Chapters 9 and 17, involve examining a number-: of combinations of values of the independent variables and drawing conclusions from the magnitude of the objective function at these, c;ombinations: _ An obvious possibility is to calculate the value of the function at all possible com~inations of, (or example, 20- values distributed through the range of interest of one parameter, each in combination with 20 of jhe second paramet.er, and so on. S!!c;fl 'a searc,h method_ is ,not very imagipative -and is
also inefficient. The search m'eth~ds of interest are those which are efficient, particularly when applied to multivariable optimiza.tion~ When applying search 'methods to continJl0us functions, since only ':qiscrete points are. examined, the ~xact optimum can only be approached, not reached, by a fmite Dumber of trials. On the other hand, when optimizing systems ,where the components are available only in finite ,steps of sizes, search methods are often superior to calculus methods, which assume an .' infinite-gradation of sizes.-
- 148
QE5JGN OF TI1ERMAL SYSTEMS
7 Js· DYNM.flC PROGRAI\1lVUNG '">
"
)
"
.)
.
. The word ~'progiammingtt her~ and in Jthe next several ,sections means optirp.i:z:ation and has no direct relationship with ~oP1pyter pJ;"ograrllining . ... for example .. This method of optimization; introduced in Chapter 10 ~d :. ' ,amplified in Chapter' 18, i$ ijnitlue in that the· result t~ an:..opti$1ti~rF_furicpob· , :' ::"_ ",rather ¢~', an op..4m~m state P9irlt.~ th~ ·.fesult:·of1!i~,~opiiiiiizatio~:draii ihe :;:~:~'~:' ':'. \?,,~ ~other~meth~ds::riiention~ '( f se(6f \i~ues', the indepeI)deni: y.arjables '.~' l' X'n : that. result i~ ~'e optimal .value 'of the objective function y .' Th·t problem atfacked by·dyn,affiic' .progrannirlng js one where the desirt?a res.ult .is' a path, e,.g.', the best route· of a gas. pipeline. The result is ·ther,efore· a function relating seve~ , vm;:i~bl~s. DYDainiC·.P~~~m.mIDg 'is relate,d. to the' calculus of variations,"'and it does in a series of.discrete processes what the calcul:u's 'of variatiqns ,does continuously. ';. l
..
,"
of
here',is
"'. x to
. 7.9 'GE01WE,TR;IC PROGRAMMING -
.
.,
Probably the youngest member of the programming family is .geometric programming, discussed in Chapter 11. Geometric' prograrruning optimizes a function that consists of a sum of polynomials wherein the variables may appear raised to integer and noninteger exponents. When the usefulness of polynomial exp:c:essions In Chapter 4' i's 'recalled, it is clear that the fomr of the function to. v.:hich geometric programIning is applicable' is one that occurs frequently in thermal systems.
7.10 LINEAR PROGRAlVIIV1JNG Chapter 12 presents an i~troduction to linear programming~ which is' a widely used 'and well-developed discipline applicable when all the equations (7.1) to (7.5) are 1inear. ·The magnitude of probletns now being so]ved by linear programming is enormous, occasionally extending into optimizations which contain several thousand varj~bl~s. .
7.11 SETTING UP THE MATHEMATICAL STATEMENT OF THE OPTIMIZATION PR'O'B LEM One of the first steps in pwonning an optimizatIon is ro translate the physical sjtuation into a mathematical statement. The desired form is that c;:omparable to Eqs. (7.1) to (7.5) because the opti.mization techniques can pick up the problem ~hen the objective func~ion and the constraints are
,
of
specified. In [he optimization thermal systems, establishing the objective function is often simple' and sometimes even a trivial task. The chaJlenge usuany arises in writing the constraints. A strategy lhatis often successful is t~,,< 1) specify all the direct constraints, such as requirements of capacity,
-
.'
0P11MlZATION
1!.J9
" ·h·I.E equat1.0n .,.. , te mpera.ture an d ' pressure " etc ." (2) \- d.. eSC[LE. rOTfC the component characteristics and prope.rties qf'working substances, and (3) 'write mass and energy balat1ces., Opera'tions (I) and (3) usuar.y provid,e a set of equa.tions contaiping more variables ,'il1.Clll 'exist in the objective .functiol1 ~ The set :of con§trairlt' equ?-tions is then reduced in nu~nber by'.efiininating . l"
' .
'
HJT~t2.tlOn. S
0,.f
':_:. " ..,'~-'y2I[i?-bl~~.' th~t go no:t ,e~?·$;,t: il~,
Between two. stag~s of aircompression. the air is to be cooled from 95 to 10°C. The facility to perfof!11 [his ,cooling, shown in Fig" 7-2v , , rust cools the air in a precooler' and . ~en iIi a refrigeration unit. .Wa~er· p.asses through the condenser of the refrig~ration ut1' it~ then inw the precooler, and finally to a cooling to\ver, where he~~ is "rejected to the atmosphere. '
is
The .flow rate of c
cost
costs are Refrigeration unit:
(7.12)
=,
.Wnler
2.3 kgls
.-------~--t---------1_.~, Cooling lower
Compression power P. kW
tI
1\ir 95°C 1.2 kg/s
,
I
Precooler
FlGURE 7-2
Reidgcration , unit
II
~I 'q)
150
DESIGN OF mERMAL SYSTEMS
Precooler:
X
f )=
.50q2 13 -
(7.13)
t~
..
.whei-e the equation is ·applicable ·wh.en . t3 > fl. .Coolin a tower: . . . .0
-l'3
,
".
0•
= ·2Sqj
,:: ..; ... ,where . th~ q '5 are·rates ~(heat trfl¥sferin kilQwatts~ ~S desi_n ..d in Pi .·7'-2. -,. . ··: . ,~·,< ··Th(coinpressjon power P k\V requi~ by~the~getation unit in 0.25
;'L : ,.~; c:~;~i~1~G;~::i'li~~:~iih:~~~~~~~~~G4tf;~~~;~~;'t~~,:~~~~n-,S::1 r . .. .. ,. . '.Develop· (0)· the objective fU:ri¢tion. and 'Cb) t:h"e constraint ·equatfons· ·fof ., . ~....~ optimization ..
to provide minimum flr.st .coSt.
'. '..
,
.. Solution~.-Th~ goal ·of '~is'exampl~ ·is only tD set up the optimiz~tion p~Obl~m ip the fonn of Eqs. (7.1) to (7:5) and ~ot to perfonn the actual optimi?ation. B·ef6ie prqceedir.lg~ h9wever, it would be· instructive to examine q~a1itatively . the optiniiz~tion features of this systep:t. Since the precooler is a simple heat exchanger,. under most .operating condition's it is lc:ss costly for a given ·heat. transfer rate .than the refrigeration unit. It would appear preferable, the.Il, to do as much cooling of the ·air as possib]e. with the precooler. However, as the ,t emperature 13 approaches the value of t}, the size of the precooler becomes very large. 'Some capacity is required ' of the refrigeratjon urut in order to cool the air below 24 °C~ The cool ing tower m~st reject alJ the heat from the system, which includes the heat from the air as we.11 as the compression power to drive the refrigeration unit. Shifting more cooling load to the refrigerat}on .unit increases · the size and cost 'o f .the cooling tower moderatel)~. (oJ The first ·assignment is to ' develop the expression for the objective fUDctjon. Since the total first cost lS to be minimized, the objective func:tion· will be the first cosr in terms of the-variables of optimization. A choice must be made of these variables; the objective function could conceivably be written in terms of the. costs of the individual components (the x's), the energy flow rates (the q's), or even the ternperatur'es (11, 12~ and TJ ). The most straightforward choice is to use the component costs
Total cost .= y =
XJ
+ .\"2 + -'"3
. (7.15)
If the q's are chosen as the variabl.es of optimization, it is necessary to stan with Eq. (7.15) and express the x's in terms of the q's., (b) The. next task is to' write the constraints. and this means developing the set of equations in tenns of the variables used in· the objective function . . Establishing the objecIive function is usually a simple process; the major challe.nge is setting up the constraints. The advice in the .. earJy part of this section was to specify the direct, constraints, the component characteristics. and finally the,energy and mass balances. This expanded set of equa,tions is ,then reduced by eliminating the variabJes that do not appear in the objective {unc.lion. A direct constraint is the requirement that the airflow rate of 1.2 kgls be cooled from, 95 'to 1Q9 C. Thjs ~equiremenl can be expressed .in two equations ql q~
= (1.2 kg/s){l.O kJ/(kg' K»)UJ =•. ( J .2 kg/s)[ 1.0 kJ/(kg . K) }(95 -
10) ·
(7.16)
I J)
(7.17)
. t;OPTH"UZATiON
151
Under the heading of component chifracteristics fall th~ expression. fOf the compressio.n .po~vef equaling 0.25ql ~~d the rela~ionships of the sizes (costj) [0 the CafJLtc·ity. ·Eqs. (7.12) to (7. J.:J). The. fin~r' category .includes energy 'and mass 'balances "
.:_:.
'~ "
B..efrige;ation unit: /q'! . + P: = (2.3! k gls) [4. 19 lcJ/(kg' K)1(i! ~~24) .
.':', .'
.
•
"~,'.~P: ,,-"'(~ " ; ,\ ~. :~!.. ,- "· r ·t.!:'/~ - . . . ,~ ~......
. . ' . ..". {,.'.~ lPr¢cooler:·:;·:-· ·.·· '.
. !Coolirtg t~~~r:
'\f.~: . -'"
'
"a
j
-~
.0;' .' : ' ,
"',
":
....•.
'
.
(1:2)(: 1·.nr(-95 ··:~ 1-3J:: ~\" f2':3){4:·'i9),t f~;; .-l.'ct1~.y'".'\; <·/ ;:···' t ·'·"·-· ··"·;."
,..
. .' '.. (2.;)( 4. i9)(~1 - :24) .'=
.
Q3' "
The complete .set of .constraint equations i;S ql = (~.2)( LO)(!3 ~ 10)
(7.18)
q"2 = (1.2)( 1.0)( 95 - .(3)
. (~.19)
P XI
= O.25q,
-'
-
X3 -
+P
. (7.21)
= 48ql
x,=
ql
(7.20)
. ' 50q~
.(7.22)
13 '. -/1
(7.23)
')-
_:1q)
= (2.3)('4.. 19)(tl - 24)
( 1. 2) ( 1. 0) (95 - t:5) = (2.3)'( 4. 19) ~ r2
-
I
d
(7.24) . (7.25)
(7.26)
(2.3)(4.19)(t2 - 24) = q)
There are nine equatio~s in the set, Eqs: (7.18) to J7 .26) and ten unknowns, q,. q2t Q3. P, · XI. -'"2 • .'(3. t), 12, and·'t3. The next operation is (0 eliminate in this set of ~q'uations all but tb.e variables of optimization, Xh Xl. and X3. As the elimination of variables and 'equations proceeds, there will always be one more. unknown than the number of equations, so when all but the three x 's are eliminated, there should be two equations remaining. These two constraint equations are Egs. (7.28) and (7.29), so the' complete mathematical state~ent of this opti.rrUzation problem ~s ns foUows:
Minimize subject to
y = O.01466xIX~
- 14.t:1
XJ
+
+.t:. + x 3
.
1.042tl =5100
7.69:C3 -:c I = 19.615
(7.27) (7.28) (7.29)
7.12 DISCUSSION ·OF EXAMPLE 7.1 The objective of this chapter
i~
to
introduceprocedu~s
for setting up
-the mathematical st~tement of the optimization problem. With one of the methods in the subsequent chapt.ers.,the execu,tion of ,the optimization process would show that the optimal valu~s of Xl,. :'2. and ..\" '3 are $1450, $496, , and $273-8. respectively_ Equation (7.13) [or Eq. (7.22)] was presented as valid if 13 > t l' and this condition could legitimately be !isted as one of the · ~straints. Ift3 < t 1_•• ..\".2 becomes negative. whlch is physically impossible.
'
. ;
152
DESIGN OF THERMAL SYSTEMS
.
.
. The constraints are an fite gral .part qf the statement of the optirniza~ : tidp. pr~blem .. ·The obJective function without the constraints ·is meaningless Oecause .."the x 's co~ld all shrink. to ' zero and ther~' would be. no..cost for . .the .·system. The cori~trairit in':Eq ~ (7. 28) ~q*e~ a: positive valu~ of Xi '.' . .: .. ' ' which' tbe same asj:·equiring. the I existen.ce 'of " a refrigerati6n·-uni~. ~rom . . . ~'~'i;~: , .:'.•. )l;~~~:-~ansfeJ,:. ::c;~nl~~d~t~tions , the.·preeoo!er can c?ol ·the..air no .lower than:. a'., '. . ".', ~temperat:Ure of24-~C<':~\Ib~~ritu~¥~·:%.~,.•};::.::O..?*~.::Pl7u (?., ~8) makes x 2 lnegative, . . . w~j~4 is p~ysic~I:y, i~PQ~~j91~."Equat.ion.:(I~~2·~f:~?e.S~p~~~~>J.::..Y:?;~!~~·~Z¢tP;:/. : in wlllch case all cooling is penonned"by .the,. I~fri.gerat.i6tl IDlit:,: :.;. -.'- .:' .: :'.' -, . - . The constra"int equation (7'.29) imposes a miDi~lJ,m VaIll~ o,f the··c~ol':: . ' '.. ing-tower size aDd cost x3.' As' the size of the refrigeration unit. and Xl . increases, x 3 al$o increa~~s because .of the compression power associated with the refrigeration unj.t..
, y .... . .
. "'
."
is
• • •••
••
••
•.
•
.....
,',
'
j
. 7.13 SUMMARY While it is', true that eng~eers have always so~ght to optimize their designs, it has been only since the widespread applicauon of the ciigital computer. that sophisticated m~thods of optimization have :become practical Jor complex systems. The application of 'optiID:izarion technique,s to large-scale thermal systems is still in its infancy, but' experience so far indicates that setting up the problenl to the poinr - where an optinllzeition method can take over tep. resents perhaps 7.0 percent of the total effort. ~e emphasis on opt.i~zation techniqpes in the next five chapters m-ay sllgge~t mat ~ngineers are home free ,once they know seve-ral nle~ods. Realistically, however, the execution' ,of the optimization can only begin when the characteristics of the physical· system have been converted into the equations for the objective function and constraints " ',. . .. !
PROBLEMS 7.1. A pair of pumps j,s available to fill a tank from a constant-level reservoir, as shown in Fig. 7-3. The pumps may be operated individually or together, J, and the objective is to fill the tank using 'a minimum total amount of energy. The pump characteristics (head-flow rate) are shown on the graph, and both
\.-
,
pumps have the same efficiency as a function of flow rate, State clearly the m.ode of operarion of the pumps over the enter filling process that- resul ts in minimum total energy. . '.-. , 7.2* Two heat exchangers in A circulating water loop, as shown in Fig, 7-4.· lransfer heat from a fluid condensing !l't SQoc '·0 8 fluid boiling at 20°C. The requi~ed rate of heal transfer. is 6S kW. the U value of both heat cxchangeI:s is 0.03 kW /(mJ • K), fir~t cost of the heat ·exchange-rs is SSO/m,2 of heattr~nsfer area, and lhe present worth of the lifetime pumping cost in dollars is, 12.00011.1. Develop the objectilve function for the total jJresent cost of the __~.YSI1malOng wilh any ~onstrainl equation(s). . . .
me
i
~&~.
~t
r'
.'.' : .. '. ,:' , -
:. ~;:", . .-:.. - ..
,
DP111v:!lZ..A.TION
153
~
"
.
. ·. . : ·~i. .
I
8
==
.' -
.
.
. '!;-~' "\ ' . . - _. =-: . \t!l.:i .:;,.
"Wv"
::-
~"',,:v.~:~~:" (·
.,~. , ...~. ...-
J -,
'~ Tan~'
o ..
t.
'
~ ~
: .;:!
20 Q.Us
10 .. .
G(
s
-~ (\-\ - 8 )
...
?S
.'
.'
-'
"
60 50 40
i> ·0 30
1
!;: ~
OJ
B
~o
Power ==
0
(Head)(Q)
. Efficiency/l 00
FIGURE 7-3 Combination of pumps in Prob. 7.1.
1'2
w kg/s
""'-""'"'!""..-+-"" ,. . ......"l,;'
Heat exchanger 2
-~.
I
~O°C
liquid
, FIGURE 7-4
Heat-exchange loop in Prob. 7.2 •
.-
, . 20° C liquid
10
20 Q.l/s
154
DESIGN OF TIIERMAL SYSTEMS .} o
"Air and
~O%
CO2
Extractor I ' Size=A - , ... .\ . . "';
:Q m3/yr";~
Ak and'.cO.,,~ ."
".
I
•
•
..
, J.
~' .,.
~ ' .• ,
; -.:.p .'.... : .. . :. '.:, ."'I
'
.
,,:
..
: -..
:
"
. .
I
~
-
. .. :7
• .:... ... ~,
I; ·.
~": '-:--
,;''' '-:.
. ~~.. ,. ,:
. .
,
.
7' -~ ';:'~~""'~:r .,::,-~ - :::-_~1 •.,!."'-': ~~id~~Ij.~'~ ::--:·:-.~C01.-."; ; ; -:\ ,' ~'~'::>_ .;""' ~': _ ..-.:; -! '
Exhaust air . , aildCQl "
F1GURE 7-5 Carbon dioxide extraction in Prob. 7.3.
'/
/ .
7.3. A plant produces C02 by ex.tracting it from a mixtUre of CO2 ,and air ~ ,as :" 'shown in Fig. '7-5. The feed rate o(the' CO2 -air mixtUre is 5 X 10 9 m 3/year. Jpe system consists of two extractors-Extractor I of low cost and lowefficiency, and ExtraCTor II of high cost and high efficiency. The CO2 , in m3/yr removed by Extractor r is (total m 3 Jyr entering)(x)(l - e- AI2OOO ), and that removed by Extractor n is (total m 3Jyr entering)(x)(] _',e BIJOOO ), wh~re the x's are the fraction of CO:! in the feed stream. Th,e annual capital costs of the extr4ctors in dollars/yr afe 30aOA and 500DB, respectively. The value of the CO 2 product is O.OlQ doUars/yr. Develop the objective function and constraint(s) in" terms of A lB. and Q. ' 7 .. ~., An optimum shell.-and-tube heat exchanger, as showri in Fig. 7-6 has a VA I value of ]500 kWIK and has a pressure drop of the tube f1uid of 300 kPa. Applicabl~ equations are ' " , ' ,
A = 'O.2N-L
V ]1 U
= 5A11/V ;:: 0.08 + 111;
;=
N = (7T D 114)J160
0.08
+
1/(0.4 V)
= 0.005 D,2
where N = number of tubes 111 = number tube passes A = heat-transfer area in m1 V = velocity in rrlJs U = overoJJ heat-transfer coefficient in kW/(m 2 • K) h = convecrion coefficient on the tube side 'in kW/(m 2 • K) )TJ she1l diameter ,m L = tube length in m
of
=
In
The pressure drop of the tube, fluid per lInit Jength of tube, kPalm, j's --.0.' y.,. The optimum h~at exchanger is one of Je~st COS[ thai meets the other
OPTIMiZATION
15S
. .
. .FI~URE 7-6.
.
SheU~and~tube heat eX,changer in Prob.
7.4-
requirements, and the cost is a "function ofL ,and D. DevelQP the constrnint(s) ln tepns of Land D. ' 7.50 The flow ~ate of raw material to the processing plant shown in Fig. 7-7 'is 1.0 kgls of m.ixture consisring of 50 percent A and 50 percent B. The separator can remov~_some of marcl1al A in pure fonn. and tnis product has a sellip,a price of $@ per kilogram. The other product from the separator sells for~ per kilogram,' and some of this ~tream may' be recycled. The cos.t of operating the separator in dollars per set;:ond is
where w = mass rate of flow, kg! s x = fraction of A in the mixture ([his .sepa(ator 'cost equation indicates that the cost becomes infinite for perfect separation.)
(a) Set up the objective fU'Qction in terms of Wl, W2, and x I to maximize the profit. The cost oftl)e iaYJ ·rnaterial is constant "
Mixture of A and B W == 10 kg/s
~----~---u-
x = 0.5,
separatO(' ··.. ~ · ··
.
WI ';X.
.(
·.··or ·,
A_ ' _ _
Procc:ss,irii .pTanr j-n Prob. 7.S.
to-
'- .
4
~--~------------~$~g
' - -_ _ _ _ _--11
fiGURE .1.:_
3
-$lO/kg
.
w),..t 3
"'4. X 4
156
DESJGN OF THERMAL SYSTEMS
..
.
(b) Set up the -constraint(s) in - te~s of WI, W2, and.x J. . " Ans.: (b) (5-wlx})(wl -li)2) d J(10- Wl)(WI X l-W2). 7,,(jp _The plapt shown schematic'ally in Fig. 7-8 'pr~duces chlorine by ,the ,r eaction of Hel ' ~nd ac.cording to ~he ,equatio.n " ' "
92
•
'
, '
~
,
,
1. 1, '
9: . .
",
-4HCI ",
I
.
+' '-0" 2 ~ 2H1 0 +, 2Cl , 2'
,.:
..
',The. 'p lant receIves 80 'molls :ofHCl and 20 'moils 'o f Oi. "The c~nvertet is , :' -' '~--'~"' :~" ~ ..' ~' : cap~bl~,~ of:achie1{i~g, pn~y, ' ~Rarti.!:{. ~~!1v:~si.qn, ', ,~~ the unreacted H~f and I
.. . .
"
,
--
" ,; .','0 2 ,ru-e', recycied"followfrig.<.rernoval ;o~,fu~~,:Wa-ter·,~'a,~(rthe~·'cli.lo~,e,;i.~:,:~~·:~,·I'::-<' , :-
, " 'The -fIrst cost of-the ~system.is to, be. -minirn.,ized~" arid' ,the' inajor"- vari',~liI~: -affecting , th~ cQst is that of the converter :vhose 'co~;t" i~ r.epresented by: ~, ' ,
Cost per
-' ' ,24 '000 of reactants at A, dollars =:== 800 + ,- -'--
,
,
100
iX
'
conversion efficiency, o/c '
Ylhere x or
fll01ls
,x = (fraction of'entering Hel and O2- that' reacts) x 100
Determine (a) the objective function ofw and x and (b) the constraint(s) that permit optimization for minimum cost: Ans~= '(b) 11o'X = 10,000. 7.7. A supersonlc wind-tunnel facility is being 'designed in which the air \vilJ flow in series through a con1pressor, storage tank_' pressure-control ' va1ve~ the wind tunnel, ,and' t.hence to exhaust., During -tests a i1o,w rate of 5 kg/s ' nlust be "vail able to the wind tunpel at a press'ure of 400 kPa. The test~ are intended to study heat transfer, and before each test 120 s of srabiJ'ization tiDle is required, during \-\'hich 5 kg/s must also flow. A. total of 3600 s of useful 'test tini.e is required during an 8-hour period. and this 3600 s can be 'subdividedjnto any nUDlber, of equal-length tests. , The mo<;le, of operation is to start the compressor and allow it to r un continuously at fulJ capacity during the 8-hour period. Each cycle consists of the foJlowing stages: (1) buildup of pressure in the storage tank from 400 LO 530 kPa, during which time there is no flow through the wind tun'nel. (2) 120 s. during which the flow is 5 kg/s _and the storage -tank pressure begins to 'drop, and (3) the useful test. during which f10w is 5 kg/s and at the end of -which [he storage-tank pressure has, dropped'to 400 kPa .
SOmalIs of HCI 20 molls of O:!
1
..
A Conye~r
Cooler
,
Separator
.......--
w moJ/s
Remoyes
., all Wnter
FIGURE 7·8 Ch1cri~.,~ ' jn Prob . 7.6.
..
w~ler
Remoycs
, all chlo6f\t
A 'p ressure 400 kPa is available in) the storage tank at ITle start of the
day. .
The compressor-storage-tank cOITJ.bination is to be selected for 111.inimum total first cost. The compressor cost in dollars is given bv . .
I "
'
I
. .
,~
.'
. -.
I ·· ,·
'", .
. ... -
Cost
.... .
.
';;'- '
".
== ' 800 + 2400S
. I
(-
Ans.: (b} S t 1 + 137S
· 1.5V·
+
J"
0.214 V = 5 .
.
.A· Si'I!lplified version ·of a comb.i ned gas- and steam-turbine plant I for a liquefied-petroleum-gas ' facility at Bushton~ Kansas is shown 'in Fig. 7- 9; the plant must meet the following minimum requiremenrs:
Power to propane compressor
3800 6500 8800
Low-pressure steam equivalent for process use. High-pressure steam .equivalent for process use
Combustor
~,kW -1--
3800 kW
Low-pressure
steam Propane
Air
6500 kW
t 8800kW
Auxili.:uy,
t - -_ _-tr--tll
Boiler
burner
FIGURE"""""''''''Y-
.'
Schematic diagram of gas· and steam-turbine pliant in Prob. 7.8.
High-pressure steam
158
DESIGN OF THERMAL SYSTEMS
perc~~t
of
..... . ' . '
". ":"' ..',,
,.,. ~ .
•
of the heating value of natural gas js con verted into mechanical ,power ari4 the lnrmaiqing 80 percent passes to the exhaust gas.-As the ,exhaust gas flows through the boil~r, 60 percent of its ,heat is converted ipto steam. ,The boiler is also equipped with an auxiliary ' " : 'burner'- which pennitS. 80 percent ·of the',heatiI;lg v'a Ine the flaWIal. gas:to, . be con,v ened int~ ~team. ~igh~.pressure· steani. :(lows to piocess,use:and/or' to the turbIne;:, where 15 percint of the 'thenn~ ', ~neigy is .'cQJlv~rt.eg. iptq
In the gas-turbine plant" 20
steam
,
.,.,. "'(~,: ,:,,~, . "'. ,:·::~::J::~E~:t~~~~;;~;:;~~r~:~.~~~:~~~:,:~?~,:::e~~~O:~f!.~:i'·.': -to
Th~ plant 'is be operated 'in' such· a -way, ti:l'at it.coI;lsumes, a.. minimt;l;Q1 .... , total quantity of natural gas. In te~s of q 1" and q2 '(a) write the objective -. .funCtion ,and' (b). ~,evelop the 'constraint ~quatjonS". Ackno.wledge by inequalities in the constraint equations 't he possibility of dumping power ,or st~am. Ans.,: (b) til + 1.176q'2 :> 28, 090 kW -
.
l
,
+ ·O.441q2
18,820 kW qJ + I .665q2 ~ 3.1,880 kW,' . ' 7.9. Nagip2 discusses improvement in · the
cycle
:>
The
by
precooling the air entering the comprc;ssor. precooler is an absor:P~ion-refrigeration unit, which is supplied ,with steam generated from the heat of the exhaust gas, as shown in Fig. 7-10. Achievement Qf this. improved efficiency 'entails additional investn1ent cost for the absorption unit " and boiler. Furthermore, there is an optimum combination of absorption un'it, boiler~ and regenerator. -. Performance data
Compressor po\ver. kV/ = 34(11 Absorpt.ion unit by the s(eam.
del~vers
+ 273).
.0.6 kJ of coohng per kilojoule of heat supplied
4
Exhaust
480°C Regenerator
Boiler
820°C Combu,l\lor
2 Src;lm loop
J05°C
Shaft
FI ,
32°C 45 kg/s
Compressor
Absorption rdriserBtion Precooler
FIGURE 7-10 Precookr of gas.turbine inlet air in Prob. 7.9.
-.
.'
-
Turbine
OPT!MlZATION b
159
The flow rate of gas through the;cycle jis 45 kg/s, and negligible 3dditiOD .of nnss at the comb~stor is assumed.' . 1, "
U value of the boiler bas.e d 'on steam-side ;)iea~ 0.15 kV!/(m~ . K). . . U value ofr~~nerator", 0.082 k'\IV /(m 2 . K}.
.cp of gas throughout the 'cycl~, 1. .0, kJ/(lcg . K), · . _ "", , •. "';:~.,. 'i T. __ ;.;:;) . : ~'- '-'.•',' ",-'~-. ' . "" ',,':- ,_',' Additf-on?-C '~e~peratur-e-s' are Jndic'ated ~9'n FIg'. 7':J U~ '- ,. , ~", ." .: ': -' ", j:"" .." , ," - , - ' , ~.- ;" .
)
':I"
.
.
.
,
'.
'
~
, I'-
, Cost data 'r'
Present worth ,of pO'wet genera[,~d during the life of the plrulL 2'25P
dollars Present worth of fuel cost for the plant life. 45Q dollars
"First cost of absorption: unit, gOA dollars· cost C?f boiler, 1 ~ OOOB doilars . First cost of regenerator, 90C dollars
, First
where Q = he;it'rate at combustor, k\V P = power generated, kW . A ,= absorption unit ~ize, kW of cooling B = boiler size. m:! steam-side .area C = regeneraror size m:?
of t
(a)' Develop the obj~ctive function for the total present worth of the profit of the sy~teIT.l in terms of A, B't C, Q', and P . (b) Develop expressiqns for the temperatures tit /2, t3. and t.), in terms 'of the variables of part (a). .~ (c) Develop [he constraint equations ,in terms. of the variables of part (a).' Ans.: (b): ') = 32 - O.0222A t'). = 262.5 - O.039A
, 13
14
= 262.5 - O.039A + = 480
-
'217.5 + O.039A S48.8/C + 1,
217.5 + O.039A 548.8/C + 1
(c): P - 4930 - O.7548A = 0
A - (27)('375 - 21?5 + O.039A)< 1 _ , 548.8/C + 1 .
"e-O,OO5~~6B) == 0 '
"
(
-Q - (45)[(557.5
+ '~.039'A
-
217.5 + O.039AJ 0 548.8/C + I =
A desaHnationplant show'n schematically in Fig. 7·11, ~cejyeg sea water having a .saline concentration by mass 'o f ,3% of the mixture. This sea water is mixed with some recycle solution and enters ,the plant which is to have a ~acLty of 300 kgls. The cost of pumping .and trea.ting the raw sea water is ..-
.'
-160
DESIGN OF THERMAL SYSTEMS
• Sea water .
x = % salt
Treatment faciliry
3% sail, Q kg/s
I
,: -
I·
w = flow rate of
Desalination plant ., '
FIGuRE 7;'11'
pe~aliIJ,atiDn
Y . == % salt
. .. .,
plant in !>rob. 7:10.
. ,$0.08 per"Mg ~f mixture, and the cost :of heat
to operate the pI~t is $0.)0 per
Mg of feed. As the s~t co~centra[ibn entering the .desalination plant in.creases! the effectiveness of the plant decreases, foilowiI1g the equation y = 3.4'..jX, 'so should the entering conc,e ntration reach 11.56% the plan.t wou'l d produce fresh , w·at~r. Develop the objectiye function and constraint to describe the ~inimum-cost operation . .Ans.: constraint is
no
Q=
(300) (11.56) (w ,.- 300) , 11.56(w - 300) - 3w
REFERENCES 1, B., G. \Vobker and C. E. Knight, ··!Y1echanj~a1 Drive Combi~ed-Cyc1e Gas and Ste.am , Turbines for Northern Gas Products:' ASME Pap. 67-GT-39, 1967. 2. M. M. Nagib, "AnaJysis of a Combined Gas Turbine and Absorption-Refrigeration Cycle." ASME Pap. 70-PWR-18, 1970.
ADDITIONAL READINGS Beveridge, G. S. G. and R. S, Schechter: Opli!11ization.~ Theory and PracTice, M,cGrawHill, New York, 1970. Denn. ~1. M.: Optimization by variational Melhods t McGraw-Hill. New York, 1969. Fox. R. L.: Optimization Methods for Engineering Design. Addison-Wesley, Reading, Mass .. 197}. ' Ray. W, H., and 1. Szekely: Process OpTimiz.ation, Wiley, New York t 1973. Rosenbrock. H. H. and C. Storey: CompwQlional. Techniques for ChemicQI Engineers. · .Pergamon, New York. 1966', Wilde, D. J .. and C. S. Beighr1er. ~oll':ldations of Op,imizaTion. Prentice~all, Englewood " , . . ,, . Cliffs. N.J., 1967. t
t
,
-
..
.'
) I
~:-:--,-,.~ ;~ '.:·2r.::J-~,'~:'-'
:' " 0,;- -. .
. ~
,
' .
. '-~
......
. '.
.'
. ..: .....
-'
""~ .~~
. :. '
...
'.
> .'
'
..
····LAGRA.NGb
MULTIPtIER~S
8~1 CALCULUS METHODS OF OPTThfiZATION Clas~ical
methods of optimization are based· on calculus and specifically determine the optimum value of a function as indicated by the natiIre of the derivatives. In order to optjmize using calculus, the function must be differentia~le and any constraints must be equality constraints. That· there is a need for any method other than calcul us, Such as linear and nonlinear '. programming, may arise from the appearance of inequality constraints. In addi,tion. the fact that the function is not £ontinuous but exists on]y at specific values of the parameters rules out calculus proc-edures and fa'vars such' techniques as search methods. On the 6the~ hand, since some of the' operations in the calculus methods appear in -slightly revised forms in the other _optimization methods, .calculus methods ar~ imp90ant ,qpth for tlie_cases that they can solve in their own rigl)t and to illuminate some of the procedures
in noncalculus methods. . . This chapter presents the ~grange multiplier equations proceeds to 'the interpretation ,and mechanics of optimization using these eqll,ations, 'and then .begins the presentation' of the background and visualization of constrained optirn.ization. Since unconstrained optimization i.s on ~ y a spe· cial ca~ of constrained optimization. lhe method of Lagrange multipliers I
~
~
~
161
162
DESIGN OF THERMAL SYSTEMS _
aU
is appljcable to situations explored in ,this chapter. this chapter also explains hOVI an optimum con~ition c~ be tested to establish whether the condition is a maximum or a minimum-:and,1 finally, introduces the concept, : of the sensitivity coefficient. , : . '
, Op' timi~e ,-
"(8.1) ': (8'.2)
, subject to.
, (8.3)
The nlethod of Lagr~nge multipliers -states that the optinlum, occurs· at ,values of the x's that. satisfy the equations V'YI V ¢'1 ~"Xm\l ¢m = O' ¢1(Xl1'. ~ ~,XJ1)
cPlI1 (x 1
!
•
•
•
,
=0
X n) ~
(8.6)
0 ..
[he remainder of the chapter' explains the meaning' of the symbols and oper~ ' . ations desjgnated in the equ,ation~, the mechanics of solving the' equations, . and appljcatlons, examples, and geometric visualization of the Lagrange multiplier equations.
8.3 THE GRADIENT VECTOR The inverted -del ta designates the gradient vecTor. A scalar is a quantity \vith a magnitude but no direction \ \vhile a \'ector has both magnitude and 'direction. By definition. the gradient of a scalar is n
.\' Y =
av . + a)" . + ax J " ax 2 -
-·-11
---)j
. ... + -ay -I
ax , n
n
(8.7)
\vhere i J, i ~, ... , i" are unit vectors, which means that they have direction .and their magnitudes are unity. . ..
Suppose, for example, that -a so1id rectangular block has a temperature distribu.tion that can be expressed in terms of the coordinates x J, X2, and "x 3 as shown in FiK. 8-1. If the temper"uture f is the following function of I
;" J • x~,
and x),
------
."
LAGRANGE MULTIPLIERS I£>
.
,f_
63
I )
. i3 " . ' • • ~ ••'
-. -1,- ;,. I~
.;
:t
. : ' ~ ... :.. ". '. ' . . . .
:"'~ ..- ,,':'! -;~~ :~l.~.r- .• _
:.
. '\
'
, .. .. .,.J,..~
X:z
.. '· .11
' Y-"
--:-"--~.
' ..
· I"'~·"." •
I
.~.
"
.~
,
"......:. ..
.,~\;l~ "; l''''I.?·:'~ ~ r::r;.p·,.I?:--~ ':'') . ~
•
•
", .
FIGURE 8 .. 1 . S~lid object in"wbic1;l ~ scalar, the temperature, is express'ed asa function of x.,
Xl,
and
.l3.
'1,
~hen the 'gradient of t, V t, following the definition. ofEq_, (8.7), is .
'
Vt ~ (2'+ x2)i l
"
+
(Xl
.
+- ~~)i~ + (2t"2Jt))i3
where if, i 2 ,' and· i3 are the unit v'e ctors ·in the Xl, Xl, and X3 directions, respectively. The gradient operatioq is one that cO(lverts a scalar quan.tity . ~to a vector quantity. - .
8.. 4 FURTHER EXPLANATION OF LAGRANGE MULTfl>LIER E·Q UATIONS 'Also appearing in Eq. (8.4) is a series of lambdas, AI, . ..• , Am. These tenus are constants and are the Lagrange mUltipliers, whose values are not ' .known until the set of e.quations is solved. Equation (8.4) is a vector equation, which means ' that for the terms '. on fie left side of the equation to equal "zero [he coefficients of each unit' vector must sum to zero. The vector equation (8.4) is thus a shorthand form for n scalar equations,
(8.8)
. .. . . . . . . . . . . . . . . . . . . . , . . .
(8.9) These n scalar .equations, along ,r 'ith the m constraint equations (8.5) to (8.6), form the se't of n + m simultaneous equatjo.ns needed to solve Jor the same number of unknowns: xT, X1 • ..... x! and )Q,A2, .•. Am. The asterisk on !he oX's is often used to indicate the values of the. variables at the ..I
~
164
D~IGN OF TIIERMAL SYSTEMS
• optimum. The optimal values of the-x 's can be substit:ated into the objective function, Eq. (8.1), to detennine the oP,~imal value of y , designated y If!. , The number of equality constraiD:ts ~ is) always. Ie's5 than the number , .of -v.ariables n . .In the limiting case where ' m = n the cons,traints (if they .. are.independent .equations) fixthe valu~s of the x's~, ~~ ' no : optirtpzationis 1
" " possi~le", . ,:;,;
.-:~ ..~.:::=-~ ~.~: '~~:.'~'£-'
.;. ••
"
" ,,'
-<-,:( ~'.': r-'~:': .'~T ' ..,::: ;.'i:" ·i.~i'j.",
J
. •, ' . '
. _'..
.
" : '.",' , ._, .. .. " " '_. . . __
8.5 . . UNCONSTRA!NED..,OPTIl\tlIZATIQN" '. . . ' . '
~'~"
~ ~
.
... . .
" : ,. . .,. .
.
! .
~
.... . .', <. ..-.., ..:.... c ·:::~~~ ...... . --
Th~ Lagrang'e multiplier equations. can be used to att~ck Gonsttain~d " 'opti mization problems, but the equatio~s apply equally well to unc0Il:str~ined optimization. The unconstrained 'optimization is a special (and simpler) of constrai~ed optimization. The objective-function,y ' is a function oflVariabIes Xl, ' - . . ,- X n
case
. ' y -:- Y (x 11
- • - , X
.
. (8.10)
n) .
are
When the. Lagrange,:multiplier equations appJied to Eq_ there are no ¢' s, the condition for 'optimum is . -' Vy .= 0
(8.1.0}~
since (8.11)
.or
ay ax}
av' .
av ·
0, --- = 0,'... , - -~-. = 0 aXi. · ax"
- - '=
'(8.12)
The ,state pojnt where the derivatives are zero is called a critical point, and it may be a maximum or minimum (one of which we are ,seeking), or it may be a saddle point or a ridge or vaIJey. Further mathematical analysis may . be necessary to detennine the type of critical point, although in physical . situations· the nature of the point will often be obvious. We shall assume jn the ~emainder of tills ch~pter that Eq. (8.12) describe a maximum or mlnlml,lm. '
A function), of two variables x 1 and X2 can be represented graphically as in Fig. 8-2. The minimum exists where . ay
= 0 = . ay
ax I
aX2
,
Example 8.1. Determine the minimum valu'e of y where oX.
)' = -
X2
,
.
1
1.,
X3
2
J6
+ - - + -x· + X JX)
2
Solution. Tbe Lagrange multiplier equations for this unconstrained op-tim'iza-
-
Vy == 0
.
..
or
oy
0)'
d)'
ax.
ax~
ax)
--=-=-=0
LAGRANG~\1ULTIPUERS
165
y
. ' ., •
'
"I.
~
.".
;
' .' ",
Minimu~
dY _ dy. oX
I
•
• ~ .: .I. ,
where = 0
ax}',
-
•
,
FIGURE 8-2 Unc'ons[n;tined optimum occurs where panial derivatives equal zero.
'lay = __ 1_ ax) :c l.x~
+
_1 = 0 16
.x; =
x;
x;
Solving simultaneo~sly gives = 0.305, 0."673, and = 7.243. , Substituting these values into the objective function yields y'" = 1.585.
Some!;imes it is convenient to convert a constrained problem' into an unconstrained optimization by solving for a variable in a constraint equation 'and, substituting that expression wherever else in the remaining constraints or-objective function u:e variable appears. ~e Du~be~ ofcons~aints is ~ro gresslvely redu.ced until only the unconstramed objective functJon remaIns. Example 8.2. A total length of 100 m of rubes must be lnSbUed ina sbell, and-rube hent exchanger (Fig. g..l) in onkr to provide the necessary heatttansferarea. The total cost of the instaJfation in dollan includes I
1. The cos't of th~ rubes, which is constant at $900 '. y 2. nie cost of the shell t:::: 1100D 2·'L
..
166
DESIGN OF THERMAL SYSTEMS
,
.
,-
.,
"
.
,. -
"
'" "
.
1":" '-'-, ',,,:.,;"
~.
~ ,~ ~ .- ~
.
-
" ', ',
.
\-
. . ./
(
o·
1 in 2 aFcommodares 200 tubes
FIGURE 8-3 Heat exchanger in Example 8.2.
3. The cost of the floor space occupied by the beat exchanger = 320D L, where L is 'the lengtJ: of the heat exchanger and D is the diameter of the sheJ], both in meters The spaciQg of the rubes is such that 200 tubes will fir in a crosssec.tiopa( area of 1 m 2 in the~helJ. Detennine the diameter and length of the heat exchanger for minimum . first cost. .
.
Solution. The objective function includes the three costs, Cost
= 900 + ] lOOD~·5L + 320DL
The constraint requires the heat exchanger to include 100 m of rubes
or
,
To convert this constrained QptimirZBtion' into an unconstrained optimization, 'Solve for L in the constraint and substitute .into the objective function . I . Cost :::: 900 +- 2200 D,o.!J
.
~
+
640
~D
- The objective function is now in terms of D only. so differentiating and
LAGRANGE !vl'L1...TIPLlERS
eq u.ating
'·0
"~ 67
zero yiel. ds :
I·
de cost) 1100 ·640 - - - = - - - -.- = 0 (feD)
.
'j'; 'D O.5 ,
. TID:'
m
D *. ='0.7 !.
.:- : ..... .......' .
:Su~stln;tin~:'b.; opU.i£;i ~vallle' of'D'B~di(tifi{d;illd'tonsrrn;n tgi ve's' ' , ~ .,. ;,- ,: :, ' ' .; . L
.;. "
"
~
...
"- "
"
-
1.3 m
Clnd ·th.~ ·minimum··cost is ..
-- Cost* ·= 900 ' + '( 1 190)(O.7):!-\ 1')) + (320)(0.7)( 1.3)
~'
'
.
.
806::', CONSTRAINED OPTIMIZATION USING
'L AGRAN(]E MiJLTIrLIER EQUATION$ ,
\
is
WhiIe,it so.~etimesposs·jble to convert a constrained optimization into 'an unconstrained one, there are severe limitations to this option'. For example" it .. may not be possible to solve the constraint equations explicitly for variables that are to be substituted into the objective function.: The more general qnd powerful technique is to use· the classic Lagrange multiplier equations (8.4) to
(8.6).
.
Example 8.3. ,$oIve Example 8.2 using the Lagrange multiplier equations. Solution. The statement of the problem is: Minimize
y = 90.0
tV/O
(8.14).
gradient vectors are
'Vy = ((2.5)(1100)DI. 3L
nnd
(8.13)
320DL
507fD 2L = 100
subject to' The
+ 1100D 2. 5L +
, 'V ¢
+ 320L]i, +
lOO7fD~'il
=
(1100D 2 ,5
+
320D)i 2
+ 50'TTD 2i 2
In component fonn the two scalar equations are
2750DI.$L
.f.
320L - J.. lOO1iDL ==0
1100D 2,s ~ 320D - J..50irD 2 =0
which along with the constfui nt
50rrD1L
=:.
loo
proVloe: three s,;'multaneous eqtlations to solve (or DO, L • and I
A. From
the,'
168
DESIGN OF THERMAL SYSTEMS b,
i 1 equation A'= 2'i50D
: "
1J
j+ 3~O
~OO1TD :.
.' 'S~l?~tituted j~to.'ilie i2 equation, ~this yi~.Id.s ..
. - -::"
..,"
~-
.
-.'-' "
..
" ."
. '
.
-
' .p'G, :== Os7 -:m .,., . .,
: .... ~.,.).
:
"
,~
.
.'
.~
:
','
"
.
: '"
.
..~ ";" "'~': ,: ~ .•.;' Subs·tifutiri'g·. thi~\\laiue ·'of .Qj~t~:~.~ c~(jJ;ls~n! give~ " . . . . . , ~ >.~.t)~~;~-~· '~~;'::~~':~'-';~'~";~ ~::~:-_;~:~'. :··::;~n(::':;.-:.~<~A:/·;:,·· -":':..'
'" ':' '. ." , .' . .'" ... ' .' .' .
.
(2750) (0. 7) 1~5 +' 3'20' ~. = . .. . IOO7f(O~7) . =,8.78 -
and
The minimum cost is .
~)'-.~'= 900 + (1100)(0.'7)2.5(1 .. 3)
+ (320)(O.7)(1.3)·:=:z $1777.45 ...
The objective"fu~~tion, Eq. (~·.13), is. sketched -in Fig. 8-4a.' So lo~g as - ei~er ~ or D is zero, . the cost is, $900 _ As Land D -increase: -the cost rises, providipg a \Yarped surface. that rises as it rn6v~s away from . the v~rticaI axis. The problem takes on meaDjng only after introd.uction , of the constraint in Fig. 8-4b, which is a curve in the. LD pla'ne. Only the vertical projections' from the constraiIH up to the objective function surfa~e are perrrDtted. The minimum value pf the ·objective function along "the projection js the constrained mlni~um. . Since Example 8.3 had only one co~stnrint equation, only one A appeared In the' Lagrange multiplier equations. There will be the same
Cost
Cost
Constrained
900
D
(0)
,
FIGURE 8-4 Objective.function surface (b) wjth constrarnt added.
(0)
---
..............
-.. "..:...:.~-.-
D
LAGRft..:l'iGE MUI.."IWUERS
169
number of ,A's as consixa.:nts . The executiGn of ·'J.e solution reouires the ' ..t. solution of a set of equations which are 'l ikely ~o be I10111inear~ In Example 8.3 it \vas .:,?ossible t'O solve them by substitUtion .. V!hen th~s carmot be done~ . the tools are noy/ available from Chapter 6 to .solve the se[ of; nonlinear . .simultaneo us equations by using such techniques as the rIewtori.
met.hod4
'.
.:J"
.
..,'
'L .-;:.
867' GRADIENT VECTOR IS.NO~l\1AlJ· ,·T'O TlIE CONTOUR. SURFACE . .
,
,
The 'purpose of ·this section and tne next is to provide a visualization of [he , . . Lagrai1ge multiplier equations. The pres~ntation \-vill not .be a proof I but a g~ometric displ.ay. Tbe first step in provid~ng the. visualization is t
,
'ay dx 1 ax 1
t
+.
'
Bv ' '.
-J-
ax]
dX 2
Further substantiation of the method is provided in Chapter 15. and a proof can be found in
the book by Wil~e and Bei.~htler given in the Additional Readings at tllc: end of the chapter.
.ll
FlGURE 8-5 Con tour Jines (curves 0 f constant y) whe n
y
::a:
I (x It :c 1).
170
DES1GN OF THERMAL SYSTEM:S .
Th~refore along
the line of con~~a1?t Y ~. say the y = 3 1in,e~
, dy ; ~'
. '
"
'
=0
>. ,'. '
' . iJy
==
•
~ dXl 'dX f
,,dXI
~
.
J
ay_
+ -dx2
aX 2
. . " .' = --',~~:-' ~~,<:.'.~, ,: "',::' ;" :-'n'. : ':"";:,' " f ' "~":'~~J: ;~,l,
:"',- '., 'or
.
',': ,
Pilly arbitrary 'vector ill Fig ..-8-.:.5 :-is
,-'"
" .
. ~ . .'
' . .1,... ,', ayla..X.',2, " , dX 2 -
-
",' ~ ";'\' "
",t'
,......
' .
-'. _dXlir + dX2i.2 ,
-
-.
"
" " , : :" '"
..
,
and any arbitrary uni~' ,:~ctor is ' " d,x I'il
. '-
(8.15)
, -",.J
I
+ dX2ii
JCdx J)2 + .( dX 2)2 The speciru unit -ve~tor T ~ the one that is tangent to the y
.
(8~16)
constant line; has dx 1 ana dX2 related accord~ng to Eg.' (8.15); substituting dx I from " . 'Eq. (8.)5) into Eq. (8.16) yields'
(8. 17)
T= aVJax -
2
( By/aXl
)'2: + 1,
aY j' 2 (- ay )' 2 ' ( BX'2, + ax} ,
Equation (8.17) is the unit vector that is tangent to the y - constant line. RetUrning to the gradjent vector and ,dividing by its magnitude to obtain the unit gradient v~ctor G, we have
ay tax J i 1 + ay/ dX2i2
--Vv' G
= IV; I -
J<8y 18x d 2
+
(8y I8x 2)2
(8.18)
The re] ationship between the vectors represented by Eqs. (8.17) and -(8.18) "that one is perpendicular to the other. If, as in Fig. 8-6, the con1ponents of vector Z are band c. the perpendicu]ar ve~tbr Zl has -components c and -b. Thus, the components are interchanged and the sign of one is reversed. ' ' The important conclusion reached at this point is that since the gradient . vector is perpendicular to the tangen~ vector. the gradient vector is normal
'is'
the line or surface of constan,.)'. A similar conclusion is' applicable in three and more dimensions. In three dimensions, for example. where y ="Y(XltX2.X3). the 'curves of const~t vallie of y become surfaces, as shown in Ag. -8-7. In this case the
)0
,,"
.'
LAG RA-l" lGE M ULT' PLiERS 'f
. )
.; '
"I ...· 1
-.
.
'.
:.: ~., ".,::..,. '
l
:
.. --
~.., -'
.. 1"",
,'.
. .. .: ..... ' .' ; "'~',
,. .
' !.
': . ... ... :~
- ',
'
. :~
.
FIGURE 8-6 c
Components of perpendicular vectors.
y=5
4
FIGURE 8-7 A gradient vector in three dimensions .
gradient vector V y. which is
· VY
=
ay •
--al
ax.
iJy •
+
- .-12
aX2
+
oy .
--I)
8X3
(8.19)
, is nonnal to the y == constant surface tJ:l .~t passes through that point. The magnitude of the gradient "vector indicates the rate of change of 'the dependent vru1able with respect to the independent variables . Thus. if I
171
172
DESIGN OF-rnERMAL SYSTEMS -
'in
.
. .the' surfaces of cons'tant y Fig. 8-T are spaced ,vide apart, the ,a bsolute :; , magnitude 'of the gradient is small .. For the Itime being, however, we 'are< interest~d OJ;lly in ,th~ fact that ~e gradient yector is jn a dire~tion nOTIni;11 " , _to the constant" curve or surface. , ' " ' _.. ,r, " , :.
.;
.'
.-
~
) ,'! ,.
'"
- I ,
.~ . ' •
'.
','
_
~
•
,
- '
,
' An optiffijzatio~ of 2,:problem with two ,variables 'of optirtlization 'ca,ll, be . displayed graphicallylo help clarify. the operation of the ~agrange multiplier , equations. Suppose that the funct?on y is to' minimized, where -
be
(8.20) subject to the constraint (8.21) ,
,The
Lagrang~
,
mulripljet equations 2·
(2 - A:X 2)11
+
,
~e
-
•
(3 - 2AX IX2)12
-=
0
and
x; ,-
,which when solved yield the solurionx~ = 3, 4 and y* ' 18. Some ,Jines of constant y ' and . ~~e constraint equation shown in Fig. 8-8. \Vjtl1 , such a graph available it . j~ possible almost instinctively to Jocate _
are
x,
8
Consmlin[ = 4&
x)x5 3 2
,
o
2
3
4
5
6
7
, 8
.1" I '
.
I'
)
I
fIGURE 8'-8 Optimum-occurs where the constrahlt and the Jines of constant), have s common normal.
173
LAGRANGE MULTIPUERS G-
the (x ], ~ .\:2) position rh3.t provides the constrarncd optlrn.lirIl~ p o~nt A. ir;. Fig. ' 8-8 .. The process of visually CLTTiving .at lth:ls s.o llq:ion migh ' be one
of foH6vving along the constraint line unti.I the co~.str~int and the line· .9f constaL.t y are '.~paralleL " Stated J110re precisely , the tangeat veCIors to the . curVes have th~ same direciion. . . , ' , ' ... . . AI-: more cQu,'enient- me,a'ns:-:~f reqn.i-~,yg JJ1.?J the , ~f-!,ng~nt ' ~~~r::tors ' ..:tile' same-direction· is",to. ~~ql)ire~ ihar ,.qle,..nonnal :vec~(~n;s.Jo, tl~eF.cllr-ves;,b~ye~Jh~~,;.~ ;'~:,..~~: 'same' direction.' Since the 'gradient vector is nOTIDfd- to -'a' conto~r lin~,' tIle'· ...... . inarI)e~.atical state'merit' 9f 'this requirerri~nt is , . ' .. . . . , '
have. .
VY '~
A-V ¢ ~ 0-
(8.22)
The' introduct~~n ~f a const"ant, "- is ,nece.ssat-y 'b~c~use the rriagrutudes of .vy: and V' ¢ may be different q,ud the vectors ·maY ~ven be pointing lin'opposite directions. The oniy -requirement that the twogfadi~pt vectorsbe. collinear. The co~straint equation is satisfied because the only points being' considered are those _ ~ong th.e cO!lsfrain~ e·qu·htiofl:. 'I
is'
809 TEST FOR MAX.IlVIUNl OR MlNThfUM In mathemati~s books that treat optimization~ -test proced~res for detenniniug the nature of the c~tical point receive a major emphasis. The.se test procedures decide Y'hethe~ the point is a maxilpum, fI!irtimum, saddle point~ ridge, ot valley. These procedures are gen~ra1Iy not so irpportant ip physical problems, which are our concern, because the engineer usually' bas an insight into whether the result is a maximum or ' rnini~um. It also seem:S' that the' occurrence of saddle points, ridges, and valleys is rare in physical problems. In addition, the test~ based on pure calculus usually bec~me prohibitive when dealing with -more than about three yariables anY"Yay~ anq we are especially interested in systems with large numbers of components. If a test is made, it usually consists of testing-points in the neighborhood. of the
,optimum. . . . On the other hand, a brief discussion of the classical tests for maximum and minimum provjde fuIther !nsight into the nature of the optimization process. The discussion will be limited to the optlffii-zation of one and 't~o' variables of .~n ·unconstrained ~unctioI1. Consider first the cas'e of one independent variable y = y (x) for which 'the minimum is sought. Suppose tha:t the point x = a t .jl$ the pos,ition at whfch the minimum is expecte(~ t9 occur. To test' whether the value y at this point y (a 1) is tru.ly a ~nimum; move slightly in both possible , directions from x = a 1 Ito see- if a lower value of y can be found. If a lower . value is available. y (at) is' not' die minimal value. For the mathematical' cheek, expand y in a 'Taylor series (see Sec. -6.9) about pOin-t X= al
of
-. y(x)--== y(al)
- ,.
'dy
-- . . ' 1 d 2y
+~ d (x ~ aJ) + - ~2 (x x..
2 dx
- G'I )2
+
(8.23)
174
DESIGN OF THERMAL SYSTEMS iJ>
First examine. moves so s-m al1 that :the (x - 01)2 tern1S and higher-rorder termS can- be ignored;. if dy ldx > .0,. -~eri 'I ~ (x) > y (a 1) f9f a value of x > ,0 1 ~nd y (a 1) is still the acknowledged minhnuffi_ -vYhen x m~ves to _ a ' v~ue less than 'aj, however, y(x) < y(al) and, yea}) will not be -a -' ,- - minimum. In ' a similar. manner, i~ caD be shown . that .when dyJdx < .0, a _ ,-;",: -' '_ ,-"-· lower value tlian y (a 1) can found for -y.(x). -The."only solution to the,.'
-be -
:·: ~:~::-~:<-.·.'~':- dilemma i~ for dy /dx' to -equal zerO, which is the claSs'l·~l<,~~?f:-'':'~:, ..·,J,· .. • · .', ... . . ,,~.:~- . -' ·: . r '. 1~.' .'.; .~ ~ ' ..: . . ' Incluqrng -the-''---influeiic~.·,: " ot:~-~,.the~ -;;' (!)(d 2y Idx 2 j(x ~ a})2 :, 'we,'observe that .a "moye 'of'x'~" either-din~ctip~:·~'·. " from ,a I·results in ~ positive value of (i - a f )2, so that the sign 'of the second. . .' '. -derivative decides whe~er the optimum is. a tn~iIiiurri'or minimum. ' t
dy-
Mininium:
-
. dx
.
\
'
---= 0
dy-
Maximum:
dr -==0
.• •. :...
.'
. .. "
, _
d 2y --; >0 dx-
and
'
.."
d 2y
and
<0
dx 2
The foregoing line of, reas,oning will now be extended to a fun~tion . of two vanables }' (x] ,'X2). Suppose .that the expected l!liDimum occurs , at, the -point (a 11 G2) ?-nd that to verify this position the point is shifted infinitesimally away -from (a 1, il2) in all possible directions: The Taylor -. ' expans-ion for a function of two variables is y (x 1 • X 2) = Y ( a 1' a 2)
+' Y 1(x I
a 1)
-
+ )':2 (X'2
-
a 2)
+ (1.1 Y i 1(x I
-
a J ):2
(8.24) where the prinle on the y refers to a partia] differentiation , with respect to the subscript. . When x I and Xi move slightly off (0 J ,02), both of the f~rs't derivatives y j and y ~ must be zero in order to avoid some position where y (x 1, X2)
<
y
(.a 1 , 02).
"
,
The second-or~Jer tenns "decide whether the optimum is a maximum .' .or minimum. If the combination -is al ways positive regardless of the slgns of x I - a 1 and x ~ - Q2, the o'ptimum is a minimum. If the conlbination is always negative. the optimum is a maximum. The test for maximum and minimum is as follows. The second deriva-' '-tives are structured in matrix form and the vaJue of the detenninant is called D, where -, .
D ,
= .
t
I
I I
YJII "
Y 2)
J I
j
Y12 II
Y22
Then if D > 0 and Y Ii > 0, the optimum is a min imu m. If D y i I < 0, [he optin1um is a maximum.
> 0 and
.
LAGRANGE ;"IULTIPUERS
175
( i'
-"
',
"
. : " - ~-."~ ~ :..
~
... ,.....
. . \: .
" .
"
..
.. ........ .
.}
. '.",S,iJIHtio'll;~: The,' first d~[i Yutiye~". ~e: _. ..: . ••
2 -:;::--. -
,.' 0)'.
ax I
'XI
,
.
a..'1d
2
.' . ,
'{:
.'
';",
~
, •
I _ : ',
'• • • •
iJv ,,' L ~. = - - - +.:1 , iJ .r 1 " ~. 2 • , .l .l -~
-
2
.
.
Equrrring these derivatives to zero and sob~i,ng simljllaneous~y yields an
d '
~ ',
x., =
'1I'
The se~ond deriva.t~ves ,e valuated nt x ~; af!d .'r; are'
2,
. a2y =
a'2y = 16
,oxi '): axi "
a.1y
------'-'-- = 2 ax 1 a~'r2
,The deteiminant
.
''2' I~ 16/:d >.o .
and
so tIlis optimal point is a minimum.
8al0 SENSITIVITY COEFFICIENTS There is often an additional valuable step beyond determination of the optimal value of the objective function and the state point at which the optimum occurs. After the optimum is found, subject to one or more constraints, the ' question that logically arises 'is: What would b.e the effect on the optimal value of slightly relaxing the 'constraint? In a physical situation this question occurs, for example. in analyzing how much the capacity of a sy~~em ·could be increased by enlarging one of the. components whose perforinance characteristic is one of the constraint equations. ' In .Example 8.3, w,here the cost of the hea't ,e~changer
'. '
+
Cost == 900- :~1100D2.5L +" 3~20DL "
was minimized subject to the constraint
,
501fD 2 £
?=
100
the question might be phrased: What wou'ld' be the increase in minimum cost, cos to • if 101 linear ·meters· of tubes' were required rather than the
.'
176
DESIGN OF ~ SYSTEMS
orjgi~a1100? Tq analyze this particular eX2mpl~~ replace the specific value of.J 00 by a gene~al symbol H .and perr,?PTI optimization by the method
thr
of Lagrange multipli' ers. The result \vould be. . . . -.
-
'
..
,
~
',.
~.
-t
' . ,D· .0.7 m" ' *"" O.013H . ':;;' ." , ' ' ' ':';'~' '-''i ;~.,;: ..'~.:.~' <" . '. -::Cosl"'" <900,,+ ··(1100)r(t:7)~~~(ft·of3H)·~+~·;'t'32(}){O:·7}(O:13H'J::-r·",.'". : .::.:::~.~~ •
•
•
r:,"' \
••
_ •
-:'_'
. "'.'-' . '. . ' ';. . ' .'. .900 f·. . .
•
+
• -
8.78H. •.•"
•
•
~
I
. .••••
. !"
'. ,'/ . 'W e ~e ~terest~d in the varia~~~ ~f c~st~"with E or, a t~rn:l c~led' the sensitivity C!oefjicient (SC), which is
mo~e' spe~ifically, '. '
.
,
·C " .. d (cost*) S - --'-._-
.
ali
. In this ~xample, SC' = 8.78; thus at the optimal proportion~ an extra meter of tp.be for the heat ~xchanger . would cost' $8.78 more than' the original cost. 'Referring back to. the solutjon of Example 8.3., we' note the remarkable fact- that the sensitivity -coefficient is precisely equal to ¢.e Lagrange multiplier A,. This equality .of t:be. SC to A is true :Got only for ' this . particular example but in general. AIsq, if there is TI10re than one cons.ttain~~ the various sensitjvity coefficients. are equal to the corresponding Lagrange . . multipliers, SC 1 == A I , .... " SCm == Am. The .optimization process by Lagrange multipllers therefore offers an a~ditiona1 piece of useful infonnat~on for possib.1e adjustment of the physical system, fo1l9wing preliminary optimization. .
8.11 INEQUALITY' CONSTRAINTS The method of Lagrange ~multjp1iers appliers only to the situations where the constraints are equalities, and the method cannot be used directly with inequality constraints. This limitation should not completely rule out the employment of the method when inequalities arise, .because a combination of intuition and several passes at the problem with the .method may YIeld a solution: . . As an example suppose that the capacity of a system is said to be equal to or greater than 150 kW. It would be a rare case when a lower-cost system· would provide 160 kW c.ompared ~ith the one providin~ 150 kW. '.This. constraint wou] d al most certainly be used as an equality constraint of . 150 kW. As a fUI}her exampJe, suppose that the temperatUre at some point. . in the process must be equ~] to or less than 320°C. In the first attempt at the probJem, ignpre the temperature constraint and after the optimal condjtions 1
are determined check to, see IfI the temperature in question is above 320°C. " If i.t is not, the constraint is not effec[ive. If the temperature is .above 320~C .. rework the problem with the equality constraint of 320°C.
----.
LAGRANGE MUT ....1lPLJERS t;.
117
1-
8. J ~ For the function G'
-
\
....
,;. '..·..;.~.c:<. : .~:-· 'I(~~; pe~e1fu.i~~· tl}e ·p?siiiv~~.Dp~~;iu:i~nrr \'all1~s :-or~i'i1~and ':r'2~ .:'..: : ~ . : /"~'~':'_":.~:l'~'~"·'~~_~-'~.'_:' '> . .
.
'{b) Us'e the ~Jhether
of -':Sec. - "809'-' t'?' 't¢st" - the· solution' 'to .detemiine"- ' :,:. the opt.imum 1$ a ·niaxir:p.lJ.m-or mi.nim~m~ ~echniques
_An. s.. a).x *I -8
1 - ':-:, - 2 . ~ ~ X 1 --:- . ' '. '.
, .
of .
are·
,Sa 2 ~ The -piIl1e-nsions ,of a fect"~"'1gUlar duc'r to . be chosen '50, mar the duct . maximum cross-sectional area ~an be placed, as -shown in Fig_ 8-9.~_ in the opening in a bar joiSL" _ ' (a) Set llP the.objective func.tiori and co~stra.int in.terms of h an~ w. (b) Using the.. method of Lagrange m'ultipliers - for ,constrain~d, optimization, determine [he optimal values of h and HI. ' . Ans.:- h = 0.3 ill. 8~3. JWo parallei pump-pipe ~semblies show.n in Fig. 8-10 deliver w~te,~ froT11 a' common source to a common d~s'tination. The total volume flow rn."te requi.re'9. . a( the ·destinarion is 0.01 'm3is~. The drops in pressure in· the t\VO lines are •
1
(-
I
'
-'II
[......~~- 0.8 m -----I!~c-II
1
0.6 m
========~~==============~ j FIGURE 8-9 Duct in bar joist in Prob. 8.2.
0.01 m.Ns
FIGURE 8-10 Paralle I, pump-pipe assemblies in Prob. 8.3.
178
DESiGN Or THERMAL .sYSTEMS
functions of the square of the
flo~ rate~,
.'
. ..-where F I and 'F~ the re~pective· . flow r~~es In. cubic meters per se'~on·d .. :,: 'J: , ... , . '. .' - . The .tWo·pumps have the same' effiFjency~ and,:the .t w'o IT,iOtors .that ,Mve ~tbe " ~': .,\ _': ' ; -'. '; . . : .. 'PlJmpS also ·have.'the same' efflciencJ- . ..... .. :. . " " '.' .' . ·. -:, >~/<~~~::-~·: ~ ...t:;··:~,Ja}JLjs_.J!esii~c! to"'iniIiiIrrize ' ;' total . ~wer ' equipme,nr. ·Set."up the :
.are
the
<..
.' <,:~-; .. .:. (i·bj~c'th~e. f#·~2~:i'on:,~. ~~!.:FR~~W.~.~::~,e~~_.:9~~~:.~:·;fP.~ f,2.~:· .t;':'<'~"'~':' ·~;':r";"':' ' ~ ,: '.:t·';:
.' (b)' Solve ·for ~e oijtin1,il·':YaI.l:ies::,,6f.~tlie~ '-f1Q~·:,''f..ates .~lliit· _.r.e~.~ltjn::ihlhiiritinj: :,-: total wat'er using ~~ method .qf.,Lagf4nge multipliers",:., '. . ~: . .. , ' '. , AnS .. :, FI = O.00567~ .:. ~
'.
power
is
8.4., A steel framework, as In Fig~" 8-1'1, to be con~tructed at a minin:rilin .cost. ' ". The c'9 st in doU~.rs of ·oJI the h~rizon.taI .,ne:mbers ·j none orien ration is', 200i 'J' and in the other horizontal orientation 300i 2: The cost in dollars of all vertical ' ' m~nibers is ,.500X3' The trame must eDdos~e -a total volutne of 900 m 3 . (a) Set up the ol;Jjective function for .total cost ' and the' constraint(s) in ..terins of Xl, X 2, and .l.'3. " . (b). Using the method of Lagrange' multipliers ' for constrained optimization, detennin~ the optirn.'aI ,;alu'es 'of the dimensions and the minimum
co.st.Ans.: Minimum cost = $9000. ' . / ' 8.5. A flov,.~ rate of 15 m 3/s of gas 'at a temperature of 50~C and a PJessp.r~ ,o f ~
./
.
.175 kPa'is to be compressed to a final-pressure of 17,500 kPa, The choice of compre's sor type is irifluenced by the fact that centrifugal con1pressors can ' handle high~vorume flow rates but develop only] ow pressure ratios p'er stage . The reciprocaring compressor, on the' other hand, lS suited IO low-volume
FIGURE 8-11 Steel frnmework in Prob. 8.4.
----.
...
LAGRANGE MtJLTT.2UERS
179
P-z = i 7, j 08 ~P3. Iorer-cooier ..,;
.
.
-,
. '_
.~
•.: , .
:'
_. I .
, . I.. · .. ~
_Centrifugal.
""
cOmpressor
p o = 175 kPa .
Qo =' 15 m 3js I =
FIGURE 8-12
SocC'
Staged 'compre~s'ion, in ,Prob. 8.S.
, flo'w rates but can devel.o p high presstir~. ratios. To combine th~ advantages of each, the compression will be carried out by a centrifugal compressor in series' with a' reciprocating compressor. as shown in Fig. 8-12. The intercooler returns the tempera.ture: of rbe gas to SO°C. Assume that the gas obeys perfect-g'ls laws. The equations for the costs .of the compressors
are
Cc where Cc Cr
= 70Qo +
1600 PI Po
Cr
.~ 200QI + '800 P? p)
compressor, dollars . first cost of reciprocating compressor, dollars
= first cost of centrifugal
=
.. Q = volume flow rate, m 3/s (a) Set up the objective function for the total first cost and the constraint equation in 'terms of the pressure ratios. (b) Using the method of Lagrange multipliers for constrained optimization, solve for the optimaJ P~_~J.?tios_ and minimum total cost. . Ans.! Minimum cost = $24,100: ~Q6. The packing rings in a distillation tower are in the shape of a torus (doughnur) as shown in Fig. 8-13. The outside diameter of the ring is to be 20 'mm, and the val ues of D I and D2 are sought that yield maximum surface 8Jea where the are.a = 7T 2D t D'}.. S~cture_Jhe problem as a constr~ined optimization nnd solve by the method of Lagrange multipliers.
Ans ..: /
D; :;: 10 mID.
8.7. In a cascade refrigeration systert) shown in Fig .. 8-14 that is used- for lowtemperature appJica,tions, the condenser of the low-stage unit is the evaporator ,. of high-stage unj,t. The area, this interstage condenser-:-evaporator is to be chosen such that minimum lifetime costS result. The costs associated with . the decision of [he size of the condenser-evaporator and the temperatures are':
the
-
.
of
\
.-"
.. '
.
LC-
ISO
DEsIGN OF
~ SYSTEMS
V
v-
~ 't!
~
C;cJ
~
•
JilJ
l ~lo-5{;'
U .. ~ -, .:-:
"---
",
, - ~:-
'.
-
'.
-
,
3~
~
4) .
-\ 'LG ~ V
. ... -:
-
I '· ···
-
~
. . ~f:,~~~~~~~;t;;-::'. " ; ~ >,;~.: ,Z,:'i" :~ •.,",.",:~,;, .:~.>: f ':, " "DQughnut;shaped. ri~g . with. ~mu~ ~~~ace .. area.in ... - .
'D'
Prob.
2 .
~L6.
. :
. cost ·of the con'denser-evapo~tor present. ·worth of the , lifet~me energy cosf:$ of the low-stage 'unit
.
=' 75 A =
. -31',000 + 150 Tc
. pres~nt worth' pI the ilfetime ~osts. of '. ' . . energy for the high-stage unit '-:- 189,000 - 6~3
T;
Heat rejection
Expansion valve
High stage
Compressor
T,. 160'kW
Tr Condenser~vapornlDr
Expansion valve
Low stage
FIGURE 8·14 Cascade refrigeration system in Prob. 8:7.
-.
~
Compressor
. where A = heat-transfer C1r~9., [nTc = cond~nsir1g tern.perature,: K Te = evaporating tegnperature, . K ')
The .Uvalue of the' C'o'ridenser~evapora.tor 'is 0.8 kV{i(~f12.. K)~ :aad the rate '. : of heat .that must be, ~a.n~~erred is·l~ ~Q_K~'>,,', ..;,. :':". i ·... ' . . , · " ':.., Ushl]:g the' ~~6.d J~l: Lagr~nge,.,rnu.l,tipliers.~:..deterilline, ..the ;,~Ja1UesJ1Df:A'·'F.~' ,';;;, ~:.: ' .. :' Ti . and . :z:;~::tbat resuit 'i~':niL~uin tatat 'present ~w.orth: of ,costs',.';·: :. ' . ':. ' :.. .. : " . : '. ' . AIls.:' A'" k 20m2 •. :,".... ",
" , 2< : .', "C.',
" Y .
,. ' .
..'
,'.
, .
..: ..
'<
P,,$v A
.
/
I
. '.
•
'sol~r ~~ll~~t?~ aod ~torage tariic~ ~hown. in Fl.g. 8-15~~' i.s· ·to be o~ti~iz:d. b~
. to achlc:ve m1ll1mU~ frrst cOSL Dunng '~he day {he temperature. of water In
.-t · c;
the storag~ vess~l i~ elev?ted from -oC (the rnin,imum useful temperamre) to t mnx • as shown in Fig. 8-15b .. The collector receiyes 260 W/m'!.. of solar en~r-. gy, bur 'there,is heat loss from .t he coHector to ambient air by convectioJ1. !The convection coefficient is .2 WI(m'2. . K), and the av~rage tern:p~rature diffei.:. .·· ence during .'the IO-hour day is (25 + t~ax)/2 minus the ambient tempera~re .
.of IOoe: . -
.
'
. The energy above the minimum useful temperamre of 25°C that is to be stored in the vessel during the day is 200., 000 kJ. ·The:. {l_~ASity of wat.er is 1000 kg/m 3 'and its specific heat is 4.19 kl/(kg .' K). ' Th~ cost of the ,solar collector in dollars is 20A. where A is the area in square meters, and the cost of the 'storage vessel in dollars is 10 l.SV, where V is the volume in cubic meters. (a) Using A and V' as the variables, set up the- objective function ap.d constraint to optimize. the first cosL, (b) Develop the Lagrange multiplier equations and ve'rify that they are satisfied by V = 1.2 m 3 an.d A = 29.2 m2 . ., ' . Ans.= (a).y = ~O~ + IOL~V, subject toA(230-4i.71V) = 5555.' t
.: .
\
" '.
"
.
----=-';.~
-
'
t.A
-0-
30
u
·0
_ I
oL.. :;:J
E
8.
,~
Storage tank
25°C, J,owest .usefullernperature
1
E.
o
I
'letoC nmbient
I
to h 36,000 s Q
Time
,
(a)
(b)
FlGURE S.lS
.
(n) Collector and' 'storage tank (b) temperacu~ variation in Prob.
-
.
'
B.8.
A
-t
'2 0 ~ if
;
182
, 2.4 ri1 3 '.
,.
DESIGN OF THERMAL SYSTEMS
/s
~
, -
,' , '"
.' .
.,
.
- . ,'
, .
.B '
A ,
. -:,. '
"
0.8 m~ls e
'. -
.;:-' ; >;C.;' ,:~;.;--: " j c .
.
. .' : . '
'FIGURE 8'--16 ~ ,
;' . \ . "
.
.. . ~
" :'0,6m?/s:; <;'). o);O;~ ~{~::':;" .~. ': ">. :"'~:c'- _;: . _•.
'. ' '
',. , : '
, ~,
Air duct 'i.n Peoh. 8,~~ . ~.
'
.:/: 8.9.
Deteumne the diameters of the circular air duct in the duct system shown , scheI!Jati~aJly .in Fig. 8-16 so that the drop in static 'pressure betWeen poilUS, A and B win be a minimum. . Fprther information'
,Quantity of sheet metal avaiIa bJ e for the system, 60 ,Iri2 . :Pressure'drop 'in a section of straight duc,t, j (LID)( V "212) p . .Use a constant friction far;:. tor f = 0.02. Air density p, 1.2 kg/m 3 . Neglect the influence of changes in velocity pressure. Neg]ect the pressure drop in the straight-through secti,on past an outlet. (a) Set .uP the objective function and constraint(s) in t~nns of D 1, D 2 , and D 3 . , (b) Using lhe method of Lagrange. mu~tjp] iers for constrained optimiza' tjon~ detertnine the optin1al "alues of the dimensions an.d the minimum co~l.
. Ans.: Minimum cost = S9000. 8.10.. Load dispatchers for electric utilities attempt fo operate the most efficient combination, of generating units [hat satisfies the total load requirement. If the input-output curves for (Wo units are as shown in Fig. 8-17, where 0 1 and O 2 are the outputs of units 1 and 2, respectively, and J J and 12 are the inputs, respectively, and where the total load, is T. use the .method of Lagrange multipliers to prove ,that the optinlum landing and occur when the slopes of the curves are equaJ. . 8.11. A cylindrical oil-storage tank is to be constructed for which the foHowing costs apply:
0;
Cost per square meter
Metal for sides Combined costs of
$30.00
concrele base and metal bottom
Top
-
37.50 7.50
.
. ...
0;
...,;." ,"
LAGRANGE M ULTIPUERS '
.. To[al~ ,
t ,'k\V ":",
,,t, ", ' 0,"' ,
.10.
,"
.
,
•
183
,
"
'
•
.FIGURE 8~i7 Optimal )o.adi~g of nvo units ,m , a pO\.yer generating station~ Prob. " 8.,10.
ot
Outpur. k\V'
The tank ,is to be ,con,s tructed wjth din1ensions such that the cost is miniq1um ' " , (a) One possible approach, to selecting the capacity is to build the tank l!ITge '" epough for an additional cubic meter of copaciry 'to cost, $8. (Note that' , this does not m~an 58 per cubic meter average for the entire ta'n k.). \Vh~t ' 'is the optilnal diameter' and optimal height of the tank? , Ans.: 15 tn', 1 I.25 rrl. , ' (b) Instead of the nppro,ach used in purr {a). the tank is to be 'o f such a size , that the cost 'will be $9 per cubic ' meter average for the entire storage capacity of the tank. Set, up the Lagrange mUltiplier equations and verify [hat they are s,atisfied by an optjmal' diameter of 20 m and. ,an optimal height of 15 m. ' .
for 'whatever capaclty is selected. ,
bene~th the beams ip a building, as shown in Fig. 8-18, is to have a cross-secrional area of 0.8 m 2 • The cost of the duct for'rhe required length is $ I 50 per meter of perimeter. The building must be heightened by tile amount of ,the 'height of the duct. and that cost is $0.80 per millimeter. After the duct has been sized to provide minimum total' cost, the possibiJ ity of enlargin'g the duct is explored. What is the additiQna~ cost per squar,e meter or cross~sectional ~ea of ,a very small increase in the area? Ans.: ,$642 per square meter. 8.13. An electric-power generating and distributjon system consists of two generating.plants and chree loads, as shown in Fig. 8-19, The loads are as follows:
110120 A rectangular, duct mounted
h
--r
DUCI
EXIra b~i1ding height
-L
, FIGURE 8-18 Duct mounted under beams in Prob. 8.12.
·
DESIG~ OF ~MAL SYSTEMS
184
Load A
2
Qenerat9r2 . .. . .. . ~
. ..
I~
•
I
4
_.
'
-"
"'" . '. .I:: :,.
'~"
.
.,:
-.
•
1
0'
~~.
. .......
.~,
4 '~ '
.•
:'I ..~J.~ '.~
..:; .~. -.":",~,~, .
~'." :'~ ,:::
. ._ .. :
'.'i':. :
,'". ~:', ,;:
'"
•
~
.
"
.
-;a. -
' I ',
.. . . .
":
.
~
"'"t
,'
.
, 3,'
,
" -
.~
:
.
.
.
.;
~------------~~.
LoadB
.
, ,-
"
'
LoadC
FIGURE S-.I9 Qenerating and distrib'ution sy~tern 'in 'Probe 8.13.
load A = 40
Mvi,. load 'B
. 60 MW, and 'load' C = 30 MW. The losses'
ill the ' Enes are given , in Table .8. t', where the ,loss in tine' i is a function 'of the power Pi in megawatts carried by the line. To"be precise,. the 'line Joss should be specified as a function of the' power at a certain point.:in t,pe line; e.g:, the entrance or exit, ',but' since'the loss will be small relative to the ' power ,carried, use p j at the point in the line !bost convenient for calculation. As a first approximation in th~ loa'd balanceS'. ~sume that Ps .leaving load A equals entering load B and rec~culate. if necessary, after the flISt complete solution. _ Assuming that the two generating plants equally effiCient) ' use the ' . m,ethod of Lagrange multipliers to compute the ,optimum anl0unt of power to be carried by' each of the Jines for the most efficient open~tion. Ans.: 24.3, 1b.3. 40~ 46.4, .4.6, 31 MW. 8.14. The power-distribution system shown in Fig. 8-20 has a source voltage of 220 V at ppin.t ] anti must supply power 10 positions 3 and 4 at 210 and 215 V, respectively, with a current' of 200 and :300 A, respectively. The electrical resistance Rfl is a function of the area and length of the conductor: R = 17.2 X ] 0- 9LI A, where L is the length conductor in meters and' A i's the area of conductor in square meters.
Ps
t
are
t
of
n
TABLE 8.1
Line losses in Prob. ' 8.13 ... Line' Loss, MW
,
O.OOlOp~
3.-
O.OOO7p~
4
' O.OOO8p~,
5
o.ooo8pl
6
. - '.,
o.OOJ2pi
2
-- .
O.OOllPl
'
y
~.
. ..
:
. LAGRAN~ MULTIPLlERS
I 220 V
2
1.0 t~ m
O.75-k.m
I
G----------
185
215 V
300 A 200 A
O.5.krn ,'
" .
, : • • ,1
r'~
:', '
r'
. :~ -: ' oJ ' '" ~ ~:.
. .'
-
.' .
"',
\
,.
.. ..~,
. Fu;tTkE 8~2ij .,I'ower-disi;ributlon ,network in Prob. 8.'14 •. .
'
.
I
.(a) Set up the objective function for the total volume of. conductor arid the : constraint(s) in lerms orA 1~:2, A 2- 4 • .and AZ-3:. '. (b) Verify, using all the Lag,range 'multipli~i equations. thar A ;~2' = .0.00273 ., m-.
ADDITIONAL READINGS Bowf!1an, F .• and F. A. Gerard: Higher Calculus, Cambridge University Press ~ London, 1967. . Brand, L.: Advanced Calculus, Wiley, New York .. 1955 . . Kaplan, W.: Adl'Qllced Calculus~ Addi$on-W~sley, Reading, Mass .• 1952. .0 Taylor. A. E.: Advanced Calculus; Gino, Boston. Mass .• 1955. Wilde, D. I .• and C. S. Beightler: Foundations of Optimi:.ation, Prenti.ce-Hall, Englewood Cliffs, N.J .• 1967.
,
-.
..
, I
CHAPTER.
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.
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.
i ·
901 OVERVIEW OF SEARCH METHODS The name search nzelhod should not suggest a helter-skelter wandering through the variables of optimization until a favorable value of the objective function, subject to ~the constr~intsJ has bet?ll found. Instead, search methods generally fall into' two categories, elinlil2atio~' and hill-clbnbing techniques. In bot~ there is a progressive impr~vement throughout the course of the search,' In one respect search methods are paradox, They are often the ulti.mate approach if other methods of optirnizabon (such as Lagrange .~ultj~ pliers, 'dynamic programming, etc.) fail. Fr~m another--standpoiht search' me~Qds are unsatisfying because there is no one systematic procedure that _ is followed and the literature is Jaden with diff~rent va,tiations of search
a
technjques. :'Nhen' a problem is found in which the standard techniques e~perience difficulty, another variarion is developed. In the. short treatment of this chapter we shall ad~ ~o several appro~shes that are qujte cla~sjcal, have broad applicability" and flow ·naturally from the calcu.lus principles , presented in Chapter 8. Chapter 17 puts the concepts of search methods as applied .to constrained prob1ems into a sharper focus through the use of peniliY techniques 'and the popular reduced .. gradient method.
-.
186
~
.,
SEARCH METHODS
187
"'"
,T he outline of the coverage in this chapter is as ,
f0I10 Vll S:
)
10 Single variable Cl.
, , -" " , ,~
' E xhaustive
h. Efficient ' iL Dichotomous iii. , Fib-ona~~i'~'{'.:'·,",
.. " .~,:, :",:~ "
'::'
I~
.-
.!~••
• .... '.1 ' \ .
~": •
'
• ,.
' ,2~ M111tiy,ah~b'le" ~u,~c~rtsiraiped a~
.
.:..... .. \ ,
'L'attice
h. Univariate c. Steepest ascent ,30 Muhivariable, constrained o. Penalty f':lDctions h. Search along a constraint In the calculus . method of optimization pre$ented in ~"
-Jating the ,numerical
Chapter 8; calcu-
valu~ of the objective ' function \vas ~~al,ly ' ~~ .last
step in the process. The major effort in the optimization ,'was determi~ing ' the values ,of the ind~pendent variables, that provide me optimiJrrL In opri-' , ~z~rion 'by m~ans of search methods. an' opposi~e sequence is folhl\ved in that' values f)f the objective ' function' are detennined and co~clusions are' drawn, from the values of the function at various combinations of independent variables.
9.2 INTERVAL OF UNCERTAINTY An accepted feature of search methods is that the precise point at which the optimum occurs will never be known" and the best. that can be achieved is to specify the interval of uncertainty. This is the ~ge of the indepe'ndent variable(s) in which the optimum lS known to exist. An interval of uncertainty prevails because the' search .method computes the value of the function only at discrete values of the independent variables. '
9.3 EXHAUSTIVE SEARCH Of the various search methods used in single-variable problems, the exbaustive search is the Jeast imaginative but most widely used, and justifiably -60. The method consists of calculating .the value of the objective function at ' values of x that are spaced unifofmly throughout the interval of interest. _1;ru;.
interval of interes.t 10' (Fig. 9-1) is divided here into eight equal intervals. Assume that V'le,val:ues of y are calculated at the -seven positions shown. In fhis exampl,e the maximum lies be.tween x A and x B; -so the fmal interval of uncertainty I is 210 10 1 =-=-
8
4
188 '
DESIGN OF THERMAL SYSTEMS
•
YA
,
JV..
;
. •y.
. ·)3
",',;'.~.~/:-~.: -~.. c•.~. '·~I'.~ . '; '" ~. ';":~;.• :;"~' :bl. •
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_ '
..
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4 .•..... •.••..•..•.
1.;:
,
.. '.
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. ~1..•..~:. " ." . •')~.,;.,,;~L,~. '.,:,','...'
•
•
--"{t-----. ...• . ------:-----.':X;o _ ' ""-:------.X - ~~J~' B
•
..... .
.. '
~' .
•
'.
FIGURE 9:'1 ExPaus~ive
search".
, If tWo observ"ations are made, the final interval of uncertainty is 210 13; if , ~'"ihree observgtions "are made, the final ~nterv.al nf,~ceftainty js 210/4, in general "
and
Fip?1
9Q4
UNIMODA~
int~n'al
qf uncert~inty
=I
2/ 0
12
(9.1)
+ 1
FUNCTIDNS"
The dichotomous and' Fibonacci methods, introduced next, are applicable to uninloda/ functions. A unimodal functlon is one having only one peak (or valley) in the "interval" of interest. Figure 9-2 shows several unimodal functions. The dichotomous and Fibon~cci methods can 'successfully handl"e not only the smooth curve like Fig. 9-2a but aJso ,the nondifferentiable function like Fig. 9-2b or eyen a discontinuous function like Fig. 9-2c. -.'V
,
)'
.~ ,I
I
~
I '
,I 1/
~
/ I
,x I
,( a)
flGURE 9·2 Unimodal (unctions.
---.
.I" I
(-/J)
.1"
(c-)
SEARG{ jviETHODS
169
In optirnizing ncnuair-nodal functjons "where there or e several pe~ll(f: (or valleys), the function can be subd~videdl '4I"tto sev.e[~. paTj.:s and each part processed separately as a unimodal function .
, 9Q.5 - ELIlVfiN:AT '~G A SEer :' .. ,J:~O, rESTs.: ': -'.. 'AY'-_ " , :, . . , , '
ON BASJiD "
,
-
, ~. ~~~
Ol\f ' ,-
, , _
' ' ' ', '
~o'tledg~ .Of · tli.~· ~alue of tlI~ Objectiyduncriollq,t t~odlife~ht .posiiirirui'O'
e(
in ,'the interval 'o f interes,t 'is sufficiept' to elitoinate a -pprtipn', the teg~o~ " -of a unimodal function. Suppose thp:t mliYJrnum 'vallJe of y, is ' sought it"]:" -the function that exists 'in Fig., .' 9-~a. ,The'Dj,agnltude of y ' is -kno~/n 'at, tWo'" , valqes of x, desigpate,4 i A and x B" :From this infonnatioD it is' possible to' elimiriate the reg{on to th'e left of XA-- The-region to the right of XA. must still I?e retaine.d. It cannot. be detennined whether'·the maximum 'lies betwe~n XA a'ad x B or to ~e right' of x B ' becaus'e th~ maximum' coul,d, reside ' in 'either' interval, as shown' in ,Fig. 9-3b. , '
a
906 ' 'DICHOTOMOUS SEARCH The, c_on~ept :of the ~fcholO/7.zqlLs s'ear~h follows closely Ute discussion of , ' placing two test p-oint$. 'It asks the following: Where ~houlc! the two points be' laced in the in tetvru of uncertain '-in -order,' to "eli .-," .. . laro-est ' .possib ,ee:ion? 'A !itt e re echori ~.ill ' s~ow ,that placiilg the pqints as near the cen,ref ,as , ossible whiI~' maintainjng qistinguis~ability 'of the y values Y'jll-resu~t in,' elimj~ation of almost half ,the 9nginal interval of uncertainty. Figure' 9:..4 shows th~ two points placed symmetric to the center, with , a ' ,spacing of E between. With the values of Y' as shown" the region ' tq the left' of x A can be eliininp.ted, s'o' the' terri~ining interval of uncertainty is (/0 +. c)/2; " ,. . -
'
y
o
o
.r
,
(0)
FIGURE 9·3 Two
lest
points
-
.
On
a unimodal function.
'( b)
DESIGN OF THERMAL SY~TEh1S
190'
,,I I
. . ~ , .: . : "
"
.
;p-~~
,-'-B
' ,X A
. -'
_____'_"~ ,, ~I~~~l~·'--~~~~ ' I, '
1-4------,---~
10
FIGURE 9-4' " 'Dichotomous 'search.
------~ ,
'
, " ','- Th~
.'
next pair of observatipn is' In' a' .si~lar ,' mann'er. 'in' ' ~e ' ' " , remaining, interval of ll:ncertainty-, resulting in the further- .-~eduction 'of the ' interval by near~y one-half. In general, the remairnng interval of uncertainty' ,1 is , ""-"', ,,' "", ' 4")" .. ' ,
made
,
-
,
~ .. 'where
11
"
0 .., -, ~Ill_
: 1
I.
+,.€(l ,
-,~), 2nl-,
(9.2).
is the number of t~sts (2, 4, 6, etc.).
9.-7 FIBONACCI 'S EARCH The most e'f ficient of the single v41i,able search techpiq~es is the Fi~onacci method. ,This method was 'first presented ' by 1Gefer;l who ,applied the , Fibonacci' number series', which was named after the thirteenth-century , mathem~tjcian: The rule 'f or detennini~g a Fibonacci nux:nber F is ,' . " .,
Fo .
~
FJ = 1
F;
, it,', ' ,
= F,.;2 + 'F'-l
for i ;::: 2,
Thus~
after the first two Fibonacci numbers are available, each number , thereafte~)s fc;>und by sumn1ing the two preceding numb~rs. The Fibonacci series starring with the ind'ex zero is therefore 1, 1,2,3,5,8,13,21,34, ". 55, FlO = 89. etc. . , Th~ steps in execu[ing a Fibonacci sear~h are as follows: <
1. Decide how n1any observations will be made and call this number
IJ •
.2. Place the first observation in;J 0 so that the djstance from one end is , J o{ F 1/ - J / F IJ ) • 3. Place the next observation in the interval of uncertainty at a position
~
191 ·
SEARCH jvlETHODS
~r
that :5 symmetric to th.e existing observa6oxL ,According to the reiative _. ~c~_:,.·' ; ''Values ·of. these obs.ervations, 'elirw?ate either the region to the right of ! the right point or to the left of the l~ft point. Continue placing a point , ;~: and eliminating a region until 'one point' renlains to be placed.: ).~t ~lis ' f.,. stage. there will be 'one observation 'directly in the center of r11e inte.rval (f . ..c . . , . • ' .' I. .. " .. " ','~"-'<'~;:::.', :;. : . _ 01 ypce,r:au1ty. " ' ,", ~c;'_', :,>~&:~:,::.RI~~e: 'th~ -· laSr::ohsl~rVation 'as' 'clos'e"as~'p6ssible to this' center pqin(" a#"?( ,' . .~ ;, " .., eliminate half the "intervEl-L -.' , : . , ., ":, ...... ... . '" ~'." -
.
.'_ ',,'
.. . ".'
;
to
Ex'a mple '9, I. 'Perform a Fibonacci search find the maximum of the ·fune.,. . ,tion Y' = -'(x)1 + 4x + 2 in: the 'interval 0 5 J~ .< 5. ,
, i
Solution 1. Arbitrn.rily choose
/1'
= 4.
...
2 .. Place the first (?bservation a distance"] o( F 31 F-l) from the left end .. as in Fig~ 9-5- This distance is 3/5 10 or (3/5)(5)~ ',The current interval bf nncer:tainty- is '0 'to 5.w·ith an observation at 3.
38 The '·next observation symrrH~uic in the intervaJ oJ lln~~rtainry to 3 locates ' " . this observation at x =.2. By rnakip.g use of the relative va1~es of y at, x = 2 and x = '3 ~ the section 3 < x :s 5 can be ~Iiminated. The interval o.f unce.rtainty is I).OW 0 :::; x -::;; .3, with 'the ~bserv'\.tion at x ~ 2 available', PlaCing the third point symmetric to the x = 2 o.bseryation locates it at ~ = 1. The relative values of y at ~'t = .~ and ~'C = 2 permit eiimh~ation~ of 0 -< x < 1.
)'
8 Last poinc
\
6
a
tl
First observation
0
.;
2
o
()
FIGURE 9·5 Fibonacci search in E.xample 9. i .
.:.
j .
~~ ,; :.
. . . . ,. '.
I ~ • , '.1
I
• ••
. ~ .. .-
. ' .. r
. .. './
~ J .'
..
.-.;' ,' -'
:'·.·1
SEARCH ~lETHODS c·
193
A;,,~SES _"flVIRrrf :JF S Il'~r'GtJE:--~rARliL3LE I
SEARCHES Th~
efficiency o f such tnethods as the di.cLoto!Uous and Fibonacci' se'an;:hes compared .\vit4 the exhaustive search is ' imp~essive. Reaiis.tically, IYo~ever,. the single-v?riable optiJ!lizationS are not- ~he. ones where ., high ~~fi~i~p sjes "";,ar.e~~ .ne.e.~ed. "- In.~nlo.s.i:,~~ip-gl~-:?ariable. .og~i~zaijoD.s,jn:..t.herrnati.systems,.... .~f~ ~:,;.: ·.the :~ s~aich·. is~ 'pert'okrned ·a . .. c6'mp i.1"i:er~ -·,:th€::.@xhauslj,y;e.·searc-h,,'-requlres .. ' :Oflly few: more dollars of compute'r tim.e and is easier to' prognlrl1.. '.Only .if the. ~~cQla[i()ns .required 'fo-r each fPoint are e~x[reme(y lengthy is . there"
a
on
. any significant advantage in one of the efficient search inethods, and such
.' . extensive' calculations for srngle~variable--optiif.ti:4at'ions are [are. ' . : Another po-tentia~ use of efficient single-:vatiable searches is as a in a muI.tivariable search.: 'An ?dditional re-StrlctiQll arises, howev~r, and . that is the _risk of the function being nOJ)unim-odal. Later in- this ch~:(pter a single-variaqle search will be shown (Fig~ 9-1 L fer example) making cutacross some contou'rs in a multivariable problem. -If those -contours are . kidney-shaped, .the. single-variable 'c'-:!t may describe a nonvnirriodal functio~ - to which-.the .dichotonlous and Fibonacci methods dO:l1ot apply ..- . \Ve move' no.w to multivaricible proble~s,' ·where exhausti,:,e searches . m.ay be prohibit~vely expensive and efficie.nt te,~hnique~ are de.m anded.
step .
a'
9'.10 MULTIVARIABLE, ·UN.CONSTRAINED OPTIMIZATION ' . High efficiency in ·.multivariable· searches jnay be crucia1. The m'ost significant optimizations in them1al'systems involve many components and thus
many variables. Furthem10re, the complexity of the equati6ns of a
~any-
-component system makes the algebra extremely tedious if a caicull:ls, rrietJ:1od. is used. This complexity breeds errbrs in .forrnulat~Qn. In many multiv~able ~ituations, ·che number of calcul.ations using an exhaustive search introduces considerations of compute.r time. Suppos~, for ex.af0ple, that .the optirnu!l1 .temperatures are 'sought for a seven-stage heat-exchanger chain. Assume that] 0 different outler temperatures for each heat exchanger will be inves-' tigared in all combinations with the outlet temperatures of the other heat exchangers. The tot~l number of combinations explored will be 10 7 , which, if the calculation is at' all complex makes computer time a definite concern . . - The single-variable' dichotomous and Fibonacci methods are elimination methods " because after each hew pojnt in the Fibonacci search (or pair of points in the dichotomous search) a portion of the interval of uncer- ' . tainty is eliminated. The three methods for optimization of an unconstrained mulJivaziable problem. [0 be explaintd next, are ' hill-climbing techniques, because the calcuhltion is always moving in a way that improves the objective function~ The ~hree ,methods are (J) lattice search. (2) univariate search, and (3) s~-&5cent.· ,..
194 -
.,
DESIGN OF THERMAL SYSTEMS
,9&11
LATTICE SEARCH
T]J.e _procedure in the lattice' search i~ :to' sWt at ,o~e point in the regi~n , 9f., jnterest -'an~ c~ec~ a n~mbe~ ?f.points ,in ~ gri~ surrounding the ' ce~tfaI ', ' " poinL_Th~ surround~g,point haYiIlg the large~t value (i~,a maXimum is _being , ,~.' ,',.sought) is 'chosen ?-s. ~e ,central point for the next search. ,If nO: : sUrrouri4~~.ig' ",'- '.' POiD:t.: ,p_royjdei','a,:'great~r~~yallle.:'~,of;:thb'~func~on-'th~!:tb!i·, '.c~ntra1",:,p·o~('C" tne{;':';~: " , r .. , " c~n tr~l: poirit: is tl;le"iiiaxinjUIiL;: A:. 'fieq~eht ,practice is, first , ~o' ,a ' ~0fl!se' " ,':':",',: __ .':grid and atter the 'maximum, has ,been found for that grid Si1bdj~ide the"gri9 ',' .'~-, -,: ' 'into, smaller ,elem,ents for. a ~er sear¢h, " ~~,~g from the IDax.lmu±n .,of
,use
'the coars,e , grid., :' '
" _ As '~ , example, of the prOgressIon to the maximmn Df a function' of
a grid is ,$uperimpo,sed over the contour lines of a' ~ction " in Fig~ , 9~ 7. The starting ,point 'can be selected near the center of the region " unless tp.ere is some' advance kn~wledge ~f where the maximum exists. tall the first p'o,int '1 '~d evall:late the function at points 1, to ,9. In this case, the maxiiQum' ,value of ,r hefunction occurs at point 5, so point 5 ,becomes'_the central point for the next'search. OUf special objective in' applying the lattice search' is to improve, , th,e efficiency compared with ; an , exhaustive search. ~inc~ efficie'ncy is a goal, it is attractive to consider saving infonnation that .has already been calculated -and can be ' used ag~jn., For ' example, ' in --Fig .. 9-7 after , D)9v,ing from point,] ,- ~o point 5, the va] u:~s of the functiQ'D,~' a't' points 't 4:, 5, and ' 6 are- needed in deciding , which \vay ·to' move, from point 5. Writing, a _computer program to accompJi,sh iliis task is tric~y, hqwBv~r~ particularly j'f '[VIO
v~a~les."
.r .. G
61
~
.,
•
•
•
•
•
•
•
., 3
• • • •
•
,
•
•
•
•
•
•
•
•
• .\' I
FIGURE-9-7 A lallice search.
-
.
SEARGP'r.I METHODS
the number of independen.t varia.bies exceeds jt"i1e number of dimensions. possible.for subscripted vilii.aqles on the .coD1puter being n~ed, It. is th~refore Ii{~ely '~hat the func-clons \NiH -be evaluated a.t :all nhie points, in the -case of two variables, for each central-point -location. . . , ~ , " .hIo.) defmire. sta~e!!;.lent ,can ~be ,m.acre about the ,coiTLparative' .e'fficieX 1C'l " .,.... of. u~e ~ s
lattice
~ ' .
. .
1~5
in
•
.
,
'
.
••
4
.
.
.
In -, '
.
-
,
9,,12 ' UNIVAiUATE SEARCH In ,the univaz:iate search?' tf:1e function is- ' optiilliie~ with 'respect to one variable at a time. The starting 'procedur~ is to substitute trial values of all but one independent ,variable in ,' t~e -functi'on' and ?pti~ize -the 'resu16n'g function in te~ of the one remaining variable. That optimal va.lue is then s~bstitute9 into the function afld the function optimized with respect to another variable. The function is optiulized with respect .19 each variable 'in sequence \~/ith the , optimal 'value, of a variable substituted into the function for, the, optiolization' of the succeeding vapables. The process is shown graph~cally in Fig. 9-8 for a function of two variables. Along the liI)e of constan~ Xl, whic,h is the . , "initi~ '~hoic~, the value of X2 .giving the ,optimal' value of y , is determined. This position is designated' as point 1. With the value o~ x 2' at point I substituted ,into the func~on9 the .function' is optimized \vith 'respect £0 ...~ 1, \vhich gives poipt 2. The process continues until the succ'essive change of the depend'eDt or independent vanaqles is less than a specified ro]erance.--
. .{ I
FlGURE 9-8 Univariate
siirCti.
,
..
196
:/
')::<~" ".
DESIGN OF TIIERMAL SYSTEMS
The ,m ethod , chosen ' for. perf6~ng tIre single-variable optUpiza'tion may be to use ca1cllh~s, where the t?Sk becom~s o.n e of solving one equation . (usually a nonlinear oJie) for v~nal?le~ It is also possible to use a single:. . ·variable 'search, exh(;lu,stive or .efficient, .,.e.g., Fibonacci or dichotomous." , ',~ ~, '.: .It: is)bis llse,:.of the e·fficient '.sll?gIe~varia,bie search. that is 'probably .th~ · .
one
'\r :":" .
':f1~< · '·/:'J:" >;:·'hi'o'~i·sig¢fi~~t ~pp!i(;';l~tion,.~(t!¥s.:typ~ ..·'
~;".,~':~".;;;'oP~r.!;t~:i~id;~"~~h,in~fail~~~:a'ridg~~~"in.:'the"Qbj~~~~ , .: .
functiQn. In. the function· in. Fig_..;.~:c9;_·,(Qr ,e~ample~:··if"a-. trhlryaIii~, : Of iris······ -... '~ ' selected ' as. shown, the' 'opt:iTIJ.aJ ~alue .of X2: 'Fes' the :ridge. ' When',this ' .. optimal:vahie of X2 "is 'SUbstltuted~ the attempt to optimize with respect to .' x l' does not dislodge the pbint from the ridge .though the optimum"has
on
. . not
been reached~' ,'From .the purely matilematicat point
even
. of ~iew, get:tillg hung up on the
. ridge would appear to b~ " it' serious, deficiency of the univariate s~.arch . .physical systems, how~ve.r,' ,the occurrence of ridges is ran~ because nature avoids disco.ntinuities of both functioris and derivatives o{ functions .. Caution is' nee~ed, .however: h~cause even though 'a true , ridge does' po.t . exist, problems 'may' ,arise '-if" 'the contours are very steep. If the intd:val chosen for the univariate sea~ch is too large; the' process rna);' stop at a nonoptimaJ, point, 'sucti as pO'j nt A in Fig_ 9-10. Another note of.caution is',that even though ridges may not occur in tbe physical sY$tem, .the eql,lations used to represent the physical syste.m m'ay' accidentally contain ridges.
. In
I'"
I IT' 1
!'/
'na
(
. XI
.1')
, FIGURE 9-9 Failure of the invariate search at a ridge.
-
.
SE.
197
) I
" . .,.
.:j ... ,: .. -: . . .".
.'
,
"'~': '
." '
• •
•
'1"',
· 0
e ·
FIGuRE 9-XD
Erroneous conclusion because . . ,. . the interval of search··is· to~. large~
.·9013 STEE.PEST.. ASCENr IVIETHOD
. As the
name implies,' this multivariable search method ' moves the' state pqint in 'such a direction that" the objective function changes a't the 'greatest favorable rate. As shown in Sec. 8. 7 ~. the gradient vector is normal to th~ contour line or s~rface and th~refore indicates 'the 'direction of maximum rate- of change: In the function of two variables wh6s.e .contour lines .~e ·shown in Fig. 9-11, the gradient vector V' y at poirit A . is nonnal to the contour lirie at that point and i~~~ates the direction in ,which y . increases at the greatest rate with respect to distance on· the x lX2 plane. The .equation
for Vy
IS
(9.7) where il and i2 are unit vectors in the Xl and X2 directions~ respectively. ... The essential steps in execu9ng the ,s teepest asJcent ·method are as . follows: 1 • . Select a trial point. 2. Evaluate the gradient at the current. point and the ,relationship of the changes of the x variables. . 3. Decide in which direction to move along tbe gradient. 4. Decide1low far to move and tllen move ·t hat ~stance.
198
.,
~
DESIGN
OF THERMAL SYSTEMS.
...
'. t
. ' ....
.- ' ,- -,
..
~ .
f: '
.' . :: • •'
. •"
- ', I
.
'
•
I
~
" •• 1
.
\ " , \':''l:: ...:." .:
.:
"
..... .
~'.,
.
,
FIGURE
9~11
. : ,Steepest-ascent method.
. 5. Test to determine whether"the opnmum h:as been achieved. If so, termi.h~te 1
otherwise return to step 2 . .
",,- .
~ Th~.
frrs.t three steps are .' standqrd, but ther~ .aremany variations of steps 4' and ? .The individ~al .steps will TI9W be. disc.lissed.
Step 1. The triaJ point should' be chosen as near to ' the optimum as possjble, but tlsualli such iI1S.i ght is nor. available and the point is selected arbitrarily. Step 2. Th,.e partial derivatives can be extracted mathematically, or it may be ·more convenient compute· the'm numericaily by resorting to the equation for the partial derivative,
to
ay
Y (.x J,
.+.!l,
...,
X
n) - )J (x 1, . . . , x (, . . . ~
X
II )
.
fl ;
aXj
where 1:1
• • .' , Xi
(9.8)
is
a ' very small value.' In order to move in the direction of the gradient," the r.elationship of the changes in the x IS, called flx' s, is (9.. 9)
. Step 3. It is understood that Ax; indicates the change in x I .. bu.t a decis'ion ~ must be made whether to increase. x I by l1x I or to ,decrease x I. That decision is controlled by whether a maximiiatilon or mi,nimization is being perfonned.
If a maximization is in progres.s and d)J lox I is negative .. Ax , must be negative ;n order for y to illCrease as a reslilt of 'the move. ..,
SEArfCH f"'lETHODS
Step ~L ~Jnly two of the numerous methods rqr deciding -hcfl.v far to move in the direction of ste~pest asceil't \'VilI be presented. The first method· is to select a rbitrarily a. step si~e for oJie ~()nable'~ say 0.X I, and COD1pute .!1.:C·2 -to .t.l~t II frQffi Eq. (9. ~)., This metD~,d .~s·ually '},.,orks .\Nell unril one or.~nlore of . ,the ,partial. 'deriyatives ?pproa~~~j..'. ze-r.p! . ..... .:.': . .~-'-: .. ~ . ~=i"'."'~ ~. ~, ~ .':- . ,:.:-.:-: -Tne $ec'ond '-method~:of'de;clding:"how"~ffu- rG
, ..
..
.' :
.'an:.
The
~".'
:vCirIable search. Example 9.2. An insulated ~t~e[ ·tank storing .ammonia, as shown. in .,Fig:. 9-12, 'is equipped with a.rec,o~cf~nsation sys'tem which can controI'the p'ressure '. Lind thus the' -temperature 'of the ammonia: J Two 'basic decisions' to rnake in m'e design of the, tank are ·th~ sheil thickness and insulation ~i~kness . . If the .tdnk operates with. a temperature pear ambient, the pre~sure in the lank will be high and a heavy expensive vessel will be required. On the othe~ hand. to maintain a loY' pressure in (h~ t~mk req;ires more operation of ' .. ' the reco.ndensarion system becau?~' the'rewill be more .~'eat transferred fr<;>m the environment unless tile insl!lation is increased. which also 'adds cost. Deterrnine the optimum operating ternp~rature and insulation thickne.ss if. the following costs and oth~r data apply:
Vessel COS£, 1000 + 2.2(p - 100) 1.2 dollars for p > 200 kPa· Insu [.arion cost for the 60 m 2 of heat-tta~sfer. area, 2Ix P·9 dollars
Recondensarion plant , ~5°C ambient
\
-. Ammoriia storage IJFiX :, Ex,omple 9.2.
200
DESIGN OF THERMAL SYSTEMS I}-
RecondensatioH COSi , 2 •.5 ' ce~ts per kilogram of ammonia Li~etime hours of ope~tiont ~O ,oobl h ,' ,Ambient temperature, 25°C , ': , '. Average l~tent heat of vaporizatipn 'o f aminouia, , ~200 kJ/kg , , ,:", ' ,~ : ~ ', ,,' ",', :, -> , Conductivity of. insulation k, 0.-04 WI(in , ~K;) , ', ' ' "..> ' '~, ~, '~,,:~> ~~',:';-': ~, " : ~,,:' . --:,~.-, ~r6~sure~~empera~ ,iclati?n ,~oi: ~monia
:
'.
'. "~,~,/~~:~~,~:,",~~:::~:-~~~-~;,,. ,/ ' ",:": ':':' ' .'''' " " . ' " ',;':::
" , : ,: ~
"
..::'O:::,.;:~
'. ' ,'" .
~ ": '
, 2800'-' ,
' , : Inp - ,
-
+ 16 33
<'
-t+ -21~ ':.<-:';,: ,;;;;, ::~-
. -
.
,~o :_':< .- :' .,; : ,. _
Soluti~n. The' total ,lifetime. :cost' is the' -'sum" of ,thr~e "in4iyidu"ai,'-CO$ts~~ 'tne vessel ', ,t.he insu]ation~ and the lif~tirrie cost of recondensation. ' All th~e' costs will b'e expressed in -tenns of tp~, o~rating ,t emperature .1 °c and 'the: insulation , thickness' x DllD. : ' , ,The iI;lsulation cost is Ie = 21xo. 9 The 's aturation press~e is, a func~on' of temperature' , ' , ' p -:-
.
,
e-2~/(/+273)+16.~3
,and 'so. th,e vessel ,cost VC is
VC
=
1000
+
2.2{e-28~/(/+273~7.16:~~>_ 100)1.2
Recopdensat-ion' c,ost'RC is','
RC
=
(w kg/s)(O.25 $~gj(3600 51h)(50,000 h)
(9.10)
~'here w js the evaporation and recondensation rate in kilograms per second.
But also
w
qkW = ---=----
(9.11)
1200 kJ/kg'
where, q is .the rate of heat transfer from the environment .to the ammonia, Assuming that only the insulation provides any significant resistance to heat transfer, we have ' .'
q, k\V
=
25 - r · (xmm)/JOOO [0.00004 kW/(m'
KJ
.. (60 m-)
(9.12)
Combining Eqs . (9.10) to (9.12) results in the expression for the recondensation cost '
RC
= 9000( 25 -
t)
x The total co.s.t is the sum ~f the individual cos,ts l
Total co'st C
:::r
Ie + VC
+ RC
The search method chosen to perfonn this optimization will be the ~'pest Oescent. The po~ition wiJl be moved along the gradient direction
201
SE,5},RCH METHODS
until :J. m i n im~m is reac-hed ; then a nevv )gr adienr wdi be established. The part'ial derivati"ves of total cost C 'wi th respect to x and tare ..
the
.
ae ~ _ (D.9)(-,-l)~1. ? ' :."70,1
-.iJ ..:c',
'-. ,.. ... . . :
'
.
,
. ' , ;-~~~ '="(2::m
:~.'
,
' ::
.
_
' :' .
,
,
9000(25 - .r) ' , .:r-
,' . )
'.
,-."
" '....
'~"" "
. .
..,
,.".: , ...:- . ',
,
where'. .
A=
.
.
.
~,
.
'
~
..
. " I., ·
2800 '
--~ +
[ +' 273
'
16.33
to
.~
.y, 1~( "
. --
. nx ,
.
=
'·-6.075
~t 77.3.47',
or'c
The minimum value ac~ieved by nl0ving in the direction of this gradient is" $667,5.18, ~here x = to1.23 rind -lO ~ 66.N~w derivatives are computed at 'this point -nnd. the posicion changed according to new gradient.' !abl~ 9.1 pres~nts a' surrimary 'of the calc.ulations. The minimum cost is $5986,04 When .the, insulation thickness is 196.8 mm and the operating temperature is' -23.3°C . Both partial derivatives are neatly .zero at this point. The steepest-descent calculation required a n appreciable number of steps ~efore finaJIy 'hon1ing in on the 'optimum,. The. reason is ,t hat· the route passes ' throug~ . a curved valley ;;md the minimUI)1 poi~.t along the gradient moved frorn one side of the vaUey to the other) as !!ndicated. QY ,t he alternation in sign of a c/ at.
partial
i .:. .
the.
I
TABLE 9.1
Steepest-descent search in .Example 9.2 lteration 0 2 -... J ..
3_
•••
x,mm -
l,oC
C
aCliJ:c
100.00 101.23 142.83
5.(X) -10.66 -7.54 -'17.78
$7237.08 6675..58 6334.98
-6.075 -19.409 -2.845 -7.16$ -1.471 -3.051
143.,62 ~ 68.92
4 5 6
169.40 182.87 ,.
•••
_
•••
&
i
_
,
'.
40
32 ,'
196.5'6 ,196.62
34
196.69 196.77
33 3S 36
-
• '196.83
-15.54 -20.80
.,
-19.50 ..............
,It
..............
6'152.85 ' 6067.68 6026.09 6005.85 ,
...........
-23'.~8
5986.07 5986.05 5986.05
-23.38 -21.3(r'
5986.Q4 5986.04
-23..24
-23.36
.. .'
For me firs~ point,' arbitrarily select ~~ = . 'iOO xI-un and ct' = 5~C. A'£ 'this pqsition.C = $7237.08~ · 'aC/.a~r- .= -6.075, and aCI8r '=.. 77.-347. (Th~ derivative with respect (0 x is ' negative and with' '. respect t is p-ositiv,e ; therefore to decrease 'C the value 'of x must be , incre.p.sed' and t must be decreased. ,Furthennore, to. move 'along the ,direction of the gradien·t. ih~ . ,changes in x and! :. design~~~d !1:r and D.r. sho'uld bear the relation
'.
'.
l
,
~o
.
I., ?~,.e,A:,~_-{cJO):1:e'\C;:8~~~;~:, __,.9~,~<*"" ~;, .,:" "~Co' ",.'~;,:~,;,,:,
_', i:
, , ~"
'
-0.749 It
•
It
.......
,:
aCI81
77.347 -1.455 37.505 - 0.634 16. 145 .... 0.294 7.477 1
..................
-0.090 -0. ,112
-0.086 -0.100
-0.076
.
0.189 -0. J23 O. J lJ: -0.143 0.088
" I
202
DESIGN OF TIIERMAL SYSTEMS -
,
9d14 SC..4.LES OF THE INDEPEJ\TDEl\IT VARIAB.L ES
~.'
-.
' .. J
_The name .steepest ascent implies the'· best possible -direction- ih .which . . ,to' move. The. m~aning 6f .this statement' is that for a -g iven distance AI"~+:U~ '+' . . . . 't he, objective ',func.ticin will expe~ence a ni~~~~ ch~ge.~:_ .::~ :.'. ;j ,...-" " ._,,-··Xs':· '.F.~g. .;9,;.1?B·· sho.w.s ;~)J.6~v.ever-;·,jd ;may-r be~~de.~ir~ble; .f~~;::' th~fe'. to. Ii#~!'farge- ?~.-f ·'.:' . ,:'. ~~h~ges .ih . . Xi ·~ompaied · .witl:I ·tr.tose Of"X2- · -Wild~ · ··~xt~~ds. '~e ,conclusions '. .... . ':;of Buehler: Shah~ and KamptboPle4 reCommend that 'the 'scales be cho. 'sen so that the. contours are as .spberi~:al a$ _ po'~,~ible in .order· to accelerate . -th~ convergence-. Ln Fig. 9-13b, for ~xample', th~ originai equation. would be _r~vis~d vyith a n~'V. yar:-iable _X.2 .replacing' X2 ·so. that x 2 = 4x 2 and the contours· \VQuld thps "cov,er 'the' same range as x I. ' . ,;.,' .. " " '-.
\"." _.' :"_'. I
to
CONS:T RAINED OP'TIMIZATlON .
9.15
.Const(ained optimiza~i
400
300.
J()()
,
:-
...
.
:..: lUU
"'7
200 ,
'
'" 100
100
JOO
300
. 400
o------------~~400 300 200 )',
( u) Oric,inal
,
scafe
FIGURE 9-13 Effecl o'f seCt/Ie of independent variabJes.
-
.
(h)
•
J
~,.
Revised scale
203
SU'RCH Ivl,E1HODS
10t
'ineq uality cons~;raiats) \-viB be cOr1sidereCl~ although the user' of a search
, COJ.ld adapt the ecjJial ity~cons.rrajI?t technique by rl1aking an linequality 'co nstrai nt . act~~e qr inactt,,'-e depend ing upon 'N he[hei-' th e con~rraiDt-. iJ violated
or not.
. . , . .... ..~
If. a function is
- -!: . .
19 be mf\4imized y
==
.'
.
-
, '
...
"
y (.x I , 'X 2~ - . - ~ .X 11) ~ ill a'ximun1
subjeer to the ' cOl1strain~ ¢l.(X I ,J,- l, .', . . , 'x 1l) -:- 0 . (
.
=0
.
a ne'w unconstrained function can be constructed
'. . y =
y ~. PJ.( ~.l)1 .- P2( ¢2)'2 ' -
. ~ . ~ Pill ( 4>m)2
..
·If the function 'is to be minimized. the Pie (hf!. terms vvould be added to the original . objec~ive funcrioP . to, ~onstrocr. the new unconstrained function. The underlying principle. of the .technique is valid, bur care must b~ exercised in.the execution to maintain proper relative influenc~' .o(the function being optimized to that .of the ,constraints", The choice of the p' .t enns provides the relative weighting of the .t wo influences and if P is too high, rh~ searc~ ~ilI ~atisfy the constraint but move, very slowly ·in optimIzing. the. function. If P is too' small~ the' search may 't erminate without satisfy.ing the constraints adequately. One suggestion i~ to start with small values of the P's' and' gradually increase the values as the m,agnitudes of th~ ,¢ 's become small. I
9.17 OPTIMIZATION· BY SEARCHING ALONG A
CONSTRAlNT~HEMSTITCHING
The next search technique for constrained opt{riiiz"ati"on ,to ,be explained "is the "search al.oJ)g the constraint(s)" Of 4~hemstj tching'H method. 5., The techniq~e consists of s,tarting at a trial point and fust driving directly toward the, constraint(s). Once on the constraint(s). the process is one of optirrllzing afong the constrain~(s). For nonlintar constr.aints . .a tangential move starting .on a constraint moves sl~ghtly off the constraint, so after each tangential move it is necessary to drive back onto the constraint.' This search meth.od is one of many that ru:e available but-is effective in most problerns'-and offers
£he further satisfaction of bU'ilding logically on the prin~iples of Lagrange multipliers. He·mstitching introduces the flavor of the generaHzed-reduced gradient method that will be explored in ,Section 17.4.
, Tmee.
J
I
"
204
.DESIGN OF TH~AL SYSTEMS
with one' constraint, and (3Y three-vari~ble problem \-vith two constraints. . . . .' I ,. .All of the constraints will b.e ~qllalities, These three cases lead us thr:ough .- the se'v~ra1 fund~menta1. operatiop.s that .appears in larger 'problems ,as' ~eil, ',' "' ." namely, (1) driving iowan:~ consttaipt(s), (2) ·movmg}ri.a favor-a,b!e direction. " :.::,... . ~ith .:respect to the objectivt- . function u~der cC)J~nplete res~ction of the '-i .,~~~:. . . 'constraint(s) and (3)" iri.a .favorable directio~ Vvlth resp~ct- to the .~:' .·~/.;,:~~j..~~l)J~~~Y~ .~~ncti6n with · ~o or .more .degrees of .freedom. - . '. . ~ ".~ ' :~: ~·.7, ~~.>. ;:·~~;:::··7·:·:;,{:~~(:~· :::~.->~:·~d'.::~~' .::/t \<,;:: ·,~j ·(~~·:·' :;7:· ,. )':',;;:,-:.;>; . '. < .' :',- "". . .: .: ..~,;~~'\::::.' ''~'> ,.....
mo'ving
•
•
. __ •
....
::
_·9.18 ~ DRIVIN.9::TOW.~"TlIE.~CO~~TRAINT(S) ... .., " '..':. . ....'. :.. ','
oi
of
m
. . The number' co.nstrai~ts WIll" ~lways be ..]es·s than the number "arl' . abIes n, so' if .In v'ariables are held constant at the cUiTeDt"location,' the remaining n - In variable(s) can b'~ ' saIyed to bring the point back 'onto the c
xi
Example 9.3. A con~trtiint in an optimizat.ion probl~m is X2 = 8. Locate a point op. the con~traint using a tria] value of (a) Xi = 2, and (b) X2 = 1.
i
= 81x = 0:5, and (b) x J = (8/x~)O.5 = 2.83'. The' purpose of the trivial,calculatioQ in Exanlp1e 9.3 lS to draw attention .to tlle proce~s of returning to the constral,nt following 'a iaJ),gentiaI move, as. illustrated in Fig . . 9-14. The choice of whether '~o h.old X,I or -'"2 ' cons!ant when retumjng to the constraint is arbitrary, but in ' certain probkIl'?-s the choice may influel!ce the rate of convergen.'c e to the optimu·m,.
SOliltiOJl . . (a)
X2
9.. 19 HEl\1STITCHIN.G .SEARCH WHEN n - m
'::=
-1
. One constraint in a two-variable problem or two constraints in a threevariable problem'. establish the vector for a tangential move along the constrajnt(s). Only the direction. of ,the move aIong the vector is open ' to ,quesri,on, and thjs direction is .~hosefl to achieve a favorable change . in the objecrive' funcdon. In the two-variable problem~ where the constraint' . ¢(x J. X2) = 0, the tangent attempts [0 mainIain ¢ = 0 during the fi10ve, so a¢ , tl¢ = - , ~ - llx 1 ax), ~o
+
,iJ¢ ·
~ I1X2
= 0
dX2 .
(9,13)
the relationship of Ilx 1 and. LlX2 in the tangential move must satisfy .
,
llx J a¢/ iX2 - - =t1 x 2 . ' " a¢/ ax J
(9.14)
Thus. to determine whether to incroase 0.( decrease x 2 t return to the objective function y (x J t x 2) ,. .where ', .>0.
n..,·
.CE:;
::-=-!lx J ax}
+
uy
-~x
')
ax]."
(9 15) i
SEARCH METHODS C;
205
(b)
(a)
FIGURE 9,-14 Hemstitching search in a two-variable probl~m \vitll (a) rerum to the constraint with fixed x (b) ~etum 10 the constraint with fi~ed Xi.
~-l?ich
I
in combination with Eq. -(9.14) gives "
A' - ' D.y =
,
"r' -,a-y - , 'ay a
ax I a¢/ aXl
ax 2
U~2
, In' a mi~imization, for example, when G, > 0" thus x 2 should be decreased.
= G UX2 A
(9.16)
-
LlX2 shou~d
be negative, and
'
E~ample
9.4. The objective functiQn associated with the constraint of Example 9.3" XTX2 - 8 = 0; is 'y -- 3-.\... 21 + •x 22
Minimize this function start,i-nK with ,a trial val ue of ..T2 = 1.6 choosing a st~p size lt1x_~1 ~
Q.DS, nnd returning to the constraint by holding X2 constant.
Sol4tiOn. With the ~aJ value Of.t2
= 1.6,
the ,constraint requires xi
= 2 .236,
so the' first point (2.236,1.6) giy,es y == 17.56 n~d 4) = o. Table 9.2 sho\1JS ,the progression of moves, s~.l1s; }~~n.8¢n,t ~9 ,tJ:le ,c~nstraint followed by Ii . move back to rhe constraint, at a constant value of X2. The initia,1 ~e of G is -6.175, but s'teadHy decreases until i,t passes through z.e,ro and attai,ns a positive val'ue of 0.0631 on 'the J7th cyc-le. The zero-value of G is critical. because this -position satisfies the L.agrange multiplier- equations that establish the opti'mum. This fact can be shown by arbitrarily defming a term A
as
-
...
..
: X = oy/.o)',1
a
206 _ DESIGN OF THERMAL SYSTa~-1S
TABLE 9'. 2
:
Heillstjtch~g seat~h in. ~xample 9 _,4 ' Before '.
'move .,
_ CYcle'
.. :' .'. tap.g~ilt ' " , ·2.236, "
·1
'..17~560 ·
1.60.:
X~. :~ >:: tetU!Dc",' ;.:.~'. ":.,2.27]" '. ,': " 11 ..65 ;-, ' tano-ent , 2.202· . . 1.65
....'- .'
~:""" . ' , ~::- ~"-? " . . , ....
: c 'Cc c c-
"~-~-, :", :" • ~
... •
.
. ","
-
c'.
. . . . .. .
.
It
~
":5
1: " c., c ': ?
"
.. ' . . . . . . . . . .
e '
........
-
. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i6:
tangent
.. ·-1.886~,
1.907
J7
.return. . tangent
18
1 .:865 1.845 '
~eturll
.l.~86
tangent .return· .
1.907
..
"
...... '-
......
15~725
0 .. ' 0.361 0
15'.272
~O.343
15.729 16.195
0
,. '. .·15-..129 16.195
2.25 2.30 .,2.30 2.25 2.25 ' 2.30
~
','
~
::
-
..............
...
-0..241 I
0.0631
-0.241
0.361
Wben. G = ' 0, A .also equals (aylaX2)/(a¢la~"t:2). From the 17th cycle in Ta.ble .9.2 an oscillatiqn about the optimum begins, and it w.ould be here that :the 'Step size should be leduced. ',
Th.e' ,three-varia~le problem where n - .In y (xJ,x2~i3)
. Ch(x 1'., X2, X3)
When on the requires that
(9.17)
0
(9.18)
=0
" (9.l9)
.
and
con~traint',
= . 1 is
optirri.ize'
-7
a I.l1ove that starts tangent to both con.straints
(9.20)
(9.2 I)
For a chosen ·step size of one of [he variables, x 3, for example, D.x I and ~x 2
must satisfy ,t he matrix equation, I
iJ¢l
I --
ax )
..
(9.22)
, The direction of the tangential move along. the vector to achieve a favorabJe '
-
.
,
, , .' ~5.5-15
"
subject to
-
Hgf :~: '~'~'_ t~~~.;.:, <:"t~~S'-- ; :-'Y'O;:~~t.~'.',,~::-4;~:' ,.~. . " "
•
.:." ',> :
~ . . - ,-6.175 :
: 0
18 ..194 " . . 0,.509 · ' -17.26'8 ': , ' '0,'
v
:
..
··. ·,·G
y.
c
SE~\RCH d ~ ! HODS
207.
I J
4y
= ,
oy .
~-WXl aXl "
av~'~{\ x'} -+ +' aX'2
-
OV
---.!l .:r ; aX3
-
,-
. dl ~,~erIB!lj~$, ~yp~4~~r 'A:~YJ s~pu.l.d~e, pO$irive or negat~ve. " "; .' ';"'."lI':':;'~;'; ;:. .. : ~. ", : '. ,'F0)lo:\iV.~ii$,~,~,1~ ::ni~ :,I:e,~.ta 11 g~,.~~~,:; ~o"::;, th:e:~ "I~~Yv9. ;;~:~r!,s;u~ir '.~s.:.~: t~;~,.~. p)=!:s, i~,~p'~} : 'I: ~'~ ,.::' ' . .,- ' .' ' brought . back" onto.":the 'constraints 'by 'siniurta:n'eoi.fs-solut~on ·'-of Eqs·. .(9::.f8.). . ," ,
'.'
an(~f (9 ~19) holding one o'f the vari~bIes con~ta'~t.
9020" r~(10VIl\IG TA.-NGENT T'O , " , .' /:\ COL JSTRc-\ll\Il' IN· TI-IREE J)I IVIENSIONS ' , A speci,al situation OCC 'U IS in a thfee-variabie optimization having one' con- . stralnt, .when the direction of the mqve must be decided such that the objec-· tive' function' chang~s in the -nlost favorable mann~r. ,The mO\,'e could be visualized aj; in Fig. 9~15, where a' po~nter (the vector) is pinned onto [he ,s,urface (the ·constraint) and is free to tum,-in a plane. The direction to, be .taken by th.e. pointer is, such that}he maximl!_~_~hanKe .of Jr' ,occurs, ~here
ily
(9.23)
, choices of direction of
I ,I'
,-I I I I I J
I I
I I
I
./
J---------,,'-..",c...------~--"""""""'-~ .t:!
FIGURE 9·15 Vector gwJrlg lJJaximum r;:lle of chan~ of y. but also tangent to 0 -constraint,
_ 208
DESIGN OF THERMAL SYSTEMS .
_
.
. J
.but even more-specifically the maximum Tflte of change of y is sought. This means· that the ~axPnum Ay is to. pe determined subject to the. consir~int of a give~ lepg~ of tbe ~ove ' I: .
. _
L1~i
+'Lixl-:t L\~.~ .= !~. " -: .
. -' . '; .. (9.'24) .
':~~;~-:- :;:'>:.')~~ wbe~~r; ·~~·. is ~;iiii;·· arb.itrCLry::je~gili~·: .iliat:"::Will:.£ultini~t~ly::/~~c~r:-~~~~:::~ci(~·~~: .~' ;, ' ': - . .. . ~ interested ~Ocl.y ill the ' r~~ativ¢ ' maglliti!d:~'s~ ' of 'Ai ~, Ax ~~ .and 11~ 3:.' .Th~ ."other requirement j~ .that 't he vecto~ ·h~· . '. .constraint, so .that
. .. .
a4> .
;
-. !1'x1
aXI
.j)
cf> :' . '.
,+ -.·-
l1x2
aX2 "
au' initial direction tangent to. the .
a¢
+ - -LlX 3 ~X3
.
.
'.'
'.. (9.25)
=.Q.
an
. We are confro~ted with optii.rrization .probler;n "to fmd the relative' 'values of the Ax's that. maxiniize Ay a~ "expressed . in Eq. (9.23), subjeet to the two constraints, Eqs. ·(9.24) and (9.25). . . Choosing the method of Lagrange "Inultipliers to pe~ontl t1;lis optimization yields t:4fee equ'a tions: (9·. 26) . (9'.27) ay
' . .. .
- .- - .- AJ(2) llx 3
.aX 3
"
'. .
-
,B4>
A2- = O·
(9~28)
aX3
Multjply Eq. (9.26) by a¢/8x}, Eq. (9.27)·by a¢laX2, and Eq: (9.28} by a¢/ BX3, th~n total the fust tenns, ·the ~econd terms, and the thhd tenns of the three equatibn~,
(9 ..29) From .E g. (9.25), 'the second group in the. sununation is zero. For convenience, ]et . . ..
, and '
----.
~.
/
..
SEARCH METHODS ti.'
I I
A
(9.30)
,\, = ~ - .B I
209
Substitute Eq. (9.30). into .ea<;h of the equations Eq. (9.26) to Eq.· (9.28). This gives the {QJIQ.\.~~~ng .~xpres~ion tQ[ tbe' 'relati ve vait~.es of Llx I, . !lx: 2, and
.- .'...'.' ~~.v ~ ..... ·r.' J.,;l. ....\.
" " : L'1 •
• • ••1. -
.~
J)
~
.
. ,.; _1_'
I _
: '2;\, ,
.',.
_ .. _ :~.~~ :._. ~ ,-
•
~
. ...
~.
•
_
-.
.
'. ... .
" •. j
.J
'. '
,
'.
•
~.
\
'
•
•
•
...
•
.,
•
"
'. :: . A~r~' .:'.': ~ ':" ;. J.~: :Jyrax."2 ~CAf!3)a¢/aX"2
a_vlJxi ~ (AfB)d .cP1dX·, ·. 4~t"3
.'.
=
Ll x I
:"
;
.
''-When me step size of one of Jhe variables in the move- has' been selected~ the' .magnitudes of the remaining ·Llx 's 'can be computed from Eq. (9.3i)..
9 .. 21 SUMMARY This ·chapte.rexplored single-variable searches as w~n as both unconstrained and constrained mu1tivariabl~ searc,qes. The types of problems for which search methods are most lik¢IY to be called into service are the difficult ones, which are probably the multiyariable constr~ined problems. Of the :rpany methods ,a vailable' for m.ultivariable ·.co:nstrai,ned optimizati.ons, the ' search along the (~onstrain!(s) was chosen because of its· wide applicabqity (alrpough it is not necessarily the most efficient) and because it follows logically from calculus ~ethods. The actual·: :eNecution, of the calculations in' complicated problems would probably .be carried out on a ,computer, and' the availabilitY of the . interactive mode· .is partkul~ly convenient. The possibility the searcher has qf cnanging such quantities as. the step size or· starting- with a' DeW trial point facilitates ~onvergence to the optim.um in"a rapid manner. The techniques presented in this chapter are tools tha~ can solve .some realistic engineering problems. The chapter also leads into extensions, such , as those that appear in Chapter 17, where' penalty methods are explored in greater depth, and the reduced-gradient technique 'is shown to be a generalization of some of the specific s.i tuations examined in this chapter.
-
PROBLEMS'"
)
9.1. The iunction y. ~ (J n .t') sin (x:!125)
,
i
-
.
x where x 2 /25 is in radians, is unimodal in [he range 1.5 S x
- .
:s 10.
.
•
210
DESIGN OFTIIERMAL SYSlEMS
(a) If ~ FiboDii~ci search is empl?yed to )deterrnine the maximum, how rua'u y :c: points will be 'n eeded for the fmal inijerval of uncertainty' of.x to be '0 .3 :: ', . or less? _ ' ", (b) ,Using the number points determin'e d irfpart (a), co]}du~t' th~ Fi150naccl ' search and determiDe the interval in Which the ,maximum oocurs; ," ,_
of
_:.~ ~- ~,2::', '" ':,'-'
:' : Ans.:.:, ,~p).,:?,~?~:.' to" , ~'.OO;:~'' : : 'l, ':' :,;,:;:,:~:~::,: :,: ':' :~<~" ~.:,~':~,;-::~~,~ ~ :./" ',:,~?<~~~: : :" :'~' :~ '~: ,:~,'~:';\ ",',:::~;':' : "~;::: ' ',:-: .
:,<, ,::", '::: ~':!J:2.~-One~~of 'the" strtii~gj~s',:_i:Il'-some"search metho~s is 'to fust, use,a·' ~arSe · ~9bdi.;. , , , vision to' determin~ the ,approximate regkm the optimum and then 'a rme ': . sllqdlvislon for a second search. For a single-:-yariable se~ch '16 points ' tqtal , "
or
. 2£~'j~.be applied. -C~mp~ ' the :ratio 'o f the initial ,to final inte,ryal of 'unceT- " nsedin one -F ibonacci s~arch and (b) ~ poi.nts in ,:: , , .. tainty if (a) aU 16 poirits ,: :'a FibonacCi search, used to dete~ne an interVal of unceru.inty of redu~ed. . ' : size on which another 8-p~int Fibonacci ,search is applied. ' 9 ~3. An .economic an-alysis of a proposed facility is being conducted to select ·an , . , operating life -such that ,the ,maximum uriiioriri- annual income, is achi~ved. : A short life results in high annual amortization costs" but the main~e~ance costs become excessive far a long life. The annual income after ded:uc~ng all . operating expenses, except maiIitenance costs, is - $180~OOO.The fitst cost of the faciltity is $500,000 borro~ed' a~ 1'0 perc~nt interes~ compounded annuaIIy. '" The mainten~ce costs are zero at the ,e nd of the first year; $10,000 at the end of the second" $20,000 at the end of the third, etc. To express these rpajnt~nance charges on an annual basis the gradient present-worth factor of Sec. 3.8 can be multiplied by the capital-recovery factor, which for the 10' percent interest is presented in Table 9 . 3 . ' . , Use a Fibonacci search for integer years benveen 0 and 2) to find the :lifc of the facility which results in the maxjrnum ,annua.1 profit. Omit th~ last cal~ulation of the Fibonacci process since \v'e are interested only in integeryear results. , Ans~: 12 years~ $62,760 annual j'nc'Ome. , 9.4. The exhaust-gas ,t emperature leaving a continuously operating furnace is 260°C, and a proposaJ is, beIng considered to install a heat exchanger in theexhaust-gas stream to generate low-pressure steam all -105°e. J1le que~ti,on to be inves.tigated is whether it is economical to instan such a heat exchanger, and, if so, to pod its optimum size. The following data apply:
are
are
TABLE 9.3
_
Factors ,for conversion of gradient series to an Year
.
Factor
Year
Fact-or
an~uaI
Y~r
cost Factor
I
J
0.000 0.476 0.937
4
1.379
5 6
1.8]0
J. 2
,
2.224 2.622
7
-
.
8 9 110
I
I
..... I
~
.-
3.008
3.376
JJ
3.730 4.060
12
4.384'
,13
14
..
•
4.696 5.002
I I
I
is
5.275
16
5.552 5.801
17 .J 8 J9 20 21
6.058 6.295 6.500 6.703
/
2 11
SEP.RCH lv'iETHODS B-
,r
I . (~" ~
'.
,'
Value 'of the heat 'in the
of steam, $1.50 ,per gigajDule. '
U value of heat exchanger based on g3ii-side area. 23 WJ(rri2• • K).
" .A: " , ,,G~qst
"
fOnTI
pf ; ~e)~~~t"
exc;4.~ng~i)pclud,ing in~tal1atio;;l b~" "on ' ga's-side
I.~'; ,-~. ~.; >'~. ~~:~;i~~!::, :~7:~ :;::::~;:~. ':.,...,>.>.', "". '".... . ;~, .. ... :.,'. t , " .' r...
Life' "of'installation;. '5" years.:'
.
~
, Satu~ated liquid , s~it~ated 'yapoL
water ent~rs
- " ',' , '
.
.:. ~ =-..
<"l.. :
0" •
.,
•
~
":
••
~
r
• '.
•
,', "
heat exchanger iJ:t Ib5°~ and leaves ,a's " . ,
(a) Develop the equation frn; the s~vjngs ~s, a functi~n ' of the area, -expre~sed ,as a uniform annual amount. .. ' . ' , ' , ' , ' " ,t :"~, ,:' r ~ (b) What is the maxjroum permitted area if the exit-gas,' t~mpeciture is to be , . above ' 120°C in order :,to prevent condensation of water vapor fro'm the ' exhaust gas? ' , , (c) Use a seven-point Fibonacci search and set up, a tabI~ to simplify calcu~ lation of the opti.m um heat-transfer area. ' 2 An.?: Optimum area' between 686 and 724 m • ,P erform a univarjate search to find ~he minimtlm value of the function y
16
,x"·,
XtX2
2
+ - - +-=
=X'l
. using ~~.:~:g= ; . a 1 : Y d X2 an~ starting Withx2
-f-
= 3.
9~6. The ~nimum val ue of the function . y
72xl
360
'
= - - + -"'----, + x lX2 + 2'(3 , X2
XIX3 ,
be sought using the ~~tbQd oZ steepest descent. If the starting point is ,: 5, X2 = 6, x 3 = '8 (y ,= 115 at this· point} and x J' is to be changed by <~,: 1.0,' what is the location of the next point? ' Ans.: Ne~ y :::; 98.1. 9.7. The functfon
, '.is to
x ~ '=
is
[0
be minimized by the stee est-descent search
point the 'direction of steepest escen
WI
'
.. From the starting
deternlined. and the search is to
move in that direction llrHiJ a m~nimum is reached. whereupon a new location is to be- ascertain.ed. If the arbi'l.rarHy chosen first po'int is (1,1.1) w'hat'is the
second point? . Ans.: y at second point -. 8.548. A pipe 'carrying high-temperature water is t.o be insulated and then mounted in a restricted space, as shown -in Fig. 9-16. The choices of dle p-ipe di'ameter D m a04Jn.e insulation thickness x m are to be such tha.t the on of the insu-,
-.
..
,.
-
.
... - lation y is 'p. miriimum~ bu.t the totalannti~l op~rating cost o.f ~e installation " .is li~ted to $40,000 .."This annual . pperati~g :cost has two ·components~ the '. ". ..water pu~ping cost .aI?-d the cost of ~e h~at loss:
.."
.
1500
:
.Heat cost = - ' - dollars
x ..
(a) Write the objective function and the constra·int.· . (b) The values of D = .0.2 ' and x = 0. 1 satisfy the ·constraint. -?tarting at this poin't move in a favorable "direction tqngent the constraint by an " . increment of fl.D = 0.005; then return in the most direct manner to -the constraint~ What are the new values of x and D' foUo"-~ng ~js hemsti.tch move? . Ans.:.v = 0",3664 m. if> = 19.86. 9.9. A refrigerati~n . pi~( water and delivers it 'to a heat exchanger' some distance away. 'as shown j'n Fig. 9-17'. The suppJy water temperature is I J°C, and the .return water temperature is 12- The flow rate is w kg/s, and the pipe diameter is D m. Th·e cooling duty at the beat exchanger is 1200 kW, and the ari·thmetic me-an temperatu.re, (1 I + I'2J/2, must be J 2°C in order to transfe( th~ 1200 kW from the air. The pump develops a pressure rise of 100,000 Pa that may be assumed independent of flow rate. The equation for pressure
to
Air ') 0
Refrigeration pJant
C M'
kgls
p_rn H~t
exchanger
Pump·
,
.
FlGURE 9·17 Refrigeration plant, heat exchanger. and interconnecting pipjng. "
-
.
S~CH METI-IODS
1::.3
drop in a circU:lar pipe is
:
:. .....:
'.~ "
"
-,' 'J '
,' .
- The 'specific l~eat of wai~r is 4.-19 kl/(kg '. ICY. ' The Wee major' costs associated whh. - the choice of i t . -.
'.
'.
.
,
. _.,
.
~
.t"
.. ,
~n.~ D 'are ' . . '
-
'
Present wo~ of lifetime pumping costs, Pr~sent worth of .lifetime' chilling costs,
.
: "'T. . . .
' .
1.0/, -
'
Cost of the pipe, 150~OOOD doit~s
....
..
~OOw -dollars
60, . Don. .
4000t 1 . doIlars
, (a) Set" up the" ~uation forthe cost as the' ob}ective filliction~ the pres~U!e ,
drop as 'constraint 1, and the heat~transfer 'requireme'n ts as constrain.t 2." (b) The poi t ,W = 26 kg/s, D = -O.lSim, and t'l = 6.5°(: essentially satisfies both ,nstraints. -What is the neW point after making a move of l-V ' = 1 kg/s tangent to both c,o nstraints 'in a direction that reduces the cost? :',, A!ls.: (a) (2.938 X I0 6 )DS'- -w 2 ,= 0 and w(24':- 2td - 286.4 = O. (b) D = 0.184, w = 25, ':1 = 6.3. 9,. iO. The optimization of the dimensions of t:J.1e steel fra~e in Prob. 8.4 is to be conducte9. by a search method 'as a thJ;"ee-yariable problem with one COD,sttaint. If the- current pos.ition is x 1= 16, X2 = 12.5; and X3 ,= 4_5 ill, what are the new values of the.x 's' following 'a step of fix I := 0.2 ffi. if the move is tang~nt " too_the constraint and in '~e dire.ction of the most -favorable rate of reduction of cost? Ans.: cO'st 'reduced $66 in ' the step. ,
,
REFERENCES 1. J. Kiefer, USequential Minimax Search for a Maximum," Proe. Am. MaIh. Soc., vol. 4. --', p. 50.2, 1953.' - " 2. D. I. Wilde. OpTimum Seeking Methods, Prentice-HaU, Englewood Oiffs" N.J .• 1964. 3. N. McCloskey, "Storage Facili[ies Associated wjth an Ammonia Pipeline," ASME Pap.
, 69·Pet-21,1969. 4. R. J. Buehler, E. V. -S hah, and C. Kempthorne: Some Propenies of Steepest Ascenl and Related Procedures for Finding Opti"!um Conc/i#on.s •. Iowa State Univessity. ~~ica:l ·... Labora~ry, pp'. 8-10, April 1961. ' , 5. G. S. G. Beveridge and R. 'S. Schechiet: Optimization: Theory afuJ Practice, ,McGraw-
HiH, New York. 1970.
/'
,
. 6. S. M. Roberts lUId H. I. I..yv,e-rs, ·The Gradient Method ,in Proces!,, CentrO,}, /nd. Eng. " Chern.; vo1. 53, pp." 877-882, 1961. ' . ~ U
,
-
...
.
)
.
.~
.-:: :
) .. -:. '. ': •. \
1
'." :~:
'~:"~' . ". " ',: ," ; .,.;:. ....: . .. "
.
•
'.
"
"
..
j
'J
....
'
.. .....
;..
i ·
•• ••• J.-
•
'.
'"
,. . . r- .... :. .'"
10.1 ,UNIQUENESS OF DYNAMIC-PROGRAMMING PROBLEMS Dynamic progr~m.ing is a m,e thod of optimization tpat lS applicable either to staged. processes or to continuous functjons that can be approximated by staged processes.. The word Udynannc" has rio connection with the frequent use of the word in engineeqng technology, \vhere dynamic 'lmplies changes with respect to time. . . As a method of optimization, dynamic programming is not usuallj' interchangeable with s~ch other fonns of optimization ~s Lagrfinge multipliers and linear and nonlinear programming. Instead, it is related to the calculus of variations, whose result is .~n optimal junction. rather than an optimal state point. An optimization problem that can be subjecte
such an appljcarion, the finite-step approach of dynamic progranuning is an , approxj.mation of the calculus-bf-variations' method. In many engineering situation.s , on the other ha,nd, the problem consists of analysis of discrete stages. such as a senle s of compressors. heat exchang'ers, or reactors. These 214
.: ' :'
' ,
'.
215
DYNAMIC P:ROGR.AJ\-lM1NG
cas~s fit dynamic prograrnrning exactly~ , and 111 e re the calculus of variations ~Nould
only ap/?foximate ,the ,result.
10~2
SYMBOLIC 'DES'C IPTION OF DYf\TAlV ~CPROG, >~~~"~G, I:',; ",' " '~ ,", : 1;:
'
"
..Fi~~e -.iO--j : ~h()\;;s.· ~·s;~holicdesGripti~nl7f . the . dYnaniic-pmg..~ing · ... probJem. The- dec;isiort i!aiiables are to be chosen ~o ,that for a specifi~d " ,'input, to stage 11 and' a ~pecif~ed output' frpm stage' 1" the summation 'of.. " .
,
11
Lr
-returns
'
i
:
.
is' opti~um (either maximum
' .
01:" n1inU1'lUrXl,
.
depending upon
{=l '
" the prob l~m). ' " ",The des<;:ription. of the 'calc.ulus, of·variatiqns . in Section' rO.l suggests ,t hat ,'a funC~l:n, for example y (x) ~ is sought. In ~ig.' 10-) that goal is to', find the opti' al state variables S for th,e var:ious stages, where the st~ge corre~poIi9 ,to the x 'variable. The decisiQn variables control the change in S through eKe stage and also 'determine the 'retum fro~ a stage. calcul;s of variations seeks a function that optimiz~s, 'a n integral. \vhile dynamic
The
,
,
11
programming seeks to "opti~ze a, sumrriation, ,
'
d~not:d
here ,as
L
-
i= I
'
rue
'
T i.
In
the.c'alculus of variations terminal points of the 'function y are specified. In dynamic programmjng, ' also, S nand Sf are sp~cified.' ' Often some insight is needed to recognize that a phy~ical problem fits into the IDO,ld qf dynamic programnling. Some hint is provided when theproblem involves sequences of stages, 's uch as a chain of h~at exchang~rs" reactors, compressors, etc.' . So far only the nature of the ,problem has been described. The next section shows how dynamic program1ming solves the problem.
Srage _ _ _n~---"Il
Stage n-I
sn'
rn _ 1
\
SI
n-I
S'2
Stage 1
•
,t,
------------------'J
~-------------~---V"
, FIGURE 10-1 Pictorial representatio'fl of problem ,that ~an be solved by dynamic prognunming; S t:I state of [he inpu!...~ ..3 stage " S' state of t~.e' outpUl from a stage, d dedsion variable. and r c retum (rom a stage. E%J
:;lJ
216
DEsIGN OF THERlYIAL SYSTEMS
•
.
.
1{t3 CHARACTEPJSTICS OF THE
D1(NAMIC..PROG~GSO~UU:ON .
. '.
:.".1 '
' . ·The trademark of dynamic progra~prni?g in arriving at aP: overall optimal, ::.. :: plan·is to . establ~sh 9ptimal plans fQT' subsections of the problem. In, suc- :: '. ':.,' :: ceeding evq;luadons the optim-aJ. p'Ians fOT the. subsections are. ~secr~ and all .' :'. :'.'- :;:.· ··nonop·~ plans are ignored~ Th,e mdchaIlics ·'caI1:.. l?e 'l1histrated by a prob- : ~~.'-~ : :~;·'·~ :. · ...lein(of.~~¢~w.B..· t4~~~pp.timaI ~01:lte qe~e~iL:~H.,.p;~m.t~""as iPt. ~xamp~e : .
, . ..:, · ::.10.1.· ~·· .: ". ' . "::
..
..
""':"" " ""~<""":'~.'~'.:' :;:j\.: . ~:.,::'~:.>.~.;~~~·>:T/: ·:: ·'· ···.'::·~···'·::-- -- 7 .' :.:~;
-<
. , T . .. •
(':r... ;.,.
<...... .
~ ~.: ~?:~./.:-:
.'.. '. " ' -.-~- Example' 10'.1. cA riruIiiti.lh~cqst pijJe1me IS to·be.constrUct~ be~~n·Pofuts · . .' .. . A and 'E ~ passmg ~u~cessively through one node of-each,,' B, 'C, and D, as : ',: " shown. in Fig .. 1-0-2 .. The cos.ts frow. A to B' and from D to E ax:e sho~ in ... .. .:. fig. '10-2, ~d. the costs between B and ' C and .between C' and 'D are ' giyen :' . in'Table 10.-1. .' . I
-
.
Solu?-~~.
The s~~tegy in dynaml'~ p:ogr~~g is to begin with (j~e' o! the . stages (either A-B or D-E m ~s ~rOblem) and. then progreSSIVelY .
/
.4
4
4
FlGURE 10-2 Dynamic programming used to minimize the ct?st between points A and E:
TABLE Cos~
l~, l
from B to C and C ,to D in Fig. 10-2 . To
, -
From
1
2
-3
4
.J .
'3
12 1'5 21
15
2
16
21 J7
17
16
28 24
4
28
24
15 .
12
-- .
15
2 7
.~ 1 .....
b: DYNAI"llC PROGRA.l '..!MING
no
analyze cumulative sets of siages. In -thiS! problem it makes difference the whether we start wi th A.-B or tNith D-E'- . In some oroblerns the fOHn existing data_will n1.ake starting at one end or the other raore ,conveuienL , In this problem we shall start at the' fight end and won~ lto\yafd th~ left. Thus the first table yviH,:be, f.Qr:, siag~ D-E the ,next [able for ili~ c~mlliative' , , s~ge:l C-D and jj~E.," ~n~, ~',o -~~' i1ntil, tae..1ast.tabte ?T:tRAi~:~i, tQ_.th~J~:i1tiJre~·£~~?ten~....~. ~,:, ~, :~~ ::: A-~ ' :: , _',' -",' " ,.. " '" .' .'> ,: ' ~ ' -: '1, ,:" , , " , .:' _ : _' .. " " , ,Tabfe :1.0.2 :s imply shows the, costs from , the v,mious' l) positions , to . ,: 'the end, \;I~iCh is at E. Table 10.2 aSsumes the' forin- that "liU be -used in ,S'il,~St::q.~ent bies; jge." it' '_des,i~n~tes_ the' Op,til1?-um. ~anner _in i~hi~~ ',to pa~~ -, from D 1 7 ' 2,7 D3~, and D4 t,o. the next stage , which here IS 1),6- e 10.2 is ' trivia! because there is only one way to pass from ,D'l to t~e terininus, and thar route must be optimum,. Future tabl,es will ,rule 'out ' nonoptimal paths, but no such selection can be made in Table lO~ 2 because it .is invalid to say, for, example, that the route from D2 is less costly than from D I, which ,would pemit' ruling out D i. The , ~onsequences of 'the choice ot' ,the rouU;s fr6'm 'D2 ov~r'tIiat from D 1 may impose overriding penalties later, ;so alLmust be kept in consi,deration. ',I t is in Table 10.3 from, C to E [hat dynamic' programming begins 'to show its, benefit. From position C 1, for example, there are four paths by which toO'reach E. thtough D 1. D2. D3, and D4 .. The total costs are shown -in the appropriate column in T~bl~ 10.3, and, the least cosily, rollte is the one passing through D 2, ,resultin,g in a c;ost of 30. "Yithin that block there IS ,a common basis of comparison: ali the paths start from C 1.: Table ,10.3.a1so denotes the optimal chokes when starting , at the other C positions. In the path from C3 there is a tie (cost of 32), so either choice be made. Now " -that the preferred route .from C I to E has been decided, aU the nonoprim-al . routes are henceforth ignored. Advancing (backward) to inClude stage B-C in the assemblY9 Table 10:4 showsilie .a ccumulation from B to, E . TIle costs are, shown for ,t he possible paths starting at all of '[he B points. From B 1 it is possible to pass th~ough C 1. and therr to {he end, th rough C2 and then to 'theen~, etc. The total costS from B I through Cl to th~ end, for example. are computed by summing the cost of 12 from B 1 to CI (frO,Jil Table 10.1) with the cost from CI to e,nd. The cost of C 1 It o the end i's,30. which is available from Table 10.3 as the minimum cost of the four- poss.ibilities in the 'c 1 block. Of the various paths from B I to the end, one is optimum , with the cost of 42. Table 10.4 shows .
-
.
"
J.
,
of
.
~'
.
?'
.~ , ' ... : .....
. , ' I."
...:.
.
t
•
E.
e
,
can
me
TABLE lO .. l
Example 10.1, D to from
,
P
Thro~gh
'
Optimum
DI ,
20
x
D2
15 16 20
,x x x
D3 D4
,
218
DESIGN OF THERMAL SYsrEMS
TABLE 10.3
.
I
Example ,10 . 1, C to E
)
Cost
.X I
D2
20 16 IS.
DI
15
20
._ 31 35
D4 D3 D2 'DI
28
20 1.6 15 20
. 37 30 32
x
D3
. CI
' 44
24. 17 16
D4
C2
21
15
12.
33
.
-
48
.
TABLE lOA
Example 10.1, B to It Cost From
ThI."ough
B to C
C to E
Total
Optimum
B4
C4 C3
12
31 32 31 -
43 47
x
15
C2 CI
B3 ~ •
0'
B2....
C4 C3 C2 Cl C4 C3 C2. Cl
, Bl
C4
.
C3 C2 --C J
24 28
30
55 58
15
31
46
]6
32
48 .
17
2J
31 30
48 51
24 17
31 3~'
55 49
3·1 30
47 45
. 16
lS 28 21 15
12".
~
31 -
59
32
5,3
31 30
46 42
x
x
x
.
.
219
DYNAMIC ifROGRAMM ING
) I
Cosi: _.-
'Total
.. ,-.;).:c,'
·' :~ --'7~
: 82 • ':
... .
"
,
_.- , . 1 " •
.:
:" ~"
16,
45, " '
20
BI '.
:;;, -.' .'~' .:~-:' <; ,.!;
-
I
' 42 , .
'<..'~ :~:' ., ~O-. '. , '~'
0"
61
;t.; ", :-' ."
'
" . ; :: :: •• '.'
' ,.', ~,
,62
,"'"
-_
'
,
"
-
_
"
fquI- state vatiab ~es, B I?' B 2~ ,B 3, and.jJ 4.,. and the optimuIn cost fr'Om each of these states to the ~',nd. ' I , The finnl step in',the ,~ollltibn is provided by'picking UP. 'th~ firs~' stat~ , (A t9 B) ,in t,he accumulation? and ,'the resul'( is Table'IO.S. Ther~--is, only one ' state variable, the position A2. From A2 me .options are' to pass thrOllgp B 1', , B2, B 3, or B4. The cost frqrn A2 t'o, B is' found in 'Pig. IO~2 and the coOst -from B to ,'E 2.is the opt,lmal oue- fr
to
l
'
The key feature of dynainic programming is that after an op~al policy has been determined from an intermediate ' state to the final state. futUre calculatio':1s passing through that state use only the optimal policy.
10<04- EFFICIENCY OF Dl7NAMIC PR'OGRAMlVIIN G The combinatipn of tables necessary to ~olve Example 10. I may seem to constitute.'a 'lengthy and tedious solution. Comparatively speaking, however, dynamic- progniniming is efficient, particularly i~ ~ arge problems. If we can, each line in Tables 10.2 to iO.s a calculation, a total of 40 calculations . was requir~d. If an exhaustive examinatio-n of aU possible routes between A and B had been made, the total number would have been the product of [he number of possibilities. From A2 tq B there are four po~si,bilities, from_ each :of theE, points there "'are, fOUf possibilities of passi,ng to Ct. and s,imi )arly from ,C to- D. From, D to E 2 there is just one possibility. The number of. possible routes jf all are coo?idered is. therefore (4)(4)(4)(J) , == 64. M
. The saving of effort would be more' impressive if the problem had included another stage consisting of four positions. The number of calculations by dyna~ic. programming, would have been the current number of 40 plus an additional 16, for a total of 56. Examining all possible routes would require (tf4)(4J -;; 256 calculations. .
i<
•
~
220
DESIGN OF THERMAL SYSTEMS
Feed 0.6% ',' ".'
- LactoSe .
( •
,J
~
•
. .-
:.~
.
" ,
, 'i . ,
--:'.
-;- ' ,\,,;
. . .. .., ",': , ",
--~
," , , . '~~ample . 10.~ ... ' ~hey is 'a by-prqdu~. of chees'e manufacture a.hd contains, ' ,among o~er cons~ituents ; protein ?TId l~~~se. The t~o subst2TIces . are .val~ab]e . when separa1ted, protein-rich whey for yogurt and la~tose for ethyl alcohol. One me~od ·of 's eparation is' ultrafiltration, w~ere. the separation o,ccurs ~n the basis of mol~cu1ar size and' shape'. A series of ultrafilters is employed , ,-'(four iIi the"charn shown i~ Fig. 10-3) to separate the protein and lactose progressively. Klinkowski 1 show'S' the ope,rating cost qf ~ stage to be a function o( inlet and outje't protein concen~tions, as shown {n Table 10.6. Use dynamic programming to solve for the concentrations leaving each st,a ge that results in the· minimum tota). cost.. :
the'
.Solution. The dynamic-programming calculations start, arb'irrarily', with stage · TV, in Table 10.7 and proceed back·through.the ~tag~s until Table lO.10~, which' includes a.11 the stiiges. The lninimum operating cost is $34.42, ,which is achieved 'by operating the' filtration plant with concentrations 0.6,
3.6
[0
0.9,
1-8,
6.0.
TABLE 10.6
Operating cost of one stage in 'a protein.;·Iactose separator in :e:xample 10.2, dollars . Entering protein concen· tration. % ,0.9 0.6
Leaving protein
. ?53
0.9 1 ,1 1.8 2..4
3.0 3.6 4.:! 4;8
1"
1.8
2.4
)0.77 ' 3.73
20.24
28.38
10.77 5.54
1 7.23
10.78 3.74
3.0
~oncentration,
3.6
4 .,.
,%
4.8
5.4
35.20
40.70 44.88 ' 47.74 49.28 49.50 28.3.8 .33.07 37.18' ,40.70 43.63 15.67 20.24 24.47 28.38 3J.95 35.20 7 ..33 ,]0.79 14.00 ]7.23 20.24 23.10 2.82 5.55 8.2 J 10.80 13.27 15.67 2.26 4.47 8.73 10.8 ) 6.63 1.89 3.75 5.56 7.33 3.2J. 4.78 1.62
23.10
1.42
5.4 6.0 ~ :.;--~
.... ' ~~: .- :.
,-
6.0
2.82 . ,1.26
22~"
DYNAMIC PRO GRJti\1MING -t'7
,___~-'--lE 10· ,,1
Exampie l{L 2, stage I I
I I
.,'' .:"""".
...
~~:~~"~,:. ;:,;;~: : -: ;:
... : . .... ".;
lO ~ 8l
TABLE 10.8
Concentration III
~ntering
1.1
Through L8 ~.4
3.0 3.6 '4.2
, :4:8
5.4
L8
~.4
5.54 ·+ 23.10 = .. $28.64 10.78 + . 15.67 = 26.45* 15.67 + 10.81 = 26.48 20.24 + 7.33 = 27.57 ·24.47 + 4.78 = 29.25 28.38 ' + 2.8Z~ 31.~O . 31.95 + 1.26 = 33.21 '
3.0
2.82 + 10.81 = 13.63 5.55 + 7.33 = ' 12.,88* 8.21 + 4.78' = 12.99 10.80 + 2.82 :;;; 13.62 13.27 + 1.26 == 14.53
3.6 4.2 4.8
S.4 3.6
Cost
3.74 + 15.67 ';::: 19.41 7.33 + lO.81 = 18.14 10.79 + 7.33 = 18.12* 14.00 + 4.78 =, 1~.•78 17.23 + 2.82 = 20.05 20.24 + 1.26 = 21.50
4.2 4.8 5.4 3.0 '
IV
2.4 3.0 3.6 -4.2 4.8 5.4
3.6
4.2 4.8 5.4
2.26 4.47
-1-.8
I
-.
2.82
= =
9.59 9.25*
::z
9.4~.
+ 1.26 = 9.99 1.89 + 4'.78 = 6.67 3.75 + 2.82 ~ 6.S7f6 '6.82 5.. 56 + t .26 lC
I
~.2
+ 7.33 + 4.78
6.63 +
'8.73
:...----~ .. '-.
~.:
.. " . . .
4. 8
1.62
SA
3.21
+ 2.82 + . 1.26
5.4
,.1.42
+ 1.26
J:I
4.44· 4.47
1:1
2.681\)
=
':
. ":,.
..
. 3I;a.d
~:' -, ;
,0': . :.:. ":, .. ~ .. .' .. . '. .' . ••. .
7..33 . . 4.78 . 2.82 1..26
. Example 10..2, stages 'nI
"
- ,-' ,
.
• • .. " . .
" ': ' ,
222
DESIGN OF THERMAL SYSTEMS
TABLE 10.9
'Exa.m,ple.lO.2, stages ll, ill, and IV ,~ .
~oncentr.ation
I
·1
'
:~ X>', ': ' ~, : :.- entering n,
1,"hrough
Cost
~ ;(t,~~<~ ~,~;9.:,2,•.:.," ".'.", .'.~'.'-~'~1~"-~ "':' ,.<,;;::~~:~lj~;~;~:~~i~~\,:~,'.,:·: ;: ,; .
,
4.~
'1'.2 "
·2.4 < 3.0
3.6 4.2' , 4.8
' 5.4 2.4· 3.'0 3.6 4.2 '
4.8
3.6 4.2 4.8
3.0
4
...."")
,
f 18'.1,2 +'12.88 + 9.25 + 6.57
=
= = =
+ 4,,44 + 2.68
=
-
--
3.74 + 12.SS 7.33 + .9.25 -JiO.. 79 + 6.57 .14.00 + 4.44 = 17.23 + 2.68 = 9.25 6.57
:::;
= =
4.44-
-
2.68
~.26
3.6
3.6
5.54 10..78 ]5.67 20'.24 24.47 28.38
2.82 + 5.55 + 8.21 + 10.80 +
3.0
,.
23.66* 23.66*, 24:92 26.81 . 28.91 31..06 16.62 l-~.58~
17.36 18.44 19.91
, 12 .07* , 12.12 12.65 13.48
4.2
+ 6.57 4.47 + 4.44 =
4.8 .
6.63 -t ' 2.68
4'") '4.8
3.75 + 2.p8
=
6.43
4.,8
1.62
+
-
4.30*
1.89 +
.8 .83* 8.91 9.31
~
=
4.44 2.68
6.33*
TABLE 10.10
Example 10.2,
s~ges
Concentra'don ,entering I
- 0.6
1.2
L8 , 2,.4
3.0
,
3.6 . 4·.2
-----' .
I to IV Cost
0..9
-'.
"\...
' 37.-18-+ ' 2.68 =, 39.86 , -' -
5.53 +,2'8.89 ]0.77 ,+ 23.66 20.24 + J6.,58
t!1
~,
'=z
28.38
12.07 8.83
.35.20 + • 40.70 +6.33 44.88
s: a=
D
a
S~34A2"
')1.43 36.82 4().45
44.03 47.03
+ 4.30 .. 49.18
DYNA\1IC PROGRAlV1MII..;'G
2.23
A TH'E prll r"' TJ~ F ~'lF XL .r~ R-"1 ~)} __ TIXP ,l', ..Lll..l'~ ~~_-l\[ ~lVII Cb·PR O,G R~IVj[pl]U\T G SO~l!l1I,O r~
10 _ nJ-
r
t
, ,- ,
Examples 10,1 and 10.,2 begin ,to. display a pattern of the n~r:-.r~ f?f ·t11S: problern, ,and tne rcharn<;:tenstics 9f the dynamic-p~ograrn.muJ.g solut!on-_ the solutions ,can be' show~ gra_p.~~~s~_I.~x~,~,J :~< ~ip,:;_ IO~~~ _a~d ,p_:- ~xar~ple, '~O:: l, " : , ~;... , __
.
'
c.l? 5
al l
0
e c:v
I
I
-'> 3
u
J,j
C
4
3
0
u
.f: 2 v
2
~
tl-
r
0
-
B
A
C
Stage (0)
D
E
I
2
3
4
Stage (b) ,
FIGURE'10-4 Graphic display of dYfl
-
-
5
224
DEsIGN OF
~ SYSTEMS
TABLE 10.11 }.-~lr§'t· taoRe of Example 1{t2
stage I . ""
','
!", • : : '.
:: . :
.'
:' '{<,':'
~
~ving stage • _.
':",
I • I. •
.
.
"
•
g:! ,,': "
>: ".": , ' ,;. ~"g~~:,;, ~ '
,:,;;.:.:'';t:;:eJtc ':,!.'6, ;'.' ;,) ~-
-
Cost, $
"Entering stage I
~:~ ;~ ' ,; .. . .. . '.
.
proceerung iOf1Vard, . '.1 I
i
._
~
. '.-
':;
, I , , ' '.
<",". c' ,
.'
.
. !' ."":'.:,.10
'1~:~~
,
"r
;~:~..:;;~.' '~:,:,:'.,o. ''~" • '\
,, '
' : "",
.,' ' ,,'
,'-:.0
~< '~.:o..
.
.
.., , .
,
TABLE 10.12 ,';"
',z~.,:,,'
-. ... ... , . ,' ,
.
_. . ' -.-
, , ' Second,table of 'Ex~m'ple ,10.2.ptoceePing forward, StageS I and' II . ~ving :' .
,stage n
'ThrQugh _, at position 2 ,
0.9 0.9
, 1.2 ,1.8
r.2
0.9
,2.4,
1.2 1.8 etc.
C~st~
$
, 3.73
+ + +
5.-53
5.53 = 10~ 77.
=
9'. 26 ' 16.30*'
10.77 5.54 = 16~31 5.53 + 17.23 = 22.76 lOr77 ,+ 10.78 ~ 2~.S.5* 20.24 + 3.74 ',= 23.98 "
used to' conlpute"a more detaileq table. The coarser grid was chosen simply to lessen the n~mber of calculations. After having established the approximate optimal p'a-th a recalculation 'could have been made using a finer grid ' but considering paths only in the neighborhood of the preliminary optimum. It has already be~n pointed out that the form of the ava,Hable data may in some prbblems make starting at the front preferable,' and in other cases' starting a~ the end and working backward. In Examples 10.1 and ~O.2 it was 'immaterial which direction was chosen. It should be realized, however, that the form of the tables will dilfer 'depending 'upon whether the progression is forward or backward, because the Sta,te 'variable will be different. If the progressi,on- in Example ID.2 were forward, the fIrst tab]e would hav~ th~ form shown in Table 10.11 and the second table would be as shown in
T2;Qle 10.12. , The state variable in TablfC 10.,12 is the protein percentage leaving . stage II, and once it has been detennined that the optimal way to achieve a . . protein concentration of 2.4. for e?Cample, through the first two stages (0.6 to 1.2 in the first stage and 1.2 to 2.4 in the 's,econd stage. as indicated by Table 10.12)., the nonoptim,al routes are ne~lecteQ. _' .
-
..
-
DYNAMIC PROGRAM!vlJNG
225
I I
i nlportan~
class of .problems i n ' dynanlic program.ming .is that of COD. strained GPti . nizati on ~ v.;here. a fu~ction y (J~:) i s' ·so ught . that J.; t. iniwizes a . 'sumrnation ::£g(y ~ x) ,but in 'addition SODle other summation is,. 'specifie'4 ' .: ""'Ih(y 7X ) . 11,' ~!here.~the. furtctj~ons gl;,.l7;-, ·and the ' uurrieri.callenn Ii are '. :~" '~:~'~"'Iu'1oV'/n ~-This' ct~.ss tif',~:onstralli~d"problen1s·;:Vi.il,rb{:'ifeate'tti"IJi'>(~~pi6F·'i8·:(·':·:··~· .;:· "".' ;'. .A.noth~Jf ~t~ss' of problems-;at':fjrst gla~ce·:·:uray~'seeni.'}c{ be '.corrs'tr~in..~!;L' 'bUt . ': they can be c'C?flve"ited il1to. a fOnTI l.denti,cal to that u'secLin Examp,le? 10.1.- : . arid 10.2. This class may 'be called (ippilJ~elltly ·coJisirained. and illustrated" by Examp~es 10.3' and lO.4~ ' . .
. . Ji.::-D
is
Exa~ple 10.. 3. ,An ~vapora.tor whIch boils liquid
inside tubes consists of faur banks of tubes. pach. bank; cOf),sists C?f a llu~ber of 'tube? in p~lle.L and the . ' banks are connected in series. as,shown in Fig. 10-5~ A mixture of liquid and yapor 'erHf~,rs' the frrst bank 'with' a fraction of. vapor x =. O.2~· and the 'fluId .' ··Ieaves the evaporator as saturated vapor.,.x ::::: 1.'0. The' flow rate is' 0.5 kg/s, and :each tube is capable of vaporizing 0.0 I kg/s' and thus of i~creaSing x by 0.02., . . '" r
Forty ,tubes are to be -arranged in the banks so that the minimum total pressure drop preva.ils in · the' evaporator. The pressure drop: in a bank is approximately proportional to.' the square of the ve10ci ty . and 'a satisfactory expression for the pressure drop flp is ' J
.. (10.1)
-
5
Bank pi OUller
.r
=
1.0 4
Bank HI
3 ..
-
--
Inlet
x
= 0.2
If
T Aow, rolte
:2
0:..5 kg/s
FIGURE .10.5
Evaporator in Example IP).
--
,..
II
Bank 11
2
Bank I
226
DESIGN OF TIiERMAL SYSTEMS
where x j = vapor fraction entering baPJe .' n = nqrnber of tubes in bank
.)
" \ ,'
,I·
. Use dyu?JI1Ic prog~arnuring to dete:rmine the distril;:mtioJl of ·. the 40 tubes S9' · ~at- the total "pressUre 9rOp in ~e evaporator "is Iilininjmn.
.·. So~OJll. Selection' 'of Ltl6·:nrimben of tubes'-ip: ' ~. stage: (bank)' as the state : .:-:~ . .:;~-., .... .: .:;;,,,:!~'~:~J' vaxi~bl~, .is_.unpp;jda.Gtil!~+',l?e~use.·.,tb~_t~onjpa1 points iri graphs comparable "to '
. ., .
.; \.~,.; ..' ~ :-'..::' '~ig~ '1'0-4 ar~ not ijXed~~ Ftiith¢rmori_~':'i~~~ ~b~ice~. o~ P~~?;f4;?£~tupes /~~,:tb~~:_ ':~" . ·d~es
. .' s'tate vatiabl e ·availab.l e. .
.l?'ot a.ccoWl t' 'for the. Cpp~trahlt · th~t-· pr~~lselY-·: 40" t~QeS.:~~e;:·~:.'· ."":" . ,'. ' . . .,
The difficulties .-are .overcome by choosmg ~as tb~-state variable CUIDU.... ' Jative· tubes c0111IIlltt~d, which resuJ;ts iri coordinates as shown -in ·Fig.· 10-6... ....After stage 0 (before stage I) no tupes hav~ been -c9mmiried, and following 'stag~ TV' 40 tubes have beeo' corrimitted. · Table lO.I3 .shows pres~ure drops 'for the first bank or stage for several ..differe.nt choic~s of tubes. Table 10.14 uses as the state vanable· -.the total number of tubes committed in the :frrst two stages and permits an optimal ". selection. For e~ample, if ~3 tubes an~ used . ~ the. first two stages, the op~m.al distribution is. to allot 5 tubes in bank I and 8 tubes in bank n to achieve a total . ' .
.
~
4O~ [I
-'_3D :.r. ~
20
'!..J
>
-
10
,'--' 1
. FIGURE 10'·6 State variable of cumulatiye IV number of lubes commirted in Ex.ample 10.3.
III
11 Aft,er stage
TAI:LE 10.13
Exanlple
~Oe;3,
Total tubes. c,o mmitted
stage 1 'lUbes in
Total IIp; kPa -
stage J .(
~
2 3
·3 4.
5
7.20 3.20 1.80 1. 15
.6
0.80
,4
S
6~
-.
DYNAMIC ~~ROGRA IYfM ING '
217
"
1 {L3~,
2xanlple
,
trtages I and ]] , '
~--------------
. "
,
12
, O~'80 -+ 2.05 . ~. 2J3~ ~ 1.15 + :L32 = 2.47* , 1.89 ,+ 0'.88 = .'2.68
{'}
'7 8
9
3.20 ,
13
7
8
.9 10 14
7 8
9 10
1.5
8
9 10
----~~------~----~--
+ 0.60
= 3.68
~ .
,0.80 +, L50' = 2.30 LIS, +' 1.01 '= 2.16* " 1.80 + ',0.73 = ~.53 3.10 + 0.:49 = 3.69 0;59 + 0.80 -j1.15 + ,1.80'+
1.70 l.tS 0.80 056'
= = = =
2.29 ' I.95'i! 1.9S::' 2.35.
0.59,+ l.JD.= 1.89 1.15 + .0.8.0 == 1.7.1 >I: L 15 + 0.65 '= 1.80
, 6.p in banks I and n of 2..16kPa. 'The pressure drop in 'stage 11 is computed
from Eq. (10.1) using x/ = 0.20 + (0.02) (number of tubes .in stage I). Table 10. 15 determines the optimum number of tubes in b,ar:U< ill for various total numbers of tubes ' in the frrst three banks. Table 10.16 is the ' ,. final table 3J1d indicates that. 17 is,$e optimum number of rubes in bank IV. The optimal distribution of tuhesis 5, 7 tIl. 17, resulting in a total preSSJJre drop of 4.71 kPa.
A· further illustration of the solution by dynamic programming of ·an apparently constrained problem is .shown in Exatpple 10.4 in the optimizaIrion of feedwat'er heating. Heating the boiler feedwater with extraction. steam, as sho'wn in Fig. 10..7. improves the efficiency of a sream·-power ' Icycle and is a common practice in l~ge ceutraI. power stations. Some plants - use, more· than half a dozen heaters, which draw off extraction steam at as many different pre'ssures. That feedwater heaters improve the efficiency of a cycle c,an 'be show.n by a calculation of a,.specific case. but a qualitative explanation may provide a betterv sense of tNs improvement. First, recall _ •
-
'
228
DESIGN OF THERMAL SYSTEMS
TABLE 10.15 ~xample 10.3, s:~ges
10 ' Jl ,
24
l
....
,12 ' 13 ,
25
10 II
12 13'
26
II 12 ,13
TABLE lO.}6
Example
I
an~
TIl, , ,;
) "
)
,.' , . 1.95 +. 1.66 '~. 3~6i ", 2.16 +.1.26 = '3.42* 2.47 + 0.97 ='3.44 2.95 +' 0.75 = 3.70 1.71 1.95 2.16 2.47
+ 1.80 = 3.51 + 1.37 = 3.32 · + 1.0'6 ~ 3.21:~ + '0.82 = 3.29
1.71 1.95
+ + +
~.16
'
1.49 = 3.20 1.15 = 3~OO* 0.90 = 3.06
,
10~3,
Total , tu,bes
stages I to IV
Tubes in IV 13 14
40
15
]6 17 18
Total Ap t kPa 2.93 + 2.3.3 = '5.26 3.00 + f.90 = 4.90 3.22 + 1.57 = 4.79 3.42 + 1.30' = 4.72 3.62 + l.09 :;;:: 4.7J· 3.86 ,+ 0.9J = 4.7.7
that in, the steam-power cycle approximately 3 J of heat is supplied ~t the ooiler for every joule of work al the turbine shatto The difference of 2 J is the am·ount rejected at the condenser, which llsuaJly represents a loss. The proposal to try to use some of ,the heat rejectled at the 'con.denser for boiler-water heating .is doomed, because jf we tried, for example. to' heat
the feedwater with ' exhaust steam' from .the turbine, there w<;>uld be no temperature difference between the exbaust system and the feedwater to provide the driving force for heat transfer.
-
-.--,
. .......
450 C, ~800 kPa 0
L.
SLfperhemer
.. ......
~'
c
o
FIGURE 10-7 ,
'Se~e,cdon
.
.'
"
....
."
'I
'"
............. '
,-.
of optirpum areas of feedwater ~eaters in Example lOA:
Extraction steam, however, has a higher temperature than exhaust steani and can be used for the heating', Concentrating on 1 kg of extraction steam leaving the hoiler"we find that it perfoIms some 'wo~k in the tUrbine" before extraction and then uses the remainder' of its energy above saturated' ", liquid -at the condensing temperature to heat th~ feedw~ter. 'I n effect,: then, . all the heat 'supplied to that kilogram C?f steam in the boiler 'is eventually converted int9 work. The practice of feedwater heating by extraction steam raises the effectiveness of the cycle compared to rejecting 2J of bo!ler heat ?er joule' of work. ' It is further to be expected that the high-pre~sure ste~ is more valulble than the low-pressure steam because the steam extracted a.t high pres:iure would have been able to deliver additional work at the. turbine shaft. Example-l0.4. An economic anruysis has determined that a total of 1000 m2 of heat-transfer are'a sbould. be used in the four feedwater he~t.ers shown ...in Fig. 10-7. This 1000 m2 can be distributed in the four heaters in lOO-m 2 increments. The overall heat .. transfer coefficient 'of all heaters is 2800 ··-tr W/(m 2 • K).The cost , of heat at the boiler is 60 cents per gigajoule. and the worth of the extraction steam determined by thennooynamiccaJculalions is . listed in Table 10.17. the flow rate offeedwater is 100 kg/sa I
Use dynamic programming to' determine the optimum dis,tribution of. .
the ares. I .•
SolutiQ:11': .~ b~ginning the SDlution j,t may be instructive to try to predict . lhe nature of the 'op'timal solution. It is desirable to lise the lowest-cost' stearn
~.
230
DESIGN ,OF THERMAL SYSTEMS
"llitilLE [email protected]"l
, E~acti6n-steani data
J
)
futra~tion point
and
.i/ ';" ,~: '. :~ "bea~er n~ber- ', '
temperafure,,
, "
,
.
Worth of e~"traction " steam, c~n~ p~r g~g,~j()itle ,' ~': ': "
Sqturation
°c
90, ,'
' 23
'
·'· 'i'·,'J:'::"!~:-:'-" " "';~') I}:;' .-,,~:. \~ '. ", 215'
47 '.
. .
,.":. .... :':.
' ..
,.
. ,/
.pOssibie which .wocld' suggest' 'a laIg~: i? st?ge_ ~'J. but each (ld~itig~~l unit.of area in that ' stag~ is less effective than the previous' unit area becaus·e,. , the temperatUre' difference between the steam and f~edwater is less. (nUS' , . there . mu'st be ~ co~promise betwe~n trying' to yse the low-cost stearn and ' . maintainwg a high ~empex:ature djffeience. . ' : ' This problem is apparently .c onstraine9 ·because the total area is, spe~ ified, hl:lt it can be convertdf iDto tHe" unconstrained form by using a~ the '. , 'state variable" the total' area co]Wriitt~d. .In this pr'oblem it- is advantage0I:is to start at the fr0iI1t (with respect to the fee~"?Iater flow), because the -in}et t~mperatu're of 'the feedwater is known there (32°C). Table 10,.18 shows the out~et temperatures from stage I for various areas in that stage. As is typical , of dynamic programming, this frrst table is ro~tine. The temperatures are computed by use ofEg. (5.10) for a condenser, and the saving is the value of the heat saved at the boiler less 'the cos,t of extraction steam used. Table jO.19 uses as the state variable. Jhe total area comminedin the first two stages. If, for examp1e, 1000 111:! 'is available for the first. two stages, the optimum distribution is to allot 400 m 2 in the first stage and 600 m 2 i,n the second, restIlting in asaving of $1.285 per second: ' 'Table 1,0.20 shows various ar'ea q.istributions in the firs,t three stages.'
area
9
TABLE iO.18
Example 10.4, stage I Tota) area coriuni. tted~ m:!
0' 100 200 300 400
SOO 600 700
o 100 ~OO
300 400 500
600 700 ' ,800
, 800 900 )000
Area for stage I
-.
900 )000
Saving per secon,d
32.00 60.27
7,4.76 82.19-
$.000 ' .438
.663 .778 .837
86.00 87.95 88,.95 89,.46
.89J
-89.72 .,89.86 , . 89.93
.895 :897 .898
.867 .883
DYNAMIC P3:0GRAMMn-.rG
.231
I
Total ~L ea
.
Area
on II
'. .
o ..
".
. :;: .;'>:1 :'~"- '~'.~_: .\.' .. ., ' . l(Hl·.:,
.
-
~.
200
:".·-'.:'.:·J2:~.6p,·..><:.:.;:.:::.::~;$;~~Db?~.·\t::·. '~-:' .,'. ..:.-:. '~.... ::": ,~ .. : ., '~ .... .. .' ,. .~
....
•
'
·~OO a!l
4:0
..
_
•
__
•• • •
1000
_
.,.
0 100 200 300 " 400 500 600 700, . 800 900 1000
, :.. , .:,
•
,..
If.
•
,.
7"
74.89 '
.• ~. '.575*'
74.76 89.38 . ·96.88"
.663 :.829 . '.870*
.....
'••
,.
_
••
_
••
_• • • •
III
"
... .
~
:
."IlS.8})' 117.S6 f lS.S7 . . 119.38 119.65 09.18 1.19.85 . 1i9..89
.• •
,
TABLE 1020' '
Example 10.4, stage I to III Total area
Area
inID
O·
Savings per seco~d
13
~OOO
0
32 ..00
100
0 100
74.89 94.39
.575* .575*
200.
0 100 200
96.88 116.37
.870 .957· .870
................... 1000
126.37
,.
.. . ....................... .. . ~
o
H'9.3.8
1.~85
JOO
138.88
1.462
~OO "·
1.49.14 154.26 157.02
300 400 500
,
$
600 700
800 90(l'
1&16
158.37 159.13
-15'9.48 J59.70 159.79 )S9.84 '"
c
1".548 I.S90· . 1.598·
1.597 1.566 " ).537 1.449' L358
1.178
'i'. "
"0
.. ~.: "::.; .".:',
:r Oo .
'
.898 .. r.094 ' 1:.194 1.245 1.270 . J •.2~2 , i.285* 1.280 . '1.267 1.237 1.178
89.93 . . 104.55 112.04
.
..a
.... ,. , 60~-2-h.; - -,', .. : .' '-A38
-:0 100· .0 100
......
. '..
...-...'
:.
.~
~
'.'
S-avings P'ci- secon.d
.....
•
••
'
....
:.
,"
;.
232
DESIGN OF THERMAL SYSTEMS
•
TABLE 10.21 . Ex'~ple 1~ ~4; .
-
•
stages I
t~
IV -
I
,:
I
,Savings --
~h,~tr~~~';f} "
,
"
J
"
I,
~r -seco~d
.157.02 : ::" :: ~ -, $L598 .'
, "
':';•.
': ~~;'\"L":'' 'I'\',n',_.~}::'(::-:» , ~ ~[{~:j':,:,~rJ;.:~~:: ~;:_' . i:~~;"~'-::S'(~',<~'~"~~1.:'i: ;:; ".,' ,.:::'i:!." "': ' ;:,,;;-;:luu. '-
: " ,. ~ '
' ,
'106.72 '" ' "
',L804*'
"
" " . ~:,'; '\ ," ,<,,-'ziO.68 " ',: ' ,,1.790'
:'i.,"~,:;!':·:;:i':,.:'.,:!:"-" \"
"
:$00 ,' ~~'- ' ,' '-,' ~
, 21-254,', ' ', '\ -I~ 773
"' ,.': :,: ,:, " .;:,: ~'§fog "< ' ;:E~ -
",,,,~,,, ,,,.,,,,,",,
, '" .- "' :' '~ ',> "": ' ,800 '
, ' " '~"," '"' ,,, 900
'214.5-3 "
.
214.66. ,, - -
!::~
,1.492 ,', 1.336
'
1000 ,
214.77
..
" ,
'.
'"
.996 '
-
...
.
.
,Finally, Table 10.21 where the full , area of 1000 m 2 'is coInm.itt~d, inclicates that ~e optimum distribution of area is 100, 300, 300,- 300 -for a .- total ~~ving <;>f $1. 804 per s~cond. '
:'
.", ' ,' ,' , ,
10.7 SIJI\1IVIA,RY , "'hen optimizing a system that ,consists
of ~ chain of·events or components
\\·h.ere the output condition from one unit forms the input to the next~ dynamic programming should be -explored. In large problems th~ amount of calculation may be extepsive , even though it repz:esents o.nly a fraction of the effort of conducting an ,exhaustive e'x ploration. The systematic nature of dynamic programming lends its~lf to d~velopment of a computer program to perform the calculations. The major challenge usuaUy appears setting up the tables" and especially' ill identifY,ing 'the, s~te variable., Chapter 18 'w ill revisit dynamic. programming and focus on its rela-,tionship to calculus, of variations. The ins,ight provided by calculus of variations sugge,sts proce~ures by whicJ1 ,dy.namic programming is able to solve constrained optimization problems. '
in
I
.'.~ .. "'!:- ,: ~.
'.
,
\.'
PROBLEMS -19,~} ..
The total pressure drop from point 1 Ito point 5 in the multi-branch duct
system shown in Fig. '10-8 ~ to- be 500 ~. Table lO.22 p~esents the costs · for various duct' sizes i~ ,each of the sections asa function of the pressure drop in the section. Use dynamic programming Ito detennine the -pre~sure drop in each section that results in the. minimum total cost of the systeIlL , Ans.: $599. 10.2. A truck ,climbs a hHI that consists of Wee sections. The fuel consumption '
-- ..
~" ~jn
,each section is a fune'don of the time required for the ' ~.
'
di~~ce
in the
DYNAlvllC P~9GRAMMl.N G
1.-<<,- - -- - - -
! I -- , -.
.
. fi}' ,
- ·-· ·~;;J;.~~..:r
..
.
.
=.. "".~~.;= ~....._; ,:~":::.L~ :_,:y~~_=_...,;.._::::_.:...:.:.~.. ·~~"l':..';::::::;J.:::::=:::.. .....::.=o;.~ =_= !".:.:.~-:- - _ .
c
'"' 2
'--~l
SOOPj - - -
J .
233
~-
';'~ ~~ ~
--
3
,
FIGURE 10-8 " Duct ,systeminProb.lO.L -;fABLE ).tL22
.Pressure drop and costs of sections 'of ,duct in Prob.' to.-1 .
drop, 'Fa
.Section
Pressure 100 150 200
" 222
,1-2 .
100
150
J80 166
200
157
,_-.J ',
Cost, $
205 193
4-5
..
135 '
100 3-4
"
150
'125
200
117
100
93
150
' 86
: 200
81
section to be covered. as ,shown in ,Table 10.23. A ,total of 25 s, is availahle' for the climb of the hill. Use dynamic' programming to determine the time allocation to ,each secEion that results in minimum tota] fuel consumption. Am.: '] 19 g. ' 10 . 3. Use'dyn.amic programming to detennine th~ fl ight plan for a .commercial airliner flying ·between two cities 1200 km apart so that the minimum. amount
of fuel is consumed during the flight. Specifically. the, altitudes at locations - . c:~ th!ough F. , ·~n F~,~., . ~O~9' ,durin~ the cOl1:fSe. of ~e fli~t are to be sp.ec!n~d . Table JO.24 shows the fuel consumptIon for 200-km ground distances as a function of the, climb or descent during that distance. Determine the flight plan and the minimum fuel cost. Ans'.: 2770 kg. ' " 10.4 A minimum-cost pipeline is t,o ' be constructed between positions A and G in Fig,. 10-1.0 and can p,a~ through. any 0'( six locations ,in the successive ..stas~LB'.,_C, D. E. and F.
-
.
~
, (.
234
DESIGN OF ~ SYSTEMS
TABLE 10.23
".
~
,
,Fuel cOIl$nmptiou' in various sections of hill clit"Ilb in Prob. 1002 . Section
Fuel cOilsunlption, g
Time7~
,7
40
8
, 34
' -: ': , -;;,': ~:' . .:~". :,;,\ . ~;,;.}, , : ""~' .,' , ,~ _9 , "-" , , :)~
' 29
.'\ " , '
.' _.
••••• •
1 "
,. ,
.": " A-B
.. '. .
9
' ,45
10
38'
7
49
8 '
41
' .
. ~ ' ';'' .~ .'
.,
,
'
, ',
30
r------.=--~-__,~~""'---'O~__,.==~--__,
8~OOO
----r------t--~~__i>-----......,j
p , ..
6.000
a-------+-_---I------:---+----.--~.
4.000 ~ I ,2.000
,,
35
9 10'
]0,000
' '
61' 52
(:-D
'
~,, ~,.:,,= ~- .\',::~; ~ ~~~~::~I; '_'.A': ' ; ~-'-: "; ,~ ;: ":':";'~:;~'I~"'~: '<:-:. ·,:t'::< ',~ f::.. ,\,>. ,'",: ,.'--"< '~ ' " ,-~~,;" , ::
7 8
" -B-C
"
_
_
_
~!!__--'--~_+_---'------ll~,~--------t
I~--~~"'""'------!f__~~~=t_---_II'
Destination
Start
I--
c
B
A
200 Ian
D
F
E
G
--I
FIGURE 10-9 ' Altitudes and distances in Prob. lO.3.
TABLE 10.24
Fuel consumption per 200 km of ground travel, kg To altitude, m
Fror:n a!t~tude, m
0 2.000
"0
300 120 0 0 0
.4.{)OO 6.000 8.000 10.000
-- .
2000
4000
1500 600 J80
300
60
120 30
840 200
0
O· ,.
6000
8000
10,000
l~OO
2070
1000
' 1500
2300 1950 1160
2300
0
2500
1730
600
1200
80
ISO
600
0
60
90
DYN AAJlC F1~OGRAI'''1 'M 1NG
. &
235
)!
5 1 .
.~ " ,
. ....
J
•
".
" , '
.....
.. ~ of , ' ,' ,
'
•
• ,'
__ '
-
_
_
• ""
. .... .
,:'
~ ~ ... . .
:'••
: '
J
• •- ....·01
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3
G
~
20
'
I }D . .
.
,:
:..
...
,
'-
,,
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, ' , ,"
,'
;
@
e C
. .B
.
G
~
@
D
E
F
FlGURE 10..10 P~pe' line route in, Prob. 10.4.
(a) If an possible combinations of roures are' 'investigated, how many different paths must be examined? (b) If dynamic p~ogramming is 'used, how ·m any cal~l~tions must be made .' .' ·jf one calcukmon·consists of one line in a. table? Ans.: 7776 and ~56. 10.. 5. The maintenance schedule for a plant is ~o be planned so ~hat a maximum ,.total profh wilI be, achieved during a 4-year span that: is part of the life, of the plant. T~e iilc
Maintenance expenditures made at beginning of year, thousands of dollars ~ncome
level
.:anied over 'roni previ.oU3
-
,
Income level during year
:ear
$30
$32
$34
$36
$38
$40
>30 32·
$.2
$4
$7 5 4
$16
$-23
' 13 10 8
14
2
$11 9 '7 5
')
'2
6
10 9
,.,0
I
4
8
34,
'2 I
36
0
38
X X
.w
-
-.
:,q...,
3 2
J 0
X
-
18
.. . . ,'
.'
.. . \
,236
DESIGN OF THERMAL SYSTEMS
.. ' ". >
37SC
.
"
,-
. "'- .
-..
.: -: ,•..--:. !'.I'<:': .. '
FIG1)RE 10~11 Chain of ·heat exchanger~ 'in Prob~ ' ~0.6 ..
maintenance expeJlses' are made is $36,-000; and the..Income.level speci5ed during:and at ,t he end of yeaY'4 is to be $34,000. The -profit for anyone, year will be the income during the year less the expenditure made for maintenance the,beginning of the year. Use ,dynamic prograrilming to determine the pl an for maintenance expen:ditu.res that results in maximum profit for the 4 )lears: , , , Ans.: Max.imum profit = $130,000. 10.6. Four beat exchangers (or fewer) in series, as shown in Fig; 10-11, are each served by steam at a different temperature and heat water from 50 to 300°C. The sun1s ,of the first costs of the heat exchangers and 'p resent worths of the lifetime s~eam costs are shown in Table 10.26. Use dynamic programming . to determine the outlet temperature from ,each heat exchaI)ger that results in
at
-the minimum total -present worth of cost:S. Ans.,: $254 ,.300.
TABLE lO.i6
Present worth of heat exchanger and lifetime ' thousands of dollars ' Heat
Inlet temp.,
exchanger
°c
100
150
200
250
300
50
0
520.8
558,. 0
X ,
X
X
50
o
23.1
62.6 36.1
5,0 100 J50
, SO tOO· 150 200·
150 300
----. ---
°c
50
200 4
of steam,
~ullet temperature,
100 150
3
cos~
0
o
2·LS 0
. £132.3
X
X
.93.,6
X
X
0
62.8
X
X
79.9
129.9 94.3 51.2 0
$),77.0
$308.3
141. 7
266.7
103.0 57.6
223.6 176.4
41.1 0
372.4
309.3 243.7 173.7 94.4
0
. .
•
. .
~.::;. .-
237
If:..
DYNAl'vlIC PROGRAi\Ill-All'\lG
j~ cooling pond serving a pO~iver p laJ~~ is Ciq1uipped with Circli}l.a.~g pumps and sprays to enhance the rate of heat J:'eje,~tion fr~nn the pond. F'Urth~rmore ~' .' the pumps are to'·be opew,tE;d so dIat the. heat rejection is accom.plished wiLf-t n1inimal pumping energy. 'On one p~1:icular day the pond' temperature is ~8aSC at 18boho~[s, and i;he P'?~~ , teinpe[ati,lre)s t
of
,,\leu.as
~,
. "; ' "
'
",
'~"'-\'
"., ~
.~ ."-
the
Ans.: 1533 mis. 10.9. Hydrazine', N2 H4t is a possible fuel for an emergency-use gas turbine' because in the presence of a catalyst it decomposes in an exothermic reaction'"' . .
TABLE 10.27
Pumping epergy remperature difference
Drop in r", in 3-h period, ,oC
It start of 3 .. h period,
" - twbt
0
.5
0
0.5
1.0
16 12 8 . ·4
') ,L.
~,O
1
:.5
0
:.0 _ :.5 ·.0
0
-.5
'.0 :.5 ·.0 I.... .b
..
, ;z temp:er;srurt: of waler if! pond'. °c = wet-bultrte~!"3l'tlre or L'Tlbient air,. °c
.. -
1.5
2.0
2.5
54
'X·X
X X
45
36
96
27 18
80
X 166
64
139
9
48
0
32
111 83
16 0
55 27 0
238
DESIGN OF THERMAL SYSTEMS ",
, TABLE 10. 28 ,
, Increase in velocity in
3i)-~
intervaJ,. r;nJs )
, Fuel b~ed
hPtial " . " ', mass of. ',: " ' fuel, kg :
" 2000
1000
5000 -
4000
3000
,
in 30 s, ~g 6000
'7000
I
.~;,.;.~.:,., !~~.~ .' . .:;"§:..: .~:~~.,.~!J~:~ <;":"' ~!~·r::.".·);·7;;;::. .,. :'' i~:~'r. . +"~ ',.:."'~~'. ·~. ,;.... :,.,::.•'..'.• 5.000 -
312
152 143 136
:,.J 6~OOO
. :7.000 ,8.000
1~1 127:
9~OOO 10~900
-124
486-
288
445
269 253 241
412 385 ' 362.
231
,;343 -
679 ';:' · · ' 895 . 61-7- - ~ '808 . ' 567 737 525 679 -491 630 462 589
, 1024 "
- 926 '
847 , 781 726
1140 .
1034 '/
947
875
of
The rea,ctof is to,consist four stages; as 's hown in Fig. 10-12. A particular reactor is limited to a tota(of 14 kg .6f cataiyst"which is available'in I-kg packages. Each stage can accommodate 0 td 5 kg of catalyst. The fraction of hydr~zine undecomposed ~' stage is given by
in
)'0 -=--= '0 . 5 "
)';
where
+ '0 .e 5 -m," ' ·'
,
= fraction of undecomposed hydrazine' 'e ntering stage , Yo = fraction of undecomposed hydrazine'leaving stage 111 = mass 9f catalyst, used ir.l _stage )'i
Use dyn'amic progra!llrning to det,ennine {he optimum distribution of the, 14 kg of catalyst. To shqrten the 'c alculation effort, construct the tables only in the I?eighborhood of the answer given below. Ans.: 2, 3, 4, 5 , " 10.10. Some' impurities in industrial waste watei can be removed 1 by passing the 'water through a series of adsorbers contairung activated carbon. In a cenain instal1~tjon the adsorb~rs are arranged in four stages with waste water having
Stage I
-
I
I I
I'
I caw,ys,li
Pure J hydrazine I ~J
I
,
.
,, I
III
I
JV
J
I I I I I
C]rD 0 '
I
J
,J
I
J
0
2
f1GURE 10·12 Four-stage hydrazine reactor.
-
11
I
, I
f NH 3 • N 2 , H] : and undecomposed I hydraz.ine
I
I I
3
4
~
DYNAM1C f~OGRAMivIJNG
239
an initial contamination of 2'000 ppm. e:nlqring the first stage. A total of 140 kg of activ(1req carbon i~ available ·to be bi~tribut~d. in the four stages in qup ntities of 10.20,30, <-/·0. or·SO .kg_iri each stage,· The ·perromlance of . each stage is. expresse~ by
.. . where x 0
~o= X;(;'~';~" +":W .66~t,:booj f. ·:. :.; ....;"..<,~:p;:..:. ..
=
contamination 3'{ oUrlet of stage. ppm .x;.= contaminarion at inlet of stage. ppm In = n~as·s ~ facti vated carbon in th~ .s.t~ge. ~g
Use ~ynamic programming to e·stab'Iish ·[he distribution. thar results in· Ithe ··· . minimum contamination of· the waste water leaving· stage 4. ·To shorten · the calCulation effort~ construct the tLlbles only in [he neighborhood of the ·answer given belpw. Ans~: 10, 30, 50, 50.
REFERENCES I, P. R. KJlinkowski. "Ultrafiltration: An Emerging Unit Operation." Chem. Eng., vol. 85~ no. ll.pp. 164-173, IvJay6.1978. 2. 1 .. L. Rizzo and A. R. Shepherd. "Treatil;lg Industria.! \Vaste\~·nt6r with AG£i\'~ted Carbon:' Chem. E.Jlg. • vol. 84, n~, 1. .pp .. 95-1b.~. Jan. 3. 1977 .
•~DITI0NAL READINGS Bellman. R·, E.: Dynamic ProgT'Qlnming, Princeton University Press. Princeton. N.J., 1957. Be]}man, R. 'E .• and So ·Dreyfus: Applied Dynamic Programming, Prince£On University Press, Princeton. N.J., 1962. . Denn, M: M.: Optimi:alion by Variariolla/ lYlethods. McGraw-Hili, New York; 1969. GJuss. B.: An Elementary IJlITOductioll10 Dynamic Programming;" a SWle EqlWTioll Approach. A.flyn and Bacon. Boston. 1972, . . .. . Hasrings, N. A. 1.: Dynamic Programming with Jrlallogemellt Applications.- Crane, New York. 1973. . ' ~emhauser. G. L ..: lmrodllction 10 Dyiulmic ·Prqgrammillg. Wiley. New York. 1960.. Roberts. S.: Dynamic Pro,gralllming in Chemical Engineering ~)Jd Process Control, Academic, New York. 1964.
I ..
,
- . ..
-
.
'
I
'J
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~~;..: :.1 ,: .'.~';
,,-
I
'
.-
11.1 INTRODUCIJON Geometric prograrnming is one of the· newest methods of optimization. Clare'n ee Zener fust recognized the significance of the geometric and arith- , metic mean in eases of unconstrained optimization. Since then others ~ave extended the methods to accommodate more general optimization problems.. ,The form of problem statement that is particularly. ~daptable to treatmen t by ,geometric programming is a sum of polynom~a1s for both the objective function and th~ constraint equation~~ These polynomials can be ,made up of combinations , of variables of either positive or negati.ve noninteger or inte, ger exponents. After seeing the usefulness of polynqrnial representations in Chapter 4, we recognize that the ability to optimize such functions is clearly of engineering impprtance. A ' feature of geometric programming is that the first stage of the solution : <) to find the optimum value of the function rather than first to determine the values of the independent varjables that give the optimum. This knowledge of the optimum value may be all that is of interest, and the . c~lculation of the values of the variables can be orpitted . I
This chapter first presents tl\e form of a geometri~ program,ming prob· lern and then defines degree of difficulty, because problems with zero degree ,o f difficulty are jdea]~y suited to geC?metric programming. Furthermore opti. mzation problems with degree of difficulty greater than zero qre prob~bJy . b~st sol ved by some other method lof optimization. Next the mechani cs of the ~ethod will be lll~strated with several examples. Gradually this chapter I
240
-.
GEOl','lETRlC PRGtJRAMMING
241
weaves in some explanation ~nd proof of geoI{lerric programming so that , the user win have confidence ip a technique that at firs.t may seem to be a dark art. . Unconstrajned optimizations are the !rrst · ones attacked, a~d later ' the -study ill.oves to constrained .optinlri.zation vilth· -equaFty .. co.p.~ traints. · ,. , . " GeometnG" ,:progiam'ming;_~s;'ca:pabl~'-8t::.optimizing<·"O'bJecti'te~'·flhictidrl's'·~s,{itr~~;:·~ ~'~';/;·'~'i :.·. , jeet to " ~~ei:lu'ality cons:t~C!inrs'~ -but this'"appli¢ariorf;isJ5eybnd",the' .o(hl:i.s,,:::' · .- . .<.,. . , -introductory chapter., ' , . '. . ;.
scope
, .In. ad~. iti9n to providing optimal values
or the objective function a nd
lndepe'ndent variables geometric . programming "supplies additional ins,igh t into th~ solution. For example, the solution by geometric prog~mming ~lso shows h.ow 'the rota~ cost is divided among the various contributors. y
11.2 FORM OF THE OBJECTIVE,' 'FUNCTION AND CONSTRAINTS
'
Geometric programming is' .adaptable' to problems vvhere the ''Objective func- , rion, and con'straints are sums of polynomials of t~e 'variables,~- 'The variables can be taken to integer or noni~teger posit,ive negative, exponents. The following examples of unconstrained objective functions can be solved by geometric programmin~: .
or
Minimize
Y = 5x
Maximize
y = 6
. Minimize
-V --
+
10
+' 3x
?:r ........ I "~y2 2
(Jl.l)
-Jx -
X
La 5
+
.JXjX2
(11.1)
+
l 'Q.\v.'O.8. I
4....v ::?'
(11.3) '
An example of a constrained optimization adapt'able to geometric pro--::,', gramming is
Minimize subject
to
y ,
11.3 DEGREE OF
=
Sx I
JX;. + 2xf .+ X~12
XIX2
Cl1 .4)
= 50
DIFFICU~TY
Duffin, Peterson, and Zener l define degree of difficulty when ,applied to geometric programming problems as T - (N + 1). where T is the number . of tenns in 'fhe objective function plus, ,those -in the cons'traints and N is the number of variables _The degree of difficulty of the objective .function in Eq. -( 11. J) is 2 - (1 + 1) == O. The degree of difficulty of the objective function in Eq. (1 1....2) is also zero,. since only 3x = x 1'.8 win be ,optimized. The degree of d.ifiicul ry of Eq. (11.3 f ,is unity. In the constrained optimi.zation
242
DESIGN OF THER1v1AL SYS1EMS
• of the objectiv~ function in Eq. (1 i.4), · the pumber of terms is 3 + "j =: 4 (three in the objective func.tion and one' in the constraint), and the number of variables is 2,. so the degree 'of difficuI.ty is 1. . .}¥hen the degree ·of QifficuIty is zero, geometric prograinming is often ·the simplest method· available for solution. For·degrees ·of difficulty greater . ..•~~ .':'.' I than· zero; .geometric : prog~rirlming ~ work~ l ~ut the method ' involves." ': ' .::...... .:. ..·fbe . so1.uti.on...9f~no.n,Jjnea.r.,e.g.!!ati9ns,.· whJch will probably be ·more time~ ' .consu~g than if·some·~.oilier )D:eilioi:f~.~:~~tich7;J as.:':)~:agra.ng~\·.Wultiplier,~;,..js ..:..~ , u~ed . :Hen'c~forth' in this ·c hapter, .. only pro.bi~~~·of . ~ero ·. degm:·:o(drfftbuttjr:~:~·. : will be .c ons] dered ~ . '. .
11.4 MEC~CS ..OF SO'L UTION FOR ONE INDEPEND·E NT VARIA~LE, UNCONSTRAINED Befo~e
providing the support for 'the method, the. PJechanics will be pre. . sented through s~veral examples. The optimal value' y ~ will be sought for . . the function (11.5) The individual terms will be designated by the symbol u; thus
Geometric .programming asserts that the optimal value y * can also be rep, r~sented in product fonn by an expression' that we shall call g *,
.
y
~ -g _ *
_ 1( Cl X,DI )WI("C2XaJ )~ "'2 .
-
.
p'r ovided that
and'
WI Q1Wl
.
+
W2 ~
+ a2w2
W2 .
WI
(11.6)
1
(11 ~ 7)
= 0
(11.8)
A consequence of Eq. -( 11.8) is that ,the x t s cancel out of Eq. (11.6) t so the solution is
y* = g* = .
where
wJ
and
W2
. A further
. CJ
-.....
( WI
JWI( -C., ). "':? '
are specified by Eqs. (11.7) and (11.8).
s.ignific~ce
of
WI'
'and
"
WJ
=
"
W,2
u*J
uj + ui
is ' that at the optimum
uT
=-
y*
,
Equations .(U.J 0) and (11.1 t) ,.may be useful in solving for x". ' ~
(l1.9)
}1J2
-
(11.10)
GEOMETIUC PROG RAi\1M!NG
243
-
,
E_~ample 11 ~ L
Deterrnine the optim.Llm pipe di ameter. which reSJ IcS in D1!n·inTuH1 first plus operating cost for 100 m Ofl gipe conveying a given water fl o\v [ate. The first cost of the installed pipe in dollars is ~,60D. where D is . the pipe dianleter in mil1imeters. T~e lifetime punlping COSKis ..(32 x 10 12 )/ D 5 doli~s. It to 'be expected that the pumping cost ~vill be p roportional·to D -5 ' because the pumpi.q.g cost for. :i specified.Jlow rate Q, n umber:'0f hours'··of ~'. '. ' .::..,: : operation',.. p~p . .a.fficiebty. ;; }i~Qf0r,"efficieucY'f,: ~arid.."electric~·f,ate':.!s-" propo(ciohaI ~.; :'.~ ~ I;·~:,: ;~·(i"/·., -~'.. to..the..p[~ssun~ drop in~tfie: ..pip~., Fu.rtheF;-' this"·piessure··droil' £rp ·...i~ ,. .' . . :, . . . , .. -'.. . '. - I. _ . .
is
~
'.,..: .~ ~
The objectiv.e function,' the cost .J'. in terms of the va.riable D is then I
Y = 160D'
32
X
lOI::!
+ -D5
(11.12)
-,
Solution 1.. To .p rovide a check' on the geometric-pr6gramnling method, opri. rnize by calculus.,. .
Then
D*
= lOOmm
y*
and
.Next optimize Eq. (11.12) by geometric
= $i9,200
p[o~ran1ming.
Solution. 2
y* = 160D*
+
32 X 10 12 (D*)5
= g*
. '160Djk·Ji32 X IOl2jW2 y* = g* = ( - wl D5 w2 provided that
WI
and
W2
are chosen so that
and
and
_Solving gives
,
.
w
2
t 6
=-
Substituting these values for WJ and w~ intO the expression for g. results in the cancellation of the D's. leaving
y. :;;
g~ .
y
$ ,
= g~
==
(160)1"6(32 X10 5/6
=,.,$19.200
1/6
12
)1/6
•
244 .
DESIGN OF TIiERMAL SYSTEl.,,1S
The value of.D* can be fo~nd by .·applying Eq. (1 flO)
=
.Wl
" .
.
uf·. · lui· = Ujl'* + U2* = y:* . =
D* = lOP mm
" and so ' r:;,,'"
5 -6
'
.
'
~
optimum
I ..
'..
160D* 1.9,20'0 d~amete(
.
....." ., .
.. " . '. , " ....:. "
o·
::::~;:·~:;~:·'-~::~/r~~::f~p.
: Example It.i. The torque T in newt9n-~eters dev~~oped by 'a ~ertain inter~ n~,
combustion engine is represented by
T
.
~ 23.6 wO. 7 -:- 3.17w
(rJ is the rotative speed .in radians per second. Detennine the 'maximum power of which this engine is capable and the rotaiivt? speed at which the . maximum occurs.
'w~ere
Solution. The power P in watts is the product of the torque and. the rota.tive . speed,
p '= T £c)
=:
23.6
ltJ1.7 -
3.17w2
Applying geometric programming, we have
.
I
_l~*=g*=
. '.
(' 23.6)W - I' ( ,
w~
-3. 17)W1
i
(l1.13)
Hl2
. provide.d that ·
and · Solving the two equations for
Wj
and
~'l
and Sub,stituting these values of
*
=
HlJ
+
2W2
=0
gives W2
= -5.667
and W2 into Eq. (11.13) gives
(_2}.6 6.667( _3.17.)'-5'0667 I 6.667)
g
L7wJ
= 122 .. 970 ·W
,
. - 5.667
= 123.0 -kW
== y.
To compute the value of w*. use the fact ,that
ut = , so
-.
w·
23.6
",.1.7'
= Y ·wll
= (122,970)(6.667)
= [ ( 122.970) . (6.'667>"1. JlJ.7 :: 469- radJs = 74.6 tIs 23.6 . .
245
GEOMETRIC PROGRAMMING Ii}
A:o, ali:eIl1ate
mEaDS
of deter.min.ing Lt:';~. \Nould be; to uSe l/~\
.
.
ui =
'. :
~
I!.
(- 3 .17) (.w~)2 = 12:2,970w2':::
(W*)2:= 219~830
(122 970) (-5.667) 1
w* = 469 radls
).
-\
-........~,~.'~
..: '
. .:- . ~
:." ~ .
..:" _ttjJ3ce. ·of.the.., ~.~g:?~iy~" cge:fPc~~nt"i'1.: ~:e; ·.qbj~.¢.tiye: flli1c.tio~...~~'f:l~e:d.li1o
" '~pecicJ 4jffic;utfy.:... ' .
.}
I
'.
'
' -
7'
: ~ '.,
.. - .
.'
" >.
.
'
. •.
•
~.
'I
.,~
: ••.' •••• ;,••~..
' : , : ' , -.
" ': ' ',"
-.<'
I
:
.A further eXfu11ple will illustr~te yet another class of probl~rll~ and . als() a siniatibn ,which at first may seem to make ·the problen"1 in$olu-b~~. . . . '. Exam.ple 11030 Find)' *'and x * for the function
~~
. I
+ x 1.6
y - -4x
.Solution
(
_4)\I'!( - 1 )W2
.:
y* =g* =. . '
.' proyided that
WI
+ l-vi
= 1
The values .. of the w's are
WI
WI
= 2.6~7
.
\."'2,
~vl + (6W2 = 0
. and·-
and
-1.667, and so
W2 ' =
. (-4 )2.~7( 1 )-1.667
y* = g*
~.-
2.667
-1.667
.
.
(11.14)
Each combination of numbers in the parentheses in Eq. (11.14) is negative t and a negative number taken to a noninteger power is a complex number. The'diffiqulty can be resolved, however. by extracting the negative sign from both te~s in parentheses, .
4
y * = (-lf~·667 - 2.667
(
)2.667(-1) -1.667(1 )-1.667 -- .
4 ) 2.6fJ7( 1 ) = (-1) - --
l
2.667
1.667
.
1.667
1.667
=
-6.90
The optimal value of x can be found next
. . ~"4.~' ~ uf '~~i~~y- * ==
(2~667)(-6.90)
x· = 4.6 ,
The optimal value of y in this example is negative, which makes the components complex numbers in Eq. (l1.14). Ie is possibl·e, however, to extract the negative terms and group [hem as .negative unity , taken to the power. of
-.
_____.
uni~~
~
·.1.;.
. •
246
DESIGN OF THERMAL SYSTEMS eo '
1105
WRX G-EOIVIETRIC PROGRAl\11\Afl\JG
WORKS;
€!~~
INDEPENDENT yAJIDtBLE
of
The previous section presented ,the m, ech~nics optimizlng a function of ,'one ,independep.t variable by llSing geom~tric prognimming but 'gave ,no , -:' " supstantiation: for . the method.. This- s,ect?on will prove the validitY , of the "". '~
':,', ~,' ~~ ' ' - :" " ,>geom~tric.~prog~ain/mi,Dg, met4Q4~~f~r.?thb",9n.e~i~;~pep'dent~~~abt~,, :pio51etn:::~;:~::~7;}:': ~:">:
",::',:',:ThE,~fo~::Qf 'the:"-obJ,ective"fUi)cti6n :is~:·the. 's~e' ,~'as 'th~it bf :Eq'.-.',< it.'5)
(11.15)
Next fabricate a func-bon
g such 'that,
where
(11.17}
A "certain 'combination of val~es of WI ,and ' }\'2 will provide ,a maximum value of g. determine these values of ,\-Vl and W;2, apply the method of Lagrange multipHers to Eg. (11.16) subject to' the <;onstraint of,Eq. (11.17). , The maximum values of g and In,g both ace'u r at 'the same val,ue,of th~ 'YV ~s; -since it is more convenjent to optimize in g:
To
11aximize 'subject
lng
t~
= WJ(lnuI +
1Vl
-In'wl)
1
l1.'2 -
+
(11.18)
w2(1 nu2 -.1n)'v2)
= ¢ .=
0
(11.19)
Use the me00d of Lagrange multipHers .V (.In g) - A V ¢ =, 0 '
¢ = 0 which provides the three equations:
\-1'",:
1n III
-
1 - In W J
-
)..
== 0
]n 1:1 2
-
1 -:- Jn H)2
-
A
=0
Wj'
The unknowns are
}-i)l, W2
f
+
H':! -
1= 0
and ~ and the solutions for I
, WJ -
II) -----------~
ul
+
WI
and
11'2
are
(II .20)
U2
"
and
(11.21)
t}
GEO&1ETRIC PROGRAMMING
' .
g = Ll u +
Thus
247
I
H2
\
Let ~J S pause at this point to. assess our statu.s . By theSfloice of l-t'J and according te Eqs. (11.20) arr~·(11.:21) the .v alue of g is 'tnade equal to
. . that of It 'I ':- U2 ' ~nd · th~~e~~~,~._. ~Iso, .~o 'Y!,~"t.7n~\,.9.0.~f.:~~.Q.mpitn1:ti.on/of. wr-'arld··::,."... . " .•• .• .-"'-v )t~2_~~~.4lt~. :~~ p_.:y?-lue ~of. ·g~ ~tl~~t ) sJes:s,~.than_.u.l <:.,±'~ l!-2':~ . >;..y' _.,' ...:.-.,;;;;::,::: ';;;., .'-(... ~'r.:'/ ~:j;: ..:.: :.:.~..';~ ..:w·!:_T: --, Sl.n~C'e:o·ut, onginal.objec:tive .is·:£o ·.rni~iInifle '.y, the ~exr·step·· is·,to "u-se .' , the value "of x' in Eq. (11'.15) that results the minimum, value ·of HI + u2, The va~ue 9f g at this c-o.ndition g:i: will therefore be the one vvhere the 1t-' 's are chosen., such. that g always equals Y ,bur also with [he value of x' chosen '. so .that y is a minimum. This va'-lue 'of x~ can be found by eq'ilating the w;2
in
deriyative of Eq. (11.15) to zero
.i
+ Q2C2X(02-1)
GICtXCa.J-1)
Mt1lt~'plying
= 0
by x gives
- )
al Cl .;rDI
.-and so
,
+ ..a2C2X02
alIlt
+
0
=0
Q2 u j
.·\vhere uT' and ui are- the values of u'J :,and From Eq. (1 1.22)
.
(11.22)
at the n1iilimum 'value of y.
Il2
which, . when substituted into Eqs~ (11.20) and (11.21)y yields -(a2/ a l )U1
-
WI
~
and
-
( /al )112* + u2* a2
uj
w"- -..
-
' * + 1I2*
~(a2/al)1l2
-a'2,
(11.23) .-
a, - Q2 °l
Gl -
Q2
When these values of WI and H"2 are substituted into Eq. (11.16) for the exponents, the expression for g'* results
.. '
(CIX,OJ) -a 2/(01-01)( C2XD1):01 / (0,-0:0
g*= ' - - .
- -- -
WJ
Of special
jmport~nce i~ th~
.
W2
fact that,x can be ,canc,eled out leaving t
-02/
g * - El , -
I ( WJ )
I
I
or
gfl = ('
El.
). 01 / (01
~al)
W2
ElJwl( C2 )1"-2· = y. WI
,
. (11.24)
' W2.
In executing:g&Jtrretrlc programming the values of w proportion themsel ve ~1 so [hat the x s cancel in the expression for g. t
248 ·
DESIGN OF TIfEIUvfAL SYSl.c.MS
11.6 SOlVIE ThIS~GHTS PROVll)ED BY GEOMETRIC' PROGRA.Iv.IJ.\11NG )
.. J
~.. Geomeu-ic proir~mming not qnly yields optimal values of the variables
arid
.. .- . objective function but can provide some insight into.lhe solution. CO!lsider, . . _. :,: for. .ins.tiUlc~, . E:ra:np1e 11.1,' .~her~ ~~ life-~y~le :'c~si .the .p~p-pipe":"'~'i'~' ,}.~~._:: '. >~ ~~~~7.~~: ~~ .op~ed:· " . . '. ' ~ ~ '-~' :-'r .: - :~ .'. :' . ~ . . '. : ..... . : .:.,....:-.. y ' :..-. pipe c.os.t.·ct·~pres.~nt ,w~rtl! ofJifetiroe eJ;1ergy _~ost .. .... ~:";::'-~~.i.'·:·;:·~~)·-.-,· ... ·i~.!''';·::.·· :, ':.','; ",:;:;::': .::.<: ,.:~" ,, ;, <:/ . ,>J;~~ :",:'::-.::-~ . . ' 32 x ·10 _ .,., .. .. '. . ' .. . . . .- 160D + 5 " . . . . ~ . 1) . .
of
..
-<
. . '...... '...
.- ...
0,'
.
-. : .
'.
_.
"
".
-
. - -• • •• , .
".',
•
• ••
: :....:. ' ;i'·.
-
..
' .
.
For this objective function the optimal.value of diameter D* =. lOO~ mm and y * = $19,200. The values of the w,eighting factors, WI and W2 of '5/6 ,and 1/6, respectively, are alsd of interest, becaus.e at the optim'um five-sixths of total cost is 'devoted . the pipe and o~e-sixth ~o the energy. Let us suppose .that die cost. or .t he energy ·in~eases. What would· be t:Qe effect on .the optimum? 'wp,at bappens tq ,the solution, for. example, if the lifetime energy cost becomes .(50 X 10[2)ID 5 ? Of particular importance is .the fact that the distribution of the total cost remains unchanged (five~ sixths and one-s'ixth) because the. exponents of.D that control 'WI and W2 remain ' unchanged. Both y * and D * increase; the optimum total cost I;l.OW. becomes $20,680, and D = I07.7 ffiffi. The increaSe.in optimal diameter pr.obaqly confonns. to our expectation that upon an increase in the ~nergy . cost the diameter responds -by ·incFeasing.· But regardless of what. happens -to the -unir cost of energy or pipe', the qi~tribution between the.fust cost of the pipe and the energy remains constant. . Suppose, howeVer, that an exponent of D changes, e.g., the cost of the pipe increases at a more rapid rate than linearly. If th~ cost of the pipe is proportional to D 1.2, for example, the distribution of costs betwee~ the pipe and energy changes because the new value of Wl = 0.806, compared \\I·i th the origin a] value of O.833~ The ·optimal condi~on :responds to the more rapid increase in pipe cost by decreasing the fraction of the total cost devoted to the pipe.
the
to
11~7 UNCONSTRAINED, MULTIVARIABLE OPTIMIZATION
The
geometric-progr:amming procedures for the one' independent v.ariable extend in a logicaJ manner. to rlIultivariable optirrilzations. If applications continue to concentra.te on problems of zero degree of difficulty. a twovariable problem. for example, wHl have an obJective function containing . three terms. . Example 11.4. The pump and piping of Example) J.1 are actuaHy pan of a_w-a.~eetreatmen( complex~. as shown in Fjg. 1) ~ 1. The sys,tem accomplishes
€:.>
2r~9
GEOMETRIC' PROGRAlviM1NG
Wasi'e :'
I I
~
Effluent
: . .:~ /, -~ :
,-
;
~;'
-;/ ':\::' . "
; of':;,! ·,' :. ' . ' ""
..•~. . .":
~lGURE
.'
....
,
.
~.
....
•
:'
'
:
-::
.~
.',:,'
.......': :
' ."
lId! ,"
, \Vaste-tr-eatment sy~teni in Example il.l. "
't he treatm.ent by a' con1bination "of dilution and' che~ical actiOJ!] so iliat tqe effluent 'meers' code req~irements. The ,size -of .the reactor can decrease, as the dilution increases. The cost of the reactor is 1501Q, where Q is the flow rate in cubic per second. The equation for the ' pumping cost ~ith 'Q b~o:k'en out of. the cOffiQined co~stant is (220 x 10 15 Q2)/ D5. Use ,g eometric programming to optimize the tqtal system.
meters
Solution. The total cost is ,th~ sum of the costs of the ,and the treatment ,plant v
.,
pipe~ the pumping power,
'
=
220 X 1015Q1 '5 D
+.
160D '
y..* "= g* =
150
+ -Q
(-160)H' 1,( 22'0 X10 15 )11'I 2(' -150)".1] I lVl
YV3
}V1
provided that
to cancel
D: Q:
Solving gives 5
and
wJ 8'
-2 8
=-
'( -160)5/8(220 X l0I5)"B(J50) _ . -- I114 -$30
Then
5/8
. }/8
5
Q
Uj
= J50
Q.
11.4
, <
uj = 16{)D == "8(30.224) I
W3
= ~(30 224)· B' "
so
•
, 224
D* = 118 rom
so .
The core of [he execution of Example 11.4 by geoffi'etric programmi'ng \vas the somtlon of three' simultane"'"ous linear equatjons for ,the w's. Had the problem been solved by Lagrange, mulripliers., two simultaneous nonlinear
,,",
250
DESlqN OF TIIERMAL SYSTEMS
•
e quations (from V y := 0) V/ould bave been solved. In general~ the solu tion ·,by geometric programming of a zero:degt~-of-djfficu1ty problem requires the soiution of one ,m9r~ equat~on in a set of simultarteo~s equations than required by Lagrap.g~, ,m ul,t ipliers, _but the equations 'aT~ - linear. -For this, sPeCial ,c lass, of pr
,to
:' ',' .
tI~an~bi~ "'prcbI-em;'<''Ex~pi~{', -.i j.'~4~:',;h~:>not: .y~t~~~t.t.· PKQXided,.,:}t:J;~'~ Q.~ '. , dev~loped by following the patte;m,:.for ~~ proof 'of :the:slilgle7:~:an'aBll{opti~~};. ~tion 'presented j.n S~c. 11.5. The steps are as 'f~llo~s; .':", " " -
.
,
.
1. Propose a g function -in the fonn of Eg. (11.16). 2. Set the sum of the .W 's equal to unity [Eq. (11.17)].
3. Maximize Ing subject to N
~ ,Wi =,1. ;=1 '
to find that the ,optmlal w' s equal the fractions that the respective u' s are of the total [Eq. (11.21)]. With those 'o ptimal values of w the function g equals the original function to be optirriized. ',' 4. Optimize tDe function. The differentiation with respect to the x ~ s is. now a parlia' differentiation, and ,the derivatives are equated to zero, following - which the equations are multiplie¢l through b~ the appropriate Xi. 5. The result will be a summation fO'f each variable T
L a1nlv/' = 0
(11.25)
i= J
where
tern) number in object .function T = rota] number of teqns in objective, function '17 = . variable, ranging from ~ to total number of variables N I =
The N equations designated by Eq. (11.25) are the conditions that result in [he cancellation of all the x's in the g-function fonnulation.
11.8 CONSTRAINED OPTIMIZATION - WITH ZERO DEGREE 'OF DIFF1CULTY l
The final type of geometric-programming problem to be ,explored is one wirh an equality constraint. On1y the zero-degree-nf-difficulty case will be considered, and the toeal number of terms T means the sum of those in' the , objective function and th~ const~int. Suppo~e ,that the objective furrction to be minimized is
-.
(11.26)
GEOMETRIC PROGRAMMiNG
subject .to the
co ns tra~ n{
+
l!.:j
.~VJhe[e
251
) )
Us
= 1
11 ~s
are polynomials in tern1S of four independent variables, x I , ,.·X2-,. x) ~ ~Jid x 4: The, right sid~ q,f thy constraint- e~llatiO? m~t be ~IJ)ty:, .. .;.._,·YV,hic.h ·pDSes. DO proQ!em. a~. loJ1g .SS OD.e p~u:e. ~n1ltC1ep.cat~ernl._apI?~~~)n.xh~.~~~-·~;·:,·. .:, .'. 'ecJ..Lt~tipn~-:~lf ~I}-~:( !Jurr~~.~-:i? not ~nifj1.-' tEe '~ni1i'~ equatiQn . c~~·b.e. ~dh'jde((by~,,:·.·.·:· .." the
the' number to convert it into unity., The ~bjective fupction can be rewritten'
.
(11.28) provided that
and
l-Vj
=
+
III
1i2
(11.30)
+ ll3
-The constraint eq~ation can also be rev/ritten as lL-t. +.U5
=
1
J.V~.
+
.
(!!.!)' W-I( ,Us W.:J.
J"
"'5
(11.31)
1Vs
provided 'that
and
Ws
(J 1.32)
== 1
and
Ws
Us = Us 1
= -
(1 L33)
Equation (11.31) can be raised to the Mth power, where M is an. arbitrary constant, and j t$ value rema~.ns .unity t
( 11.34) Next mUltiply Eq. (11.28) by Eq. (11.34)
.
Y = g
",.
()IVI( !!.2 )Wl( !2)Wl(' u4 )MW..I( Us )MW.3 =!!J. w1
W2
,
'V3 .
W4
Ws
(11.35)
.
Momentarily set aside the representations of Eqs. (11.28) to (11.35) and rerum to Eqs. (11.26) and (11.27) and solve by Lagrange mUltipliers
,
Y' CUI
+ u2 +
U3) -
A[V (U4
+ U5)]
= 0
(1 1.36)
(11.37) The vectoLeQ18,tion (11.36) represents four scalar equations. the terms of which 'ire- the parti al derivatives with respect to .x I 'to x 4. By taking
DESIGN OF TIIERMAL SYSTEMS
..
are polynomials each of the scalar with re~pect to
1.36) can Iby differentiated. The resUlt
.. .III
............
"
. . . . . . . .,
.....
111
. . . . 41 "
'"
..
III •
'" '" . . . . . . \l1li AI . . . . . . . '" . . . . . . . . . . . . .
fII
.....
&
•
.II
.......
'"
_
........
41 •
~et
_
.........
<6
+
a'J4'WJ
..
0lil
...
..
..
...
..
..
G24W2
..
..
..
+
.....
~
..
"
..
fII
.......
'"
....
+
a34wJ
..
..
..
..
..
...
..
..
..
Ma44l1'4
"
..
~
+
..
..
..
...
II . '
it equal - AI)' *.
..............................
~
..
Ma54w5 = 0
(J 1
Along with the . \.1.'1
+
\<1·'2
+
\1/),
= I
(11
(11.
=M
. (11.40)
They provide six- linear simultaneous equations Can sol \'ed the .. unknowns \.1:1. H'2. H1J. M... It is especially ( 1 L 39) all the x terms in . (11.35). fac't a convenjent evaluation of value of y
11 A water across 4l nation plant at the seacoast to a city. The pipeline. as In J J J6 water. costs of the pipeline
+ O.OOO32t\pl.:!
1)'
are
dollars
where /.lp = pressure drop iri =
Assume a frlction Use geometric
,
m ve
[0
results in the minimum
-
..
r
---:-------'2l
-- 30 Ian - . -
Seacoast
".
'
or
".. :, ' .•'
.
B.GURE 11-2 \Varer pipeline ·in E~ample' 11.5.
Solution. If . n designates t:J:le number -of.pump-and-pipe sections, the total cost y
~s
y" = n(lSOO
d00032~p~.1)
+
2~560,OOODl-5
30.000.m L
=
n
+
where L is the length of each pipe sec[ion in meters. The pressure drop in each' pipe sectio~ is. .
L V2
!1p =
f D2 p
- L . 0.16 )2} 3 (0.02)D (7rD~14 Z( 1000 .kg/m ) '. -
=
.
andso
/).iD
The statement of the problem
rs .
Minimize
Y
= 75,0.00,000
L
2.410LlpD
subject to
5
$
=
0.4150 '
.9.6~pL2
+
L
(l1.41)
2 560' OODD L5
+.
·
= 1
(l1.43)
L y. ' =
(11.42)
(75,OOO,OOO)WI( 9.6J""2( 2,S60.QOO)WJ [ 2.41)M'W4
l.
WI
l
W;2
l
Wl
W4
provided that
L:
MW4
+
trp:
=
0
MY!4 = 0
1.5w3 +5Mw4 = 0
D:
SMW 4 == M wI = 0.0385, -0.2308_ .
so
- .
W2
= 0.1923.
WJ
C
0.7692. M = - 0.2308, and
MW4
=
..
DESIGN OF THERMAL SYSTEMS
-0.2308
y*= -
'-V * =
$410 150
,
L* = 4750 m
D~
so
*..=
. L* .2.410D"s -
= 0.246 m 188,000 Fa
=2
k.Pa
sections is six pumps ~ould be used; which L to 5000 fit can be (11.42) and (l ],43) a reo-pnnrnZ2l!l0J and D.
11.9 SENSITIVITY In . ] 0 rhe multiplier A as the sensitivjty ~ . . n~ cient, which \A'as "A numerical term on the· the constraint equation. the development of the procedure fcir conuSIng -: .A /y * \vas replaced by the tenn 1M I in optiis complete· y * and M are kn.own, the sensitivity coefficient can be detennined . In Example I] 1M J = )' * = 410~150. so the sensitivity coefficient A = -My* == 660. value of the tivity applicable to the constraint equation i.n the .... ,.
A .. JA .......... J...........
of
. (11.43). The value
sen~jtivjty
coefficient for the . (11.41) is 94,660/0.4·1 = ,100.
, of
the
form
coefficient in Example rate the total first cost with respect to the change in constant 0.4150. If, for example , the water flow rate \\'ere from 0.160 to O. J 61 m 3 /s I the constan t WOll Id ~".~""'" 0..4150 [0 (0.4150)(0.161/0.160)2 = increase in cost of the _ system is (218 100)(0.4202 150) = ·$1 J 87. of
1
11 . 10 HIGHER DIFFICULTY AND EXTENSIONS OF GEOMETRlC only problems to appl i~ in this chapter were lems-geometric programnli
is
GEOMITRIC,:PROGRA~L\'!lNG
G eomeL'_ic pro gramrni ng a lso -p rovides SOlTIC phvs iell!, .Tn s~ ght Rnto the sol u[ion not usun.lly offered by other ~e[h?ds of'optirnizatio·" . '-
'-'
•
).....
<-J
•
Geometric progra.mming is not 'Jimited to zero-degree-of-difficulty prob lems: it can accomlTIodate higher degrees of diffic~l ty. but the need '. to 'solve a set of nonlinear simultaneq-us· equations. arises in s~cJ).. problems .
. <'J.J\1-8Cfr..DI- th.'e ,'effort~'by:_.,operatioos:reseq.rch vvo.rkers" ..who~.are ...conce.ntr:atirig,~ .... . ........... ,._ , , /.... ... .. . on geomet;ric,.:~p~ogrammi~'t.'~" d~vote~ Jt6',:de\i~rqpliig, effici~rit..and . .[eliab-le, --:.: : ',;,'i,
.
.
"
_,,"..-
....
~
t'J
".'~
.......:
.·. · 'r~.'."··
~
,~
~
computer programS tha·r· include the soIu'tion of: these nonlinear eqi.iat'ions ... ,,' ., These computer programs can also handle inequality ~onst~jnts, as .well as the' ~qualjty cOIlSt(aints pr~sent~d in this chapter. ' , : ·Geometric programming is useful .too[ to ~arry ::1 the optimization kit for,tf.1ose spe~ial :situatio,~ t~ ,which it i~ . partic~larly adaptable. ' .. .i
a
Pl\{OBLEMS So1ve the following problems by geometric programming . . ' 11-1. The thickne'ss of the ·insulation of a hot-water tank is to be' s~lected so thar the total cost of the insulation and standby heating for the IO-year life of the
facil-iry will
b~
minimlim.
'
Data
A verage water temperature, 60°C Average ambient temperature ,24°C Conductivity of insulation. 0.036 W/(m . K) Cost of hear ene'rgy. $4 per gigajoule Cost of insulation, where,X = insulation thickness: mm, per square meter
0.5:,(0.8
dollars
The operation is continuous. Assume that the only resistance to heat transfer is [he insulation. (a) What is the minimum' total' cost of insulation plus standby heat loss per square meter of heal-transfer area for 10 years, neglecting interest charges? (b) What is the optimu"m insulation thickness? .Aris.: (b) 99.3 mm. 11.2. A hydraulic power system must provide 300 W of power. where the power is the product of. [he volume flow rate Q m~/s and the pressure buildup 6p Pa. The cost of the hydratilic pump is a function of both the Dow rate find pressure buildup: ' COS[
= 12000°.-, ",-'10 + (Ap
X
10- 4 ) dollars
I
C,onvert to a single-variable, unconstrained problem and use geometric programming to determine the minimum cost of the pump and the optimum values of Q and fl p . ..:- -Afls.: D. po = 400 k.Pcr:
256
DES~G:N OF TIIERM.AL SYSTEMS
11 .. 3 .. Schlichting.3 presents the following eqt.IatioD for the velocity jet that issues into a large space, as il1u.stratdq Fig. 11-3:
a
. ~
.
u= . xII
+ 57.5(rlx)2F
.where u =' v~10city .in!, the di,n!c~bri,,, in/s . r = radia,l 'dis~ce measured from the ,."",..,,'"""'.,..11 ..... "" m ~.~~Uo'· .~ ':'oll tfet"Neloci~~.JD..is;:;".';{ '/~:'" .... .. , . A ;;;: o.ti~tet ..' ~2":' ". -:. . :- ': . ''': ,.:-.:.;:....., ':.i:. ;" .~~~ .··.;.2.~:;""Y>"
x:
~
.x
.=
of
. The position m~~ u at a radial distance of 0.5 m the ..................". . . . . ,'"' be .B . jet wlthgiven values cj[uo and A. Sp~cifitally~ use' programming .w· find the value of x at which u is maximUm r = 0.5 m. HilrL- The problem can be structured as one of zero degree oI difficulty. Ari.s.: 6.57 m. The. total annual co~t an insulated facility is. the sum of .................... , cost . .insulation plus the aru:mal cost of the ,energy. Specifically,
to
11
~ diiectlon measured- from ·wall ... m': ,.' ... .
..
$/m 2 =
0.5
+
C1
:x.
whe:re x = lnsu]ation thickness, mm If the unit energy cost increases by 10%, what must be the percent decrease in unit insulation cost so that total annual cost remains constant? 11.5. A hot.-w'!-ter boiler consists of a combustion chamber and a exchanger arranged as shown in Fig .. 11-4. Fue] at a flow rate of 0.0025 kg/s with a heatlng val ue of 42,000 kJ/kg bums in the combustion chamber. Thereafter , the flue gases are cooled to 150°C the process of heating the water. combust jon efficiency with an in the rate of llirflow according to the equation Efficiency
I
= 100 _ .
0.023 (mil + '0.0025)2
,
,
It----
of
..
I FIGURE 11 ..,3 Free~stream
jet in hob. ) 1.3.
:.r..
GEO.\1ETRlt::-· PROGRA~L\'l!NG
Ho( \ \
257
\.\:Llt~r
,
.-1.'.-------t AiL 2JOC 1111/ kg/s
So.ck gas
Combu.<;rion
lSOOC
.·f
:,,",'," .' ,., :.:.~- i,";::~e~7q2~~,:c,~(:,,~:;.~;:?H!_~~S_, ' ':. ·e:~c~~~
,.", .:
!':g/~ \V.:tl~r
·FIGURE 11 1-' Ho[-w
gas mixrure is 1.05 kJl(kg ..K,). Dettrmirie the.val~e of
of h~at "transfer to the water. Suggestion: be optimized. Ans.: 0.0695 kg/so . 11.6. In a .three-~tLlge compression system air enters the first stage ar a pressure of , 100 (Pn and temperature of 20°C. The exit pressure from the th.ird stage is 6400 kPa. Beiween stages the air is passed through an ·intercooler rhl£ brings [he temperature back to 20 9 C.. The expression. for ~he work of compression . per uni[ mass in an "idea) process is lila
Use
thaI results in. ~he m~xjmurp mre JJ7 0
+ 0.0025' as the variable
[0
a
k
- ' Rtl k - 1
[
, . (k -
1-
(P:!.) ·.PI
I )1 J:
j1
where subscript 1 refers to the en tering conditions and subscript 2 to the leaving conditions. (0) With the intennediate pressures chosen so that the total work of compres- . . sion is a minimum. what is the minimum work required to compress I kg of air, assuming that the.compressions are reversible and adiaba.tic? (b) What"are the intennediate pressufes that resull in this minimum work of compression? ' Ans.: 429 kJ: 400 and 1600 kPa.
11. 7. The hear-rejection system for a condenser of a steam power plant (Fig. 11-5) is to be designed for minimum first plus pumping cost. The heatrejection rate from the condenser is 14 MW. The following costs in dollars must be included:
,
or
First cost cooling tower. 800Ao. 6 • where A ~ area, m 2 Lifetime pumping cost, O.OOO.5w l where w = flow rate of water, kgls Li fetime penalty in power ~roduction due to. elevation of t~,mpera.rure of coo~ing water, 2701 where I = temperature of water entering the eona denser. 0(.": , j
t
~
rnte·--()f heat transfer from rhe cooling tower can be represented ndequareJy by the expression'"' q. W = 3.7(w 1.2)rA.
258 . DI;.SJGN Of TIiE.R}AAL SYSTEMS
. Cooling tower
,:;';:'~-:-·a~: a::~TaJ:. a ,
.. . -
,~.
. . • &,
.
a-a .:~.:.1. ;:aa~~':'~,~ ".'f':::~~~a:;._; . -:~;: . :. ~::: '::'~':':
~
- -__--:--.i
:-'1 ..,~~: ._ ._ ,.a<:.~--~'~,. :., _:j
'.' . . : ......-.
wkg/s
re, FlGURE 11-5 Heat-rejection system in Prob.·1 1-.7. a
a
_
.. (a) Set up an untonstrained objective function in terms of the ..variables, A and w. . . (b) Determine the minimum lifetime cost.·· (c) CaJcu~ate the optimal values of A and w. " Ans..: (c) w = .202.6 kg/so 11.8. The total cost of a rectangular building shell and the land it occupies is to be minimized for. a :building that roust have a :volume .]4,000 m 3 • The following costs' per sguare meter apply: land $90; roof, $-14; floor, $8; and walls $] 1. Ser up the problem as one of constrained optimization and detenTllne the minin1um cost and the optimal di.mensions of [he building. Ans.: Height.= 71.4 m. 11.9. Newly harvested grain often has a high moisture content and must be dried to prevent spoilage. This drying can be achieved by warning ,ambient air and blowing it tJu-ough a bed of. the grain. The seasonal operating cost in doJIars per square meter of grain ~ed for such a dryer consists 'of the cost of heating the air
of
I
1
Heating cost
= O.002Q~f
.
and' [he blower operating cost Blower cost
where Q
= =
10- 9 Q3
air quantity delivered through the bed during season, m 3/m 2 ~
Af
= 2.6 X
area
.
rise in tempera\ure through heater.
°c
J'11e values of Q and 6.1 also jnf1uen~ the time required for adequate drying of the grain 2ccording to the equation . a
Drying time
-----..
=
gO X ]0 6
Q'2llr
days
GEU :·.! FTRtc PROGR;~ 1,
m':G
~59
U~ing
the gc.orncL:lc-pr0gra:mIni.i. ~g. tD.~tll0d of cODstrnint:G c.-~t~.x:~l~:n,"u~ "on. corn.pute the mini~um operating cosi'at-ld the! optimum value of Q and ~r rha[ will aC,hieve adeqL:ate drying i.n 60 days. Ans.: 111 = 2.28°C.
," ~ .; .~ .' 1J~' ~ .REr
-:
York~
E.
1967.
:L~' Petersbu',
'-'
'''!',:
.. ...
' , ... .. -
I
'.~' -
......
I
•
and · c~,)y[;Z.e.rjef~, GeOlJletric prog'i-am~li~!i>\1iil~)~. " ''X,e\\~ ' . , ' . ' . 2. W. F. Stoecker, Design of Thermal Systl!ms. ls~ ed .• McGraw-Hill. Ne""· York. 1971. 3. H. Schlichting, }:jOl;udary loj'er Theory, 5th ed., McGra\Y,.'-HiIL NeVI Yolk. 1958. 1.
~: i ' D~ttn,
.'. :
i.
B!BLIOGRAPHY
C. Zener", Engineering Design by G~omerric-Progr(1';l1nillg.'. ~i~~y'-fnter:sc'~p.~. Nc\\~ \ork. 1971..
C , Beightler and
I
--
p.
.
'
.
.
.. .. . '. . ..
' . '.
"
'.
T. Phillips, Applied Geomerric Programming. \Viley, New York. J 9-:0.
.~,
1
I·
.,'
.
~ \"~
.-., _....-. . ~ _ . .
' ".
".
,J. • • ',
'-''\.."
• t...
....:.
~'.
- "
LrNEAA"'~_; :oY"-"~' 5';~'~""" ,~. ','",: ';,.Y. '::-:.,
~...
•
•
. : .t:-.~ ... ~ ...:-
'_./~ .~.~ ":.~~r~~
..
...-=~:......... ' .: I
.
. "
··PROGRAMIVIINO··
\"
12.1 THE ORIGINS OF LlNEAR
p!ib'GRAl\IIMING Linear programming is an optimization method applicable ,vhere both .the objective function and t~e' con'straints can be 'expressed as .linear combinations of the variables. The' constraint equations may be equalities or inequalities& Linear programming flrst appeared in Europe in the 1930s. when economist$ and 1!lathematicians began working on economic models. During World War II· the United States Air Force sought more effecti ve procedures for al10cating resources and turned' t6 linear prograrnrDing. In L947 a member of the group working on the Air Force prob1em, George Dantzig, reported the shnplex mefhOd for linear programming \ which was a significant step in bringing linear progranuning into wider use. Economists and jndustrial engine~rs have app]jed linear programming . more 'their fields of work than' most other technical groups. Decisj~ns
in
about time allocations of. machines to various products in a manufactur- ing plant. for example, lend themselves neatly to linear pr0grammjng. In thermal .systems linear progrartuning has become an important too] in the petroleum industry and is now being appJied in other thermal industries. Most large oil comp.anjes use linear-progra.mmjng- models -to detennine the ,quantities of the various product~ that will result in optimum profit for the entire operation. Within the refinery itself, Hnear programming helps determine where the bottlenecks to production exist and how much the tota]
260
-
.
LlNEAl;: PROGRAMMING
2 6=~
i.:~
,,'<
•
'
outpu.t of the plant could be increas~4, f())~ instance, by enlarging a heat exchanger or by jncreasing- a certRw_' rat~ o.f. flov.;. ' , Entire bo oks are devc~':ed , to J..Tnear pro g;,-anun.ing ~' but j t is passi. DIe Ln o~e chapter to ,expi~ the physical situation 'that results-',in a' . linea!-progriu11Iaing:, structure, and to, solve pr~~,tjcal ,prob~ein.s' \/liu7'-- the'· ' -,~,"technique. T,Qe emp'~asis,:};v;ilfhe;,.(Q}i'0btali;iiig~,a..gebID~tric:feel~foi~the1me~ar~-~;'~i,'.'· 'protr'amnling~; sitli~tio.n7-·ZDd-, the, .simplex- algorithm: 'fo.i~. sbl vrng both _r6axl- ', ' '" '-mization and, minimjz?-tion prob}enis -iIi Iln~ar'pr~gr~g ~Till be applied. '
12.. 2 SOl\1E E)(P.J:V[PLES OF ,LRffiAR. JPROGRAMlVUNG . Some ,classic ,us~s of lip-ear prograrnriting are, to- 'solve (1) rpe blending pr,oblem, (2) machine' ailoe;ation, (3) inventory and pr~duction planning, and (4) the transport,ation problem_ The oil-company application, mentioned' in' Sec. 12.1 is typical of the blending application. ,The oil ~ompany has a choice of buying crude from several diff~rent sourc,es with different c-ompositions and at differing 'prices. It has a ,choiGe of manufaGturing' various quantities of aviation fueL aU,tomobile gasoline,' diesel fuel, and oil for heating. The combinations of these products are restricted by material balances, Gased on the .incoming crude and by the capacity of sllch ~~mponenis in the -refinery as the cracking. unit.' A mix of purchased crude and manufactured products is sought tpat gives maximu~ profit. , The machine-aJlocation problem occurs where a manufacturing plant has a choice of making several different products, each of which requires . varying machine times of different machines such as lathes, screw machines, and grinders. The machine time of some or all of the~~ different machines is limited. The goal of the analysis is to-determine the production quantities, ' of each 'product that resuIt in maximum profits. The sales of some manufacturing fIrms fluctuate, often acco~g to a' seasonal pattern. The company can build up an inventory of manuip.crured products to carry it through. the period of peak sales but carrying an inventory costs money. Or it can pay overtime rates in order to step up its production during the period of peak sales, which also entails an additional expense. Finally the company can simply plan oh losing some sales because it does not meet the sales demand at the times that it exists. thus losing a potential profit. Linear progranuhlng c~ incorporate the various cost and loss factors' and arrive at the most profitable ,production plan. , The fourth application. the transportation problem. occurs when an organization has severa] production plants distributed throughout a geographical area and number of differently distributed warehouses. Each plant has a certain production capabili ty and each warehouse has a certain reqrrirclflent. Anyone warehouse may receive the production from t
I
a
I
262
DES1GN OF TJiERMAL SYSTEMS
.'
o'ne. or more plants. The object is to determine how much of each plant's : production should be ~hipped -to w~ehouse in order to' minimize the total' manufact1.Uing .and ·transportation. co.st.·. . (, . Sj.mple Iinear-programm~g problems can be done i~.a hit-or..miss fashion, .but those with thr~e or..~ore v~a~l~s .require systematic procedures . . Even when us~g,.·rrietho~i.~~ t¢c;bp~qu~~! . . ~e .magnitude.of·a problem·¢at
.
.~.vanables)
each
.': '. .~ . :.~~~.~'~e:. sqIye'~~:'?Y':~~d ~J.~) ~~it~lI:)~~~~~ .:pt?b1~ms~{W~,~.e~~ ~:f~O~t~~,~ - .
requlIe cO'mputer programs, WblCh .are_. . .
currentlY""avaIlabnt~:as-~:l
li.bniry routines.
lZa3" MATHEMATICAL STATEMENT OF THE LINEAR-P~OGRAMI\{ING PROBLEM . I,
.
,
The fCJT!Tl of the statement is typical, of the. optimization probJem in that it consists an objective function and constraints .. Tpe objective func~or;t which is to be ~inimized (or maximizedi is
of
y
.
C1XI
+ C2 X 2 + ... +
c}Jx~
( 12.1)
and "the constraints are ¢l =QIIXI +012-1.'"2 _
..
•
•
..
..
•
_
_
..
•
;
_
+ .. .. '
•
•
·+OlnXn >- Tl ill
•
•
_
•
•
•
•
•
(12.2) ,
Furthermore~
if the x's represent phy~ical quantities, th~y are likely to qe non-negative, so that x I , . . . ~ X n . > o. The c values and Q values are all constants', which make both the objective function and the constraints linear; hence the name linear programming. The ~a]ues of c and a may be positive negative, or zero. The inequalities in ,the constraints can be in either direct jon anq can even be strict ~qualities. At first glance, this problem lJ1ight seem readily soluble by Lagrange multipliers, but we recall' that the method of Lagrange multipliers is applicable ,where equality constraints exist.. Furthermore, Lagrange mul tipJiers apply where n > JJl, but in linear programming n can be greater than, e9ua1 to,' or less. than In. The significance of J2 < ]]1 wiJI be discussed In Sec. 1-2.14. I"
1264 DEVELOPING THE MATHEMATICAL STATEMENT The translation of the physical conditions into a mathematica1 statement of linear-programming fonn wHl be illustrated by an example. ,
I
ExampJe 12.1. A simple power plant consists of Dn extraction rurbine that drives a generator. as shown in Fig. 12·1. The turbine receives 3.2 kg/oS --or-srom. and the plant can selJ either electricity or extrlJction steam for
263
LINEAR PROGRAMMiNG (II-
3.2
k:~js
. ,:"::"
••
-
,
/.
FIGURE 12-1 Power plant
in
ExampJ~ 12.1 .
. .~
. .procdsing purposes. The revel}ue rntes are . . Electricity, $0.'03 per kil?wmthour Low-pressure st~dJ!l, 10 per m~gi1gram High-pressure steam, $ f .65 per mcgagram
$/ .
The gene~ation rL1[e of electric power'depends upon the flow rate of steam passing through each of the sections A, .B. and C; these flow rates are WA, WB. and We. respectively., The relations.hips are
PA , kW '- 48wA
PB , k\V
= 56ws
Pe , kW = 80wc where the w's are in kilograms per second. The plant can sell as much electricity as it generates. but there are other restrictions . . To prevent overheating the low':pressure section of the turbine. no less than 0.6 kgJs·musc .always flow through section C. Furthermore, to prevent unequal loading on [he shaft. the pennissible combination of exmcrion rales is such tha r if ;r I = O. then ~t.2 ::S L B kg/s and for each kilogrum of x I extracted 0.25 kg le~s can be extracted "of.'C2' The customer ofthe process steam is prim-arily interested in total energy '. and wilJ purch~se no more thah I
4x I +. 3.x.2 :s 9.6 Develop the objective function for the tota1 revenue from the .plant and also [he constraint equations. . SO~fion.
The revenue per bour is the summAte revenues from selling the steam"and the electricity. ...
,
::... I . , '
.
," .; -
264
DESIGN OF THERMAL SYSTEMS t;>
1.65 1.10 r :. Revenue = 1000 (3600x 1) + 1000 (3600x2) .
..
. .
Since WA = 3.2 kg/s an4 .from mass balances
:. '
+ O.OJ( 48wA + 56wB .+. 8p wc )
! . I
."
,
.
WB
= 3.2 - x 1 an~ .w~ = 3.2-
,.:'.... '..... ~~.~~:~~~:.;.._1?.>~. :" ~.~.6x ~. +
the
1.5.~~2 . ·
,.( 12.3)
.: .
. :' . ..... _j. .".::.:. .' ..:. . . 'B~cause 'the coPS.tant: .has ~no effecr.~ori. state 'point at.' vihlch ' the, ~'~:-!:.:~,~.":~~: :~7"', ·.:;_~~optiIDuni::.O·9~~.,::,~~_ ~1?j~~.~ive ~.nction to'· be ~aX~zed is ' . . . ". . . . 'h ".:,. ~';L;;", ~t::86;;<~.i:L56£i/',
• . .'
:. :. '::~_,:' :-!.,: .~ :::~ :(. UA'r"
.' The ·three constraints ..are .. ". . .' " .' X I
+.'
x 2 .< 2.6
.XI
+ 4X2::; 7.2
4Xl
+ 3X2 -< 9.6
( 12.5) I
(12.6)"
(12.7)' -
12 . 5
GEOI\1ETRIC VISUALIZATION OF. THE LINEAR.. PROG~G PROBLEM Since it mvol yes only the two variables x 1 and x 2'J Example 1~.1 can be 'illustrated geometrically as ' in Fig. 12-2. The constraint of Eq':. (12.5), for example, states thar only the' region on and to the ieft of the line Xl +
r----
4x I + ~X2 $ 9.6 .\",5
=0
..r--
.x1+.r2~2.6 x)
=0
A
o
2
FIGURE 12·2 Constraints and lines of constant profir in Exlmple 12.1.
-.
3
2~55
LINEAR PROGRAMMING ,~
x~
= /.,6 is pennitted. Placing the other two cOl1slrain"tS in Fig. 12-2 further restricts the perrQ.itted,regJ,on to ABDFG., !! ' I\Text the lines of cons,~ant revenpe yare plotted 011 Fi.g~,12-2~ ,!.nspection shows tnat the great~st profit can be achieved,'bY,lTIovrng to point D~ ~:vhere , _ Xl = 1. 8 'and X1 ' O.B. ~"1,jmp?~t g~Iie,(,i.dization "is that t'be optimu.rn ,', ' ': solu1.iqn li~s, at ~. COj~;'1:e~~ 'j:. sp'e~i,al ~ase ofJh.is, g~ne!aJ),2;:atjonjs J\~hereJ:.he.. ...;,,-. , ',', ,"J~~s' pf .cons~~t' profit is' ~panil1.el 'a co'rj~tfaint lirie,,' in \.vhi,ch case ~y._,'~",. -, 'point tl1e 'ljn~ betv/¢en the is 'equallY' f~vora,ble~'··' : . If the objective' function depends ~pon three variables, a three,dimensional graph is requjred, aDd i:h~n the constraint ,equations are represente.d by planes. The-corner ~here tJ?e optimUlTI occurs is' formed ~y the ' intersection of. three planes. '
on
12.6
to,
consrramt
INTRODUCT~ON
I,
cQmers
OF SLACK VARIABLES
The constraint ~quations (12.5) t~ (12.7) are inequalities, but they cari be conv~rted into equ~lities by the introduction of another variable In each equation X I
Xl
+ X2 + x) = ~.6 '+ 4X2 + X-t =,7.2 + 3X2 + X5'= 9'.6 ."."
4x I
~is substitution is valid provided' that
(12.8)
( 12~9)
, .
X3
>- O,X4
( 12.1b) > 0, and Xs >- a.'These,
ne\v variables are called slack variables. Reference [0 Fig. 12-2 pennits a geometric interpretation of the slack yariables x 3, X 4, and x 5. Any point on the graph defines speGific values of X)'X4' and X.s. Along the Xl + X2 = 2.6 line) for example, X3 ~ O. The value of x 3 in the region to the right of the line is less than' zero and is thus prohibited, while on the line and to the left of it X3 :> 0, and thus this region is pennitted. l.
12,,7 PREPARATION FOR THE SIMPLEX ALGORlTHIVf The simplex algorithm has a mathematical basis, but the mechanics will be presented before the theory. No rigorous proof will be given, but a geometric explanation will' be provided to gjve insight. inco the the simplex algorithm is performing. , The first step in preparing for the simplex algorithm is to write the equations in table or tableau form. The reason is that equations will be
functions
written many times; instead of repealing Xl, for example, in all the equations Xl wilJ be used as a column headjng, as jn Table 12.1. The double Line is interpre[e~ as the equality sign and only the coefficients of [he x terms appear inrneooxes. ~.
'266
DES1GN OF THERMAL SYSTEMS
TABLE '12.1
.
Constraint equations'in'" tableau form , . , '
,. 1
1
I
"1
'. 11 :.. :. '.,' "' ..
1 -'
7.2
'r.
---+-----;------'-+---:--~t_'_-------'t__-_____ir__.,._-_____i~..____--;--;-~
. , ,:..
;>:.: ''"-:':·;'~':'"'i:..':;,~~:. ~'~~_~:'. ' ."J.!'~';':
'. _', ".
1" .... :
.3.""".
,-I'~ ., ') .
'"J.'oj ; '.:
~
:
•...•• r-
,9.6
. ' :.~>' ,~~:., . ":~, <·t~,:~o:'~:?'.~' ~ ;.:·t:;,;'; ~;:~\ ;.~~' ~":iT.:-.;~X. <,
12.8 mCLpDING THE OBJECTIVE FUNCTION. IN THE TABLEAU ". .
,
.1
,
The bo"ttom line' of Table '12.1 was left plank anticipating the inclusion"'of~ the o~jective function, Eq. (12.4). T.he forin of the equation is ~evise.~·, . however, before' inserting the numbers' so that all the x tenns are move.d to ,the left' side .of theequatio:Q' .
(12.11) Insel1ion of these terms into the tableau yields the results" shown in Table 12.2. The coefficients of the x terms in the bottom line for the objective' function are called diffe.rellc~ coefficient;. 6
, 12.9
' , "
•
STARTING AT THE ORIGIN
The progression in linear-programming solutions is to move from one corner to'the next comer, acrueving an improvement in the. objective function with each move. When no further improvement is possible, the optimum has bee'n reached. The starting point lS always the origin, namely, the position \vhere the physical variables are zero. In Example 12.1 the starting point is where .~~.I. = 0 and .x 2 = O. One teGhnique of indicating these vaJues is to TADLE 12.2
Constraint equations and objective function
in
tabJ~au
form Xl
XJ
1
1
1
4
4
3
-1.86
.-1.56
:tJ
x,.
x$
q
2.6
7.2
J
,
1
9.6 -
0
-
-1.86
. - t.56 .
o
note them in the column heading;, thi~ converts the tableau .into lhe forin of. Table 12.3, wbi.ch.is n.ow the ~Q~plete tableau 1. . .: A property of all tableau?'. throughout the.linear-programming proce- ' . dur"e is that the current values of all the x's and the objective function' can' be'!read immediately from the tableau. In Table 12.3 t for eXaJ?1p~e7 x 1 and X2 are zero; since the first line corresponds ~o Eq . .(12_8)~ the value of X3 is 2.6 .. Similarly, x 4 '= 7.2 and X5 = 9.6. The bottom line is the objective function. iIi the fonn· of ·Eq. '(12.11), so y = O. The number in the lower right comer of each tableau·.is the current value of the objective function. . .. .,
12 .. 10 THE SIMPLEX ALGORITHlVl . The simplex algorithm is a procedure wh~reby the successive tableaux be developed f~oh1 the first ta~leau. Tl?e steps are as follows:
can
1 .. Decide which of the variables that currently are zero should be progranuned next. In a' maximization p~oblem the variable with the largest negative difference coef{ic.ient is chosen; in minimization the variable with the large'st positive difference coefficient is chosen. . ._ ..' . . 2 .. Determine which is the controlling constraint by selecting the constraint with the most restrictive (the s~allest) quotient of the numerical term on the right side of the equality divided by the coefficient of the variable being programmed: . 3. Transfer the controlling constraint to the new tableau by dividing all - tenns by the coefficient of the variable being programmed. 4 .. For all other boxes in the ne~ tableau (including noncontrolling constraints and difference coefficients) use the following procedure: . a. Select a box in the new tableau. Call the value in the same box of , me old tableau 'V. b. Move .sid~ways in .the pld . tableau to the coefficient of the variable bemg programmed. Call thls value w.
268
DES1GN OF TI-lERMAL SYSTEMS 8-
C"
An
the new tableau ,move from the box being calculate~ up or down to the row . which contains the:previ;O~s controlli~g equation. Call the . . J . . value in .that box z. '. '" . '.... ,
d. The vallie of the box in ~the new taqleau 1S v -""w z . .
'.
- ':. .-:~.-.
~i:~. ~:-:
.
. ,. :1 ... ·'r.
12.11 SOLUTIOI'fOF- EXAMPLE
>rr!
.II",'
;..~
""';:.
•
•
•
12~~'" •
, _
_
~
•
•
~
·The·.simp.1e:~ ·a1.gorithm,.p.~,~~nP.e? ~~. . S;~C;.., }~.~.~Q ~in .n
T"
Step "I. The' firs~' s't~p 'is to "decide' VwJhich of "the 'variables ~urrently' noted. in the col urnn heading 4S iero (x] .or x 2) should be programmed first ..The l~gest negative 'difference is -1.86 in the x { column, which indica.tes·Jhat. . x 1 'sho~ld be programmed (increased from' its zero value). Th~ vertical ~rrow ~ indicates tha~ x 1 is .being progranuned. "
Step 2. 'The next step is to ,determine to what extent x 1 can be incre~ed. The numerical te~s to the nght of the equality sign are divided by ·the c'oefficients of the variable being programmed in the same li.ne~· 2.6/1, 7.2/1, and 9.6/4. The smallest of the resulting quotien.t~ i$ 2.4, as shown jn the left column of Table 12.4, so the third constraint is the controlling one. In Fig. 12-2 [he operation in moving from. tableau 1 to tablea'u' 2 is that of nloving along the x I axis from A to' B.' The quotients' jn the left column denote values of .X J that the respective constraints permit. The first constraint would pennir x 1 to increase to 2.6. but the most restrictive is the third constraint, which permits x I to increase to 2.4. . The construction of 'tableau 2 in Table 12&5 can now .begin with the designation of· the column headings. T~e' variable x J was zero but is no longer and . .,-.2 remalns z.ero. The variable x which was nonzero) now
5:
I
TABLE 12.4
Tableau 1 of Example 12.1 ",ith indication of variable being programmed and controlling equation
.u ~I=O
:2.6/1 = 2.6
J
x~::;o ~
XJ
.%'.-4
-
X.s
I,
I
'2.6
..
7.111
,
~.6/4
-
.
= 7.2
J
.4
~A
4
, .3
-1.86
- 1.56
::;
7.2
I I
9.6 0
LrNEAR ~.~OGRAMM1NG
~bleau.
2 of Exam p le 12.1
? 59
) !
{It
}~-:::-(l)(D. :==q~:. -~_J.~·(!)(O~7~»·: · ·I:":-(l)(~)-·-· ~O-(l)(O). \. t
.~ .. ::
........
~
I'
-
,fA·"
=
O··.~~l_.":.
.:._.. =
~.:-·-(1)(O.75) . =3.25·
i
3:2
:-
.
.
.0.75
"':"'1.86-
-1.56-
(-1.86)(1)
(-1.86)(0.75)
=0
= -0.165
~
, ...... ::::=.?: :,: ~,;: .. ;...
·0·-(1)(0) ==:0
.0 0-:--
(- 1.• 86)(0) ·=0
0....:(1)(0.25) _; _.
2.6-{l){1~~·L
,.;:::...:-.Q~2~ :.~: . ~_.. . ,;~g.:~?:·;,..,~.~~;:i·~::"~
1~(i)(Or\·· '"Q"-(I)(CL1$j ~ = l = -0.25·
-~7.2:;("i Mi4) =4.8
O~25
2.4
0-
0"":'-
0- .
(-1.86)(0)
(- 1.86)(0.25) =0.465
( -1.86)(2.4)
0
=0
I
=4.464
.
becomes ze[o~ as indicated both by. the geometry in F~g. 12-2 an~- by the third constraint in t~bleau 1. If that'constraint is thought ·of as an eql.latiop" . the only. v~ables taking part in the action of the move ate x I and x ~. All ~he other varia~les .are .either zero or have zero ~oefficienis. The variable x I "increases until x 5 has dropped to zero. S~~p 3~ The first numbers inse!1eq.in the boxes of ~ableall 2 come from -the controlling equation of tableau· 1. Dividing all the coefficients of the thjrd constraint by 4, which i~ the coefficient of the variable being programme(t the numbers in the boxes of table~u 2 for the third _constr~int become-'1, 0.75, 0,0, 0.25, and 2.4 ..
Step 4. For all the o~er boxes in tableau 2 the v - w z routine is fpllowed~ The individual calculations are shown in the boxes of tableau 2. Tableau 2 is now complete. It contains the v~lues of all the x's and the value· of the objective function. Two of the x's, namely x 2 and x.5. are zero. as shown by the column ·heading. for each of the other x's a constraint equarion will provide its value. In the fU'St constraint all··the variables are eiEher zero or have a zero coefficient, eX~~Fr. for x 3. ~o the equat~on has reduced to· X3 = 0.20. From the other two constraints, X4 = 4.8 and Xl = 2.4. The box in the bottom rign~ corner indicates the CJ,1rren[ vaiue of the objective function· [0 be 4.464. .. . The e'D.tire simplex '~goriuJn is repeated to transform tableau 2 into tableau 3. ~e largest negative difference coefficient,'~ fact the only nega-· programmed next. This lime the first tive one. is -0.165, under Xi, so X2 is . .
constraint is most restrictive; it therefore becomes ·the controlling equation. and as X2 increases to its pmitt XJ drops to zero) 8S shoWn by·the column heading in tableau 3 (Table 12.6). The coefficients in the ~t constrainr of tabloio.2 'rue divided by O~25 and the tenns transferred to tableau 3,
.
270 ~
DES1GN OF THERMAL SYSTEMS
TABLE 12.6
'1;ableau 3 of Exampie 12.1. I
x,]
o '
"' 1.
=0
' Xs
o
4
'. ';f'
. _, ;-:.13
,
=0 . 0.8
-) "! '
,
J,' , 3
'," 2.2, '
, -
'"
<:0
:
. ... ... "
\-;j~.'.:.<~(. )~.:.t::'~ :;. ':.~:~:::~'~" . "~".:~~:.' ':.'..'~3;:"-" :~7i-~~,.:?:f }~:v:;,;.4;:~.·:..c~!: :j{f.'~'h~;(> •
•
-
'
",&.
.
,"
...
are
'The remaining' boxes computed according to",step 4 in Se~. l~ '.lO. Th'e' complete tableau .3 shows :x J =, 1.8, 'X2 = O.8~ .:f3 = ,0, X 4 = , 2~2) .;\;,~ " 0. " ', and}' = 4.596. . , search for ' which vanable to program next discloses that there no 'negative differen~e coefficients, and this condition indicates that no ' further improvement is possible and the optiml)ID has been reached. The' transformatiqn from'tableau 2 to tableau 3 'was'a rt;love 01) Fig. 12-2 frqrn point B to point D. At the solution two of the slack variables, x 3 and x 5, are zero, but x 4 = 2.2. These values indicate that the second constraint has no influence on the optimum ~ although jt might not have been pos~ible to rea] ize this fact in advance.
, , .The
are
,
,
12.12 ANOTHER GEOMETR):C INTERPRETATION OF THE TABLEAU TRANSFORMATION Figure 12-~ showed one geometric interpretation of the linear-programming problem and th~ progressive ' moves from one comer to another. Another geometric interpretation can be achieved by changing the coordinates with each transfonnation so that the current point is always ar the origin.' Since the first tableau start~ at the origin of the physi ca1 variables. the coordinates x I and X2 in Fig, 12-2 represent the first tableau. In moving from UlbJeau 1, to tableau 2 the coordinate x I is replaced by x 5. The procedure in making this rep.!ac~ment is, to solve for x I in the third constraint xl
= 2.'4 -
0.75);2 - 0.25,\'5
(12.12)
- and substitute this expression for .x 1 into the first two constraints a,nd the objective function to obtain l Consrrain-t 1: , Constraint 2:
Constraint 3: Objeellve.-!unction: ,Y
X '2
+
4X3 -
x5=O.8'
13.r3 + .x4 +
3.xs
+
).: 5
">:1-
+ O.66x:\ ,. .
3xJ
+ O.3x;«i
= 2.2 = 1.8 = 4.596
(12.16) (12.17)
(12. f8) (12.19)
LlNEAR PROGR,\yIM!NG
171
"
, i,
J )
.' 3
... r'l\'.
'~
-,
'
PeJ,lllined region
o ~~--~----~~------~------~------~--~
o
2
FIGURE 12-3 . . Tableau
~
expressed on .r SX J coordinates.
The three constraints, Egs. (12.12) to (12.14), are shown on the XSX1 coordinates of Fig. 12-3, as we]] as a fe\v lines of constant objective ftinction from Eg.· (12.15). An examination of Fig. 12.3 indicates that to improve (increase) the objective funccion .r~ should be increased until a constraint limits the advance. The restriction is imposed by constraint 1. which occurs when .x 3 == O. so [he coordinate X2 wi]! now be replaced by XJ. That replacement wjll be p~rfonned by solving for X2 in Eq. (12.13) anti substituting into Eqs; (12~t2)r (12.14). and (12 ..15). The resulting set of equations is . Cons~int
J:
X2
CQnsrrain[ 2:
- 13x
Constraint 3: Objectiv~n:
3"
xI y
+
+ 0.66-\.3
4X3 -
X~
+
-, 3X3
+
X.s = 0.8
(12.16)
= 2.2
(12.17)
1.8
(12.18)
= 4.596
(12.19)
3x.s
X5 =
+ O.3xs
DESIGN OF THERMAL SYSTEMS
I J
1.0
-p"o"
"'Y,...,."'P<:~l"rl
on
XSx}
,,-oJ
:.
":
coordinates.
onglTI because an j
is
or
the optimum f\1rther· improvement . about th.e transformations where replaced in Figs. and 1 the equations on which ba~ed. (12.12) to (12.15) are placed in a tableau column 7 is A Table 12.7 tableau 2 12.5) shows the two ~ (12.16) to (1 19) could also tabulated to show tableau 3. conclusion that ....... "............... as a progressive transformation coordinates replaced L<.... ..., ...
TABLE 12.7 (12~ Xl
Xl
0
-
Xl
J
0
3.25
J
0.75
0
x,"
X-t
I
1
-0.25
0.2
-0.25
4.8
O.:!S
2.4
0.465
4.464
I
0
1
-
...
....
27:)
LINEAR PROG RA!'viM ING
1 2~8
TABLE
C ~ ..11P:'] .~
L,.. rn of . abJe 1. 2 44 {Y
) )
.I .
2.6
I I
" ,:4..... }.j;!.
>
;"-7:l~L ~'~';X"-::::.:::;,.::y::.;",,,":,,.~.~ .. O~,.;~<:,~ :
~.
~",
.
~.6/4=2_4. ~.~
-'
.'(5' ..... :,-,.,\
. . . .. . .,;4 ~
".,
.;,,,"L:L,,. . ;'!·~.'·"·i.",:··"-' .'., ;;~,-.:"" .~,
"
~
- 1. 86
:.:.. 1.56 .
.".
:
~
. 9.6
", .. , .D.3·· '.' I
.0
. TABLE 12.;9
Compact form of Table 12.6 . . XJ
'XS
."(2
4
-1
0;8
."('"
--:-13
3
2.2
-3
I
1.8
0.3
4.596
I
XI
~
I !.'
0.66 --
12.13 COMPACT FORM OF TABLEAUX Examinarion of Tables 12.4 and 12.6 shows that the coefficients 'in the ·columns headed by nonzero values of x are all zero except for a value , of 1· in the position that keys to the numbers in the righe-hand column .. This pattern suggests that just as much information can be supplied by the .
tableau even if arranged in a more· compact form. Tables "1~. 8 and 12.9 are conversions of Tables 12.4 and 12.6_ respectively. In Tab,le 12.8, for example, X3 = 2..6,x4 = 7.2, and X5'= 9.6. Because XI is':the variable' to be programmed next, the -right-hand colu~' values· are divided by the coefficients in the x I column to find the most restricitve constraint.
1).14 NUMBER OF VARIABLES AND NUMBER OF CONSTRAINTS
of
The relationship of [he number of physical variables a.nd the number ~onstrain~s gives an indication' of the number of variables that are zero in lbe solution. Let the number of physical variables be denQted by n and the number of constraints (and therefore ~e number of slack variables) be denoted by m. There wjlJ aJways be n variabJes (physical pi us slack) equal
- -
274
DESIGN OF.THERMAL SYSTEMS I;
m>n
m
(~)
(0)
.
BGURE 12-5 of number of physicaJ and slack ~~abl~.
~elatioD
to zero at the optimum, or' for that J?attei ~t any corneL When ~ > ~, as :fig. 12.,.5a shows,. at least m - n constraints play no role in the solu~ion_ In Fig. 12-5b ,where m < J1! at least n - m physical variables are zero.
12 . 15 MINIMIZATION WITH ~ GREATER-THAN CONSTRAINTS Solution of the maximization problem with less-than' constraints consisted of moving from one comer in ¢e feasjble region to whichever adjacent comer showed the most improvement in the objective function. Since linear programming always starts at the physical origin, the origin is in the . feasible region wi.th Jess-than constraints. In the minimization problem with greater-than constraints locating the first feasible point may be difficult. Admittedly, in 'simple prob1ems involving a smaJl number of 'variables, co.mbinations of variables could -be set to zero in 'the constraint equations and the other variables solved until a combination )s found that via] ates no constraints. In large probJems this method is prohibitive and a more systenlatic', procedure must- be employed. The introquction of artificial \'ariables facilitates this procedure. I
12.16 ARTIFICIAL VAluABLES Suppose that an inequa1ity constraint with a greater-than sense is to be
, convened into ~n equality but, also must permit the physical variabJes .rake on zero values. The inequality
-
.
3x J
+ 4X2 ~ 12 . ....:.. .
to
(12.20)
Cl
be cOTI1/erred into an equality
3x I , +
by introducing a slack variable x 3, ' 4X2 -":'X3 -L 112,
(i2.21)
'Th e slack v-3Jiable takes on a negative sign t so that the constraint is sati~fied when x) > O. . , ........ .
,
, ,The next requi~ement is::the., ability :to' set Xl
and X2 equar to, zero. 'Eq,u.ation' (12.,12) wilI·, noi~~peJ:!nit,.,ier6 "~vall_h~s- "'jf, .,'c{. :,.and{-c :h\~lli,:d~ 's,d -anoth6F':,·,,:,,:< " vanabi~",r.h~ ~cial variable" is'Jinserte~',:::'::,~,, .~:_~; . _. -.~:. -, . > ..," ( 1~.,22) ,
The geofIletric mterpretation taken on by' the slack aq,d artificiat variaples with respect to the constraint is shown in ,Fig. 12-6. Along the constraint 3x 1 + 4.x2 = 12, x3 == 0, 'a nd X4 = D. When moving to the, .rig~t of the constraint into the fe?Sible region, X3 rakes on positive vallJe~ and X-I remains zero. When rnovin,g to the left of the constraint into tpe ~feaslble regioD~ X4 takes on positive values ,and :,\) remains zero. The result of the , ,inrioduction of both the slack and artificial 'variable is that the position may be located anywhere on the graphs but all the variables are--s~ill abiding by the requirement that .x j :> O. i\.o uneasy feeling should prevail at this point because the arb~trary addition of terms to equations is no( orthodox m().rhemati~s_ Further treatmen [ ,o f the artificial y~ab]e is necessary, and this treatJ!1ent will be explained O-.? part of Example 12.2.
4
3
2.
5
FlGURE 12a.6 -
-
Slack acd art.i.fici31 vuiables.
-
276
DESIGN OF THERMAL SYSTEMS
12.17 SIMPLEX ALGORITHM APPLIED TO lVill.aMIZATION WI H ' '. 'GREATER-THAN CONSTRAINTS) I Ex'ample 12 ..i. Detennine the
arid x 2 at this mlnjrnwn',
minim~~ value of y an4 the magnitudes of Xl
~here
-, '
}
~
• ....c.... .
~~~:.~ .>.::-;.... ~,' " , c
,
subj ect. tb·~'the~ coristrain ts '\ 'S; t: .;:.""; ,~, ,';I~, ...J,:" ".',',., ~' . '.;.j:'.,..: ;. ~ '~....,...; .' '.' .'.' ". 5~1~:;:'~~I~-:"';' _' ';..0-
<'/";" , " . .
9x I
+
13.i~ :>
-
".,;><, .;,/;ji£ ' '':»:. Y
,
':.F
74, ,
Solution. ,Because iQjs problem inyolves ',pnly tyJo- physical variables, the constraints and lines of constant y can be graphed,; 'a s in Fig. 12-7. " For the sqlution by , li~ear .prograrnn;:iing, first write the constraint inequal ities as equations by introducing the slack vari abIes x J, X 4, and x 5" 5Xl
9x I XJ
10
\ X
+ X2 ,+ 13x2 +, 3X2
X3
=
10
=
74
-x.s =,9
,
\
'\
/,,33
~
\
\
,
\
\
\
\
6
."=27
\
,
\ \
\
\
.\
\
\
\
\
\
\
\
\
\ \ \
\
o ~----~------~------~------~~----~--~
o
2
4
F1GURE 12·7 '-1inimi2.ariol1 in Example 12.2. ,
----.
8
10
277
LINEAR PROGRAMMING ~
l'-Jext ;:ile lli-rlficiai varia bIes. x 6 , X 7, a,nd x 8 ~) are inserted In each 'eq uation
+ . x~ ~ X3 9X l + i3x2 -
+ -'.:6
5x ~
" -.' 3
.x,\ .+ ~::"
..
,f' ': .:.
.
......
~Tc,w
-J"
."X2. ' . .
X4
".
: ••-
+
.
I., ~ ~X5.,-·. •
I
••
=
•
X7
~C
'= 74
(12.24>'
,~
.·(12.-'J~)
' . '+"~-O • • : •• , .
~
_7. ':/",
the ·'remhiri1t.ig' c'dn'di'tio'hS su~oH~dT#g·tlle"artificiarv~abl~'s '~e-"sp:~~ified ""~ .,' :
ill·that the,o:bjecti~e,furi~tforl ~s'a1so r:~:yiSed'as folIows':' '.'
....
. ( 12.26)
,TI::e coefficient P never assu.mes a numerical. 'JaIue but is only' considered [o.be extremely ·large. The' existence of the products of P and the artificial variables in Eg.· (12.26) is so penalizing to the minimization 'attempt th~n no' satisfactory minimization will OCCUI- until the values of the-artificial variables have' been driven to' z e r o . ' ".' The linear-programroing process can now ~ at the origin in-Fig. 12-7 with all the slack varia,bles equal to zero but all the artificial variables having , positive values. the simplex algoritl?m applies~ but Qne further op~~tion is performed on the objective function before writing the first tableau_ From' Eqs~ (12_23) t9 (12.25)
-:JJ
Subs~ituting
X7
= 74 -'- '9x I
~'(' g
= 9-
-
13x2
Xl -
3x 2
these values of the artificial variables into Eq. (l-?26),and group-
ing gives
y = (6 - 15P)x,
+
(3 - 17P)x2
+
PX3
+
PX4
+ Px~ +
93P .
The first tableau can now be constructed. In the minimization operation the variable with. the largest positive difference coefficient. is chosen to be programmed. In tableau 1 (Table 12.10) this choice wjll be X2 b~cnuse P appears with the largest coefficient and the P values dominate over any pureJy numerical tenns, . TABLE 12.10
Tableau 1 Qf ExampJe 12.2
-
.
.'
.ij.
,
~-
Xt
......
.ra
",--,
XJ
-1
1()
-1
0
74
0
0
-1
9
-p
-p
-p
93P
I
, 74113
x,
9
1,1
Q
~,3
x,
1
3
15P - 6
~
.
J7P -3
x! 0
5
-- ..
I
0
. X6
10
%4
" .
278
DESlGN OF THERMAL SYSTEMS
TABLE u.li
Tableau 2 of Example 12.2
, =>
,', ~ 6 .
312
~
;
I
,
•
"
9'
'-1.
..14/3 ......~: .. .. . '
" , t;..
.
, Xs
. X,3
f"
: '-:<" :,,~,' ,~f65/i.4<,':', ':~'7 " ..' ;:'i'4.t3,' ,;' , ,.
l'
{It
"
....
4
: .....
~
0, ...
~
....
'. -0 ~ ,;- r
~
113
. . . .. . .
'
~~- .",
':
.
~}L~, ,~,~i:'
',1"~'
13/3 ' "
:" .;,
.~, ?: ~ ',~;' of: , ",~:~, .:,~,;y::~~: ,~.T
''-:1313 ~ :' ,'. ':'35
':'; ""'<~
~ :. "';;"~' :.' •
1/3
.t')
.~
O
·
-113
0
3
.·113 "
____~~ , I _-- +________~--~----~----~----~--_*~----
28P -,15
i4P - 1_ .~_::r7p 3, 3
'-P
-p '
'3
+3
42P +9
I .
TAELE 11.1i T~bleau 3 of Example 12.~ , II
X3
I ~1
XI
~7
X7
-7 . .r ~
-3/14
'4-
1I14 14P - 15
14
0
-p
..1'8
3114
1114
0 -1
I
X6
Xs
X.t
.
-]
J.
-11l4
3/2
-4
28
-5/1~
- 1114
56? - 9 J4
-'28P + 15
-70P +9
14
J-1
5!J4
512
56? + 33 2
TABLE 12.13
Tableau 4 of Example 12.2 x.\
x,
-13/56
.1',
x,
13/56
- 11.56
x~
1/56
x{
1M
- 114
-1/~
.\' :
9/56
-5/56
-9156
-51/56
-9i56
-
-.
5/56
+ 51156
'-p .
,
J/4
,
:...p + 9/56
Xs
0
j
-)
7
0
5
-p
2.J
.
~~
,
279
LINEAR fROGRAMML~G
~, ~1f~' ~t'
~~"~\e
~_w,~ ~;,~,r', ;,)." kW'
V' ' ( ;"~'-'
:':'-'
-"~
, "
next question is this: \Vhat ,i~ the ~imit \::) whic ~'( 2 can be increased? J liSt as in the m:lXin1ization process, the varlabloe x 2 is incr~ased to the most lin-::,~tjng constraint. which in this caSe is the rhird ODe. ,The procedure in . ' trLDsforrriing to tableau 2 (Table 12.11) is the standard simplex algorithm~ The differenc~ coefficient with the) largest po~itive' value is (28P -:- 1,5)/3 ~, 'so X'l is pro gran1fl;ted nex;: until' "lim;ted by.' tr)e ,point ,where'~"C 6" starts' to '~O ',negatlv.,e i-9-.-J l?:e=:,fi rs,~' cons'v.~jnt. ")n~t'ableau;, 3':..'Tabje'-12~'f2 yi ~':"n'hsL"~[he<'-rfu:g{~~':"':'~::' t positive' difference'coefficien1:,"so,Xj" is'increas,e;d 'untjf it fI!a'ches tIie"limi"t'ini: "'~ , value of 7 beyond wh'lc~ X7' would 'be~~~~ '~~gative.' , , ~ In ,tableau 4 (Tab Ie 12. 13) all difference coefficients are ne ga rive 5C no fuI1he~ ~eduction in the objective function is possible. The solu[i~n is
',: <,"
y
q
••
'-
,
•
~
••• _
•
"= 1
Xl
and
I
.....
,x; = 5
at which point y'" = 21.
12,,18 REVIEW OF lVIINIlViIZATION CA~LCULATI0N ' , Now that the I!linimization process in Example 12.2 has been completed .. a reexamination of the successive tableaux in the problem will present a IDC?re complete picture of the operation. The introduction of the artificial variables in~ the objective function 'and, the constraint .equations pennirs a tempor'hry violation .of the constraints but only at the expense of an enormously l~ge value oftbe objective function. The solution will cert~inJy'not be satisfactgIY ,until all ~~."f terms are r~moved from the expression for the objective
functioD_, ' In contrast to the maxiI1!ization problem, e.g., Example 12.1, vv'here the constraints were like solid walls tpat could not be surmounted, in the' minimization probJem the constraints are like stiff rubber bands that can be violated temporarily, but "with a severe penalty in ,the magnitude of the .objective function. Looking first at tableau I, we see that the position represented in Fig_ ' 12-8 is the origin, because XI = X2 = O. The slack variables X3, X4, and x 5 are also zero, but the artificial variables have nonzero values: x 6 = 10, X7 = 74, a'nd x 8 == 9. The value of the objective function is 93P, which is
prohibitive]y large. In tableau 2 the nonzero values of variables are X2 = 3. x 6 = 7 t and X7 = 35. In moving from point 1 'to poine 2 in Fig. 12-8 t the magnirude of the objective function drops from' 93P to 42P + 9. In other words. the pure l1umerical value increases by 9, but the staggering P term decreases from 93P to 42P. The next shife is to point 3, 'represented by tableau 3. where the nonzero values are Xl == Jh X2 = '12, X7 :::: 28. The magnitude of the objective funccion---is (56P = 33)/2, .., showing a continued increase of the t
280
DE.S1GN OF TIfERMAL SYSTEMS
} J
. " .. .. .~:, .
. : '>'.' •
.'
j - '. ~ "'" i·
'l. - ~-,
.'
,:. ':-:~"::' ~. ~~ ~ ~ :.' :&.~.. : . ':.. ~'..
.
,X4
=
'~~. • .~;~~.'
},t"
, ... ~.
"'~,:~;- . ..\
'\.1, ;.:
~ .
.
I, = 0 X.s=Xg=O
12..8 Points
repre.~=:n[f~a
by successive tableaux in .hx2l.tnple 12.2.
numerical portion
a
x 8.
At this point and x 6 PO-int 3 requires a positive \lal ue of the -x 4 + X 7 a zero of and a positive value point 3 to point 4, where the are xj = All the artificial variables tableau 4 is
point .........,......,..,. . . . .,. to move around
Lll"'\'EAR PROGRA.t\-fM1NG
One :SL,-rher cornn1ent should :be··made:· regardin~ taoleau 3, \vhere, f6novv.""ing 4?E: .decision to program .:C 5, th~ ]iriijt~tio'n for increasing .7F 5: ~l1:~. 'bY.J9.e . . .
': ;·,:·...~·~tlrst. ,
-
.....
1- .
-.
.' .
..
\ '; .... . f
... ~ .,'~ , .. '
•
,
equation, \'!a.s 2Irby··the· ~se'Co'nd -eciua~9n' 7;ahd 'by t~e 'it1iifd :" ~qy,~/16jr""
· --: 7 .:~ Strict sonsistency' vlould have' req'uired us to' cl1Qo?e' 'the 'rbiter a's ~ the.·· '.' controliing equation_ Geometrically, in Fig_ 12-8~ the t~st to find the ' COTI,trolling ,equation consists of moving along me -X:3 + x 6' = 0 line. Beyon~. point fl" \vhere X5 = 21, the value of Xl \vQuld' go negative'. Beyond 4, where Xs :=;: 7, X7 would go neg~tive. The linlitation C?f Xs = ~7 is represented by point fJ'~ which would resu~t in a step backward, so the'limitatiops with negative values are ignored. .
12,,20 EQUALITITES AND SENSES OF INEQUALITIES' · The two classes of linear-programming probleT?ls considered so far are (1) maximization \vith less-than constrai~!$ and (2) minimization with gre,atert~an constraints. Physical problems ,somerimes require mixtures sense.s · o( the constraints,' i.e:, some less-than and the rema.inder greater-than constraints. One or more equality constraints may also appear in combination with the inequalities. The next several sections explore these combinations and certain other siruations introduced by such physical requirements as mass balances.
0=
12.21 . MIXTURES OF EQUALITIES ,AND INEQUALITIES AND OF SENSES OF INEQUALITillS \. A few general guidelines are applicable to se,tting up problems where the constraints are mixed. These guideiines control the introduction of slack and artificial variables. In [he two classes of problems treated so far, where the constraints were all inequalities with the same sense, the insertion of slack and artificial variables accomplished two functions: (1) converting all inequalities into equalities an~ (2) permitdng 'programming to begin at the origin of the physicaJ variables. The same objective prevails if the ,constraints are mixed. One additional variation is that when an artificial 'Variable is introduced, it must appear in the objective function as + P x I if the objective funccion is being minifnized and - P X j if the function is being
maximized. EXDlnpJt 12.3. Find the maximum of y, where y = 3.r J + 2t' ~
+
4x 1
OF TIiERMAL SYSTEMS
J..J.r;:;...:J.IU1"
subject to- .
+
3X l +
(
= .:
+
.
9
(12.28) ,(
2+
is aITeady an' equality, to start at into Eq., (12.28) ...,......J,p;;;, ... 4JLU.,...........u,.f'o
,
'Xl +X2
"
+x3+ x s=9
( 12.30)
, f u n c t i o n ~ust also be revised so that x 5 will ultimately be driven to zero to "
+
y =
(
1)
a the function and [he maximum should not be reached IS
until X.s has shrunk to zero. a variable. The slack variable
-X6
an
converts the inequaJity into ~ equality. to .start al: origin. Equation (1
with
,tem revises Eq. (12.3]) further )' = 3x 1
the an~,.t7
+
X7
- P.t oS
int9 -
P X7
step before x .5 from Eq. (12.32) into the objective function. Eq.
y =
+(
1
+ (
,.,., ... ,-n"..,.
function (I
02.30)
ur'fEAR
-:{
1 2.~ 4
I
Tableau 1 of Exanlple
~OGRAMMING
. J.
{tXl
.. ' I I
.x J
'. 4013 .1 ;". , .0:'_ ,: •.•
'x s·
·····: ·1 ··
v -' . . : ..:.:. :'Y-;:'
7
. .. ,~: ~
'.
4
-8P.-3
.
-1 .
4
-sp -
.. .
42
-sp -4
2
TABLE 12.15
Tableau 2 of Exampl~ 12.3 It;
.r7
23/7
317
-3n
22
317.
In
.. -In
3
411'
-117
In
6
-3P - f6
-I: -
SP +3 7
-3P
Xl
XJ . '
:?154/23
7
x~
1617
Xj
3n ..
4214
- 4J7
XI
-3P - 2 7
3
7
7
+
18
TABLE 12.16
Tableau 3 of Example 12 .. 3 . {} r ..
Xtj
7n3
3123
-3fl3
154123 '·
-2f23
3/23
Sn.3
50123
Xl
16J23 .
154J16
Xl
~1
Xj
3/23
-3/23
2123
XI
4123
-4123
::-50?3.
2512
X7
'
..
.'
'.'
-3P
+ 30
23
......
3P
+' 16 23
-2P - 3 23
25P
+J
23
-3P
+ 766 23
284
DESIGN OF THERMAL Sy~S
TABLE 12.17
Tableau 4 of Example 12.3 . x,
.r
: ' -18
' --1613" 23/3 '
":"':6 "
,XI"
"
}),
~1J3
,.,113
213'
-2)3
6
::"~~13'~'·;..' ,~ .J'.[ ~~~~:if~';X.': ':~"::' ;';'~ iI~:>':.}:'.
" .. ;::2.-t:~~;~Vl ~.~ ':' ~;;"~~ ;" "-!:-"~', ';. ~: ;.;~' .. .' :
3f' - 30
3 ' .
l'ABLE 12.18
,
-]
P
+
..
I'
••
-
••
~ " '.
~..
• ••
32
1
.
Tableau 5' of Example 12.3·' Xl .\,3
112
.t6
3/2 112
J: I
J/2
X~
.XS
X7
'
,
-
-
1/2
-3/2
-312
23/2·
-112
5/2
]/2
I
2P + 3 2
0
1312
-I
312
0
512
p
33.5
I
..t';
tableau 4 is now in the feasibje region. The optimal soJution, then, is = = O.x)• = 6.5, an d y .. = 33.5. , "
2.5,x 2•
12.22 MATERIAL BALANCES AS CONSTRAINTS In the application of optimization methods, jncluding linear programming, to (hermal systems. materiaJ balances often impose constraints.' The inputoutput materia}' balance ofren results in a unique siruation during the simplex procedure that should be recognized in order ro' provjde the interpreration that allows the procedure to continue. The situation will be illustrate,d ,by an exan1 pJ e ,
Example 12.4. In rhe pro~essing plant shown in Fig. 12.. 9, the operatjon is essentiaJJy one of concentmting material A. The concentrator receives a raw matedal consisting of 40 percent A by mass nnd elln supply two products of 60 nnd 80 percent A. ·respectively. The flow rate of the raw material is designated x I metric tons per' day and rhe 60 percent and 80 ptrcent products are designated .r~ nnd :t';\. respectively. The ptjce$ are:
-.
~.
LO-lEAR PPrIJOR.<\MMING
.1.";::.60% Processing plan[
40%A
285
, 1
.'(3;80 %
nGuirnri~9
• <.. ,
..
'. ~ Discbarge ~'~'" .. ··.cpnc~ritt&or in ""Ex..aInp!e " l1'.4~ .;. :
Amoun[ Price per metric ton
T~e
$4Q '
S80
S120
capacitY of th~ . loading facility. imposes the constraint 2r;!
+
3x J
:s 60
betermine the combination 'of raw 'mat"eri~ and products that results in maximum profit for the plant. Solution. The objective function is the difference between the income of the productS and the cos t·of [he raw materiaJs
y = 80X2 + 120-'3 - 40-, 1 In addition to the constraint of .the loading facility , the mass balance of marerial A imposes another constra in t that can be expressed by either Eq. (12.34) or (12.35)
+ 0.8-'3
= 0.4,t 1
(12.3 4)
O.6.r, + 0 .8,t3:$ 0.4,r ,
(I 2.35)
0.6x ,
Equation (12.34) is • strict equality for material, while Eq . (12.35) allows for the pcssibility of dumping some of material A during the processing . In some rare instances the most economical solution can be achieved by dumping some material of value , so in the solution presented bel ow the possibi lity of dumping will be used and Eq. (12.35) will be chosen. The mathematical statement of-the problem becomes:
Maximize
,y
subject to
-
80-'2 + 120-'3 -40x, 2.<)
+
3X3
s 60
+ 0.6x, + 0 .8x, s 0 Taole i 2. 19. The yanabloto
- 0.4-'1
The first tableau is shown in be prognuruned for which the most ",strictiy. constraint is the second one. Using first is the simplex algorithm gives tableau 2 (Table 12.20) .
x,
,D~IGN OF THER.MAL SYSTEMS
286
• . TABLE 12.19 Tabl~au
)
1 of Example 1.2.4
)
----~----------~--------_r~------~~~--~_ff-----.
20
12
,.1" ~ .
.
.:<: .. ··.·-~,~~~Oe,~.O-·'.;.·)'''-~x.5~.~~·~
"
"~60
"< ~":':~~~ .~~'"
o
·3
.
.
·::r·T~4 ~.~: :~~~;:;";,q'~P~"T.o ... ,
.. .-120 ":
'. -80 "
40
'
The difference coefficient of x i in tableau -2 has the 'largest negative value. so is programmed next. The straighrforward procedure would indicate that th~ < 40. The key featUre'in solving this . class of problems~ however, is the inrerprcration of the indicators of control. \Ve shall. interpret the zero as a "negative zero and, as is the practice, ignore constraint associated with a negative number in the left column. The . fi~st constraint is used as the controlling one and 'results in tableau 3 (Table 12.21)- No negative difference coefficients re~lain, so the maximum has been :. reached: x~ = 40, = 20, andy· = 800. .' .X J
, second 'constraint controls, beca~.se 0
H
any
x;
Let us now return to tableau .2 and e,xplor::e what, would have .occurred if the second equation had been chosen as the controlljng ·one. That tableau -\vould have been' as shown in Table 12.22. The· values indicated by r)1e TABLE 12.20
Tableau 2 of Examp~e 12.4
.vx~
XJ
:::;:'40
-O.~5
1.5
XJ
[I
Xs
-3.75
60
\
0/-0.5 == 0
X.l
-0.5
0.75
-20
10
TAnLE 11.21
Tableau 3 of Example 12.4 Xl
-D. J 667
Xl
0.833
.r .'
-
13.3)
.
X.a
0.6667
0.33-3 J3.~J
, x!-
-2.5 0
)00
f
40
20 800
1.25 150
a 0
I~JLBr • I ="i1 Incor ""cf rf a lea'il 3 of Exam.pIe 12 .4·
o
60
'... , ,~ ... ,.. ~..:';"" x t' _" -. ~ ;'"'~ ~.5 ..;...;~::.~;:-:- ?-:"':': .'.;.•.. >~.~..:.,
.....~,.
.
'~:io ~·<-f"··~40.:··,·:,.·;iOO· ·:;·~
I I
.:~
•
'to
:,,~Q ~": ,. ;, .... -..:..-.......,:"~':.':~:r~~.~:;,''''''''::' --, "-':"~ ,::: '. ,. .: .: . . ~.: :>:. ,"' ':":~'" :"0 ..... ,.:...... . .-
4.
t~bleau in Table 12~22 are Xl = X2 = X3 = X5 = Q,.andx4 = 60. 'These "XE precisely the starting conditions of tableau I, so the process is in an infi~re loop. The interpretation of the negative zero breaks" open' [he loop. . : The phys'ical explanation of what :occurs between tableaux 1 and 2
is as follow's _ Most .profit can be made by selling:.x 3, so this variable is .understandably .the one .to be iIi~reas~d flISt. .But tab~eau 2 show that x 3 is still zero~ and the reason is that the second constraint, the material Qalance~ will not anow x J, to increase until some raw material .x I has also been brought in_ By the shift oJ the slack variable x 5 to zerO' 'in the column heading~ the next prograrruning operation wilJ assure that .x I is increa.sed in proper proportion to the product x 3.
12.23 PRIlVIAL-DUAL PROBLEMS J '.
The primal/dual pattern \-vill be illustrated by two examples, EX;iroples f4.5 and 12.6. . EX3mple·12.5. Maximize subject
y = 4_tl
2'( ~
to
+
+
3X2
x:!::s;'
8
and
Solution. The' initial and final (third) tableaux are shown in Table 12.23 yie,lding [he oprimal resuhs of-x; = 3. = 2. and y ~ = 18_
x;
Example 12.6. subject to and
Minimize
y = 8wJ + 12\1,/2 2w .. · +<2W2'~ -4 -.-.Wj
+ 3Wl
2: 3
Solution. The ftrst tableau is shown in Table 12.24. and the fmal tableau in .-.- Table 12.25. The optimal vaJue~ DIe = 3/2, wi = Jl2, and Y· = 18.
w;
The Jinks between the two "probJem' statements are the following: (I) the..:coefflclent matrices of- [he constraints are transposes, and (2) the
.'
288
DESIGN OF THERMAL SYSTEMS
TABLE'U.23 ,
'
Initial and final tablea11X in
,
~xairiple .12.S ~ ' X4
,Xj'
2·..
x,
,-,
8
1
Ii .x: 2 :~',,;••:' .\, '_~. ,. .-4,-,".:.:.:. .~ ~_2_'+--'...:...:.",_3_'f-I-r---:---:~ -=- 3 ~: O~.~: -,:~:~~. ~.(;~: o J•
• ,:,
..
_ - I . - _---'--;--'-------l....L-----:-. ,
.. ' ,"
.
3/4
-114
3
112':
,'2·
I
. . . \-I:l""·
'-112
"
~-"'.'
I'
.
,:, ••
) ..
t
-
"
,'312:.:,.. ,· . ,l~' 'c ' ..." ....: '," :~,'_:::.,~ .:.':.~-':
. 18 .• ,-. "
. ';, ,~:
TABLE "U.24
.Initial tableau in' Exa~ple 12.;6
'2
2
1\16'
1
3
-
W... ~
U's
3P
WJ
WJ,
WI
8
..
-1
4
'
":""1
3
-p
7P
..
5P
-
12
-p
TABLE 12.25
Initia1 tableau in Example 12.6 WJ
W'"
W5
W.s
WI
·-3/4
·}/2
314
-112
3n
\1'2
114
-1/2
-114
]12
112
-2
-p + 3
-3
-p + 2
J8
right-side ·constants of one problem are the coefficients In the objectiye function of the other. There are three relationships between the two so1utions: (1) the oprimal val ues of the 0 bjecrive function are the same, (2) the' solution of the x's 1n the primal are the numerical va]ues of the differenc~ coefficients under the artificiaJ .variables of the dual, and (3) the sol.utions or- the w's in the dual are the difference coefficients the primal. . The primal and dual problems in Examples 12.5 and 12.9 were twovariable, two-constraint optimizations in which both the constraints were . active. There was no amblguity in identifying the solution of the dual based on the final tableau of the primal. When the number of constraints differs from the number of variables, some interpretation is needed to select the
tn
--
"
~
','. .'.. '. ':;::- .... ~. <.:~:.:~: ~. -.. ;f.:.:,;i;. :-:. ?~.;'::,',:
L1.NEAR PR06RAl..,!MING
.
.
applicable numbers. _. 0 the .maximiz.:ltion prob J.err,. in duce adclitioD2'!' constraint. I !
an
+
E xar.11ple: 2.7. lvIaximize)' = 4x I
289
Ey.~.mp1.e
~nt[o-
12 .5,
3x~
subj~ct to.
..
.:
'. I
.. ;... ..
:, ,'
.
....
l
.
..,"... ir r' .*:' .x ~.__~ .s.~... ".....~::, . : ;;:".i"":":' :'.', ": ' .-
~'-
2'-( 1
The new· constraint turns out that,appeared in Tabl.e 12.23. SollLtion~ ' Th~
' . ' .J
• •• :
+ 3x~ < 12 '. ::"" . ..... ".; '.: .....:..:... ~ ., .' '."
be inactive which permits tracing the te11l?s .
[0
firs[ and third tableaux appear in Ta.ble 12.26.
ExaTHple 12.8. The dual ·of the problem in
'f
Exampl~ 12.7 IS
" minimize
subject to
,W l
and
4Wt
+ _-2w:!
+ . 2W3 > .4
+
+
w,:!
Solution. The ini,tial and final tableaux are' showri
3 HI)
.
Initial and final tableaux in Example U.7 XJ
Xl
I
X-'
I
.t5
xJ
1
4
12
XJ
514
1-7/4
1
.r J
2
1
8"
XI
3/4
-114
3
Xj
2
3
12
X2
- tl2
112
2
-4
-)
0
312
l~
18
TABLE U.27
Initia1 tableau in Example 12 .. 8 w. \116
)1.'7
. w)
Wl
"4
WJ
-1
1
2
2
4
J
3
5P - 12
3P - 8
5P - 12
W~rf W6 I1I\d ~ mthcial 't'ariablcs.
, ,
...
-p
3
in Tabies 1·i~17";~a·12.28.
respectively .
. TABLE 12.26.
:::>
4
-I
3
-p
7P
290
DESIGN OF ruERMAL SYSTEMS " . . ..
TABLE 12.28
F~IiaJ' tableau in EXaIpple 12.8
....Wl ' :
"
'. J
.~•
•
,
W4
-5/4
-3/4
w,
Ws
112
"
..... 112
3/4
'312
.. "~] :
,, •
WI
).
7/4
:.
o-~' "'7":
,1/4
':"114
-.:..112
J
\.'
. . 112
'. ~.p +: ,3~.
,. - 1_: .,. ' "~"': ~' ~~ r:!.:'~'
.
;- .:7l'!
: ' .; il2
+.,1, ." I~:,..":'-'oi L :"",,:',::J' -'.'
Let us propose ' using the results of .the pIjInal in' Example -12.'7- to ',' predjct' the optimal SOlp.Don in , EXaplpJe 1-2.8. For the tlrree vv·,' s -there 'only .two. values', 3/2 and 1/2, avaiiable :in Example 12.7_ Oile of the w's will ' be zero. because there are three v,ariables and two constraints in Ex~pl~ ' 12.8, but which ope? The natural order is 'follo'wed: x] =rf 0, so Wl = O. The difference coefficient 'of X4 '= 3/2 = W2, and the difference coefficjent of Xs = 1/2 =, W3.
are ."
~2024
POST-O~TIMAL
ANALYSIS
Post-optimal analysis is the activjty that begins afte;r the original problem has been sob/ed, and asks the follo\ving question: ·'What jf a const.ant or coefficient is changed slightly in the statement o~ the original problem?" Three of the sensitivity coefficients applicable to linear programming are to evaluate the change in y" with respect to (l) a coefficient in the objective function, (2) a right-side constant in p constraint, and (3) a coefficient in one of the constraint ·equations. The first two of these sensitivity coefficients will be examined now. The sensitivity coefficient with respect to a coefficien,t in the objective fu~c[ion is readily available. If, for example, the objec[ive function is y = Ax J
+
BX2
then at the optimum
y. = Ax; + Bx;
x;
x;.
so the s'ensitiviry coefficients are: dy-/aA = and oy-laB = The prim'aI/dual TeJ atjonship offers the key to expressing the sensitivity coefficient wit1!, respect to a right-side term in a constraint. In the maximization problem of Example 12.5 the constraints are
2x 1 '+ x2< · 8 ,=C
lx, +
3X2 ~ 12 == D , The dual of this problef!l is Examp]e 12.6 ha,~in~ .an o~jective function
---..
Y = CWJ
+
DW2
,
~/-S( JltJl ~jj ~ . .. , , , /I." ULlLj )..~ ~ / " .7 \ 1 ~' ~ / ./ .-"
C
,(
)
./ /"
,
G
I~ ,
~
'
/
....
/"
C
,/
/.~y./.) (' \~)~ L..v ,/ C · .----, . -
/"
•
.
~\" J /'
UllEAR PROGRAiVfMING
.....
.......
/.
"..../
"
)
. and at the optimum, y
so
- '1=', -' . ---
.
.-->.
"
'"
ay / ae' -=. ~1' i'l ~E, By
-:tiY..f'ly.; -'-;.,
= '-Y
~,
'. •
.
~
*
= CW 1
~"
*
+ D"'v2~
-
J J:.? (.;' ..~r ~, .,-~he~~~ v~~~~s , ~!;~ 3/2 and
:- " ,
. ~. ::~,~';:.:\:.&-:-.-'
.~.'
. " ' -.
-:.,..~;' .,.' ~\ ...
1/2, ,resp~c .. ,:
'
- ':~.~"
',.:
,1-
,-1
....,. '_:": 0.:'. ,',
'12 ~25 'EXTEN~I,Ql\JS :OF LINEAR 'PROGR.A1\1J\ID\T G' "
The cq.apter illustrateg hand calculations to solve linear-programming prob,.. '- lems, and there.is SOIJle benefit- to,understanding the techniques~ Hereafter ' ~ 1ibrary 'routines will certainly- be used, ,particUlarly JOJ larg~-sca]e problems. ';' Two classes of sensjtivity-.coefficients ~vere explored; one remaining '. is that ~ith respect to ', a coefficient in ,a con~traint equation_' The sol utian to a body of problem,s opens when an opt,imization no~ completely linec;tr,.i$ , line~e~ and subjected to linear programming. PRO~LEMS
12.1. A farmer wishes to choose the proportions of com "and soybeans to plant, to achieve maximum net return~ He has 240 hectares, available for planting although he may ele-ct to operate less than 240 hectares. The ~apital ~9-d labor requirements and the net return on each of the two crops ,a re
Crop
Capital cost per hectare
Labor) workerhours per hectare
Com Soybeans,
$120 40
10
5
Net return
per hectare $50 75
The farmer has a maximum of 2000 worker-hours of labor a\'ailable and maximum capital of $19,2<:)0. (0) Set up the objective function and constraints. (b) U'se the simplex algorithm of linear programming to de[erinirie"" the
optimaJ number of hectares of com and soybeans and the maximum net return.
' Ans.: Maximum net r~tu'rn = $16-.000. 12.2 .. During a summer sessjon B srudent enroJis in two courses, psychology for
four hours and engineering for three hours, The student wants to use the available time to amass the, largest number of grade points (numerical g::rade multiplied by the number of hours). Students who have taken these courses pr.eviously indicate that probable grades "are funct-ions of the time spent:
~
G I
= !l 5 '.
and
G::: P
X2
,1
DESIGN OF
G
,
, ,:~ ~
(4.0 = B. hours per week spent outside engineering .;= number
.. :.
.
.... "';1 ,.~
~.:
u .....
UJ,1.J ........
&.
#
•
Also ·it is
. J . .)0.
.
. .
.
-
ofhbiirS.~ avaitable:-fo~,tnitsid~ <:'h~lrh1'-··...... ",.,.-t'17·&i"'''V- ,.'~""'f.,,..r most englneerillg studentS can tolerate the two
ne(:essary to
to-
earn an
lJs.e the simplex to
courses. is planning construction of a plant that co~d manufacture a combina,tion pot TV dinners, ""'.-T·.....-.""'r.. cos~ c
cost
Food
Pot pies TV dinners Pizzas
..., LAJ OJ •
1
500.000x: 400.000x)
where x J. X 2 .x 3 dinners. and pizzas. I
"'P""T'p4~pn
$100.ooox I 400 IOOOX 2 200 ,OOOx 3
the hourJy rate in thousands of units.
hourly
m
donars in It 30X21 and 40X3,' respectively, X;:I are to be The values
profit
linear programming to determine optimal XJI
• X:3.
:- Optimum
profit
= cemeht mix with clay or shale, and then in 1 1 A cement:
are to the mixture in a rotary
Jimestone, , ~s
ASTM types of
,..
t)-
LI1'.r1:AR PROGRAJviMING )
. ICoarse-ground·
.
1Estone ,- .• =,~=-kJ ·' -----i<.~l .Gnnaer ..
. '.
.,'
.
storage
.. - = -==.~
. >I ·F~
..<.,; ......., . t, • ..
..
~
~
.
~
.
~.~
Cemeu[
lYilxer
. · /- /: . ""storage '..',. . "" . . .
-J
~, 1=. ~~:~
..
.
'[ ~l'.~. . : '•. ' >·<1" ~' . .
- .
293
,·_··..'r:<..····,· _:.
..
,,;.,,:
.-
Kiln ~.
-:.
-
'
".,
~
Gay
. F:GURE 12,-1(' Cement plant jp P~b. 12~4_
the
-The profit, C?f each type (lOd cap~bilities of. the gringe~ and kiln in , proce.ssi.{lg 'these cements are shown iIi the ta~l~~ ,
Cement type
Profit, per megagn1m.
Grinder capacity, GgJday
I II
$ 6 10 9
Coarse 10 5· Fine Fine 5
'm
Kiln capacity
Gg/day 8
. 4.8 6
The grinder capacity shown in the taqle of 10 Gglday for type I means. for example, that the grinder could grind the limestone for lOG g of type I if it operated. aJJ day solely on limestqne for type I. The grinder and kiln operate 24 h/day and can switch from one cement tjpe to mother instantaneously. The limestone storage space and mixer capacity are more than adequate for any rates that the grinder and kiln will penniL . Use the simplex algorithm of linear .programming to determine what daily production of the various types of cement will result ip maximum
profit.
,
Ans.: Maxhnum da~y profit = $51 ~OOO. 12.5. Three materials A, ill and C of varying thicknesses are available for combining into a builcling .\:V~) as shown in Fig. 12-11. The charac~ristics and costs of the materials are .
,Th~,
Material
resistance, , ceDtirnelt~ thtckness
A B C
Load bearing c:spsdty; units/an
Cost per c-entlmeter
30
7
$8
20
2 6
4 3
units per
10
,"
0• •
,.J
~ermaJ ·resistplc~. of the wall must be 120 or greater, and the _ total load-bearing capacity must be 42 or greater. The minimum-cost wall
i3 sought
294
DES1GN ,OF THER.MAL SYSTEMS
B
A
,
,x em
!>liIil
'
cm X3C
1>1 ,
FiGURE U-l1 ' ', '. ,.. COinpo~·jte wall in Prob. 12.5. '.:.. , " .
". •
I
·
.
~
'.1
:
::
- ,
~ .~: ~
....
'
"
.
. -.
~
...
,' .;,
~
I
...
~
: ••••
;'. '''.-~'' ~~'.." .... "(a}·::S·~tPP)9.~~Qbj,~~ve:fun~tion·~d c·onsn:ainis .: .' '... .' ':" .- .." ..'(b) .Use the'· -simpI~~~,~,!£g~rithyJ ;~h·~~?-~9f;.~J?!.P.~.~R¥.'-~:~~: ,-~,9.~~~~e. ~~:~:,._ ~ optimal thiclqle.s_~es of eacir ~ate~~L. :" .... '... ' .. - .. .- - . " ".. ':: . ' ':.'. . Ansa: $30 minimum cost: .. " . . U .. 6. ' The optimization of the ~ombiDed gas- and steam-turbine plant in Prpb. 7.4 resulted in a linear objective ~nction and three linear constraints. Use the simplex algorithm to determine the optimum value of ql and q'2~ To simplify mathematIcal manipulation, use the following equations instead of those in Prob. 7.4: Objective function:
q'=.qJ
Subjec~ to
+ Q2'-
ql
+
1.2q2
:>
28MW
g]
+
0.4q2 >-
19MW
qJ
+ L7q2.
>-
32J-..1W
Optimal q = 25.75 MW. 12.7. The furnace serving a certajn steam-generating plant is capabJe of burning coal, oil. and gas simultaneously_ The heat-release rate of the furnace must be 2400 kW, which wirh 'the 75 percent combustion effi'ciency of this furnace requires a combined thennai.lnpu[ rate in the fuel of 3200 k W. Ordinances in certain cities impose a limit on the average sulfur content of the foe] mixture, and in the city where this pJant is located the limit is 2 perceJ1t or 1~ss. The sulfur contents, costs, and heating values of the fuels are shown in the table. Ans.~
, Using the simplex. algorithm of Jinear programming, determine the c.ombinalion of fuel rates that results in minirrIum costs· and yet meets all constraints. Ans.: Minimum cost $0.00231 per second. 12.8. A manufacturer of cattle food mixes a combjnalion of wheat and soylxans to (orm a product which has minimum 'requirements of 24 percen't prol~in _ ond J.2 percent minerals by mass ..,The composition and prices o(the whent - -and soybenns are given in the table.
=
295
LINEAR PRO®RAMM1NG
Constituent \Vt1eat . S.~yb~~.ns
'.j:".~.'"
PeJ.""ca~t
CO.Hl.position by mass
P r otp-in
J\1inertlIs
. l6 48
1.5 J " ~ i" .Q' .'
P r ice per 100 kg $-10 .
"'"20'" ;" .,,' ;'.... :.
~:u
_ _----:-_ _ _ _ _ _ _.....,......~:_______:_----~-.. :-.- - - - = : : . . .......... ~~ :" ...:..~ .,
w"o
......
:f'.~' .. :.
Use"the sim'plex algodthin of ljnear ~rogr~rl:iing :to .d~ieiTIii~e· 'the ;n~s of each of the constituel1ts of 100 kg of product s.lich t..hnt'the cost of. tbe raw . . materials is nli n Imuf.n an d [he n utri tionnI. requireme~1:s a..-re. meL Ans.: 75. kg of wheat 'and 2~ kg of soybeans .. 12.9. A wax concentrating plt:J.nt (Fig. 12-12) receives feedsto'ck with a low cor{-' centlLltion of wax and refin'es .it into a product with a high concentrarion of wax. The selling prices of th~ products are x I. $8 per megagram anQ .t 2, $6 per megagram. The raw material costs are X3~ $'1.5 per megagram and $3 per megagrarn. The- plant operates under the following constr~ints: . . ..... "'. .
X.h
'
I. No' more wax leaves the pb~t than enteis~ ') The receiving faciJities qf the 'plant are 'limited to
atotal of 1600 Mg/h .
.3. The packaging facllities can accommodate a maximum of 1200 MgIb of _f,2 or 1000 Mg/h of .\" I and. can switch from one to the pther with no loss of time. If (he operating cost of. the plant is constant, use the sUnplex algorithm of linear programming to determine the purchase and production' plan that results in the maximum profit. . Ans.= Profit = $3650 per ho~r.. 12.10a A dairy operating on the flow diagram shown in Fig. 12-13 can buy raw milk from ei[her or both of two sources and can produce skim milk, homog-' enized milk. and half-and-half cream. The costs and bUlterfm contents of the sources and products are
Item
Designation, Uday
Source 1 Source 2 Hal f-and-half Skim milk
Xol
Homogenized milk
Xj
XI
Xl ;( )
,
Butterfat content,
Sale or: purchase cost $0.23 per lieer 0.24 per liter 0.48 per.. half·IHer
0.60 per two-liter 0.68 per lwo-litu
vol % -. 4.0 4 ..5 r,
. -~10.0 ~ 1.0 ~
3.0
The dnily quantities of sources 3hd prol:!ucts ore to be determined so that rhe plant operntes with maximum profit. All units in the dairy can operate for a maximum of 8 hJdny. (a)~fte-r1le equalion for the profit and the constraint ~quations in tenns of [he x variables.
.::.~
296
DES1GN OF TIiERMAL SYS'fEMS
wax
X3
o ,
,no
>"'"
.1'4
' _. '
60% wax
Wax
wax
~.
F~GURJE 12--_~,2
~
concen~ting
80%
pJaI}t
-
0.
XI
Wax:(2 ..
Wax concentratjon plant ,in Prob.
I I
12.9~
.
'.
....
"
0.68 Lis"
'
.-
,
"
.;
_
~
J
:.
"
Slcim milk
Crearn 40 % bunerfat 1
'
1% bu nt:rfa
.
1
Adeq'uate storage
Adequate storage
,
.. .
~pa'ralOr' : ~ ;-
..:., . ,
,
,
,
Pasteurizer
(one'fluid at a time) ] .4 LIs of skim milk 0.1 Us of cream
-"
--.'
.~
.'
,cr~am
Adequate skim milk
srqrage
storage
Adequate
. "~
"..-
~
J:iomogenizcr ·1.1 LIs
, .. .
r
, Packager
3000 packages per hour
(either l/l-L Or 2-L)
JXl
-
..
~ x,
I~.x,
Skim
Hom08enJud
milk
milk
FIGURE 12 ..13 Dairy in Prob. 12. 10.
-
llN'"""c.A.R PROGRAMMlNG
b) Use fine;}[ progra!J.1mi.ng to sclvr for the
2.)7
pl an that res ul ts 1'1 maximUln
profit.
s.: (a).
",:.--. .-
Separator.-- .. :
~
r ..
&
•
. x (-+'\X"1·~·-1.g."584~':·· ",
"" •
J
•• ':
\
Past~uri zer:
X I
+ L08x:!. ::; 20, 150 X5 ~ ..
31,680 . ::-':,....:: .. ,..
Homogenizer: '=',
.'
.Packager: . Butterfat: Total Mass:
. -XI
. (b) . Maximum profit
+ x:2
-
",
,',;...
+ X4 + Xs"
<
48,000
IOx3 + x.{ + 3xs
-<
0
.4xJ --:-4x 1'.- 4.S:r2
.'
+ x) +.Lol + Xs :S 0 .
= $5908 per ~ay:
"
, 12,,11~ . A chemical plant .whose flow diagram is shown in Fig. l~- 14 manuf~ctures ammonia, hydro~hloric acid urea, ammonimn 'c'hrbonate, ~d ammoniu:m' chloride from carbon dioxide, niu-ogen, hydrogen . .aJ1Q. chlorine. The x . values iri Fig_ 12- 14 indicate tJow' tates. in' moles per secend .. . The costs o.f t,he feed stocks are Cl. C"2, C3, . and C4 dollars per mo!e. and ·the values of the products are P5~P6, P7, . and 'Ps dollars per mok. where the subscript corresponds to that of the x v~ue. In reactor 3 the ratios of molal flow raCes are nl = 3X7 and ·x 1 = 2x 7 , and in the other reactors straightforward material balances apply. The capacity of reactor '1 is equal to or less chan 2 molls of NH j , and th~ capacity of reactor 2 is equal to or 'less than 1.5 moUs. . (a) Develop the expression for the profit. (b) Write the constraint equations for this pJant. y
XI
Xl
x)
X"
CO 2
Urea and ammonium carbonale
N2
H2 Cl,
.r,
Xli
NH3 Hel , FlGURE 12-14 F10w diagram...of chemical plaot in Prob. 12.11.
- .
298
DESIGN OF THERMAL SYSTEMS
•
ADS.: (b)
' xl-<2x 7 =0 I .
2r2 _ ·xs -
3X7
-xS ~ 0
2x, - 3;c, -;C6 - 9;C7 - 4;c, = 0 ~ '.;
..............~ '~.:::
.....
=e
U4 . -X6 . -XS
, -~ . .' "- . ~
.... ·.:t .... ~. '....
x . + x, S 1.5 .~~
x, + 3",- +·x,
"'-i .o. c'
...••.
..-' -..... ' .,
" ",
12.12 .. 'When large fabric filter inst.allaticins. called baghouse fluers, fil ler hightemperature gases, e.g., -from a smeJter; temperature of 'g ases must be teducted to .265°C or Jess, even whon using glass-fiber filte". Three methods l of reducing the temperature may be used' singly or ' in ' combinati on , as shown in Fig. 12-] 5: (1 ) reject he!n to 3D11;lient air lhrough the use of a heat exchanger, (2) dilute the bOl gas with ambien t rur thal is at a temperature of 25°C. and (3) inject water for evaporative cooling , Ii either d iluti on air or water.injection "is used. the baghouse must be enlarged to accommodate the additional mass fl ow. Designate
the
the
x I = area of heat e..xchanger·, ru 2 X2
.
= mass
flo~
rate of dilUljon air. kg/s .
x~ = mass now rate of spray water, kg/s
Data Cost of baghouse is $2000 for each kilogram per second of capacity. The hcaHransfer swface costs S15 per square meler. There i$ no cos t for the dilu ti on air and spray water other than that due to the enlargement of the baghouse.
x)
kgls
--r------------1--~.--~ ,
.
Heal exchanger
Water
XI
Spr::ly
m'
a: Are.e
fIGURE J2· JS Ct'oling ~ ~ns before il e nlerl 8 baghou~ 'fdler .
-.
.~
"""'?} I S 265°C
Baghouse tilter
299
LINEAR PR:f)G.RAJv:i.YUNG
The er)tering tlow rate o.f gas' -:- 16 kgls ai: 500°=" . Each Sql:~-e meter of hep-l exchanger lreduces the terD_p eratllie of gas' . ~~C.
.
.
The evap.oration 9f spray water cools. the. gas,.-ajr· n1ixture equi~~alent :' . . to. 'a sensi ble..;h,eat .removaL from Hte' gas. entering the water. spray of 200{)
.-idfkg Df -spray '.Y$..ter. . ' . '. . ' . ' . . . ~. ...-'.. -.~ ., . ~. . .. -':":' . . ~'-: .-- The sp.ecific... heat.:o£',the. gaSr'dilut:i6n"aii;"ana;:'ga:s'~cir"n1i~illi~' is''I.:O-· ". :.~ 1·.:.IcJ/(kg: .. ·K):·· < ' , . . . . .... . " . :
." to:"'<:>
..-,:~...
"'.
. , "J.
III
'i' -:: ~
". ,J~ ~~.:
•
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••
•
•••••
•
.".
•
To avoid corrosion, the .mass· i-ate of flow of spray \vater must. be 5 . .percent or less of the combine~ flow o( gas and dilution aiL ' (a) Set up -the matbematic~ statement'of the opt.imizaIion problem'in tenns of x 1 to -~3 to· minim.ize the' cas[, subject [0 the appropriate .C00- .
straints. (b) 'Develop the llist tableau of the simplex algoriilim of the' linear.' programmin.g. solution of this optimization and indicate which v~able to program f~SL
.
J\ns.:< (a) y = 32,000 subject to
.
+, ISx I + 2000x 2 + 20OOX3
2t 1 + 15x., ••
1-
+
125x3 >- 235
.".
12.13. Some perrochemical plants take a large t10w rate [.com a natural-gas tr~s mission line, remove ethane and propane from it, and return the methane to-'"the pipeline, as shown in Fig_ 12-16.
Cost data Cost per cubic tneter $0.06
Feed Price of x I returned to pipelinet
0.0595
X2
0.08
x]
0.'10
Operating cost: Extractor. per cubic meter of total flow Separator. per cubic meter of pure propane
0.003
t Since the methanl: has a lower hearing
vaJu~
0.001
th..an the original feed, it is 1.es.s valuabl~.
Restrictions ~
The composition of the feed if 900/0 methane l 8% ethane, and 20/0 propane. The maximum capacity of the exu-actor is 200 m 3/s of feed. The maximum capacity "f the separator is 3 rn 3Js of pure propane£ Set up the objective· function and constraints to maxhnize the profit of this plant. ..
-
..
300
DESIGN OF THERMAL SYSTEMS
Pipeline
Feed 90% 8%C 2 H 6 2%'C 3 Ji: 8 ,.:
. ..
;,'
~.'
:
..
:".'.,: "···r-. I _ ........._ _
..w........""'Io#- • • • •
I.:',"
'. .
: :'.' .. :; '....: "
t.'"\o~
••
.
.. ".c.'.
I,
11---+----'-------11-
.
~2
':t"~~~ __ ~=_: __ ~~_~~~~_,-- __ ~_~J- '- .
FIGURE 12-16 pipeline. The units of the flow
Removal of propane and ethane a and XlX3. are in cubic meters per second_
Aus.: Constrail'lts are
. 12.14. Consider as the
+ 3x 1 + 5x 2 + 6..'( 3 ~ 4x J + 2.r 2 + Sr 3 :;::: .r
subject to
=2x}
3
2J
16
+
2tl
the , and (b) by solution of the dual determine the optima] values of the x's in the primal. . = \' = ]0.667 ]5. (0)
Y to
=
2'( 1
+
.'(J'+
;s;
21
2..t J
+
.2 .$
At
Xl
+
X2 ::S
(b) From information in
,
A =
final ....................... ...
Ans~: part (b), 2/3. 16. A simplified version of a certain oil consists of one distil].arion unit which
----..--
4x:!
is shown in Figure I the
17 and into
of
naDmas, oils. and bOttOr..1S~ an oil crnckirg 1rrit LhQ.~ corrvE:..1i:S SOffle tile oil~ into naptha; a bottom cracIcing u:nil: Lt~alt converts some' of the bottoms into oi13; a gasoline blender; ~.nd a fueI"oil .blender. . . . prices.are The composition of the feed·and products and their . , )
'-"
.
. .
~.
:. l\1apth-a;:, _ _. ()i'~.,~· '':.._, -:- .. ·B.ottnms" ..:.. ,'. "_¥'It':iee,i'::': .' ...... :.J' , ••••,,; •• •:.~.r:::, : ';.~.~".'(::' ':"'~ ::: '~••'!~:' ., p ef'Ce,i'l t-· ',percen~' percent. $/bariet .~', .... " . . . .. 0........
Crude 1
Crude 2 Fuel oil Gasoline
10 15
75 65 .
30
70
80
20.00 24.00
15 20 20
25.00 36.00
The output of Cracking Unit'l is 4.o.%·-R-~ oils. Tbe output.· of Cra9king Unit 2 is 70% oils and 30% bottoms. The following capaciw liI,TIitations ap'ply: . . barre~s!day
naptha output fr.om distillarion unit
-<
3,000
Cr~cJcing
Unit 1 input
:$
8,000 barrels!day
Cracking Unit 2 input
;$
.10,000 barrels/day
In terms of x II
Xl. X3,
a1).d x 4, where th~ ~r'~ have mrit.s of thousands of
barrels per day, (4) write rhe objective function of the diffe'rence in income from the praducts' and costs of the crude. . (b) construct the constraints. (c) solveJor the optimal vaI~es. Ans.: part (c) $377,333 per day_ 12~17 .. A plant jn a process industry is capable of generating steam at 4,000 kPa and at ],100 kPa." This steam is available for heat exchangers or for o
Naprhas 'rude 1 _
..
Cr.1Ck.ing Unit I
I
Distillation
I
2
Oils
Gasoline Blender'
Gasoline .%')
Oils \
Unir ~rud~
Napthas
L . . -_
Oils
BottoI'M
2
FlGURE 12-17 011 refinery'ln frob. 12.16.
~
~rxkin8
-Unit 2-
B.ononu
Oil
Blender
Fuel Oil ..f4
...
302
DESIGN OF l1IER.MAL SYSTEMS
.
generating power in turbines. Power also may be purchased from the local electric utility. The, plant regui!es 30 M\iV of shaft power wNch may , be , obtained ~y combinatfon of the three sources shown in Fig. 12-18. There are'thr:ee prQcess ' heating requ~e~ents and ,the · flow rates at' the steam required for these need.s are shown in Fig." 12-19. ", ': The flow rates of the 4000 and 1100 kPa steam may be combined . ' ljnearly to meet 'tJle requii~ments,lf~r example, 15 kg/s of 4000 'Jd?l ~team' ,'~ \ : ,'",--' . -~'.,"-, plus,.20 .kg/~ ,of k?a steam coul9 se~re th~ ,175°C 'need. ", :' ' ~" ',':. - " . '- a 'IirnHatiort ,.:,Qn',! .th~;,}?,oiler~ .'~P-~8,o/~;. ~-;,I?,?f..~~,~ , C!f. :kg{s ,of.,4000 kPa' " , . '.. , stearn cap qe "gerierated::"'Thefily~~~ept~1??~a",o.t.th~F:ie, a riexchangersjs~1~:', ,>::,~~,~ , . '. . . .
1'100.
/9
~
Because at:'-
.
. ' ,' " , (O.l)(c'a pacjry in legis) mvestmeht cost, dollars ,per s = ' _ .. ,.. , " (c - 1)
where t c c = c =
outlet fluid temperature,
°c
250°C for 4000 kPa steam 185°C for 1100 kPa steam a
I50 C for 480 kPa steam
The costs of the utilites are 4000 kPa steam, $3_20 per Mg 1100 kPa steam. 52.30 per Mg
..
Purchased electric power, 50.0] ~ per MJ In (eons of (he ;c' s which are rates per second. (a) wrir·e the expression for total operating and investment cost per second of operation, (b) write the constraints, and (c) use linear programming (0 de!ennine the operating conditions that result in minimum cost- , ' , Ans.: (c) minimum cost is $0.694 per second.
4000 k,Pa 200 k.1/kg
kPa
Work.
125 U/kg \Vork
"
I J 00, kPs
~80
From
Motor 75% efficient
ulililY
FIGURE 12·18 Means of providing shaf, power in Prob.
-------.
~.
J 2. J 7.
lPl.I
1..-------1....
.1'."
LINEAR PRoQRA.:,,2M1NG
303
4.000 kPa Full ·lo.:Jd handle(~ by 30 kg/s of 4.000 kPa 0-
40 kg/s of I . ~ 00 ld'~ sre",m or Hne.1.r combinarions tl t::rer.;f
.J '--'~. ", ~
~.
-
.)
'-~
..
.'
• •J.. ~.~ I,.
. -:
.'
~
.
"••• "
.X3.---.:....-)::1 r-----""'-~
140 0 e ---1-
.. Full load handkd by 20 kgJs of 4.000 kPa or
.
24 kg/s of 1. I 00 kPa stearn or linear combin:uions thf:reof.
l.100 kPa ..1"5 - - - : ; -
FIJI! I03d handled by
24 kgJs of l.IOO kPa or 27 kgls of 480 kPa S[~3JT1 or·
linear combinations
TIGURE
lhe~of.
12~19
Combinations of steam flow rates to meet the three process he~tjng requirements in Prob.
12.17.
REFERENCE J. P. Vandenboeck, "Cooling HOl Gases before Baghouse Filtrarion;" Chern. Eng .. vol. 79. pp. 67-?~,. May t. 1972.
ADDITIONAL READINGS Chames. A .• W. W. Cooper. and A. HendeI3on~ An Introduction to linear Programming, Wiley, New York 1953. ,. . Dano, S.: Unear Programming in Indu.rtry. 3d ed., Springer. New York. 1%5. Dantzig. G. B.: Linear Programming and Extension, Princeton Unjveniry Press. Princ~toD, N. l .• 1963. GMYin, W. W.: Introdllc/ion 10 Untar Progrpmmlng, McGraw-Hili, New York. 1960. Greenwald. D. U.: Linear Programming, Ron~d Pre", New York. 1957.
Hadley. G.: U~Q' ProgrammIng, Addison"We,Jey. Rearnng. Mass .• 1962.· LleweHyn, R. W.: Lintar Programming. Holr. New York. 1964.
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.
13.1 NEED FOR"MATHEMATICAL l\10DELING A necessfu-Y preliminary step before a simulation or optimization of a ther-
- inal system can begin is almost invariably that of mode]jng some chruacteristic(s) 9f the equipment or processes. The simulation and optimization . operations "almost a] ways use data in equation form, and the conversion of . data to equation form is called mathemati~a] modeling. The groundwork for modeling, has been laid in Chapters 4 and 5, and in ~any .cases the techniques covered tpere are adequate for our needs. Chapter 4, "qn equation fitting, approached the task as a number-processing operation and showed how the method of least squares could apply to. polynomial and other forms of equations. Chapter 5, on the modeJing ofthennal equipmen~1 took advantage of physical relationships that prevaiJ for heat exchangers, distillation towers, and turbomachines to choose effective fonns of the equations. This chapter carries on the work of matbematicaJ modeling by con~ centrating on non-linear as as linear regression, and on expressing thermodynamic propertjes in equation form. Before settling on those topics the chapter briefly introduces several criteria for the fidelity of the model comments on the choice of the fonn of the equation.
weV
and
The ability to express themibdynamic properties in equation form is valuable in work with thermal systems. It is true that property equations
304
---.-.
tvL"'.THE~'IA.TICA.L ~IOOEL1NG- THERMODYNA.;\UC PROPERTIES
305
abound, bu'- the acc ur~Le ones art; usually cornplex. Our 0- je.c~i ye vliE be to develop equations that are simpler, although slightly l~ss accurate. In. many C4Ses it\.vill be poss ible to use some classical therrn~ dynamic ptoperty . relationships to suggest a il . ajdiio~al term or t1l~O that c~ be ,add::d ~o a simpI~
or ideal ,relation. '
---:,'.1"'_1.:.
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:'302: - THE FORlVf:OF ,THE , EQ:U~t~IOt~r ~,',~.
~.
.
.
~
.
...
"
p.-". : / _ . ,Section 4.12 ,described the ","'art, of equation', fitting," and suggested that having sorp.e in~ight into the reliltion~hip nKly fe,sult in a ,choice of equation torm, that gives a' 'good representation with a sjnlple equ.B.!:ion.' Problem:. .13.1 illustrates two forms of 'e:quation that both require the determination of two coeffic.ients. One of the representLlrions is a far better match of the data ho\vever. While 'ti).ere ,are some expectations of how, given individual functions behave, it is difficult to set rules for choosing the form of the equation. Whe,n no, iosl~t exists, polynomial -representations might be chosen. Transcendental functions should frequently,be explored, and'c~rt~in rechniques of nonlinear regres~ion allow the detenn'ination of exponents orher man integers which provide a better fir of the data to the equation. (.
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13.3 CRITERIA FOR FIDELITY OF REPRESENTATION There needs to be some measu.(e of the ~ffectiveness of the equation fitting the data in order to have some basis for selecting the constants and coefficients in a given form of equation and also in choosing between avail~ble f.ol)l1s, of equations. There are numerous qlteria in use, but we shall-limit ourselves to three: (1) sum of the deviations squared (SDS)) (2) average percent absolute deviation CAPD) and (3) goodness of fit (GOF). The SDS criteria, used in the method of least squares and first presented in Sec. 4.10, is the sum of the squares of the deviations, I
n
SDS
= L(Yi -
Yi)1
(13.1)
i:=> 1. _
where Yi :::::: value of the dependent variable computed from the equation Y,. = value of the dependent variable from the ori~inal data n = total number of data ~oints Squaring the deviations prohibits the negalive qeviations from canceling
out the positive ones and giving a Jow value of the summation for what wou1d be a poor representation. AI characteristic of the', SDS is that since the deviations are squared, the data points that are far off of the u lrimate equation_contribute more than a proportional influence to the summation.
The mimrrlizatlon, therefore
~arks
toward reducing [he large deviations.
306
DESIGN OF THERMAL SYSTEMS
The second
While
third eri terion i
GOF., is · SDS % = 100 1(
where
~
SDS has some . the ,
sum
0.5·
(1
'Y j from
the n
the mean value
Y, ;=1
results a zero a perfect fit of points a GOF of 1000/C'. OOF is more demanding flat curves than of CUD'es. In Fig. 1 1 ~ for example, the same devia~ions of the curve from the original for both the nearly-horizontal and IS for the curve The GOF is less The
.3
_ _----.~..,£-.....c:r":_::::-::--:-:-:-::-.- - - - - - - - - - - - - - - - -
~y rne.a..n
of horizontal of
for both curves. G is greater
the
'~p
curve:.
iviATHB.'vLATICAL MOOEUNG- THERMODYNAlvlIC PROPERTIES
)
,
3C"7
,
Regression analysis as ,the term win ,be ~sed 11en~, applies ~o the 'process' of detef[nining constants and coefficients (call~d parameters .. in equations , that represent a dependent variable 'as .a ' fty:lct~on.' of one more indepen", de nt '~/ariables. In "'lineat r~gre'ssion 'anqly$is~,~~ the partial deriva~ves of the 'parameters ~e' ~dep'e:nd-eht ' ~f the .:_ , ,,' , fid~lity criterion"v.ii tli" ,'~especf" td al l 'the < :-~~:-"'t ci th~l~1 pirrame't~rs.' :Artother\·way'-or'jaent.ifYm'g'~'a'; '-Iilfear·"tegiessi6n: ': si tUatloij'" ' .',' , ~s to ask whether th~ , paraine~ers. .be soI\ied ,uSing 'a, set' of 'linear siiniiI~ ::, ~ Lan~ous eql:lations.',l\Ton'l inear regression "viU require the sol1!-tion of one or ", , more nonl~near equations.' " '~e method least squares presented in Sec. 4.10 is aD. example, of lipear 'regression. ,Values th,e co'ns-tantS ,~d cO,efficients 'in ~e equatipn )t' = jex) are sought that minimize the sum of the squares of the deyjations, , the SDS. Section 4.,ll'e,m phasized ,th'a t the designation linear does not refer to the form in which the independent vari?ble(s) appear, but rather 'that the P£1!ameters appear in lin~ar fonn. In the minimization ,process of the' SDS with respect -to the parameters, 'the partial , derivatives are extracted and equated to zero. Even non-linear fODDS of the independent variable can provide a set. of linear equations with re,speet 'to the parameters. Some of the example of fOITI1S soluble by linear regression are: 7
ot
,can
'<
or
of
,y = a
Y
=
+ bx + bx +' ex"
a
'J
,y =a+blx+c"(L5
y y 13~5
= ae -x + b sin(2x) == a + bx 2 - c la( x)
NONLINEAR REGRESSION ANALYSIS
When the p~~p1eters do not appear in a linear fonn in the equation, determining their values usually consists of an optimization process, perhaps using a search method. The task is to find in the unconstrained optimization problem the optimal values of the parameters, ao, a I, . . . an th,at provide the most favorable value of the fidelity criteria, C. One possibility is the method of steepest descent (Sec. 9. J 3) which starts with trial values of the parameters and revises them in each step according to the relation
/lao a C/ aao
-~
Lid J
-
aCI aa)
-
!J.a n aCI aa n
The partial derivatives must be extracted numerically ,
ac
C(a(J, ... a,
+ s, ... all) 8
because
I
- C(ao • ... tail • ~. an)
---a summation, not a ,single mathematical equation.
-C IS
(13·41
( 13.5)
308
DESIGN OF THERMAL SYSTEMS
• !
Provjde trial values of
)
- ~,!!...... ' •
Yes
at
Yes
~~~~~~~~~~~
.I
.
~~~~~~~~~~
Other parameters
l----------r----~--~-
Terminate
13·2 Flow diagram of a search ............ "", ....
of
a nonlinear regresslon ..
While the
one
3 {~9 '
MATHE!\4ATlCAL MODEUNG-THERMODYNAllllIC PROPERTIES
'ff.LBLE 13 .1.
.
S ' arch fer Parameters "n . ~ xample J3.1 .
I
J
l\.:verage Pel ce:-:; ,: Devi~tion ,', 4
. Itera tion ·
I '. ,'., ,~;'!:', 2-' '. 3 I
42 43
.......
b
a-step
.
.b-step
3.00000 . O.~40PD~t ,: :" ' ...~_:?~..~~Q~ ~_ ".~'.' '",1, _:." O~.~_Q_9qo . ,_'.:. . · )J:-P2:QQ90~,. 3.10000 :. '0.42000' , 49.500.6 0_10000.. . ~. 11020000.',. 3:2000~O·->~.;<'~O.44000~'::-"~· - "!:~'44A l17 -,~ .....s""";"" "~'O"~'lOOQ{r'- :' :'·~':O~020000~.'3''''~' '. . 3--30000 0.46000-' . - 38.4851- " .. : "·O~ lobon ' .. ((020006··· .. ··· ,', .
o·
l1li
a
J.,
,
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2.58753 2_58753
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O.60062~.~
0.1360213
0.6006298
Op 1360212
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a
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....
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7.62942-7 9.5367E-8
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1_90735E-8 1_]9209£-9 I
.
1,. Incre.ase Lhe parameter by rue, initia11y~assigned step. size_ If the fidelity c.qterion is Improved, go on to the. next parameter~ 2 .. If the crite.qpn w'as not improved in (1)', decre.ase the parameter by the ." . st~p size. If th~ criterion is improved, go on to the ,next parameter. 3 .. If neither (1) nor (2) improve return to (1).
the criterion,
decrease the step 'size and
A technique that often increases the speed of convergence is to increase the step size after each operation, because there could have been an occasion where the step size had to be reduced, but a larger step size may work for some future operations. .
When the step sizes of the parameters have been reduced signed levels, the search terminates. .
~o
preas.
Example 13.1 .. The following pairs of (x, y) points are to be fit to the equation, y = ae bx : (1, 4.7), (2, 8.6), (3, 15.7), (4, 28.7), (5, 52.2), (6 95.2), (7, 173.4), (8,316), (9, 576), and 00, 1049). Use a search technique 7
to determine the Q' aqd b pararneter:s thiit result in the minimum averagepercent deviatiol?-'
Solution. Trial values of a = 3 and b = 0.4 are chosen, and the iilltial step sizes selected are 0.1 and 0.02 for a and b, respectively. The early and final values are shown in Table 13.1. For purposes of comparison, the.same search was conducted except that the regular increa.ses in step size were omitted. ,That se.8rch.required more ~an 1600 itef1l:tions to a~ve at the tina] values . . shown In Table 13.1.
13 . 6 THEIDriODYNAMIC PROPERTIES The need for equations to represent thermodynamic properties arises continually Ln...,!,Qrk with thermal sts~ms. Those properties of particular interest are the temperatlJre pressure. enthalpy. density or specific volume, entropy. I
,DES1GN OF THERMAL ~YSTEMS
310
• J
'
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'-
. ::1 U1
.~ 0.. ..
.
'. '.' . . .. J
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. 'J
~--~--~--~
En~py
" . J.,
alll4J'
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and 'occasionaily the internal energy. The app'roach, that vvi11 be ' fol1.o~ed ' .in the rem'a inder of this chapter is a compromise between e:xbaus~ve ,_e~rt, to achieve precise represe,ntations . on the one hand, and ro'u~ . idealizat,ions on the other. We will be satisfied if the accuracy of the equation can be ,improved, from 5%, for example, ,~o 'an accuracy of, say, 1 ,or 2% . . ~epre~en~ation of the,thennpd.ynarruc properties requiFes both property rel~tions 'fr9ffi classical. thermodynamics ~d experimental data. 'T here ate a number of possible starting points and paths 'i n arTIving at desired ,property 'e quatjons, but a frequently used teclmique U is to seek fro'm experimental 'data the following four equations: 1. p-F-T equations for vapor
specific heat at zero pressure
2.
cpo, the
3~
p-T relation for saturated conditlons
4. Pj,
th~ den~jty
of liquid . .
Based on the above four eguati.ons. it is possible, by the judicious use of classical thermodynamics to compute. two 'nonmeasurab1e , properties that 'are of great importa~ce in thermal system work-enthalpy and entropy. It is .no accident that some of the most useful property charts, such as those . shown in skeleton form in Fig. 13-3, relate the enthalpy and entropy to qthcI properties. The next several sections will introduc'e some relations fron) classica1 rhermodynamlcs which wilI later build on the four equations to con1pute"other properties. . I '
13.7 INTERNAL ENERGY AND ENTHALPY .
(
,
Computation of rates of energy flow js a key operation in analyzing themal systems. This requirement con tributes to [he importance" of the internal energy and enthalpy properties. The iI~ony is that there are no meters and
instruments .to measure these propehies; instead they have to be deduced from measurable properties, such as pressure, temperature, and specific volume.
--
-:---.
311
MA1HEMATfCAL MODE.LING-THERMODYNAlv_~!C PROPERTIES
Th·~ first
law, when applied
a c!osefl system consisting of cylinde., such as Fig. 13-4, yields ·the equatibn, . . [0
-d q = d. u .+ d~v
piston-
.( 1 ~.6)
, . , ;':~Yv'.here~·( ~...r2.te of ·h.eat - transf~(. pe; · -Jnii; ;rna~_~~, i:l1. _\:~$seJ"..,.,kIfl~g .', , "I ..,. . ",. _,... ... . ,
. "'
Cl
" ,
-
~
,
""
.I
, ; ' u -.-:
.'
'j'
VII ' =
0' -.r.. • uitemhl-. enerO'y lG71co· ' · : ~.j·:.,.·,:;,·~"" ", ,·. 'r' ' ,:: :.: . , .' b'.::l· . ext~ma/ v';ork .per unit 'mass ~,'in ' ves'sel; kJll<:g . ,;.:.;; ........ .':... : '. :' .~ ..... ;' . io
" , • • • "...
.I: •• "\",."
..
'--...
.. ..
•
•• ,
~-
.~
'.'
If the :heat is transferred to the· syste'm in ,a therrh6~ynamically' :-eversible . process,. dq}
·rev
" T ds ..
.' (~3.7)
where T = temp~aturel K . s . ."enrropy, kJ/kg - K and if the .work is perfonne<;t reve.rsibly -. dw')
. rev
~ 'p
dv
. (13.8)
where p '= pressure, kPa . v = specjfic volume, -qt3Jkg Substitute Eqs. (13.7) and (13.8) into (13.6) to arrive at
T ds = du
.+ P dv
(13.9)
Even though Eq. (13.9) was arrived at ~ough the use of a nonflow process
with the assumption of reversibility, thequodYD:amic logic permits us to state that Eq. (13.9) is perfectly general and is applicable to all processes whether flow or no~ow, reversible or irrev~rsible. The reason is that Eg. (13.9) is. exclusivelY'a relationship of properties. and the procedure by which \Ve arrived at the equation is jrnmaterial. Classical thennodynamic is replete with partial derivatives of properties, and some of these relationships will prove to be valuable in building a full set of thermodynamic propertie~ . .I~:CLuarion (13.9) offers the basis for the first two of these relationships. If v is held constant, dv = 0, and the
du Ma5.$ m
¢:=t===
dq
nGURE 13-4 EnerBY blla[)ce in
8
closed system.
DESIGN OF THERMAl SYSTEMS
h=u
'+
V
v
.1
(13.1
v
h
1
II
..
--
':;--
MATHENtATICAL MODEUNG- THERMODYNAJ\!{i'C PROPERTIES
313
region wher. p is const~.T)t the temperature i,& also constant, so in this region the line.is straigliJ. ""hen the line moves f;;m:he_" into the superheated vaJ!oi" region T progressively increases., so the line curves upward. l
;,
13~8 '
.
~-'
•
•
.'
J
THE CLplP i'~ N ~QUA . 0
•
"':_ . -_0" _ .
I
.....
I
. 'A'.~eful
".
.
•
-
' .
•
s:xpiessi.6i:L~: (o{.:.~biafjrig '~some -therriiodyp:~c :.: li-q ilid ~ :'~d '.:.Y~P9r _:. ..~.:
properties 'at saturateo. conditions is the Clapeyron .e quation. This equ:atioG.·"· can' be developed from pure,l y .thennodynamic principles and is applicabl'e [0 all pure substa:lces .. The developm.ent begjns 'vilith rhe expression of the . Gibbs function F :
F, kl/kg = h - T s Because it is
qased on propertie's', F
~s also a property. In differentiru form~
d F = dh :.-
and
~ubstituting
{13.16) .
T d s ,- ' s dT
. (13.17)
v dp - T ds .- s, dT
(13.18)
Eq. '(13.12),
dF = du +
P dv +
Next apply' Eq. (13.18) to a change between saturated liquid and saturated vapor at the sam~ : pressure. In the mixture region the temperature reEains constant in a constaht-'pressure process, so b~th dT and dp jn Eq. (13.18) are zero,
dF
dll
:=
Since fro'm Eq. (13.9) T ds = du
+ P dv - T ds
+ pd~', dF ='0
so the Gibbs functions of satUrated liquid and saturated vapor are egual at the same temperature and pressure, Fj = Fg • The n.ext operation will be to change slightly the pressure and temperature of a 'sample of liquid/vapor at saturation as shown in Fig. 13-6. Since 'Fu = ·Fg.r,.aJ?d Fl."!. = Fg .2t . dFg = dFr
dh, - T dS g
-
s,.dT = dh( - T ds( - StdT .
or
and
so
\1,
dp -
58
dp(v J
-
(
dT = y( dp.- s(dT }I()
= dT
(St -
Sf)
dp = sir -' s( = T(s, - Sf) dT va - v( T(v'l - v() .
Equatron·{t3.20) is [he
Cla~ron
_
h. . . ;. (~8_
__
(13.20)
T(v, - v,) .
equation and relates the derivati.ve of the
DESI~N OF THERM.~ SYSTEJlAS
314
..dFg
Fg • 1
[ ;0-
:
I
Fg.",:".,
~
..J.. J..
dFj
fj .1
.
..
FIGURE 13-6 .
Pi .1
III
~ ~hanging the, pr~ur~' and temperature of a
'"'
IiquicVvapor ·mixture.
,',
,
:·;.·r~ :'.-' -.;.... ~ 1',
: . " ;!. _"'.:' .;"
•• .• • : ':.' '-:' . .. ..
"
~.
'.
'"
','
~
•
• ' •
• •
,
.: _
I,',
. {'"
• . '
'.
.
'
. '
.
. , .safura~oD··:pres~." wit!ti~;I#~',to_-.~~~:~s_a~~.On~:~:p'~~*el:t~ ~e~,p..~~t~ prop~ni~s a:t · saturatep_ .Go~di~~,~,
:... :_~.
.... '-:"''-. _;. __ _
,"
'.
Example 13~3. At 50 a C "and 51°C, respectively. water has the, following properties at sa.turated conditions, 1'( == 0,00.10121, 0.0010]26 · rp31kg, 3 Vg = .12.05. 11.50 m Jkg; h( = 209,26, 213,44 U!kg; and lzg = 2592.2; 2593.9 kJ/kg. Use .the Clapeyron equation to predic~ the ch~ge in saturation ' pressure between ' 50 ~d' 51°C. . Sol1itiDn. S~bstituie the property values af50.SoC into E.q .. (13.20)
," (2593.1 - '211.4) . I1p - ( 1 K ) - - - - - - - - - ' - - - (50.5 +. 273.15)(11.775 - O,OOlO~24) 0.62.5 kPa . The s[eam tables give the following cbange in the .saturation pressure, =
~
12,961 - 12.335 = 0.626 kPa
13.9 PRESSURE-TEMPERATURE RELATIONSHIPS AT SATURATED' CONDITIONS By making two bold approximatlons the Clapeyron equation can be used to suggest the fonn of an equation to relate the saturation pressure to jts corresponding saturation temperature. Assume (l) that v( ~ v~, and (2) that the ideal gas equation applles, even for saturated vapOr, pVg. = RT, Equation (13.20) then beconles
dp _ __J~2(8"'---_ dT
T(RT/p),
and
Integrating,
or
~.-
In.p
= -hr/(RT) + constant of integration
1n p
=A
+"'B IT
(13.21)
MATI-!E1VfATICAL MODELJNG- THERMODYNAMj,.C PROPERTIES
315
where A and B are· pos i ti~e and negative cor~stants, respec tively. rI'he form suggested by i::q. (13 .2l) is . . videly used an d indicates that on a In p I.:S 1.IT graph a .straight J.in~ represents the cha'ractelistics of a given substance . . Figure 13-7 shoVJS such a graph for several fluids. A g()~J,. of ~is "94apter is' to· identify. mefu~S of making slight refme111ents
... ,"'to ~~" rOM. of an·.~q.q_atX~.n.~.6~):l.ch!~Ye. a.noti~eable.iriJ.p~i~1y~(oenLilJ._the..fideJilY.. ",,;'"·: of ~ep(~~~ntal1oIL~Arr ~x.aipp'l~ of this P!"Q~.~$·:··'is·thf"·A~.tqine equation'-C 18.88) ...' "~ .. which introd,uces one additional pai'amet.er,.;~··~ "' . ,:: .' "" ... ,....... '.' :., '. , .. .
50.000 ------1-------.. _..._-._.... -
.... _- .. -,--- .__ ._ .... -_ .. - ......
. , - - -.. - .. ----"---'-~I--'----'-.---., .. - .
·-·---I--"'-----,~----l----I--.--- -
-:\~-' ~:-' - - - ' --.:....--. ~ .'
.
1
~
(')
.....
7.
_ _~_ _ _~_~__~____~______~______L __ _ _ _~I
0.001
0.002
0.003
0.004
0.005
nGUR~
Display of ttrms in Eq. (13.21) for ;evera!' substances.
0.006
O.OC)7
0.008
-'"
TAIlI.1( 1.1.2
0\
.
T1lcrmodymlJllic Properties of Wuter -
:(
Temperature t,OC
o
,
20 40 60 80 100 ,120 140 160 180 200 220 240 260 280 300 320 340 360 374. 15
T, K
273 . 15 293 . 15 313 . 15 333 ,1 5 353 ,1 5 373. 15 393. 15 413. 15 4).1.15 453, 15 4]J,15 493.15 513. 15 sn 15
~S3,15
573.1.1 ,I<}3. 15 6 1.1 . 15 6.13. 15 ' 647.30
LiCJuid Pressure
lJenslty
p, kP-.l
kglnr'
0.6 108 2.337 1.375 19.Y21J 47.360 101 ..13 19R.54 · 361,40 6IX . IO I(XI2.7 1554.9 2319.8 ' 3347.8 4694 .} 6420.2 8592.7 11289, 14605. 18675 , 22120 ,
009.S00 998.303 092,260 983, 187 971.(,28 958. 130 9·12.R6) 925.H,O 9U7.276 RH (,,9IH 864.678 B40.336 3D.fiO, ·
783 .94.1 75 0.S2 5
7·12 .200 666.B89 6/0 .2_0 528.011 3 15.457
Sp. Vol. or V:lpor
Entha
ln~/kg
hr
206.3 57,84 IY, 55 7.679 3,409 1.673 0.8915 0,5085 ' 0.3U68 O.1 93H 0,1 272 0,08604 0 .05965 0.0421 3 0.030 13 ' .O,Ol I65 0.01548 0.0 1078 0.00694 0.003 17
- 0.04 83.86 167.45 25 1. 0 334.92 419, 503.72 589.10 .675.47 763.12 852.3 943.67 10)7.6 1134,9 1236.8 1345,0 1462.6 1595 .5 1764.2 2107 .4
MATIiEM.';'TIC.AL ". (ODELlNG - nIERMODYNAl... ~ PROPERTIES
+ B.I (T
in p = A
(I3.12)
t- -::') )
~s
A pre) Ds al ::or the estimate) of C
+ O. :.9T
. C, K = -18
.r.; .0: .
317
. " ... Fo_·."the,: saturation pr;~ss~es ,0', \y~(et' sho~vl1 in 'Table L3.2,. the . av~rage p~rcent .d e.Vi.a rion :.(frOID J2.rp.b·..) ,3. :.4)...u.sing..,¢e .An.toine..tor.ixds,j.. 39% ;::.W hi ch,::r... ,', .nlay' .he~comp.ate.d".,to 6.53 %Ior. tfie. ~Qi-.rp..of E(f~tJ3.. il f used.j~LPr0b .···{-3-£2 .., .... . . •
,
,. ..".
. '
•
t,."
•
_"
• • 1.
.
of
A set of equations that 'a ssists in achieving the goals developing c6nvc:ruent, yet" reasonably"ac(:urate} property rehriG.ns are the Maxwell re1arions. The Maxwel1 relations are·
aT):' _ - OP) av s as v
-
{13.23) (13.2.4 )
aV)
_ .
as)'
aT· p
-
-
ap T
:~),.
= -
~~t
(13.25) (13.26)
The Maxvv'ell relations provide .n link between entropy.and the properties ill the p-\t-T equation of stare and are derived from four differential equations:
T ds = du
+ pdv
= dh
- v dp
T ds dF
.=
(l3~9)
(13.13)
dh - T ds - s dT
(13.17)
and from the deflnition of the Helnlho1tz functiQn
. A = u - Ts
the differential form is
(13.27)
dA = du - T ds - s dT
. The same pa~tern is used in developing each.of the Maxwell relations, ,
and Eq. (13.9), which is the basis for Eq. (13.23), will be used as an illustration. The internal energy u can be specified by two independent properties, s and v) so
u =/(s,v) and
-
-..
4
du
;;;1S.
au) as
~
ds
+ OU) dV
dv I
(13.28)
DES] GN OF TH.E.RJvtAL
du
Equating the two
.9)
ds - p dv
"
".
t·
.
"~. :.,,:.... '.
equal, so
:., .:-.; ... j
'.'. .
':. . "
dv
dv must
.coefficients of ds
(13.28) gives
-'
.~
,"
"
relations are -
a ., - aT) av . a1'.
= _
aP)
is
the
l'
MaxweU
'SPECIFIC HEATS
to.o15 listed properti~s
is the specific those at constant vo]uD1e and at co.nstant ......."'... Their definitions are .I.J.u.J..J..I...I. ......
C'111r'O
constant va] orne ~
constant pressure
(13.29)
(13.30)
l
en~alpies, and Specific heats are means of base are alo/ays chosen for those proper1~est so the is more precisely defined as compllti,ng the differences in enthal PY entropy. internal between two points . . ]n ,. heats vary both with respect to pressure and temalrhough we shall find that the temperature has dominant I
I
of a ~hus
entails an tion for the ,heat as a funcrion of p to be followed js to first
Cp
with respect to
deveJoped by solving
...
~~ ,
",iATHEMATICAL MODEUNG-"":RERMODYNAMIC PROPERTIES
~r ~,
319
lj~'.
~
.cor "rfs 1'"'1 1 J.- ':-:::n J ..... ,:."
i
(13 \. ',
13)' Lt..!.. ::!nc~ ...c:hO·lf .lr ' ·.r.: Vv J·~~"'~ 1 (o..r thp. .....
>.
,
~ ah)'
~.
,. . ., arr 1 . ,',
~~
0 r er"':C'/'~l1/' v'-ec;:,"llT"e D-, r'h~t J. ..-0 '_ __ !. 1 v~, c1J ... ......
r>f"lDI _,'U . , ,sL--:.~_L K.
I
S)
P
= ' .J ::I T .0 _
I
P
;t,-.
f:
SubstitutiLg c; from :Eq'. "(13.~O) .anq differen tiating. \vitb. respect
~..,,~~~,~~".;;itT giv.es:,", c,;~,,: ~,' :
~
-,' . :. -t ::....:::..E!. 'c'< ~)' ,.." "'a'
1
ap ., T
Substi~ution of the MaXwell . . .
.;.' .". ;: : ~,':'" .'" ..
. . '. ,". .
"
'" '.·iJ~-"[ v.:t-S:j ~'j .. u . ." = aTap = iJ T '~a p . T P .~ . ,.... . ,. .'
~2s··,
'
to p at
.0
,
relation, Eg. (13.25) results. in . . '..,...
~)'
"ap
,
"
" . VJ -T aT2 p
_. T -
~
iJ2
An important feature of Eg. (13.31) is that Jepf ap)T is expressed in terms of the P-V-T ·pfoperties,. which are assumed tq be .available in eq~aiion . form. The flrst use Eq. (13.31) is .t~ apply it.to a perfect gas .w hich has . the equatio.n ~f state .
of
(~3.32)
The second derivative, a2 v/oT2)p, is zero, so with a perf~ct gas there js no change of cp with r~spect to pressure. It i~,acknowledged, however,. that even for. a perfett gas cp can vary with temperature. The lack of influence of pressure on cp of a perfect gas encourages the ~oncept of c~ at zero pressurey c.po, because all ga~es ~ehave as perfect gases ~s their pressure ,approaches zero. Thus, if cpo can be expressed in terms of 'temperature, anq if the equation of state is Jsnown, with the p.elp of Eq. (13.31) cp can be expressed over range of ~emperatures and pres~ures. Values of cpo determined- from experimental data can be. expressed as functions of temperature using simple polynomials. An ~xaniple of these representatiops for several common gases is presented in Table 13.3. ~ " Some' of the ent~usiasm for the use of Eq.' (13.31) wjll be blunted by attempts to app~y the equation. Most u~eful equations of state implicitly incorporate the specific volume v, so the expression for a2 v/ aT2)p is complex. The· ultimate use. of c p is l*ely to be in computing changes in enthaJpy' or entropy over a change in p~essure and temper'1ture which then requires an integration of some complex expressions. All of this represents a departure fro~ the overridipg objective of this chapter of. achieving
a
an improvement over the ideal representations 'with a
rnod~s~
incre'ase in
complexhy. Returning to this objective, we may wish to pr~eed no further ~n expressing Cp in a form shown graphically in ,Fig. 13-~. The zeropr~ssure values of a given 5~bstance' are functions of temperature, such as from Table '13.3. The slope of any curve is a.vai~able from Eq. (13.31) using the equation of st~tet perhaps by numerical ra~er than analytical differentia'ti9IT, ~see Example 13.4~· If the curVe for a ~yen temperature is
320
DESIGN OF THERMAL SYSTEMS
K,
'-0.0849 + 0.69380°.5
the equation
pressure from' 0 to 2000 k.Pa at of applies,
of oxygen as a
a temperature
of 60.0 K
the van der Waals
a
p. kPa = - - - --:;v - b v-·
where R = 0.2598 kJ!(kg . K) a = ] 346 b = 0.000995 m 3Jkg
From Table 13.3, = compute aCplap)T at O. numeric.ally by .computing v at 595, 600 , and
So]ut~on.
=
=0
o
. 500
1000
t"relssure. kPa
.asB
----.
1500
",""",H~~'
__
j\-IATH£MATJC\L :-'-{ODEU~G -'fliER~10DYNAM ~ PROPERflES
O.07'74274~~3033
VS9S
=
V 6011
= 0.0780840914874-
V6~~'"= 9.07874059~1?40' ',,' :..J. -~' -.. :. " 'The'~s~~'Olid ,de rjv_~qye ~w.nrf'respect)o' tetnper,~t~re ~(e '=
, ~ '.
32 1
, ,.: _' - :..;...',
. ....: ..:' [,t~~5-'~'~600-) :.~' -('~;600 ~~~~595)~J ' . .
;209
'l.,j-:-~ i~, .
. ..
. ... • ' -----------'--'--- = 0.470 ~< 10- 8 5 .
5'"
"5
'. ' .:.. ~",
Then from Eq_ (13.31)
,yep I
ap
" At 600 K
~
2_844 X 10- 6
T,
oe
~}
J
OP'T
ac
"and
= 2.844
.::..::.t!..j
Jp
kl/(kg' ,K) per kPa
'
X
'T'
=Oatp"=O,
"
10- 6 at p =2000 kPa ,",
Proposing rr second degree vari:Hion with pressure as suggested in Fig.
13-8~
At 600 K~ 'where 'a I il"2
=
0
= (2_844 x
=
IO-6)/2p
1.4~2 x 10- 9
.
Thus cp at 600 K and 2,000 kPa is ~P
= 0.9865 +
(1.422.X 10- 9 )(2000)2
= 0.9865 + 0.0057 =
0.992 kJ/(kg . K)
In Example 13.4 [he correction Jor pressure revises cp by 0.60/0. In general, the influences of pressure on cp are much less than those of temperarure. The tools are now available to assess whether a. correction fo~ pressure on cp needs to be made at all.
13.12 p-v-T EQUATIONS Many fanns of equations are in usc for p~v-T relations, and there are still more versions' when the various m'odifications of the original fonns are included. This section jdentifies severa] widely used equations of state tMr require knowledge of only a small number of constants. Furiliennore, in several artfie equa[ions presented below these constants can be computed
~.
DES:1GN OF TH.E.RM.AL SYSl'EMS
only
the ,.. . . . '-H·"'·"
mCllec:UI:ar weight,"
sure of 'the . The simplest
teD[]pe~rature
and
.;)u,!..,,,;) .. c;.....u. ............
D.rc~bablv
most
'".
refinement factor, PV.=
RT
f3
\vhere data.
Ct
+
3
and l' are
OP) - - 0 -0\1 Tr where When
'= O·
(13.35)
critical temperature,
van
(13.
RT p (
v
b
a
(13.
v2
b=
(13.3
£vlATHEMATIC!-1- MODEUNG- THERlvl0DYNAMIC PROPERTiES .
".
.
'
.'
3.73
I
:D.e development of.5q. ( 13.37) is addressed in Problem 13.7. The values of the a and b constants in Example 13.4 were cOQ1puted frorn·£q. (13.37). The vail dei Waals equ.a tion uses t\VO constants ~n additio'!") to R., and its accura<;::y extends over a wider range:' than .Jthe perfect gas equa~on. A .still more . ' .. . .• . . " p.CQ~Jiat~ two-constan.t equ·ai~on .i:5~6ne·· proposecl·.iYIf Redl-ch~ lliJ.d· I( N'OD2::'!
~.,, '
. ,'
...
"
• •
'."
','
• •
,._
~
-
.
p =
'.
•
••
.'
.J
...
, _ • " ' - ... , ..
R·T ··....,,: .. ,.- ··:·' ·a.~ .: ':..: ..:.:.. .. .
v'- b
- , . ' .,
.
TU-v(v + b)
. ..: . .' ':""
. "
.
.' ·(13.38)
Soav~ made even further impro~ements in the accuracy by expres.sing the a constant as a function of temperature .. The simplicity of the RedI1c.h-K~'ong . equation of state in Eq'. (13-.38) is, ·boweY~r, .consistent with the npproach
of this.·chapter. Applying Eq. ' (13.35) to 'the Redlic.h-·Kwong equarion yjelds a =
R2r5/1 c .
O.42748~·- -
(13.39)
Pc . RTc b = O.08664Pc '
(.13.40)
ExampJe 13.5. Compute the specific volume of ~uperheated water vnpor ar 5000 kPa and 350°C using (a) the perfect gas 'equation, (b) the van def. Waals equation, and (c) the Redlich-Kwong eq~arion.
Solution. (a) R of w~ter vapor v
= 8314/18 = = =
0.461889kJ/(kg . K)
(OA61889}(3'sO + 273.15) 5000 k.Pa
0.057565 m 31kg
(b) Using the values of critical pressUI1: and temperature from Table 13.1. the van der Waals constants can be computed, a = 1.7048
and
b = 0 .0016895
The van der Waals equ·ation. -
5000
(0.461889)(623.15) = ---~--~ ,
y -
1.7048
O.OO16~95
can be solved for v by an iterative process, such as Newton-Raphson. to find
.
v = 0.05303 mJ/kg (c) For warer) the RedJich·Kwong constants from Eqs_ (13.39) and (l3.40)
are
-
.
a
= 4~.95J
and
b
= 0.001171
DESIGN OF THERMAL SYSTEMS
Substituting Ulese constants into- Eq. (13.38) i3..nd solving
v yields
v= "
... -
'""
.
. ;L-.
-
....::..
FULL SET
I'~ ~',l ~~
~/' '~":.:' ~ ~ I."
..
.' .:" ....#1: ......
• ' •
DATA
of saturat.ed liquid. superheated. saturated vapor have are enthalpy entropy. -n.'I'"l·rn'!:··Ir> to developIng 'a' of property conditions of entil£lJpy . 13-9. and entropy are usually are in reference to . values assigned to saturated liquid at a chosen. Clapeyron equation then jumping the enthalpy and entropy of From this . condjtion enthal pjes and saturated and superheared vapor can be the ..... .J-!...I ..... JU. ...
to
h and oS of superheated vapor at all temperatures
11 and .r of saTUrated vapor using
Saturated
h and s at base temperature
equarion
h .::md s of :\.alura~cO liquid
h.and s of salu rated
C1apeyron
. J f I
--~""----,
I Ii and s of I saturated
:
vapor
al
all
I
I I •, j I __ J
polynomial
I fif L _______ _
s.equence
10
enthalpies a.nd entropies.
'I..","
: :; ".
I'
I~.~:;.';.
325
MATIiEMATICAL MOD EUNG-:- THERMODYNA M!C PROPERTIES
~
equations outlined below. The enthalpy aUld entropy of saturated liquid
alread~l,
at one temperature have been' assurhed, and the -'ualues at othe( could be computed in one of I~WO ,,vays. it would be possibie . t-;. to ju~.p back from saturated vaFor to saturated iiquid u.sing the ~1~peyrof1' . S'.'- . R· :'··~,,,,,, equation, or itmight be as, sifl1ple tCD 'p~.qorrn a polynomial fit of tabUlar ;.' ~ :;'~data for the' eptheJpy and . . eriH::opy;~6i.satIiIate(rliq~i~r;:;J.;,:·. .t.;·",: .;. :.~..:.'-::: ';:""":';'<;"':':':~'<~';>J"', " . , "'. Figure' 13-9, im·i;lles ':--~thaLspecific::.y',alues-~of. enth~lpyJ ,ana, enti-opy--are'" , " -, be computed. 'ip '~allY' such cases a' :C'illcuIa'tion will 'no't be needed, how~ver, because 'only a differ,ence 'of e~thalpy (during ~f process) needs to b¢ comp~ted. Sirilllarly, equal entropi~s before and 8.f~er a cOlnpression D;1ay need to he assur~d so the specific value of entropy is of :,10 consequence_ The change in enthalpy in the vapor regjon ca,n be fouhd by integrating an ey;pressioo for dh which is developed as follows. For vapor~ h c'an . be represented as' a' f\tnction of p and ,T and then the' expression for cp ,' substituted. .•.
~~
:~emper.atures
. to
1
dh =
Also, since T ds
ah)' dT aT p
= dh
+ ~h)'
dp = ' CpdT +
dp T
iJh)
dp
ap T
- v dp
-Dh)
op
- v
T -
'as T-)
...L
,ap T
I
Substitute the expression for OS/Jp)T from the Maxwell relation, (13.25),. to obtain
,~q.
(13.41) The expression for a differential change in entropy comes from a parallel development
as)
ds
aT
ap T ds
and from
as)
dT + p
dp T
= dh
- v cfp
:;;p :;)p - i = :
Then
ds == cptf!... - dV)' dp T aT p
(13.42)
Equations (13.41) and (13.42) can be integrated through a change in p and Fto fInd the change in ~nthalpy and entropy. respectively. with cp
DESIGN OF
yields P
so
+
+
x1
I
--
.
':--
so h at
(13
>rATHEMATlCAL MODELING - D-JERi....tOD'(l\' )'I1C PROPERT1ES
,
327
using the Rediid.l-t(wong equatiofL,Unf0f}unately v cannot be soived explicitly in this equation~ so a .1umerical c'ompu~atiori 'Of av/ aT)~ must be ~nade. For example. 4000 KPD. the following specific vol umes can be computed,: .at. , .
='0.085963245748 '
Vl
····f .. ~ •
, '. " t ~.,
\"'
:
-" • •
I
.'
500.0 ".'.'
J
. r'·":
~
.
1".-'
.
,
':
'"
.. '
=·500;.-1-":; " -'
'''-
"Nb:·~·.o~0859757:;:,J31 1 ",·"." ,v·'" ~'...'. -. " . . .
"",vc
.
.
.=~O~08~9881.8054.l- ,
•
.
v -
ay . '.
r-) a.T
=
p
Vb -" .
773.15(vc
= "-0.010
.
O~2
.-
v3 )
'
m3/kg
The br.ick~ted term in Eq, (13.43) is essenti,ally c.onstant at a value of -0.010 ·throughollt the 'p~ssure rnnge .of 0.6108 to 4QQO kPa so t.he, integnil is C~O.010)(4000 ~ 0.61) = -40.0 kJlkg." Finally. h at 4000 kPa and 500°C is 3488.5 -"40~9 kJlk:g = 3448.5 kJ/kg.
The agreement of the ca.Iculations of Example 13.6 with tabular valpes is good, as Table 13.4 shows. . One action that helped provide a~curacy was .. to use a p-T equati on for saturation applicable to the low pressure range, rather thz·,n the ace deve]op~d in Prob. 13.4, which is a best fit to data of the \vide range of O°C to (he critical point. . When the techniques ,presented, in this chapter are used, it is unusual to' start at one point, as in Example 13.6, and to build property values over the wide range of conditions based on equatf Jns. alone. Instead, the tools pro'vided here are most often used for more ,:~onfined regions 1 such as condensation, evaporation, or heating or cooling.a vapor. Using several tabular val ues of properties as reference values 'at various "positions on the property map wi11 prevent the accumulation of errors. When simulating or optimizing systems the need often arises to perform property Galc~lations. This ~hapter has presented several tools which TABLE 13.4
Comparison of Example 13.6 calculations with tabular values' . Vapor enthalples, kJ/kg
I,OC
pJk.Pa
Table
0
SOO
0.6108 0.6108
2501.6 3489.2
500
4000
3445.0
. Example 13.6 I
2501.5 3488.5 3448.5
328
DESIGN OF TIIERMAL SYSTEM'S
iulpr~::rve ' the
•
accuracy of these' calculations by a modest increase in com- _
pl~xity.
. .... ".
~ROBLEMS :..... ,: .. ',13.1. Use ~e m~thod ofleast:s~uar~ to .find the tbre.e param,eters' in the fo1l9win~ ., . . ,,:.' ,' e~:j.1-laf:i6~s:~at :giv~ the -best fit tile.' li'lui4 density"orwat~r"to ', the ,absolute . . : : J.~'" .,~" <~ ;~. ,7 '!emp'e~~ ~ JJ.s~. the ~,~ fr.9}I?~ Ta~Je ~~: 2. . " ~ . '... . ' : ' . '- .', _.' :.. ' (a')'" dens ' rl~ty'" ",,::,:~'.~. '+"'a' ,;,T:+·"·a"2. .. T·:'!:'. ,V:.'t~·:: '.~' . .:-~ . ;:_;.i ':. b ··f'-~;·" . '. ' . ;~,:':'.'. : .,.-. '. ' T ' . ':"'.' . . . '~" . .. .~. "." .I'~ ~O.,... .J....1..: ~ >-i::. . (b) density'· = Co .+' CI (l.~~ 'i'rI:e: i~~~·':+-.~;( > . ;~)'~3:"." ....;~ . . ",. . , ' ....... '..,: .. ~.
of
,.
i <1'/i
"
where'Tc ='~ritical te~pe~ture1 647.30 K
,
'1, 1'
•
."
.
r
• • :,.'.
....
.
,-'.
..
. Cdnip~te the sum of the deviations squared. for harp representations'.·, Ans4: (a) 30,885, (b) 174/. -,' . ' . , 13.2 . .· Using the 20 temperature-pressure data points fi?r water at saturqfed condi- . . tioris fro'm T~.ble ~3_21 write a computer'progTarll or use one from .a comp~t'et _ library tO'determine tbe values of A ~d B in the equation . p
=
e(A . ~·E!T)
. that give the rnin.imum average ..p~rcent deviation. Select the trial yalues of A and B by substituting two, different'p-T. points jnto the equation, (0) What are the values o( A and B? (b) What is the 'minimum average-percent devi~tion? Ans.: (b) 6.61 % or 6.530/0 depending on trial values. 13.3. A vapor pa.s ses through a venturi-type flow rn~asuring device, as in Fig. 13-]0, havjng an area ratio A tlA2 = J.8. The process from point] ,to point 2 is usually assumed to be isentropic. The specific volumes are VI = 0.06 m3/lq~, v:! == 0.068 m 3Jkg; the arithrnetk mean of these two values may be used in Eq. (13. 15), If I1p {-2 := 40 kJ!3. what is' Vl? Ans.: 82.1 mls. 13.4. Examine the 20 temperature-pressure pajrs for water at saturated conditions from Table 13.2. (0) Detennine the values of ~ B , C in the revised form of Eq. (13.22) P = e{A+BI(7-C)] I
and'
that give the minimum average percent deviation of p. (b) Determine the minimum average percent deviation. and compare this . yaJu~ with the results from Prob. '1 3.2, Ans.: (0) A 16.577, B = -4023.qS, and C 37.20
=
(b) 1.02%
~aJ
------~~----------------
------.
=
-
FIGURE 'I 3-10 Venturi flow meter in Prob. 13.3.
13 . 5 ~
Starting with the di,ffeTenti:u of the 'Gibbs functio n ; E g. (13.17), de'Jeiop
.r11e l'l:la.xweU re arion, Eg. {lJ.2.5).,
I
13. J~ , Vhea cp is knov/il, C\', ray, be determi0ed! if a..'1 ,expression for' cp - Ci' ca n be devdoped .. Us;ri,g s = f l( T ,p) and's = f lCT, ~: :') equate the t\.)-"o , exp[ess~ons for ds to d~v~jop' 4Tl equation for dT _ '
'
, _~tz) Equate ~:his d~", t~ th~ e~pre~s.~.~t ~oi ~T ~r~r.:- T, r f 3(.V ~ p)', and make , ': apprapnate suI?stIlutIOT'L". i to gevdoP an equatlOf!_ :wf cp ~ "C'.)". "".,:'..!, •. ,,' .. _, : -: ~'(8) -:~Nhat. is"cp~ ~7"'"
c~:'-foi
a~pelfecr'g'as'?, 7,..;:,~.'".,_:.,~...,,~,:,
" ,:'Cc) ~W~ai is'
• "'"''
',;, .• ".. { ", ~" :;:
,.:,' .. :.. ,
tpe magriitude of. tp'- - c fq(.~,flter vapor at '300~C p~d".7000 " ' , kPa. ,~i!l,g the Redlich-Kwong eqLlatio~ in Example 1 3.51 --, '.' l ',
'., ' Ans.: (a) T (av) lap) ---;;: l ,
,iF
pI
aT,'"
(c) 0.920 kJ/(kg : K)
J.3.7. Sh,ow·that because of tJ:1e, required conditions at the critical pomt (expressed in Eq. (13.35)), the a and b consUlnrs in ~he van der Waals equation:are as represented in Eq., (13.37).' , 13.8'. Several properties of Re frige ram": 12 (dichlorodifl~or~methane. CChF2 ) are as follows: molecular mass of 120.93, a critical temperature of 112.0°C~ and ~ critical pressure of 4113 kPa, C~mput~ tb~ specific volume of R-12 superheated vapor at 100°C a,rid 2350 kPa using (a) the perfect gas equation. (b) the van def Waals equation~"and (c) [he Redlich-Kwong equation. An$.~ property tables show v = 0,0079888 m 3Jkg. 13.9. Using as the starling point the entropy of saturated 'water v'apor of 8.9767 kJ(kg . K) at 1 kP-a', (6.98°C), compute the entropy of superhea[ed water vapor at 400°C and 4500 kPa. Use the Redlich-Kwong equation and the specific heat data from Example 13.6" Ans.: fr9m tables, at 400°C and I kPa, s = 10.671 I kJ/(kg . K) at
0
400 and 4500 kPa, s = 6.7093 kJ/(kg - K).
13.1 C. Refrigeranc-12 has a molecular mass of 120.93, a critical temperature of II2.0°C, and a critical pressure of 4113 kPa. An expression for the zero pressure cp is 0.64496 + (0.1783 X 10- 3)( + (2.1384 X IO-6)r 2 • Starting from saturated vapor at aOc (p = 308.61 kPa) where the enthalpy is 351.477 kIlkg, compute with the aid of the Redlich-Kwong eguaLion the enthrupy of superheated vapor at 100°C and p = 800 kPa. Ans.: from ta bles, at 100°C and 308.61 kPa, h = 417.591 kJlkg' at 100°C an? 800 kPa. h = 413.38B kJlkg.
REFERENCES
.
1. Handbook of FundarMnLab. Che.pler J. Am~rica.a Society of Heating, Rc:fr{gerating, and
Air-Conditioning Engineers. Atl~nt.a. GA, 198.5. 2. W. C. Reynolds. Thermodynamic PrOptTlits in 51, Stanford University. Palo Alto, CA. 1979 . .-- ..
DESIGN OF THERMAL SYSTEMS
.1 ~
----.--
."
••
..
w
•
•:
.v.~!.·
a..
j
•• ~( ')I,' ~
l
•
•
. I, ,', '-:', ',,;c '
~'.'
""=",'.
~..
• •
••
" ~': ~'
;'~ .,,!~.,
.
'. ,', ."",
: ',
,' •• ~
. . .. ,. ..:c.,-'!.
'.-
. . . ._ .
It:.,. . . . .
•
" . • Ii',
_~.: , • ...; ' .
It
.
-
';.
.. '
':
...
:-
~TE
J'
OF
LARGE SYSTEMS
14Ji DEVELOPJ\1ENTS SIMULATION
~
SYSTEl\1 '
Chapter 6 defined system simulation as the· prediction of the performance of a system when ,the following information is known: performance charac- , teristics of components, properties of working substances, and conditions 'imposed by the surroundings. 'The value of simulating thermal systems lies in exploring the operating starns at off-design conditions as a step in optimization and to detect potential operating problems such as excessive1y higp .... or low pressures, temperatures, and flow rates. The mathematical description of the system simulation process is the simultaneous solution of a set of algebraic equations that may be nonlinear. ' ' Chapter 6 provided sever~ tools for simulating systems including suc-
cessive substitution and the Newto~ ..Raph.son technique. Successive SU~St1. tution is straighdorward to program and does not demand large c~mputer memory. ,The Newton-Raphson method is powerful and CODvenient, particularly when using generalized computer pro grams I that extract the partial derivativ,es numerically. For moderately sized problems (less than 50-100 equatiEms) mete is not much J1~ed for more sophisticated approaches. The
331
DESlGN OF THERMAL SYSTEMS
setting 'up. the equations to properly' represent" chapter 00 simulation to large problem~ and memory re!quirernents are ' succ~ssive f:nJ.g?l inmemoty method on additional importance',. Chapter,6 already ppinted Qut several Qf . s~bstitl.1rion jn Jh~t is slow, . .... : :-and . calculations the IS divergent. substitution b~n a:.popW¥ early simUlation programs
.
. 10
:·appeais;\that>N~W~9,h~~.J:~ir?~:;
..
and
Raphso~· similar are'.no~~"g~~~riing,&9~.n"d.:·. d~,apret"wJlf'· , ",. look dee~er iri.to:'-rhe·,- divergence problem, jncluding ways to avoid the" ' problem and correc;ting it it occur. to large problems, . the computer the Newton-R~phson m~thod . roughlyl as the square . . . u. .. This refiriernents of the Ne\vton-Raphson method that .L..L~L ~ problems. r·,.'.. r· ..·T\r that important in ,Jarge systems is the modular approach, in which considered to be a col1~ctjon of subsyste~s. each having Two final topics covered in chapter applicabJe to both small and , are "(I) coefficients and (2) correction of component' models when experimental or field data become avajlable to verify the j.uJ.J..,).
.L..........
.......
1
simulation.
Section 6-8 emphasized dictated {"ure of ~he inforrnation-fiow diagram deterrriines whether the sequence will converge or diverge. In multi-equation simulations there may be a number of information-flow and we can hope that some better approach can. found than blindly trying one after other until a one is found. I poinr out that techniques are available to a without acrually perfonning the successive substiturion. Unfonunarely. the test often requires more effoI1 than the ve . We , consequently devote our effort to of ho,w to A clue to the converg'en'ce criterion 'can' be Seidel method solving 3. set of Hnear, simultaneous equations. The Seidel method starts with values of alJ except one uses first to unknown. next equation for next unknown, and so on. rime p new able is computed it is used in subsequent ca]cuJatjons. After one set of unknowns has been' computed the returns to the first the until l
I
I
...
STEADY~STATE
JFIGURE 14-1 ' , The Ga,uss-Seidel rhe'Lhod ,as a successive
sub~tirLltion
SIMULATION' Ot;'
A.RGE SYSTEMS
process.
Exainple 14.1. ,Using' the Gauss-Seidel method, solve for the x ~s following set of s,iro.ultaneOlls - .J~
ill
the
Un~ar eq~ations:
A:
4~
I -
3X2
+
B:
x
I -
21' 1
+ 2t' 3
X3
= 12 ~
6 ',
Solution. The, Gauss-Seidel method is a form of successive
substi~~t:ion.
as shown by the infor:mation-flow diagram~ Fig. 14-L If trial values of Xl = 0 anq X3 = 0 are chosen and the diagram of Fig. 14-1 is executed, the results are as shown in Table 14.1.
The Gauss-Seidel method in Example 14.1 converged to the correct result: 'X 'I =2, X2 = -1, andx) = 1. If the order in which the equations were S91ved changed. however, ~e calculation sequence may not have converged. Its for example, the A-C-B order wer,e chosen such th,at the foJlowing equations were soI'ved in sequence:
the calculation,
A;
x 1 = (12
C:
X2
=6 -
B:
X3
=(6 -
as shown in T4b1e
+ 3x 2 2'(1 Xl
Gauss-Seidel Solution of Exarnpl~ 14.1 Cycle
XI
Xl
1
3.0
2
1.75
-1.5 -1.625
. . . . . . . . . . . .
10 :lO
-- .
.
.
.
---l.045 2
. . . . . . . . . .
XJ
•
I
-1.021 . . -1
0.5 1.375
••••••
3X3
+ 2.xi)/2
14.2, piyerges.
~
TABLE 14.1 '
x 3) 14
1
...
0 .977 1
334
D~IGN OF THERMAL SYSTEMS
• ~pJe .14~1 with in the A. . C . .B seVIUe.![lCe
:" .. 1 -: :" ,2
~"' ...... C"
solved' ,
.r,2
'f0vcle
"
a9' ..
.-: ·3
. XJ" I.
,,' 0
, " '2.625 . \' ....
~-"
,
'~
3.75':
',"'r,
-
.L;.~U.L......J
.
-
~
:". '", .
0.703
of eqn~tions. of ,the ,~ix po~~ib~e s'eque~ce~ 9}1~Y . For ~s particu1,~ , the A-E-C "'''''rIf'-'''''' One characteristic of the differen'ce b~tween, provided by anexa.D:llnation.,of ~e ~oefficieI:it matrices: 1nr'''''
A-C-B. Djvergent
A-B-C, Cqnyergent
-3
-3
[I
[r
iJ
"1
1
:-2'
i]
,lp the convergent s,eguenc:~ the large-magnitude coefficients appear in the'
djagonal position of the matrix. ' , , , More precise me~s are available, to test the c,oefficient matrix to ',detennine whether the sequence that if repr~sents is c,~nvergent or divergent. ' One technique 1 is to multiply the lower-triangular terms by the constant A~ set the determinant' of the resulting matrix eq'ual to zero, and solve for the values of A. If the.A ~ s an absolute magnitude less than unity the convergent. to A-B-C
an
I
4,,\
1
;\ 2;\ A = 0, 0.125
-2), A
+
'2 3;\
=0
O.696i; 0.125 - O.696i,
so A-B-C is convergent
but for sequence 4A 2;\ A . 1 A' -2A
1
3' 2,,\
=0
A = 0.0.2713. -4.] 46 so J\-C-B is divergent wi}]
The translation of Gauss-Seidel criterion to nonlinear equatjons throu gh '\P e.
STEADY-STATE SIMULATION OF iAftGr:. SYSTEl'vlS
is a function of
E,. '':l j,1ple 14.1 _ The Do'ver required ' by a certain ' automobi e its speed~
,p wh~re , ~. ~~) •
•r ' ·
,P = power, leW', ,, "/ ,=:=7.. ,$.P~~~tof
" ,,;, "-
•
= 4.2 + 0.45V +' O_0015V~ , 3
_ _
'.
."
'.
. "
(14.1)
,
::~~a~,t(f! _~s "
. '"
-.
~.'
"
',' ,', _.. ,
'. • •
•••
•
- - ,. •
•
.'
•
- , 1be P~~$~-.. ~Ji~~[~d by a',d1rect-drive 'engrne at 'Speeds above '12 ~
•
•
•
.
. ',
=:' 60 +
p'
•
.
=
-.
: '
8V " ~ ;O.'i6V2
.
",,, '
"
' .:
...
r:n/s'~is' •
•
, (~4.2)
' By means of successive subs,ti.tution" determine the speed ,of wh'ich the auto ;;) capable and d1e p()wer detiver:ed by the engjn,e at that speed_ So/utiOJ1. Two different infoflT!.a[i~n-i1ow diagrams are possible, as shown j'n Fig. 14-2. Ro'w' diagram a, with a trial v~loc~ty of 50 m1s\ yields a set " values nbstracted in Tab1e 14.3 and converges to P = 112.39 'k\V and V = , , 42.25 m/s_ Solution of Eq_' (14. 1). whico is a nonlinear equ'ation. requires an " iteratl ve process,
or
r __'
AUla
~' ,--_Eq_ , (_14_.'_)-----J
_E_q,_(_14_ , 1_)-----'
Vl
pi V
I
Allra
p
Eng.in~
Engine
Eq. (14.2)
Eq. (14.2) (b)
(0)
FIGURE 14~2 Information-JJow diagrams for Example 14.2. TABLE 14.3
Abstract of successive substitution calculatioris nith flow diagram a in Example 14.2 '
0 1
151.9144 76.6163
3,
18 79 8()
50.0000 32.0188 47.8273
60.0000
2 a
Y
P
Iteration
•••••
It
•••
t
•••••
---.
--
,
•••
JI
.....
112.3907 112.3893 112.3905
35.84,g7 l
•••
,
••
I
••
~
•
,
42.2501 42.2499
42.2501
336
DESIGN OF 11IERMAL SYSITMS
•
TABLE 14.4
Abstract of successive substitution ailc1ilatio~ with Dow diagram b in EXaJl1.ple 14.2
- --- - - ,
." ' ~
.'
.
'
;
.... ... . ." ....... .. -. ' ... .. .
",
29.431~ 156.8580 . 51.2382 ·. 49.8492 .. Squa,re root of negati ve flIunberin Eq'. (14.2)
. 24 25
26
--
Flow diagram b diverges, however , as Table .14A shows, even though the trial value of V = 42.0 was close to the solution. The insight provided by the coefficient matrices in the Gauss·Seidd method will now be translated to the set of nonlinear equations. The clements that make up the coe,fficieot matrix ', ""j~ the set of linear equations are comparable to the elemeDts of a matrix of partial derivatives:
al, ax, al. ax)
al, aXn
al. ax.
and we propose that the partial derh'atives be used fo( the nonlinear equ ations. Tho arrangement is th at if variable.T , is computed fromiJ ' the afJ I ax, must be one of the diagonal elements. 'At th e converged values of v = 42.25 and P = 112.39, the matrices for the two flow diagra m are as shown in Fig. 14·3. For convergence .we seek a coefficient malIix thai has large magnirudes down the diagonal. Figure 14·3 shows that Flow Diagrar:' a, which is convergent, best meets thai requirement.
v Eq. (14.1) Eq . (IA .2)
[
6.36 -5.52
p -I -I
J
·FJow diagram o (coo'olcrgcOl)
Eq. (14.1) Eq. (1~ . 2)
f
p
-I -I
V 6.36 ] -5.52
Flowd iapom b (divergen!)
FIGURE 14-3 Matrices of parti al derivelives in Example 14.2 .
---:.,---
.
,
.,
114.2082
2
. . . ' ." . ' ....
.110.8078 ",': "'i <"
',::.'~
'f
v
p
Iteration
"
ST""c.AOY·STATE SIMULATION OF LA.R~E SY~fS
..
337
'The test of the absolute values OL.A. desfli.bcd p[\.:. via ···s!.y is vu).id for Example -1.4.2 and con ceivably could be used ' for calculation loops with many. egyations ..vV irh l~ge num,bers of equations, ho'wever, ' the method b~comes cumbersome and is quite. impracticaL OUf approach ·wil..' be to use the -:oncept in d, qualltative m~pn~r ,;:iJ. trying to arrange · ~e caIcu.la-:
.". . . ./ ..: "ti.o.ry, . ~~q}~S~F,f :~~rfh:~.~. I-~f~9:!a~.9·qar )~Jetpeh~ ~~~)P"'~ Jru-ge.-:,~qll~t~4~ E<+~.i,al . , '.. ·derivatlY.~s~.~~Ail()~flet "vi'ay ,'of st,atin.f'tfie"sa'rpe o~jectlve','it~,to: ·lIy...ti:t s'truc~ :.' ,..
tun~ the sequence so that the vanabl~ computed through use of;~ equa't~o'n'
chnnges only ,a.·small ,amount for large change's in the othl?f vaJ.iables'in that
equntioTI_.
,
It will not 'always be, possible to choo~e a calculating sequence such , that the largest-magnitude partial derivatives appear in the diagonal positi,on , for all equations. Choosing the yariable to be solved by one equabon' may force into the diagonafposition another ~quation a 'partial derivi:ltive that, .is not ~he largest for that equat,ion.. A crude rule is to resolve the cdnfJ,ic{ by placing ill the diagqnal position t,he partial den vative tha! has the' greatest ratio in comparison to (he other denyatives of its equation:
of
14.3 PARTIAL SUBSTITUTION IN SUCCESSIVE SUBSTITUTIO N Example 1~.2, illustrated' two unde,sirable characteristic,s of successive substirution, F10w diagram a produced slow convergence, while flow diagram b produced divergence. Partial. substitution may be effective in correcti:ng both of these problems. The equ.arion that d~fines t1!e partial substitut"ion process) to be applied EO each of the variables i!i x j. ,. + 1 = {3 x j. ; + 1*
where
+. (1
- (3)
X j,i
, (14.3)
f3
= partial subsritution factor , x j = variable being computed
i = subscript indicLlting previous value 1 = subscript indicating new value
i + +, 1* = subscript indicating new value computed directly from the equation Choosing f3 = 1 gives the u~altered successive substitution process and for a set of linear equations defines the Gauss-Seidel method. Reducing f3 shortens the steps during the. iterations and in general do ves the calculation 'roward a more convergent process., No general statement can be made about the influence of {j on the speed of convergence. Figures 14-4a nnd 14-4b show the effect of {3 on the number of i'Lerations needed for convergence of flow dMFa m a and flow ding!am b, respectively, of Example 14.2.
DESIGN OF THERMAL SYSTEM:S
"
'. ,J
:.. ~ :, -'.',: ~ " .",,\-
~. ....
".'.
'
' ,
, ,
::;' '::-,' }~':'I!
:1': ,'.;,;" :.... -,
I ,
.. -, ,.::.,-.-,
,:
.
~'; "':
,
'::":' ).
~,'
,
",',
"
,
-
,'. :'; " ",1..::' jf~ ~
:..
'"
~...
..:
t.:..
o
.
,
l00~----~----~----~~--~----~--~
~I
.'~~ __
-0, L-_-=---t--.:.'...:.., " ---:~C-==::I:::=:=-----=---L o 0.2 '0.4 0.8 -
.:......L..-","":"",,,,----l
1.0
1.2
600
500
400
13 300
200
~
100
J.,~
o
'0
J
~
0.2
----
~ '0.6
O.B
j3 b
flow
-----
:--..
....
1.0
1.2
:
STEADY-STATE SIMULATION OF ~GE SYSTEMS
.'
339
Table 14 .3 sp-owed that Vi/hen Lhe va1u~ of 1f3 was unity thai: the ch&.i.1ges in V 8n d P during an iteration ~Nere 'too ·la1rge" resulting in oscillations aro und the ultirnaie solution. Vlhen f3 js set ?-t '0.55; Cc.1.v'ergenc:e ~.o the criterion list~d in the caption :of Fig. f4-4 occurred in 8 i~~ra[ionS. Increasing f3 . 'beyo1;ld 1.06 results jl '). div.erien'2e. Fl,aw.:diagram b \A/~·th f3 '. 1, .is divergent; . 'but, ~ 'Pig. 14-4q shows, choOSI!l.g.f3 less' tl1.?rf 0"~95 perinitS· the siriluI2_tioD.
.
~:,:~,; ..,=.;-td co~y~rse': ~'~'~;-::_'~:~;'~
.' ",._...' .'.<' i.-' ':'. >-"
. '::'~"~'::':;.' :. .::. -:"', ":',: " : . '.>' ..." :.
~ v.~.~;
:'
..
I
, • •_ . " .
":''''. :'.' ..... :'.':...' : ~.
.~
.14 .. 4 ,lEVA·L UAT ON OF THE NE\VT011 ..
.RAPHsON T~C][Il\Ji6UE "VI1Ef-~ -lOVIPLJEMENTRO ,AS A GENERALIZED PROGRAlvI '. . .
, «
Two methods for' soJving a s~t of simultaneous nonlinear algebraic equ~~ tions (the mathematical description of system simulation) that are presented . in this book a,re succes.sive substitution ~nd N~~ton-Raphson~ 'Some relative. advantages .of these two methods ar.e as follows:. t
sllr;.qessive. substitution. -?tr.aighttorward "to .pro,gram· for th~ computer~ and spmng in the ~se of co~puter me~ory .. Newton-Rap}zson. More reljable and more rapidly convergent. Also, it is not necessary to list the equations in any special order. The above comparison probably translates jnto the following approach. If a person forsees [he need of simulating a small system with no likelihood' of simulating other systems, succe:ssive substitution is probably. the :best choice. The flow diagram should be constructed follo\ving the recommendations .listed in Sec. 14.2. Shoulq the sequence diverge or converge too slowly partial sUQstitution as described in Sec. 14.3 can be applied. Ii the' . worker anticipates a frequent need of perfonning simulations, gaining access and developing a familiarity with a computer library routine for solving simultaneous nonlinear equations is recoD;1D1ended. Some library routines have extensive capabilities and can accommodate a broad range of assignments, but this characteristic may also be a disadvantage. The additional and non~se~ ~apabilities may only complicate and add to the expense of using the program ~hen applied to the single purpose of system simulation. An alternate approach 1S to write pne's own generalized program which offers ~e possibility of bU,n.ding ~n such conveniences 'as input requirements matching the system simularjon assignment. providing intermediate printouts following each iteration, and building in diagnostics for the. types of 'problems anticipated in sim.ulating thermal systems. The genenU.ized system simul ation program cited in Reference --1 iS'less than 200 lines of FORmAN code. and is reproduced in Appendix II. Certain operations are standard in the Newton-Raphson solutjon such as the computation of the! -functions that are to ie·dnven to zero. the eompuration of partial derivatives of all funcJ
340
DES1GN OF THERMAL SYSTEMS
tions with respect to all variables, arid tb.~ solution of a set of simultaneous linear equ~tlons to compute the coqectio~ I9 each of ~e variables. The " user has the responsibility, 3:Dd indeed will demand the 'right~ to, describe ", ,', the system by spec;ifying the equations, ~d pro,viding trial. valti¢s of ~e .' ,'< :'" variables. ~ . ,. ' , . ' ",: ' ., '. .', . . ~~ ~': ',~', 'The p
: I. : ..
<'
I ,
1.-
,., '
"
"
'program·
•
1 ' "
. '
•
...
•
•
MAm progrpm Designa'tes: number of variables (and equations) convergence criteria, for example.
the. fraction of change o(al I van abies' i nd icati ng convergence trial values of all variable.s
Use" provides
Call subroutine SIMUL
Sut?routine EQNS Ihal provides the simultaneous equations
-'
S"ubrou!in~ SIMUL that performs the Ne...ytonRiJphson solution of the equations in subroutine EQNS' with appropriate applic.:.rion of subroulin~s PARDIF and GAUSSY . ~
Subroutine PARDIF that extracts pnniaJ derivatives Subroutine GAUSSY rhat solves simuhaneous equarioM
B
In Jibr:lry routine
set of linear
FlGURE 14-5 . Struclure of s genern!ized program for sleady-stBte simulation of
-.
systems.
STEADY-STATE SIMtJUTI9N OF L ttRGE SYSTEJAS
341
&-
i ;,
~
lJ ~nC>l'al; ot: ~ 1.:.
zea' l:r)··o .,- -a'·....,1 g,t iLL
s-::l\/p.~ _ ~ 1..'1 1(.. "", 'U.:"" c, ~a.. "".:J CO"lSi 11 Cil,,,, LLU!..J -.; C,.... (·i~'n l..L..)" 'I\,,, " L
J''1('
=-~ rro:1 C 'l-m-'t;> (I f;'l:~ ~ J. '---'
'" ..-..{,. W . O. '
partial derivatives numerically, rather 'than r~quiring ,mathem atical expressions for the derivative~. rI'he partial d~rivative o:1? fu:nctiun j i iNit>. respect to variable Xj ; for 'exarnple ~" is' ' . ,
,
'J. ' "
.". _!:,.:
.~:
-..,
-!>L-.
•
~ ~
j
".
~.
~ ... '''-'
/,
-'
-. ':
'
! ., _
.
I.
.' _.
r.~;
..~'. .'
/1. (14.4)
. 'S~?routille ,PAROIF makes' a c~l ~f-each equatioI;l in EQNS 11 tlTI)es, so (n ~ n) equations are calculated.; , ,. A generalized 'program 'such as this', ~s ,useful for frequent or even " occasional uses. ,The 'program can be executed·either on a mainframe ~'om, purer or a microcornputer~ and the execution time and memory requirements not severe for simulations of systems represented by., for example; 50 equations. If the system to l?e, simulated is of ,this size or smaller, it is qu~s~onable whether, effort in developing a program of. greater speed or sophistica,tion can be j~stified.
.are
14.5 SOME CHARAClERISTICS OF THE' NEW1~ON .. RA1'HSON TECHNIQUE In comparison to successive substitution, the Newton-Raphson techn,ique has several advantages. Newton-Raphson is not subjec;~ to divergence that results from an improper choice ·of the information-flow d.iagram or calculation seq~ence. In fact; the sequence of equations 1isted in Newton-Raphson is jmmaterial) and the probability of convergence is high ..The NewtollRaphson technique can diverge, however, if the trial values are too far from the final solution. If the iterations drive the values of the variables too far from the,physical solution, the computer program may blow up, or it may also be possible that the calculation converges· to a nonphysical solution. This nonphysical solution may satisfy all the"equations, but one or more of (hem may be in a range beyond that intended for the4 use. One of the characteristics of Newton-Raphson is that the variables
may change a large amount on the first iteration. It is ~so possible that the variables foHowing the first iteration are farther from the solution than the . trial values, as illustrated for one equation in Fig. 14-6. otten after th,e ,big jump in the fust iteration the process then coqverges to the correct solution. , On some other occasions the variables are thrown so far from realistic values that ¢e program eventually fails. It is possible to damp the rust iteration
-- .
. DESlGN OF
HL.. .."Y'ncL.
SYSTEMS
0.4
0.19 )
I
0.3 )1
II. !.(
.~ I ZI· ·1 .
Ne\\"U)n-Kapn!;On iteration may move the values
the variabJes
the solution
a
one _-.,__ ~._ ..
f=
----..
=0
(]
3 3
StEO\DY·STATE SHvIUL4.TlON OF LARGE SYST B .;IS
. 0.15
r. J
. 0.10
r. .
0.05 !- ~' ~~
.. ~: ·I .':. · ' ; . . .
'~
l
'
"
,,'-
":"2' " '
-J
-0.1
flGURE 14-8 Behavior afJin 'E q. (14,.5) o\:er a range of values of the wat~r flow rate
\Y.
of f in Eq. (14.5) is shown in Fig. 14-8. which is ipformation not normally av~ilable
when performing a simulation. The solution is:
'W
= ,0.6 kg/so
"
, The Newton-:Raphson technique converges to the solution if the 'trial value w t is
o<
wt
< 0.92
If 0.92 < We < ) .27, the first iteration jumps to a nei.;ative value of HI and from t)1ere drives to -co. Tri3.I values of w greater than 1.27 drive w to + x, An example will illustrate several other pitfall,s that in' rare instances may arise when simulating systems that include heat exchangers.
Example 14.4. A counterllow heat exchanger shown in Fig. 14-9 has a VA value of 6.5 kW fK. and uses a stream of water to cool oil from 80°C to a temperarure " regulated by a proportional controller set at CfJoC. The temperature of the entering water is 3:>'°C and the controller regulates the,
Oil. wo::z 2.1 legis ...-_ _ _ _--;
,
- '-ndfJRE 14-9 Oil
.
cool~i1;1"tared
in Example 14.4. ~
c= ,4.19 kJJ1cgel( kg/s
MI.
. 344
.
.
flow
of water
. wh~ to
. .
.
DESIGN OF TIIERM.AL SYSTEM-S
Ww
according to the eq~ation'l .
.
. outlet oil temperature,
, " '"':»'~~~"".'
°e.
Simulate thi~'system to:de~rmine to
. . .. . . . I . · · . .. . .... :~;p,~:' ."~" ~ ~'. '. :. . . . ;. ~~. '_SDlu1ion~ ~jn~!"SoJuti~n is 'to .' . 62~ 70°C ~a:D.9. ww ' .' ,0."8095 k!is u~der: whIch ." ."
' .. ~ ". :~'O~dJii'ODS'· t~;. in~';6.~·fl~t: w.ate~t~tQ.p~.W!~;i\~·:q~.~"C;:.J. lP._~)!,?!p_?,~~-.of.Jhj~ . ~,~
: ," example,' however,. is .~o' snow ·seve~"~peDerices·Jbat,:occuf·~~~.,·qi.ft~r(;b~~:.~\' .. equation formulations' and ~al. valueS: Fll'St-·usectb~ equatiOJlS~ __ :,.... .
J.l
= O"3(to'-60Y-w~" ..... .-:
-
.
(14.5).' .
. The Iogarithrnic-.m~an-tempera~e difference (LM1D). may be compu~d by . the equation, . LMTD = [(80 - tw) - (to - 35)]l1n[(80 - t~)I(to - 35)J
./2
= 6.5(lMTI?) -.(3.2)(2.1)(80 - to)
/3
~ (3.2) (2.1)(80 - to) - ww( 4.·19)(t~ - 35) .
. (14.6)
(14.7)
A -simulation attempt. ~ith the trial vaJues S~ow'fl"ln Table '14.5 fails." One approach to circumventing the problem of me negative logarithm is to express "Eq. (14.6) in terms of an exponent' of e which~has no' limitation on lts argumeryt. Tpe executjon may be able to pass temporarily through the physi..: cally impossible siruation that terminated the execution in Table 14.5. Revlse Eq. (14.6) by taking rhe antilog, j, = -
(80 -r w ) ~ EXP{ [6.5(80 - t~.) (to - 35)
-
(f~ ~ 3Srl]
. [2.1(3.2)(80 - lo)J
.
(J4.8)
and use in combination with Egs. (14.5) and (14.7). As Table 14.6 shows, the simulatjon is successful. Lest it be concluded that the exponential form of the rate equation is a pan'acea choose trial values as shoWn in Tabl~ 14. 7 . . The form converges. bur incorrect vaJue~. The simulation has driven the water flow rate to a vaJue such that WCwal.cT = weal}. with the consequence t
'.to
to
•
TABLE 14.5
Simulation of Example 14.4 using log-mean.;. temperature difference in th~ rate equa tiOD w. TriaJ values Iteration I It.eration 2 w=
5.000 -0.667 l.B07 ExccutJ on
t. 70.00 57.18 66.02 ~mtina1.ed I
t..
48.00 56.82 82.26 nesative Jog
STEADY-STATE SlMULATlQN OF l..Ai?GE SY5fEMS
1:~BLE
14.6
.
345
J
Successful simuJ.ation f .0X:-: j-n r Ie 14.4 u~ngJ t ti.. l 'fornI of tl2e · ate (::quation t,.. .. :,,=', ".-.
-Trtal val.ues '., .. _ . . 5.000 70.00 48.00. ' .' . . " . "~'. " '. Tteratiort I .~ ~ ''':''',fj~5{)7 -,'" - . 71:69,:-·--1,:1,:,_- 4:r.'50~·'·i./:-<';~-·';' .-.... ,'" ." .. ,.. , /, ",~ .,',.!:-,.. . Itera~o[] 2 ~T:OOO '56_61".'" -_. 53:'~~ -.... ";... " IreIiJtion 3 4,563 75.20' 132.87 . ,i .", .":-" :'.' ,.'. Iteration 4 2.958 69~S6 72.85 . Iteration 8
. 0.8095
69_21
62_70
. I.
TABLE 14.7 .
,
.
Unsuccessful attempt at Simulating . Example 14.4 USing the 'expQD~tial fonD of the ~ate equation
Trial values Ireration 1 . Iteration 2
5.000 1.184
Ite'ration 5
1.577 .
61.00 63.95 65,25
51.00 53.31 ,49_ 18
1.604
65.35
49.60
that the rise in water temperature equals the drop' in oil temperature (this situation' was first explored in Sec. 5.4). The mean temperatu.re difference is 80 - 49,6 :::; 30.4°C = 65.4 -' 35. Thus the rate of heat transfer = (2.1)(3.2)(80 - 65.35) = 98.45 kW. The required VA value is 98.45n0.4 = 3.24 kWIK which does not agree with the specified value of 6.5 kWfK. The source of the difficulty lies in Eq. (14.8), where the numerator of the argl.!ment of the exponent drives to zero, so that the control e;r:erted by the VA value in the numerator is wiped our. .Incidentally. the use of the logarirhmic form of rhe rate equat'io~ is successfut' wi th the trial values of Table 14.7. A third form of the rate equation is availabte-that of Eq. (5.6), which, along with Eq. (14.5), provides (\.Vo independent equations for »'w and 10 , Unfortunately, with·ehher sea of trial values used so far, (5 kg/so 61°C) and (5 kg/s, 70°C) the simulation converges to impossible values of Ww = 3.:24 kgJs and 10 =' 49. 14°C. '
Example 14.4 showed some difficulties that can arise when simulating systems that contajn heat exchangers, and most thermal systems incorporate heat exchangers. To place the situation in perspective. it shouJd be pointed out that "the trial values cho~en jn Example 14.4 were rather far from the
346
',-,
'correct solution. When the trial values are close to the final solution, any of the formulations should work. The occuirenCei of any.of.the problems cited in Example ,14.4, is' but can be tr011~lesome, so i~ is beneficial!o he , ,alert to this possibility.
."
:
,
DESIGN OF THERMAL SYSTEMS
':
~.I
,i.
1 _
,'".
' "
-J ' ,
.
,
,
. , " "'~-:'),4.,6. ACCEL~MtING THE SOLIryJ:ON:S "':'~", ": :OF LINE]\1r'·E,QQA"TI()~~:·~~;·>'?·:,~.,,-:·.~,.~::~f' I
,
~
' : '
•
~
~
....
~'.~".~' .:~..
•
"~::"" t._\:-}~ ~:,: -
~
••• ,'.
',j"
, " " , "_ ' :, :, .!~:.;~~~~~::~: :/.~: ~" A
a.Fr~~i}:;: ~;~~~~,. ~ .
•••
Because the Neviton-Raphson te~b.nique.,pQS"sess~s" many,,:,stferig~S":;·:jt'~can' serve as the basis for further improvements. The targets ,for these" iefi'ne-" ments are the reduction of e~ecutio~ time' and ,memory requirements when' simulating large systems. The fI.r::;t,' means of addressing these'targets that ' will be described in this section is to improve the procedure for solving the , set of linear siml,l1tan~ouseguations. A faster m~r.hod for extracting partial deri vatives 'is explain'ed in Sec. 14.7. and Sec. 14~8 treats ,a change in the basic technique-the modifie~ Newton,method. " , Improvements should be 'possible'in the Star1dard Gaussian elimination routine for solving the set of linear simu,1taneous equations; since most of the elements in, the coefficient matrix 'are zero. One specific ex'ample of this class of sparse matrix [echniques is· a routine caHed' XGAUSS4 .along with an associated routine NZERO, both of which are listed in Appendix ill. . Sparse matrix techniques'- often consist of storing and handling ele-' ments outside of the matrix to which the elements' belong. The row and column numbers and the value of each nonzero elemen[ are identified 1n the sllbroutine ,NZERO. Three arrays, A( ), !ROW( ), and JCOL( ) ) are set up by a caU of NZERO for each nonzero element in the coefficient matI"ix. Suppose that there are 6 nonzero elements in the set of equatjons, Xl
+ b' 2 = 7 (14.9)
1n matrix form the equations are:
1 2 2 0 [ o 3
(14.10)
or
AX =B
(14.11)
The first step is to proceed tf;troughthe matrix. row by row. numbering each nonzero element in sequence, as in, Fig. 14-10. CaU the encircled number the "designator.·· The A array expresses the vah.,les of the elements jn tht...~ce of the desiGnator, A(!) = 1J A(2) = 2, A(3) == 2,
t-r-~·:" f~
~:
~,~,
STEADY~STATE
~~:
~: ~w: ~K ~.
,
SlMULATION OF lARGE SYSTEMS
341
...,
! ..)' '
K i' f.j!." ~~i~,
~
~
l.f &',
nGURE 1~10
",' G) .. :.,(0
L. ~':'::"~' _~-"~~,}
:
,
"
-. '• .
,'-: _ ' :"'"
~~~u~~e~~g_dien.OPZ~{o.e:ep1e~~;~ilie mauj~;.:. ' ... _ " ...... - .~
l ..
= -1,
."
A(S), = 3?' and !\(6)' ="1. Inste~d :of the, A' aIT~y heing:, hvO- ',',' dime~sional as jt is in co~vention~ G.a~ssian elimination, ~r has ,no\v become ~ ~ one-dimens ionaI- alTay, of nonze,ro values. The IROW array prov!des, the designator of ,the' frrst nqnzero' ~lement in,eacb rov~". Thus, IROW(l) ,=' I.
: ~(4)
~OW(~)
= 3
IROW(3) , =', 5. There are two JeOL arrays: JCO~(lt I
3Jld
.
','
I,
.
) and JCOL(2, '). The' JCOL(1 ,de'signator) in9icates the coliunn nuniqer of each' norizero elemenr~ Thus JCOL(I,)) = 1, J~0I:(1,2) = 2, JCOL(1~3)..-' 1, JCOL(1,1) =' 3,," .;reOL(l,5) ::;:: 2, and JCOL(I.~) = 3. The' ]COL(2, ) array sp'ecifies the , ' ,designator of the next element in a row after the 'fIrst nonzero element. Iri the ftfst row for example, the irrst element' is design'ated 1 and the .next element that row is designated 2 •.so J~OL(2, 1) = 2. Designator 2 is last one in that row" so there is no element after 2 and lCOL(2,2J.- .~ . ~ , O. Continuing, JCOL(2,3) = 4, JCOL.(2,4) = 0, JCOL(2,5) = 6) and JCOL(2.6) = 0, , , After the subroutine NZERO has established the A, !ROW, and leOL arrays, these are fed to the XGAUSS subroutine which is a modified version of the standard Gaussian eliminatioiJ.. The' structure of XGAUSS is as follows: I
,' the
in
" ••
1
I. Formation of the upper triangular matrix. A .. The outer loop operates on each row, designated 'K, from 1 to the, total number of equations N. For each K the rows, des'ignated J, are processed from K to N. ' Form the upper triangular matrix by fust determining whiCh row. called IMAX 'contains this largest coefficient. (If all coefficients in this coluITill are zero, ,the eq~atjoO$ are dependent). Then exchange the l( row terms with those in Th1AX. . BA Produce zeros in the K column below th~ diagonal term by subtracti~g from at] teiw$ ~~.tpe l"equ~,tion the quantity A(/,K)/A(K,N) , where! runs from K + 1 to N. . I
ll. Back substitution. Back substitution is performed to solve for the X values. The operations perfonned by XGAUSS on the set of equations (14.10) are su~zed in Table i4.8. The basic Gaussian elimination processes are identifithle. bur most operations involving a zero are omitted.
DESIGN OF THER..iv1J\i. SYSTEMS
348
TABLE 14.8
Oper~tions perlormed by XGAUSS ' (14.10). ' .
)
·n
". . .
Revision , M a t r i x
. ' :-. :..', '0:."
'Oricrinal .'
II
: .Arr.ay
:1; t<~.:, ,0.
".J ..
1
2
3 '.
.4 ' 5
J- ,0: ,·>lROW(n}.,_ .' , l' ' 3
5,
6
'7
0
. . ..
·.'~:r,: :": . ,~~~:,',--','" ,_..'~3~> ,,- if: :'~~:,,·:";~g~f,ii:-F~~:_:~' g.:
'"
•
.
;
, 23
0 '. -l~ ,8 -2:! ',: '0' 1 O' 3.5 '.. 16 .. ' ·1
Swirch ,.' rows J , and 2'
]1
,
:
0
, 7'
" B(n)
,"8' , . " · 1'-'
·1ROW(n) 3. JCOL(i-;n) . . ·1 . ,2 . .1COL(2,n) ) " .AO. '1 . A(n) 1 . JB(n) 7
a
3
2 0
Add
-QI2)(Row to Row
n n
~
-.1~
22 1127 16 0 35
-' ··8 3 1:
·lROW(n) . JCOL(l ', n) JCOL(2',I1) A(J1)
·Swirefl Rows .2 'and 3
2~
0
o 35 o '2'2
-]'"
8
,IRO,\V(J7);... .
3
JCOL( 1. J1) JCOL(2, 12)
}6
1127
A(n) i~o
o o
~~
0
-I..!
]6
8
~ 1/6' 7/3
By back substitution_ x(3) = - J--I, x(2)
2
oJ-.
3
3 0
'$
0
5 ' 0 0
AI")
0
0
Bln)
8
3
IRO\V(JJ)
1
O· , ')
2 0 7 0' 2 .. 8 I
.. B(n).
Add -(1J:1)(Row 2) to Row 3
4
8
, JCOL( J . )1) Jeot{?.11)
= 5 _ and _'t'( I} =
(J
-2 6 3
.0 0. , ,"
3 0 ·1
·0
]
3 '2 0 '2 '0 7 ,
·B(n)
5 1- . ' 3 4 0 -'J. ·2
5 ] .
2 1 4 2
3
2
-1
6 3
o·
3 . 2 0 6 -:-1 ·3
3'
o
1.
3 O· 112.
3'
3
0
0 1/2
1
:7 4 :! 7/3
3 0
~
·6 -1' 3
3 0
.3 0 -1/6
- 3
.So'me, elements that wer.e originaJIy nonzero may reduce to zero, and SODle that were original1y zero may ' take on nonzero val ues. Revision 2 in the third line of Table 14.8 reduces A( 1) to zer'o at which value it rC1Dains. Revision 2 also introduced a new nonzero element designated 7_ The economies provided by XGAUSS while not obvious in the sma]] -example shown in Table 14.8. occur in the following ways: t
a
1. In searching down coll\fTIn for the largest element, only the no.nzero. values indicated by JCOL( 1 t-) are checked. 2. When rows are inrer~hanged, only [he nonzerQ, elements are switched. 3. In [he triangul arion process, of producing zeros in the column below .the
diagonal term. computations are performed on the nonzero. elements. 4. In the back substitution proces.s only the nonzero tenns in'a row are ~nsnttred.
~
STEADY-STATE SNULATfON Of LARCffi SYSTEMS
349
/\ test of }CGi~_USS VJClS condu'cted on a 2)95 x 295 InatrL~-l ill vvhich most equations contained only 2 or 3 nonzero variaples.: ,'1"'h.e menlG.ry requireraents on a mainframe ,compute:( ':0 solve this 'set of sirnultalleous ~in- ' eC!,,~ 'equations using conventional :GB.Ussian ,elimination 360 ,000 bytes:. , . ~,d tbe, execution'"time' :VIas in ,:~icess:,bf SL~ mjnutes . .TJ,sir.,~ X3'AYS,s on ' :. .'!i;ilie· .same" :.o:fo,Wt?m,~-~t;q~re,qvirements~"w.er.e..,' ,f'1? qqo ;,b y~t~s._ ;~.~ ,~J., ~,?,~::sec,?p.~';, ~~ J_~;
was
resp~ctjvejy.
•
l
• •
•
':
:', ~: . -'. " :,', ,'""
;T
••
,.'
.:
,',
_~
~.
,':,
i.(.·
I
I
'
' ,,'
,"
,"
' , ' , ~:,
.
. ", .... ','
.
.'
•
~
A tilLE-corsuming process in the generalized,Newton-Raphson simulationI is , " the computation of the partial deriv'atives. which are computed numerically using Eq_ (14.4). Each equation is corripuied once for 'each "ariabie, thus 11 X n ,equations. Most of the equations 'db not ,contam a'given variable., so [he result of most of the computations is zero. ' An approach that saves computer ,time is to ~xecute the foll<)wing steps:' (1) compute the complete 'set of partial derivatives in' the usual f~shion, (2) ide'ntify the .no~ero partialderivatives, ,and(3)'wben computing partial derivatives, for ~ucceeding iterations, compute only those equations associated with nonzero partial derivatives. Applying the cOIDputed-go- to command, .is one approach executing st,ep 3. Application of a fast partial derivarive routine on an 81-'equation simulation 6 resulted in a 15% increase in compilation rime, bur in a 40% decrease in execution time.
to
1408 QUASI-NEWTON lVlEl'f10D In the Newton-Rapnson tec!mjque the corrections of the variables. the 6x 's~ are found by solution of the set of linear equations whose coefficient matrix J is c,omposed of the panial derivatives, as in Eq. (6.31)_ !n the quasiNewton technique w.ith the Broyden update 7 '
x
= -H F
(14.11)
where X '= the column vector of the f1x'5 that are additive to correct the variables. X is rhus the negative of the LlX ' s used in, Sec. 6.1l. " H, = the inverse of the 'partial derivative matrix = J- 1 F = the column vector of the values of the functions fl. 12. etc. ' Furthermore, H is updated for each new iteration not by recomputation of me, partial derivatives and in~er1ing the matrix, but by an operation' e~pressed symbolically as
-
.. (14.12)
350
DESIGN OF THERMAL SYSTEMS
where Yk == Fk+i - Fk subscript ,k indicate$ current 'values . . subscript k' + 1 indicates values for next .' : '. ' :.... T 'indicates the transpose,.' .'
iteratio~
;-.'; .<'~ '~~SimUIlltion of a sin;;ple sy~tet.ii. will ill~trate the quasi-Newton,
..'
'. " "method .~d p~ovide~:~'''c6mpaiis9i1 wi.th. the' :coliventiori~l NeWt.on-R.3phson.,' _ ,:.:- .;~C:"~?tec~q~e.;~:'r.h.~.:'.~:~Y.~l~;,~~9~,1ists'. of two .components--a ,lap: and a. ~llct, . . ,as' shown in Fig .. ' '14-t i.a~:~,·~\Villi .~~Pf.eS·S.~~7n~v.(::~dh'ari¢t~§1j~':·;§:9-~Wlt jp.... :;;, :,. .Fig~ 14-1 lb. The .component. eqp..~UoDS aie." :. " . : '. . -,-.:' ....... ''':....'"1 .. ,.. "'';'.:~'''' .
.. ' f
.' ~lf.ct. "
fan
I
.
= O~06~5 + O.653Q1.8 - P
12 =
0.3 ~
~.
0'.2 Q2 --: ~
" 'Tabte 14.9 ..shows the Newton~Raphson soluti<;lll including' th~ D)a.trix' of . 'partial derivatiyes. and 'the inverse of that matrix' ~t each iteration. The .' , , soIiltion is P = .'0.25 kPa' and Q' = 0.5 m,3/S. . The quasi-Newton metqod starts' with trial values, and a .temporary inverse H ~ For this jlhistration we use the same trial values of P =' 0.1 ?lld Q = 1 ~O~ compute the partial derivative matrix, an'd invert it to obtajn the initial inverse. With the above assumptions,
Fk =
['fan ductequatlon' equ~tjon] =.[O.6J55] 0
and .
x _=' - [ -0.25392 -O.7460~] [0.6155] = f 0,63449
l
-0.63449
0
b_15629] -0.39053 Duc!
0.3
/
.p
Duel
"'"
0.1
Fan
O'~--~----~----~--~~d_'-'~~
o
0.2
0.6
004
Aow rate, m) Is
-.
(b)
(tI)
FIGURE 14-11 A f.!,!!-duCl sysltm, and lb) lhe
(0)
-.
.
p~ssure·now
characteri!tics.
0.8
.
STEADY-ST/,TE SrMULATION OF LARGE SYSTEJ'IJS
35'
wkgls
'S',J-: >u.. ":' "
, ' J,:, '
,
~.
_ '. ' , ' .
\' , '
.'
..
.
.,.
'
F
"
... "1- " ..-. . ~
..
•• :
(h)
(0)
FIGURE !4-~5.
.'
.
"
'
.
(a) AL.l.:0.lonla compressor-condenser (bLinfo~on,-flow diagram ,in.l\·oblem 1<.3.': ' I
in borderline stability. and (b) determine the valu,e of f3 thn~ provides the most rapid convergence. :' 14.4. A counrerflow hea~ ~xchanger having a VA value of 12.3 kWIK and shc,--'lll in Fig. 14-16 is part of a system being simulated.' Within the: simularion the heat exchang~r is described by two equations: . ,
(ri~' t2) _EXP.[l.blJUi-,~)-(t(\-.tl)]} .
.in = (t
Q
r.i)
-
,[wl(3.9)(/i - (0)J
1,,+1 = w~(3.6)((:!
.
- II) - lV)(3.9)(rj - 10 )
b~pending"upoir the trial values chosen for the simulation, one of the solutions. shown in Table 14.12 results. (a) Do In 'and/n + I = 0 for both soJutio.n s? (b) Which of the ' solutions is valid?
CP
= 3-9
k!/(kg K) 0
/ .I
UA
= 12.3 kW/K C P ::;
3.6 kJ/(kg" K)
F1GURE 14-16 H~nt exchanger in Problem 14.4.
TABLE 14.12
..
(
Two different solutions to the heafexchanger simulation in Prob. 14.4 5
2
'.
VII .;. : " " ::••:' ' . ~ .' . ~ ",
'
WI
I
WI
loY]
J.
to,
'I
t)
2.95
2 ..52
38.60
29.B3
20.39
31.51
--26J .
2.85
37.15
26.5J
J9.86
30.50
,
t"
364
D£SION OPlli£R.MAL SYSTV-I$
,
)T~bin< ___
BoikT Au~
.' :;
."
'.- .
~~~ ;;;"i90 i.WJ1( ~
gas
'5O"C ' 90I:;!ft: ~
cp ..
-
Power .. q,
C_=
. -,.
W'''f
."io'oc
l."i"
•
;'7f~~.~ .
"
,',..'. 15l~ 96HW1!l;~ ' -.'
c , .. 4.19:- '
.-
"'mp
,.--
... . .
::,
::~.:.;.
- --
kg/>
n GURE 14-18 Stf..lUn
PO""~
planl in P rob. 14 .8.
Po ... t:r dclivutd by [urb!?t
.
f~
""
- _(tb +273. I· + 273 ,1 -
(280)('11) Ie
) 0 .5 - qt
Enthalpy dungt
."
EnthJlpy dl Jnge through the boiler is 1900 kJlkg, so
h ""
1900
I<' -
qt>
Enugy balance J6 ~ ql+qc-'lb
The solution is Ib ~ tl07 .1"C. Ie'" 38 .69°C, W = 10.774] kg/s, 'It> = 20, 471 kW,qc= 15,39Jk W, itndqc -=5078.6kW. The turbine is enlarged 1.0 percent in the sense thaI the flow rate I<" iol l is incre ased 1.0 ptrccnr for givc n values of It> and f e . Use Eq. (14.17} to compulc the . influence coeffi cients and determine the new IV and the newq,. Ans.: Fro m a simulation new w = 1O.82t12 kg/s, and the new q, ::z 5089 .02 ),:W. 14.9. A helium liquefier opera.ling: on the now d ialram shown in Fig. 14· 19 r~ceives high pressure he hu m vapor, liquefies I frac tion of the vapor, and returns the remainder to be recycled . The conditions of hel ium v.lpOt entering lhe' liquefier a t point I arc as (ollows: p .. 1.000 kPa, W ,., 4.6 gis, T ... 18 K, and h .. 101.8 kJ/kg . The ~ep8rator and the entire lo w-pressu re side operate at J00 kPl. So~ helium propenies are At 100 ),:,Pi
utuntion lempe.ratUrt enlhalp), of utunted liquid enthalpy of salUnt ~d vapor enthlilpy of su"perhded Vllpor, kJA:&
.. ...
4.2 K 9.74 kJlkg 30 .31 Ulkg 2.81 + 6 .79 T - 0.0578 T ~
At 1,000 kf. - ' 0 - - enthalpy of su~thealed \·Qpor. kllka ... - ::!9.67 nlC v~
value of both hea t
c:xch~ngcrs
is 70 W/K .
+ 8.79 T - 0 .0826
T~
' 351
STEADY·STATE SIMULATION OF LA,J?GE S¥STH,1S
TABLE 14.9
)
Jewiol1~·Ranhso:i] A _
siInuiation of fan~dil'tt syst em. "
'1Vla trix of F--df"'"J al
Her a tion -
·_.At ~-iar
'.
Derivatiy~
c·
. • ..
p':;::::: 0.::. --.:.' "_ ;'-
.~-::--LO-'~""'\.1:7i59
,-.<',"'.
·.~O,,2539 -o.l46~·
'.
.
.~ .-, .:~ .' ":. ": Ilaliies -'. ""., :-~.,. .... ,·. . .·ft ",~.i, EO{.·~.-j5''';. :~ ...b...:!~ ~. r:o"'~ ·::"':::'0~-4008-:",,·:·:.:~c.-J:'~:!"';':·~:~ O:,,
••
','
•
.,"
,
•• ,
_.,
J
';",'
"?
...';
.. , "
'.
?'
-
....' . .
0.508
'-1.0
0.6836' . -0.2031
0.250 0.500
'':'''1.0
0.6748 ' .
p'= 0.2.50 Q = 0.500
-1.0 -1.0
'2
P = '0.250
3
= P = Q= Q
'. ,-J.O
~6.2285
-J.O· -0.1999
. 1.1432
: o(
V k+I
= [o~_i.].: + [_ 0.156~91 ,1.0
~
-.'
:.....0.2291· -0.77D9 ~.1276 -L1276
so .
'.."
, ' _~O.23-?6 .' --:-0.7644" .• ·0.9674 ....:.0.9674
. 0.7QO-.. ,. .,' -1.0' -0.2435
Q = 0.608 .'
4
c"~
.p, =".0..25,6. ._ ..... ,-1 .. 0
{.
,-.0:-,.
[0.25629].'
=
O.39053 J ,
-0.7715 -1.1432
0.60947
.
/
At these values of the variables the new function matrix is
F_
= [
k+I
0.07402] 'd y. = [ 0 07402] _ -0.03058 an k -0.03058 0
[0.6155]0 . ['-0.5414'8] -0.03058
Applying Eq. 14.12 yields the new inverse.
H
- [-0.24630 0.76030
k+ I -
-0.74956] -0.69190
The complete set of quasi-Newton calculations for this example is shown in Table 14.10. . . TABLE 14.l0
Quasi-Newton simulation of the fan-duct system Variables,
Iteration
Y
Functions, F
. 0.25629 0.60947 2
, 0.074024
~verse,
H
-0.030579
-0.24630 0.76030
-0.74956
-0.69190 -0.76370 -0.97139
0.25160
0.020603
-0.23200
0.53203
-0.008211
1.04286
0.25011 0.50257
0.00 1656 -0.000624
-0.22850
0.000042
-0.22744
-0.76.816
-0·000016
1.16053
-] .08636
I
3 4
c=a-
0
-0.2.5001 0.50004
¥
i.13130
-0.76713 -1.05802
--
352
.
DESIGN OF THERMAL SYSTEMS
The quasi-Newton .algorithm e~pressyd in Eq. (14.12) is a rather clevet; . procedure. that uses the ~xperience 'of"'a prev,ious inverse with the results of its corrections on ~e 'variables and fubc~ons to arrive. at' an improved . irive~e. CompariSon of ·the inverses cqmputed in Table 14~lD with that ·.of _ the Newto.n-R~pbs.on in Table 14.9··s~ows that the quasi-Newto~ technique - . .:': ,-.'::,::. ~·.dri:ves toward :the: eorrect IDverse lanA the correct ~oltinon. -YJ:1e' ex.am.ple·~ . 1. ,<~'.. :.-. ~:mi~t baye s~owed favoritisrp. -t6~ard'the qiiasi~'Ne~on method~' because .the ".. '.
~·~i~'~·:.~·:· .fiis~.~~,e~e;w~.:NQYi9:.~~.:b:y:~~.~.a;~~~s . ~~~;~~PJY: ~~Ji.~~~~~_~~~ph.~~9..n ~?~C~?;~._."" .
. . , . The .,fact.-1s that quasl-Newton- pas: ·.sorp~2':t'.:"
m
moti.o.g" . -'. .' ,. . What advantages~ if any, does the quasi-Newton' me.thod' offer .in comparison to Newton-Raphson'for the large simulations which are th~ foc,:!s of this chapter? The extrac'tion of the parual derivatives and the solutio.n ·of a set of . linear, simultaneous. equations. that are.'·integral operations in Newton-RaphsoI?- ~e eliminated. In their places the quasi-Newton method substitutes a number of ma~ additions subtractions, and multiplications. The'poteI!tial benefit of qu~i-NewtoD in large systems app~ars not so" much .in reducing computation rime -as in allowing greater tolerance in s'electing the trial values of the var.-:iables. This advantage could be jrnportan t in large systems where selecting trial value.s close to the solutions may be difficult. J{en]ey Md Ro~en8 conc]uded that the Ne~ton-Raphson. Il).etbod is rapid 'when convergence does occur. The quasi-Newton technique is reasonably rapid and may require ,many iterations if the trial values are not' good) but will converge to a solution in some situations with trial values that do not permit Newton-Raphson to converg~. . J
14.9 INFLUENCE COEFFICIENTS Influence coefficients seek to answer the following question: UWhat happens jf . _ . 7" For example, IIWhat happens to the capacJry of a system if a heat exchanger is enlarged by 5 percent?~' InfJuence coefficients have been used in some form f~r many years. For example, a traditiona1 rule of thumb applicable to refrigeration systems js shown in Table 14.11. The wble indicates the effect on the refrigeration capacity of a 10 percent increase. in capacity of one componen~ at a time while the capacitles 9f the other two components remain fixed. The dominating infiuence. is exerted by the compressor. If the capacities of all the. components are increased by 10 perce~t the system is simply 10 ~ercen[ larger, and so is the system capacity. Influence coefficients provide insjght about the" system as part of . simulation but may also' be a partjal derivative that provides data for I
opt.imizarion. Influence coefficients, therefore, build a bridge between system ·simulation and oprlrnizarion. The direc~ approach to finding .an influence coefficient is to rerun a simulation program with appropriate changes
-.
;'
:;53
STEADY-STATE SIMUU_TlON OfL.·-LARGE SYSTEMS .
TABLE 14.11
EL!..rreCt ,"P
It.
if
.
fOJL-I-D '10 O!. • - . Inc eases liD COill pOnent
.Pen::e.nt .. ~, .is. ~nciCrC~ Se(~
:; 'rl., ~:
.. ,. .... ~"
.
.
jncre~e'
I
'J
. .'
condenser
in . ....
._
.
~
. ' .' :--;~.~,. '~" :.- :".:~~ . '6-""' " , ..--.. _.....~':~.>. '... ':; '; ....... ',--:: . : . .. .... ..,-
.. ; ~~p~SQr . "I'_:~ -.~_ eyap~rr:.~or
j
10-% . : I
..
I
-.
a11
~-
_ •
' ",."
.-
-"
..
.
"•• _._J
-
, '" I.• ~ •'.
.
10 I
'.
jn ~he equation(s) repre~~I?-ti~g the perforr;';ailce of the (:orr.ipo·ne~t(s) b~ing exainined4 The· vqlue of the variable.of inrerest,in the rerun :is then tompare~ , to ·the vallie in the base f u n . ' . . . . . '. Vlhen numerous influence coefficients are to be co.mpu~ed, it is prob-' ably more efficient "to' apply th.e principles' outlined below. which use the . matrix of partial deri vati ves developed in the ftnal i teratipn of the N'ewto~ Raphson solution._ Mos,f influence coeffic~ents fall)nto one, t\vo.fo~· \vhich lequire SOD1e\vhat ditferent treatment. FOrTn 1 ·asks. for the change in a certain var'(a:ble of the. ~imula[iori occurri.rig with a' given change in a constant that appears in ·one or I1).ore equations. In Form 2 the chang~ in avariable of the 's'imularion is sought when the· capacity. of a certain c0rr:tPO~ nent is altered by a given fraction. The influence· coefficients applicable to _ the~e two fOnDS \vill be treated individually in the I?-ext hvo sections.
of
14.. 10 lNFLUENCE COEFFICIENT WITH RESPECT TO A PARAlVIETER-FORM 1 An example of Form 1 is the complJ~arion for a steam power plant of the change in power P when the area of the condenser A changes. The influence coefficient sOl)ght is aP / aA which has units of kW1m2 •. The Newton-Raphson compptation c:;onsists of the· repeated application of the matrix e.quation. J !1
= -F
where
J
afl
all
aXJ.
aXn
.
-
ojn
ajn
ax,
aXn
.X J.new
XJ,~ld
, ~=
and! = X n..ne:w
:c n,old
At the wlution, F = Ot !1 = 0. and J is known. An approach appUcable to both Form J Jnd Fonn 2 influence coefficients is to make a slight change
DESIGN OF THERMAL SYSTEMS
..
.
,,'.,
..
:'
'
J- :..' ~
-, ;
·1
•
t.,: •
.~,~.~
~': *
• < • I.'
:.
.... :""
~
.;
-:".
"H,'"
ti
~.
..... "... ,,!
t
..
"
.-
",
th
0 0 • >
row"
+-
Xj,old =
-
.
(14.14)
J ij is the cofactor 14) can
.i, ---" . . -.in
C{.
llc;. (
8; from
. (14.15)
~
04.14).
a -a
. Ax) ::::::
ACi
approaches
=-
,·or. (1
1
__
,
I
..•
...)..:.J~
STEADY-STATE SrMU(ATION OF L-!.RGE SYSTEMS
its eptering temperature j.s - JOC. ~he steam coil receives saturated Si;e.am at 230~C, and Lhe enthalpy or' evaporation a.t 230°C is [812 kJ/kg. The 'condensate Leaves the st~arn cO=, ~ at 230°C ~ which js a high enou.gh an
temp erature to preheat 'the ajr, il1 a condensate..cQ·i~ ,;:hat is ~sumed to operate' as a COUllteIflOW heat exch~ger. The U-yalif~. and ?Teas of :i1,e c~i'ls are:
-
~;
... ,-;"'-........ "." ,
.,
.. ',.:
_ 1\' _ •• , .
!
I,.
• __
,
.
; _ ..... _.
'r-~-.' .~
..
-4 ,..
,
~
Vni-~ •
•~ ~
.
....
•
..
"
-"("",
Condensate ' 0·. 04 . Steau::i " 0.'055
.
'
,
'0.5 1.21
12.5
22
(a) Simulate 'this system to ,determi2e the st~am flow rate w. and the air ,
I
temperatures leaving the condensate coil t l and leaving the stearn. coil fl. (b) If .there is .the possibility 'Of increasjng the area- of either- coil 1;>y a. gi yen ~ount, whicb coil should be increas,e,d in size to provide the greater incre~e
in
12?
.SOLition_' 'The three. equations that 4esc£ibe the system are as follqws: Energy balance on the steam coi11 Il = (0.8)(1.0)(t2 - II) - }~(1812) ' Rate equation, steam cOlI,
12
= -'(/2 - rj)
+ (230
- tL)[l -
e-(O.OS.5)(A s )/O.8]
.R,ate equ.ation, conden~ate coi.l, from Eq~' (5~6),
. [ :1 -(I I
-
( - 5))
+ [230 -
where As = 22 m 2 and Ac
= 12.5 m2
-----iJ>o-
-5°C. 0-8 tg/s
~
-
CondenSBte coil
-
U=O,04 ,Ae= 12..5
-
-
-
-
--t-
tI ~
~
w kg/s
J
eO. 04Ac ( 1/0.8- Uol .
•
I
J9~)
~ _e OAA ,( 1I0,8- J/4.19 w). 4.19w .
(-5)]
-*" -
Air
~
.Condensate
Steam 230 C trap D
flGURIfl4=12 Series of heat exchaoge~ in Example 14.5 (or heating air.
SallJ.Iilled steam, 230 0 C Sleam
coil U= 0.055 A J, = 22
1
DESIGN OF
'. [-J81~
- ~1
yields
.- -I -1812 . -0.8]" -1 . 69
at"J _A-
,2
as
!
det [-"
=-1
1
] o -0.8 0.22 2169
given amount of heat-transfer area -as if to
to
=
,0 2 0.309 Clm, steam coil is ten urnes
SIMULATION-FORM 2 . The
section explored influence constant a parameter normally a comthe must somesimulation. to one of a pump is one of.the-components in the a rate cW'Ve as illustrated 1 13. What is meant by a 1 of the pump? it mean a J flow
35i
STEADY-STATE S - fiJL AnON OF IMRGE SYSTEMS
- FIGURE 14-13 , . L-;-------------~s_.
-
T'}!~ rntewretations. of. CI. 1.pe~·cent
.
I
- in
Eq~, (14.13). the ·matrix of partial derivatives from ~e Newton-Raphson simul~tion wip ·once ag~ provide the additional .data for calculating the _'
iililllence. coefficients. What~ then, is the' ~:xpression for 0; it: the functi.on·, . ii is reVised so that variable x k increases by the fractidn p? To revise_Ii so' '. that x k increases by the .fraction P., divide x k wherever it.app~ars by (1 t p). An alternate procedure is to multiply x k by (1 - p) wherever-it appears. The .results. are essentially the same for small values of p, since from 'the '. . geometric series
I
1 1
) neglect
~ p3 . ..
1 - p +p2
+P
A positive p' indicates' that x k increases. by a fraction p. Multiplying x k at all appearances by (1 - p) distorts the original Ji equation by OJ'j fi,nc.w -
fi long
=
0;
Express Ji as a series of p, r----------~
/;.ncw =
~J
I
j)J;
iJp
/;1 p=O '.
I
+ (Jail 1 P
-(p)
p~O
o2~ I'peO (p2)
up
a[x k (1
aif
p ::::: p [
up
a[x,t(l-p)]
p=ro
. a[xk~l - p)J ~P
and.
+ ~21
=
neglect
- p)
J] , p=o
.
-Xk
since all leans in the followi,ng differentiation disappear upon differ-
'entiatioD' and setting p to Ot
- ..
oh
oil
I
:=--
DESIGN OF ~~AL SYSTEMS
. 358
..1bell
aft·
<'
B; -
-PXk - . -
".,
i)Xk .
:':! '':s;hstituting into Eq. (l 4.14) yields - " -" - . :, i'••·,_;' 1, '~:L:;';7:"·:.~;·.'· . ',.: . . , '.•:.: -'. :7 --:'-·A~f'';-p·.~~:f~' -_dj~l:G;~~· \~~~, .;~ '~}:,(.~~;JJ)< . ,' •
_.
•,
•
fu.m.ple 14.6.: In the ·fan-duct. system shown in Fig.'. 14-11 the fan capac~ty '...' 'is -increased by' l.percent (1 p'ercent greater flow ·at a-given static pfess~e). .. . . What is the. increas.e .. in system flow? ' r
Solution. J71e' equatiQns
~e.
.
duct
.
11 =
0.0625 .'.
,12 = 0.3
fan
+' 0.653 X~·8 ..")
--:-
-"r'
.
-.O.2x~ -XI.
where x J = the press.ure P . Xl ~ the flow race Q
At the final Newton-Raphson iterarion' xJ=O.25 :r~
= 0.5 -I
J =[ -1
The ~omponent in which the capacity in.crease occurs is the fan. so i = 2. The riature. of the capacity incr~as~ is that Q increases while other vanable~ remain constant, so k = 2. FinaJly, the effect on the system flow js sought, soj = 2. Applying Eg. (14,]7), .
~x')
/1X"l
=
~
.
of1
J . . .,
px.., _.. ----. - ax.:!
del J
(0.01)' (0.5)(
-..
-(}.~). ~ 0.875
= 0.00] 14 m3Js
So a J petcent increase in fan capacity results in a 0.22 percent increase in flow.rate when Ih.e fa~ is a part of a lar.8er system.
14.12 CONTINUING DEVELOPMENTS IN , SIMULATING LARGE SYSTEMS One of the directions taken in recent ~ears by. developers of computer programs for simulating Jarge systems is the modular approach. In this concept numerous subroutines are provjded which the user can call in the ..
-. ~
STEADY-STATE SIMUlATION OF LARGE SYSTEMS ,
appIicatjDD to a given sy?tem. f\
;~rop~rty
IV' -
rqutine
VIOD~d :Je, provided,
359
for
ex.?J Dp le" that relates press~re" tern,perature~ q..hd enthalpy ~ The user could
call this subroutine providing: two,of the pi'operties in order to calculate the third. The user selects from lhe large 2c~sortrne~t of subroutines' the' compo, ,rient pe.rfo~~nce and "pr9P~rty,: fu.j1ctI9I!s needed) organizing ll-te, variables "ill_an apprppqa.te:,ni~in'ner for the sy,s~eln:'being,si~uI~ted~ .; -, ." ;:> ",-' :,: ,"", ,','. " ~ ~ "<;;This chapter,h~:--exp16re'd~ sol-fie .technit:[l:l~!f~a(are': ti~~fuL'·a~d.'~same~': . times·~even nec~ssaryf' when siinuhi.tihg large ,systemsd . . Thes~ topic~ .might ~iinu1ta-' . be c6nsld~red 'an extension of techniques of solution of the set ,neoqs ,Poruinear equations a'nd can be very usefuL 'It should be emphasized~' however~' that describing the sYStem' (setting up, me equations) )JUlY cOhtinue ,to b~ the significant· chaJlenge ~n sir:.J1ulating large ,sys.teins. ' . J
of
PROBLEivIS ·14.1. The sy.stem consis~ing of' the two nir-heating coils in ,Example 14.5 i$ to be simulated using successive subsritu[ion. The three equations are to be solved for the three unkno\yns t t. fl.. and w., ' (a) Construct all the possible information-f]ow diagrams for a successive' , substjtution solution of the equations (there are three different qiagrams).· (b) Using the partial derivatives from Exampie 14.5 ahd'the test on the matrix of partial derivatives described in Sec~ 14.2, detenn~e which of the information-flow diagrams ,of part (a) are convergent. and which are divergent.. , (c) Solve by successive subsritu~io~ the flow diagram(s) that are shown.to be convergent. 1402a One of the methods by which some of the water from the blowdown froIj1 a boiler can be recovered is ~o throttle ir and condense the steam .that flashes into vnpor, as shown in Fig. 14-14. The system is to be simulated, prim~ly to compute the flow rate recovered~ We'
Slowdown. 2.8 kgJs h
Throttle valve,
Satur.l.ttd vapor
= 11 85 kl/kg Condenser Flash tank
VA 70 legis
I
,() C Wd
k&l,s We
Discharged
FlGURE i4-1·f" " Recoveri~" ill. poi1 ion of water blown ..down from a boiler.
Ret;overed condensate
k.g/!
DESlGN OF
TH.EJ'.U...1AL SYS'fE:1v1S
14.. 5 .. The following
are to be solved by the Sp4:u"Cf~-rnlatr:lX routine"
...... ......, ...AL.I,U.'U.l.lo.>
.
14.6):
!
'1'"
+ .. ,. '".',: ' ..... ::.
....;
.
.: ".
'.
';'~'-: ,:.'
~,.,.
,.'
J"
0
,=
,$.;,"!'
''"
.
. ,
-", 4,'
~_i
',', .':: :
'.
.~.
•
I
.......~ ..... ::.~.:J
~-.~~.~.
.
. ,.'"
2 CD
-1°
0
·0
1@)'
0
0
-5@
6
·7
0
3
3
0070
0
0
-5
3n
2
3
~OW(n)
4
6
JCOL(lyn)
3·
0
~3,
-5
14 .. 6. Equations (1
4· 5
n
2 A(n)
2
-1
0
B(n)
-3
9/2
5'
2
I
0
(14.2) are to
S
,P using the g'uasi-Newton
update •. as q,escribed in /1
8. \Virh functions arranged as
= 4.2 + = 60
s
O.4SS + O.0025S 2.H _p
+ 8S - 0 .16S 2 - P = V (1)
=
.",.,..'''''. . P .= V
= 70,
= -10. Also, the inverse of the Partial derivative matrix
is
= [ 0.06078 -O~061 -0.4862
078] .
inverse yields additive
of
~eHvers
energy from .a a UAa
s with a P!~uct, . ~~~_._
evaporator as
8
,..
is fU'flction
36':)
STEADY-STATE SIMULATION OF LARGE SYSTEMS f1I'
,________
11
12
~
__
~
La _
9
J----I----.-- - -
\/\N\A UA ! =70 v..'IK \!\/W\fvV\,.-
-
4
2 "
..
~
' ..': .;
....
")
......;
.
', ...
Work
FIGURE ,_A-19 ,
__ '
Helium ,liquefier in Frob. 14.9.
'"
,
-,
T6 ,=4.6
Turbine characteristics:
and
W,
+ O.lT3
"
,g/s =3.75 - 0.125 T3
,
'
Simulate this ~ystem an4 compute th~ following quantities: WJ
W9,
=
W6,
g/s;
g/s~ ~V4
W10
=
=
W11
t
Ws ,::= w?~ g/s~
gJs~ T~ '=
Wg
g/s;
73 = TJ, K:
T5 K; T6 K; TlO K; and Til K. = 2.00 gis, Ws = 0.782 gis, Ttl = 11.28 K. 14.10. Figure 14-20 shows a d iagrarn of a cenrral chilled water plant that consists of a primary loop and two secondary loops, each with its own secondary pump. , Pump characteristics:
Ans.:
W3
,
"
A-B np, k.Pa = 300 C-D 6.p. k.Pa
+ 2.464Q - O.04648Q2
'-
= IBO + 3.294Q - 0.1 i63Q2
F - G tJ.P, k.Pa = 180
+ 2.107Q -
O.0131Q2
where Q =flow rate in m 3Jhr.
Control valve
>~~~----------------------+--------~~---------~------~H
I
FI GUR E ) +l2~
ChiJled.wa~r distribution system in fStob. 14.10.
366
DES1GN OF THERMAL SYS"1B.1S
'ibe pressure drop ~!1 each of piping or <:oil sectioDs"is given -bi the equation: Ap~ kPa =: C Q2. J The v~lles of C for the various sectio~s are:
c
Section' , .,; ';:.~;
B-C , ' ,
....'.: '
,
,
, .. :
..... J
,.' ,
.,
r !
,0.017 ,:'"!"' O;O..i4J"IT'."':-'I
, ' . '~!)"~E.~. ~:'~"':.~·'}' " ,
t
~
E-C
::"~
'
~,'
"
J
:"
. ':0'.546' 0:089, 0.0359 . , 0.0302
E-I
C-F G-H. ·H-I"
0:.0089
-J.":"A .'
0-0199 0.162
'H-F,
Simulat~ this system. specifying the pressp.rf:s in k.Pa absolute at each)ettered point and the flow rates in m 3Jhr in each of the pipe sections. Ans.: Po ;:::::: 401.8, and ·Q£l. = 84 m 3/kg . 14.11. The purpose of the multiple-flash evaporator shown in Fig. 14-21 is to concentrate a solution of NaOH that enters with'a 20 percent NaQ~ concentration by mass. The entering temperarure of the solution is 20°C, and e stearn is' supplied to coil of the first stage at a temperature of llS e. The pe.rfonnance characteristks of severa] of the co~ponents ~e
the
.
.
VA of heat excbangers in both stages I and II = 65 kWfK Cond~nser capacity, ml kgls = 0.03(1 s2 - 25)
Equations for the enthalpy of sarurated liquid and saturated water vapor are hI kJ/kg = 4.184 r
.h,. kl/k.g = 2524
where
,.=
water temperature in
Ste.am. 115 0 C
,---
+
1.68
I
Cle.
m J Warer vapor Conden
20s(' NaOH I :::
20" C
";-
.
w!::?
Stage'r
.,..----...,
.",-.
Condensate. 115" C
Flow regulated
"
FIGURE 1-4-21 Multi~21poralor
in Prob. ~}4. J I.
Condensale. 'lt
W2
to maintain
h:!.
x:!
l::l
0.38
STEADY-STATE SLvWLATION OF
~-\RGE SYSTEMS
36'i
The f.'1thalpy of the NaOH solutioll1has £he chatLlcterist~c c~rves shown . in Fig . 14-12. An approximate equation representing these curves is
= ·111.3 + 77.49x - 2556x ~ +- 9592x J - 6579x .J. . . ... '+ (4.249 - 7. 932~ +" 38. 97x~. - 85. l5x? + 61.38x -l) (t ~
h .kJ/kQ
wht';re x =.NaOH mass concentdtion
expressed . IIS.~ {ractio!!_ . '.
- 26.6)
•
.
. .
. '. The rel~tjons1irp" of 'sahlratio'n ,te:rripeia'tur~ to ~olutlon "temp~-I1l:tll'~e is .... ::": . ~howrt'.by .the. ,Ouhrii1g. lin~s in Fig. ·t~~13 .and .·exp·ressed ..,: ,~ . by the' . eqliati~'~~" ,:-:', _ .
'.
'.
f ::::.
(1
+
O.05x)r s
+
58 .3x
v/here t =- solui.i.on temperatu,re 'f satUration temperature of water. · both in cc.
s':=
Simulate t!tis syst~m) $p~c.ifying the following:
solution nQw rates w. \Vh ~od \\/2, kg/s wat~r vapor -f1?W rates m 1 and m~'-kgJs saturati6n temperatures solucion temperatures solution concentration x I
;'600 ...,
~
~
E~
W 400
o
~
___-I-_--I----+-"IC..--+--
L -__
o
i1GUUE Enth~lpy
-L--~----~--~--~~--~----~--~
0.2
0.4 0.6 Mas~ (merion of NaOH
atJ n1
of NaOH solutions.
0.8
\.
368
DESIGN 'OF THERMAL SYSTL'v1S
l~ ~--~~---.~---.-----.-----.-----.-----.-----.
.·140
..... o Q.)
L..
~ ctI
~
80 \
E Q.) '
c 0
. r.i '-
.::3
C U') ..
20
o L. ____ 40
.
~
____~~----L-----L-----~----~----~~__~ ______~____~____
60
50
70
90 .
80
100
110
.· .120
130
1..f0
Solution lempernture, 0 C
. FIGURE 14 23 Relationship of sarunHion temperature w
[0
solutiofl temperature for NaOH solutions .
. so]ulion enthalpies h, and 112, kJJkg vapor enthalpies hg} and h8?, kJlkg
Ans.: w
=
1. 783 kg/so
RE'F ERENCES 1. W, F. Stoecker, "A Generalized Program for Steady·State System Simulation:' ASHRA£ Transactions. vol. 77. Part 1. pp. 140-J48, ]971 . 2. B. Carnahan, H. A. Luthe.r. and J. O. 'Wilkes, Applied NIIJllericnl Methods, John Wi.ley .
.New York, 1969. 3. R. W. Hornbeck, Numeric-a} Methodl, Quantum Publishers Inc., New York, 1975. 4. 1.1.' AnseJrnino, "Computer 'Program to Simulate Central Chilled Water Syslems," M3ster of Science Thesis, University' of Jllthois at Urbana-Champaign, 1977. 5. R. P. Tewarson. Spars! Malrjc~l. Acad~mic Press. New York, ] 973. 6, W. F. Stoecker, "Computer: Simulntion of rhe Performance of an Aqua-Ammonia Absorption Refrigeration System:' Paper B2.30, InternationaJ Congress of RefngefQtion, ()nlcrnalional Institute of Refrigeration). Moscow. J975. 7. C. G. Broydtn. irA Class of Methods for Solving Nonlinear Simultaneous Equarions," Malhtmotics of Comptl/mioll. '101. J9, pp 577-593, J965 , . , 8. E. J Jienlcy end E. M. Rosen, Mtiurial mId EIJ~rgJ' B"IOllct ComplJlariollJ. John Wiky. NewYdrk. J969, ,. I
-
t:.'''-,·· :.
.
-.t"··
...
-
--- - - - - - - -
' .. - .:..
~~.
-;.
-... -DlTNAMIC
T.'
. BElr-J-Lt\VIOR· -OF THERMAL -·_SYSTEMS ./
15.1 IN -WHAT SITUATIONS IS DYNANlIC ANALYSIS IMPORTANT? ·The previous chapter concentrat~d on steady-state simulation; [his one focllses on dynamic behavior where there are changes with respect [0 time. Both steady~state and dynamic simulations are important, but for different reasons. Steady-state simulations are needed and performed more frequently than dynamic simulations; in the majority of cases a steady-state simulation can be justjfied-in the· design stage of a plant in .order to explore partload efficiency and potential opeh~,ling problems." Dypamic simulations, on the other hand. address transient problems that could possibly cause shutdown, damage to the plant, or at least imprecise control. Problems related
to dynamic behavior of the system may be infrequent. but could be_criticaJ when they occur. All too often in engineering practice it is assumed that, [here will be no ~oblerns and a .dynamic analysis is omitted. If a transient
370
.
DESIGN OF TI:IERMAL 5YSTEMS
....,'-' A......,.......
by an a In some' other cases case correction· analyses. are· st£Utup 1
occurs,. it is
controller in a . . .
J-j.
v......, ....... ...., ........ •U
.l.l..U.L.-4~""'U
.
........ LJLU.......
.: . ~
..
. .,:, .cHAPTER· -. that any one :chapter. It
are related to automatic control, and' it· .this cannot be 8$'sumed, that the f(::ader· subjeccts th~t. are to :mat field then is the: inten,t pf this chapter? . on thermal (2) the I'
'-"-IA.""....., .•Urr"<'T.r. .....
emphasis of (3) . si.tuatiorts into symbolic or mathematical :representation. The,underlying objective this chapter is to app~y the principles of dynanljc to physical hardware. Many of the texts on automatic tro"1s unfqrtunately choose few examples from the thermal in_stead mechanical or systems. One objective, is more comfortable dynamic analyses of thefU)o-fluid components and S~condly,
the of automatic controls can a .highly mathematV{ith the challenge being· the development of somea process or system. While the field some in skills) there is. a far greater interpreting a somewhat lower level of mathematics into the physical sjtuation. of the techniques that provide this physical interpretation is ~ as .. of perfonnance in the time domain. This cont-rasts the practice of many automatic control specialists who work primarily in the transformed ical
one'~
The third·
contribution of
is to more U~~''''''''L.''U''' into a control diagram. We . automatlc control highly r""'1""""~C·A""''''''''J"I
as a block
~'U'4'~~. ~~,'H~'&U.~~
[he
first pJ -namely. the symbolic fonn.
some
in
a bridge from the work of the
--
:---..
chapter
simu1ation of this chapter the nex t t
14 to the· a hybrid
DYNAMlC BEHAVIOR OF TJ-IEru.~AL SYSTEMS
371
:i' ..}allGll YVi]e[c. the 111aj o . part o :~ dIe sl.mllbtlo i IS adeQ -a tely ;:ei-.reseZJ.'LE d I . . , as ·a steady 'state one, but one element .intro~uces changes "\:vi&. respect to fjDle ~
.
AMIC ELElVJJ]NT . U'~r:A ' ~r ~~A.::ij ~~Sl:Al, .'.;.SIritWLATIONU:' ~;.;t~ ,.~.;",~,/.'~~;-.,:::,;::.::~~: 15 ..3
';0
.j.
t
AO
•
•
:'":'::'. : ....
:
,< :;', .f~; ~.~ t.
:.'-' ',"'.,'
. '
•
'.. lj.. ":?1odesf extension frQ m ·st~dy-state. to dyn'!mic SilTIulation is 'appropriate ..' for' some syst~ms, ·where, for ~xainple, the thermal' capacity of all' compo-' D:eLtS except one car. be neglectedA An example 'will illustrate' the, approach.5· . ··TIle refrigeration system shown' .in ,f ig·... ,IS-1 is to maintaiDl. a low ternpera'lure in the' chamber. The evaporator that is located In the cbamber removes the heat from the space that is copducted inJ1u:'Ol1gh the \V~.s·.·and. also' that heat.Temoved from the thermal 'capacity ot the' contentS of the chamber, .' First"coIL~idet the case where the cpnditions have s[abilize9- throughout the' system such that a steady-state simulation""is appropriate. Th~ controlling'. equations · a r e ·
\
=f
Compressor refrigeration capacity:
q~
Compressor power: .
P =
. qc
Condenser heat transfer:
f
,Tc)
(15.1)
2(Te1 Tc)
(15.2)
I (Te,
= (~VA)(cpo)(Tc - Tamb)
{I - eUAc/[(WA)(~pa)]} Evaporator heD:t transfer:
, qe
= (Ts - Te)( UAe)
)
Tr;
Con.denser
Air
UAr;
11jennal capuci!y. M t:
FIGURE 15·1 with on~ dynamic element-refrigeration plant serving a cold room_
S'ys~m
--
-
~
(15.3)
(15.4)
DESIGN OF THER.MJ\L SYSTE.M:S
I
...-..-. . . . . . . ~,.,,- to chamber:
:=:
'VA ch ()Tamh -- Ts)
The of the system: '
aJ...II.,UAJ....""..,., "
.
' " p.
\'.9
. . .~............. rate at the evaporator, kW, .' are
fUDctions
and area of
UA c , proquct
of ~vaporator, kWJK
of U yalue
lfAe7
~,-,." ____ .. u.... ",, ...
)1..'Q~.
kg/s
Suppose that instead for example, in pull-d~wn of ~_A4"""'~" a wann condition. Furthennore . . . "" ............................ of the no longer equal to Eq. (15.6), . . . . . . . . . . . . . ....u ...............""',
t
Tarnb'- Ts) Also assume that heat and the objects good
the space internal conduction
objects is the same as
of
the
-a
temperature Ts 1;ligh so that balance within
equation. , (1
= product
mass and simulation a difficul (dTJ Idf).
-----..
that two
DYNAMIC BEHAVIOR OF TIiERM.AL SYSTEMS '
3"73
.
equat~ o12 .
To resolve the diUicuhY 9 relllo ve ) ~-~ d.S a variah'e; of the steadystate sim ulation. The initial v(]Iue of 'Ts 'mIls! be ~cno ~lIl just as is usu'a lly required in the' so; ltion of d:"fferential equations. If Ts = 3JoC i~i. tiaJ.iy the ~et of 7 equations ca~ be solve.d for the: 7 variables, :.ncludirig .. (~Ts I dt) o:~§.~RI??.se ~a~. ,,(~r~_!.t:!~~._ ~~T~.s..~l!t. .~~.OO~~.4 .oC/~ •. If a ~irll~. ~t·ep .· ot JOQQ. s l~··.·arbltranly- ~hoseD. '} tne··:value· ·o,f.Ts after. 1000 ·s "vould be Ju -{~) ~ 6. Of)Q.24. ~C/s)·' == ',;29',FI6- .~€~:::.Thi~ val tie' ,of 'Ti""worild"~tlien \'be," hsb:r~ ~;,y:, :, I
_:
d. .
•
:. c.Tonq ( for the next ::;te~dy-s'tate sjmllf~tiR~~ ". ".''
. -.' ~ .' ./ "...:
" . ,'.
.'
.. ' .. ,.:. ,-
. The foreg~ing apprC?ach , of alt~mating ,between. a sre2.dy-s'tate and a dynamic'simulation applies to> 'cas~s where the resp'Onse of the steady-state processes and Gompone:'lts is r~pid. e,:ompa.rison to rhe elerllent(s) treated dynamically'. In. the above example ¢e ne\~ sr.eady-stat~ val~es of Tc~ Te , P , qc, and qe were assumed to respond very qui.clcly to a change in Ts 't'~lhich is 'the slo\vly changing variabl~.
in
15u4 ' LAPLACE TRANSFORMS Because of their special relationship' to differential equations~ Laplace transforms are a powerful tool in pre9~cting the behavior of d)llamic processes . and of control systems. For control systems there is the added ~~ility. of , predicting stability or instability of the control loop by characterization. of the loop a? a Laplace transform without inverting to the time domain.. , It- . . is assumed that th~ re?-der l]as already studied Lap,l ace transforms as well as control systems, so this section and the next two should be considered ' be' bypassed by anyone'·wi~ the subject at their fin. refreshers which gertips. ' . The deftnition of the, Lapla~e transfoITI1 of a function' of time. F(f.) is
can
l' {F(r)}
= {" F (t)e
-Jl~t
-
fCs)
(15.8)
Example IS.1. What is the Laplace transform of the constant c?
Solutioll
..L'{c} = .
Joo ce-J1dt
=
_::e-Jtja: = s
0
,
S
0
S
.
Exa.mple 15.2. What is the Lnplace tronsfonn of bl?
SolutWn .L{bt}
--
:-.
= fo':m hle-Jld, ~ -b~l;z eds
0
ll
dt = _bd(lls)
dJ
~ b/s 2
374
DESIGN OF TH£R)..fAL SYSTEMS
..
The Laplace transform of e Q1 is a bujlding block in developing. ~e transforms of hype'rbol~c fun~ti9n:s: . . .I I
{ea~
. To exte~d ilie~hyperbolic relation's . . the' id es e ix
=)1 (s - a) .
to trigonometIjc functioDS, make use' 9f .
= cos)." + i sini-
= cosx :-. i sinx
·and .
wherei=R.
so
e ix. + e- ix cosx = - - - -
and
Slnx
2
~{
Then
=
cosh(ix)
sinh( ix) ;" ~1
,1
sin(al)}
a (sin(at)) =
s2 +a'2
Table 15. J is a compilation of transforrns of "some frequently encountered fUl)C[jons of rime. The expression for the transform of a derivative F' (r) can be found by subsriru FI(I) as crion ln the definition of Laplace
Eq. (15.8),
~ jF'(t)f = L'" F(t)[S/ 01 F'(! )dt as
ax
and e- l l as),. and integrar.e by parts:
(I)) = [JI
----..
(~) 1;-
D'
F (t)( - sV" dl
TABL ... . 1.5 ,.1.
Table of Laplace transform s f(s)
p(t)
J(s)
IF (t) I
1
-
~,'
_ ,j
_~.
1.
..
, '1
-
.
~:
~~
...
.:,;~
.
'
[
"jt":'
S2
. .... ,
,~
-.
. .-
r.
-. I ... ..... ... ...
,.'
:-
.\(
..
.J.
:.;:.~.
cosh(ar)
r2 :-- aL .
........ '.'V'
/I .. . • ... j
- ...'.
"
,' .
',
~
··S···· .. ·
'. ';.).
'.;, ~-
.
a2
S2"+
t~-I
.. ..... , ...
..
~
(s
:
:
..
" '
..
'.
.'
'=o~.(at)
,.~
..
~
e-.Qr·sin(bi)
a)2,+'b?
+
.','
.'
b
(n· -:- 1) !
s"
• • • -1.'
= 1.2.3.
n
.........
: ._~:,~'4.'~' •• "'
, , t .. ~
1
-
·5 ,-
1
s
I
..
-
r fo
1
..
--
is
...
,
1
--
e
s -a
+a
5
~
lll
' (s
,.. af + b
.
\
'
e-·J( cos(bt)
2 \
r
1
i~(l-
I
teDr
(s - a)l
Q"2)
5(S2
-
CQS
Q-
al) ."
r
t""!"'e DI
'(s - a)n
n
Sl(S~
(n - 1) !
1 , a 2)
1 a
-(01 J
-
sIn ar)
1, 2,
-
I
1 (5
-...!.
1
-
'.
s
s'
+a
2•
52 -
.
+ (c-a)e hr
(ae
lJI
-
- .. ..
-.
.
-
+
2a
s
(.s2
+
-
a~)2
t
20.
J
- be b ,)
1 sin(Q/) a
..
1 ' -3(sio at - at cos at)
Q2)2
sin(ar)
(a - b)(b - c)(c - a)
(a - b)
1 0 2
(S1
(a - b)e"']
1
(s --: a)(s -b)
.
[(b - c)e nt
(5 - a)(s - b)(s - c)
1
e1>l)
_J_(e!1( _
a-b
a)(s - b)
I
1
(5 + a)(s2 +
I
1 b~)
2
a +
+
·/a 2 +
I
-1 sinh(ar) a,·- ..
where
.~
.
".
b~ [e-
b
e
D1
b2
sin( hI
l ~)b' I
c
.tan-
-
1
8) ]
D,ESIGN OF
THE.R.M.AL SYSTEMS
l ' {P'(t)}
so
. sf(s) - F (0)
sciwly
F'(f)) - f'CO)
)} •
I
~"
j
.
'.
the
.....,L.I...JIII...J.L.LL.. \oJ
'.' '. ' " s2f(sJ) ~,'
'.'
(0): ~ p·'{O).
:~.
.
, •
:.
.~:
; 'f'
~.. :
p.. .' ~~;
'
•
:
~
b
0'
.~ dynami~proce~s' or
The
by determining
(t)
l'-l{j(S)}' =
(15.11)
.
.
transform appears on a table, such as 1 1, the inverse not found on table, objective [0 decompose complicated into ones t}:1at can found the. . key to the denominator of the' into factors, A 'B ------------ - ---+ ---(s
+
a)(s
After A an d B are available from tables . Example
+
b)
+
S
the inverses
s + b
Q
the ·simple tenns are surely
+ 2).
. Invert s/(s2 -
Solutl'on
A s-2
--+
B s-1
find A and B, 'equale the numerators, s =
A(s - 1)
+ Bes -
2)
two
" 0 =
constants:
and, s: Then A
1 = A
=
---.....--
and B
=-
+B
1.
- ~'
DYNAMIC BEHAVIOR OF THEP....'VIP.~ SYSTEMS
A r'J.
·1"
• -..
t..
1
-
l
377 1
1
sp ec2J SltU9.tlOD. .~revm. s \Vnen t!Jere are repeatea ,0 ts_ "J! '.~7. at ·:,ase tIlE:
s-r dep.ominators r.Bust be proYlded in 'dest~ndjng powers from the !?o\ver of l.c repeated :".'oot.
'
•
•
,
•
•
... .,,;; ... ~
.. , ....: .>,' ....
•
.'..
.
'
",
:':",
.: . Sc(ution~ ,T he attempt to find ·A. and B in the partial fracu:Jon i"epresentarion . bel,ow is doqrned: ' .
+
(s ,~st~nd,
~~
-;- 10
A
,B
'=---i-'. 1) (s - 2) 2 5 + 1 ' ( s -- 2) 2
use this .expfession: ,· :
s + 10 '. A B B' . . .,-= , + ") + - - . (s ' + l)(s -2).s + 1 (5 :-2)-' . s-:-2 .
Th·i.s · supplies t~e,e "simu,"~eou,s equations , . 4A + B -.:.. 2B'
10 =
constants:
1"'= -4A
s:
0=
A
from which A = I, B = 4, and B' p-l [
-
(s
+B '+
= -1.
S + 10 ] = + l)(s - 2)1
B" B'
-
Thus, -I
e
+
4
"21 _ e"11
Ie
.
Another technique for finding the A., B , and. C C011St-aJlts in the partial fraction decomposirion of ~. ~
. .'
.~ ~
N(s)
A B +--' + s-a s-b
D(s) is~ for lion-repeated roots.
B = N(s)(s -
b)j
lX.s) If b,' for example, is a repeated root
.....~. NCs)
--"~~"'-:'A ~
D(s)
s - a
Band
-.
J
J~b
l
+
B
.+
(~ - b)2
N(s){s - b)2~
D(s)
Is-b
':"'d{ N(s)'(s - b) 2] ds D(s) s-b
B=-
B" . S -
b
, (15.13a) (15.13b)
378
DES1GN OF THER..\1AL SYSTEMS
'EXample 15.5. Rework. Example 15.111sing Eqs"
(f5~'3a) and (lS.13b).
I
.Solution
.)
• • J. ~
..
~?
~
••
•
':"i~~'~i~>;j:::~,:·.:·:,",;:,/~.j,'~ '~~~}' .~.:.~:~i_·. '.',r:s (,Sst.~O)I~~~2:,.:~:"i;'~:,.:,:,,' ,....~ ;~, : ;',';, t'::~,,~" •.. ;
:/. ' ·:"~:'::":~"15:"6"~': s~tiriI6~ ' .' .·'EQUATI9NS The standard steps k
;'oi:.···oRi)mARv DIFFOONTiAL" solving .a· 'differential.'equ:atiQ.n using Laplace trans-_···
; forms are the followIng: (1) invert the' differential equation, (2) sol~e for'f (s) ~ and '(3) 'invert f(s). TJ:le bou:adary. condition(sY may. be substituted, . . 'into the transformed equation .fol~owing step. 1 ,if,'the' specific .vaiues 'of 'F(D), FI(O) , etc~, arelmoWD. Otherwise cons,tants may b~ wserted for F(O) ~ , -. F'(O), etc., after step 1 and the boundary condltion(s),substituted into the in~erted equatio.n following step 3. . . ' , Example 15.6. Solve fhe differential eguation
+ k2Y(r) ,= ,0.
Y"(t)
subje~t to the boundary conditions: Y( 0) = A,
and
== 'R.
Y' (0)
Solution. Transform the differen"riaJ equation, s:! r ( s) - s y (0) - Y' (0) + k 2y (s) = 0 ,,
Insert the
~oundary
conditions and solve for y (s) t As yes) =
+'k".2 +
S2
B
s2
+
k'2.
Finally ~ invert y (s,) Y(!) = A cos(kr)
+
(Blk) sin(kr)
Example 15.7. Solve the differential equ'atioM 2
.:; .:. ,
.,:-...... " ......- -
,
"
d " . + )'-= x -" dx~
. with the boundary conditionS: dy I dx = J .·\\ihen x' =
,Solution. Transform the
-.
e~u~~i.on,
-O~
and)'
= -0 when x
=
.
s'y (s) - s Y(O) - y/(O) + Y (or) = l/Sl,
7f
DYNM"lIC BEHAVIOR OF TiLE...l"{j\.1AL S YSTEMS
379
f'(O) = L but :J'(O) is not yet known, so assign it the symbol C . Solving [ory(s) )
~s
1
1
=. s-(s.- + I) + s~ + -.-)- +. 1 . s- -+ l .. ' . 0': f _,~ _/: . . . '.'.!f(~:~..'.
.'\." (s)
"I
-JpC'l inversion
').
-'J
-
'!
: " . ,-
.":,,;"" . : - :: :1·Y~"""~i:··.-=;·sin_"i·'+··C~6s·~i-'::~·~;i~·x~~-~·'~~"+"C:~~~·_;~: .o.~.:.""'_<'" '~.'., ~-
~
.
. . ~e ..final.~ou~a~·~~ondi~ion ·may .0 =
r:t~~"b~' substitEi:~d~ JT + C(-l)
.so
y
Then
=x +
1TCOSX
15.7 BLOCKS, BLOC~'DIACRAMs, ANll . TRJ~NSFER" FUNC·TIONS The symbolic representation of system and control ch~cteristics'is typically .'. ~ccomplished through the ~se of block diagrams. These block _~iag.f~_lms-· are infon:n,arjon-fiow diagr~ms such as those 'used for system simulation in Chapters 6 and ·14, except' that in cpntroI diagrams th.e .variables are usually expressed in the s domain, rather than in the time d.omain. The diagrallls are composed of bLocks 'which have precisely one input .and one ·output. and surn.m.lng points, as shown ill Fig. 15-2. Blocks may represent eith.er processes that are instantaneolls or time dependent. The summing points are. algebraic additions of two more inputs to yield an output. . The Utransfer function" of an element is the ratio of the output of . the block to the input, both expressed in the s domain, as shown in Fi,g.
or
lS-3a. transfer fun'ction = T F
=
l' { 0 (-t)} O(s) = .£(J(t)} I (s).
when initial values of input and
outp~t
are zero
A property .of transfer functions is that they can be cascaded, as illustrated in Fig. lS-3b. Thus,- if G(s) is the transfer function of the frrsr block and H(s) of the second,. O(s) = J(s)G(s)H(s), ."' ...... "'l.., .:; ..:.~~n:.-~ .",
+
lnsrantaneow -~
'.
~
",. -[... ~
.
FIGURE 15.. 2 Symbol~ ~--"""'I-n-b"'roclr:: diagrJ.ITls.
~.
~
.,.--'
-
380
I(s)
DES1GN OF TIIERMAL SYSTEMS
~I
~-
GIS)
G X
I(s)
-- ~-
__ G (~~
.
- - , -'"
- -
H (S)
I
.
'
.
,
IF = ,
~ , .-
.. .-. '- . ..
,
: O(S)
,~(5)
= G(S)
.
O{S) '-J' . (S)
,
"
)
-
,
= G (s)H (.S) (b)
. -_ (a)
. . :,. - '-" ~
, , -, - ELn
CCS)'
I-----I!""'i ".
Des)
~~
' ,~
; '
'.
,
.
~
·~. I ' j~1"" I ~ :~
.
.. -- +'
G .( s)
I, '
(b) ,
(a) FIGURE 15-4 (a). Unity feedback loop' (b) nonu~ity feedb~ck loop. '
,.(
,
15.:8 ~EDBACK ,C ONTROL LOOPS A' feed~ack control' loop is one in which th~ controlled' variable is sensed, , and this ' information is -returned -to ', re~djust the controI]er to ,reduce, the en:-or betwe~n the, sensed 'and the set values. ~yv'o basic forms of feedback loops. are shown in' 'Fig. 15-4. In the unity feedback loop of Fig. 15-4a the magnitude of the controJled variable C(s) is compared directly to the , setting~ R(s). The difference is th~ error, E (s) which regulates the actuator to which the prot;ess, responds .. The n6nuruty feedback loop of Fig. 15-4b typically results \vhen the response of the sens'or is time dependent. Of special interest to a designer is the transfer function 'of the entire control loop, C,(s)/R(s) , -which is useful in stability analysis, of the loop and in predicting the response in the time domain of disturbances to the system. w
=
unity feedback
TF
_G_(_s_)_ 1 + G(s)
non-unity feedback.
TF - - - - - - 1 + G(s)H (5)
G(s)
(15.14) , (15.15)
1$.,9 TIME-CONSTANT BLOCKS· O~e of the frequently occurring pr~cesses in physical (includjng th,ermaJ) . systems is .il response \vhere -the rate of change of a variable is ,propoI1ic~)DaJ to the -difference between the magnitude of a driving force and. the va]ue of [he variable. Such a process can be characterized by a "time constant.·'
;--.--
DYl'it\.tvdC BEHAVIOR Of1l-lER1\iAL SYSTEMS
381
} ,D e~' arDple "-viII ill ustrote a proce:'is of thjs cJ~.ss'J the deve",o_ w.en.t Df the transfer fu ncriqn for the proce,ss1 ~.nd the in~.ersion into the time 'domain .
The response of the tem,pera~ure T of a ~eD.sing bul b ',.'lill be sought as the fluid ternperature' l j in contact ,with ~he,bDlb. )ch~.ges, as illustrated , in Fl,gs _• .~5-5a ~d6 15-~~,., ~T~~: sens~,?p,,;~~,b. h~ ,~. m~s ~l~ ~pe~.~~ :..ea~ '~,J.: .co.u;vectlon c::; cific:.ellt 11.. 'and an area A ror· heat transfer, befwee:·~.. the fl uId . :.:..-;. 'and th~ b'u!b.·' Tbe-standi'rd-techniquc;'i'of a~velopiilg1lie-'n-a:nstei"furicfion: is"" : ~I,
:
I
:',.
as ,f611ows,: .
. "."
'., . "
"
.me
=-~. Write lhe clif(eten~aI 'egu~tlqn for process 20 T~s~orIn the equa'tion , .and 3. Solve for the transfer function, .£ {O}/.J:' {I}.
"
. , '. "" . , ;, '. '.
y
The f1rs~ attempt at these three steps·will give an awl;cward transfer functi'ori, .. .so after seeing tliat result, a revised approach will' be taken. An. energy . bal~ce pro'vides the differ~ntial equation': . .
dT·
.nzcdt
=
(Tf - T)hA .
'.·whlch \vhen transformed becomes
(;;J
[s1' {T)
..
'
,
,-
~ TeO)] = l ' {T!} - 1 {T}
Finally ~ divide by ..£ {Tf} a.nd solve for the TF,
hA)]
1 + T(O)[ (mel .£{ Tj} TF =.---------------(Jncl hA)s .+ 1
(15.16)
Equation (15.16) is noe in a useful form because of the second term in the numerator, except for the special case where T(O) = 0.. next ,: pass at expressing the transfer function will revise the differential equation,
The
Mass. Fluid
m'
. S~cific hear, c .Convection coefficient, h Area, A T{I) (b)
(0)
FIGURE 15-5 ResMTlSe of a ~mpcrature-sensing bulb to func t i on tf\ is ti me-cons! ant block. ... (a)
or
8
change in fluid .temperature (b) trnJlsfer
382
DESIGN OF THERMAL·SYSTEMS
'normali~'g it with tespe~t to the yalue of T(O), which~will To.' )
eo
.' ./""r:":::'"':: " ' '-:'. '. ~ .'
me
d(T - To), '. ,dt.. '
:"~'"
.
< ....
.. .
I
•
-,
.
'.'
: . . :<~{~.~>~ w1),ic~, after.. tran~fo~atioD ~d, ~y:i~;o~ by'''£{ Tf .:{:~;:~ ::'7~f~~t:;· ~"',,:T ",. ~,C ," .~ '.~ ~j: To;;,:' i'~,.' 1."
. : ,'" " .IT -,-
des'ignat-
(T -:- .To)JhA
:=. [(Tf - To) .', •• '
.
be
',
. " '• • ,.
"
' ;, '
• • ,.
,
r,o,} yields tb~ '~~,sfer'
.','.:' :;.: :_'~ .'.' .,. '.,. '.', ,','
. , .'.-" "', ,. 'h {y'.- ~ T."}~""-,., .'(!Z!f.)'s.'_:+:-~1-'~' :>.:~~,t"."~.~ .:: _:::::.(1,5) 7J.:,~~~-· f, o· hA. -.., , '. .. . . .Th~ 'definitio~ of th~ tr~sfer ~'cti:o~~ j~ ~ay .be ~~~~(f, 'had fu~·stipulati'~~.:'" . " -
..
~
.L. .
.~
, that:: all' iJ;lltial conditions were' zero. ,ThiS)5 the frrs~ appearan.ce' of· the. ':" normalizing process which can be accomplisheq.by arbitrarily settlng rcO) ". to zero in Eq. C15.16) if tbe transfer fun~tion is .n·ever ~o be inverted. 1f the ' ,.response in th~ ,ttme dam'ain ~i~l ultim~teiy be deter:mined, the redefin'ition . .' of variables (T 'to T ~ To, for ex~ple) is necessary ~ , ,'. . The power' of .the transfer ~nction, Eq. (15.17), is ·th~t the. response of to a 'variety changes in Tf can be' determined. If .Tf experiences. , a step increase ·of.!l from To ~. illustrated in Fig. 15-6, the transform of. '(T - To) is, from Eq. (15.17).
r'
of
1
~
{T - To}- -
fi {
-
Tf - T. }
-
0
-1 (~;) s
+
1
-.
ml'
A·
,,:s [( M ) S
.
+
1]
Inversion yields the equation for T - To which is an exponentiaJ expression
T - To = !l[ 1-
(15: J 8)
e-.II(mclhA) ]
The group m c/ hA has the units' of time an~ is the- IItim~ constant" T for the process) from .which th~ name utime-CoDstant block" derives. In response' to a s'tep change in the inp'ut, 7 lias the geometric .significance shown in .
Fig. 15-6. .
,
.
.
The responses of a time-cons.tant block to several other fonus of inputs' 'are explored in Problems "15.9 and 15.10. .
1.
•
/
/
I
/
I I
// / -I
/
/
/
/.
o
t
Time
FIGURE 15-6 Step increASe in fluid ~mper sture 1J and response of the bu! b tempe~ruri!,
-'''"-,_,~ _ _ ....c~__" ,
AlI pressure, P
S !
Volu. m~
.
A.rea
,/
·LIP}.
1""
. ts+ 1 " . '. . .I " , ' "~ '. ,,' . '. '. :"':'" .', ~;...;: ..........i:,.: ... ~h; ·.r. .•..... .. '-': "'~ '.. ~.•:'.. ' '~"':". ~~ .:" '." (p).., .. :.". ,:. :;.0 ...
..
"
} . • ",
. .
K ~~--;~'I' L I ~-l
,
.
'.
~
.
''-' . ." .... ,'..... : ',,'
....:
'
.
. .
, , '
....
".
. ~
.
" :Force, F '. : ".
'(a)
FIGU".RZ 15-7 .. Time co~~t block wirh gi~ k.
Th, the heat-transfer process to the tempe!"a~e-se~ing bulb in Fig. , ,15-5! ,the rp.ultiplying factor in the transfer function'.. is UQ.jty. In many . . proce~ses represented by a time ~onstant, such as the. one illllstr~ted in Fig. i5-7 a mUltiplying factor k, called the gain, appeaps. Fig. 15-7 k has ~' . ' the uni ts of
In
J
.
force, N .
pres~~.~
.'
Fa
=
") =
N/m-
.
t -
2
area,-m·
CASCADED TTh1E .. CONSTANT BLO.:CKS
15.10
A first approximation of'actual contiolloops can often be made by repl~e senting the elements by tim~-constant ·blocks. To gain a bit of insigh~ in~o cascading time-constant bJocks within a control loop, ~e behavior o(two cascaded blocks wil1' be exantined. Suppose that the temperature-sensing bulb of Fig. 15-5 contains a liquid· whose 'saturation pressure is transmitted to an actuator which regulates the position of a v~lve. As, shown 1n Fig. 15-80, Ta' acts on the bulb temperature Tb which in turn affects' TL. Whjle the representation of ilie pIoce'ss in Fig. 15-8a by the block diagram. of two cascaded time constants may seem reason~ble, there is an impO"rtant implicit assumption. This assumption becomes apparent by wriring the heat balance aOOllt the bulb:
,
(Tn - Tb)h 1A 1
=
I
dTb dJ me
+ (Tb
)neglect
- T0h2A2
where subscript 1 refers to h and 'A be"tween the air and bulb,' and subscript 2 to h aQd A between the bulb and liquid. Notice that in order for the be represent~ by the time heat-tnmsfer process from the air to the bulb -
-
-to
0
,
.
. .. of....'
384 '
OF THERMAL SYSTElvfS
DESIGN
~----t
1-------.,...--- '
I
Bulb lemperawre.'7:" b
.Air -l>-
.
T~
.' 1,'
, : . : - ',
•
;;;;'>0:': ", >:~:' ~f"~' -;:? ,". ' . "'. ,.':'~.'.~.:!.
'-:-iquid .1erhpernture.TL
1,
J
-' :.>" :',~
0, ,. ;::,', '
,;,
", ':,.:- ':'.' ."":}' .--~~~. '..... .:. '.
. ',-
..... .
I
,';,~
•
I
," " ; ; : , ( Q) :"
- .
'; '
~-,;;" i, ",'" ';t 1~" :.'::; ,; :,:> ,"
.-
,,,',' ".,; ',:' ".<:'
'
r--'' ----,
'..£1 Th-:--Tul
'· .!'IT~-.TaJ .::::. .:. - " 1
,J
.. .. : :£ITL-Tl»)' , .;, ., '
'~uib
Air to bulb ·
.... .
to l,Iquid '
(h) .',
":<-'. ~
. ',
-
'. FIGURE 15-8 . '(a)
Response
of
I~quid teJ!iperatu~
TL to a
cbange,'~
CO.fl~tant blocks ,to represent the dynamic pfoce~s~
air'te.rnpe'r anire Til (b) cascaded tim·e- · , ... . . .
'constant, ill~/h'lA :l' 1 the'rate ofh~a(transferred to the liquid from th~ bulb 'must be negligible. When those relative. rates of ,heat transfer do prevail, the product of the individual transfer functions'js the transfer function of the cascaded process . .Suppose that. Ta experiences a step increase of magnitude j, fro~ To- .,Vhat is the response of th~ liqujd temper.arure TL? (15.19) The inversion of Eq. (15.19) is ( 15..20)
The response of Tb and TL to a step change 11 of Ta is shown in . Fig. J 5-9. Several conclusions can be reached by appropriate proces~ing of Eq. (15.20): ;
~
:::I
"''§
g, £:
'~
T
"
.
A
i;
10-
FIGURE 15·9 Response of single and Tfi
-- .
0
Time
'. :
•
...
., ._...;......-~-_
;", '
I
"
CBS,
caded Ji"me-consr'am" blocks a step chan'ge in the input.
10
DYNAMIC BEH)\VIOR OF rr-1ER; ...l P...L SYSTS.....iS
1. TL - To 2 . d ( TL . 3,. ,P:1.S one
=:
385
0 at t = O.
:'"'0) /d t = 0
at t . J. I 'f becomes very short relnt,ive' to the other"
TL -: To
7'2 ~ T J
for exa:tL~.Jle~
'.1.( 1. -. ~ -:-(17"1)
\~{b'~~t !l. . ,T2, Eq. (.~?/ ~ 9) ,. P1!lst ~e,r~il~"e'rte0 '(se~ pr,ob~' J5D'1.3) ~~ ~bi~n ,"' " .~:: ~, 'A'~ Tr; .. .....• ,:.: ["; '.c-.. i~e~I/' ' , . , ' : •. , ' ...' ( 15,2 1) ." :
, , , 'I.t
,"...
".•.>, .':, '. ... .
-
.,
'15,,11' STABll.J:T ( J~AL "SIS OF, A 'CONT ,O L L.OOP USIN G FREQUEI~CY. ,RESPONSE AND THE PRINCIPLES OF ,THE BODE
D~GRArvI
.
Two [ec~iq,ues will be presented for analyzing a control loop' to predit;:t , , whether or ~ot the contI-ol loop' will be' stab l~. nex.~ section' the 'narure of the transfer function will be analyzed to indicate ,stability ~ In this section the Bode, diagram" which.uses the frequerycy response concept, will be' the , ,,,tool' to pre.diet ,stability. \Vhile, \ve speak of the. Bode di~grarri, \ve will nonn·ally calculate and not draw a graph to make the analysis. , , ' The Bode diagram is a practicul method of determin'ing stability, but the concept iHls another advantage. The 'logic leading up to the criteria for ,stability when shown on the Bode diagram offers an ,excellent technique for ,explaining the mechanics of instability to a person not well-versed in ,
'm the
Laplace' transforms.'~ Frequency response, which -is the idea on which the Bode
___D.___ sin[21rft - tan- t (27TjT))
-J 1 + (2nj T) 2 so the amp1,ifieation ratio, output
to'
input, is 1 '
Jd
+(27rjT)2
and the phase lag is tan -1 ( 21Tj T). The amp 1i fi cati on ratio decrea ses and the , ,phase la.g increases as the frequency of the input sine wave increases. The principle that Jeads to the Bode criterion for stability is based on transmission of sine wnves throughout the 1~.9P· !,he sum of the phase lags
around the lo·op and' die 'prochict of the' amplification ratios are computed. Consi~ .dle eJtampJe of the ..ai r heater in Fjg. 15-1 J in which an electric
386
-,
•
•
1
r·
~ .~j
,. .-::
, ,. • ",
'.'
•
- .. .
",
':J'.. ~.
DESIGN OF THERMAL SYSTEMS
_, I
~. '
:. :
" .
:. "
.
;' "
,
"~;;":'
":
.. .
~~~~"'~:~
, ,-
. ,,'
, '.
...... ' .
" .
.",
~. :.~~:;
'".
J'-
':"
~ '0
.
:'
' • ••• • }
...! '
.
~
•
. :
,
-"i', _ ..
. ~ , '. 2 ', . .':' ",
.
J
•
... .. ;':~~.;:..'.>,,'? ': .... ~.
::I .
.'
~
"
.l .
•
•• ,
. J,
..
1:
,
..
"
-, •
"
, '
6>"
.. _', , : .
.
I'·
•
.
-
I
j
••
c". ' ..
:',
',-, FIGURE:', l5--JO'" ,"
"
, , Frequenqy. reSplmse in which a, :' sinusoidal input provides an Ollt,- put lagging the input and afso , exhibit,irig a different ~pljtude , . than ~e input. .
Time
. he~ter is controlled by 'a loop that senses the outlet air ~einperature Ta and' converts this sensed temperature'Tb to a ' {:ontrol voltage ·Ve·• The elec~c ,'power provided. to.the air stream is proportion~ to .the differeDc~ _V set - Ve. The block diagram 'is shown .in .F~g_ l~-llb, .a nd it i~ assumed that ,both ¢~ , tempera,ture s~ns~r and 'the heater possess themal capacity that introduce, phase lags at those t~JO elements. The visuaJ lzat'ion suggested, i?Y the principle OJ the Bode diagram is, that for SOD1C reason Ta experien~es a disturbance that .js the top half of a sine. 'Nav.e , as shown ,in Fig. 15-12. The sensed temperature Tb lags the variation in Ta by an, 'angle .¢l and also varies with a smaller ampJitude. The variation jn Tb translates to a half sine wave of Vset - V CI but reversed in sign, as shown by the 'd ashed line. This reversal is needeg because as Tb increases the power input should decrease. This reversal is equivalent to a phase lag of 18Ct. Because of the thermal capacity in the heater a phase lag of 4>2 exists, provid.ing a variation of T.a, that is' a continuation of the half
Controller
\ 'SCI
and
+
healer'
Transducer
(b) F1GURE J5-11 ' Air'healer (b) control block diaBram.
(0)
-.
~.,,:.
~,
". .:: ;: '.:::'~"~/'~'/ <~<, ".:"::,~,.•. ,~; .••. '::~ .
. 5:.
\ ''''''r
•
:<.
".
J
337
DYNAMIC BEHAVIOR OF THERJ..iAL SYSITMS w
,
.~
I'
"J• .'.'
{ ' .....
~
.;·1......
.j
&
' ...
.,\.-:.•
.
'
-"T,-. ... :. ..... ~
' - - - - - - - - - - - - - - - - - - - . , - - - - - - - - - 0..'-
. ·F~GUi:E 1.5-12 Perpetuation of-:sinusoidal disturbances· throughout the control loop. of :th'e air heater in Fig.~ ,IS-Ii.' . 0
.- ·"Yo
.
'
sine wave that started the oscll1ation.· From the position marked'x a bottom-· ~ half sine w~ve hegi.Ils ano"ther sequerice, and 'the'oscillations ~ohtinue. . , The Bode diagram prQposes th~t a ~rucial frequency is that. whi.ch causes the sum of the phaSe lags. to be 1-80° •. as ilh.~stra[ed in Fig. 15-13a. In combinati6n wi th the buil t-1D 180 reyersaI, a 360 0 displacement i~ .provided· which continues the oscillation.. and begins to appear unstable' . .Another element is invol ved, ·ho~ever.J "because if the oscillation' continues but at progressively lower amplitude ratios, the loop would b~ stable. The amplitude ratios of the elements in the loop which are additive on the'logarithmic scale of Fig. 15-13b. If this product js greater than 1.0. then the new amplitUde 'is' greater tha~ the 'pa$t one. The above visllaljzation leads . to the 'following .c.rite~~D for s~abjlity from the Bode diagram: .
0
If at the frequency where the sum of {he phase lags in the loop is ] 80° the product of the ampJirude ratios is greater than 1.0, then the loop is unstable.
The diagram, as in Fig. 15.-13) shows graphical behavior of the trends, and indeed it would be possible to solve problems by adding phase lags and the logarithms of amplitude ratios graphically, but it is usually easier to . perform the operations numerically. A few comments abou'[ the concepts 'on which the Bode djagrarn is based' provide more understanding. about the physical nature of instabilHy. The fIrst suspicion might be tbat it would be highly unlikely to get the , sine wave disturbance of the precise frequency that would.perpetuate itseJf. , When considering the mathematica1 represeotatibn of arbitrsry functions by a Fourier analysis, however, it ,can be realized that harmonics of many fr'equencies are embedded in disrurbances. The loop can select a sine wave having a frequency close to the critical one; the loop is a form of frequency filter ~ identifies i[s critic~J frequency. When instability doe$ occur [he o~cilla[ions will take place at the critical frequency.
388
DESlGN OF TIlERMAL SYSTEMS
•
.
.
... . ~.
,.
:~.
',.';'
-
.. ... :..,:.
o '-'--=--" O.QI
,.;
_~-,-Lc---~-'----,----...J
0.1
I·
10
.100
. ,Frtque ncy, lis
,.
(0) Ratio
J
.
~ombined
10.i-=--.,--:"''''''',....,.-''I.J--- - t - - Ratio ,2
C.I ' - - - -' -0.1 0.0 1
---'
10
100
Frequency, lIs (b )
FIGURE )5·13 Sum of the phase Jags in [he Bode diagram (b ) product of the amplifica(0)
tion ralios.
The concepts· of the Bode diagram offer a practical method of predict· ing loop stability without the use of Laplace rransforms, and furthermore ·suggesl a means of visualizing how instability occurs and what might be done 10 correct or preven! instability .
15.12 STABILITY ANALYSIS USING THE LOOP ~NSFER EVNCTlON One of the strengths of. the Laplace transform representation of a co ntrol loop is that stability analyses can be performed without the need of inverting backto the lime domain. The reason for Ihis ease of analysis is thaI the loop ironsfer function. for example.
TF
-.
_ loop -
.
cs 1
as + b + ds 2 + es
+f
. ..
:~
D';'~'lA.J.'v!IC BEHAVIOR OF THERMAL SYSTEMS
F (t ) il
3::9
F (t)
F(E)
F (t) ;::: e -,01
F (t) == ~+Ol
r< 0
r>.o
, (a)
(b)
(c)
,FlLGmu: 15- ~4 " ~ve.rses F(t)
,\
are dependeh.[ upon' ' the 'slgn of.rhe ro()~ i~ (const)/(s -
r). t~.
"
can 'be decoinpo~~d ,int~ components'"
----'-A....,....-- + :. -'B.~
- r
I
C +,,,--
S --:- r2
:, S -
T3 .
and
11"l:e valu~s of r 11 '2, T3 are the roots of what-is called the ch'aratteristic 3 2 . equation) cs + d s + es + f.' The natt)re 0 f the roots indicates stab iljty 'or ~stabiliry, because the ll1verses of the terms in Eq .. (15.22) are exponents as showU 'in Fig. 1~-14~' If a root is less than zero (he contribution of that' . term decays indicating a stable response. If, on th~ other hand, the'roo{'is positive as shown in Fig. f5-I4c t1le magnitude grows with time' so the response is unstable. When ~eroot is zero"as in Fig. 15-14b, the respon'se' is a cOI?:.stant value, so the loop response is neutral. . Figure 15-14 implies ·that the roots are' re~ numbers, bur roots that are imaginary numbers appear frequently .as well. If the root is, a + ib. for. example, . the inverse is ear e~bl. Since eibr = cos( ht) + i sine ht) 'this coritribution is oscillatory, neither growing nor decaying with time. 'The foregoing observati"oDs lead to the Nyquist criterion, ~ which states that a control loop is. unstable jf the real part of any of the roots of the characteristic equation IS positive as shown in Fig. 15-15. t
,
'
I
I
•
t.......----ti20-e-
Roots indicate ....~~-.-,.......;.. stabililY
-.
I
• Roou indicate instabi Ji ty
FIGURE 1~ .. 15 Location of roots of the loop tranSfer function in di cate stab j1icy or instability .
390
DESIGN OF THERMAL SYSTEMS .
A convenien't technique for determining the numbet of positiye roots without actually calculating them "is -the Routh-Hurwitz criterion. 7 If the ch?Iacteristic equation has the form '.':. .
,C..,
.
. . ' : ' .'
"
. ans
n·
+ an-IS n -1 +
an -2 S
n
-2
+ ' ... +
..
als
+ aO .= 0 .'
(l5.23) ·
·:-:~~·~-::-.· :<·~~·~-~-j}1 .~ ~ c~efficie.~~ ..are -'ammged in the form ~hOWD l.~· Fig. ' is-J6>
.~ '::.~ ;.i'~' :~~·, .'_~;,.'ihe :ROU~1-HunYi~ c~qq.J;l._.·.:state~ · tha~. the .nurnber of.·~oC?~ .·With 'positiv~ - ~'
;'.;:--~? :, :~~re'4~-~~fi '~£i ·eqJ~.tP. :.rp·.~·D:u~be~- :~f ~ ~P..~g~~~i!~~\.~~~~~ -i.~ft- ·coh.i.~· o("~ ' .' .,".' ~ :' ':'-'.th,e.,arr~y. . . . '
-.
_. . ...... . .,. . .:-' :' . .. '..!,: :
~ ~
-
...
~.L .-:-\;.;r:-':\-'-'~~~".'.::'./ :t~~?
'
Example 15.8 .. How' many 'roots with positive: real paru ;ue there ' 'of' the .
_ cb~a~~r{stic 'equation . '
. .
. Solution, Using Routh~Hurwitz~ ' Q.i, .and ao
..- .
= ].
Q3 '
'
.. ' .. ,
=. 7. a~' = -10, a J ==
= -:-80. -1
bJ
1 - 10
=7
7 - ' 116 = 6.57
, ' -1 1 ~ 80 b n - 3 = hi::::: 7 , . 0 = -8l3
'7
.
:-17 -116 '. 30 8 6.5) 6.57 - 80 -.- .'
c = -----". 3 .. ·
,
so [he array is 1
-10
7'
-116 -80
6.57 -30.8
-80
The one sign change down the left column indicates one root with a positive real part. (The roots are 4; -5, -3 ±·2i).
,--,
I
1011
J 0,,_1 :
Array
o I~:"~ 0
/1 _)
(JII~ .
(J n-~
I' I h /._ t I h /l _}
r I I en.' I
bn _,
C JI-.\
('n .. ,
Where
b 1 1_- 1- -I-
L __
J
--
- .
-I }"
/0"
.' ,
O"_~I
"lJ-IO,I_1 O".J
.. ' .
'- .~
1. _ "n ,- .... I
-.
,
lJ II.,
I
:
\
tJ 11 . 0
all_I
H-41
lJ n-!>
FlGURE 15·16 The RO\JLh·Hur'-"iu array
derived. from 8 characttristic ~uation of the [oim of Ecj, (15,32).
.
DYN.~'vIlC BEHAVIOR OF THERMAL $YSTEMS
S j~me
techniaues are .,
8_'.'9 i lable
391
for deteIDl.i.ning the. ..SC"2cific v2.Iue of ,d"J.e ), t '
roots sho uld that be required. H ere 'we 'will dilly discuss'second and third ~eg,~ee
polynomials. Tne rOOfS of tIie characteristic equation that is a second , degree poly[iomi~l can readily be'fq und .by use. of the quadratic forn-.. :ula. ~ .If the. charac~'e~stic equation is.:a 'cubif one~' .the f
. . cl::li:..ser--,"."
";"<:~'. }'~.: .::s,p~.'" I
:-
'::'_
..•. ,':'''':~''_''~'_''.' ':'-":
.'
',~
.. - .~..,,,,,,,,,,,""_~. .
r.JJ-,•
•
.
~.,.
".
.,
•
~~
..
••
I
•
."
.
1. o~e real root and, ·~o roots ·~ri.a:t· .ar~·a conjugate 'pair :'.,. ' . .", 2. .three :re.al root~ ,tWo' of \vhich 'are equ~l'" . 3~' ·thre~ ~e~l r~ot~ ~ aIr different I'
'to
Subjecting the characteristic eq~ation the Newton-Raphson tech-' . nique, perhaps w.irh some different trial values, sho.uld le~d to one teal 'root. There~fter ,'the q uadr.Ll:tic formllia can be ~sed to find £he ocher rdots~ . The Routh.-HurWitz criterion ca.n detern1ine how many roots have positive,: real parts; which may assist in choosing? trial value for the applicati<;>'n of Newton-Rnphs~n. .
-
15~13 'NORMALIZING THE VARIABLES. FOR INVERSION TO THE TliV1E DOMAIN
The next several sections address the ch.allenge of trans[a~ing the characteristics .of the physicaJ equipment into the symb~Is of t0e control blqck· d.jagram ()JC?ng with the specification of th~ ·transfer functions: WheIl 'the loop transfer function must be inverted to determine the response of one or more
variables in the loop in the time domain, a normalization of me vmables is usuaIJy necessary. Considec for example~ the controller in Fig. 15-17a that regulates the air pressure in a reservoir .by admitting a stream of air from a high-pressure source. The spring attached to the piston is under tension, ~d the displacement of the valve stem is proportional to the pressure difference:
x, mm = K(Psct - P, kPa) The flow rate admined valve stem:
[0
the reservoir is proportional to the position of the
Win'
kgls
=
Lx
The block- diagram of the loop with ·PSl:( as the reference value and the reservoir pressLlre p as the controt1ed variable is shown in Fig. 15-17b. If the ~ly analysis planned is that of determining stability based on tl,le loop transfer function. some short cuts can be taken. If Ws is constant, thnt summing point can be ignored, and the loop transfer function TF is written directly as
-.
TF _ "V
K LG{s) 1 + K LG(s)
,
.
392
DESIGN OF TI-IERMAL SYSTEMS
Pressure
~_p_~_r____~~,_.
~
.;:- ,
]!.
~ass,M kg
.' ~
.
, p
4, -
, (b) .
FIGURE 15-17 '
,
0"
"
' ••
~(a)- ,'Th~ hard~are for regula~ng the au: P~$ure in :a re~erV~ir (b) the control block diagram. . . . . ~
'
.
' .
" . If, 'eli the other h~d, the respons~ 9f varjables is to be' deterrined through an inversion, the variables must be normalized. Suppose' -~at the equatiop of th'e control] ed pressur~ p is sought in response to step change , in Pset while Ws remains cOI1stant. The n~ed' for the 'n ormalization becomes . apparent when developing the transfer function of tl?e .process iI1corporatin,g the reservoir. ,T,h e difference -of flow rates in and o-qt:of the 're~eIVoir) ·.L\w = d!vf / dt. Use the ideal gas equation, M = p YIRT, with constant values of teinperature T, gas constaJ1t R, and vol,1;.lme V.
a
'V J dp Llw:= ( RT dt' .
Transfomling h9th sides of the equation,
1'1 Awl
:T)l' I dr) = (:T )r l'lPl -
=r
S
p(O)j
To avoid the dangling term of the pressure at time := zero, p(O). or Po, the variable must be redefmed as was fust, done in Sec. 15.9. In this case when the pressufe variable is revised to p - Po,
RT G(s) = CVs)
But the consequence of reyisin~ p to p - Po IS that the other variables must be restructured as well. In many instances th,e procedure "is simply ane of redefining all variables as their tran,s ient values minus their values at the initial steady state. as in Fig, 15-18a. At the initial steady state p = Po PlI:.t = Piet.O. etc., so all variables are zero. The normaliZAtion .of Ag. 15-18a now justifies the following treatment of the w summing point. Since the pro~s to.be examined is one where w, remains constanL w. = W, 0 I
I
•
" .
I
DYN}lJ':"UC BEHAVIOR OFTH.5.-,;(MAL SYSTEMS
393
I·
. ..
-=--
~:.
.'.
.
.
J .
~:
..... :... ••• , ••• J • •
(a J~~"" .'
I'
.: J.."
~
."
.J.'~
: .
..:
·,·Psr:I-:-P .,
..
P-Po:
(b) F1GuRE.15~lB ' (~) .Norrnnlized .v~~bles in' pre:s5~re c~ntrol loop (b) w'ith ~V summipg ,point ()m.it~ed.
So \'Vin' - win.O ·p~s.ses dj~ec1;ly to the RTJ.( ~s). transfer 15-18b; and the sumriling poin~ d.isappears.
~cti
as .in Fig.' .
..
. Exa~ple .15 .. 9. In' the pressure controller of Fig. 15-17~' V ~ 3 m\ T = 300 K, R =' 0.287 kl/(kg . K), K =.0.2 mrnJkPa, and L' 0-.3 ~kgls)Jrrun . .(a) If PSl:t.O = 200 kPa and ws.o = 1.2 kg/~,. wh~t is Po? (!') If at zero time Pset is abruptly changed to 225 kPa, what is the .equation for p as a function 'of time?·
Solution (tl) Ws.D = '180 lr...Pa. . (b) G(s)
win,O
= O. 3x o = (O.3}(O.2)(200 - Po) llnd since
.
Ws.o
= J.2, Po =
= (O.3)(O.2}(O.287)(300)/(3s) = 1.722Js p
-
Po
=
.-e- 1[
P = 180
+
25[ 1. 722/s l} s 1 + 1. 722/s .
25( 1 - e -110.3807)
The new steady·state pressure is ~05 kPa. so the same pressure difference, PSI!t - p, is restored because the same steady-state flow rate of 1.2 kgls once agaiii prevail~. .
Of .more practical mterest .might be the choice "Of Wt as the reference variable rather thirfp~t ~causep~ like1y to be adjusted rarely, while w, may change frequently during operation. The fust pass at the block diagram .with the variables normalized is sho\VIl in Fig. 15-19a and an examination of this figure shows two potentiaJ problems. One problem is that the reference value is not entering the summing pojnt as a positive value, as is standard, and the other concern i~ that a .sign reyersal at the summing point of the pressuu wjH frustrate the attempt to 'drop out this summing point for the constant P!d case. Figure 15- 19b shows changes in the variables that still
is
394
, Ws
DESIGN OF THER.};1AL SYSTEMS
-r'
P-PCJ ~ Vii.S' O~_'.~------------\---.-. ~.in - Ws fa' ,: ~ l4··r-!I------------------~-
.
-(Wjn.o-
+
~s.,t:I
_
•
. .' ,I psel . . ,~ .. - .
~)
r:" -,
'.'
"
)
-
.
.
.
,
- '-t':J, '-"
...
I
.
r
. \"
.
.
,:!.....:;: :. ~~;~~.:~ :.; .:::,-~.
.~ >'4
~~.
'
"
;-:: .. ~ . )':'::. ,.
.'
Ps.et, 0
IV + . ,
. ,.' .
,'--.----:-:-.-.-'-.--:-------:---1
.
IV.\)\:~~;:;
Pset-P
"'in""" ,Win.o
--
,
..
~
~.
~
•
•
. : . :'-.'
. . .:.'
~
J
~
..
.
t.J'-,.
'.'
. .'
Win -:- Ws' .
.' .'.- ( H!in. ·O~
W S,
,
~
..
~
::;'
_
-
' . '1
~(po -Psel.O) . (b) '.
FIGURE 15-19 the reference value rransla.ted to (a) the ,preliminary block . Pressure controller with w, dl~g~am '(b) with- sign revisj.ons to provide the stapdar~ fo~. . .
as
the
abide by .physical requir~ments, but eliminate the :two above-m,entiqned problems. The transfer function of this loop .is· ~ .. RT .
"J,
:
(Vs)
TF == ----=-----=...~[ RT J l+KL-.-, .
(Vs)
If the initial steady-state conditions ofE~ample 15.9 prevail and Ws increases . to 1.4 from 1.2 kg/s af t =:; 0,
_ p
.=
Po
/,-1 [
-
-O.2[ 28. 7 s s + 1.722
1]
p =. 180 - 3.333( 1 - e -tIO.5807)
The new steady-state tank pressure is 176.67 ~. The pressure' changes with the same time constant as when the setpoint w.as changed but this nlay not be the case for loops having more than one dynamic element. l
15.14. RESTRUCTURING DIAGRAM
THE }3LOCK
In' the attempt to develop a trans~er .f!-!nc.tion of a loop. the task may be facilitated by converting a complex brock diagram into a unity or nonunity feedback loop. Some of the elementary restructuring operations that' may be useful in..sjmuHfying the loop are:·
-
~
~
~
-
•
. :.~.
' .. .........,;_ _....... •
-
e"
.'
P-Psel·
-'
.
~
,"
) ." . . ' , ' r----"-..... P.-Po·····~· "."', 1-:-----------....,...---:--:------:-,------.-, .. :. _
.',f,!Sd -;SCL~ ..
.
•
,
.~,.
"
,
".
and
- '.
. ~ ~··---.f;~i;·,,· :-; •. '.~
9
••
TIlERi~1AL SYS/b'vlS
DYNAMIC BEHA'/lOR OF
3.7 5
FIGiJI~. 1~w20 -" . '.. Combining. two tfa.l1Sfer; fuOCtlDDSin ~e:ries; -
-' -~"'-:. \.;-~ '-'.:.- .. ;:<. _........
':. ":"'::" .. -: . . . ., '. '. . . .:. .' . ,
;. -
. . ." . 4
.
~
.
.,
•
;..".....
~ C(s)
'.1
_ ..
. ..•..... .~ .
A(s)
.~'f Bt~).; :
. I
FIGURE 15-21 . E~cruinglng two adja.Cent si.uriming points.,: .
1. Combine ~o transfer functions in series, as in ,Fig. 15-20. _2. Exchange two adjacent sum~jng points, as in Fig. '15-:-21. : - 3~.' Mo.v~ a summing' poine upstre~m or d~wnstrean1 of a. block? as i~ Fig. 15-22.
. '
- 4~ Move a takeoff poin~, as in Fig_ 15-23. Example 15.10. Convert the lo,?pll shown
In
Fig. 15-24' into a nonunity
feedback loop.
So,lution. First move the summing' point and combine it with the surnming point for the referen~e variable, and move the r.a.k:eoff .of the lower reverse path, as in Fig. 15-25a. Finally, combine the upper and lower reverse paths . to obtain the nonunity feedback loop shown in Fig. -15-25b.
B (s)
A (s) +
FlGURE~
".
Moving a summing point around a bJoc.k.
C(J)
,,'
396
DESIGN OF TIiERMAL SYSTEMS
B(s)
: .;.. '
R(s)
+
C(.r)
FIG1JRE 15-24 . COl!ITol ~lock d~agnUTl in Example 15.10.
R(s) + ~
'-------------11-
H, / GJ
(0) C{s) ~
t
GI G1 G .1,
. H2-.." .!:!J. .+ GI
r ..
.-
Gl i
(b) FlCURE 15-25 Modifications of Joop in Example JS.l0. . '.,
-.
...'
DYNAMIC BEHAVIOR OF THERMAL SYSTE.i'yjS
AniUustrai:,ion "will· shov\'. hOVl, to: [.epresent ~ phys;caf ~ysLern in .b ~ock diagran1 symbols such that. the fOITa is ready ~or ail. i.ti version .lnt!? · ~he t~Ine ~,8T?aiD. The ,~ysteIIl :is: ~ .~!f)ciri.c, h~~ter . i~.·The7·cont1:dHer-'pl~~-~i~le$ .~9Y!~i. t~i)fi{· :~·-:· ·h~a~er 4cco~~g·~t·9 .the ·equation·/qh· : '·.K (Ts~~' ~ 'is) ·,vheie 'Ys IS. ~he sensor" .: i .. t~mpeIatUre. The rate ~f·heat 'traJ;lSfer from the' ·J:1ea~er to the flov/i'ng ~ir qa ... ; is r~p·r~$ented by the rate equatioD 1
.
,
qa . ··'··wc(··Th ~ 7j)( ~ - e-:-J~JwC) 'where
.
.
. W.(~"
'.
~ Ti)€
= rate of air flow· . c. = th.e spe~iflc' heat of the a~ , Tb = heater temperal1Lre }v
T j = t.empeiature of eTl.teri~g .air . h = convection coefficient· A .= heat-transfer area
F9f convenience, let H'e = W, and the effectiveness of the heat exchanger is rep.re~ented by "E. The initial block d.iagram of the control ·sys~em is sho\vn in Fig. 15-27 whe·re the two dynamic el,~merits .are. those associated wirh the thermal capacity of t~e, heater M M.d the 'time' constant of the seQsor . 'Ts. The structure chooses T set as the reference vaiiable and anticlpates ,an analysis in whic~ the entering' air te.m.ee.rature. remains constant. Table 15.2 sho\vs variables corresponding to the numbered positions in F~g. 15-27. Also shown in Table 15.2 is a column of variabl.es non:ruilized by subtracting steady-state values from each. A next, minor revision of a small se9tion is shown in Fig. 15-28 in which, to avoid two positive values entering the. ~umming point, the input (variable 10) is changed to !ita - Tj. .
Heater Air
~
Ti we::;
~mc;;:::M
-=!---hA
W ~Th
Tt.et
F1GURE 15-26
Air he;)[jrTg ~srem ar.: its control.
Sensor
Te
398
DEsIGN OF THERMAL SYSTEMS
~ " "
.,'
' " f - , '.
-' ,
:
..-
. .
.'
'
",
K
,·3-
'
..
.:
.
, J:,'
,.,.'
2
, -, '
'.
. Ms .' .
.+
.
.
,".
.
:.
..
-
. .',
.1. '" .
.~
I , 1: s S+
;
.'~
-. '::
',_.
_W.'
_
;.
~,
J -."
.'
•
,
1 :.
'"
• .".
..,.~.
i ...' .
:.
• ••
~
~
FIGIJRE is-27 ' , Air hc:atiJ;tg ~st~m and its con.tI:ol ..
TABLE l S A l · . D~igi:lations .
_
of variables in . 'block'. diagram of ,Fig.; 15-27. N6nn orm ali zed
Position
NonuaJized
Ts,d
1 ....
T!.d -:-
3
t}h .
4
q,h - qa Th
5 6 7 8 9 10 1J
Ti Th - T; ql T~
- TI
(Tc -
T·,
Td -
(Tc • 0
-
0)
7i. 0)
Ti - Ti• O Te - Te.o .,T, - T"o
Tc
T,
]2
Tsl!'I... 0
. (T~, - T~) - (Tsc.1. 0 - Ts.. qh - qh.O (q~ - q"J - (9h. 0 - q~. 0) To'-' Th• O Ti - Ti• O (Tn - Tj ) - (Th. O -]i.D) q~ - q:J.. 0
T1.e,·- T s,·
..£..
. If the variation of the .controlled temperature is sought in response to srep .change in Tf-t-h the ·next .stage of consolidation is shown in Fig. [5-29. At the initial. steady state qa,O == qh,O. T,- = Ti,o. and Tc.o = T5.0T\.vo summing points call be eliminated. The final restructuring derives from recognizing an interior feedback loop that has qh - qh.O as the reference '\.'
_+7}o-Tr, '. ( Tc - T, ) L . . . . . - _ - - I - ...:.. (
FlGURE 15-28 i!.ineu.!..,to summing point.
Ch8n~e
-.
~
' 1
Tt.. 0- TI. oJ
+
1c. - Tc: 0
I
OY~.-\M!C BEHAVIOR OfTHERJ'-:iAL SYSTE.J.\;!S
.
399
.
. FIGURE 15-.29. -.' . '. . Diagram ~.fter e.1imlllation .. of tWo summing po~nts_
.1
Tr:-Tc.n
KE/UvIs) t
I + ~VE/(A'/5r
. 1. :rs5 +
}
FI G URE 15-30
Simplified nom.hii.ty feedback "loop for air heate~ conrroller.
value and T h' - Th,o' as the output_ When that loop transfer function. -:is . substituted, the final diagram in the form of a nonunity feedback Ioop:-is that spown, in Fig. 15-30. The expression for Tc - Tc,o in the time domain can be obtained by inverting the product of the loop transfer function and' the transform of the Tset - Tse"Q disturbance.
15.16. PROPORTIONAL CONTROL The pressure controller in Fig. 15-17 and the temperature controller in Fig. 15-26 employ the proportional mode of control. In both systems the actuator· (the valve position x in Fig .. 15.-17 and [he power qh in Fig. 15-26) are proportional to the error between the set and sensed values of the controlled variables. The proportional controller simple, which helps keep its cost ·low I and provides adequate control in a wide variety of applications. The .proportional controller is characte6zed by an offset of th.e va] ue of th~. controlled variable from the setpoint at all but one value of the load. In the pressure COIHroller of Fig. 15-17, the offset is shown in Fig. 15-31 to
is
be a function
of the demand Ws fO.r a given gain. The offset experienced using a low gain in Fig. 15-31a can be red~ced by increasing the gain as in Fig. 15~3 Jb. A -further influence of the high -gain is that" it may introduc~tion as the systsm adjusts to a disturbance. Some control systems become unstable when the gain is increased beyond a certa~n valu€. I
400
DESIGN OF THERMAL SYSTEMS
Psct
Pset" ---------------------
~----------------~---
\
; ~ '.
:/'-/
Controlled F
'.'~
-,., . .
Step increas.e iI1~~
,"
,-
. ..:.
.!;.'..
: •• 1"
-.'' ' ..;... ,::". - ' -
..
~
.. ~ .... ~~ .. ';." . :
.
~------------~----~--~~
.
.. Time
.
. -,.'. ,
~
....
~,
',' TI~~'~
'.--
'(~,)
""
.
.
..... '
.
:.
.
..
(0) : : ,
F1GURE
'
. ,.
~J"~,
.".
"."
.
"
'~'
.'
.'1:":~'. ~r
._.
II
:
v
: ••
15~31
Pressure controller (a) with low gain (b) Ylith -high, gain.. .
Adjustment of the prop'ortional ~dnstant ~o achieve an adequate comp:ro~se between low offset and stabilitY is called "ninIlfg'; the control. ' , M~tenance of the desirable quali'ties ot'the pioportion~l controller but deemph~si~~ng its--dFawbacks can be achieved by combining the proportional mode with- the integral and derivative modes as explajned in the next two sections.
15.17
PROPORTIONAL~INTEGRAL
(PI)
CONTROL The purpose of the iptegral mode is to eliminate 't11e offset associated with the proportional mode. The contribution to the control signal is ,
KJ
J (error)dr
(15.24)
so continued operation with an offset bujlds a contribution to the control signaJ that tends to eJjmin~te the offset. The transfer function of the I-mode is K 1Js which may be jllustrated as in Fig. 15'-32. In, the time domain, if the input experiences a step change of .1 at time zero, the integr.ation gives the ., output Kl(ll)!. The transfer function is the rransform- of -the output djvided by the transform of the input
1F . KI(j,)/s2
=="
jjs
K 1-
=-s-,
- (15.25)
In the control block diagram t.he combination of the" P- and I-modes are as 'shown in Fig. 1.?-33.' Each combination of system characteristics, K P. and K I exhibits a unique response, but to show the behavior of a PI control, consider the example of the air heater shown in' Fig. 15-26. The block diagram of this control loop. sho\vn ·in Fig. 15 ..30. must, be revised by
..
~.
",.
......
"
.. .
401
DYNAMIC BEHAVIOR OF THERMAL SYSTEMS fi7
A) .
J"
I
~:.
' . ..
.---~
f
J
·~~~:in~l · . .,. r .' ., :.},,~ '.: .
.
,-'.~ ~~:'-": ..
"" . "
_~"-
.. ! I -
.
--L-i_____~
. '," .. '"
__
D"
0.
Time .
' . .:. i J " .:.
t,-" ...
'"
-
.
-, ' , :
'
•
- ,
..
....:
-.
'
,
.
.- . . -.
..
Transformed ~omain I ,
"
..
flGURE 15-32 Transfer ~ric~ton of.tpe)-m. od~·. :
cl1angi~g K ,in the forwprd' path to (K p ' + K I Is). Intrqduce :the following , 'numerical values:, E == 0.7, '/1,11 , 50 kJ/K, W '=, 0.6 kWjK, -7 == 4 s, and , ,..th~ entering temperatur~ .of the air,. Ti == 22°,C. The loop tran~fer function ' ' then becomes'
TF ' '
-=-
~ooP , -
'
53
+
+
+
1) . ,. (0.0021·+ .Q.0035Kp)s + O.0035KI
O.0035(K p s + 'KI)(4s
O.258s 2
(15.2~)
The response of the controlled temperature to a step change in Tset ' from 42 to 47°C ,will be examined for four different combinations of K p andKI:::' ' ~
Case 1.
Kp == 2.4,
Case ll.
K"p· == 2.4, K I
C'ase ill. '
Kp=2.4, KJ==O.l
Case N.
K'p == 2.4, KI :...- 1.0
'KI ' ~
0
= 0.02
The transfonn of the input, ..£{ Tset - Tset .o } = SIs, so the transform of the output, L{Te - Tc •o } is the product of Eq. (15.26) and SIs. This transform
Erro~
Error
f
~
(0)
FIGURE 15·33 Block dia~ esYtirbols of the PI conir&'
K,,_ Kp+T , ,------
(b)
,
. :~~~" '~~ ; : . ~:-;~.~~.:~~.,
:.. .' . . .>...... ,'."..'
Time· ". ~ . OW
I
.
~ ',
~.
DESIGN OF TIlERJIAAL SY~TEMS
402
is the one to' invert for all four cases: " ' .
's[s3
+
.'
,
.
O.258s 2. .+ (0.0021
. (15.27} ,: -:,.'
+ O.0035Kp)s + 0.0035[(1]'
' :
"
-
"
\
.-: ..
>:::,;f:caseJ:: .". ,;' , K 2A,K] 0., : . '.' . .... .' ..., . "'.::. ~....",-:<>' :"....... :;. .": ",'..:1!..-1{':.' : .O.16Ss +. ,0..042. .'.:. ." J': ' .,:."(1''5' .'28),' ',; ", p
.:~ :::.
~~
'.: .,..
..... .'
•
', ' ".'.
•
.
'
,
'
:, ·~'".·7:c;.!-:-:-;~rc,p. -, ' •
, ', . :, '
'-'
.'
;
.-
3
'~.' ;.' ~,;
.:-
• •
•
S,: ',
-
t .-
:.
" , : ... ', +,..o.258s 2:. j-· ..'OrQ-.10~s ·y..Jl '.. '. .. ;•
•
',
.~
_
, "
. . ....:
•
•
~"
,-
~~.I';
... ,.
. . . . ... t ••
-~':.
:
.....
.' . :. .-§-',.
I.,~. " ~s·
"~
.:-.... ,.':~~~", ", '.,'"
" . from' the 'steady-state .calcuhlt10n. Tc.o. . .: ' ~8~C;,-'~d lrorli'.the·inverSiolf' of~ ' =-':, "" ,:.' ':~
,,'. . ' . Eq::-.'-(-{S.28),
' ......,.. '''- '
"
.
"
' : . ",'.
,.'-
Tc . 42 ~ 4."220e -:-.{j~9.74 '.+' O~~20e:-tl~.823 ',' A graph of :Tc wjth P-mode 'control is 'shown ' in Fig:, ,15-34. The . ini~.ial . " , , offset of Tc is 4°C, and the 'offset ,after the change in setpoint is SoC. T.qis . - .' . ': ' ."",response will be the basis of comparison ,with 'the PI mode which will now" . ," . be developed by cOIDQiping the I~mode with the P-mo.de ~ Cases II, ill an d'-IV, are all PI control with. varying magnitudes of KI ' :".and shov.;ing ~espoJ;lses 'as.ln' Fig~ '15-35..' " , - ~ . .. " '
I
.
'
Kp == 2.4, KI
Case II: "
= 0.02.
c
Instead of there being an offset between Tset and. Tc at t == 0 as occurs in 0e p~oportional controller of Fig. 15-34, it is assumed ,the I-mode in Cases'II to IV has brought Tc equal to T~[ when ttle step change'is made. \Vith the low. value,ofK J in Case I the I-mode slo~ily corrects the offset and Tc approaches T set asymptotically.
48
,-----------------------
u -:46
I I
o
2
r T~,
44
I
:J
I
g.E 4"').~ 40
38~,- - - - '
36
L -_ _ _ _~______~~_ _ _ _~~----~------~~
Time,
',S
FIGURE 15-34 Response of the air heating system wirh the proponional-<>nJy contro.t. K JI Ii:a: 204 (0 a slep chnnge in setpoi,nt. . ,
-.
~nd
XJ ~ 0
40'3
DYNA ;lie BEHAVIOR OF THERMAL SYSTEMS
'1
,FIGURE 15-3$ J:<~spons~ of the air hearing sysrem ,wh~n ~sing PI modes with various values ·of K / .
_
Caselli: , With the increased value of KI the correction of the offset't*es: plaee rapidly' andTc experiences a slight oscillation around the final'steady state of 47°C with the excursions 'q uickly "dying out.
CaseN:
](p
· ,
= 2.4, KJ = 1.0
This case shows that when Kr is set -too high the response is,,,unstable. The illustrations in Fig. 15-35 apply to only one value of K p, ?TId it is clear that the two constants must be selected in conjunction with one another (tuning the controller) to give the" desired control characteristics for the particular process being regulated. An active field of 'd evelopment is [he application of an on-llne computer program to self-tune the control.
15D18 'P ROPDR,110NAL-INTEGRAL-DERIVATIVE (PID) CONTROL .A skilled operator controlling a process manu~]y will instinctively change
the actuator position more if the deviation of the controlled variable from the setpoint is increasing rapidly. In a similar manner, jf the controlled variable is currently deviating from the setpoint but approaching the setpoint
404
DESIGN OF TIIERMAL SYSTEMS
rapidly, the operator will back off on the co~ection in an&effort to home the ,controlled variable in on the setpoIDL In the above situations the operator )s ~aking an adjustment baseq. the rate of change of the CO~,tr~l1ed . v~able, or .more precisely on the' derivatiy~ of the .error "with respect to·· : ; :.' '.'.~. ~time. The differentiation 'process , in the time .doma,m and, jn· the'.s .· domain '· ': , '·:· ,··' . . ·:·:'>:;:are 'shown,in Fig. 15~36a .and 15-3.6b., respecqv~.ly. ·Dividing the . ollq)·~t-6f . . ·.· . ,',
on
l
....":. : ·.:·~:',. · : .,·:·:;:~:~,~:~t4.e{~bldckjn.~Elg~., j~:~.3.pp.;.by_·.ih~..:iiJp.~;,. spows .t1;lat '.tb.~ trails(er function of ':. ; .:', . ':".;' ~. ~:···· : :·,: · ·· · tfie·~"diff~rentiationj)JC)cess,:: .i~>s~>f(.di( ·~1.'6c~-:'~rept~~:~P:!:inR)~~;J~I. ~o~.tro~le~r··. , '.--:.:'" .. .. : .)n 'Fig~ t5-~3 ' is extended. to incorporate .deijyative· cOP"tror\£iili~~·:'~c6nsbuitr..,~;,;~~::,;. ' . " ; .. r K D "': the ' transfer.function of the ·]~:>.Ib controller 'b~cor.nes- · .. " . ,., ".',. ,. ·:··..··t :..:: . - . ... . . .. . ---:
' ,'
.... "XI '"
~
.
:
..
,·K p +'.+ Kn s " S -
,
.
.
..
.
-
"
(1.5 .29). '.
,
.
"
"
. . .
. ,'. .' To illustrate the i~uenc~ . of the D-mode of cODtrQl, apply PID control·' . . ' ·to the pressure regulator of Fig. 15-17. Examine the case where p ~- p.o'· is to be determined in respons'e to a ,change in th~ outlet airflow rate Ws ",:-ws,o , as represented by the block 'diagnim in Fig. 15-19. The same parameters as . _. '. in .Example 15.. 9.·will be used, namely, V = 3 m 3 , ·T == 300 K, R ~ 0.287 ·, kl/(kg ". K), and L ' = 0.3 (kg/s)/rnm. The pressure setting Ps~t is 200 kPa' \vhich remains ·unchanged. Replacing the proport'i onal constant K~ by :Eq. (15.29) yields the 'loop transfer fun'c tion .' " .
"TFloo~ =
86.1s (3
+
25.83K n )s2
.(15.30)
+ 25 .8·3K p s + 25.83K J
If steady state has be.en pennitted to develop~ the I-mode \vill be assumed to drive the controller 'pressure to the set ,value of 200 kPa. For a step increase , of l~'s from 1.2 to 1.4 kg/s the transform to. invert 'is p
~ 200 =
-
("-l{ '
(3
,
.
.' -17.22
+ 2~.83KD)s2 + 2S.83Kps +
] 25.83K J
,(15'.31)
Figure 15-37 shows the response of the pressure with and without the D-mode for values of K p = 0.2 and KI = 0.05. Following the step increase in '1'5- the pressure drops rapidly and sinks to 2.7 kPa below the set value when
Fen
, dF(I)
F'( I)
J(S). [
TF .'" f
(1/
I
.....-=0
sj( $) + F(
~)
, diffdu:ntialion
differenlialion
(b)
(0 )
F1GURE J5·36
Th~ differe~tialion process in
(0)
the 'rime domain. (b) the transformed domain.
DYNfu',:IC BEHAVIGR OF THI;R1'.1.AL SYSTEMS
:: .. ~
, . ~ ,: J99 ~
~
0... ,
.!<:~
.
v
.. .!
<05
~
",
,
'
' . ',' .
- 0
" 'Kp=O.2
L-.
' ;J ~
-
[(, '= 0.05 , ,
:,I]
o
a..""', 198
.; 197 , ..! \
'-______l'-___'__---L..._---L-_--I----!....--I-_~_....l__.l......_______l'_____'__
Q'
4
2
8
10
__L....i>_
'
12
, Time. s
FIGURE J!5-37 Influence b'r the D-m,ode on the response 'of pressure controller of Fig. 15-17 to a step ~Crease In the outlet flow rate, "
,
, '-no, D-mode is appEed~ Using PID control with KD == 0.08 t~e maximum deviation is , 2~46 kPa. In, this case the 'PID control experiences a short interval wh~re the deviation is greater than for the PI control. This difference' is attributable to the,I-mode, which was accumulating a greater corree'tion' during 'the 'penod of high deviat'ion occuning \vith the PI control. ' 15~.19
FEEDFORWARD CONTR:OL
An additional mode that can be combined with PI or PID in ,some 'situations to provide superior characteristics is feedforward control. Byjtself, feed- ' forward is open loop control that supplies the gross contribution, whjch is then supplemented by the fine tuning of PI or PID control. An example of feedforward control is the steam pressure and excess-oxygen regulator' for a fumace/boiler, as shown in Fig. 15-38. The intent of the controller is to maintain a constant steam pressure by regulating the rate of fuel 'supplied to '¢e burnet. To ~aintain efficient combustion th~ percent ,of excess oxyge'n in ~e stack gas 15 ~lso regulated. The excess oxygen is a function of the combination of flow rates of air and fuel, so the t1o.w rate of comb,ustion air must be adjusted ~ the flow rate of fuel changes', Two possible control concepts for the air fl~w rate are (1) PI control to regulate the desired'percent excess oxygen, or (2) an openJoop controller ,that sets rate of. combus~on air baSed-'o~ the rate of fuel ,
the
.
I
.... . \".
"
.
I
J
-
.
406
DES1GN OF THERMAL SYSTEMS .
I
stack e.as
t
~
I
. .02
Stearp pr~ssure'- '.'; . ' .
.
'-:;'- (". >/... .,rrnnsducer '- ~" " '.'
'J
. .
. . sel15or· '. -
y! - _.
J:J
.
.. "':' •. ~ ...
- .
. .
-.
.
.
.: 'Boiler " . '.......~
conrroJier
._'
' .' ,
:~ ~.~ ,
.\1 ' -. .... -
. Feed water ~:_:. ',
~urne~
..-.._ _-.-----=-_----:._ _---1 Fu~ace
Combustion. air
Fuel
FIGURE -15-38 Controller. of fuel ~nd air Dow rate to a furnace using a combination of feedforward and PI modes.
.
flow-perhaps in direct .proportion. Mode 1 using PI control alone is likely to suffer large excursions when an abrupt change· in fuel flow occurs~ or alternately it may require a long time for a very stable controller with low. settings of K p and K I to bring the rate ffow' of combustion air back into range. The open-loop control of Mode 2 will react in concert with .the fuel controJ ~ but the percent of excess oxygen is not Ijke]y to be precisely helq be~ause of th~ nonlinearirles in the actuators or the combustion process. The
"of
combination feedfor:ward and PI control pennits a rapid response of the w rate of combustLon air in the proper range, air coritroller to keep the .flo_ and still pennits the PI controller to constantly refine th~ .setting. .
,
Example 15.11. An electric hearer. shown in Fjg. 15-39. raJses the lemper~ ature of a stream of water having a flow rate of 1.2 kgls from T j to TJet which
is 6SoC. The temperoture of water in the tank. which js the some as the outlet . temperature Too has reached a steady value of 65°C with the in]et .temperature Ti of ~O°C when at I = 0 Tj changes to J5°C. Determine responses' of Ta as
-.
.'
. . ';;- .... . .~~, .
. .... .
Air flow
... . .,. ..
,
,-
. :.> ... . .
- -PI comroller
regu~a.tor .
,"": ' \'
407
DYi'JAlyHC BEHAVIOR OF THERt'v[AL SYSTE\-IS (!}
r-
>t-==~~
FIGURE, ~5~39 _'~lectric water heater an~ storage '.tank
"
,
in Example',15 ~ 1L
funct;ions of time if the modes 'of control are, -(a) ·PI control 'aion~ ';'i'th K p ~ i5'· " 'kWtC an:d Kr = -O~02 kW/(oC . s), and '{b) if the"'PI,controller of part (a) is ' combined with a feedforWard control keyed to Ii such that its -c ontribution is q ff ,= .{ .1.0) ( Tset _-, Tj) . ,
.
'$olution. (a) The block di,agram incorporatlng both the, PI and feedfonvard control modes is shown 'in 'Fig . .15-40. For PI control ~lone the ' loop transfer function is
,: .. (1:2)(4.19)s :.£ {To -:- 65} (15.32) ..L'{Ti - 6S}, - (750)(4.19)S2 + [K p + (1.2)(4~19)]s + Kr which when multiplied by - 51 s and inverted yields the results labeled PI in
Fig. 15-41.
'
(b) If rhe feedforward control had been set for 1.2 kg/s, its contri,bution would have be,e n '
qff
= (1.2 kgls) (4.19)( Tset
-
Ti)
and this open-loop control would keep the ou~et temperature at precise'l y 65°C
I T!.et =65
------------------------------------, I Fecdforward section ,
I
, +
/ " ...., . ,
,
\
J
, - - - - - , qrr + 1.0(4.19) 1-----1> I J
r----~
+-
------
/
I
TI
I
FlGURE 15-40 Block diagram of outiet water temperature in Example 15, II . ."..
0
408
DESIGN OF THERMAL SYSTEJv1S
~ r---~~----~----~~----~------~-----~----~r---~
.
," ""
'i'
- - I
:':::~-: J;~. ;~=2~,::',:' '~/·~f/:"·~: ~:~);,; ~.:":.i.:' ;', ~::.:~;f,:/:·~: :-;: - ._:.·,: " <:i~·" .>~ . .. . . ..
o
400
.
-
,800
600
1000
1200
1400
.. _.
~
]600
.· Time~s
, FIGURE 15-41 R:espon,s e. of the outlet 'te~perature of the water heat~r in Ex~~l~ 15. j 1 to PI and PI-plus-' , f~forward
con~L ,
'
' "
,
..
"
,
,
,
is "
without' the 'need of the PI modes" The flow ' rate for this particular peater , ~ormally 1.0 kg/s whlch 'explains the setting', and at this moment the flow , ' rate is different than expected. The loop transfer function of the complete " 'loop shown in Fig. 15-40 that includes th.e feedforward control is precisdy : , 1./6 , of that express~ in Eq.' (15.32). Proble"m 15:23 asks for the execution , of this 59] ution .
15.20 NbNL~ARI~IES 'The analyses presented so far in this chapter have "been based bn ideal -- performance of components and systems. Actual equipment often functions with losses, friction, and delays. Most valves and othe:r"mechanicaI actuators suffer from hysteresis so that a rev~rsal in the direction that . the control signal is changing does not immediately reverse the direction, of motion of the actua.t()r. In thermal processes there may be some heat transfer to from unaccounted, 'extraneo·us sources. When fluid flows through pipes there is a finite time required for a gjven mass of fluid to move from one location to another. Th.~ existence of noniinearities does not repudiate the linear control analysis, becal.lse the solutions for the linear cases indicate the dire~tions
or
to strive for good ~c.ontrol. Sometimes the extent of nonlinearity is small enough that the linear analysis s~ffi'ces. ' One nonlinearity that will be explored is dead time or transportation lag. which is often combined witb a time-constant representation' to adequately model certain complex processes. An example of the existence of a transportaTion iag is shown in Fig. 15-42 where the sensor of outler air temp-
erarure is located some distance downstream from the coil. If the, temperature at the coil is a function of time X (I), the sensor receives that function
-.
......
,~;
..
409
DYNAMIC EEHAVIOR OF THERMAL SYSTEMS ~ -
-
.-:---------'r::";=~l7l' . C~-fi ,
,~ I
. " /<'::' " ';'. , ;' '.
-
. ',--___
-~=~o-----'-----
.. -
•
I . ) _. , . : .-
• -
- .~
• -
..
'
<, :<._,;:::' ':. '._.:'~: _'.: - -. ~ .' ~ .:__ ::~ . . ,.....,..:. ::; ~. :~;-..,;~.: i.:\.J•• ;_-.: - ,'~ / ;:... .': - Vel6~i ~)f-t4'l';;\:'»:~':~"':~s~·:~· .--::', ~~""" .>_~ :':: l::: .r-!,>~-.._ ..~", ~<.'.~':~;1' - _Ai!'JJo\}~>.-,{ '.'." Coil ,; .." -XC)' '. __::-_ . f '.- '.' ..--.- ,:;.--;.;,.'_:u':\~ ".
-.-~ .. '.... -- . ' ,,:.!y.(r)7' ".' ---'·: ·S~n"..~cir-·. '. .... ..
.- _ ',.-.
-TrJ1l.."porru[ion ;ag~ T. s
.. .
J. }
FIGURE 15-42
'
,
-
'Transportation lag' due to the distance 'b~tween the coil outl~t a~4 .the tempei-atur.:e - ~ensor· in - a~ air-tempe'rature coo,trOller. . ,_ . . -
T··. L/~ secondsJate~ ...T~e Lq.pJac.e transform 10 -.of the !e:n:1pe~atur~seen'
by the sensor 'is
. . - ,
-
1 {Y(t)} = y (s) = ("" X (t
Jo
The -integrand is
ze~o
.
-
for
.a < t <
. 'y{s)
=
--'
~T)e
-sf
, .. '
dr
-
T, so
J'" X (t T
T)e~SI dt '
Let t * . t - T, then dt = dt tt which chang~s the lower litnit of integration to
o.
(15.33) Multipfication of the transfonn of a function by e- sT is equivalent- to imposing a delay of T seconds on the original function. Some physical ~.oIPponents can be modeled, at least roughly, by combining .a time delay with a time ,constant. -In a heat exchanger ~here air is' cooled or heated by water flowing wough the tubes t there will be a transportation lag attributable··to -the- time required for' the water to pass thr9ugh the water circuit. ~~.lThus, the aGtual response of the outlet air temperature to a step change in water flow rate or water temperature can be approximated by [he combination of a de1ay and a time constant, as shown -'in ·Fig. 15-43. The transfer function of this coil is
-----.
ke- T1
:'.-:
(15.34)
410
DESIGN OF TIIERMAL SYSTEMS
')
---------------~~------~-
Il.) l-.
-~
it ",:'" ,:,/c~. .',...' , ,~
.: -.
l' ,:, ,.:.:•.
"... , ' -.
~
,I
Time.con-stant·,
'. .. .
.'
.
-,
<,
'-; " '
,
~::. ';~Read:~e", "' 2~_~~::':':':"'; 'C \~,~,~~l~j~k:r~~:~::<';'F:';;/
J-----.:~___,_.J_-----!--'--_ _-
,response,to·a s.t~p., in,q~s,e·!n,' :,' .' _, . :. " -, ' . water iempe~tnfe or 'w ater flow,_ -rate. " ., -
_ _-----:-___:__~...... - ..... ~ -,
0,
.~
il '
Ie e~TS
Time ,.
.
.'
te= introduces nonlinearity into the transfer function of.the;oop :ich often precludes a closed fonn' for the inversion': Some 'a pproximatIons ' , y ·be possible, ' as .Jor example if the coil with ' the -transfer function of {15.34) is in.. the forward' path and the ~sfer.functi9n ~f a se'nsor: ·' Tss.. + 1), is' in, the reverse pa¢~. The transfer functi,oP' of the ,loop is
1,
~I
.
~~
TF]oop
==
+
1)
1) (TsS
+
k(TSS Ts
,e ('res
+
.
1)
+
k .
(15.35)
I'
!.l exponential, e Ts Can be expressed ,:1\
I
e
\ •j
Ts
'
=
1
+
Ts
+
as a series. " (TS)2
+ ...
2
nclusion of the frrst two or even the first three tenus of the series gives 1sfer function that can 'still"be inverted conveniently. Also, if the value is small, the resulting inverse is reasonably accurate.
l
SELECTING CONTROL VALVES
'm now from somewhat mathematical considerations to issues of pracardw~e. although 'a point that will be made in this section flows naturom analytical insight. Suppose that a control valve is to be selected to
te the hot or chilled water flow through a coil, as shown schematically , 15-44. In selecting a valve. essentially three items ~~ specified: , ze, ,C v v'alue. and chara2teristics (quick-opening, linear. or equal:age).
pipe 'size is often specified 'to match that of the pipe to and from the ~. aJ though sometimes the pipe size of the valve is smaller in order ' d uce the cost of the va] ve. ' --
t~11
DYNNAIC BEHAVIOR OF THERMAL SYSTEMS C7
~~~~ 'Ii i
.
~"__
_ _" ..
! .
Valve
_.~;. -,-", .
-
"
.
, ", ., '.: ': J.. '-
·. Coil ", . ') ~
. (
---
:,t.,.
j', '
.
· . T~~
' ~P
. ., ,
. ,' .
.>t
.
. '<:" '\.."';. .: :.":.-," :.,' ~: "
,"
~r-'- - - - : - - - - ' - - -
fj,p:::;
80 ~cPa
.~.- .. ------l~~ .
.
'.
~
..
'. .-
, ~' , "
_. :_~ . ' : "
"
.
:~:~.l~~
-
I .
-.. '
.' .
.
. ... -.
"
FIGURE '15-4-1 Valve and coil combination.'
. '''.. : ····(15·.36)' where' !1p IS the pressure drop in kilopasGals across the .v.a.rve ~hen th~ ·valye is 'in its wide open posi tio~. ' . 3 .. The three different valve characteristics c4?~QJ11y availabl~ are -shown , .in Fig. ~5-45., . The valve manufacturer provides a certain characteri.stlc by means of the design.cbQsen for the seat and the plug. The influences of the' choice of valve characteristic and C..., value are demonstrat~d by two ,9ifferent choic:es of C v ~ designated case I and case n. Suppose that the coil has a pressure drop LlPcoil'
= 2.5 Q2 leRi "
where Q =' flow rate in LIs' and the available 'pressure difference across both the coil and valve is constant at 80 kPa, as shown in Fig. 15-44. A valve with linear characteristics is used in both cases, but the ell in case I is 0.6 and in case.11 is 1.2 . . For the valve with linear characteristics,
Q=
percent stem stroke r;:.. .. 100 Cy yt"p
percent stem stroke. ='
or
,
·100 Q ,
Cv -Jflp
(15.371
For a given flow rate say. 2 Us, the percent stem stroke can be computed for the valve. The pressure drop through the coil would' be 2.5(2.0 2) = 10 kPd, requiring 80 - 10 = 70 kPa to be dissipated in the valve. In case I, with C." = 0.6, Eg. (15.37) indicates the percent stem stroke to be I
. ---
-.
•
( .. ~!'.-.::~. .~ .'.: .. -..-::-~
.
"412
DESIGN OF THERMAL SYSTEMS
lOol . o j
r
0
• •
,': :.', . I
. r::: .
,
;,.-{:;:: , "
.. . . .:' t', ,-':
i
~
,"
" .. . •
100
_
.
;,:'.~
.
'~.::1 '
: :..:-.:
t..-.
'
~.
~,." ,; "':', '.Q \ :'. "
c ::"
_.::>: ,' ;~:..
",
.
.
.
.,,'....
, ' l ',, ;
"
,
.
/:,:~' : ::>':··:]-X:L ." ... . ~:
, -:
:
.'
"
....
:-.
.- .: 0,
- " _ .• -,
. . -."
.0.
" .0 ·
' ioa
.
, COpen)
, . Percent of stem stroke
, I
FIGURE' 15-45 " Thr.~ valve characteristics,. · . percent stem stro k'e == tOO(?) _- ·39.,8at. 7(. o
O.~J76
.The" relationshIp of the percent stem stroke to the flow :rate through the coil-valve combinati9n for the nvo cases is shown in Fig. 15-46. from which .two '.obse.r vations can be made: (l) even though the flow-stem stroke
~
E
ffi21---f--k---+---7"---t--t------1I---+--t-----I
20
40
60
80
100 (Open)
Percent of Slem stroke>
FIGURE 15-46
.
Flow-sl~-stTo~e ~
relationship of control vaJve 8nd coiJ combinarions. ,.
:~
DYNA!'vlIC BEHAViOR Of TI-IERlviAL SYSTEMS
L}13
relationships of the valves are Ijnea[~ the flo-w-stem, stroke relation of the cornbination is not Iinear~ (2) the cOlnbination 'If/ith the valve of the high C'v yields the. characteristic 'with less linea~ity. I. . 'i te c urves D.~ Fig. 15-46 .are significant, because the' slopes of the ., curves .rep.!,"esent gt ·p.ortiol1 of the gain of the control loop., The curvatu~e .9f , , ' ,the 'C v . 1.2 cHrve.-in Fi-g':: ~5-4~ '~s ~e~s_~esir3.t'lf: th~,n that of the valve \vhose .',: ',:, .' .. :~ ..,;·," ·:~C~., == o'.~,6~ Wirli:.qle':·(~,:_y,aVl.e;·".9(,{~2.-..the g~.ip-j$JQ;o/>~~,:J;le.n.thfkYaIy.e)~:near~~~L",~·::. ~lly~ .opecied~ v!h.i~11 w'ou-ld' c-ause: high~;-throttlin-g~.' ~ange; .if:-.th:e:.Goritr{;ner:;.is ··,:::,:;>:·:~' . onlY ' pr~poriionaL Wheh the ,valve is iiearIy closed the .gaiI{is hign:·.· and,:the:,·:::':'\<, loop !pay b.~ yn$.~a.ble. This behavior may explain $orn~ syst~ms being-stable ' ar moderate land' heavy loads and un'stable':~t Iovj·' loads. Another mean's of approaching.. a .linear flo\v-stem stroke relation .''is. 20 choose a valve WiLh ~qual-percentage c~aracteristics' as. -is don.e in .·Prah.',J ? .~5. !'.
lS.2i SOLVING PART~AL DIFFERENtL4L' .. EQUATIONS U~ING :LAPLACE TRANSFORMS An important class of dynamic thermal. system p~obler:ns is the one ·where there are changes \vith respect to both 'time and some additional variable'} the such as distance. Transient 'changes iD; t:emperature along a tube 'pressure of a vapor: as it flows in a pipe Me examples of ca,ses where ' the fundamental relationship i? a 'partial differe.nti~ ~qlJ..9-ti.on .. A short ex:tens~on of the use of Laplace transforms provides a convenient means of solving ' this type of differential equation. An illustration will be the .pre~.iction of temperatures in a wall, shown in 'Fig. · 15-47 ,in whic'h originally the temperatures ~lt both surfaces 'and the interior are To. At time t =='0 [he teinp~r ature of the surface where x :::: b is changed to Ts , 'while the temperature of the left surface remains constant at To- The govenring differential equation
.or
is k
at
1
aT
a
at
=--
subject to the T(x, t) boundary conditions:
and Originally To • then
Constant at
,
. 'changed to Ts ."- ."
70
Wall
.. ;.. :.--- ~.
T(b, t)
= Ts
. .
T
b
FI G URE 15-47 , Transient temperature distribution in a
-
.
waH, '
.
,
414
DESIGN OF THERMAL SYSTEMS
.
.
For convenience normalize the t,e mperatyre by defining U = T -
To~ so~at : --.- .~.
, " (1563:~)', '
---,-
,a
:-.' ~ :
,at
.
'.'
.'.
-'-'
.,;
- '
- ,-
~
.
.
. X .
.-
L
-.,
~
.
. T- To = x + :2 (_l)n e-Cl:I,2r,1,tib'sin(n?l.,) . .... (i5~40r , Ts - ! 0 ',b ' 7f n =:= 1 n b I ·
,Th~' in~ersion -techniqu'~ is', a general' state~en( of 'one of th~ IP~thods :cited '
In 'Se¢'.,
15. S:
where aj is .a simple {Do~ep~ated) pole 'of, D(s). An example of the application of Eg. (15.~1) is to invert (s + 1)/(s2 + 2~). D(s) == 5,2 + 2s so D'es) == 2s + 2. Thepples arroots of Des) are·s '= 0 ands = ~2, so " t
-
r
- I[
S
+
1
s2 + 2s
l'=
J
0 + 1 01 e · 2(0) + 2
2 + 1 _., ,-1 ' - 21 + 2( ' --2) e - = - 1 +e + 2 2( ) I
The steps in the solution of a ,partial differential equations are as follows: 1. Transform one of the independent variables in the differential equation; for example, transfonn t to s. IntrDduce a boundary condition, if helpful. The resul t is an ordinary differenti,al eguation relating u(;;, s) to x and s.
.
2. Solve the ordinary differential equation. If the form i5 a familiar one" sin1ply write the solution. O.therwise, transfonn x to its own Laplace variable, perhaps t1, to derive a function U (4 ,s). Solve for U (d ,s) and introduce either constants· or -transfonned 'boundary conditions directly, then invert d· to .x • 3. By either of the techniques in Step 2, an ordinary differential equation for u(x, s) is now available.- The, final step is to invert s to f - a step that usually entails the most /!ffort of all. The individual steps will now be executed on the prob1em at hand.
-.
. Step
~.
Transforrn
t
to.S:
,l ' . . '. 'Ino(.l:, s)= ,( ![s 'a(x, s) ~ U(x, 0) ]
a
. One of
the boGnda.ry sond~d6~s in Eg. · (15.3 9) indicites ~h.ar U (x ~ 0) ',;':'". :~,:. :~' :"." .:~.. ;.~":: " :~'1. ;~
...'; . ·::;'-:'.ISO"" . ...~.. ,~ '~ ... :.:.:.~~."I_ :~, :":'...:. :::'/:'>::;;"l. l:.~f,
:::!..
... '.
.
: ". :' .•. .'.:... . " -
'
. : . :.', ,", " .. . ,..
;'
:' . . , , LL::.~·C~'·:;~) r~ T~'111'(~~ .:.;). :~ 0'"
~
"
~-
. . .. ~ .
.
•.... "
... .
','
7
.' " 0') "
.: . . " ': ,'. ' j .
.
.'
"
'.
.· (1~·.4~) "·,
' {]! -'
Step· 2. ~ .
. The ordinary differential. equa.tion, Eq.: (15.42),. is . a .familiar. one ~vhose '
solution 'is'
.
' .
.
to the transformed ·boundary conditions U. (O~ s) =
.
0 and u(b~.s) = (Ts - .To)/ s. Application of. the ·fIrst of these two boundary conditiC?DS yields ..
sU.bject
".
-
"
u(x, s) = C~sinh(x vis! a)" .
and from the second boundary condition u(x, s)
Ts - To
-.
(15.44)
s(sinhj ;b)
\vhich is the function that will be inverted using the form ofEq. (15.41). Step 3
A series representation of the hyperbolic functions will repeatedly appear in what" follow.s so for future reference, recall that I
.
slnh y
.
1
= y + 3! Y
3
cosh Y
and
+...
', .
If s
= 0 is
D'(s)
- ' .... - '.
sinh.jfJC
s~o = 51anhjfJ- b . +~. --cos s · .
.
.~-
substituted into N(s)!D'(S). from (15.44)
Ne s) I .-
.',
1
= I + ,),2 + 2.
..a
2;,fU:
/"
hJfh- s~o a
... (15.45)
,416
DESIGN OF THERMAL SYSTEMS
If the SiWl and ,cosh functions are 'e xpressed in the seri-t;s of Eqs. (15.45) and ' .JsJ a canceled, the result is ".' I '. . .' ' .--,' .'
'
.',
~
..
'
.
p O.
,:..::;: .: ::,·;:.~~hich:·r~ay.look"rdaso~able,.-b~t is V\rro~g. 'The":reaso.rifot"the.. error .is. ~,~( :, ' :~~~'« '<~~~;:;.:.; () ~.~. a .. ~ompou~q._:P91~'J:. riot.:·a,: ~Imp~~~-pqJ~
' :J' ','
.
.....
":'.' .
'.
.- ) ,S
-
x
.
3.
+ 3"r;x ,+." '.
, -' ."
. ' ' ., ...L':' 3 " S (b +" 3! a b . + . - .) ~. is' a simple pole and
Now
N (s) 0/
'e DI(s) . - .
X
=,-
b·
.
-
, which is the expected ' steady-state ·valu~.. ." ' . , The other poles exist.when sinJJ.(b .Js/ a) = O. .,Recalling that i sinz == sinh i z, and that sin z == 0 when z = 0, '7T, 27I, .. _ , n 7T, then poles exist . . ""'hen b-vsl a = 11111" for 12 == 0., I, 2, ... Thus, , . ~
0'
S
n 21T2 Ct b2 for n == 0, 1, 2, ...
=
but the 11 = 0 poTe has already ·been accqunted for, so the n values of . interest are 1, 2, 3, etc. Applying Eq. (15.41) to perfornl' the inversion,
Substituting sla
-
r- 1f } l
=
;
In '!fIb
_ .~
lranSlent -
'.,
L
11=)
'
sinh( in7T t)
s~nh(in7f)
+
.
.
(~)cosh(iJ21T)
e
-Jl
cc
p_ J{ -
- .
}
•
Iranslent
=~
i s j n(12 1T X I b)
+'J (. n 1T12) cos (n 'l1' ) e liD } L
0
-/1
217 1at / bl
2 7T\XJlb'1"
DYN}-.~AIC BEHAVIOR OF THER..vlL\L SYSTE;\IS
,.
h
l...
1 -
•
-
l'
,:WHIC!1 'IN ,en COWDH1:::G 'VIr.1 HJ.~ steaClv-st~\~e DJ..' O[U(,f.J •
I
•-
."
j
.,..--
u
-
/::"(J1..' _
f ~ ~
Q i..J
-1 C\
':-1"0]. I'
.417
21ves [nc 7
""-
complete solution" Eq. (1.5.40). l One of the streng'ths of the Laplace tranSfOfo.1 approach' to ,solving prlrtial di~ferentiaI equations is that. as was ,true for the' sol u.tion of ordinary , . dltlere:nt~al eguatib:1s; the s9Iut'ion is a methodic31 process that qoes not , " , delnand special i~1~igh~.. PP:~,o~' .t~~"'-~fi~~V\l{Ltl+~_~ :n -P1,~ ;)UP.1HlatioIl terrn,.fGl1o\~J
"\"qlr,ectIy. from'Jhe 'z~r6s' .g~'teiniine'CfihJ-rn ttlgnornetric fUEc,tiOriS~' - "-.. ,";' ",": ",: . .. , '. . . . -
...
-
.
.
~.
' .
.'
'..
'.
;'" "
15~24
.~
.:' .
- SUIVIIVIAR.Y
This ~hapter 'addressed ,numerous' 'topics related' to 'the dyna,mic b~ha\;ior andcontrot of thermal· systems. 'Th'e intent has been remain close 'to thenn-aI hardware and use' mathematical tools of onlv:moderate difficGlrv to provide more ' ~nderstanding bf the ~esp'onse of actual ~Yst~ms." The chp.pt;r', also sought t'o provi.d.e pract~ce in' :t ranslating physica.1 equipment and systenlS ,t o a symbolic fonn ,that- can 'be attac;ked by the mathemati"cs .of control
to,
to
s'ysren1s.
PROBLEIVlS lS.L Compute the time necessary to raise the pressure of air in the storage vessel· , from 200 kPa to'900 kPa 'with the two-stage compre~siori system sho\vn in Fig. 15-48. Data and conditions in addition to those shown in Fig. 15--1-8 are as follows: Adiabatic compres.sion efficiency of both compressors 7]c
=
isentropic work , (100) = 75% actual work
Volumetric efficiepcy of both compressors i]v.
volume rate measured at compressor suction %=. . (1.00) displacement rate of compressor '
~ 100 - 12[ (p'd~~h/Psuclion) - I] Displacement rates
low-stage compressor. 0.12 mJ/s
high-stage c~mpressort,O.07S m3/s Properties of aJr
R:Jir = 287 J/(kg . K)
- ..
cp
= 1.00 kl/(kg
CV
= 0.7J2 kJ/(kg . k)
. K)
418
DESIGN OF TBERMAL SYSTEMS
VA =052kW/K.
0.15 kgJ$
Water
2· . 3 . " InterCooler. ' ,
:. / "
~
. 4 " "--_ _._-....J . Aftercoofer .
,
. 5 -
HIgh stage
. .. .
•
-
I
'
, ~- Storage
.. . -.' :.; ',. '-vessel ' ',' ',-, '.' "
' . .' ,
;~:,\: " \ ~,-, . < . .~ g~fr:~~~a: £/<;:':'·r~ . F~nal p
FIGURE 15-4$ .
Tw~stage air , comp~ssor i'n Problem .15.].
=900 kPa
..
1
."
AssmD.'e no heat transfer between the environrrlertt· and the compressor, ' ' , intercooler and storage vesseL , The principles of fillmg processes applY "to air entering the·, tank~ namely, the temperature of air after entering is 'greater than before entering, ',the 'tank. ' The air at point 1 is drj enough' that no moisture condense's at either heat exchanger. ' Use a time ' st~p of 4 s. A.ns.: Some values ·whe.n the tank pressure 'has reached 900 kPa: T~ == 130.90'C, 13 ~ 27.4°C, T t.a.nX = ] 44.,4°C, P2 = 21& kPa. 15.2. A recompr~ssion unit servjcing a liquid amnlonia stoql,ge vessel, Fig. 15-49 consists of a compressor, condenser, and expan~ion valve. The purpose of the recqrnpressiop unit is to draw off vapor and thereby cool ' , the stored liquid. The vapor is condensed and passed through a thr.o'ttling val ve back to the tank. 1
9
UA=12kW/K
Water.30 D e
Flow ratc:
AmmoniJ Liquid 15.150 kg T rrn&, = 35°C T (in;IJ = 10°C
=5.6 kgls
Condenser
Power Compressor ~ ,-.
Throttling
3
Saturated 'liquid
valve
FIGURE 15-49 unit for reducing the temperature of Jiquid ammonia in a storage vesser. '
Re,Cl)mpres~jon
- .
.::i';.9
DYNAMIC BEHAVIOR OF THeRMAL SYSTEMS I
:n ~he process under consideratio'n j the temperatu.fe of the 'at1unoni:1 ,Jiquid is to be hrought down from ,35 9 C 'to' IOoe. D,sing a, stez.d.y st~tc , simulation plus one dynamic element, deteolline the tilDe required ,for [hi::; red~c~on in temperature. : ,Y'
;:,.,~ ,:" -'" ,"
,'",;' :; " .!
. './", ,.- . Co m pres.()r,,~cte~~~';;' ":'''.''_ ".' , .. ,..~;" ,~~,~,,~' ';,.',: ~~,':',:;~,;,~:,' ....•~... .; ".''-
',"
" VoI~me"rate.Jlo~"measll:fed at)he ,~6mpress6r suctiqn' 81315:'
0';"
';":.".'
.' . , "
:",
, ' [(' ) 1.2 O_OI8':"O_003;~ . - I]
V/ork required compressIon: '
.
by
.
the comprc;ssor which LS absorbed by the vapqr durin'2 . , ' . ,
'
~
"
-
Condenser' heat transfer
Compute the log-mean temperature difference in the condenser as though the ammonia were at a constant t~mperature (the condensing temper:lrure). , Ammonia properties as a function of temperature,
h of saturated
liquid, kl/kg
h of saturated vapor, kl/kg
°c
= 200. + 4.5991 .+ 0.004177t 2 = 1461.5 + 1.125t - 0.0103(2
specific volume of saturated vapor, . m 3 /kg ,=0.2854 - O.00826t
saturation pressure-temperature relationship, where
In
= 6 1_
pl.
Suggested time step
+ O.0000776t~
p is In kJ!l: 2740
(t
+ 273.15)
= 200 s .
.,' Ans.: Two variables for checking at final state: TJ = 37. 7°C. mass flow rate through the compressor, 0.0596 kg/so . t'S.3. Substitute the following functions of time into the definition of the Laplace transfonn, Eq. (15.8), and in\egrate to verify the transforms shown for these functions on Table 15.1: (a) ~/, (b) cosh(at), and (c) cos(at), making ~se of coshUat). 15.4. Use trigonometric identities and transforms from Table 15.1 where possible
to verify the following transfonns:
(4 . C~sin I + 2 cos I}
1 + 2.r + J
- ~2
420
PESIGN OF THERMAL 'SYSTEMS
" 1 (b)- ...c{sin(t) COS(t)} -=" 4
' s- +
s2 +,~2
J: {co,s '\at)} ~ s~
(c),
+.4sa 1 '
; '::: ."F.', 1~ ...5~ U.se p~rtia1 fractions rujd Table i5.~'llwhere possible' to sQow 'th~t:~.:':- ':',
~;; ~· '
:>::.::-.,.. - ••" '(a) ·.•
'p~i.[ js~ ~ 4;;t~~~::;.;~~J: : .,~:t;.:t;:;;e~:' ::f~:~~;~:~',]. ~':-~; '· " ':" <~ . •.
,(;).L,-l{ 65 2++ )~s +~, 20} = 4+2e~1 c~S(2~)
,.-
.
. s3
2s-
.
5s
' . '-~-
" ,
.
'
15.6. S~lve the following diff~rential ' equations using Laplace tr~siormations:. ' . (a) Y'\t)
+
2kY~(t)
+
k2Y(t) :.-.- 0 '
.
: (l?)
:flier) ' + ,4Y(t)
+ C3 t )
= sint with
Ans.: . Y = (113) sin 1 (c) yll(t) -2Y'(t)
Ans.:
I '
"
.. ' = ' e~kl (C1
boun~~ 'conditlons:'Y(O):~~ 'Y"CO) ~' O
(1l6)sin(2t)
-
+ yet) =0 widl boun~ary condit jon;: YeO) =0,
Yet) -:- 21 e(h1)
Y( 1) =2 '
'.
15.7. Solve the follov..ing differential equatjons using Laplace transfQnnations: (a) Y'(t) - Y(r) = e- I with boundary conditions: YeO) = 2
Ans.: Y(r)
+
(b) Y"(X)
sinh
1
Y'(x) = 2 whh boundary conqitions: YeO) '= 0 and Y'(O) = 4
-
,
Ans.: Y(x) (c) fll(r)
= 2t! + = 21' +
+,Y(t) =
f
Ans.: Y (I) = I
2 - 2e- x
.
with boundary conditions: Y(O) = 0 and Y( '/T12)
+
=2
(2 - rr/2) sin I
15.8. Electric current passes through a small cylinder of semiconductor material. shown in Fig. 15-50. The following conditions prevail: ~,
. '
"",,---...-- T=' 20°C
t ----.
C7'"'
K = 1.5 W1m •K . p = lO-'sn·m, 11 = 12 Wjm2·K Ambient temperature = ;Wo C T= 20°C
FIGURE 15·50 Semiconductor element in Prob. 15.8.
421
DYNAMiC BEHAVIOR OF THERMAL SYSTEMS
d.i:llUeter o:~ cylinder. 10
111m
length of cylinder, 40' mm ,: thermol conductiviry, k _= ' ·1.5 '/Il/m - k
current, J =:: 12 A , eleci!ical fe~,i_s~yi.o/.' .p. ':>-::.-•.\/ ./":':,:,. :""\ I,
-. '
= }lQ-s ,Q-I!l .
.
,
.
conve~.iiQZl
-'cyllndel
coefficient fo-r heat transfer-.frotn::,ext6narsiIrfa.ce..~6.f.~the ,: ,aiI ~.-iz':'·~'~12~·-WJiiJ?' -;."!-.. ..:. . ~:.. -:.....,: ...". -( , ". '..:. ,,:-:' ., ~ . ~ . J •• :
:. ;' The' teu;ipetatllr~s.__,o{both ·ehd~
,
"
2.mbi~nt·
.:
....
cylinder· . a;,.~ · 20°C7 ~arid, ·the·. .. of Lhe material is p (length in .
.of·the
aIr is also at 20°C
' The electrical Tesistance 'of a ,length· d-irection of flow)/( c~oss..:.s~ctional area). ,
.
'
(0) What is the' differential' equation. (including numeri<:-al coeffi€ienrS) and ' . boundary conditions relating the teJl1pera~ure to the distance x .along . the element?: ,' ", " . (b) Laplace tran'sfor~s, solve th'e differe~tiai e.qllati~n to develop the expression for T as a function of x., . . Ans.: At the ffiidlength, T = 40.2°C . . '15g9. The temper~ture-sensing bulb i~: Fig. IS-Sa is s~.bjected to ~ fluid tempe~ ature: Tf that varies sinusoidally according tq the equation
r
Using
Tf - Tm = !1 sin(21Tft) , where Tm is constant. 11 .. = amplitude'
f
=
frequency
'.,
(a) Determine the transfer function .L'{T - Tm}I....tCTf - Tm} if at t = O. T = Trn ... (b) What is the equation for th'e temperature T of the block after the process has been in operation for a long time? (c) Develop two graphs-one of the phase shift in degrees and the other of (T - Tm)/i1-against 271fr where 'T = mclhA. Graph 27TfT as the abscissa on a log scale over the range of 0.1 to 100. · Ans.: to (b):
T - Tm
=
.!l
)1 +
sin[27rft -
tan-l(21Tf~)J
(27Tfr)2
15.10. The temperature-sensing bulb illustrated in Fig, 15-Sa is SUbjected to a fluid tempeI1i;ture .Tr that varies linearly 'according to the equation. T, = To + g t , . where g is a constant. , (a) Detennine the expression for T if T = To when / = O. (b) Sketch the variation of T, and T with respect to time.
Ans.: T - T. =
8[
I
-'(~)f1 - e-'/CmclMlJ}
15.11. A room heated by warm air as shown in Fig. 15~51 has a volume of
..:n:s
ttrl and is supplied wjth 0.36 kgls of air. The air in the system is at
,"
,:. " . ,: . . :
-422
DESIGN OF THERMAL SYSTEMS
Fan .
3.8~W -
"
0.36 kgfs~ 38°. ~
':<:,,::~, '=-: ·.;·:~r.:::· "
O,r.
7
Rbom -:'...j.. ;. .'
Ii.
,~,~? _~~in~~~a!¥r~' r
.--.. ....
'J~)'i:;'l}{,~':j},ff::c~;.: , .: : : , .''c::; ..•.":'.'.~.',., ': :';" ".•',;i':; '.~~':~
•."',' ,; . . :::.' ':: . ',:,•.' "
., '.:
. :.:,.,:.':Heating syste~' in' piob:.. IS.J I.' -
•
--
'
t
, ' -/- -
. '
-
.'
j
"
'-'
. '. a pre~sure of 101 kPa its cp = 1.0 kll(kg . K), ,~d R, -:- 0.287 kJJ(kg· K). " The heat is ·supplied in an on/off maimer in that'a thermostat switches OD. an ' . electrlc h~8:ter w~eri the _room temp'e rature "drbps to '24°C wher~upon 3SoC' . '. air immediately b~gins entering the room .. When ,the room temp'eratun! rises " , to 26°C tpe thennostat turns. off the heater and the temperature of the supply . 'air drops immediately 'to .th~t of the re~rn air~' ,1qe ~eat loss from the room is constant at 3:8 k\V. ' (a) How long 'a period the 'heater on? (b) How long a period is the h~ater off? 15.12. For purposes·of dynamic analysls, a steam coil that heat~ air is considered to be a mass of metal that is at a uniforID temperature throughout at any given time, as shown in Fig. 15-52. The magnitudes of some of the parameters are as follows: -
is
hA = product of heat-transfer coefficient ;:md area between steam and coil. 5.7 kW/K .
C = product of mass and specific heat of coil, 95 kJlK D = product of mass rate of flow and specific he.at of air. 1.9 kWfK . The effectiveness of the coil in transferring heat to the air is 0.5, thus (To - Tj)/( Tc - T,) = 0.5
Tc = Coil temperature
Air
To
=Outlet air'temperalure FIGURE 15·52 Steam coil in Prob. loS.l:!.
-.
r
DYNA0.IlC BEHAVIOR OF r.\SR~1.:\L SYSTE\lS
423
If at time ( = 0 i:be steam temperatun~ T$ experiences a step ,::hange from . 65°C to 95°C, what is [be equati'on fori the ourlet air tcmper:J:l..ure ~) as ' 3. function of tin1e'? . ,'1 ,~·s ~ T. = 46_.. ,.13 - I'...... .. 86";-0.071 -CJ... 0 . I'll
0 ,"
~
t..
,
fl,5D13·g · Inven the transfer fU?f.rio~. fOf the. cascaded ti.n~~.~ CO:JS!:arit blocks . Eq_ (!? ..19.), subJ~~t;lQ \~¢ .~·?~P.j,i)P.':lt .t.o .:de\f~,loR. .Eq 'J>J;i,5.:.2J)·~:fo.t·.tb.e.ca.se...\v.htre,,4"" ':"'.' t ;·TI ·· ~7i·."···.~· . .. ...:.. . ::. . . . ,..... :.. . .. , : . ' .. ':: ... . .; ". ,,::,
;.:._.'
: ' , :•
•
:.
"
: ,•• _
."
...J
i5~f4 ..· ·The'. h~.~i.djty is c~ntrop~ci in a 'labor~torY' that '.' . . ,
uses. IOO~ ,p~ld~or ai'~
by ' . adding water vapor as ·needed~ In the .syst(;m shown in Fig. ,1..5-53. [he humidity sensor in the 'exh:..lusr air measures the humidit}, rario.H . which is J:he kg o(\vJter 'lap'of Liss.ociated with a kg ' of dry air.' The humidity sensor. has a time constant of 30 s. The controller regulates the power suppl.icd to the electric heater iri the humidifier and th'us the Tate of water vaporiz~d inw '
'
me air stream.
w~w .. according h\",\.•
[0
kgls =
the equation .' 300(H;~;
. .'
- ' Hsensed)
'-Due to the themlaI capacity of the heater and the ' \vater reservoir. the ." res'p onse the "humidifier is nor instantaneous, bu[ has ' a time,qJnstnnt of
of
'4. min. 'Assun1e ~h;i the e~ fering .ou·tdo01: a!i contains neiii·gibie:··:~~;~·te.~· ~·i.l·por.
'"\ .
and [hat ('he mixing of v,:ater .vapor throughout the laboratoI)p is: unifom1. Use ·.trequ.ency response and the principles of the Bode diagrJm (Q detem1ine- (a) the frequency resulting in the loop phase lag of 180°, and (bJ the combined loop gain when the loop phase lag is 180°. (c) Is the control stable or unstable?" . -Ans.: (b) 1.2 . 15.15. The expansion valve regulating the flow rate of refrigerant to an eVLlpo~ator in a refrigeration system is the type that controls the number of degrees of superheat at the outlet of the evaporator~ The control loop is represented ~y a combination of time-constant blocks. as shown in. Fig. 15-54. The
Laboratory Hour
MilSS of .:lir::::: 4500 kg plus
~
Outdoor air_--'L..-..........
negligible humidity . Hsen!led
FIGURE!
1~-S3
J5.1 . t
:lSSOCial~d w~ler
vJpor
1
Flow r.lte 7.5 kg/s
=:
of nir plus __-H-i.... n------ lCisociated I'..J'-"-~rl-J'o-~""""'" Humidifier water vapor
424
DESIGN OF THERMAL SYSTEMS
~
Superheat, 0 C setting
Superheat,OC
kJ :::: loer.e "
.--,>
"+
1:"1
differepce
) Bulb .
= 3.s
sppem~at, DC
k2 kg/s perce 12 s . ~.:
-
1:2 ::::;
Flow, kg/s
r---:t
G = = _ " = " ' _....... "_~
2. :pquid in ~ulb. "" .
.. L Bulb ofv~ve
. _" $tem~seat
of valve -'.
.. / .
. " Block .
.
di.airam of ~vaporntor-exp~sion.v;i}ve control loop in ~~1:>. '.
'..
.
'
"'.
..
15.15.
-
. .
. characteristics ·o f the evaporator and certain characteristics of the valve' a{e> " fixed ·and :designated"·in Fig". IS~54. . . '.. , "U se · freq·uency response and the . principles of the Bode diagram to·· determine the maximum permissible gain of the valve, k 2 , jn orqer to make " trusloop .stable. This gain kg/s °c is pred()~nant1y dependent on th~
are
in
per
"s ize'·of the yalVe port. Ans.: 0.028 kg/s per °e. 15.16. The transfer function of a feedback contro'11oop js .
S3
14s 2 - 2s + 3 . .+ 6s 2 - 59s + 156 .
(a) Use th~ . R6uth-Hurwitz criterion to determine the n~ber of roots of the
charac;teristic equation that have positive real parts.
.
(b) What are the roots of the characteristic equation? . Ans~: (a) 2
15.17. The control loop shown in fi'g. 15-55 consists of three time-constant bl~ks. (a) If '1"} = 72 = 7"3, use the Routh-Hurwitz· critedon to detennine the product . klk2k3 resulting in the loop being on the borderline between stabHity
and .instability ~ . (b) Choose any arbhrary combination of nonequal values of the 'j's and
compute the k Ik2k3 for b
. FIGURE 15·55 Control loop of three time·constant blocks in Prob. ) 5. J 7.
"-
.
DYNAI,-,IiC BEHAVTOR OF THERivlA_ SYST6-'lS ~
ca,~
425
combiJlacioD is when all 7'5 ,,-re up.a!. S",melimes a loop be maae more .1 stable by spreading apan the values' of the 'T~S. ~~5~18_ The ten1perature "Tc of an ,environmental chamber and if:s. interior ·[BaSS., ,' .1 sh'own 'schematically· in F.rg . .··.R5-56 v . is regulated by a controller ilia[ adjusr~ . elecLr1c :power. input, qhb The theffi1al ' capaciry Lh~ interior ' , the rate mass of the.' cham~er is j il ~dfI(J~· m1d. .its. te~flpe~~f.!-f~~\,.Tc"~$ ' :J;~S~.i~~p.; :.t.o . be . . " .- .~" ':--'un ifami thIOH ghou·t~the. interior.:of...the."chamb.ep;,: bu bvariesd.vim:·~ timet ··The: . i ';," ,!, 'j • . . J .. i nst?:J.itan~·o· ~s .~t~'~' af)~~'~t -.:tmnsfer~'from [he cha.mb·e r .to"the, ?-iJnb~enr~ '.q ~'~ is ···, -" . represented by' the' equa'tion ' ' .< - "'. , : ',' ,._ ' . . .
of
of
1'
.
,
....
'~
'
,
'. . '
0
where T'J. = _a~bient [emperature~ . The temperature of. -{he' sensor is T's . and ti~e ,c~nstant is Ts. The ~h~acteri,stic of (he controller is tJh B (Tset .:.... Ts) . (a), (~ons~ruct the block diagram of th~ ' cont~o1. inse~g appropriate transfer functions and. using the ambient teinperatJ-lre Ta as the input and . th'e chamber temperature Tc the output. . ' , (b) Normalize the variables associated wi[h the block diagram in anticipation of determining the response ~f -T;: to a c~ange in T:l while T renlains
irs
toop,
as
Sf:.(
constant. (c) Simplify the loop to a unity or nonunity feedback loop. (d) 'Some numerical va,Iues are: A1 = 300 kllK. B = 5 kWJK~ H = 0.5 kW/I(~ and- Ts ,= 10 s. If Ta = 30°C. and Tsc' = 60°C~' what are stead\'state values of Ts and T/!- " . (e) If the am9ient temperature T'J. changes abruptly to 20°C~ what is the equation for .the response of Tc with respect to time? Ans.: Tc = 56.36 -' O.085e-D.078:4r + O.9ge- D.02344t 15 .. 19. The pressure in the evaporator of Fig. J5-57 is regulated by controlling the rate of vapor pumped by the compressor. We_ The purpose of the evaporator is to cool a fluid that flows through tubes immersed inlhe liquid. The control, is of the proportional type following the equation ' We
= K(p - P~t)
Controller
Chamber
FIGURE 15 ..56 Environmenral chamber in Prob. 15.1 B.
T~!
'1
426
DESIGN OF THERMAL SYSTEMS
. Mass of vapor . ..:
.. . .. . .
' . ".
.
' .
-
' S~t~ ·
._ "
,' Iiqu:id ' rube ~em'perature. TI ' Tf
FIGURE 15-57
,!hermal, cap~ci(y
Dr Jiq:uid,' J1 ~
.. ,
E\'apc)raror wi~ ~ontrolled pressure 'in ,Prah. 15_,~9. -- ,
, Some other applicable ,ielariQTIships 'are :as,follows: ~f
vapor in vessel, mv = 'Bp latent heat of evaporating fluid = h fg
Dlass
evaporation rate,
W ev ·
=
C(Pf - P!
vapor pressure of liquid, Pf ,= DTf thermal capacity of Jjquid in evaporator
=M
rate of heat transf~r from tube 't o liquid =hA(Tr - T f )
the state of liquid entering the evaporator is e,s sentially saturated liquid at
ternper~ture
T{
(0) ,Construct the block diagram of the cOhtrolloop with the tube temperature T, ,as the ,reference variable and the pressure p as the controlled variable. ~ndicate the ' variables (nonnonnalized) and the transfer fupctlons of the
components. (b) Nonnalize the variables. (c) Restructure the block diagram into a unity or nonunhy feedbac~. loop, and specify the transfer ~unctjon of the loop. Ans.: (c) 'co "
BMs2
+ (BhA +
D,C ~A " BCDh{g + KM +.CM)s + KhA
+ KCDh[g + ChA
15.20. The air-heating system of Fi~. 15-26 is subjected to a step change in TJ.e! from 42 to 47°C which yi~lds the loop tran~'~,r function shown in Eq. (15.27) ., The 'control chosen is integral only. with Kp ': : 0 and XJ = 0.02. (a) Develop the equation for the response of the controlled temperature T, as a function of time, (b) PJot T, in the 0-400 s time interval. . Ans.: (b) to check r~sults, T, first reaches 47°C in 113.2 s. ~.
.#-0 . ,
DYNAMIC BEHAVIOR OF THfRl'v1AL SYSlE'\,t$
, 427
of
15.:2..1~
?igure is-35 shows rhe response the controlled. temperLllure 7'~t: to ;} step change in Tset from' 42 to 47°C fo~ several di.fferent combinations of K ~ and K r • Invert Eq. (15.21) for d~e case where: !(p = 2.4 and j({ = 0'.1 tind c.ompare with Fig .. 15-35. . AIlS.: to check results, peak of overshoot is 48.377~C 'at 61. 1 '5. ' : .' -.15022.1' i\)ligh.~[yalue; 9f the d~ri\iativel-cons.tanE I(D ' thair us~d ,In. 'Fjg~ J 5~37 for(";:: ~: 'press ure- ·cE)ntrollei'f:·js.,:to,:be~explbred::'Compu.te:<-iuld-iprOi the' pres~'u re 'fbi :~:':: ; -.' -20 sJollow.hii~(the -O.:J'kg1s step i~creasej¢":6'utli=tllo,w)-:ate\vh~'n 1(p..- ~:.:n:~'2.~· "-, '-,' Ki = 0.05, and .KD = 0.4. ' ' .Ans.: to' check resulrs, ,max~irium drop ' in pres,S ,"Ire' .is 2.?3 kP=l"~':"
.'. ' ;'-',- -: . ;the
, 3.~7 s.- : I5~~3o (a) D,erive the loop ,transfer function~ Eq: .(15.32), for the ·PI-only cO.ntrol
. . of the water heater in Example is.I-I. ' .. '.(b),,11ultiply the. transf~rm of Eq. (15.32) by ,the transfonn of the input .. .-:-:-5/s, and then invert toyerifY the response of the PI-only connot shown . .9n Fig. 15-4l. (c) RestrUcrure the complete block diagran:I of Fig. ' j 5-40 (b~rh PI and .. feedforward control) into a unity or Qonunity feedb;~~ :Ioop -to verify " that 'the loo.p . transfer function is 1/6 mat given in ~q ~ (1S-.31)~ . - Ans.: a check,v.alue in part .(b) is 64.02°C C?-t 300 s~, ]5.24. A ~ating coil is mQdeled in Fig. 15-43 as a 'combination of;:1 time const:tnr and a transportation lag,_ In the Bode diagram' procec;iure for assessing, s~a bility of ~ loop, as described in ,Sec. 15.11, a time-'constanr block. is shown _ to have a phase lag that only -approaches 90°. When combined with dead time, however ~ the phase lag can exceed 90°. If the heating coil has a time constant of 20 s and a dead time of 3 s, above woat frequency does [he 0 pb.ase lag exceed 90 -? 15.25. Section 15.21 described the fIo\v characteristics of a coil regulated py a valve with linear characteristics. 'IJ1e ~quadon of the flow-stem position for another type of val ve mentioned in Sec. 15.21, the equal-percentage valve', is
~.:....c.Q_
= AX where
x
percent of stem stroke 100 -:- 1
=-
If such a vaJve with A = 20 and C y = 1.2 is applied to contiolling me coil' in Fig. 15-44 with tlpcoiJ = 2.5Q2 and the toral pressure drop across the valve and coil of 80 kRl what is [he flow rate when the val ve stem stroke is at the halfway position? (Compare with a linear-characteristic valve in t
Fig. 15-46.) . Ans.: 2.21 Us. ..., 15.26. An underground electric cable, as in Fig. 15-58. that is 5,000 m long is cooled by pumping oil through a pipe that surrounds the cable. Tbe oil has a density of 800 kg/m 3-. a sp~cific heat of 2.0 kl/kg . K. its flow rote is 8 kg/s or 0.01 m 3ls. and it enters the pipe with a tempenlture of 15°C. The . cross-sectional area for oil flow is 0.02 m 2 • A It the oil in the pipe is initially at ] 5°C when at time zero an electricaJ load of 50 W/m is imposed on the ,f ibre· wl1ich must be absorbed by the oiL
DESIGN OF THERMAL SYSTEJvIS
"'!!·.r .::;
Ok 1 on a
T - x
o
iJ
: (l/) -
q=
+
au dl
------~---------
20
".
() i ..:Trt1lt"trnm I ube i n
=0
+
+
U=T-
DYNAI\'UC BEHAVIOR OF Th~RMAL SySTEtviS
i12J
1~FE 11EI r ---'-'1'S
A; Lock, Transient Responses in a Large Steam Syslem. Petroleum MechanicaLEngi- , neer,ing ~Vorkshop· and Conference. ASlYiE. 1982. . 2. C. L. "Wi~son. uFan-ivlotor T~me-Torqlle Relations .• '· Heating. Piping. ,DIu} Air Condirloll, ins. vol. 4:3, no. 5. pp" 75,777~,.May19~~ ... : ";','-'" ,.~-:.... .~ ,.' , , .. , .' ·3. p: F: Baker. "Surge :Co,iltroi fot' Ivfultistate:.Centrifugal :Corripre~s.or;s:,:7:, .c!~~V!icql . .i::~~g,i:_"
1.
.
-~
.~ :" ~
."T,
. . ,,:,·,;/~;,;·"h~ering>J"·y'o1-:"'B9~; ·ne';:·· rt" ·PIN"l-1.7~:f21'::"Ma:f31: 'I'982:'~"":, ':",:':' ~. ;,; ;..:,:,.::~........,',.:",., ,,'
, ...... ? ... -". .',
,""
.,: ~:" .... 4~ ' F:.G.: Shins~~y' ~ 'Pjo(:,ess-r:~litfdi~ Sj/sretns, '24~' e~:';' Mc.dra\v':1:fill'~ N~~;';-·yoj-k. 197'~~'" .: .~. , ',' .. ,' :5. P. O. Danig nncfB. Teislev, <&A Generalized 'P rogram for Dynainic Analysis ofI\fon-line;:lI~ . S yst¢ms," RepQrt ' F29-77.0 I. Refrigeration Laboratory, Danish .Techni~al University. ',1977. ·6. 1-( ~'yqursr, '"Regenerari,on ' Theory." . Bell' S~\'~tem Technical Journal, JanuillY 193;~ pp:' ~2&-147: als'o ,in AatomaticCcntrol: Classical Lhlear Theory, G.l. ThaIe~9 .ed .• ,.' pp, l05-~26.D,owden, Hutch~nson and Ross, Stroudsburg, PA, 1974. , 7. E. 1. Routh, Dynamics'of a System 0/ Rigid Bodies, IVlacrnillan. New 'York. 1892'. 8. H. L: Harris~n and J.G. Bollinger~ /nlToduction to Automatic Controls, Inte~ationai, , Textbook Co., Scranton, Penn.~ 1969. _ " ,.9. G. &. Sapienza. "Using Fixed and Dynamic Performance Curves to Optimize' HVAC EqlJ.ip'ment Operation," paper from seminar: '-'Using Your nDC for More Than P1D Control, ". American Society of Heating. l3-efrigefLlting~ and Air-Conditioning Engine:ers, . ' . . . . New Yo~k, January (9?7. ·10. D. L. Auslander. Y. 'Takahashi, and 1-.1.1., Rabins. IntrodLlcing Sypems ,alld ,Control. McGraw-Hili, New York, 1974. , 11. C. G. Nesler, "Direct t~igitarControI of Discharge Air Ter:nperature 'Using a PrOportionul~,/ 'Integral Controller," M,·S. Thesis in Mec.h::lOical, Engineering~ University. of ll1inois' at Urbana-Champaign, 1983.
-
..'
,
•
o
~
'.
.
,
0.0 . \ I
:.. '.
.'
,
0. .
~
•
"
~ " ;:
.
,
. •
~
I
."
: .....
."
""
•
:.
" .
'.,
~ ~l
°
"""
. I'
16.1 CONTINUED EXPLORATION OF CALCULUS METHODS The introduction 'to calculus methods of optimization was provided in Chapter 8£'$ coverage of Lagrange multipliers. The function to which Lagrange. nlultip1ier optimization directlY appJies is (16.1)
,
subject to the equality constraints, . , cf>1(X.,X2 • .•. xn) = 0
(16.2)
. (J 6.3)
-. .......,
430
baa
':',":
:"
",."
•
.
I
.
"l .
."~.
CALCULL;S METHODS SF OfTIN!IZA1l0~J
optirnal values of the x"s are foui1~ by the simultaneous .+ }~ SCalClI equations~ -some of \i1nich mdy be non-J.inear:
Th~ 12
sol ~'_ 60n
43 1
J;f the .
ill
.
.
'
~
'1y - / .
. "';.'
' . ' :"
_.
0" . ·
._(. ~~
.
..
.
j "", ""." ... ,......
.._-_ ._.-, _.___. ' " _.' .f-1.-.=_1
~
Ai" ¢i
:"'.- ~."-.: . : .. "
=0 :
.
"( 16.4) .
,-
, :-~
/.
.
'
'
.
.
-
-
I
_ . The ·method of Lagrange multipliers is us~fuI and. powerful and 'could .. ,very- well .b e the best ·c.hoice for many practical optimization p~oblems~ For large -problems the explicit solution .pf I;:qs. (L6.4) to "(16.6) may be prohibitive~'-but 'since' the task is one of solyin'g a se't .of nonlinear simuhaneQus equations, .the.methods of Cha,pter 14 applY'. This chapter also' suggests the stru~tur~ .
16.2 THE NATURE· OF THE GRADIENT VECTOR The condition for optimality expressed by the Lagrange-multiplier equation~ Eq. (16.4), calJs for gradient vectors. of the' obejctive function y and the ¢ equations. Three characteristics of the gradient vector are (1) that it is nonna! to the ~urface of constant y. (2). that it indicates the direction of maxi~um rate of change of y with respect to the x distance, and (3) that it points in the direction of increasing y. , . Seation 8.7 showed for two-dimensional space that Vy is perpendicular to the line Dc" constant y p'assing through the point. 'A more general proof is available from vector algebra using the fact that the dot product of . two normal vectors is zero . .An arbitrary vector of sma111ength with d;c's as
the components and [he fs as the'unit vectors is . I
-.
( 16~ 7)
DESIGN OF THERMAL SYSTEMS
(1
, r
Vy = 0
vectors are
to one
........................... at
.. +
----.. ---
(1
CALCULUS METHODS OF OPT1M1ZA.TION
(h)
- (0) -
433
'
, FIGURE 16-1 _:~' Ma.x:i~~rp.. chnn~e in y ~:he~ moving a di,stance r in (a)
hvO
diTl1ensions (b) ¢ree- dime~ions_ .-
om
multipliers yiil1 be' chos~n ·to perf the optilnizarion-- the same iuethod -' : that \vill b~ proven ip this chapter. ' , The multiplier equation' in the i di-rection is , -dv ' , -"-,- -2A..dxi == 0
ax. ",'
,
so 1 t3v dx-;=--"" , , 2"\' ax;
which w!Ien expressed in vector fonn is
dx li 1
+ dX2i2 +
or 1 [ -a-y1• 2A ax} 1
-
+
ay .
--1"> aX2 -.
+ ... +
1 By • J == -Vy 2;\ t3xn II
--I
Of interest here is that V y indicates the direction of maximum change for a given 'distance in the space. The third characteristic an~wers a question that. often arises in the -execution of an optimization problem: Does Vy point in the direction of increasing or decreasing y, and indeed is there any predictability? In a small move in the x I -'- X 2 space the change in y is represented by J
(16.11)
DES1GN OF THERMAL ,sYSTEMS
y
.-.:..
y "" +
J
(I
1
a 2 1
(
)
)- (
1• • •
n
----.
( 16.1
CALCULUS METHODS. OF OPTLVIIZATION
435
e-
and nest alv/3.)'s be posit ive if y* is a)JJjni:cnu.m, or alvvays negative if]?=;: :is:a. : mmcimum. Section 8.9. shov:;ed the conditi61J,S ne,cessa.ry for a 2 x 2 I-Iessian . fnatrix to provide a positive s tun .in Eq: (16.14). A Inor~ general description .of the condition ·is that' the Hessiar~ matrix QJ.ust ,be positive. definite. . ,·Eq. (16.-14) to be. uniformly positive', or negath'e' .deji'nile for Lh~ Slim, to);;e . . .. . . .,! . " . :. D.eg~tiv~. .... . .... ,.... . . \' . . <. ....... ' . , ' .. _.~ '"
'for
I
..' ..... : 1l1'e tests fo{',the ":e'q~~ti·v~e-qefii1ite
"
.
'" .
'
.
Gorid.itio~
"
,;:
of a syIn;metP9. matrix")' .
'alf 'the ·diagonal ·elements . an~ positive', ·arid. (2)· ' , , .-:thai.. all ·lecl{iing .principal d,i;;'gonal .determinants .are positi ve~' ·The. leading "- ' . s'uch .'as [Q], are '. (1)'- 'that principal diagonal
deteullin~nq~ .of~he rn~trix '
.[
-~ ],
i~
. .-1 ,2
4.
are .' ; ~
....
J2l
I: ; I and -I 2, 4
is
and since all of the determinants·are positive, the matrix positive definite . . ' The test for .a neg~~ive-definite matrix is to reverse the' signs of all the elements and test whether the revised matrix is positive definite. E'Xample 16.1. For positive values of the x·s, determine whether the optimum of the function
is a maximum, minimum. or not definite.
Solution . . By setting Vy = 0, the opcimal values are found to be: xf = 0.5, xj = 2.828, and .xj = 1.414. The Hessian matrix is 1 0.354
o
2 ] 0' 1.414
whjch is a positive 'defU1ite matrix, since 011 of the diagonnl elem~nts are positive, and ' th~ leading principal diagonal delerrrunants are 11.31, 3.0, and .2.831-all positive. Because the m'atri;t: of the second derivarives js positive ~e optimum is ~minimum. .
436
DES1GN OF THERMAL SYSTEMS
16,,4 CONSTRAINED OPTTh1IZATION
'0
Most important engineering optimiz~tion problems are ,of the constrained " type. While it is possible in principle to solye for on.e variable in each con,«,-> ", ,straiIit eq\lation and 'reduce' the ,problem to an micons~ed' 'optirrtiz~tion, -;, ,',';' ,': exec~ti6n this ,s~ategy may n-ot always be ,possible if the varIables' appear ': ':":,::' , ~',"',~:"',::,,' in, anirriplicit fomi In the "cons,~i~t$. FuTthem1o~e~ 'ilie-' aJ.gebra m'aY,becQrhe.:'::":~", ~.~:;.\~,~~#o.r;mi~ap~e;,.~bringj~K-fr~~}t ¢e:,, ~~~sibility ' of er:rorSA "The ',La~g~ ', muIii- .',:,:",:"'< ':.~' :',",;' .,plier"~quations·, ,' Eqs~"~~16:~A"Jp'~~ltf-~~: .~tHu&~~p-te,s.~YJ.~th~, -P'?$.js:.fo~ !he ..~olgt.io~, " ,~ . ,'
of
,<,
:, ·;'·:~,· .. ,,:·;::,;'-', of'p~acticai constrained gptiqrizaupn -p'ioblems'.:::~~,::·~~:. ::< :'.: .~, ;~:::{~::: ':'~;~;; :~:',~,~:'~":;',-::' :':'(~~:'·(~Y'.~>' ,~ : ,Proof~ of 'the v~lidity '.of ~e,tagcinge mult:iplier.equ~tlqns,~ppear_ori.ly ~....~'- " . 'rarely in the l1terarure& The next several sections .present',not a proof in the math.e matical sense but a de:r:nonstiation of lhe.,legit!illacy . the equations. ," The tru-get proof is that ca:ll~d the Jacobian method developed~ by ·Wi'ld.e ,and ' .' Beightler.l Rather than prese:nting the veritlcatioIl-,iri full generality whiGh' risks bogging down ilJ d~tails, the next several sections.procee'd 'Step-by-~tep , to progressively more complex situations-each time sh'o wing the soundness , of the, Lagrange mUltiplier equations. 'There are 'at least two reasons· for establishing the truth of the ' Lagrange multiplier equations. Fjjst, they are -', such classjcaJ relations that ':lnderstanding their background is a broadening experience. Second, these demonstrations offer the means of sqlving certain Pf:.?S.ticaJ. problems related to optimization. , , The sequence 'will be to show that the Lagrange muItip~ier 'equat.io.o:s apply.' to . the following cases: t\\'O variables with· o~e cons'trairit, three variables with one constraint, and three variables with two constraints.' By that point the direction to\vard full generality will have been established.
.. :,"
of
16.5 TWO VARIABLES AND ONE CONSTRAINT The procedure in the next seve:ral sections will be to analyze the first-degree tenns in the Taylor series expansion of the y functioD t shown in Eq. (16&13). When optin1izing y (x 1 ~ X2) subject to ch(x 1, )."2) = O~ the first-degree terms arc
ay- )LlXl + (aY'J - L1x~ (ax) aX2
(16.15)
The base point about which the expansion takes place is on the constraint and is the presumed optima] p6int. In this cons't rained problem I1x J and Ax 2 are nor free to change independenrly \ but must change jn such a way . that the new position conrinues to satisfy the constraint. Thus,
--.
439
CALCVLUS METHODS OF ~OPTIl\tilZATION
and .J¢ "(1 6~25) ,
'
..,~,., •
. ':-
:
-:~ .~- ~ "
I
1 '.
Defin~ .'
."_
,','
'7 '.~~;- .. '
""
.: .,. . ' ,
,
,"
-. "
.~
,
. . '
\
.
~y
A·..:.. " ax'}
,, ' .J4J aX1
or
Then fr.9ID Eq. , (16-.24) .'. . .
"
lnd from Eq. (16:25)
a.v _ A 81> ' 0 ,aX3
aX3
The above three equations are the scalar fonn, of the :quation "
Lagrang~ multipl~er
'
Vy -AV¢=O fhe Lagrange multiplier relatiop, 'therefore, holds for the case of three
'miables and one constraint. ,6~, 7
THREE VARIABLES AND TWO : o NSTRAINTS ~his
case continues the patterns set by the previQUs two cases, but introduces new situation as ,well. The assignn;ent is, to optimize y. where
Jbj~ct
to
y = y (x ltX2,X3)
( 16.26)
=0
( 16.27)
2. X3) = 0
( 16.28)
" ,:-4>1 (x I ,X2'-~ 3)
¢l(x J
f
X
.:.
If a point (xT.xj,xj) has been located which lies on the two coo'raints and-=". Ul .be tested 'at the ~optimumt the first~degree terms in the \pansion of y around that· -point are
"
,
DESIGN OF 11lERMAL S.YSTE1viS
(1
' . ; .]
(1 "
(
2 -
-----=-
-----...
'
(1
CALCULUS METHODS
df
441
OP'P}.1lZ.ATION
The ~xp;:-essions for LlX 1 and LlX2 c.an pe sdqstitut~d into the Sllm mLl:i:iop' o f firs>degree . terms, Eg. , (16#29), and regrouped. Then, D.Xj .~s factored ou t:.
r
,
, :,,:< •... :.. .... J
3
api
·~:X'3· .': -'~'. ' . r.pf.;:. q:>2 : ::i(;cjY, .~" ,
l
•
.
'.
. -:
I,.
1
-
'
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•
••
~
.'
~
'.
. '. . ' .' "." ' \.,, ' .. f:.. Xf3\~ ,.; ::....~ · .,:.:;. (J.6~··.zy!5' Qlv.".' ).. ..:
~i'~" ')V2 ";OX)':" .~:'-',-~ . ,:,
: .;, L , ..... , '" ~'(~ 1.', --riY ; ,;:~::<: . ': .: ',-a (x 1, X 2 ) '
" '.
1
l~ -.ay.·
J'
a.(y ¢d ' q(x1,.x·d ~a¢2. -. a( ";t.:..: ... ;-I:... )/......~'Y ,:J' ,,'
. a (y
'.
" ' > '.
.
. ' .•. , ..
. ,
.
~..
. ,
,
Defm-e tht:: co~fficients ,of a ¢1/aX3~and a'¢>2 i.a X 3 as (L i and>A.2;. 'r espectively_ The'x3:variable,is free:'to move" so: only \~ay,in which.~X3 ca~ change
the
either negatively ocpositi'vely, and ' yet prevent t~e entire expr~ssion frorrL , .. chlli~giIig y in ~ favorab~.e .rn.;inner.is fo[. the tenns enc;loseQ in,the bracfcets to be 'zero ~ ,.th1..is '
oy "
-" -
-
a
~l A ,! '-'-·"P._ ,
iJx3 '. ,'
' Eqll~tion (16.36)
IT XI
a '/"2-' "
ax}'
( 16.36)
ax,} ,
was the r~s~lt'when X3 was'decl~edthe decision ,variable.
were chosen as"the' dec]sion .variable. the resulting equ£:1.tion is "
'. and, when
X2
.
,"
"
, ' ",
av
.
.
_J_ =
.. ax I
is, the " ,
, c"
'
'~Xi-
"-,
" J ".I.. A i-''f/_l
,
..
.
a
-
ax I . ,
-;I.. J
A~-'fJ--
aXl'
.
= o·
(16.37)
decision·vari~ible.
iJy. _;\ 'i a
'
aX2
,
a¢~. - 0
(16.38)
8X 2
Equations (16.36) to (16.38) are the Lagrange m,ultiplier equations, except that it still must be sh~'\Vn that ~ A1
= .A i = A iI' . and
( 16.39)
Concentrate ,on showing just one equality, namely
AI
=:;
Ai
Equation (16.35), with x 3 as the decision variable, r~sulted analogy with x I as the decision variable it results in At:
In }lJ;
by
9
8(y,4>2) '
a (y , ¢2)
O(Xt,X2) a(¢lt¢2) a(Xl,X2) ,
a(X2,X3)
a(4)1. 4>2) a(X2,X3)
For two functions F(x,y,z) and G(x.y,z), a theorem from advanced calculus ~tates
that
dz
- ' "= dx .-
.
rJ(F .-G) a(x,y) a(F G) a(y ,z) t
",
:
DESIGN OF
.(l.L:.J.'UVAru...
SYSTEMS
:'
.~
':; ,"
J~
t'--.-/ t
': ,::
--------~
an manner..
a
y
1, x
(1
=b
(1
as an
can
1,X
j,X
= ...
h
(1
= 0-
(1
-
.occurs
--
- A[
)
CALCULUS METHODS Op1JQPTIMlZATION
t1,:.:~3
.1
.(
ana
(16.45) .. '!long' w:i.·dJ.··conforinity .,
to the ·cOIls·traint .. :,.,.:,.....:'. .
..;.. :.' ,-' :.. ,_M .:"
<.:
, " «/:." ';~c'~.'-"';_:_ .~ : -.·~'.> ~. . '.:»--" ...··;·¢(x l·:~~(~Y . :::.;"13". . "'. .' ...:'0"....<. . ,':. . ..,.::.:.' ~. -
. , ;.,,- ; -.. ..
",
,,,).'
... :.- ., . . .. . . .
"::(.16'.46)-,: . :..... ~ . . . ~ -' "
:.'
.'~ ~
. . :'
Equ~~io~s (16i~44)'Ao' (i6.46). aie··ti1~ · Lagra~g·e· · m·i.Iltipiie~equati~ns:~·
. but the fonn of the probfem as ·expressed in. Eg .. (16.43) ~/ili b~· convenient. · . in.· develop~ng' tests for· ·~.axlma and minima as .welL as developing the . . KUhn-.TuFker con,ditions· in .Chapter i7. c . .. .
in
I •
. .'
-
l
'. '
•
16'09 INTERPRETATION OF··A is · THE ·SEN.SITIVITY. COEFFICIENT.
'of
Section ·8'.10 desc.ribed· ~the sensitivity" coefficient as ,the.r~te of change the optin1a1. value of the ~unctio~,' y *, with respect to the change in the .. constraint constant b as it appears in Eg.· (I6.42)~- For a special case Sec .. 8.10 sho:w~d that the sensitivity coefficieni SC Qr a y.*l a b equals A; A mor~ general v·erification, at least fo[ tWQ variables a~d one constraint, as. in Eqs. (16.41) and (16.42), is provided through the chcdn rule: .
(16.4.7) Also,.
cJ¢ = a¢ oXt..+ ab ' ax 1. .QJ;
tJ¢ JX2 _ 1 ~ 0 ax 2 ab
Multiply Eq. (16.48) by A where in one -case
A = (a y */a x i) (a 4>18xf) ..' "
and -in the other (ay*JJx!) (a¢rax~)
Then
,
.
By ax,·+ oy aX2 ~ A = 0 ax) ab aX2 ab . '.vruch when compared to Eq. (16.47) shows that
SC = A
( 16.48)
.
DESIGN OF THER..t\1AL SYSTEMS
(1
o
a" ' a
- - ' -A a~r i
)
, and
...... --
CALCULUS METHODS Of- Ol'T11Vl:IZATION
~!-L~-5
SYSTlElVl has 'been -adc;lressed by several a~thorsl.,J.~ _fqT , _\vhich L agrange intiltipliers seerh~ to be a npiuraI :hok:e 'is, the selection , , -_of dU,~i diaIT1et~rs for a muitibJ;aIlch I'air"supply ' :system:: In' ::~":'Sys.t~ni ,s,L1;¢h~ ,~ , ::: :,.-,.as,';o:s'ho:p.Yll: ' iIi." }?ig :_:~:t6-:.2 y!·,/the,~" .gep,me.tric,--,Jay()llt;';~itp.e~' re'qqife-a": flb~' "ia'te~" .a:t ;.',"',;'~ -:each quaet~ a~dithe static -pres'sure','at'the Jatl '.p,utlet,'afe':sp~cli}ed. , 'The taSk is; to -,s~1.ect the duct diameters ,ot' (0-:-1,- ,'.1:"'2; 1-3~' ~-4; " ~d , 3-~) s'uch that the total' first c'o st is' a minimum~ Afirst app~oximation is that '-t~e cost :of a section '- of du'ct is proportional t
, "A,D, opLiInization problem -thsx
... .. ~
'each se<;tion
C = k O-
1 DO-l
+ k l - 2 D [-2 -' , ", : ' -t-'k 1- J D l - 3 +
k3-J,'D3~4
+
k 3-
(16.50),'
5 D3'-5
, \vhere the k"s embodY ,the"length of theg~ven s'ectiofl. 'The constraints must _ sOrl).eholv ,i~sure that th-e required f1o~ rates, QA, QB, and-'Qc are suppliecL The underlying philpsopJ"iY-'of the se.t of constraint, equations, listed a.s Eq-s. ,(16.51) iis ,that al(ofthe available stati~ pre~sure, SP,,-is used to overcQme friction in the straight duct. an~ fittings: " , ' ,
)
I1PO-l
+ ~p 1-'1 == S p, ,'. + Dp 1-3 + -IlP3-4 "=, S P
D.PO-l
+ bPl-3 +
tlpo:- '1
, (16.5~)
D.P3-S =;. -sp
The physical implication of Eqs. (16.51) is that even though dampers would certainly be installed in the section leading,..w each outlet, the intent is tha,t these dampers could be left wide open. A collateral advantage of this optimjzed d~ct network is, then, that it should be perfectly b.a1anced and not need final damper adjustment.
2
Atmospheric
pressure
SP 'It:ric
o
FIGURE 16-2 Optimizatiau at a- InuJeibranch air ..supply system.
3
4
DES1GN OF TIi.ERJ\1AL SYSTEMS
can· -115 -115
11.5
+
are
==
- .. ~
Oi-j ....... 11
",.
.447
CALCULUS M..EITiODS OF OF'TIMIZA.TION
1,-r./m3 .. Q - I = · 2.Lc · · ." . 3 1. f,·r- ev ::;mY) t .1~., '\}Jhe"1 1.... 'L"he ""p C . •' p-O·""·-1 5. values are' substituted? the staterI?-ent of the ~p1irrrization problem is as follows: . ,
Th~ 0._1 "1j.L.
"
. _
den _ t . <;"1" -hV ,J tn) -
.Cost
·. J
;.j ?
1 ""-
J'\.5.~
_O~
G . ..
'J
J .1V_
...... .... . .
~-
•. 1.
~
j \I
J.
_.,.
}~.i
r
= "3383CDP~-:'t) -u~ .+ 72J.~( flp l ~2) -Uji. + 5868(~pl-3) -lis . . + 37 1 3(~P3~4) -.!l5 + .7545(Ap3-5Y- us
.: sllbj~::l:t{1 p)~(;Gnst~ai;;1ts:, .,.,,' _" •• ,1 _. ,.' .... .'c
j.-
I. ,
ro:..
.... -.
•
....,r. . ....
':. '='500':
J ....:
.+ flPl-3 .+ Apj~~1 '= 500 .. t1P~'-l ,'+ ~P.J-3 + llP3~j = 500 .' L.\PO-t
:. Application of: the Lagrange .mulqplier .eqriatio~ to the abov~' proqlem .s[aiem~nr and, the. subsequ,ent sohi'rion: )rlelds. ~~ foiloWin,g optimal vaiues: _.. ilpo-t == 137.8, ~PI~2' ::;:: '362"2~ 11';1'-3.. .133.1J 'Il'P3-4 = '229.1,. and ~P3-~ = 22901 Pa. J:be diameters of the duct seciions corresponding to pressure drops are,·fr~m Eq. (16"53) .~ DO-1 ~O.5.27,- b 1-2 = 0.281, D 1~3 = O.M6, D3 - 4 = 0.253, and'D3~5 ~ O.367m. The to~i fIrst cost .o fthe optimal .. svstehl' "s·ii,3S'6. . . . . .
,these ;.',
y
'-,
is
.Example . 16~4 is a.n· l1Iusftati'lfn of. the .general· class of problems'" directed "to\vard the optimi,zation of .flow' networks. The tas~ iUustrated in Ex. 16.4 could be considered an optimization within an opti,mization: The static pressure. ~as specified as 500 Pa, but a more global problem would be to. choose the static pressure and the optimal du~t system that resulted in minimum present cost of the combination of the first cost of the duct sys'tern with the lifetime energy costsfor the fan. ~ k Fig. 16.-3 .illustrates , ·the Present wonh of total costs
/
/' Present wonh of ,,----- lifetime energy costs
First cost of optimizt;.d ~
dUCl
-------~-===> Static pres5u~ at fan outler'
FIGURE 16-3 TOIaj owni~~
a function of statjc pressure with optimized
dUCl
;.
system.
system
DESKiN OF TH.ER.h1AL SYSTEMS
-'
y l - A1
,I -, • .• -
Am
t/
( 16. ( 16.
-
---..
..
&I
•
449
CALCULUS METHODS OS- OPTIMIZATION •
t~
•
.vlhc},-c; the shortha:.id nOlation is
~
•
...
,',
', '
"
..JY
.,h"
,"a
.. .;. ;
2
, ;::.{ ! ,
=
, 'A-.' O/i,.j ~
!Vhen any of Eqs. (lp.55a) is expres.se,d as 'q. Taylor s,eries ·the
'a ' ,
+,~,,~ , ax 1~ ,
,'.
- : ... .
-
,
, ' ~.
..
. . . .. .
"
a2,'Pi l
I. 'J"
~
(y 1! - AI'+' 'I "P
,
'" 7
,
"", '
1. ' -:- • • • -:.
Am, ¥'r:n ,/.,' ,I'.) L\, x 1
+,' . .
result is
• . . •.,J. ;': ~
l
. -.
1
' + '~(y; ~"Al¢i ~:.~<, .-,. ', ' OJ,'n
'
_
~
'Am'¢:n i )D.xn "
•
a
+-'(
(16.56)
)L\A 1 .
aA[ ,
a
+-(
aAm "
where the x /s and .A c '5 are the correct values, !:l,x j == x j
-
xj c
and
I1Ak
= Akc
Since both the x's .and the A ·s are variables the differentiation of a A;'¢',.-k teon ,m ust be treated as differentiation of a product. The Taylor series expansion of any of Eqs. (16.S5b) is
( 16.57) Since
and
--
0
: .,
. • • ) 1 .-
DESIGN OF TI:lERMAL SYSTEMS
" II I
"\ ~l
--,.
1 -
1\ 1
'1 .n
.:'
•1.
11
·x
n Xm·
11
.. I1Al
I
-. X 1'l.
11Z
X rn
'.
A
1.
Ak r/>k:ij.
m k
i -:- row I to n j I
B
1
.
11
IS
[-
.1
1'''- ,'1]
.n
... - ¢m.n
is ,I ,II
J
.n
.1
)= are I,X:h
"it
.i}
....- ' .,
......
-
. xn) - y
8
)
.
45 ~~
CALCULUS METHODS OP<:lPTIMIZATION - I
whef.e
y -; = y (x 1 7 X 2. , . • • X i -I-
.
Ii ,__-. .-"'x n) - Y (x 1 ~ X 2' , . ' . /J.
."
••••1: lZ;
. _ _-P.,.3 ·is .typical of the _l're~!tpn.-:i:la:p~~~:)l?; p;r~s~~~r.~~~ }4~~l yal~es b.e, -',' ':' . , entered.,.. ~ope[lJil y~:,the.-,:~s.er..,. 9f .:-j1e..~ gerieraliZed,;. progrmn{," ·\vill:
nlu'st
"".'''.
-
of
x's
16014 . THE· UTILITY OF CALCULUS ·lVffiTHODS
are
Calcuhls metho~s of optimi~~ltjon both instructive and practical. They are ' 'i~-struc~ive in providing insight into certain -other methods .of optimization, such ~ search methods. CalcuI'us optimization wIll' often tum out-to be the ms>s't workable rttethod, particularly when the constraints are of the equality ,'type, .but ey~n when one or more of the ~o~straints ate .inequalities.
PROBLEMS 16.1~ The equation f~r the two-di~ensional temperature distribution in the metal plate. shown in Fig. 16-4 is ' ' 2 T, o C = 4.x} -
ill
+ 6Xt X. 2 +. X22 -
4X2
+
25
The magn~tude of thermal stress is influenced by the change of temperature
OL-____J -_ _ _ _- L_ _ _ _
o
2
~
______
3
FlGURE IG4 .. Two-dimensional ternperarure distribution.
~
4
____
~-----
.s
6
DESIGN OF TIIERMAL SYSTEMS
+ to
(b)
y'
+
1
~.s3
CALCULUS METIlODS 0or OPTIMIZATION .
].6J L SpaCe is 11D:U.ted in the region tlEough which duct ~eclion 1-3 in Ex. 16.4. passes, such that D 1-3 can be no ·:gre;3.ter than 0.4 T']. D'etennine the new ~ptimaI. diameters for a minimum-cost duct system . . ' ' .' AnsG~ for checking results~ total cost ==$1 C52L ;
-; F£l'i.\ ~r~l'T:r>:~ .
w,.
<. ,·,:· )t:~,,~:>r -;;'.' "
,.: " '. ':::... >:..... ».:'<, :.:.~:;;.';.:." ':. " :','; ." '. "" ,: ., '~ ~.~ ..-'.~' ......, . <:: ';-' . ~:" :'~- ".' '. :.' . . ' : .. :' . . , "'.~ .'::... :, ~' :., '1. Db J. Wilde ruid Cl ,S. 'Beightler. 'Foundatir;>ns of Oprimization, Prentice-11a11, ·E nglewood. .
Cliffs, N.J., 1967.' ' . . . ,. . 2. M.·· I<;ovariI<:; UADtomatic Design of Optirnai Duct Systems," Use of Computers fo~' Em~i-. ronmeJ.1tal EngiJ1e~ring Reiared10 Buildings, Bl:lil~ing Science Series' 39. 'Narional.B~reau of Sl:atidards, 1971. , "' . . 3. VI. f ~oecker, R. C. Winn,. and C; q. Ped~rsen, "Optimization of an Air-Sllppl~buct . System, n . Use of Computers for Enyironmenral Engineering Related to'Buildings ,. Buil0ing .Science.Se:ries 39, National Bureau of Standards, . 19~ l~ .-.-. . 4. H. Aikin and A. Schitler, "Stlidy .o f:Engineering ailej' ECODOnU.C Parameters Related to , . . ~e .C<;Jst' of an. Optima) Air-Supply Duct System," ASHRAE Transactions, voL 85, part 2. ' pp. 363-374, 1981. ' . ' " . ' . .
,.
---..
"""
~.
.
.....
CHAPTER
""1'7 ..
.: .-
-.
.
. - ~ .I.': -_. .-
. '., : •
~
.
~ -,
.
." : , :.' ~ . . ::'.. .'
,
".
'.
-"0 . " .
"
-
,
.'.t
'.
I .
.:
-:
~
.
-.-'
17.1 SEARCH METHODS AND AN . EV.A.LUATION OF THEIR' USEFULNESS Ch.apter 9 introduced search methods of optimizarion and presented several techniques for both unconstrained and constrained problems. Se\'eral classical techniques for single-variable searches were explained, but because
most of tlfe efficjent single-variable search methods apply only to unimodal functions:' 'the use of them is risky. The intent of this chapter is to push further into practical search methods that apply ~o constrained oprimization problen1s. A challenge is to select one, two, or three methods from the . extensive variety that is ava.i]ab~e. Optimization problems fitting ~e mold of linear or dynamic programming are -usually identifiable. and are usu.ally best attacked by those meth6ds~ The method of Lagrange multip1iers and search methods compete especially on constrained prob]ems~ \V.hen the objective function and constraints are capable of expression in equatjon 454
VECTOR AND REDUCE!? GR."",[)IEi'ff SEARCHES
455
fornI, Lagrange 111ultipliers may be p~eferl-ed. particularly if a generalized progralTI for the solution ~s avaiLable.'~ v\~heh~he objective function andic:r consLaints ,arc:, ' not· in ,explicit form. a search me[hod '1\lay be, the 'most pr2.ctical choice. The' most difficulfclass of problerns.is li.kely to be that of " ," . .' constrained optimizlltions .\vi.rh ineq~ality constraints. J\s ' discpssep in ,Sec. ",~:";' :'.:,.~. '::'.'-~ ::'.,~:,~ ~J 6; (:C',i~i ~'..pr~~61~rh ':rna¥:.~s.o"fQ.~~.times':. be,,'at'tacI(edr~~by,;,re-pearedj:ap'pl:it'4db:iF6t::':~': tt'e 'l?ethod " ot:~agrange".rr)Jilti-p{ie~s, .. :yjJ~h "~ip~~i"~~gr"irnge 'inuftipli~·rs . o~' a' ... ~!_ search meth'od?' a C,bmputer pro'gram with 'sO"me 'human 'interacti9D is' often. . .- 'required. ' ' . ' " , . , '. ' '7
,'. .' This.chapter limits its etIbrts in solvjr.zg constrained optiin~zatiori prob-'" }ems [0' , thre~ oJerhods: (1) -pen~lty functions', (2Y. vec[o~ algebra, nlethods~ and (3) the gen~ralized 'reduced gradient D1ethod. ,Converting a constrained optimiza~ion process to ,an uDc.onstrained one through th~ use of penalry functions . was ' introduced in '-Sec. 9.16., The .concept- is .simple,·, but 'for ', ,:,. ,', , .the method to b~ effective, the mag.nitude Qf the' p'e naity multiplying con- '. 'stanrs must be adjusted 'throughout the op'timizatiou'. process. Secti9n ,i 7.1 . : address~s t~!~Jask. Vector al.ge~~a reiations provide extremely useful nle~.ns· of deteITnfni~g"s'earch di~ect-jons, i.1t least for tne limited class of two-, threeand, four-variable problems. No surveys are available' tD indicate which ,method is nlost \videly llsed for constrained optimization problen1s ~ but .one of the front runn.ers sur:e1y is, [he generalized reduced gradient, technique. , ,which \vill, be explained. The Kuhn- T~cker conditions will be presente'd b~cause.of their classical importance and also their o~casional usefulness.in the ' solution of problems. '"
17.2 PENALTY FUNCTIONS The straigtltforward manner of )ntroducing penalty functions, as 'outlined in Sec: 9.16, is to convert the constrained problem, illustrated here for " maxi miza tiOA I
V =y(x), x2, .. x 17 )
subject to
-7>
maXImum
cPl(X I, Xi • .. xu) =0 ¢m(XI, X2, . . Xn) ==0
to the uneoo.strained problem
y = y - P J (¢J)2 - P"!.(.¢2) 2 - ... - Pm(¢m)2 ~ maximum Squnring" the
DESIGN OF 1HER.l\1AL SYSTEMS
/:."
+X ,
y.
tP a.,
J
0
.. /
to
y
10 JOO 1000
.x
(1
/,-1-57
'VECTOR AND REDUCED GRADIE\",~ SEARCHES w
.3
H
P
4rur+--T----~-----r_----lr----4
=i
P = 100
T.rue minimum .
3
I+------"ri..------:::o-+---+------'----i-----i
'
t o .••"
Y::::30
o~--~-~~---~--~---~
o~--~--~~~--~--~----~
o
2
3
, ,5
4"
,0
(b)
, . (a)
FIGURE ]7-1 . .,Contour lines .
.3 .-
2
In the penalized fu~ctjon qfExampl~ ', ' . -
.'
17.1 when (a) P','~ ' J. :(b}.P. ";:: : 100.' . . .
If Zahradnik's strategy ,were applied. the ' search' with' P = , 1 would first be perfonned and, the st.aning value'S for' the' P -:- ,100 search would ~e " , x I = 0.8303' a,nd X2 =,'2.2,329. That second series stops, after one search direction at X I == 1.0447 ~ X2 = 2.3923, and Y = 19_5697 _ This resuit is much closer to the expected optimum for P == 100 listed in Table 17.1 t but still may be -unsatisfactory. Bu~ now sinc~ there are likely to be a modest number of se4fch directions to execute~ the step size could ~ reduced. With tI!e ,step size reduced to o.ooocfi, seven search directions wou.ld, bring the result to X 1 == 1.0246. X2 ,- 2.4157, and Y - 19.559. ' I'
1703
SEARCH DIRECTIONS FROM·-VECTOR
ALGEBRA For two- and three-variable constrained problems vector algebra' using dot products and cross products provides straightfof\N~d means of specifying , search directions. The basic operations are alternately moving towar'd the ,. consu-aint(s) in a favorable direction with re'specr to y . Thus the three classes of moyes for 'a three-vari~bl~ constrained problem are as follows:
1.. Moving freely in the ' direction, of greatest fate of change of cp. 'Ibis operation occurs when mOVing from the trial point to a constraint plane, or whe'n moving back to a constraint after a move tangent to a ¢-surface. 2.' Moving tangent to one ¢-surface, in the most favorable ' direction with respect to. another ¢>-surface or with respect to y. 3. Moving tangent to two cp...surfaces simultaneously in the direction of impro~nlf ~
~'
5
458
-.DESIGN OF THERMAL SYSTEMS &
The. first class' of moves is s~p-~y ,in the] direction of V cP' or - V 4;, depending 9ll whether 4> is negative or pos.itive, ;respectively. ' '. ', , ' .. ' 'T he s~cond class of inoves coyers. 'the gdneral' situation of moving: . tangent to Olle 4rsurface in ~e most {~vorable diJ;ection ,with 'r~spect to the . objective, function y or wirp' 'respect to,the other c.o:r;tstrairi.! ,->surfa¢e, ~.ajVing reac;hed that surface tlrrou gh a . ,,'Chiss'·'1}nov6·. Next ' deteIniine '\7 4>.and V y "at"' the"point- 6n the co~~ttaint,.· . ':': ~ a$.,.~!1: fig'~~..J.7 ~ 2,~ ~ Th~ cr~~s, ·pT.oduct. of .v:. ¢ and :Yy ';, G.alled.fl., .:
'" ,-
,... .
:
..
~:
..
,
~ ~ .
:
•
.~
I
.
.... .. . .1
t ,:' ,_
..
': .: ..... -:.
•
•
f!.: "
..... ,,', ,L'
: :,.~
~:_ .
.
'
••
"':-
~-
,is ."
-
',:
' { ',-"
. " , ...
, --?>'" ' :. ,":~: ;. ,," ":", . , .. . ,.'" .' :. 'I-!' , , ,'. :-- <. '" '.-".-i. 2~ >.,' ':, ..: '.:i3' ...:-..... -'. ":,::,,;.,',,.-::.': ". " '. ' , . . . : .
•
- -
, ':,·~ ... ···'·V¢,X Vy'=
',. . .
.
.a¢r.axl" :a"¢/f/X 2" {J¢lax'~".·ay (aXl'" a.y la X2 . By I ax 3
is
'v.
.I
,-:-:::.: -; •
. : .. ,- .: .
B . is
is
the cross product, , B
=
A X
.\1 cJ> .
,
(17.3)'
to
·S-o' B is ' the ges~red vector that poin.t s. in the direction tangent the _¢surface, yet also moving in the di.r:ectjon of maximum 't ate of change of y . The other necessary inJonnation is whether B points in the direction ' , of an increase or a decrease of y. Because Vy points in the direction of' "increasing y, as was. shown in Sec. 16.2, .B when computed :from the , seque~ces 'i n Eqs. (17.2)' and (17.3) will also point in the direction Qf . in-c'reasing y 't and should be used for a maximization process. If y is being minimized, the direction should be .opposite. to B • The third class of moves for which vector' algebra gives. ~ simple' indication of direction is that .where the curreptpO.lPt H~,s on bqth. th,~ :.!P.-1rand 4>2- surfaces and a move tangent to both constraints is desired. ' As
.~'
.'
FIGURE 17-2 Vector that is tangent to a constraint and points in the most favorabJe direction wjth respecr to y.
--.--.
-'
' ,.. :, (17-.2)- '.. ·
The A vector normal to both' V ¢ apd y " and bec~~se .V ¢ normal to . the ¢-suXtace A- is tangent -to thi~ ·~wface,.,,,The next task is to frnd a vector that is th~ rotation of A back to the.V 1> -:\ly plan~,. but still t~ge~t to ¢~
That vector
~
~ .~.~.
.. -
VECTOR AND REDUCED GRA~'Ei\'IT SE.::\RCHES
-
4-59
"""" , "-,~- -'.'--" _
.:,.....
-" •
.FIGURE 17-3 - -The cross product of V
sho,wn in
.Fig~
17-3, tJJ.e Cross product' .
c·= V ¢l-X ._ V 4>2
( 17.4)
-is norrilat -to both V 2. From a point that satisfies the two constraints. the vector is sought that indicates the most favorable change of y while f:lIso remainingtahgent to the two co9straints. The procedure is first to compute a vector '[) by crossing V y with V 4>1 and V ch. as follows: '
il
i2
i3
~
ay
ay
oy
ay
ax,
JX2
aX3
ax~
'04;1
,tJ¢l
. ~X2
aX3
UX4
ach
ath.
a~
cJX2
aX3
aXil"
'0= iJ¢l
"aXt o¢r2 ax!
a 1>1. -----'-
( 17.5)
,
This vector'D which has components of d J I d 2t d 31 and d'4 is crossed again with cPl and ¢2 [0 determine the vector E which indicates the direction sought. -
0
•
:
t
I
DESIGN OF THERMAL SYSTEMS
i3
l
to .2
2
.'
. , .0
=0 2=
1.
(
1
VECTOR AND REDUCED GR.A.DU:J. ".'T SEARCHES b
"
,(61
the cO::-lstraints by adjusti'ng, llLl of the variables through appli, cation of Newton-R,aphson to the JJZsirquItaneolis equations provided by the cons traints. ' 5. If in the process ofretuming to the co~stt3_in,ts the value of y is degraded :relative to 'its 'value.?-t th~ s-tartil!g\point;' r.etun1 to Step.3 with a srp2111eT' : ':,.-.: ,.-,.:st~p:,~'siz'~,~,,~:~:~. :,:',~,,';;,::~;,:' ,::'~:'~~,~~'~";"..::,;:'.'~;,':'>:'~': ',\' ~, ',' :0::':>: :,;<,;~~; 1:~<~;",~' , ': " , , :: ~.; .~,_ : ,I, .. .:, ' ' .....,'" ,..'-~~:. " ; : /'" RetUlTl to
',' "', '~ '.~
., -
-",
o·
. 6.:·.Jfno inlpP?''Yemenris possibIe·with.-a _su.fficj_ently-~~nan"step,'size';- termi-, nate, ·.o thenvlse, ·rep.~m to Step 2. ' _ , . ... . , ,'.' '. '. . '
••
'_th.~· centTal'feat~re ,of the' GRG ,meth~d is. $tep, 3 ~Jlihich d~termiries 9
the direction tIiat is ' tangent to th,e constraints' and at the same time, jndicates . a favorable direction -for y. Consider an exampl~ ' of optinllzi~g t]. tltree-, . vadable fu~ction sllbje~'t ,to . two ',co~straints~ .a case' also eonside'red'in Sec .. 16.7 usJng calculus methods. 'The GRG 'development parallels the calculus method of Chapter •
16 ..' A differential change in y 7.1i_~: is
•
.'
I
Ay = (ay1r3xl)Ax, +
(oylaX2)llx2
+
(a}'laX])i1xl
(17.11)
.\Vhen the constr~ints are applied to Eg, (17.11) ily equrus the grouping .s hownJp... Eq. (16.35). That grouping reappears in' the GRG meiliocr-with two modifications. The first difference is that in the calculus method D.)j is equated to zero to give the conditions for the optiinum~ whil~ in the GRG search the process ,is one of continually improving y. The sec9nd difference "is in the form that tJ:1e grouping appears. The typical conventions used in , the GRG method for the example in question are , .
.
---
==. Yold + Y i L1.X . 1 + Y 2 ~x 2 + Y.3 Ll.X 3 ' n¢l _ tP11 L\x j + '0/12 IlX2 +, ¢b f1x 3 == 0 n¢2 == ¢21 L1X l' + ¢22 /J.X'2 + ¢23 Llx J = 0
Ynew
(17.12)
.~
wh~re .
x)
as the decision variable and denote
) =[~: The solu(ion for LlxJ and
fiX2
~~l
and
Ynew=Yold+{-[Y'
K
=[~J
from Eqs. (17.13) and (17.14) is
LlX I] '· , [ LlX2 = -)
Of.
(17.14)
== acPi I a~t'j
Arbitrarily choose
so
(17.13)
--1 K
!J.
x)
Y2)]-1J( +Y3}!l:c)
( 17.15) - . (17.16)
in the GRG symbols Jnew
=
Yold
+ {_[Vy]T }
-11<.
+ 'V'y} Ilx)
( 17.1 7)
DESIGN OF THER.MAL SYSTEMS
..,
~.
,x)
+ X3
, Xl~'-2X3
8 ' "
=($
iJ
a X3
an
as
)
) -:1).,:
==
-
.,
-J ~
== -
• so
'.1"
VECTOR AND REDUCED GRADIENT SEARCHES
:..
4.::53
...q~s~·~-·-:-j:.·7-S0· . .- ·: o·: 9?i3.::·-:·.;-}._ -.~.~~::.-.;:,9~j7.b[5.,~:.::~L..8.lL,..:::.:q.9.46K.:.:~,.3
. ~,-. ~~·..J}-.lSA.65,_~. ,:.-,... :..~'..' ;; ,
" -- " 2' " , ,: 0.5<> 1:614·~:· :"O:8828~';':-·'4.U~:" · ·· 6.0060' ·":'1':721 '" · ·0.'8716·' .. ..4.0 · ,, -···6~'O678'\ ·-· ,' .' .' ". '" ' 3-" '. Q.5~;~ .. ·C679· O.784! · \4.5::' ····.. 5~5J6{J :-. .l.742·· .'.O:7656. ··· 4·.:f:~ ':' ~'~57b2: ' . , '" ~ .. ' 'r
~
.-:.4 .... - 0.2 L
5 ' - . . oj. 0.02 .-· 6 .7 _0.02 g . 0.02'
. •
L803 . 1.871 1-.922 . J.955 2.006 -_
fo.':
'r ~ -.:.'
: ..... .
0.7069 0.6666 0.6475 0.6340 0.6152
-,.....
•
•
. .
,
_
~
•"
. 4.7 '.: 5.3645.
-1.814 0.7039 4.7 - 5_3672 4.8 . 5.2547 - L896 -. O.659:? 4.8 5.2589 . , 4.'82' . 5,2~60 _ 1.923 - O.64~f .. 4.82 _. 5.2361 ·4.84 . 5~21l9 - ,1.96.0 0.6323 . 4.84 - '5.2116 . 4.86 5.1868 2. 0260-.60'94 4.86 " 5_1880
'\yhjle holding x3 c~nstant at 3.5 .yields the 'p oint (1.'8106,. O.94~81 3.5) fo.f .w Nch Y ==.6-.64565.: Table 17.2 shows t~e progression. .
X2
Equations (17.1.:5).. to' (17.'17) and ExampI.e . 1'7.2 address ' ,a "thieevariable, two-constraint prob,l em ill \vhich t~e reduced gradient ·indicates a '. specific vector, and the. only decision is which .of the two directions offered by the vector should be chosen., In problems ~here n - m is' greater than I, ' there are more degrees of freedom of the resulting ' reduced gradient. For example, in a. four-variable .problem with two constrajnts, the equation comparable to Eg. (17.17) with X3 and x 4 arbitrarily chosen the deci~.lon yanables. would be
as
(17.18) The gradient vector principle now applies such that in de-tenllJnIng the relative values ,of ~X3 ~x 4 -
and
L\ X 3
GRG3
L\X4
GRG4
--=
1705
KUHN .. TUCKER CONDITIONS'. ,
the
work in Chapter 16 and this chapter has been devoted to optimization with equality constraints, although Chapter 16 did illustrate how to attack problems wi~ ine~uality. constraints by repeated application Most of
of calculus methods. The Kuhri-Thcker Conditions (KTC) apply to problemswhere. there may be both equality and inequality constraints; they also express some fundamental insight into optimization problems.
. If a function y = y (x 1.
. .. , X n )
TIIERMAL SYSTEMS
to
to
a k
1,
K.
= O·
o
Y =
J-
+ -2>0
to
---..
g=xJ+X,2-4>O
2
(
)
4 f.5
VECfOR AND REDUCED GRADIEl'iT SEARO-lES
, (a)From tt~c diagram. i ~l F ig. J7-4 it c9·n be seen .th3.t rhe :D.V.:'~li.GU.It1!. 'IS tbe , unconstrv.ined minimum occurring at (1,2)). The scalar equations conespond=Dg to Ec::. (17.19) are ". . . ,,2'( 1 - 2
:8.f~:d
. I
'
.. . ... -, >.. ~'
..
: ~.:
-}l
.,'.: ., , . ...:, .:.:"..:;.: ."~.:'. : ..~_ ~·c,~·,.~. 4.:~.(t,t~,~.;?,
\vhi
- ,
-.
.:
,.!",
.
..• '
,~
..
" j:- 'l
+
-
;(2
~. 2
.and ~
.."
... ':"
.
. '.L,,'. _
b,~ ~"c?~bi~ed-v~ith· .~e. s~c6n~".:~f~upp "~d"'flfcli' I(TC~"-"':'-: . . .. ' -. . ,
.
=0
Il
.
'
.
.
' .
'
-
'
.
_" • .
'.
, . ,
>'.0'
>. 0 : :
'
.
"
I
: ' '-Knowing that the Ct= 1~ ~t 2) solution is (1,2) 'leads .to g = 1 and u -:- 0', (b) With th~ 2"(2 ~ 4 - u· =:0 con~trai~t, the' constraint is itcti';e so g = .1:- 1 + X2 - 4 = O. The other variables at [he optimum x I = 1.5~ X2 = 2.5, .and , - Ii = 1.' So in this-case Ii > 0 and g = '0. . . Suppose that in' part (d) ·ilieconstraint had been assl:tmed [0' be active .. Then' g = a and the optimunl would occur at x 1·= 0.5 ~ 'X2 " 1 ~5, 'and u .= -1. ~e ~. ~~a.t~.~~~~alue of LL violates Eq. (17.23), indicating that the assumption of the' ~onstrai~t being active was .il1,v~lid.
are
·A possible .approach to using Lagrange multiplier? to solve . an optimization problem where 'one or· more· constraints are inequalities was to assume that inequalities were not active. After determining the solution, substitute into the inequalities to see if any are violated. Those that are Y·iolated _a~e then reinserted into the problem as equality constramts. An alternative approach using KTC is to assum~ that the inequalities are ,active. If the u value associated with the ineqvality is negative, that constraint is . inactive and can be omitted jn a recalculation.'
the
3
..
,
2
o
FlGURE 11-4
-.
L-_----lL--_~
o
-&
_ _--L_ _
2
3
~
4
~
_ _ _ _..p.
.r I
Constrain~ and contours of y in
Example 17.3.
' ..
DESIGN OF
'~J....L'--'YLru...,
SYSTEMS
16
y =y +
2
a
= X3
+
..r2X,3
= 0
o
1
4 67
VECTOR AND REDUCED GRADIB-IL SEA.."(CHES 2>
_
.
:/. ~;,:., =-' .~', oM
.. .
".'
.',
i
. :.Jl.:.
,
..'
'.~"':'••
• :,
' : ' t.
....
- .... : .... ..... , ...
~
.J
FIGURE 17-5" ,Engine cooling system .in ~~ob. J 7.4,_
has a UA ,'value of 12 kWIK. ,The equations for power in kW are I
' .
'
•
pump power
•
.
•
= ·o-.02lV~
fan p'ower ~ 0.9~ ~;
and
For simplification, assume that the arithmetic-m~ari-temperature difference is sufficiently accurate in ' the heat-transfer rate .equation, thus . . 12[( 100 + lw)/2 - (35
+
la)/2J
= 300 kW
(a) Develop the objective ,function and constraint in terms of Ww and wa' (b) Convert this' consu-ained· problem into an l!J.1constrained ~oDe ~sing penalty functions, and solve by a steepest descent search. (Suggest first P = 10 and "the final P = 1000):' Ans.: constraint in (a), 3S.8/ww + lS0/wa =7= 40. 17.5.. An objective function to be maximized is .
y = 3x 1
+ X 2 + .."C 3 2
subject to the constraints
20x 12
+
and
5xz 2
+ 3X32
= 127
x
= 12
IXZX3
The current point is (1,4.3) which satisfies both constraints. The next move in the search is to be tangerft to the two conS traints with a change of x 1 of 0.1' such that 'the objective function increases. Use vector methods to determine the values of the x ts at the new point. ADs.: new value of y = .17.8. 17.6. The function .
-.
Y = (1I3)x,
3
2
+2x2 +2..t 3
OF ~RMAL SYSTEMS
is
to
1.5
./."..
.
Steam
in
VECTOR AND REDUCED GRADIEl'H SEARCHES
469
fl}-
(c) If the p;:oLJleITl wc.:ce insleJ.li_to be:, so!ve:d v,sing the GE.G sea..rch I.'ilethod ~ . make one complete GRG ope:ratio:..111 starting on the constraints \-vhere A = 120 rn 2. Choose a s~ep size of A = 5 m2 • and return to the cO"iSLraints after the stepv bolding 'A constant. Vi,/lj,at ~e the values 'of A:' t17 i)., ana y? , , ' .
new
. '3.75)0 Section 17.4· p,l1d.E.xfunp1.e J.1.'. 2jIlusu~ated the application of the generalized ' , ':>""; ',~'red u~eq _graciienr " '('Q~9").t,. p.1~,thocL., tQ,:~ a. _.~~~:v.ari~~I~:~· pi09 ~~t9-!,:,yvilth:[w.o.,.,;: ,:. ~·, I
' .'
' ,'
'•. " 'constraints', De,,:,~lop .tl:;t.e_.~x'p~~ssions .tQ~..the .GROs. ~\yhe.rr,-..optimizing . ~.: .> .
"
'.
~
'
/ .
.' , _
" ::
_
". . . ' ,
,~
,I
.
.subjeet to
Choose 'X3 .and X.r as decision variables', '1701tt ~ ,building is ~eing constructed on a 30 x 50 m lot ·and requ~s serl?ack as shawl? in Fig .. 17-7 ,T~e zoning rule.s re~trict the height of the btiilct'in 8 h
a
..I:b.
<::
5 (a.
+ k) '.
.
.and another restriCtion is that t.he building can occupy no ,m ore than 509'0 . of ' the lor area. The owner wishes· to c.o nstruct a building with maximum· , volume. (a) Detennjne the optimum diIl!ensioris of the building by assuming that both ,constraints are aC,ti4l~,(b) use the Kuhn-Tucker conditions to detennine whether either of ~he constraints is nonbinding, and (c) guided by the l~ values in part (b) reoptimize psing the active co.n straint(s) as equ~ity constraints. Ans.: optimum volume = 94,836 m 3 •
_~I
L
Street
-r ~~-~----------t a
E o
n
Building
1
L.-_ _ _ _ _ _
: I
b:
-=-_~
__'-,:_---if--
Jt-4e- - - - 30 m ------..j'"I.
FlGURE 17·' Setback requirement3 for construction of the buildipg in Prob. 17.10.
-. ---.,
;,.a
.~
. .'
DESIGN OF 'THERMAL SYSTEMS
:---..- ...
-- ---_._-------.----- - - - - -
.. j . ! ~'- ,::.:- ..:.. . . ... ..<.
•
•
•
"
...
.--
.-CALCULU,S . '.
' ..:'OP
VARIATIONS · ····ANl)
DYNAMIC, PROGRAMMING
1801 '1-"1IE RELATIONSffiP BETWEEN CALCULUS _OF VARIATIONS AND DYNAN.ITCrROGRA~NG
Chapter 19 introduced dynamic programming and emphasized that the method applies .to· problems s~eking an optimal junction, rather than an optimal point.· In Fig. 18~1 the function y. = y (x) between point 1 and 2 is ·to -be found that gives a minimum (or maxirnum) value of the integi-aJ of F (x) 4~. Calculus of variations (COV) and dynamic programming (DP) are companion methods since both are technlques to detennine _the optimal y (x). A difference between COY and DP is that DP breaks the function into discrete stages, while COV treats it as a continuous function . . . Which method is precise and which is an approxim4tion depends on the problem. 'If the velocity of a vehicle is continuously adjusted during a
trip to minimize the total energy, CpV is a precise representation and DP is an approximation. pynamic programming in that problem 'woulq represent
the varying speed'
as a series
of steps, while COY inhttrently assumes a
continuQUS fUl}ct19n with continuous derivatives. In a different problem of ~
0
~
471
DESIGN OF TIIER.MAL SYSTEMS
y
/'
I
(1
1)
473
CALCULUS ' OF VARIATIONS AND DYNAMIC PROGRAM.Mil'-JG .
~
accqmrnogate rhein vv'hen they are prese!:L Thel~-L equation to be developed IS
, , . .
1,.aF '].
'.J1:;· d
.
. . - - - - . - - - , ==0 " ... ay . :'. dx \ ay , /' .~ ,
' . . . .... :. .-;..'," ." . .J ..
'.. '-::,,'" .:
.- , _( 1~L2) " . "
,
,
_"
. .. ,. ~' . .:\vhichj g':2...diffe:q:;~ti~.;,e.q.J.WJiqrl:,j-9-a~.e.~e,i1tu~.l Y.}P:U?'~·J.b,e; ,sql;~~d;it() .:p_~t.~~~~,~~,~ ,;:,>:',~; ·,.:funGtio*·. y.~ 1: ~' . . ~ " ' .• :' ."" " :""<~:'.~ .. _ '... :.:. ':,.' .' ....,. ......:. .-: ' . . ........ :~ .,".:~::.:.: : ." :'. ;.' .. The· development. of the E":L equ.atiph parallels the. strategy used' j~ .t:q. ."., . , '(16~13) to verjfy rhe existence of an optimalpoirir. In that case 'the functiolt' '.:was- expan.de~ in a Taylor serIes, about me point ,b'eing tested. To :preveni tb,e pqssibility of moving .one or more of.. the 'x? s and findi~g·a inore-favora.b le , . ·p.oin~, it was necessary for \1 y 'equal zero. . '. . ' ,, . ' In' the developme'nt of the E...:~, equation a. function y {x). is. sho'wn 'in · . Fig~ 18-2 that is expected :to yield the' miniq.1um value of l'(called I o)~ Define another fu'nctib~ ,in ·F ig. 1.8-2" Y (x), .that deviates only slightly.frorp.· " y (x) '~n~ is .express~d by the' equation,
to
y (x) . y (x)
+
ETJ(X)
. \vh.ere E is a l1!l~ericaJ. .value, and 7](x) is a function of .x . SU.bsti.tute Y for y, and y' == dY /dx, for y' in. the integral, Eq .. (18.1), · '.
. I(E) =
.'
(2
F
C:x, Y ,y') dx
(18.4)
of
The value of the integral in Eq. (18.4) is represented as a function E, namely J-(E), while the value of I in Eq. (18.J) is simply a numerical value. In Eq. (18.4) the, value . of. the integral incorporates the optimal value of the integral, 1 0 , as well as the £ term. In fact, I (E) can be expressed as
........, ..
_-----~ - - _ . . . . . . :....... . .-g,
Y(x)
2
~ y(.t) That gives minimum!. called 10
I·
~----~----------------~~----~~x
FIGURE 18-2 Function tbJir gi itS the optimal value o! the integral and a function Y that devi!tes slightly from y.
.
.
DESIGN OF 1HERMAL SYSTE1v.iS
a
E+
I(
(
=0
"Y ..
(
0,
can
are constants,
a
( 18.
BE iJY
=0
aE
( 18.
0
not a yJ = Y
can
)
E.
( 18 . . (1
to
€
(1
=0 '.
y
Y. two terms :=
-0
(1 1
CALCULUS Of VARJATIONS AND DYNAMIC PROGRAiv1MING
,r
"
rJ F
2
f
,
-J-
O}'
xI
in Eq.
The ,s~cond tetQ1
YJ dx
+'
1"X~'
J X1
aF
1 '
-,"-, TJ/jdx oy ,
(18.11), can be iJ;ttegrated
= 0
oy'parts,
'~:,;_.'-.;':~'~E' '-~;~7~th:~~:;:.j --~; ·- : :-'='::i~ -:,·;L;~ ':Jf,.,~~~~j/J:2_, d:~. -."." ;"., :,:,,"';' '" ',• J]
. . - "t ..
.
: '
~
. ,
(18~
'.;--
'.'
12)
,
Both 1](x 2) 'and ,1J(X ,} .== 0, so subst,i~tii1g Eg" (18: 12) back i,uto Eq. (18.fJ) restih~
in " ' " ' '
"
r-~2'f,aF
J.r I.
av ,"
y' )] _ d(aF la,'T}( .,' x) dx . 'dx
'
"
~
- 0
, The li~e of reason~ng' used now is t~at since Tj(x) , can be arbitrary, ' , the, o~~y ,w ay to' assure that the integr?l is ,always ,zero' is for the term in the' ,brac~ets to be zero, ,
'..,
~; -~ (~;) =
(- 18.2)
0
This ,is the Euler-Lagrange equation which is d. basic, tool in the calcull}s of 'vari~tions. It is applicable where no ~onstraint x) y, and y'. The function F will be known, and the task will be to determine}" which is a function, not a specific value. Since the E-L equation is a differential equation, it will be necessary to solve this differential equation to determlne y. , The E-L equation can be restructured into an alternate general form) and also into siinplified fonns for special' cases. The chain rule may applied to the second term in Eq, (18.2) to yield
re'lates
be
aF
a(aF lay') dx
ax
ay
acaF lay') dy' _ d(aF lay') dy == By' dx, ay dx
dx
°
For brevity let
Fy
L aF lay,
F, y
= oFla}"•
F ' , = 'a 2 F l(ay,)2 1Y
,
'
Then ~e alcemate fonn of the Euler-Lagrange equation is the secondorder equ.!,tion
-
..
....
Fy - F xy '
-
Fy'y'Y" - Fyy'Y' = 0
(18.14)
DESIGN OF THERMAL SYSTE1Y.tS
can
=0
.= 0
so
'.
~ ~l
( .(1
a can (1 (18. to
are
.x
(1
1
CALCULUS OF VARIATIONS :·\..1"(D DYNAMIC PROGR.A;.Y1MING
1.;:77
---------_.
Back fe3.~c~ge .
=0.2 (p I ~
.
100)" kg/s
III
= 180 when t = 120
I
P2=
!OO+O~m
.
__.__ ?lr__ ~_ .
.
~71 kg •
)
1
•
•
'.
.'
.,... .
liqL~id
.
~ ..;. ··'lOO-kP.:t : ...;. .. 'P..'i·::.';/:..,~ ,;:'::':~'~~:":;.: ;,~:~,,~:~~:: ,-,::,,:,"--m~={) when~~~~~D._,-____----,..;._'""""---....
..~.
..
\
~
.'
_to, laminar flo~ ~b(the hydfaulic fluid, 'PI ~ j;2 : ~ 15 m'~ .wh~re rll.' .=,f;tow......:~- . . rate ... kg/so The vessel ,pressure P"2' 7= :I(lq' '0.5111. ·The pumpIng .pow.er: is ·mp'(PI .- 100), whe~e nIp' is·the. ideal pump flow rate .. There is back Ie.akage in the pump such that · . '. . . . .in' =. mp~ - 0.2(p 1 -:- ' 100)
+
.. Determine the eq uation for ]}'z' a.s a function of. time 't such that minimum pumping energy is ~equire~ ' dui-t...ng the process.
Solution. The integral to minimize is
f120
120
1= fo. mp'(p j - lOO)dr =
0
+ P2
PI - 100 = 15m'
. [m'
+
O.2(p 1
- 100
=
lOO)](p.f - lOO)dt
-
15m'
+
O.5m
So the integral becomes
1
~ Jor
120
.
[3.5mm'
.
+ 60(m ')1 + 0'-05m 2 ]dt
Subjec[ing F to the·E-L equation of the fonn in Eq. (18.14), In
11 -
O.0008333m = 0
The sol urian of this differential equmion is
m ~ C1 sinh .JO.0008333 t + C2 cosh(O.02887t) The boundary condition m(O)
= 0 requires that
C2
= 0,
m J: C. sinh(O.02887t) . The second boundary condition, m(120)
=
IBOgives
CJ = 18Q!(sinh'3.464)
so an..Q...____
= 11.28
. m = 11.28 sinh(O.02887t) . m'
= 0.·3257 cosh(O.02887/)
so
,DESIGN OF
UiERMAL SYSTEMS o· J.
6 ',·5, '
•
ot
,•
." 1
OL·~~~~--~~~~~--~~~~~~--~~~
o
s ,
::,
.
rate to buildup
in
J:,xalffiPle
y
( .1
1 to
,Y ,
=J
(1
(18
- .. IIIIiIIIIUIMpr
twa
..
....'
1
47"'5:
CALCULUS OF VARlATIONS AND DYNAMIC. PROGRAlvIMING
' . r ,0 dfu o.eVCloprncnt or [D.e J~. -L eqU3..o.o:n .lOL~ a conSCCd"tne J .i. 1DCllon is a short extension of that for the :J~constraiDed fUTIctiorl that \vas' described Sec. 1 8~2. Th~ c~ncept to be, us~d l.s sixIular t6 that used in Chapter 16 in. optllTlizing a function by calc'r},us methods to. find an optimal point subject ' . . , , 'to constraints.' :':. " PrQPos~ a.test ftln~tion y, (:i.) as tb.e,"one thatminirrlizes,:]. iri,.~,q. (18. 18) , ' . .b~t'~a.rso : .~~iti~fie~';:J:.- 'ill;" E(D~'~-(f8'] 9J: 'J~'t';',f" "'b~":~-'ri'~'arby ~ s~i ut~6~;'.?'-~~h{~ie·\'>'·~'::·' ~', ~;.,~':" " '"T1
J.[~C
1
I~·'
-;
-..,
"
(.'
1."
•
in '
"
and
"
.....
',
:
are
. ' . '"
D!. _ f3 are c.onstants. These functions show:q ,in~ig. 18-5 ~ III order to keep th~-,terminal' point~ fixed positions '
, where,
in
7](X 1) ~
7](X2) , ~ 0 '
! .
, 'and
, The deviati'on functions 'T}Ct) and l'(x) ~e arbitrary, but if ,7] is selected, , Y ,must ~e cho~en spch' tha~ $e integ~al i~ Eq. (t"8.19) continues' to equal J. The 'revised ' function Y, and its derivative will r~place y and y"~ the . .. . integrals to determine if.,411 improvement of! is possible.
IJX' F ex, Y ,Y')dx .r 1
'
Next express J and K in Taylor series of a. and f3, which dent variables. at aI' I (a,{3) == l (0, 0) .+ ' - ' ,.- a + f3 + aa ex.{3 =0 af3 a.f3 =0 .v,
are ,two indepe~- ', .....:. "~--
(18.~2)
.2
CUt (x),
~y(x)
.xl
x
FlGURE 18-5 Graph 'of the fUnction y ",that' gives minimum value of 1 subject to constraint J. and functions arl.,.t j f3 j'(x) that are deviations from y.
and--.-...-
-
..
DESIGN OF THER..~ SYSTEMS
)
.
' ..
~
. (18
=0,-
. ,.. )
".
. == 0
(18.
8 (18 to a constant
--.......,..=A
..
'":'"
can
- it
-A
aa a
aa
=0
,y ,
=0
o A.
(18.
(18
"'~.
!~ 31
CALCULUS OF VAFUATIONS AND DYNAJ.A1C PROGRAMMING
Pump , :
:
. , ' . l
• •
~
' .
_
, ;
'." ":'. _:
~~--"
" flGURE- 18-6
,
- ",",,. ,. ,
j
, Pur:nI?i~g edible oil from a - ~a~ -th.(ough a filter in Example 18.2. _
,
"
_ ,
"
, '"
'
r
pressure d}fference developed by the pUInp~ I1p ,l(Pa, iI?-creases wi~ 'time due , _ , to the progressi,ve inc.re~se -in ,-r esistance: in~ ,the filter' the thick,ening at the ' oiL Tbe pressur~ difference ,is represented by the equation
and
ow
.
•
-
- /1p = (360.0 + ,6t) V" where t =tirne
'
.
fiom start, s. -'
, TI:ie flow rate is to b~ p'r ogrammed so that minimuf!1 pu~ping energy is required during the 20 _min- pumpout operation. A restriction 'on the program bec~~~,e of operating characteristics , of the filter is that ¢e mean Ap must : e9ua( ~lO kPa. Thus, -
Solntion. Since pump power, kW= !1p V' the ,integral representing the J
pumping-energy that
IS
to be minimized is" ""
(1200 . -, )0' (3600 + 6/)(V/)2 dt (J2OO
subject to
J
-
+ 6t) V'
dt
= 252,000
Therefore
F = (3600
+
61) (y,)2
and
G
0
(18.28)
(3600
( 18.29)
= (3600 1- 61) V'
Before solving the cOI,lstrained problem. it would be interesting to solve for the minimum energy function,without the constraint. Application of the E-L equation to F yjelds . '
V, rh)
= 36.41
In(6J
V'. ml/s = 218'.46/(61 Ap
+ 3600)
= 218.46 kPa
aiidenergy for the operation
+ 3600)
= 8738 kJ.
- 298.15
DESIGN OF TI1ER.MAL SYSTEMS G'
rate so
"
';-'.,
v
6
v:.-..
+
¥'=
+
...
.
CALCULUS OF VARIATIONS AND DYNAMIC ?J<-OGRAMMING
DYNAIVIIC PROGRP~W{G lLV\ID CALC.JLUS OF VARIATIOf\TS ' I
433
1804
.
.
. Dynamic px:ogr3.J.--nming and COVare m¢nibers ,o f the-,sairlej·grrfriY:JJYriaro.1c prograrru.-mng .,yields series o.f P9inrs that 'when conp.edtecl- describe a" func~" . . . .. : jon ~~t optimizes ~ .sUInrl1a,ti6u.~.' .as ~jllil~tnit~d . ip:.:;~~tP;: ;J8~~7,-: . .C~lcuIllS of . ': '.: '. ;. " 'Valla~ions .develops. arr···:equati6n -~ f5i.-·:·tlie '''fu:~ctl'dit't!1:a( ' is<-·,conMtibii."s:"willi~ ./; :.~: .' continu6iis', This~ .fiulction integrai" ·.hased',.'on ':ihe .:. ': .furictioBl_ ' . ' . . .. ',. _ ', ' ..". ... .. . , .' :Ariother c~IJlIl1on' characterisdc·. qf both DP ,and C·OV :is' th.at .t he':terini'n31 pomts, points 1 ,atjd '2 in' Fig. ,18-:i, are)mow-xi..- The' exp~ct~tiojJ ·'of . knowing" the' te~al points 'often he~ps resolve it difficulty ,that waS dis- . cussed··ln Sec. 10.6 surrounding appare~tly constrain~d'''pf~IDlgms.· In ·E xampIe' 18.~· the expression for. ml is' sought, and the (ITst inclination 'may be. to 'denote m' as the variable being s
a
denvatives:'
:'optfrriizes "an .
pllmp
.y
B
y
E·
A
4
Oprimize
L 0 (y. y', x)
, .. I
F(y. y'. z) d:x
(b)
(a)
. flGURE 18-7 (a) ~ra.mnUng (b)
f;
Oplimize
x
calculus of variations.
484
DESIGN OF tHERMAL SYSTEMS
bas no doubt been observed that the s.olu~on ~f a ·differential ·equation. is a .necessary step in rmding the optimal function Jusmg COV. In so~e .cases_· " the analytical solution :of this differential equ~tion may be extremely djffi-· .' '. cult, 'and perhaps impossiple. When co.nfrqriting this situation a. numerica) . , . ~::. ~:' . ~ ..~ohitio.D. ·
:. '. :"
18 .. 5- , STAGED.OP~RATIONS·WITH~RECYCLE . . .. .'
,
" ..
"
..~
{
'
. .
.
.
.
. Som'e chemicai operations are char~"cte~zed by recycled streams, as-.-ShO.Wll . . ' in ,Fig. -18-8. The 'symbols in Fig. 18-8 are simil~' to those first used in . ' Fig. lO~l iIi ,the introduction to DP,- wher~ the objec~ve is to maximize the summation I
when ' the inlet. cOI)centz~gi.on S, the· entering flow rate Q. and. the recycle rate:':R are known. A complication oecuning -in Pig. '18-8 that 'SN and S '] are not lnj't.ially kno\vn. The values 'of Sn and S' 1 can. be -as$umed~ however, ' and DP applied to the serie.s of .s tages. With the vi{ues of SN and S'I detennined from the first DP solution, more.accurate values SN and S l] can be us~d in the pext iterati,on.
is
of
18.6 DYN-AMIC PROGRA.MMIN:G FOR . CONSTRAINED OPTI!Y.1;IZA,TIONS All the problems to which DP has been applied so far in Chapter 10 and i~
this chapter have been truly unconstrained. Constrained problems can be R =Recycle flow rate d,,'
ldN Q. S
SN
N
SN
------
rN
FlGURE 18-8 Recycle
-
&.1re ar;n
.
in a staged process.
5,.
dJ
S'n
n
r
lt
..-..---
5J
Sj
Sj
Q
r,
CALCULUS OF VARIATIONS AND DYNAlvlIC PROGRAMMiNG
c:t85
0-
enCOl2:J.tered, ho\vever just as has been the lease in CO:'Vo j\ dilen1T.oa 3.xlsesin m,aking the 1ecision of Vlhich is 't he optimal choice within one block of a table., l\To longer is it possible to select the lTlinimum, (or m.a"Xin1um) of the summation beirig optimized, but,:a sacrifice n;1ust be rnade jr,t- order 'to ke'ep control the c0l1straint which in DP is a Sll1Th-natiori~ _ . -, ' -- , ;:,: ~<:-': '-', ''the qu~s-do~Cl~: ho-'v!:,,,m~ctit~to,:,&~de:,off JJ-~.t~~~~~n-,;tl].e.,,~J1p)i:n0~i9p.',.-he.::ng,<, ' -optir~zed~d -'-tbe ~ohstraint"surpmatiolf. )It'is here -,that iris-1-ght·-.~fr~ni:\=,6_y~~;U'·f:' -providesthe answer. The value of A has proven to b~ the s~ns,itivity coef-,_ ' .ficient~ ~d the DP'process ' wil1 ,be 'conducted ,with interpal choices ~o ': hold a constant value'.'of A tlu;"ough the s~ccessive tables~:, ' 9
,under
:/
7
made ,
"
- Example 1~t3. Solve Exarnpl~ J8.2 -by tre~ting it as a four-,stage DP probI~m. each, sta~e consisting of 300 -sof pD.mping· ,p~eiati,o~. ' Solution_ Some understanding of the p~oblem and i~Sllit~ ~'~ - ~i;~~d~': ~'~~kii~~' able from the COVsolution of Exampie 18'.2; ,that infonnatiop'wilJ,be applied here. The first need is ~ _set of tables that express , pump energy arid Dlean pressure for each stage between a given ,inlet and outlet pressure. To- com.pute rsuch a table an assumption must be mad~ about how 'the pump'i:ng is t~take . ' place-constant flow rate, constant Ap, etc. Since a constant t1p 'prqcess ' was the optimal one - in Example '1 8.2, it will be chosen for given stage'_ No,tloubt the Jlp, while' constant for a stage, will be d,ifferent for each stage_ ; Table' l8.1 provides pumping energy ,data, and Tab~e.l8.- 2 the cGntribu[ion tq ,the lip suinmation for various combinations of inlet and outlet volumes. Start at the final pumping stage. 900 to 1200 s,' for' Which the'"energy and Clp values come directly from Tables 18.1 and' 18.2 for the available . .., ' 3 .. : pumpmg plans of 33.8,34.2,34.6, .)5.0, and 35.4 to 40 m . The ne~t table~ ',. TabJe 18.3, applies to the 600 to 1200-second time interval and shows the cumulative energy and !1p contributions'., . . The extra dimension- of A is required in Table 18.3 to make the tr~deoff in expending more energy in order to reduce the /1p contribution: In Example 18.3 the sensitivity coefficient is the difference between two energy quantities diviqed l;Jy {he difference between tVia /lp c~ntributions on the line just above and below within a given inlet volume block. Thus; for tfie inlet volume of 26.4 m3 , the A between the first two lines is -0.008 and betweery the second and third lines jt is -0.034. From Example 18.2 the value of A was found to be-O.02616 which wll1 be the value chosen as the target jn Table
a
18.3. Because of the steps in [he DP solution, precise values of A c~nnot be reproduced, and instead the choice of volumes pumped is the one closest to the desired value ,of A. The best situation is where three lines give tv-Io values of A that just straddle -0.02616, because the middle line then nearly reflects the desired A . Two more calculation tables that are omitted here would follow Table 18.3 giving the optimal result shown in Table ·18,4. ' The results are close to those obtained, by COV. with the difference explained by the ·discrete steps used in the DP solution. In the COY solution --Lhe value of ). was inherently constant throughout the entire process, and the -
•
w
~
)0
"
~i
Volume at end Volum~
at sturt IS.B 16.2 16.6 17.0
n.4
'!'
, 15.8
16.2
16.6
3694.1
'3&83.5
4077.7
17.0
l7.4
'27.2 . .27'.6 '
26.4
26.8
2343,4
2523.6 2710.5 2343.4 252~.6 2169.9 2003.0, 2169.9
4480.2
2109.9 2003.0 1842.9
2904.0 27105
2169.9' 2343.4
21,."1
1412.
1'•. It 21.2 n.6 28.0"
Bn.S tl1L
34.2
)5.0
1033
"
TABLE 18.2 Ap Contribution
:1
in n Singe, Example 18.3 Votl1r1W
~t
,c'nel '
Volume 15.8
16.2
16.6 ,
17.0
17.4
70142
71911
73693
7,5.469
77245
26.8
27.2
68826
16.6
66323 . '63R21 61318 '
17,.0
58815
" 613'18
71329 688_2.6 663'23 6382}
ot'start
0.0 15.8 16.2
17,4 26.4 26.8' 27.2 21.6 28.0 . 33.8 34:2 34.6 3!kO
26.4
66323
63821
' 56312 . 58815 ' 61318
27~6
'
7633 '7383 68826 - . -7-13 ,66323 ' 6882 73831
' 71319
- 6632
63821
"
' ,'
-
3SA . i!
1~
,
~
' 28.0,
:;~
. DESIGN OF TI-lERMAL SYSTEMS
....
~ .........r ......~
18.3
s
. Cumulative
, Cumulative
s '. CUl1:lulati:~e energy
=:: ?-737.4 . I
1472.4
=
.
33~8
'1265~O .
1107.0'
35.0.' . 35.4
822.7
,33.8
17.6
34.6 ," 35.0 35.4
959.6 822.7
1808.0
=
+.
~171.3 ,=
+ 1472.4 + '1 + ] 808.0
+
f
~~. 2630.7
+
+
] 0.0043 ] '-0.022 + 62919 ~ 11 .1 -0.049 . :+ ~ 1]5,509 ]
.
+ ·1472.4 = ,+ 1635~9 =
34.2 . '.' ·34_6
27.2
~ 57261
.45414
2436.3
'+ - ,+. .-
:=.2424.5
61211 57261
= 243'2.0
53312
+
+
=
+
33.8 .:?4.2 34.6 35.0 35.4 .
61211 1265,.0 + 1033.6 = 2298.6 1107.0 + 1171.3 = 2278.,3 . 57261 959.6 -r 1317.5 = 2277.1 53312 822.7 j- 1472.4 = 2295.1 49363 = 2332.2 45414 693.3 + 1
+ +
33.8 34.2
1265.0 T 1107.0 + 1033.6 + 1171.3 822.7 + 1317.5 693.6 + J472.4
35.0 35.4
~/
= 2504.3
= 2169.5 = 21.40.8
=
2130.9 = 2140.2 = 2168.7
=
+
45414
28.0
.A
,~ 114~450 ]
0.0163. ' . J -0.010 = 113. 005 -0.037 ~29.19 = 1 , ] -0.063 ~6~46:;:: 111;560 = 1
1
50013 = 111,224
i
+
0.0281 0.0017 564"66 = 109,778 J -,0.025 = 10.9,056.f -0.051 9 =
612] 1 + '57261 + 53312 + 49363 + 45414 +
46786 ,= 107.997 j . 0.040 50013 = 107,274 ] 0.0134 = 1 J -0.013 56466 =: IO~,8291 -0.039 59693 = ]07.
+ +
53239
=
11 0,500
CALCULUS OF VARIATIONS AND DYl\lAlvlIC PROGRAMMING
t~-69
€k>
.. :-'
...
•
: . w.! ;.::".:...:~ .;
D P solution :Jso st.'i \red to use the"sarfle lvaJuc of ;\ in malaDg the decisions at " each stage. A less-tha.T1-optima! result ),vould .ha.ve occurred had a high value of i bee1} chosen ' i~" .t he early stages and'a -low one later a;3 nece~sary to Ineet t.h e constraint. . . . 'r !l solving Example 13.3 F/e had ad1f~-,ce".infQrrna"tion L1at A SflOUld be ' " .:... ~O . Q2616 5 btlt if the problerit h~d.ilot b~n '· solv~d·rrrst-~by ;COV; 'this infor: ' . .' "Kil."ati6n ·-· vJould p'of h~ve··b.~en-·:. av8:ila-bIe:r"Wh~ii"startingrfieslto-n:; ~i u"s:ttaiiie:d >.,Oi: , j)P p~obI~m~ it is'l:iece~sary ;to'·: eSrimat.~:· a ·-v~u~ '! qfX ',aiid"talc,ll1)iie 'a"p"re~i;n- ., )J.ary solution; The arbi~--iJy chosen :valll~ of A wplJld prqbably .r~~tilt.in Lhe . . co'nstraint either being exceeded or undetused. B~ed on the results ·~/i thoLhe trial A; ~ new A would ber chosen ~ ' " . . .
to
. PROBLEMS "~18~1~ The equation for the path· through air 'is tob~ found such that-minimum work ."
.results in tra versing betwee~ points 1 and 2 . in. Fig. 18-~ ~ The work js "' the . ". product of the resistance and the distance, R ds, where R i~ dependent on the air density. The equatio.n fOf: R. ·is ' R = 15001(y
+
10)
Apply the Euler-Lagrange eqmition to develop .the differenti~ equation for y . . (h) .Integrate the differential equation to determine the path b~tween . l and .'2.
, (a)
lO)y' = )2, 250, OOOJ~l - (y + 10)2 . 18.20 A refngerated warehouse is ·to ·be cooled downJrom a temp~rature of 25~C to O°C i~ a time interval of 12,000 s. The refrigeration equipment is to be operated in such a manner that it requires minimum compressor energy · for the cool~own._Specifical1Yt we see~ a function T = y(t), as in Fig. 18-10, where T is' the room temperat~re in °c and 1 is the time in seconds from the start of cooldown. The expression for 'the ratio of compressor energy [0
Ans.: . (a). (y
+
50
JO
DistAnce, kIn.
FIGURE 18-9 Path through air resulting in minimum work.
--
DESIGN OF nrERMAL SYSTEMS
'.:-,
.'
evaporator
'"'&.-' ... ti." ......
ternp~:ra,rur·e·progra.rn for minirnu.;m
""'Y"H"''f''"C'l'''U
is
-------------------- -
+
$==
=
S
as a rH·...... '......,...
- .. .........
..... ...
,
4::.~ 1
CALCULUS OF VARLA.TIONS AND DYN:\MIC PROGRA.Ml'AING &
18JL i\ rocket sta.rLs with zero velocity. and travels on a horizcrszl path. ag2,inst ilegligible i-esistance, The mass of th:e mdtal of the rocket is w which 2$ very ,
"
)
,
'
large relative to the.02.ss of the fuel. ' , ' " . ,DeIe,u,nine' the velocity program" V(f);' Ulat results' in themax;imurn . . , distance traversed in ,time f) whiI~ using lip all ,tJ;:te fu~l, of 'mass Iv!. TP.~ thrust. is' represented .by , the equation ,'. ", " J • .", ' ,. . .... ,: ,'. ,:-,,'!,',: ," , .' ~',,'- ,'( ' '-'' :; " ",-: '.'
~. ,
'A"~" .'
. .',,:':' • '
.'.~ . ",~"'"
. '.
t~st" ~I;'~:~~Ajd::-"-":'
. ,' . dM: . . ' . .: "'where '~dt = fuel burnmg rate
. . ,.•... " " '. "
'
.
..
e
Also deteI1l1ine' the di$tance trave:led in time when follo'wing' 'rlIe' oprim.um .,plan. . .. ' . I .•
" ,,' Ans
a "
d'l'stan'" ce
,0&
,
:AJ •
W
'.
U
3
8 M 3 ' ".
,1845,. 'rhe distance betweeq two pumping stations of a natural ' gas (me~hane) . .pipeline is 20b km. The pipeline cani~s 30 kg/s and the pressure entering the pip~line is 4200 kPa and the leaving pressure is ,7000 kPa. Instead ,of ,cboosin'g a pip~ of constant diameter, ' the diamet~r can be ' ,changed every 40 kIn. because of the expectation that t~e optimum diarn'eier may change 'as the pressure decreases, as shown in Fig. 18-.11" " (a) Compute the d'iameter that re~ults in 'a press~e' drop from 3000 kPa to 2200 kPa jn a' 40 kill 'length, and 'th}js' fill in the gap in Tabl~ 18.5, The basis for the calculation of the pressure drop in 'compressible, isotherm~1 flow in' a long gas pipeline comes from the combination of the following
equations: momentum rel4tionship 9f ~g to the pressure ~o'p equation , continuity
perfect gas (constant temperature of 20°C)
D.
,
40km
200km
FIGURE 18-11 Pipeline i!!,.Poob. 18.5 where diameter can be changed every 40 km, -=-
0
· .DESIGN OF TIiERMAL SYSTE1\1S
IS
~
(b)
5
CALCULUS Of YARIAI10NS AND DY"NAM1C PROGR..-\iYEv1L.,\fG
493
length is 800 m._ The evaporating temperature is held constant du ri.ng each of the five two-hour periods during the bl!ildup_ Dynamic progro.rnlill~lg is· to be .• used to determine each of these evaporati.ng rernperatures such that m.inimuD1 refrigeration cc~npresso[ energ)' .is reg uired 'during the IO-hour btl ildup tirhe. A_simple nlode! that ·yields real-i~tic res·ults is ·tqat the ice freezes at. the
·o0~tSide. -Surface at O°C- and · t~e :.atent neat of fusion .must paSs ·by conduction . ... ,,:... :..;- . .,- ... ' .tb,rouglt ~he ice tothf; r~fngerant.I,..as iHv>strfli~d:)n .. FIg.~·J ~.:-12~'·The· o~llY.:·. ~~.,:
.
~'/_~. :.:
.
.::;, •.. •
~ •.• :
>'
f;~' ".~ res~tance to'·hear: tr~nsfer' that :ne~e·(r·De··;'cbilsid~r~d ·lS· thai: ·' of'the·Tce~·'·s:in~~ the·--: :~ .. ·resistanc,es' ·of the .tube ·apd· the' qoiling ·c6effiCj~n.t are relp.tiv·f,::Ij(~maIL .l1ie>·: :., ,:.. . <;on~llction r~lations fot" a ~hlck .cyllnder must be··u~ed. ., . '.. , Jr·r . •
Ice properties·
def.l~i ty -:- 900 kg/rn,3 l~tent heat of fusion = 340 ·kJJkg
.th:ennal conductivity =0.0023 kW/(m '· K). · .~
~,
Coefficient of perfoIfllance (COf) of the refrigeration system is 600/0 ,that of ,tb~ Camot COP, .-at the prevailing evaporating and conqensing temperature$. The condensing rem perat~re conSiant at 35°C. (9) Table 18.6 shows the energy in kJ required··by the refrigerati9h compres, sor fQr two-hour p~riods in building tfle ice from one thickness toanother. . ., . . . . Verify one value on the table, for example, the building of ice. from ·a . thickness of 20 mm to 30 mm. (b} Us~ng dynamic progranuning- with the energy costs shown in Table 18.6, determine the thicknesses a~ ,the end of each of the five two-hour pe.nods that builc;l the total thickness of 50 nun · with minimum total energy reqJlued of the refrigeration plant. . >
••••
is
.
~
AIlS.: 8231 M1 18.7. The constrained optimization for the startup of the turbine in Problem 18.3 is 250, 500 • to be"solved by dynamic programming. The rotative speeds w
at
.,,------
...........
ict surface 0 C
/=
0
rt = 15 mm ~ RndiU! to ice surface T, = Refrigerant tempemrure
I
finnl thickness I :;;50mm
rj
"
\ \ \ \.
""
......
'
.........
------."",.-
./
F1GURE 18-12 Frcezin~be
in an ice buildtr.
Tube
•
. ' .b. \0 A.
TAnLE 18.6
Energy'in
:. .
',
rvu required by the compressor of fhe refrigeration' system ·in prpbl
for freezing ice in hvo-hour periods .fronl initial to 'final icc thickl)esses' shown
~I
mm 0
10 12.5 IS 17.5 20 ' 22.5
2.5 27.5 30 31.5
35 37.5' 40
42.S 45 47.5
,IIlnn t til
'
lnltlnl thlcknt!S,
Ick~e."", .
,~
mOl
.
• -..t
to
12.5
68.0 '
'92.2
15 :.120.4
17.5 .
153.0
20
22.5
191.0
23'5.3
111.0
150.9 t 23.1
30 .. ..-~:. 32.5 '
25
27."5
t97.S 167.7
252.0 220,0
2B I."
IJ~.8
185.5
244.(j. ,
102.3
149.1 . 11 1.5
35
JI'.
3'12.9
204:~ .
269.5
[6J ..~·
.
"' 346
~4.1 . .~96. 121·.a . 1"77.6' '245 130,9 . "-1'92 }"~..
/
!-'::
::,:\.:.
'.~?
~~,:~ ~
"f" \
,
~ 141
TADLE 13.7
.
.
Cost in a ·stage, Problem ·18.7
:j
Speed ut end
or stngc
1t rp9
SpcctI ut encI uf Hinge 21 rps
Speed at stllri., rp§
lQ.O
19.5
175.2
184.0
37.0
147.1
ISS.D
164.3
173 ..3'
147.2 \39.1.
21.0
, 116.5
[23.~
155.6_ 147.2 139.1 I J 1.5
164.3 155.7'
2O.S
139.0 . '[31.2 f· ''23.7
0.0
10.0 1~3.0
20.5
21.0
202,4
212.0.-
, 37.$ "
3fi.S
19.0 19.5 20.0
.
~
131.3
36.5 37.0 37.5 3g~O
3&.S 49.5 ~.O
5O.S 51.0 51.5
~:,
~
\0 011
38.0
'
.
'38
' 1-47:.4
l \ -\ 1
1'~39A
1
a "
stnl1,
:(
0.0 19.0 19.5 20.0 20.5 21.0
37.0
37.S ~
38.0 ~49.S
50.0
SO.S Sl.0
Sl.S
170Dlcm
.,
.
~. ~
till rp!J
19.0
19.5
10.0
20.5
21.0
2521
2563
2604
2646
2688
37.0
37_5
7021
7073 7156 7240
-38.5 --
6969
7104 7135
7l8R 7271
7354
732J. 7406
. 7125 7208 7292 . 7375 '
7343 7421
--
CALCULUS OF VARIATIONS AND DYNAlvHC PROGRAMMll'JG - (-~)1 &
and 7.50 s from the start {;I.re to be dete.ffi1ined such that the sU.mrnation 4-
COSI,
$
:=
.2:
1
_Costj
i.=== 1 .-
-~
~
~
-
..
•' '.~ -..-'.~ .~ =~ ..-
".is a minimUlTI.; where._Costi 'js the integral of 120 {u/f~ :in a pille stage~ The - consu-aintrequiies ~he. total n~mbf!r revotuti~Ds it"1 fiae LOOO.s startup 'to be _. '.'" ':. :35))qO~.~.<...:' ~ ..,,:-:;.-,-.-.:;.'~.. ~:.;,:,.<,.:_. ~: -:",'.--: :".' '. _~: :~".'.'
of ,< ,-
. , , / ' ,:-",:,.;:-
,., •• -.-
, .......
:
'
--
l_ .... -,
-
4-
L
Re~i
-= ,35~ 000'
-i= 1
_where Revi i.S the' D.llIDbe::· of revolutions in' a time 'stage~' The- solution': ot-Problem 18.'3 by calculus 'of v~~.tioris de-teimined thaJ A , 0.01,44 ,$/revolution, -·ano. this value may be .used in .the' dynamic programming soJutio'n' tOI gl;!icie. th~ tradeoff between cost' '~nd number of revolut~ons. Table '18-.7 provid~s ,values' of Costj and Table' 18.8 the number of 'revqlutions Rev'j in a 250 s .s~age bel~ee~ sele~ted initial ,'and. fip.aCvalues of W~ ':{a) -Verify by integration of 120( w,)2dt 'and w dr. ope- entry in each 'of Taqle 18.7 and 18.8, -respectively (for ~xamp~e between the initial w of 20 rev/s at.t50 s. and the finaJ.l1? of 37._5 .rev/s at 500 s). Use a quadratic function . for iv, namely £r) = At 2 ,+ Bt., where t = time in ~ from the starftip. (b) Perform th~ dynamic programrr11ng optimization to determine values of w at the end of each 25"0 s intervaL Ans.: total cost = -$467.20 with -34,938 revolutions'. ,
----..
CHAPTER
.
-.
- '
. .,
~ \.
~
.,.
.-
, I . .
.
.
..
.
,
.
, ,<" ' _:.
.
'.:. '1. "
,- '
..
. ,,
,', '.,
I
"
,,:~
"
:.
;~
-
..
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. ,: 'P ROBABmrSTfC-.·':'>'. ",~,.{
.... •·. APPROACHES ." 'TO 'D,ESIGN:',:"
'. ..
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:':'::'~':".'''.. '.:· ~C--:
... ~ .;'..
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#
19.1 PROBABILISTIC VS. DETERMINISTIC Most design's and analyses are made und~r the. assumption, that nUIpeiical magnitudes 'on which the- determinations are mad~ are kno\vn precisely. Any ' projection into the future cannot be certaln~ however, so the best that cali be predicted i~ the most likely s~tuation_ O'nce it is admitted that' the prediction is not certain, the next quest jon is the degree of uncertainty. It is valuable for the, decision maker to' know whether high confidence can be placed in data or whether the distribution cux:ve is flat (see Fig. 1-4). .' ,The application of probabilis.ti9 approaches to,therrna1 systems is an effort still in "its infancy" even though probability theory and numerical applications of the theory are well established. Also certain engineering discipJines, such structural engineering and reliability analysis, have progresse'd j!l Jlsing these, tools. The purpose of this chapter' is to present a few of the principles and then show some situations where probabilistic approaches apply to thermal systems. . , One of t.h~ major challengps .is to alert thought proGesses, so that situ'a'tions that are probabilistic can be recogniz~d. The next challenge is to transl ate the problem to a form that can be addressed by probabili ty theory. An intended contribution of this chapter is to' present some situations and their solutions in order to illustrate the distinction between deterministic and probabilistic approaches.
as
498
--.
1
. ' ..
. .'; . •. . .- .
.. ,,- :- ,'~ '·.. '- .r..... - . .
.
-. " i
PROBAB1LISTIC APPROACHES?"D DES1GN
reS~9
19 ~2 'DISCRJETE E, lEr'JT:: lSo COpTTlf\I1JOlUS' D1SIT
.
'Jne of the c1assificatio::Js, fnto which' Dlany 'of the applications divj.de :3 \ivhether they are dis,tTete ,or, con~lr llD US. The distinctio~ bet~..veeD ,' the· two, "'Gp-tegories l1~ay be ,cnidely e:xpressed as the' riumberor density of events.: As the ,de'l1sjty i:r.creas·es :the f~pr,esen1:ation shifts [rpm 9i~Gtet~ 'to ,c'dntirilious
o' ,
,
",>:'~Tl).e ¢har~cteVz~tibn.'o.f'dis'cr6te' 'eY'epci ~ppe'ars in' i;io bability' texci.' 'in' e,~ain~ ', ::,: :" ,," ple~' Qf rofiihg 'dIce'" selection of cards fro~ ,:a 'deck, ,an(r.~-aw$.g,l?·a:U~',,~f '. " , ,··various cQlors from urns. The~e applications should noi' Suggest ,'that ~;~~1e , use:fuIness of the ppncii;les .,o (discreie 'pfo,bability is limited to games .. ~ '\ ,"" ", . ~", If a disti.t!ction between the"analvsis discrete events and contin ubus,.,
of
represen~ations' is the d~nsity ,of. ,e,v~ts, it .should be possible
to
prog;~ss
a
.. from' th~ discrete':event equations to continuo'us"'o'nes. Indeed,. su'c h devel' opinent will be explained' ~tarting.in Sec. ,19.12. First, ~ow~y~r~ the p~ub-: , .ability: of ,several elementarY' situations, involvi~g ratidomevents vlili. ,b e , , explored. .... "
. ~.:.
....~- . ..... "0:-
19. 3
PROBABILITY
No~ ~~ll pla~s fot the future involve"probabilitY. PredIcting 'the ' day o,f the week on which July 4 'will fall in the year 2004 projects into ,the future, - but lies o,utside the .realm of probability because we know with certainry (barring a change in the calendar) on what day' of t.he week July 4, will fall. We are concerned about the situation where there is both the possibility of an ,event bappening and Dot happening. ' '~,;; The probability is defmed as the number of favorable ways an event can occur divided by the number of different ways the event occurs .. What is the probability of drawing the ace of spaces from a 52-card deck? The probability p is 1/52: 1 -
number of favorable ways
p = 52 = number of different
way~
Probabilities can vary from zero to u·nity, with zero probability repre- ' senting the limiting case of. no possibility of a favorable occurren,ce t an',d th~ probability of 1.0 indicating certainty of the favorable occurrence.
19.4 INDEPENDENT EVENTS A set of events is independent if the occurrence of anyone is not influenced by the occurrence of [he others. The probability of a specific combination of two, indepen~ent events is , the' product of the probabilities of the separate
events.' .-
..
500
DESIGN OF'IHERMAL SYSTEMS
Example 19 .. 1" \V~t is the probability of acr-tieving whep. a ~oin is flipped ,~d a die roIled?- ,: I.
th~_ bead-2~combination
J
. ' ·,,'Solution. ,Theprobabjlity P1.ofthe -bead ap,pearing is 1/2 an4 the prol?ability, , " ::': P').' of,~e 2 appearing o~ ,the 'die is 1!6, so the probability ,of. the combination .. ' . . ;'
" .. ' ~i//"
'".,. ,':. '-:', js"(1/2)(1/6) = 1112.
.
': "
_", ' .: ' .'
.
,'.
..
';
..:-:~·~-~,~~.:0g~~gb~s~'E~N!~-·<"·, ;~.:F" ;";-'_C;_~~;."~,./~ -_, .':_;~ :'<:';_ " ~":-;;' _;:"~.' ' ;•. " ' " ,An.::'.~~t~~~io!i' ':~f:', ~he. concept
~f ' fu.d~pe~de~t':~ ~~e~,ts'._'J..s'-: Qlat :' of ' s~c·ce~~iv~ ' ·, . , " ,'"
eventS', where ,.the flfS~ ev~nt a~t,~rs ,the prqo~biI~tY of tl).e, ~econd. event,. • •
•
' •• '
-.- .:
.~-- -~
. '
-
•
• &
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•
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,~
•
~.
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•
EXample 19.2. ' What.-is Jhe probability ,of dra.wing. two aces._ in succession 'a .52-c~d 'deck'? ' , ' . , :, :, :,' , " ', ' '
fI:Q~
~ ' Solu~o"n_ ' The probabiIity'of the fi~t ev'e nt p ~
of
is 4/52 since th~re are 'origii-u~ll:i
,', four aces. 'The probability Jhe second everitis '3/51 if the first event is' ([Tawing ,two aces in s.llcces'sjon ' ,' favorable. Therefore, the probability (4/52)(3/51) = 1/221. .
of
is
I
•
•
•
19.6 ' 'D ISTINGUISHABILITY OF X TIllNGS " 'TAKEN Y AT ..~ TIME Suppose that'~ejght objects. are present and ,t hese' objects are numbered 1 through 8 --:- thus they are ·djsringuishable . What is the prqbability of making 3 extractions and drawing the objects 2t 5, and 7? At first it may ,seem that , ,the probability of drawing the, 2~5-7;, OOfllbination is (1/8)(1/7)( 1/6), ~ince the _probability of drawing the 2 is 1/8, 'then of drawing the, S 'the probability is In, and finally of drawing the 7 the probabilIty is 1/6. This approach overlooks the possibility of drawing'.a 2 first, ,next a 7, and finally a 5. In fact there are 31 combinations of 2-5~7, each of which have the (1/8)(ln)(1/6) probabil ity appearing. These combinations are 2-5-7, 2-7-5, 5-2-7, 5-7-2, 7-2-~. and 7-5-2. The pr.obabiliry of the result of three drawings including .the 2-5-7 objects, then. is
,P ,=
(1)(1)(1) 5" 8" 7 6" = 3~!. 1
( 3 !)
Another way of thinking of the pro~ability in the above sltuation is . to say that the pro.bability is the reciprocal of the number of ways tha,t 8 distinguishable objects can be taken 3 at a time, .1 p=n
-.
C,
PROBABILISTIC APPROACI-IES TO DESIGN
50
, and v/here ' 12! ' (n -.r)!r.!
:, '. ,:::'.'-".:':". ".-: :. ·E~~l~pi~. ~i5t3a;.,;What,.,~:the~pTdbab}Ety:; 0J. dfa.v;inw,4':~\:vhitb )J~S ;:fiOri{~n~:~.,.i~,:·: ·~ : . ", ', .. ' '.cop.raining··.}O "wl1ite'~~. 4 : bhick;· _ 'and -3 red' bal1s? ' .,- " . . ;: . -,:-. "
~~
,', .:
,,"
'-"
."'
',.'
,"
Solution. The number' of ways ,that 4- ~/hiie'- balls' dm be-.chOsen' fro.m .the 10 -. \.Yhite :b~l1s is equal to .the. D.umbe.r:- of ,combinations.-
.
·time, , . "
c··=~ 10 4
'.
4( 6!
, The fotal nl1~ber of ways in which' 4' balls can be·c hosen from the 1Ta~ailabl~ . is -.
The probability of success is
..
'.
lOC4 ' lO! 41 13! '3 p .= -C ::=:: ·4' 6 17' ,~ 34' 17 4, . .1 . . ' ,
19. 7 NON-OCCURRENCE ,OF AN EVENT What is the probability of the ace appearing at least once in' n throws of a die? The probability of the ace appearing iXl one rpll is '116, but ' the probability of rol1~ng the , ~ce least once two rolls is DOt (1/6)(1/6), because the chances ' should be improved, not reduced. Nor is the probability of rolling the ace in two rolls 1/6 + 1/6 as the absurd sit~ltlt:iori after 7 rolls indicates." The key to. the correct procedure i~ to work with "non~currence~" The probability of ace not occuning on one roll is 5/6. The ,probability of an ace not occurring in either of tWo. rolls is (5/6)(5/6), o~ the probability of the ace appearing is f - (5/6)(5/6).
at
in
the
Example 19.4. The probability' that Paul will solve a problem ,is 112 and the probability that John ,will solve 'it is 1I3. Whatls the probability that the problem will be '~olved if both work at it i~dependentJy? ' ." Solution p= ...
0
= I-Pr';J = 1 -
(~)(~) = ')J3
-
DESIGN OF
J..o..t:..n..l ...............
SYSTEMS D
x
1 ".
:
....,V'...A ...... V ! J
P =Pl
+
if p}, were
~.,
PROBABILISTIC APPROAG-iES ..TO DESIGN
. Vi. W' . ' ,
.'
VI
X
. ~'~ B}' . ~~ .~ B
'.
-..
. '.
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503
. ' .: .
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A
X
I
.
.
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."
.
One vJhit.e,
\
J!1GURE 19.. 2 " Prob~b~litY o(4ravving one wruu~' and 'one blac~ b,?ll. " '
-
....
the ,probabili,ty of dr~~ing a bl~ck' b,all rust and then ?whlte 'ball, 'whl~h' is, route ' Y,.is (3/5)(2/4) = 3110 also. These ar~ mlituhlly'exclusive events, so . , th~,'probability 'of one wbite and one bla~k,baH as the' resUlt-is the sU1J1" C?f th~, . , in,dep~rident probabiliti1:S~ . " ' .3/10
+ 3/10='3/5 ;'7".•
'19.9 REPEATED AND INDEPENDENT'
TRiALs '
.' "
This section deals ' with the ques1;ion of the probability. of an 'event occurring ,exactly once Ii trials when the probability of the occurrence bf that e~e'nt ,in a single trial-is.. k n own.' ., . .
in
,
;
, Example,19.7 What is the,probabil.ity of.rolling exactly one"l in one throw, " tWo ~ruows, three throws; etc., of a single die?
Solution. The probability of rolling exactly one 1 in one roll is .1/6 .. Two tqrows:
of rolling a 1 on the first throw p = 5/6 of no 1 on the s.econd throw PI = (1/6)(5/6) = §/36' (b) p = ,5/6 of rolling no I, on the flISt throw 'p = 1/6 of rolling a 1 on the second throw P2 = (5/6) (1/6) =' 5/36 (a) p = 1/6
Sequences (a) and (b) are mutually exclusive, so the probability of rolling exactly 'one 1 in-two throws'i~'5136 + 5136 = 5J18.
Three throws: The three mutually exclusive events each have a probability of (1/6)(5/6)(5/6) = 251216, so the probability of rolling exactly one 1 in three throws is 3(251216) =
25n2 Four throws:
'
p = 4[(.11-6)(516)(516)(516)]
= 250/648
~,
Table 19.1 shows probabilities fur one to ten throws.
'
504
DES1GN OF
THERMAL SYSTE¥S
TABLE 19.1 .
. Probabilities of Irollmg One J in"a .. <.;<" ': .' .giv~~ nuiriber throws .
.... . ...' exactly
·of
.4 .
, .
..; 5
. .
' ... ~;'.. 0.3858
' '. ':.' . . /..... .
.. . 0 .. 4019 .; ... _-.
, 6. .7 ,
..
.:8 , . ,9 10
1 1
: 0.40 19' .. :. : .',0.3907 .
, '0.3721
,
.'
I
.
I
.
'
~' -
'
.
"
0~3489
" 0.3230
r. , _
.
,
.19.10 . THE BINOMIA~
,.
'
LAw
..The I:Ule appiicable to. the situation in Example 19.7 and to certain more general probabilities as \yell is, called the' "binomial1aw" and is as follows:. If . the probability 'of artevent'occumng in a single tpal i~ p;·then'4e'~bility , . ,~ ., :,:!c< ' ~at -it will, occur exactly r times in. n ind~pendent trials is .
'
Pr =n C r
pr
(n~r)
(1 - p), .'
~
l'~ " ,
(19.2)
•
Example 19.8 . . Use the 'binomial law to compute the probability of rolling exactly one 1 in 8 thIows of a die.,
, SOZutio11 p = 1/6, . n , "C r ' =
so
p,
= (8)( 116) J
= 8,
r =.]
8!
8
CJ
= 7!
(1 - 1/6)
l! = 8 (8-J)
=
0.372J
Example 19.9. What is the probability that a 1 will appear exactly four times jn the course of 10 throws of a'die? , Solution .p
---.--
so::
116.' n:= 10,
r
=4
PROBABILISTIC APPRO.-~CHES TO DESIGN
50S
·'9.,11 THE ~RO'BA1B1Lll'-j{ .D ISTR~BUl'JOr~ clJ.1c~lati\ie tool that leads 'to the developrneJlt of ' the probabi.lity d~sllibut~ori curve ~ Consider ari exatnple an electric utiJity , , ',' th~t knO\VS th,a t 40% of the l~eat 'puI(lps in its area of ;Brand A and 6{)% '. ', .... '-_..--~ .. -QLB:r.a,nd·~r -hH, t.-dQ~s-,'nQt,kDovJ··, ~n .i.vhich: l1oine~ a gi vetA br~J)d is'·i'nstall.ed.jr .
·The bii1oD1ial Jaw is fhe
are
of
. ,, ""·:."\v.isbes, to rg*.~:a~ 'i~~:pectioii o{1~e:·'h~~fp~4·tf.ip-~~. ~~h~(w.6til.(r~i~~~~~,:·~'~~~i:~i"in". · · ~-'· fand{)m sampling whai.its·prohabilitY~is'·of.cho.6sing. one brand OJ[ .the·: oili·e~L ·"; · Figure' 19-3 shows the· predIctions ' from the. binomiai ,law of the -
•
I
.
"
'0.077,8
2 -3 4' 5
0.3456
••
'
number' o.f "
,
I
, 0.3 , f-
pr -
0
'
0.2592
n=5
-<:' ~
max
.0.2304 .
0,1
'0.0768 0.0102
f-
--
rL'
O~~~=-----------~~----~-----
012345
~
r
0.3
0
0.0060
2
0.0403 0.1209
n= 10. 3
0.2150 0.2508 0.2007
4 5
r-
-<1----
max
.....
10
r
P,
3
0.0003
4
0.0012
.,.
..
to
5
10
fJ·2
0.1396 0.1474 0.1360
0
0,3
.............
.,.
II n = 30 12 13
~
max 0.1
. . . . . . ...
22 ,
0
k ,
r
"
0,0016 0.0001
r-
0.1
.. - ......... ... 9
1-
r-
. 0.2 -
0.0002 ....
I
30
ill
I."
..
a
0.0000
0
0
S
10
IS
20
25
FlGtmE-19-3Probability of selecting Brand A hear pumps if the probability of a single event is 0.40.
DESIGN. OF
p
1 2
3 4
0.201
5 6
0.0264/0.0189
7 8
9 10 0.4
SYSTEMS
n = 10 .
r
0~2
TI:iERM~
5 6 7
8 9
JO II 12 13
110.0904
1 0.000810.0001
n = 20
n
=
30
·n
40
507
PROBABILISTIC APPROACHES TO DESIGN
( i ) the agreement between the two equations becomes progress,lve:'y bette,'~ \-villi -large values of n arid r? and elY the peak value 'of P r in each of the ei.ght. blos:ks of, Table 1902 ocCurs ,when r = np. Thijs, for'p ~ O ~2, ille ,maximum valu,e of Pr occurs at T == 2, 4, 6, and 8 for n' =;= 1 0~ 20 ~ 30~ and , 4G, respecti\~~ly., , ,:, '} , ," " ", , ,"
" : ,",'~, " ~,,> ~" , .. ' )"
:" Dle_ ·t.$1~:whoJs~"ijs,ed,~-in, ~ ,~iJ?,e .;:.,llorrnaL.p~op,a bili ty ;~ distriburion~,cug;;e,Nlr.e(-::j-~;~ , us~aUy _ 4i,lf~i~nt tha·n . [fH~~.~;" pl}~~Jn j?J,~J;:q~ . :,(lSLJ.}.~-.. and~ while ,~-there-:,' is', no:~.- ,',: " ',' con1plet~ standardization~ a s.et of 'symbols often 'see'h .is, ' " '. '
.
Pr '=
,1 " '
e "-(..r -
t'
)'2/J,....'2.
'0
, (-,1.-9.4)
_'-'
, ', u,/2 rr'.
When .Eq. (19.4) is , graphed; as in Fig'. 19-4,- the . curve takes 011 the' , characteJ;istiG' bell 's hape 'with the peak value'of Pr = 11 a.j2Tf' when x '.= x o. The cUrve is .h r9ad for large values of (J ~~ is ~l£tttow '"vith a high- peak ' 'when u'is small. " , 'It sh'o uld 'be P9'i.nted out that the ordi:p.ate of Fig'. 19;.4 is no.t probabiiity, , but ptoba.bility density. In ,other.words,- the oIdinat~ ,alone does nat g~n~rally .provide practi~al inforrpation. The area unde~ the Pr . ,,~yrY·e indica~es ,t he, probability; for example, the area between x I and X '2 In Fig. 19~4 represents' the probability of the event occurring between these x values. Even ':~for' the 'discr~te probabilities._shown ih Fig. 19-3, areas were :impIied became .. ..
0.2
L_~ ___ _
?;>
.:;;; t::
4)
'0
.q
:g
0.1
.D
o ~
-10 F1GURS-I~ 1
o
20
...
Normal probability distribution curve.
-
-f)ESJGN OF THERMAL
~ySTEMS
- 1
- <. 0" to x 0
Z u.
a ...
""'A, ...........
PROBABJLlSTIC APPROACHES TO DESIGN
T/li5lli
:;1.).3
Pr' .ba.bHity P (art.;a :unae not :r3lru p:!;obabpity cui"vej ,~ '''Yee,11 x " - zo- and.· x o + z a .~
, _ .: Z
p
-f
-_. z ·
.p
-.-., ,, .~ .. q-oO. ,; ~.~~ : O~QQqO:·,J)·~~· ~J:,99:i!:':': -:;u9..&9"-$.i7.~,~.:" , ."t oO:') ..'. "_.: Op9~4'~: .;' '.
.' _ '" .'::' .:. .:.- -. ._. " ._ .
. . ·'-0;05" . '.0.0399'- .:. .... -k~(JS'~ ..... 0.7063 .. 2.'05-- :" ""-"0:9396-' :'" F' ·'.· . . . 0.10 :'0:0797' ':'. Lfo '-": ' "O~'72'g7 ~ . -::: 2JO" '. ' '0.9643 ' '. ,: . .. .' ."-:- -':" , '-,'. 0.15 -: -' .0:1192 - .' L 'IS 0.7499 '- '2.15 '.',-' 0.9684 ' , ' -, . " .: , , . .- -'" 0.20 . ,0 .:1585 1020, ' ' 0.7699 2.20' '.: 0.9722 , .- 0.25 .' . 0.1974 ,1.25 ' - - -·O.7~3'87· -'-2.25 O'~9756 '. ' 0.30 . 0.2358 _rt .30 0.806..1 ' , 2.30 ' 0.9786 . - 035 . O.i737 1.35 - 0.8230 2.35 0.9812 p.40 .- O~3108 .· IA.Q OJ~_385 ' . 2~40 · 0.9836 0.45 '0.3473 '. - ' 1.45 ' 0.8529 . 2.45 : '-. 0.9857 ' 0.50 0.3829' . t~~ :· , '0 .8664 ' 2.50 0 ..9876 , - .0'.55 0.41i( 1.55 ri.fn89. :2~55 '., 0.9892 O~·.60 0.4515 · 1.60· 0.8904 2.60 . 0.9907 . 0.65 0.48:43 . 1.6.5 :', -- o.90~i . 2.65 0.9920 '" . 0.70 0.5161 1.70 0.9109 . 2~70 "0.9931 ' .:. -. 0.75 0.5467 I. 75 ' - . '0:9 ~99. 2.75. '. 0.9940 0.80 0.57.63 - 1.80 0..9281 2.80 . 0.9,949 . 0.6047 1 ~85 Q:_9357 2.90 '0.9963 ' .0.85 0.99 0.6319 ' 1.90 0.9426 3.00 . 0.9973 0.95 0.6579 1.95 0.9488 . ·4.00 '0.99994 . 1.00 :0.6827 2.00 0.9545 ' 5.00 . 0.9999994 '
0.06
- 0.04
0.02
o~----~--~~~~--~~~~--~--------
20
24
22
. 28
30
32
Temperature. 0 c FIGURE 19-5
.
Area showing P of temperature exceeding 30°C in E~ample 19.10.
---.
509
~ ~~~~~~~~~~~~~~~rr~~hrt+~~rrTi~Hrr++i~HrrrtTTi~Hrr+Ti~rr++~~t+~~~t+~~~~~
;~----------------
-~ ~ · ~~++~~~~~++~~+f~H-h+~~tf~r'~++~rt~-r~~-rrtt;~rtf1~rttti1Hrrt~~tt~Hrt7~rt.~~rt++~~~~~ ~. l
-.
., I
~ ~~Hh~~~HhHHHHhH~44~~~~~4H~~~~~~~~++++++++rrrr~~rr~~~~~~~HH~~
~~r+~-r~~~~~+;-rr+~-r+;-r++~~~~+;-r~~~~-rr+;-r+~Hr~~rr~;-~~~~;-r+~rr~-rr+;-rr~~r++4~ ' ~
c
C(.
>, j
L
, I
f
c
r
c
v
= 510
PROBASfLlsrIC APPROACHES TO DES1GN '7
'r
\ ._- - - - , ,.,
I
).
I
.
.
. . . .. .. '.: : 7 ~ . ~. ' ~.
:
-".
,'.
'
ri1'<,-",.< ¢ c ; P ' "
B '"
,
I
'
.
.
'
1__
:
..
-- '.
:
;.-
. .
.> ,,,~
;. I ~
,"..
:.. : .. '.: ' :'. . "~":'il ' "
:-
#-'"
.'"
. '
". ...
~
.
A. ~
I I
.
511
"
i5.87%
"
I
':
I
:. 1
.
',A- :,'
. ' ,' ..
yafor B
' _ .. '
,
. ... . ,
84~J4~
. .-'
·50
, C:~~la~ye percenr
.-
FIGURE 19~7 - . T\\~o differ~nt normal disuibutions sho\\:n on the arithrneric probabl1ity chan.
s·id~.qf..tbis·central area is 1 - 0.6043 = 0.395'7," and 'one-half of this 7,_or . D~1~79, represents the probabiliry of the temperature exceedjng 30°C~
The a tenn appears so frequently in p~obability work that it is gi'(en a special name. It is called the "standard deviation." The area under. '~e ' normal . probability . curve extending one standard deviation below to "ope standard .deviation above the most probable value of x is 0.6827, tbns embracing 68.270/0 of the possibilities .. The process of m6ving in tbe opposite direction t namely from physical data from previous experience to a nonnal probability curve or' eql!a-tion~ is based on the relation·ship betw~en z and P as presented in TC:ible'-19.3.' ..~ graphic to.ol for the translation is arithmetic· probability paper,-;a sample of . which is shown in Fig. 19-6. _The cu~ulative percentages are computed and plotted as shown on the skeleton diagram in Fig. 19-7 for two differen t distriQutions. If the line is straight, then the distribution is a normal probability. The point :.at which the line crosses the 50% cumulative percentage gives th.e Xo v'tilue. The standard deviation ,is the difference in x 'vaIues from 50'% to 15.87% or 84.14%. The line with the greater slope has the' large,r standard deviation. , If arithmetic probability paper is Dot available. it can be constructed ~y scal~g lOO( P /2) from Table 19.3 in both directions from the. ,50% cumula~ve percentage. I.,
FlG~
Arithmetic probability paper.
512
DESJGN OF THERMAL SYSTE..MS
,.
F
19013 Ai)DITION AND SUBTRACTION PROBABiLITY FUNCnONS' '
frequent assi;~eh~ is to ,predi'ct a p~obabilitY' expression that is a fuilc-, ' , ;" t:19D of other probabilisti,c variables. A simple exampfe is to preditt the, ' ,'):;:~': ,:". ';,: ', probability ,-fup.~tion for pr~fit of an _enterp~e ~here the profit is the _ '. ::~,\,~~,:-,> _" ,':,'_"':'difference between tb~ ,ip.cDP1e,:and,.)e.~~p.ses9,. bo,th-,of+whj.ch 'are ptobabili,s uc ' ,
A.
the
.• .,. .
'-~:E5:,-~-t;~p'~~Si~>~; .. :, ~ " ,,::~:~;,( '~;~~'-"~in~~k~:,~~~p'~~e_~fih:_~~~;,;,:;~."~:~:,~",'~ ~:~.:.;:., " , , 'A more
diffi~uli sitUationis~hen~-several~Poscl variables oria system
, ,' , ,', slmulatio'n -~e probabilistic in ,nature'
and ~ave ~'coIl!pl,f?x ,influ~nce oIft~e,
- ' simulation. We frrst consider the -:mo~t fundan;:H~ntal and simples(case where ,, :two probabilistic functlon:s are _combmed through ,addi~iort or s,ubtractio~. , Even though thi~ case falls short 6fansw'enng the -chailenge' o~ a sy~'tem ,simulation, -the '-results indlc~te some characteristios ',of ' combinations ofprobabi1i'ties, The objective ,will be, to detel1l1ln~z in the equation ' z
-'-Y'-'w
(19.6)
y' 'a nd
w are probabilis,t ic functions, as shown in Fjg. 19-8. More ',specifically', we seek the probability densitY PI. as a function of two nonnal
where
probability density qistributions py and 'Pw.
r
pw (Tw
-
.,
')
e -(w-w.,0 )-/2a;
\4'
•
"
.J211'
~d
1
Py CTy
For a difference between y and
or in general "
'so
e -l\--". 0 )2/2u..'r
2'
.J2'TT H'
of ZIt as shown in Fig. 19-8,
PROBABILISTIC APPR~ACHES TO DESIGN
w. y.::
(~ame
513
units)
FIGURE 19-8 .'.~ '. , .,Differ~nce of two'prob~bility functi(),~~_,
-. .. ..
.
-.~
For .co~venience~ let W == . p_ = -
"1 ';'wUy
w ~
fa
. -~-;!' ': .
V21t
e
Wo-
d14'
= dW , and 11 '==
-w 'nO"~ ' e -(W +=-il)'/2O:
T
2
Yo ,-
Wo-
dW
-co .
Next .perform. the following steps: (1) combine . the ' exponents of' e .. (2) complete the square for W, and (3) move the nonvanable tenns outside the . integral sign to ot?tain, -
P: =
I
exp
(z 2
1l)2]
[ 2(o:v +
.
2
U,.)
f
co
.
-
. 2O'y
-co exp
uwO:v .J27r
(0-.• 2 +2 y
.
a w 2) ')
£Tw -
~
(W +B)-
.
.
dW
' ( 19.9)
Let (z - 6)uw 2
.B = - 2- - - 2 Uy
+ Uw
'
.">:
" -
For the tenns within the integnll, let W * = W + B, so dW =dW*wilh no change of limits. This integral is of the form
.
which~hen .. .....,
-.
.
: ...
integrated yields the desired expression, ~
DES! ON OF 11iERMAL SYSTEMS
(
(
Z
ue
z
----..
=
Yl't.'
occurs
PROBABILISTIC APPROACT;fcs TO DESIGN
515
)
in some fields 1,2.3,4 ·to assist in . making de!Zision~ when available infoITl.Jation · is uncertain or at best pi-obabilistic_ In struct:.ral systems, civil engineers k110\V that the probabillry of a f~i1tL-e always e~{ists, but the designer must " assure tI-lat this probabnity.js prohibit-i'vely ·:low:' ·-The· primary' concern of ~ : ".'~,~sig;.~~.- .of t~ml·al systems~,j's~to ~pf6vitte ·-·acieqli~le' c.~pac~~.ty :i.n p.p,,,~;c9~p.!ni.~ ...,..;. .'rii~~~r ~d a fa~~ure'·:bfJhiS'··.g qal. ' is·:,u~p.~lY~}~Qt ~catas trop.!ll C.o'~_The..\designe.J;Sj,;,.; "~ " of a thermal, systerq, for a· builq:.ing.'.c9tl~d-.-be':vvell ~se,rved' :by'" the-.,ability,:·to :·:, ,predict the :probabil}ry ihe''-loac( exc,eedl#g ~'tiie capabilitY· of.·the· syst~m~-:, . '.' . If sever21 choices bf system' s'izes ,w ere available having differing capacities and costs' ~ the owner'would ·.be in. a pos{tion'to balance actd:itional cost agq.~~-1st, ' . frequency that a.ir temperature in the space exceeded a comfortable' leveL .The rate at which· applications 'of probabilistic ' ~pproaches to design of '. thennal systernsadvarices·\vill be inflll~riced by .t he pace at \vhich designers, recogmze· that they ar~ dealing 'with prob~bilistic situ.at~oris, and by the pace !',
'ot
ar which calcul~tive, to'91s develop.'
19.1. Two sou'fees of natural ga? shown as A and B in Fig. 1 .9-9~' each have adequate eapq.city to supply the citj. Sources A and B are each outfitted with , a compressor, .1 and 2, re.s pectively. In addition~ compressor 3 is located 'at the junction, and this compr~ssor must operate along. with either both ~o~p!essors 1 and 2. The probab,ilides of the compressors' faillTI'g ·are: . 'compressor (," '0.008; c~mpressor 2. 0.0012; and compressor 3, ..0.,00002 . .·What js~ the probability of the city bejng witho1ut 'ga'S'? ," ',.' '. ; .,. ~ ,:'
or
19.2. A ~hain measuring 30 lengths is composed of 10 sjngle links and some links that are not as strong, so they will be doubled in each length, as in Fig .. , 19-10. During an, unu.suaIly heavy 'load the probability. of any of the slngle links failing is 0.0002, and the probabHity of a weaker link failing if it were used as a single element is 0.012. The ' double link doeSD't divide the load? . but one link either sustains the load or breaks and transfers the load to its partner. What is the probability of the chain breaking during the heavy load?
00 Q)
'-_lkllto-------./\ Ciry
FIGURE 19-9 Gas-suppJrs-y-,rte-m-ln Prob. 19.1.
~ ~, -
' S
'x\!' '
. 516
DESIGN OF TIffiRMAL SYSTEMS '.
] 0 SjngJe links
20 Double links "
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19'.3 .. 'A condens~te "iine,' Fig. 19-11a. ~~ty~s lU',steam ~oils whlch at'design" " .. ' , ~onditiop~ have' a ~te~ flow rat~ 'of (L03 ki/s ·'each. 'The' condensate ~ps', ' . . . on each ofth,esecoiIs can 'dI+IDp'at th~'iate of 0.15 kg/s; so tPat the flow rate " " from one trap has th~ fonn shown 'in ,Fig. ,19-1 lb'.' What are ,the probabilities' 'of the simultaneous d.llmpi~g 0(0 "tr~p" i ,tI?1P9 2 traps,' ....... 10 traps? {FUrther refeJ;"ence: "Sizing a 'Condensate Return System ,by ,Monte Carlo S~mulation," by"\V~ C."Huang and Y. K. Lee, H~atl1?gJ Piping a:rd"Air Condition.lng, voL, , 45, no. 7, pp'- 44--46-," July 19~3'., ,"-", : :"":"':'~,~:,",'~
19..4~ In l~boratory buildings equipped with ftIme hoo'ds, the air conditioning system' must be designed to treat the mak~up air ,entering the building when th~ fume,hoods operate. In a ce~n Iab0I11:tory building there are)O hoods" and experience sh~"Ys, that any hood is in operation approximately' ,113 of the time "and in a ran90m pattern. What' is' the probability ~at 15 or mor~ bo'6'ds' win be operating at a given time? ." Ans.:,D.0435 19.5. A refrigerated warehouse has 12 large doors through which prod U. ct is transported. It is h:npo~nt when designing the refrigeration plant ,·t-o know' how doors ;;ire open at a time in order to provide adequate refrigeration capacity. The operating pattern at this w.arehouse is that any given door is opened on the average every 10 minutes. When opened, there is a 0.4 probability of it staying open 15 s and a 0.6 probability of it being opened for 20, s. What is the probabili~y of '2 or more doors bein~, open at a t~me?
many
.- ~ '.
Coil
~~
'.
Coil
'Coil
Coil
f - - - - - r - - - - - - - __- r -
Condens~te
(0)
FIGl!RE 19-11 Steam traps (b)
(0)
b
-t-,
-
a
f10w
Time
(b)
profile for condensare line in Prob. 19.3.
PROBABll..ISTIC APPROACHES TO DESIGN
l"'ABLE 19<-4
.
517
I
DiStribution of monthly eiect:r·c bins ~n P:roQlem 19 ~ 6 . r*.-1Di1thlY b?111 , fionars
'NUf"ilber of
f./lonth1y t~lll~· dollars ·
N urnber of customers
custom ers
e-t:i6tllilJ1~l~:';""6::';,:- :,:,':c ;-' -: ,;:~:':::'~:: ~'::: :~,,; -'~' '....,.-: . " . 36--40. 40-44 . ,44-48 . 48-52 52-56, 56-60 . 60-64 ~6g
(i8-72 72-76 - 76-80-
~ .... -":.:' " ., 1. '.
.' . ..' : 88~92 .
:1
92-96.
1
. ' ,96-100
2 3 3.
j
4-
~D9-104
4 ..
. 104-108
3 2.
. i i'2~t 16 ) l&.l~O. '.,_ . 120-124 124-128
I :I .. 1 I .
Over 1 ~8
. ··7
J,.~.
5 ,
10E-ll2'
5 .' 5 5 6
. " .. :7 ·"".·'.... ,
-y
0
19a6.. The':D'umber' of cu~[omers i.~ each .bf the ranges o{ monthly electric bills is ' ~ .shown in Table 19.4. Plot the cumulative distribution arithmetic nonnai paper and (a) determine whether th.e distribution is normal. and (b). if normal.: determine' x 0 and the stanc.Iard deviation. 19~ 7 At-a certain tim~ of the day there is a 30% probability tha~ the electric kitchen range in any of the apartments in an 80-unit building will. be turned on. (a) Denve the continuous distribu,tion equation the pro.l?ability density ~ a "functio'J1' of the total number of rane:es turned on at one time. (b) What is '~the prob.ability of more than 35 ranges at anyone time?
'on
a
of
•
........
~
J~ , . .
operatin£
~
. Ans.! (b) 0.00367 . . 19JL A manufacturer' in one month.:·anticipates th~ most probable numb'er oruni~ . that it can manufacture is 2200 with a standard deviat~on of 300. The most . probable selling price $380. with a standard deviation of $40 and the most probaQle manufacturing plus sales cost is $275 with a standard d~viatjon of . $25. \Vhat is the most ·probable profit for ~e month' on this prod~ct and the
is
. standatd 'devjation when expre~sjng the function as (a) (number of units)(price - cost), and (b) (number of units)(price) - (number of units)(cost). (c) Propose ~iifferent operations that represent .the two different expressions of parts (a) and (b).
REFERENCES L E. B. Haugen. Probabilistic Approaches to Design, John Wiley, New York.
1968.
2. D. F. Rudd and C. C. Watson, Strategy of Process Engineering. John Wiley, New York ..
. 1968 . . 3. A. H-S. AQg and W. H. Tang. ProbabilitY Concepts In Engineering Planning and Design. John Wiley, New York, 1915. 4. W. H. King, Probability for Managcrrunt Decisjon.!, lohn Wiley, New York. 1968.
~.
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,
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'
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:,
This "section 'o f the" appendix "presents some 'sample Projects whi~h apply principles studied in the text, ~. g., economics, equation fitting, simulation, a combination of them. These problems m'ay b~ used as optimization, projects accompanying the study of the text material and running as a parttime effort all term, Many instructors devise the~ own similar groblems based on their own engineering experiences'. Some of the problems 'may carry over from one teIm 'to the next,. with one team of students picking up the work where the precedipg group left off. Engineering students become proficient in solving short problems such as homework problems which require 45 min I but most professional engineering probl~p?s ~e long-tew, requiring weeks or mon~s'for cqrnpletion. It is therefore appropriate for ' senior-level or graduate-student engineers to gain some experience with comprehen~ive problems which require discipline to maintain' progress over a longer period of time., Also, 'at .the beginning of , aay long-term project there is the pefiod of deliberation ' on how to start the
or
problem-how to find the handle. Inexperienced engineers spend considerable- time spinning their wheels and making false starts before focusing on a valid solution. Experience with comprehensive projects is the best means of deveJoping proficiency in thought and' work habits. Written or oral reports mak~ good targets for completion and have their own benefit 8S welJ. 518
-.
5~i~9
COMPREHENSIVE -PROBLEMS "i~ .'
\'ha.l' fO n {'.~Vtr {'- o n~~q'- of-......? st<;q'PTce.nt of_.....the "'orob~ern \;\ ,birl:, ..1·....- . \ _. contains s'ome or aU of the re quireddata~: In cerrtain cases property data vril1 , have to 'be. extracted· from ha.ndhooks. Vlith some of the proble!TI staternents -,vill, .1;Je LT).chH~e~, SOfDe bJjeJ COITnuents on: the ~xper:iences cf groups \:t~ho . .have'vJorked on the problelTI . . .. ' " ,'~" . . . 7bE .1,1nTQ;,<>rtr-; J . 1., •• - ".
".
J _ .... _
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, . . :.: "..:" A--1l:., -€)pti!numt.te~I1p~rature.~..diSlJ:i~u,tion,j1}.:~a:"muitistage;'·fi.asb:ievapdrat~un·<·:/··:·;· :-" -:,:,'..', ~ . . ".. ~". '~ :-,des,~ma~ion' plant . " ":'" :_.: '~'•. ' .': ... :.....:, . : ~ :"~', ...: . '. '. '. : " ~ ' ...... . . . ' '.'. ,·.A~2,: :'Hea~ recovery frorn 'exhaust" air with ·an ethylene 'glYGol 'r unaround . . ' . ' . . "--: :' "',' " ,::" system '.' :"' A~3~ ·O.~sign of a.fire-\7vatergpd' : . ' ..
I.; ':"'.::_:'" :.' ',.
.
-
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A-4o. qPtirnum thic~ness. of. insulatio'~ in 4 refrigerated \varehou'se . . 'A~5 .. Sitnulation, of a liquefied-narura.l-.gas facility, .' A-6 .. Optinllzari~n of a natur~~onvectiori''air-c601ed .con~~~ser A~7 .. EclJancing the '-he~t-traTI:sfe~ co~.ftis!eilt of boiling in. tUbes Recovery o'f~eat from exP,aust·air using a heat puu:p .
".A-So
A~9., Simulation. of a dehu.m.~~ifier for industrial dryi.ng
":A. :.10.n Optimum gas pipeline when . pipeline
.r~covering power at terminus of the
'
A.,.11 .. ,Conserving energy_by using a refrigerant mixture ,A-12o Optimum air ejector for a steam-power-plant cond~nser A-13~,
Optimizing a hote-oil loop jn a petrochemical plant . . A-14 .. Optimum heat pump for pasteurizing nlilk
A.,i OPTI1\1UIv.r TEMPERATURE DISTRIBUTION IN A lVIULTISTAGE FLASH-EVAPORATION DESALINATION PLAl'fT , Flash Desalination One of the methods for water desalination is a distillation process usingmultistage flash eyaporators, as shown schematically in Fig. A~ 1. Seawater flows fust through heat exchangers, on willeh vapor cond~nses to form fresh water. After this preliminary heating of the seawater in the condensers, , a stetm;l heater elevates the temperature of the seawater to the maximum 'permitted by corrosion Jimitations. In passing through· the frrst throtf.lfl1g
Valve some of the hot brine vaPOrizes, and this water vapor condenses on the tubes of the condenser and drops to the freshwater collection pan. From the fu-st stage the brine and the fresh water flow through pressure-reducing valves into the next stage. Figure A-I shows a two-stage plant, but conunerciaJ plants have many more stages .. A"·Poster-Wheelet-plant in San Diego, for example, has-rune
stages.'
----
-=-
..
· DESIGN OF T.HER.MA.I; SYSTR1vfS
Condenser
,valve
1
same as \vatef'.
----..
COMPREHEN,S' ~ VE PROBLEMS
.JD~IS5~f~lrh;;;;;li~~t}~
:
,
)
,
One approach is to start \vith a single-st,age p lab!., \vb,i~h provide~ experien.ce' in settiytg up tb.e th~ffi1odynamic a~1d heat-transfer equations',. An in.teresti~g" , '~ 6bseAVation is that the ,tni1jimliID stage.temp~rarm:e ' is the ,average of 'the. ' , , " Sef\'l/ater teID.t'~rature, 'a.T1d-;Jh~;)nax:ir.qum:","QIjJle. JemperaulAe~ " tb.u~, (15 _~:""':":"", ': ',;12C)L2),,,',,. ",~67:5~C;:":~:~TIS,~,n1'ihjmll~..tetq.p,eci~e,,occUf$~\;w.i.tl1:,!,~~;t;,0 n'd~rise~'l.0,fj"~~~':~~·;~ : itlfmite ~ti(ei:.' The 'opti~1um ,cp.pq~nser.'areaTc:sultstTaction of a 'degree higher IDlli'J:," 67oSoe, , ' . " .' ,'" " ' : ' " ', ," , For a two-stage plant~ infilli,~e ":~e~ in, the two con~e~$ers , n~sult~ "in . stage ,temperatures ¢.a~ again :diili<;le the 15 to 120a C Hinge equally. but ,tbjs , tin1e into' thirds~ Thus ,the' TIl111injuni .'t emperature of the 'frrst stage' is, 85°C ~d 41' th~ ~e'cop.d s~ge 50~C. : . " , .' This problem can be extended into a two-J~vel opti;fl1iz~tion. 'Th¢' ftis't level -is' to determine the' optimum te~peratilres i?'~ say;- a two-stage .plant. The :q.ext level ,is, to optimize ,the nu'mqer ,of stages and for each different number of stages 'perforrri a new temp~ra,ture , optimization.
,'+,', ,:' .'
,
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I,
/Ao2 HEAT RECOVERY 'FROM EXHAUS~~ __" AIR WITH Nj,.ETHYLENE GLYCOL ' RUNAROUND SYSTEM
Introduction -, M9st large bUIldings ,have ventilating requirements whic~ bring .in olltdo.or' air and exhaust an equal amount. \Vhen the outdoor temperat~re is low, the cost of heating the outdoor air before introducing it' irito, the bt}.ilcllng "is appreciable. This heating'cost can be reduced by recovering sOn;te,.heat from the exhaust aiJ., , One method ofrecovering heat is to place a fifuled-,coil heat exchanger in the exhaust airstream, another in the outdoor aiistream and to pump a fluid between the' coils, as shown in Fig. A-2. Water would be the fIrst· choice for a heat-transfer fluid, but to guard against freezeup at low outdoor temperature, an antifreeze such as ethylene glycol Diust be added. , If the system"serves "any purpose .at'a?l, 'it must s.a~e ,more ,money ~ he~ting costs than it requires for its own amortization 'and operation. The first costs that are to be amortized ' include the ' costs of the coils, pump, piping, wiring, additional ductwork or revisions thereof, and additio'nal cost ~f, l.arger f.~~< 9r . Plqt9rs, if needed .. The operating costs ·include the ,power for the 'pump and addi~ional fan p
1.
L~I?gth, of ,tl:l~ _~~~ ~n t!l~ , cpil
2. Height (or number of tubes high) of the coil 3. Number 'Of rows of tubes deep (parallel to the path of the airflow)
522
DESIGN OF THERMAL SYSTEMS
~doorrur
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F1GuREA-2 .. Heat-recovery ~ystem.
4. Fin spacing 5. Gly~ol flow rate
': '."
are
The qualitative effects of these five variables as follows. Increasing'· the length and number of tubes high incre.a.s.es the. heat-tqmsfer area but . also the cost. Furthermore, increqsing the length and number of tubes hlgh increases the cross-sectional area for airflow, which reduces the velocity . and decreases the air-side heat-tran.s fer· coefficient·· but also decreases the power required of the fan to force the, ,air through the coil.. Increasing the ; Dumber of rows of tubes deep increases the heat-transfer area but increases the ftrst cost of the coil as well as both the air and the glycol pumping cost .. Spacing the fins closer together increases the· cost of the coil and the air-, pressure drop but also increases the heat-transfer area. Finally t a high flow rate of the 'e thylene glycol increases the glycol-sjde heat-transfer coefficient but also increases the pumping cost. ' . t •••
Further Data and Assumptions The size of the optimum system will pertain to a given airflow rate, and 3.0 m 3/s has been chosen for .hoth the outdoor-a.ir and exhaust-air flow rate. The coil circuiting chosen is thar of vertical headers feeding horizontal tube . circuits in parallel, as shown in Fig. A-2. The flow of the glycol through the U bends is counter to the flow of air. Assume pure counterflow in the coils; thus the equ~tio~s that represent this ninaround ]oop are those used in Prob. 5.14.
i.,1
,',
.
COly1PREHEt'tS1VE PROBLEMS
-Further sp-e-cHkations
. '-". -
I'he copper tubes h~:ve an 00 ' o( 1 mm. Ti~-e
.' ..'. . >-
523
mra and ··a wall thickness :-ot
16
.
-
vertically £lnd.horiz6n~ny b The aver~ge outdoor-t~rrjperaturf .is 'jOC" fc:~' 250 ' days .ot·. 24~hr, ~}J~P- " tllbe·.spacl (rg ·is 41
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.
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-m.ITl
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·, ,
·
.
, . ".:.-.;. ..:' . ~. The·.'iif~' :is: I 0' .yeai~~' ·.~n~t th~' ,i~~ere:st . ~.te s. 8 pe~cent '.
.'
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:.' .. The electri~lty cost 'is .4 ce,nts p·er'·kilo~.a~~ur~ . . ..-' ,'. .. . " 'The 'fan ~d PU!Dp:- ino~or ~fficiencies ate·.7? percent.' :': . ,. ' .
. . ' - The- '-ne t sa'v ing to ·be. maxuPized .is .the diff~rence'. between .the: .red ucition . .'. :-~. in .h~atit1g COst' and the' .ahn:ual cost '(ainq~zed, ' fuSt plus . QP~r~ting . .'cost) of the recoyery'.system . · ~t? b.uildin,g "is.'h~ted .electrically, 'so as far " ",,' .. '.' -as the ventilation aIr is ' coric'emed~ electric-reststance heat must \v.ann :the . ._ '. ':, j-n~omirigou~d6or air· to t~e'b~-H~i~g tem:p'e~~re of 24°C. The · fi~st·'6os· r:t:b· be -, . : 'a.mortited i.nclud~-s that bf the· coils, rhe.purnp ' ~d mo~or (3:s'stimed constant' _ . .' at. $400)., 'and ~e .iIlterconrte.cting piping (assumed cOQ.srant · 3,( ·$150). No :·alIQwa..nGe ,faJ c;apital c.ost for the fan' is ,provided' because fans already ·ex-~~t .. in. the system, and' it,is a,ssuined ,that th€y · w·o\lId · no.~,h.ave . ,to be enlarged to . " ovet~ome .the additional pre' ssu're drop of rbe"'heat-rec,Qvery coils. '. . . ..
'cost
-
-"-.'
.:...Cost .an'~ Perror.mance ' ~qllati,ons : The equation for the fIrst cost o( each coil.roughly reflects the proportionalitY of the cost to. th~ mass 9f the coil; \ Cost ~ O.26[fS where NF ,
,+ (0.024)(NF +
'.fm spacing along
500)~V] [L(NR
+ l)J
. dollars ..
the tu.be, flOs/m
W '.= number of rows of tubes deep (in direction of aiIfIow) L' = tube length" in . , NR = number of layers of tubes high
" The-U· value of the coil
1 . 0;0226 =' + 0.0032 'U -. V 0",,8 , ' o . a " .. ,
-
'area is O.OO88NF + 0.185 + . .' .. . . -' .
based 'on outside. (air-side)
"
43' OV ~.~ _' ~g~_
where U o = U value, W/(m 2 • K) Va = face v~locity of·air through coil, m/s Veg =. velocity of ethylene glycol in tubes mJs (
.
t
The thennal resistance ·of the rubes and flns is 0.0032 (m 2 • K)fW. Ai:..pressure drop (DPA) in pascals is . '. I.
DPA == (4.1 W )"(0.25 + O.OO16NF)( V ~.7S)
. 524
DESIGN OF
THERMAL Sy~
.
, and the pre~s~~ drop 'of'ethylene · gly~oI. (DPEG) ~:.~opasdls·i~
+ O;0875(W
DPEO' = 5,2[O.15W L
.'
,V +0.3](V!r
75
)
.', : :.... . :· ·::. . .lD' ·~d4ition -to·the· pr.ess·.~~·
'd:r~p :of · fh~· ·~thyle·ne. glycol' 'through ··the ~oil~ ... '/,." "~ .i." -::; .;:;:~s~me: alIow~ce .,~bould ·be iQade for:the intercoImecting piping:.' .is kPa~ .for : .->',. '~:·:;~r ~.i.~:.>;r.~:·~~ampIe. .' '.... ,.' .'.' . ,..' . ... " , .' . ,,; ::.: \~,:.:.
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v·a riables·for '\Yhi~h .opdmaI
, ..: their' values
ire',. "
'~·a1~e~-:~,·~~~r:, ~d.·ili~~'~g~S ~f ', ' ' . . ' . . ' , "-"'.' '.::,,\,' ",' ' '.
'.
300. 400. SOO. 600 fins/m pn sp~cing ',0.,4, 0.6, _, ~ ..~ 3_4,- 3:6 m,:: , L" rube Je'ngth " " " , , W" number' rows of t;ubes 'deep ~ , 2, 4,. 6~ 8, 10,. , .4, '5" 6'•.. '. to. LIO.06 " 'NR, number of rows 'of rubes high ,NF,
of
.... V~g. velodty 'o f ethylc::ne 'glycol'
.' ·.o.4!· '0.8" 'I ~2, 1.6,' 2.0'mls
.
.
"
," .A..ssi OnIn en t .. ' , " . b, , Dete.rrnin~ rhe"comb~,atioi1 of the above 'variables rl).atresul~ ~'U1e 'o ptimum . . econoII).ic solution.' . "
l
. .J .A.3 .~
.
-
.
DESIGN, OF A FIRE.. WATER GRiD
,
·F ire Control ,
,
Refineries and other cbemicaJ plants that process f1amm~ble substances rpust provide elaborate measures to prevent .and fight fires. 1 Almost ail such plants must be eguippe~' 'with a fi~-Ylater distI)butioil system that.is generally constructed underground to supply hydrants throughout, the plant. Typically, ' the plant is su1?divided into vanous areas, r'emote from,'each ' other so ¢at if " a fire breaks out in one area.it can be contained in that area. serve the ' entir:e plant, the 'fire-wat~r grid should be capable of providing a specified rate of water fl9w to anyone', area at a time. ' Two challenges fa~e the designer. For a given pressure at the outlet of PU,fI!P ,". .,
To
the
".' 1. Select the pipe sizes"'sO' th~t th~ flow' through the gnd 'Yilf supply fue . hydnin~s that surr~und a plant afea w'ith the 'required rate of water flow 2. Select the 'minimul11 flfst-cost 'combination of pipe sizes that meets the above requiremen,t I
,
'
Most designers of industrial fire-water grids are ,satisfied to achieve . task 1', and even. this assignment -is a challenging one. Some designers use
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Area A
Area .fi
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nGu:RE,A-3 'Fi.re-water grid.': .
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, ' spec,iaHy con$~cted e:1ect;rical' ana19gs', i~, cOfnbin'atlQll ' with a cut-and-~" " '" . .'IDethdd of.ent~giI9.g: pipe sizes until '~ach ate'a-iru.1ividuaJlY c,all'. be 'hlanketed ' , wjth specified 'w~tei flow. lhe analog must" be :'a ~special 9ne, beca~se "the " " fluid-flow conductor does 'not 'Ohm's law. ' " , . follow, . '
.
-
'
,'
.
':" O'bjective, ' ,
~
,. .
,
,
, TPe "goal of this p~oje~r ,is 'to select, pjp~' ~I~es ':for a fi~e:':y;~ater grid to' achieye 1-minirnuro. '[ust cost .. whe~ 'the 'following conditio~s are 'sp~cified: .
,' 10 Gebmetric, l~yo'tit of ar~as , ',2 . 'Locatfo~ of'hydrapt5, 3. ' Wat~r pressure at'the pump olltle,t , ~". ,11inimum flow !"ate: reqqired by ea~h operating,area ' , . Although in aC,tua! plaOtS there may be a dozen operating areas "with hYdrantS 'dis~bp.ted along ~e pipe.s, a 's uggested grid for '~his, type, of problem is ,'shown in Fig. A":3.. ' ' ..
Ope,ra,ting and Cost Data .
of
Th~ suppiy' press~ the P4mp is 800, kPa. The grid ~USt suppJy a minimum of 0.7 ,m3/s to area A and 0.8 m 3/s to area B', but pot sImultaneously,,' Area A is served by hydrants 1 r , 2, and 3, while area B is served by hydrants " 2, 3~ 4, and 5. The flow characteristics of the, hydrants , ~_ ',; ~."--':f¥"'-".--'<,,, ~:o:\l"'1
•
, "'flOW rate, m 3/s
,.~
..L~ 'O.ol
•
•
"r'"
Jpres'sme, kPa" ,'"
'.
•
". "
of
The installed cost ,the p~pe is' 30 cents per meter' of length. for each rrullimeter of diam~ter. ,The trenc'hing cost is $80 per meter, 'llut since it is
independent of the pipe size, it is a constant and dOes'not affect the optimum pipe-size selection. NegleCt o~rating cost, because one hopes the ·number of hours of pump .operation, will be negligible. . .
--
DESiGN OF THERMAL SYSTEMS
_-
-- . ......
5,27
COJ\IPREHS'-lSi VE PROBL£:YIS
,The insulation board is 'avD.·ibble
,ii1-50< ' 75< IOO-T
(,md., 12.5-,mr[l
,r'o
deve.lop thickne~ses gre~tti!,r ,~~_an, 1.25 , mm? ' one or add i~i ona[ layer~ of inswlati~g i?oard are requ}red.' thic:rnesses.
,
-
£\VO
'
-, Cost of i~suI':lt,ion, 5 C~r]~3)o.?~r-.gqu,ar~,_ ,rnetr.:r Qf.~r¢~ !9r each ran-lime,rer ",
", .;~,!:..~.,,:,.q:-t"-, ~tck.n~ss ""';'" :':, ", ;:~.,v_""",,';'-:'~",-'·
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, ',' ,I;q$taIla'tion cost!,,,firs[ )a'yer, '$2.50,per square m¢ter",second a~d~~hird, , : l~ye~ $1.5.0 per square meter ": ' ' Fii-sr cost of refrigeration ~,quipment.' $600 per:,kiioyvau"of refrige'ttition . .- ,cLl:pacir-y , " ' >,:',', ,"po\.v~r -required byief.tjgeiat~on plant, 0.6 1<:W of 'ele;ctric pOVv~er pe~" _kilo~~tt of refrige~a~i,bn , ,Cost 'of el~ctiic energy (depends i.lpon ' local,,~ates),. ,.usualli bet\veen 3 ' , and 6 c~nts per 'kilow,atthour' : "' ,' " ,Conducti~ity of i~sulation' k ,,' 0.04
vi/em·
K) ,
" ,ne ,expected 'life 'of tbe, plant is' IS, 'years:
th~, ra~e 'o f interest is 9
" ,percent,; , and for tax purposes [he facility may be "written, :~ff iIi .10 years (sugges,t ,straightline ,depreciatlO~). Federal ' i~come tax 50 percent. ' " , D~reImi~e ,the ,optimum thic~ness .. '
is
.It'
:
To, det~rmine tbe-: jnflueric~ ' of the po\ver cost and th~ ~verage' ,outdoor ' t~~m~ pe~ature the optimum thickness . .it i~ suggested that the optimizati~~ be perroITIled for several values in'the complete, range of outdoor tempe'r arures and ,power costs. The cost associated with the application of an additional . layer of insulation is likely to inhibit the optiI1)al thj~kness from exceeding' 250 ri1m~ or two full-size layers of insulation.
on'
A .. 5 SIMULATION OF 'A LIQUEFIED NATURAL GAS FACILITY' Need for: Ljquefying Natural Gas Tl1e principal reasons for, liq'uefying n~tural gas are for shipping and for storage. When natural gas is shipped across an ocean, "''if'Is.' e~ohomicaJ to conyert it into, Hquid [ann since '600 ·'times more gas can 'be co'nta,Ined in a
gi yen voJume than in me gaseous fonn at standard armosph,eric pressure. To hold liquefied natura] gas (LNG) at (lrmospheric pressure . the liquid -temperature must he approximateJy -155°C. The need for storage of natura) gas often arise~ in. the folIowing manne~. Gas u[ili[y companies are distribution c6mp~nies'. Md 'uley purchase natural ~as""'from transmissio.n companies. The transmission companies
DESIGN OF THERMAL SYSTEMS
1
o
x
114
Inp
p=
T=
---..
...
==
1
+ 13.8
1
1OOQakPu 'tr~n'slllis!)j().n ,·jn:e :
l
:"
Propano
290 l<
5 330 K
.... \
, 170·K .·
HC:ll c.xchungc:r I. .
\, '.
1.1
'o 0.. ;;:I
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...
"0 t..J
'tL ·
a '. ::J
121
14
. He-at cxchnlYge~ 2.
til
~
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F1GURE:A-4 LNG pla~(.
'. DESIGN OF TIiERMAL SYSTEMS
530
, 'In ' the' ;eihylene_~vaponitotassumethaithe temperature of methane"is at the " 's'aitlfati-~n temperal1J~e throughout. V1'.hen determinIng 't.he, ri)ean-l~~perature . :- ... difference'. ' , , -
,
.
-
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;Jli~:i~:'.;~~~~;~r,~rl::: .•·.' .':;~;:. , .·.'7.5::.:.15;5.(W'.•:;~~~I
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. '.:""
.. ,' .. Pdisch
'. " . :. .'
Cw, f
,'" . ,
'.:~mpressor',,0~:::': ,;iniei' ::':~~6,~~142_$:~"." :",:Pi~I~~ ~~: ':,f.:'~:~ ;<~j'>,;:,-y:::,;x7: • , .'..
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, '.'. ,Discussion' .- ', : " , USi:pg,-the 'syst~ili'~slriiuhitl'On pr:ocedures ,-of ,~hap·. ,6 'with the' ~id of 'a c~m ,puter pro,g nmi, a team found several ke)i ,re~uits. " - '
•
•
I
.
-
'
" Liq'uefacti6n ',i-~te" 0:64. kgjs',' , -, Higl;l, pressu~~, 3~,19 kPa" , P~essure e~teri,Dg compre~sor, 2400 :kPa'
A:6 ,OPTIMIZATION OF ANA.TURAL~ 'CONYECTIO,N, , AIR-COQ~,ED CONDENSER, Air-coo]ed con,densers that use pr(ip~lI~r fans [0' blo\v air '"over finned ,heat exchangers are comn1only ~sed in process industries, such a,S refi,n eries. The. electric motor$~ fans"be.arings, ~d gears or belts t ' if they ar,e used, ,requjr~ " maintenance and , SOD1etimes fail if nor properly maintained. Furthe'nnore, the fa~ motors requir~ energy_ and, natural-'c~nvection ' air-cooled 'c ondensers' are now often used. 3 Fjgure A-5 shows a 'sketch' 'of such a condenser, and Fig. A-6 is 'a sketch of the finned coil. '
, Objective Design for mininlun1 'first cost an air-coo1ed' condenser that rejects 140 kW . of heat. Sp~cifically, select the heighf h the dimensions' of 'the square coir nnd stack b.- 'a'nd the numbe~ of rows of tubes high in the heat-.exchanger section. ' t
, Performance'data
Temperat'ure of ambient 3jr, 35°C. Te,mpetatl)re q,f condensing fluid. 90~C. ' 'Choices of number 6f rows of tubes high. 1 to 6. Chbjces of b dimension. 1.4 to 3.0 m in O.2-mincrements.
-. --,
baa
: ~\ .'
"
: .' .... ~ ....
FIGUREA-5
Natural~convection ,a.ir-coole'd condenser.
\. '
.
Four rows of tubes high "
FlOURE A-6 Condenser coil.
532
DE.5IGN OF THERMAL SYSTEMS
Heat-transfer area' 'on ' condensing-fluid:,'siqe ~ of heat -exchanger per, . squa;e meter of face area per. row of.tt.ibe~ high, 1.3 m 2 • . . ' • . I ' . , . ~fectlve .heat--transfer area-.on the a~r sIde. of. fPe heat exchanger per y, :i< . ' ';sq:uaremet~r' of .fa~e are~ per row of tubes ~igh, '12.5 rri 2 . t. ~:.A',::C>.:·~ ··'.':·; .,~'; . ; : .:Neglect.the resistance to heat :trimsfer tbro~gh the ·tube.:.' " ..',::.. ;. :,: . . :' .~''<'<>. ~:.' ;;'/'::.i:';:!::';<\ .?>',: :~' : Hea~~transfer coefficient.·of' condehsing fluid,..2.0 ·k W/(m 2 • K)'. :.'." " ':\;" '::~>:""~">~~: ;:.~',:~'.:",.'.",..~~:.~··>:-( .~.~.·Heat-trarisfer coefficient on a'ir side,. O·.042V 0.4 kW/(Iri 2 •.. K),~,,·whej~':/:<:J~::':·:~: ,~ ·:>'.,01.
, . : . ,. ....
·. ' ..... .'.'V is~~Jace: vd~city'of a:ir~~.:, i '. Pressure dr0P',WQugh the 'cod, IIp;·~·= · 2~. 2V . '. (1 :5 '"!,··(J.7n-):,~.:'.wp~re. ';.:.. ::" :. number
the~
,n is
.
.
-.
.
-
.
.
.
.
. Pressure drop due to friction orthe' air flo'wing thr9ugb "the stack'must be consIdered; s~gge~t a fricrlOh ',factor of 0.02. ' ,
c .
. 'Costs
. Stack, including cost of' supports, ·$45 Coil, (190)(b J · 2 )( /1 '0.6). dollars' ,
.
per squar~, meter' of area
;Discussio.n
An increase in' ¢~ height h iIp.proves: th~' ~tack -effect and thu·s increa~es the ,}
airflow rate but adds to the cost. 'lin increas'e j'n the' tOll 'and stack :dimen$ion b reduces the press~re drop through the coil q.nd stack bur increases' the co·st. An increase in ,the number of rows of tuhes jncreases the heat-transfer area and .outlet temperature air t () but increases rh-e pressure drop. An .incre~se in air v~lotity over th~ COlI, incre~ses the h~at-transfer coefficient on the air . side'· .but reduces t (>. . ' Not all comb,ipations of h. and ',i may be .w orkable. The stack effect due to [he difference in densities -of the anJbjent air and the air jn the stack !s responsible for airflow and com'pensates for the ,aiJ;"-pressure drop through the coil and iIi stack itself. .
of
the
.
.
A.7 :ENHANCING. THE HEAT-tRANSFER COEFFlCIENT OF ~OILING IN 'TUBE'S The heat-tra~sfer coefficient of a boiling f1~id i~sjd~ a tube v~rie$ throughout the l~ngth of the tube as a function of the quaHty, (fraction vapor). A typical profile is shown in ·Fig . .A-7. In order to improve the overall . . ':performance .of the hear exchanger ,a modified evaporator, as ' sh.own in Fig . .
of·
1 The acrual surface area is' greater thJfi indicated. but the effective arelt, accounts {or the fin efficiency_ .
-.
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-
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~
. t ·.
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7.000 .
9.000
,/' ,-~ ~
.~ ' 5.,aoo·-·-· ~
o Cl'
, f '
'. J
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'-
.
'-tooo
.3.000 · I~__~__~r~~~__~r__' __-+__~~__~I~._ _~_ _~~~ t.2 . 0.3 . ,0.4' ..., ·0.5 o.~ · '0.7 0.8 , - 0.9 . J.Q ". O' oj',QuaHt}'x
~
'.
' FIGURE A-1.. Boiling hear.:.tra·nsfer coefficient inside a .tube. ' (F~m 1 ..M. Cha"Y1a, '''A Refrigeration Sysrer.ri ",.,;ith Auxllliary.Liquid and Vapour Circuits,''' lnternJrio,naI Instirute of Refrigeration~ Meeting' . " , co~. · ·n and III: ,L ondon, 1970: used by permission.) :., : .
A-8, extracts vapor at two positions ' and injects it ~Isewhere along the tvbe . ip order to. mmntain the quality near 0.7 ·qr 0.8 .apd .thus take advantage of . .the high heat-transfer coefficient that re~uJts.
Objective Determin~
the. po~itions.of vapor extraction
~nd
the flow rates of vapor that result in the optimum mean heat-transfer coefficient if the local coefficients . are as- sho.wn in Fig. A-8 . Sp.ecifically, determine Wl/W, _w2Iw, .,vIL, and' zlL for the ..maximum' mean coeffi,cient.' 'Assu~e. that the vapor and liquid move at the same velocity.
DiscusSion',: . The analysis is complicated by the fact that the ,heat flux changes along
the tube because the coefficjent is not Furthermore', me heat f1\lx , constant. . js influenced by the local, temperature of the fluid being cooled. To simplify, ?Ssume that ,,t,he rate of evapo'ration is uniform along the length; thus, the quality vEIieUinearly with distance from 0 to 1.0. ,
-.
~
.
~
DES1GN OF l'l-iERMAl. SYSTEMS
;,'1
"--":,........~. ilir
1(' .
Motor
ajr
- ..
... ,.. .........J. sir
I,.
J.~~ ,
COMPREHE.:."\!S!I.'E PROBLE?vIS (:>
that if the heo.t pump does not bring the remperntu:-e. Df the Incoming' air .... <; °C .( I . ! '~Hr . t,;.nlpcfJlUre). , \ jt!yC tr!c-reSISl~H1 ' . '"" Cc -neat " - use d to up , j"~ O .)..J L1C suppJ:Y IS .
1
sup pten1en[ (he heat pun1Q.·
.
, . O bjective I'
, .. :~ ,:Deterrl?Jne 1:h~,~9PiiiTI.u.1u'Jlea.t.)",p..l~:~P ;'~1nurnely', ,.ihe.. :s~ze~.~f. . compJ-i:ss.o( .....con~._..:" ':_~ . dens_er~ 'QDd ev·aporator that providt;:'s.a -miIiimuDl total·presept- tvorth-of cosrS "(fir,st C-~sr p'll).s p~esenr '\-vorth of po\ver costs). . . :'. ." .. . ,. :'. . . '. . . '.1
.
'.
.
.
.
'~'
- '
.
..
. U v~lltes of condenser. and evaporator cdil3', 25 .W !(m- - K) based on air-side area I
.
.
'
~.
.
.
..
'
.
.
Cost of.coils, $50, per square meter of air:-side area . 'Compress~r cost,. incl uding. motoL $1 ~O per n1otor kilo\vatt Po\ver cost,' 3 '~ents pe,r, kilowarrhour . Ifire~est rate. 1.0 .percent . . Economic] ife. J0 years . NUD1ber hours of operatton per year~ 40.00 , .- ..'Ave'rage 'o utdoor temperature during _the ~OOO h~ DoC'. Air. flow rate' through both coils. 5 kg-Is 1
.
'
of
~"
~
The pei-fonnanc~ characteristics of the compressor: can be, expre~sed a's a coefficient-of performanc;e (C.OP), where .
fcfri£erarion rate kW . ··COP = . . .... . ' , . . electric power to compressor n10tOf, kW 1
the COP is a fllnction of the evaporating and condensing temperatures. t/C, and Ie o~, nnd can be, represented by COP = 7~24
+ O. 352t~ -·O.096/ c
-
O.DOSS/etc
Assignm~nt
Detennine'the optimal combination of evaporator area, condenser area, and compr~s,sor motor input power that results in [he minimum ' total, present wo~ of ,cos~.;. Aiso · det~rmine 1ft Ie' tit and to at thi.s optimum. .
Discussion If the heat pump· 'can be justified· for heating operation alone, it has the, . possibility of providing additional s?1vings by operating during the summer. The temperature of the supply air entering the building is specified as 35°C, ~g some periods.Df rhe heating season it may be desirable for
DESIGN OF THERMAL SYSTEMS
l!A 7 10.5 k\V/K
frum . 0.02
Air 10
-
J
..
5Z7
CO,'v!PREHEl-JSflJE PROBLEMS 1)- '
,
'
,
I I , A.irflow rate~ iO kg/s " Rate of addi(on of v!al:e.~'. v·apOr? OD02 kgls ' Temperature of wateT : ~nte.ring ,water-coole~ ' heat ,exch.auge[~. 20°(: , Flow rate of water.·entering h4':at.exch.anger, ·4.0 kg/s :, . , , ( 4" - . ...... ,"~
~; '. !Ao :.". "••
·-t1A· of ~N.atei~·b'0i~d..-:J.1,.heat.·;.. ~XChiiiaeI(;t0,·3 ,·ckWnt;~:·.:·.'b •
..
'.VA •
--:.
' ... .. -
'i
J..
bf condenser, rs::'3-kWIK.:· , . •
'
"
.,'
..
, .
...
".
,
.
~
#
..'
-I
~..
It . ...
' /:c' " ,-
_' ,
:,'. ~
~'"
;: . . :,;
-":~,::,,:,,.j.'.:~'-::';~~ ..
'.
-
, .J
or
COTapressor·. per'fOrmaJICeo . The pumping. capacity .the compressor exriessed ..in term~. of'heat-tral1sfer [ute at the ev.aporator- .~s .
.... .
and
qe = l2q.9 + 7.683t e - . L134tc ~ qdO?516tt:ic leW' the'· p6y:er' (equir~ment of the compressor is . " .: •
\
. p .= ,
. Yth~re
.~5~Q5 -.1.~66te ,· +
I
~
•
+ O.0299!etc
O. 28~tc
;.
kW·
t; ==
evap'orating te1p.perature.,- DC . .. . le.= c.ondensing temperature, °C. "
.
Evaporator The 'e vaporator is 'a heat exch~ger' \\'pen~ .both sensible 'he~t ',·and··mass are trans.f erred from the air to the surf~ce of the' heat exchanger~ . which .is wer be~ause of the dehu~idificat1on: The heat then flows throUQh the metal and thepce to the refrigerant. Data on 'the .evaporator.,are . 0
•
•
• • •
-
'
. '
••
-
•
•
&
•
• •
•
- .
•
4..;...
•
A;,
"E,efrigenmt-,side .area ~5 m 2 . .Refrige~ant-·.side heat-transfer co~fficient ·hr·.~ .:2.2 kW/(m 1 ..KY ' . ~egJ~ct the heat-transfer'resistance of. the' metal Air-sid~ ar:~~ A a , 120 m 2 ·-Air-s1.de convection heat-transfer coefficient, ha == O.09,·kW/(m 2 • K) Heat ·a nd.mass transfer in the evaporator. The processes occuning in the evaporator coil can be Visualized"a's shown in Fig. A-I1.- The driving force for the transfer of sensible heat the difference in dry-bulb ~,emperatures of the air and the wetted surface. If the arithmetic-mean difference is assumed
is
.to be sufficiently accurate, qs
= h A a
a
(13 + 14 _ 2
Air.
.,1, . . . . . . . . . . , . . . . . . . . . ,
FIGURE A-lI t, ___ =- ..
Rehiger.mt
Hea[ and mas! transfer at the evaporator • t,
coi1.
•
:'1..)0
DESIGN OF THERMAL SYSTE.lv1S
rate
----..
...
.r: HllJ k Pa - .
. "120 km -----I~J
'.J 30 c
. -
0
. ~-
. '. ", ,,,,!
l OO kPa
50 kg/$
. ~,
~'1. ~~~~ , .. -." " . . · ..····l-Ao[or ·-/
.
f\"
.
~,
.~''-'' .. . . . ".
..
• .:. . " . - ; : : i'--'
.. .
"
.
-
.~ ' , "
. '. . 11
FIGURE A~lZ ' , N~ruI?1-g~ pipeline. where :V9~k
= 75%
i$ recovered at terrninaJ position .
.to 109 kPa. 5 Compression to a high pressure permjrs .·.a 'high-PJ"e~s~re 'dr~p " :throllgh the pipeli.ne 'a nd' also r~.sults in deqse ga~: both reduce the.pipe SIze. O~ the other hand . .there is additiQual energy require
.
at
. Metha.ne ent~rs, the compre~soi 100 kPa. 'M ethane leaves .the turbine 'at '100 kPa.. ..::",.All the pow~r generated by ·the turbin.e-generator can be used. G~ ent~;~" the .compressor .at ·30°C,:. is c?oIed after [he compression) and remains at 30°C throughout the pipeline.' . Methane flow rate js ,50 .kg/s .. . . The efficiencies of the' electric motor and generator are 95. per~ent.· The efficiency (with respect to the' 'isentropic pro~ess) o~ the compres-. !sor is 80 ·percent and of the turbine is 75 p.ercent. . _
Costs
. E~ectric motor and generator 'first cost, $50 per kilowatt output Compressor first' cost, $125 per kilowatt input (input designated We) V~u~ . _o t ~lec¢city at compressor end' of pipeline, 3 cepts per kilowatthour
'
"
.
Value of electricity at turbine end of pjpe1ine~A cenIs per: kilowatmour. . Turbine first cost, $150 per kilowatthour output (output designated W,) . ' .
,
.
Pipe cost in dolJais per meter lengrh 300D 1.6, where D js ·the pipe. diameter in meters '. ~I f i
-
...
•
,.
II·,
DESIGN OF
,I'
,
•
Data
--
":;--...
\
l
I
'
t 4 :.
5~ 1
CO;ViPREH5'1SrVE PROfH..E?IS
\-Vater 2S:JC _ H'
= 0.8 ·kg/s.
l."p::': -l.,!:; Id/( k.g .. I()
'. . _~~" ...:_f ." ": . .. I/~
. ··Satu·;a[~d .. l
-., '
-<; - ,'.....;~ .... ' '. ' "..........
. liqui~
S~wr.Jred
i· ,
L
vapor
Hear c;(ch~til!:!er J
• • ,1
"'-
Fluid 6
Eyapor.Jtor .-
-15!)C li' ~ 0.57 k .!!.I: . .s (p= 3.5 kJ/lkg .. K)
..-.;-.: .
Thronlin" .c
.
v~\lv~
.'
.
. FIGURE A-I3 .-R.efrigefarion 'system llsing ... -.
'-
:' :'
;
il
m.ixt~re
: .
b.f refrigerants . .
. " The ~on'denser' i~" ' water-cooled; \vater enters at 25°C wit;h a flow' 'r~te of -D.S kg/sa . ·The. evapo~ato.r 'fluid i~. coolecf1rom' ~ 15 to -25~'C_ Its flow ,~ate is 0.57 kg/s~ and its specific h.ea~ 'is 3.5 kl/(kg " K) .. · The refrigerant 'saturated liquid at point 3' and saturated vapor at pain.! 1. " __-The compressor has ,adj~stable ·capacity. which is regulate~. to provide the 'spe~ified refrigeration rate in the evaporator.. : .~-
-,'f
' . . ...
is
For £aturated pressure
.R-12:
In p ==
R-114:
In [2.
=
J4.8~1
- 2498.3/T
15'.407 - 2993.21T
where p' = 'pressure, kPa ' .T . , temperature, K . For e~rha1py' of saturated liquid
.
I '.
{200' =t. 0.9251 + 0.00081-/
'J
= 200 + 0.9545t +' 0.001161 2
where '2t ~ enthalpy of liquid, lsl/kg - r:= temperature, °c "'.
2
'
R~12 R-114
pESiGN.OF THERM.AL SYSTEMS
I
effi~iency. Provisions are roade to' ren10ve air cont'i nuously by the use of su.ch devices · 2.S an dir-:ejector sysrern, shPY-JP in Fig. 1-\- . 4Q 1~11e f unction of the ejection systeol- is to 'extract , (1 sanlp[e of air and warer v aj.fl"Jf out of , the nJ.r.lifl condenser and oIrinl;1te:y i:'J reject t~e Qil in the mixture [0, rhe atmosphere. ,The remova,: of air is to beacconiplished , \v[rh a IQv,f loss of
steam.' .' ," ", ' " J' "', ,. ,-, .- ' , ', " " :, '. , " ' . .' ... '- '., ' 'lrr·u'.ye·s'Ysterr{S'ho,wn1jn:.:Fig>'A~r4: a'~ r2'r'kgJ~""Crf~Ir~\irip'o?/in;kiur~ ' thaf;' coritaib's , .5j~ercent nir by,.~·i'iss, fs,, '~xtincted from the" !1lain condenser . ,ariq " .. the mixture that , is vented to atmosphere should contain 50 percenI aiL by mass. The .main condenser ope~~ltes \vi,rh a toral pressure of 6'.6 kPa and the aft~rcondenseroperares at .101 kPa. ,C ooling \.vater·.\virh a fIO:'lilJ cine of 7 .kg/s 'enters the intercondel}.·ser· at a ,temperature .of 35°C 'and passes in serie~ thiough the intercondenser an,d then the ·aftercondenser. ' ' q .-
Performance of ''Condensers . ~
,
~
.
.
~fhe perf~nn4n.ce of the intercoJ?denser and the afterc~ndenser can be approx~ . iI1).ated by assun1ing ' that they are couritertlow heat exchangers, \v"h.ere me temperature the con~ensing side is that. of the satunltlon temperature 'o f · water ~t the partial pres'sure of the \vater vap~r. Thus; if the presstir~ at one · ,poin~ 'i~, [he intercondenser is 20 kPa. anp the parti.aI pressure of the air i,s ,
on
-
~.
'.
Vent 50CJ.. air by TI1:tSS
Ejector
"'I
Steam
StC,-a.m_~~~~I ~.:.~.-----,J.. ! ·./',1 ____
....cc=------.~ ~
------t>.-
.
p == 10 ,1 kPa '
~~2a~g!S
+
-Afrercondenser
by m~ss Condensare
~
- . 6.6 kPa Main condenser
F1GURE A-14 Two-stag"e aij' ~lOr.
~
,
...
Turbine exhaust
Condens~l[e
544
DESIGN
OF TH:I!RMAL SYSTEMS
2 ~, the. condensing 'pressure will be 't he saturation _ pressun~ at 18 kPa or "57 ,-goC. Use the log~me?n t~mperature -differe'n ce the conde'n ser with .all ' 2 applicable , U value. of l'.4 ,kW/(m , . K) ..'- ,: , " .
in
"
.
.- .'
" .; Assignment :"
_,->Select ,
,',
, ,'
,
J
,. '
"
',: "
',
'.
" " ,-
1
dle' ~heat-tran$f~r-- ~~as oflh~.' mrerc~nden~eran,cCaft~~~~~d~h~er; fu~' :.::-,: ) -: . "
,
'
.
-
"
,
,
"
"
.
-
:
'-'-!'.'-,_'flow'rates of motive steam WI and ~2, and .tile iritennediate .pre_S~UIe ,pi so .-' ','. ;"'::~;:;~tha~ ~ th~ ',total" pre5erit,VoI~~p(~1l ,costs' .i~;~' ~!limU01;(~'::~': ,, ;.,,~. '< ' " .'~::' 0
,. <.>: '"
' Cond,e nset .cost; ~ ·$15q per ,square met~i- of ~eat~f!a11sfe'~' ~ea Cost of motive steam, 60' -cents' per.-megagntm' " , :' Int~'rest"ratey lO·percent. ,
,
'" ,Life of facility for eC?ono.mjc' evaluatioD 6 yeats.' - , , Hours ~f,op~ratjon . per ye~, '-4000." . ' y
..... .
: The ratios of moti~e steam fl ow, t.o,pumped .f)ow for. , -. given b),· - ' ' -~~ . _: .. ,'"
, Ratio (kg/s)/(kg/s) ,.' o. I9(:~t2
: ~ow 's tage:
,
the eJ.ectors· ,~e-..~. ,,::.' :.
,
. ' -' ', '. ( 101 \ 2.2 Ra.tio (kg/s)/(kg/s) = 0.14 ) . ,.
High stage: '
lpi
The combined .first CQst of the ejectors is approximately , constant in the expected range of sizes to be ,e xamined '
The. re~ults of Prob. 4.18 may be useful.
Discussion .
'
, The opti,mal gesign is likelY', ,to ·.be one w'he~e ' more heat-transfer area is placed in the intercondenser tha.n in the afiercondenser, since whatever steam (both f~onl the main condens~r and "from, the Jow-stage ejector) can . be cO,nde-nsed the intercondenser will not have to be pUf!1ped by 'the highstage ejector. Often lTI n1ultisrage compressions rne intermediate pressure is " the geoinerric mean. of the suction .and disch~ge pr~ssures. The economics C?f this system wil1 shift the optimum intermedi'ate pressure to a value lower
at
-' .' than,' the geo'm'etric'
mean: '," .', ' .
'·' ~'A.13
.
~
.
,
OPTIMIZING A'HOT-OIL LOOP IN- A PETROCHEMICAL PLANT'
..
~,
One method of distrlburipg heat in ·refinenes and' petrochemical plants is-' to heat the oil in a centraJ furnace and pump j[ t.o . yarious heat exchangers where heatirg is ·requiyed. Figure A- J 5 shows a hot,-oil loop that serves
-
.
, ,
.....
COj-'.~PREHB.fS[,/E PROBLEMS . "
~ . "I er,-:::,. ~ lenOI
545
q""' hr-. fw.· ~ I "1 eLv, anu ....l - ., <::'F- i r-. ~""T.•~ J'J'-':\ r,'_.b -n~, ce,- .~eOOJ t1U n ...1.p. crT. t,--0 b·-,,' C ~"... .:. .... ,L~. '- ,.!. JG' ~ the totai present worth of costs fot the eoo.Bpmic rj.fe of the faci lity IS a . .'l ..11ee --.-
-"I;
<",
'1- '
t' . ,.
;;:
!; , .... l.
m·1·fUffiUD1.
Oil d2.i!a
" ~'';, .... ~,'.. :.. .- ~
.
Oit,~h§,rn!caJly,.~~~1~,,~~?-.l;_/3~htC::· ,.' .':': ~,.. , :
. ·;Specific~he·~t,~-2p 1 .lc]/(kg. . K j '
-.;
........... .
'1 -·-·-1. '
-
: .',', ";' 1
Density, .850. kg/mJ
~,
"::
..
~
.-
'.
Reboucr data
Duty, k\V Reba irer 1 . 2 3
.. .
1800 '1600 2400
'
vallJl~.? ' J W-!(m-" K)
Temperature of fluid-being heated; .oC
'U
200
450
150 175
450 450
Tempera.ture of'fluid being boile'd' in a glveI?- r~poiler is essentially
. constant. Co's t of the re:tJoilers is $90 per, square meter of heat-exchanger aretL ., .
. Combustion air, 25°C NaturiI ~1'
gas, fuel
Oil
kg/s f'h
Flm.,y regulator
Combusdofi
chamber
200°C_~_
3
Furnace .
-
-,
I·
F1uegas
to sUlck '.
Pump nnd motor
FIGURE A-IS Hot-oil ':;;p in
B
petrochemical planl.
DESIGN' Of' TIrER.M.AL SYs:rE1viS e"
data .
1 .. 1 \.
.
1.8
l:.COne)nllC
data
rate, 10 """"".,..,..."',.,
"
,
'.
,
5 7
COlvlPREHENSIVE PROBLE:olS tD
-systems
·consider is a regen.erntive heat excha n.ger to neat fro m the flue. gases and thUs reduce lthe fuei cost. \-VOllld
rec~~fer ' mor~
A , 14 O:?T1n{IUM HEAT p:mwp .
FOR ".PASTEQRIZJ1%~.G lVllLKi
'"
.
-::.
.
.
.'.,,: .'-.~e· :ess~fitial'~ -reqHirem~n;t~ 'ii( '~i~"'p-as~euiiia1:jorf pro~esS' -ot .nullc' is -to biing the -temp~raJp.re upJo '73°C.. and: hold it for c:tpproximalely20 ·s-.The .milk arTIves at the dairy from ' the tErnk truck at .a temperature of 7°C and :~s to' be delivered from the pasteurizing pintH tq -the packaging- operati~rA :J.r a : temperature of 4°C. .. . The tniditional pasteurizing cycle uses a _s(ear~ or hot- water h~arer to . bring the temperature of the milk to 73°C; then the milk flows rhrou£h a wate~-co61ed heat exchanger and a r~frigeraDt evaporator. The heat fro;; the refrigeration 'plant' is r~jected to the atmospbe(e' ~y a cooling tower ' or an . air-co'oled 'conderiser~ : In the 'interest of conserving en~rgy., the possjbility of using a heat . pump is noyv sometimes considered4 8 One· possible 'cycle is shown in Fig . .A.,."l6 _. The incomIng milk flows first through ~_regenef.?1ti.ve heat exch~ger .. and then.to the heater, which is aLSo .the fore~ondenser of the·"·heat pump. This·forecondehser eleyate$ tpe 't empenlture of the milk to 73°C'. Thereafter -the mills flows through the otner side of the'- regeneratiye heat exc;panger an9 .. fiI?-ally through the evaporator of the he~t'p?~p. . .
. . Compressor .
,
Forecondenser
I"
Regeneratjve heJ t ·ex changer
. Evapor.llor
.~
Codling wll(er.30°C
Expansion V;]Jvc;
__~______?-~~v-+-~~.~oC
.Refrigerant (
FIGURE A-!t! Milk pasteurizing system using
~--'L-..·_A_fterc...,.on~enser
3
heat pump.
DESIGN OF , . THERMAL SYSTEMS
.........
water enters water set to remove
~ ....... ..< ........ =-
-
..
COMPREHENS IV E PROB LEI'.iS
:S,:g~
Ii)-
3. P. T. Doyle and G. 'J. ,Benkly, "Use Fanless ~ir Coolers~'o Hydrocarbon Prl?cess,.p July 1)'73 PI? .g 1.-86. . ~} • •4.JHRE Handbook Fundamenfols Volume,' chap.Gl P.<:~ericM Society of Heating. RefTigcrating. apd Air-Conditioning Engineers, AtLant~... '985, 5. CL ::. 7...:a.tll'icki. L. A. Repin •. and v, .,A. EIemao dRerrigc',Cltion by U~ijizing Jh~. ?ressure 'JfT<:IaruraI Gas Pumped ~rougt Pipelines:~' Cholod. Tech.~ no. 6 "1'974. pp. 27-29. 6. G. _G . HaseJden and L. Klimek •. :hAn EXBenmenta.}.StlJdy of' the Use 'of-lvfix~d Refrigera~rs {'; ~y T_ h alR' c·..,. " "J r.· t I I A 8'7 ' OO 1\ f(' J' ~ on'.(;8' ··- ' , . ,.,.,_oL J.-.-..:on-.LS-O;.~ erm : c:rng~,~tlC;Hlo :., .. ., ;'.l,f!J.J..'lK·.,,~.9. .. :_.nQ·,_.~.,,-PR,.~ .. J,-:-o.:;.' ,~LI~lay- une.;t ;;·...,;t., -,\_~ .. <7".' yo kaireri C.~·, Becd~.ievre/ and D~ Gilbo'tirrH~~ ""'lJrriea [femgera,:;.t for Ethy.tene.~~ '.Hydn}~ "carbon P~Dcess.> 'Yo!. 55~ 10~ pp.' 129-1jt.octobe(·j~76. ~ ' --. " 8. D. ~_ "Lascelles and R. S. Jebson. "Some Process Applications of H~at Pumps," lilt. .:Tnsr. Refrig,- Comm. ill/eel.. Melbourne. 1976. \~ " 0
0
i,,)
no.
I
..... .....
,
,
~
'
APPENDIX" ,
'
'II' "
..
}
-.
-
.
. ' : :'
. t ';
.' .' I :', )
J
.
'
,
-. '
'.
•
.
~'
- ,
" ,'; ,:-.-
.,
• .
) .
.
Structure,
Main
Pro~ that PT?vidcs input ruita.'and calls SlM1JL
.I
Subroutine EQNS that provides the simultaneous equations (many of wnich ·are probably non-linear). The required ronn is specifi.c d on the ne~t page
Subroutine SIMUL that perfonns the Ncwton-Raphsori . solution of me equations in Subroutine -EQNS with app~opria1c :appJkation of SUbroutines PARDlF and
" . .. .
,
GAUSSY
.
.
~----------------------~~------------------------~ Subrou~e
PARDW that' extracts partial derivatives . "".. '
-,:.,-":"
550
,"
Subroutine GA USSy that soJves a set of linear ., simultaneous equations
-
- .
. ' ~-:..:~ •!.. ,), '.
;.~!,~
-,"
1" '. ' ,~ .,/ . .
PROGRAM
,.-
i!t~
,:
SIMUL.ATION
.
.:,'
' .'
. :,-.' ..
.',
" : ',,
:s:>: :'OENiRAiiIzED~. SYSTEM- '"', ~ , '~"~" ,,'" ">~~':~'''.':t",::,,:·,· : ,
"
!
.. ".-', ..
I
$ -',
j-.: '. ~., -.- .... ";" :" \
GE.'=ER::>S... iZED SYSTDi Sl\j l4-XnO~ PROGR.--\..'\l
\
I
55 ~'
.
PROGRAl\1 EXAM.? (INPUT, OUTPUT. T A PE6=OrJT.OUT) 0IM:EN SrON V( J. DES( ), R( ). VCORR( ),'VDl J" RUe ), PD( • } NVAR = . Nw".her ol Equliiiofli ' ;"rLR..!~CE:::l . _ Suggest 0.001 ,'.!.
. •• : ......
' ' :" . ' . • :
•
"
lTrvl A>; c: ' .:.DATj... V/.a ..e. '"
I _ ~'." !:- ~ .. ,~ .~.~'.c> f .. ~;~_..J.,Tt:iqJ.·.'fta1wJ.. ott!!!. t'a.~iahle$......•..,,~:•.~;:,: ,:,!.,;::;,<;;:{~r:: / .. , .':;'
Suggest 10 <0.0.0 ;.:"" Eo . . . .
.. .' -DATA D£s!"TdU'r·:· . . 'Tj}J ...,.·.:~~:' \\(14·"1 ·-}:-Cf;qiacJ'er 0(- Less Alphanumerii ·· ~ .:. .. ".: CALL SINHJL:(NYA"t~,.JLRNtE., lTMAX. v~ DES.. R. PD;, VCORR;VU. RD) .
STOP .
.
END,
Equa!~Qns Subroutine
SUBROUTINE EQNS ,(NV AR,V.R) , Dlli-1ENSfON V(NVAR),R~V AR) . R(I)=V(10) - v(ll) >i<~ V(5) ' 0
-.
1 7: •• , ; ,
'.
.
..
DESIGN OF THERMAL Sy~S
c c
WIlli .
c c
c
c
C· C C C
c
c
INPUT DATA
C II
=• .21
22 "23
C C
C C C 30
WmALIZrNG ITER":::: 1 CAll..ING DERlV A TIYES Al\t1)
VALUES OF ."-L......... VARIABLES
u..JLJ ...
33 "
c
c
PRINTING
,,-.L....L._.. ,,'--' VALUES OF PARTIAL DERrVATIVES
.
- ..
,
~~---"_
PARTIAL
GE.'tERAUZED SYSTEM S!;"iULATION PROGRA),:
c 005<:;
. ...... : ~ .~
... -
:-'.-
,..
..
r ==
,:):;:y
r,l'-,fVAR
C
D053 J == 1. NVAR } I ~ == ABS(pD(I)) RBI/XQVE THE C fRONt THE NEXT THrtEE CARDS 1F PRINTOUT DESIRED
C
IF(Z- 0.000000(1) 53, 53, 51
C 51 \!VRITE(6,52) J. ],PD(f) C 52 FORMATe' P.oC',I2''''''iI2,',') =", 8°110) 53 , CONTINUE ' -' ,: ,}.f" ,,' COl'rmvrJE" :~-,,<' ,,:~~,: ;,,;,i!,~~'t::,,-~rl" " -' "- , , "- ' ,-," " CALL.GAUSSY(pDp 'R" VCORR" NV AR}' . c', '.COR.REefING TI-fE V ALVES OF TIrE V ARLABLES '0044 L= l,NVAR 44 VeL) = V(L) - VCORR(L)
, ',: _,' ,o~-,,'
,
-,
": ,, ,
-c ,C
WRITING OUT RESULTS OFTI-H5 ITERATION
,' , C
'WRITE (6.32) ITER
34
' ,--.''c 36 ,, ' C C
,
'FORMATC'ORESULTS AFTER", I4. ,. ITERATIONS") '. WR.UE(6,34) , " -_ , ' ' _ FORMA T(,,0",4X."V ARIABLE··, 16X,··VALVE".-9X."CHANGE. FROM PREVIOUS") WRITE (6,36) (I. DESer)• .Y(O, VCORR(1), I = !,l'"'JVAR) " ' "
32
FQR1.1A r
(" VC', 12,") == ", A4:" = ",2F2'J.5) , .. " " ' ,TERM1NATIN'G n= MAXThtfUM NL'1'.1BER OF ITERA TrONS REACHED OR OTHER\VlSE
INCRE1vfENTlNG THE ITERATION COUN1ER' "'- ', -' , , " IF(1TER - ITMAx) 38~ 99; 99 3& ITER = ITER + I 'C', C 6ri:CK TO SEE 'IE CHANGE OF VARIABLE 15 L'E$S THAN SPECIFIED TOLER,A ;\:CE
.(;
J
K= 1
VAL = ABSCV'CORR(K») - ABS(TLR.I\!CE-*Y(K)) "IF(VAL) 41, 30, 30 ' '41 , " tF(K - NY AR) ,42, 99. 99 ' 42 ' K == K + I GQT040 99 RETlJR.1{ 40
8\11)
' . '.' .... ... . .
• j'.
"
DESIGN OF THERMAL
c
.c
C 560
RETURN
c c t
BY GAUSS ELl1vlINATION
SL\1ULTANEOUS
c INTO DlAGO~AL rosmoN
·C
2 lMAX=I 4
·CONTI:l\.1lJE
.. "
.~
• J
-- ..
..
\
c
TESTil':G ;:=OR JNDEPEi'ruE~CE OF EQUATiOi.\i'S ;- IF(ABS(A~'vjAX) -'b.IE-1S} 10, 10.14 \VRlTE (6.12) , 10 12 FOR.'vfA T ("0 EQ U A Tl O~S ARE NOT fNDEPE~DENT') I:~E1:-u K.N
EXCHP.NGING ROW L\,IAX AND ROVi ~( B1EY~r = B(K) '
_ l~
I3(IQ = B(IMAX) B(tv1AX) = B7~M? , ' DO t 8 .. j= K N· .. ·-·
..·· - · ··:: "ATE.~~1\{KJ) 18~
C
.<.. :. . ,.
:. ',-
~.
"A(KJ)::: A(J.MAX. J) A(Ilv1AX,;) = A TEMP : . ' SUBTRACTING A (I,IC)i A(K.K) TIlV'(ES TER1vl IN flRST EQ FRO:\l OTHERS, . KP,LUS = K of: I ' 'I
IF(K - N) 22, 18, 28 DO 14 1 = KPLUS,N '. BCIY =B(I) - B(K)~A(I,K)/A(K.K) AeON = A(I,K) . D024 J= K,N 24 A(l,J)::: A(I,J) - A(K,J)';'ACO~/A(K,Iq " 28 CONTINUE. C '. :BACK SUBSTITUTION . 22
L=N 32
SUM =0.0 IF(L - N) 34, ,3 8,38
34· LPLUS = L + 1 , D036' J = LPLUS, N ·36 'SUM =:= SUM -t A(LJ)*X(l) : 3.8
'. 40 42
.
. CONTh'fUE ' " X(L) = (BeL) - SUM)IA (L.L)
n=(L - 1) 42, 42; 40 L= L-I GO TO 32 . RETURN
. Th'D
'
,
-.
.
v
.:, ' . '
..
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....
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~
"· 1
.APPENDIX .
.
.TI · ... . .;:~':>,. ·•.·J""'.", ~. •
•
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~ZERd{I,J,R , , 'JA)
SDBRoUYINE C C
C
c.
I
..
:- ."
. TH IS' SUBROUT tNE' 'STORES ' THE .' ~pNZERO ' ~LEMENTS' ~ IN.·.· ..- . , '. ' THE PROPER ORDER TO·BE.USEDWITH SUBROUTINE XGAUSS ' . ' . . . IV
- ."
,
~
~
'.
C R . C ,'JA.
'.
SUB~OUTINS
C ..GLOSSARY FOR
C r C '",-. ~..)
.-
.-
~
' .
NZERO'
.
·ROW 'NUMBER OF NEW , NONZE-RO ELEMENT. := COLUMN NUMBER' ,OF NEW NONZERO ELEMENT. :;: COEFFICIENT 'Of, NEW NONZERO' E~EM:ENT . ' ... =' ·.TEST VAR I ABLE., 'SET', EQUAL· TO ZERO AT' :=
START O(
'C
CALLIH~ ~UaROUTINE
,C
COMMO~./AREA3lj MA~,MT)N\(AR COMMON IAREA411 lRDW(3), . .:', . COMMO~ IA·REA421 JCOLC2, 7)",
'COMMON
c
IAR~A431
A(7)
.
.. ..
"..
.,.
FIRST TIME ZERO"IS CALLED M.ATRl-CES A~E INITIALIZED
C
C IF (JA ' . NE •. ' 0) GQTO 15 . ~
JAf.)A+ 1 .MT=1 , . . . pO 5 L = 1 ) NVAR __ ,5 1 RO~( L):: 0 . ." DO '1 0 L c ,- , MAX , :". '. ., .
o.a' •
... .'
• . t..-
10 ACL)=O
15 556
IF(I~OW(I)
-.
~
•
."
;'", ."
.NE. 0) GOTO
" ' :-. :"" ,,' 2~~ '
'
~ ',
--
..
JCOL( 1 ,L~ .cO JCOL(2~~~=L~1
•
....
,
:
.
. •
': '
. .
~',;
I .
.
;"IEI,:,J Ei_EIYlENT I S THE FIRST' I'I:ONZERO ELEMENT , F.D
(' V '
'-.
c
-
IN ROW
r
/"'-.
'oJ ' .
. "" ' , r ROtAK I }== MT
. . ,.', ;' ~:' ~" ~<:~:~ ,.,.-:'. .,: ~ :',: " ~" JC.pL~3::·7,~!:~~:-~~;-~.; . ',..;- . '
.,...,. . . .
~",A ('MT) =R
· 4,'
:"_
.. ," ... :. , . MT=jCOL{ 2)~T )
:
' JCO~(2,I~OW ~ I) } =O
. RETURN
'c ' . . - C' SEARCH TO FIND PROPER LOCATION ,OF NEW, ELEMEN ,C ' .C
ROW ' I
' ..
,20 LCT == I ROW( r ) ' .LCTO~D~P
25 rFCJ .LT. .
GOTO 35
JCOL(1,LC~)
~f(JCOLC2~LCT) .EQ: ·: LCTOLD=~CT
.
0) '
GOTO 30
" .,. : LCT==JQO.~ .(2,,~,kC,T) , . 'GOTO 2S : " ,' 30 ' JCOLC2,LCTl=MT '~COL(1 ~MT)~J
ACMT),='R , MT=JCOL'( 2,MT)
JCOL (2, JCOL(2, LeT) )=0 :
I
-RETURN 35 IFCLCTOLD e~EQ · .. 0) GOTO 40 ' - JCOL(1,MT)==J . A(MT)~R
JCOL(2,LCTOLD)=MT
MT=JCOL(2,MT) ' . . , JCOL(2;JCOL(2)LCTOLD»)=LCT RETURN 40 JCOL(1)MT);:J
ACMT)=R I ROW
, '.
. JCOL('2; I'RQWt: I ) )=L.CT RETUR~
END' ,
--
. .
, F mERMAL SYSTEMS .
.
'. '"jUBROUT I NE XGAUSS -- ( ,I ,'
'
.
.: I
,
1
- ' ."
.
:J ION OF SIMULTANEOUS· LINEAR EQUATIONS B~ . . .
; \ ';
'
· ~L 1MI NATION
:/' i'. SUBROUT I NE
.
tROW,} JCOL > A, B, X , N ,MAX ,MT)
.: ". '::
.
"
}
..
STORES ,.' ONL Y '. NONZERO ELEMENTS
,·;::/G.:. LINKED STORAGE::,';··' .. :' , ';:' . ' . ". , . .-:. ; '. .!.~ " ' !E'.'O$ED - ON"·'SPEr~5E ~·~~eT.R.lft:S .. iN · .CONJUN(trIDN::~.·'.'· ~'. ,··><::,.·i::', . ~':: ;:~J:';' ; i SUBROUT I HE ' NZ[RO . ~ " . ~. '-,<:. ..,!~., "".:;" .,:': "' "::-i· <:,~'~~ 'i: .;.· ·. ~ . ;.:.:·{;·:,:'~.;·:~<~.:·": .r .:.:,:.·. > :: ·· .'." :':' . . . '.'
: I
.
';' -
._
. . :
.'('!.;ARY FOR SUBROUTI HE XGAUSS ' = NUMBER OF VAR I A,BLES . , . =
~
: '~ ~ : _ .4
.'. , . ...,.,.. .:' . . ," ,', . .
" , " '. .
MAX NUMf3E'R. OF' NO~ZERb ELEf\1ENTS " AT' .'. . '.
··.ANY·TIME · '. . = NUMBER O'F ' F I R5T ." EMPTY LOCATION= tAL¢ULATED VALUt OF VA~IABLE~ '" = LO,CATION'OF'FIRSr NONZERO ELEMENT ' OF "
JW() ,
EACH ROW
'
;
,, :
COL NO OF ' NONZE~O ELEMENT ' OR O ~ IF ' LOCATION EMPTY . OLC2, ' )= LOCATION ,OF NEXT NONZERO ELEMENT DR NO OF NEXT EM~TY LOCATION~ '= ,VALUE OF COEFFICIENT ,
'JL(1)
)=
, '.') ,. I MENS ION · I ROWCN) , JCOL (2; MA"'X), ACMAX) ,B(N), XeN)
e1TMAX=O' . "00 50 K=1) N
"
'.
J
IOVING . LARGEST . ~OEFFICIENT
4
•
•
•
•
INTO DtAGONAL .POSITION
AMAX=O DO -5 1;::~,N IF ,CJCOL (1 ) I ROW( I ) ) . NE.'." K) GOTO 5 IFCABSCAMAX) ..• GE. ABS(A( lRQW( r » ) j GDTO 5 ,AMA X== ACI ROW ( I ) ) 1MAX =I
CONTINUE .
,
-
:STING fOR · INDEPENDENCEOF . , 'EQUATIDNS - · ,
~
.
.
,
IF CABSCAMAX) .• GT. 1.0E-16) 'GOTO ' 10 :'. :,' .,...: WRITE(6~JOO)
.
FORMATC '0 EQyAT I ONS",Ar?E ··~OT · I NDEPENDE~T")- ', RETURN " .
----.
.
,
..:
.,.".' . . ~
G
..r 1'1. ROf~ - r
c c . " .'
,~'
' . . I ROf;J( I J=fiJT' . '" ) . :. : ' .' JCOLCJ.'4"].t1.T.";__7:.J,y. _ ,. ' " .,. ..' ". A l' "''"IT)'~ R""
--' ..... ."' ., . '
.
,
.~ ..
,
J'
....
•
..
. • .
, .•, .
:
-.s
-
.....
': . .....
.' '
. MT=jCOL(2 , MT) . . ~ JCO~(21I~OW~I)=O
". '
,
RETURN
.C '
\. !' A a
·'-b "
SEARCH TO FI.ND PROPER LOCATION OF NEW, ELEMENT ' IN
C-
. ..c
ROW ' I
-
.' "
'. .c 20 LCT=IROW(IJ _.
' LCTO~D~O
25 IFeJ .LT. JCOLC1 lLCT)) I,f.C)COL (2 ~ LeT) EQ. 0) a
, '-LCTOLD=LCT - LCT=JGOL.:~?. J,-,l.C!) . GO TO 25 .
~OTO
'
' 30 ' JCOL(2,LCT)=MT ~COL(1 ~MT)=:::J
'
" ,ACMT),='R, MT=JCOl(2,MT) '. ', JCOL(2)JCOL(~)LCT)),,=O:
I
·RETURN 35 IFCLCTOLD O~EQ'D 0) GOTO 40'-
- JCOL(1,MT)=J , ACMT)=R JCOL(2)LCTOLD)~Mr
MT=JCOL(2,MT) , " JCOL(2,JCOL(2)LCTDLD))=LCT RETURN 40 JCOL(1,MT)=J ACMT)==R IROW( I )==MT ' MT=JCOl(2,MT)
t '.
'JCOL(2~IRQW(I»)=~CT RETUR~
END ',' ,
35
GOTO 30
f
DESIGN OF THERMAL SYSTEMS
I
)
c c
XC) I
)
(1
-c cc
:::: 1
)::::
AC)
C
),
I
N),X
) -
K= 1 ,N
c
J
c
leI
r
I
c
ITr
I::::~,N
(1 , I )
5
I» 5
10
-I
lOO
--
..
':;---
APPENDIX ill
55;
c EXCHANGING ROW IMAX AND
RO~ , K
c l O ETEf"1P=:,B( K) BCK ) =B CIMAXJ " , ~" ' :'
B f I MA X)=BTEfr1P " 'r 'TEf'x1,P':::':i 'ROW ( 1< )
!
..,.
'.
-I
. ,
._.. .
_..... ,,,":7""
..., . .
-
,'
.( .'
, ~':'
.... ~ , - - ::.: - '". ~
'. ", •
"
.IROW{K)=IRDW(IMAX) IROW( IMAX)= I TEMP
c'' c ' SUBrRACTING : A(I,K)JA(K~K) " C EQ FROM OTHERS .
-
-
.
TIMES , TERM ,
IN FIRST '
' C ,,', KPLUS~K+1 '
, tF (K ,.. EQ 'o N) 'GOTO "50 , DO '4S' ,I,;;:KPLU5}N, ' , IF (JCo.L (1 , 1 ROW( 1)) . NE,,, : K) GOTO 45
Ll=JCOL(2,IROW~I)
' ~"
LK=JCOL(2,lRDW(K)) ,', ' LIOLD= IROW( I) ,-",: , :'' , " B(I)=E(~) - (A~IROW(I))/A(I~DW(~)))*E(K): ,
,
15 , IFCLK -.EQ'.
0) GDTO' 40
20 IFCLI .EQ. 0) GOTO ' 25
,,
~F(JCOL
(1 ',LI)
, .~Q.
JCOL(1 , ~LK))
,
IF(JCD~
(1,LI) .GT. JCOL(1.)LK)) GOTO 25 '"
GDTd 35
LIOLD=LI LI=JCOL(2~LI) "
'GOTO 20
c C
NO COR,RESPONDI NG ELE;MENT I,N ROW I AS IN RON K
c
CREATE
NE~
NONZERO ELEMENT
C
25 LCT=MT '~T=JCOL(2,MT)
-~
.. .
... ..
-.
.
.
~
.'
, ,I F CMTMAX • LT. MT) MTMflX' ~MT ,
IF (MT .LT. MAX) GOTO 30 WRITE (6,200) MAX
. ' ,200 .FnR~AT(1H, ,,~ALLQCATEP STORAGE EXCEEDED, 1 MAX~") 15/1 H) ., I NCREA'SE MAX ,) THE ARRAYS
2 JCOl(2)MAX) AND A(MAX) AND RERUN'} RETURN
- ..
, 560
DESIGN OF TI!ERM.AL SYSTEMS • 1""
. 30 . JCOLC1' '; . LCT)=/jCDL( 1 ,LK}. · 9. ."..:;.. . .' "A(LCT);=:~'( A( I ROW(I ) ) I A{ I ROwe K'))) ~?A (LK) )',> .:: "' <~'" · .: ,.:,·~J·COL(2 7.li OLD)~LCT " . ' ,: ,'. '!. ' . :,' · .. · ,· '·.':. ,::-···JcOL. r2 ·',· LC'T)=LI· .,·,~' . . ': ' _.
• • : _,'
.. •
••' •
• •
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.' - . -' , .: ...
:- ":~-
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3S . ACLI) ~A(L I) ~(ACI Rowe I) )/AC IRDW(Kl)) *'A (LK)
.... LI {]LD'~.l~·I.
. .'
'. _. - .' . ..
. ... - --
LK=JCQL(2,LJ() : L I ::: J COL ( 2 , L I) ..-. .QOTO .1 5' ,
I
L.DCATION
,TO
•
40 L¢T=IROW(I) .. ~COL( 1 , LCT)=O· A'(LCT)=Q .' IROW(I);::JCOL(2>~CT)
.... J .COL (2, LeT) ==MT
MT==.LCT
45 CONTINUE 50 CONTINUE
C
BACK SUBSTITUTION
C DO . 6'0 I;:: 1 ) N ' , . PA,R T=B{N.+1-·r )/ACIROW(N+1-I»)
. LCT=JCOL(2,' IRDW(N+1- I))
55 IF
.'
(~CT ~EQ.
b)
~DTO
60
PARTFPA~T-A(LCT)*X(JCOL(1 LqT'?~COL LeT)' .. /..' .
(2,
. GOTO' 55 •
, I
•:
•
~ -.".. , . I . ~
. '
LIST ..
c
e ..
".
'
..C . ,C ELEM:ENT '1 7 K .IS··NOW ZERO, ADD . C . OF. EMPTY SPACES ' . .
. . 6 .0.····X(N-t:,1-1 )=PART " 'RETURN END ' .
'" ",. :
" .. ,
...
.
I.
,LCT»)/AC1RDW(N+1-I) .
,
.
\ ~-
' •• ; ' -.
" ',' :
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.:.....
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