Fluid Flow Through Packed Columns, Ergun

1 FLUID FLOW THROUGH PACKED COLUMN S SABRI ERGU N Carnegie Institute of Technologyt, Pittsburgh, Pennsylvania

The existing information an the flow of fluids through beds of granular Osborne flryadds (23) uas hest to formusolids has been critically reviewed . It has been found that pressure late the resi ante offered by Iriceiat to theo losses are caused by simultanstous kinetic and viscous energy losses, n'otiun ofofthe flu,I as the sum ternts, itioal respectively to of dertw first and that the following comprehensive equation is applicable to all types power of t!sr fluid velocity and to the of flow, product of the density of the fluid with AP Cl - e) t PU . 1 - s GU„ second power of its velocity :

L p, = 150 a + 1 .75 .a Da AP/L =all + beV (1) where AP is the pressure on alon th e length L, a the density of the fluid, 11 its linear velocity, and a and b are factor s

The equation has been examined from the point of view of its dependence upon flow rate, properties of the fluids, and fractional void volume, orientation, size, shape, and surface of the granular solids . Whenever possible, conditions were chosen so that the effect of one variable at a time could be considered . A transformation of the general equation indicates that the Blake-type friction factor has the following form :

which are functions of the , system . A transformation of Equation (1) which yields a linear expression is :

AP/LU = a+ bG (2 ) where 0,U has been replaced by G, the mass Sow rate . The above two-term preswrefa a 1 .75+ 150 drop equation has been found to be astir art over the range of flow rates enA new concept of friction factor,/. representing the ratio of pressure factory countered in pecked columns. Lindquist drop to the viscous energy term is discussed . Experimental results ob- (19), Morcom (20) . and Ergun and tained for the purpose of testing the validity of the equation are reported . Orning (7) have platted AP/LV ag inst Numerous other data taken from the literature have been included in G and obtained straight lines as expected from Equation (2) . The former two authe discussions. thors have included in their plots factor s hick w ptra in to the p,s rties of th• s aT HE pressure loss accompanying the utilize some of the general equations ten These : factors are re important and will flow of fluids through column s representing the forces exerted by the be discussed later, but they are irreic :ant packed with granular material has been fluids in motion (molecular, viscous, for the purpose of testing the linearit .• of (2) . As a typical plot. der. obthe subject of theoretical analysis and kinetic, static, etc.) to arrive at a useful Equation for gas Sow through a be of factors.porous A ,.rained experimental investigation . The pur- expression correlating ffthese crushed solids are shown in N. ;tar e pose of the present paper is to smnmar . survey of the literature reveals various I. The experimental results of the present investigation and those mentioned ax,ve ice the existing information, to verify expressions derived from - different ( :, 19, 20), as well as the data ott:inmd further experimentally a theoretical de- assumptions, correlating the particular the literature (3, 22) . militate that retopment presented earlier, and to experimental data obtained with or with- . from the two-term equation accurately cite tiara . discuss practical applications of this new out sonic of the data published earlier the relation between flow rate and ptasure approach . The experimental studies These correlations differ in many redrop. have been confined to gas flow through spects ; some are to be used only at low 2. Viscosity and Density • of Fluid. crushed porous solids. This case is the fluid flow rates. while others are apFrom Equation (2) it is seen that as the velocity a'sproaches zero as a lien t. the one usually encountered in practice, but plicable only at higher rates . A separate ratio of pressure drop to velocity ad, beis not identical with the case most thor- survey of all these various correlations come constant : oughly studied by previous investiga- is not included here. iAPC/tors, viz ., the flow of fluid through bells As most authorities agree, the factors L m s (3 )U+ e 0 of nonporo(s solids, and more particu- to be considered are : (1) rate of fluid whi ch is a coalit i on for v iscous flnv, . Act larly. througi solids having uniform flow, (2) viscosity and density of cording to the Poiseuille equation and geometric shapes . the fluid. (3) closeness and orientation Dar 'a law, the factor a is propcr:ional Factors determining the energy loss of packing, and (4) site . shape, and to the viscosity of the fluid . The xher limiting condition is reached at hign flow (pressure drop) in the packed beds are surface ai the particles. The first two when the constant a is negtigil It in numerous and some of than are not variables concern the fluid, while the rates comparison to bG. This is a condition for susceptible to complete and exact mathe- last two the solids, completely turbulent flow where k-aetic matical analysis . Various workers in 1. Rate of Fluid Flow, It is known energy losses constitute the whole vainthe field have made simplifying assump- that pressure drop through a granular tance. The effect of density is already in G. Equation (2) as be tions or analogies so that they could bed is propor)ional to the fluid velocity at contained low flow rates, and approximately to the rewritten : C o a l Resecrch Laboratory. square of the velocity at high tests. API' -a a'pU+b9(P (4 ) 1 -6

Vol . 48, No. 2

NY 62592 '_°p` Ossc,iption

Chemical Engineering Progres s

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1. Total cmrgy Irwses in hoot IM,I> ran I .e erraloi us Its,, solo of viarr.ra AM kinetic energy tosses.

3. Cleavers (Fractimul Void Volume) and Orientation of Packing . Free . li•,n„I y. Md v.duunv has bete one of the na,-I :,oIr,n•rr.cd fads,. h . i 4'1 .'d 073, .n,s. ulc lu,•,rvlical Ircauucnls wire n,n a„r. ,,rod m -'al li,hinR the ticpendair •d )fir pr . .surc ,.rq. niece iradmual vied v .dnnn It was sheet who first our *hllly ,rrtu,l Ile reecho by an appr„orh ;mnh . ;wu lu I lull ' if Stanton and I'aun,•11 I .5) In gwr„urt drop in circular pile,- . lliake .dnaiwsl Olt- foll.,wlpg dimes . ,and.•s- grw1P.ti

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nhrrr t is the fractkmal void volume . p . ll .' ar,,.vitatioINl ca.NtanA and D. the dia.ndarr of the solid particles. The first of hcxw aruaps is recognised as the md(iliy.l rfcl .an (actor aid the second as the n .,slifuvl Reynolds number . Blake sugp,-a,l Il.at the inner of these groups be I11• .it.s1 against the latter. Since both di - group, ro,tafn the fractiona l , .id e,Aume. it can be deduced that pres.ure drop is not a function of a single Group aluMsc The failure of lux earlier attempts to arrive at a useful expression can be attn . hard to the want of recognition of the fact that pressure drop is caused by simultanew, kinetic and viscous energy losses ,

the garter needling tilt fractional vow ndunIe Icing (I - r)/e' . This range of tl .e plot at Blake tae generally 4m over . la .ke,L Basel no the theory of Reynolds for resfstanet to doid flow and the method of K.rsoy, a gateral ol• .atirnl was developed by Ergun and Orniol; fur pressure drop thr.wgb fixed beds, In summary the folluwiou raxlusknra can he drawn from U .eir w.xk :

shred„ o«, One law ., 4 Fig . 1 . Typfiwl phsrs of the On... hew of pe . .tar.wlrop .quill. . rot . cynic pochsd 'a diltonmt hoetlit of veld salver.., 0g.. .'1.. (2). aurae.. cow ihrwak 1670 ere, high 1.w pin .NN pen cake. P.nki domitp ra 1 .046 g./m Crowsnuik..w.l oron of rob. 7.74 wtse . fall .1 724 ms Ms . and 21' C .

Theoretical corsitkratwns of later workers (3, 7) indicate that dependency of each energy loss upon fractional void volume is different. Burke and Plummer proposed the theory that the low resistance of the packed bed can be treated as the stun of the separate resistances of the individual particles in it. Accordingly, via coos energy loss was found to be woporuonal to (1-r)/, and kinetic loss to (1 -The authors, however, failed to recognise the additive nature of these losses and correlated the pressure drop by the use of dimensionless groups similar to those of Blake . For viscous flow . Koarny(14) arrived at an equation widely used later (4 . 10, 11 . 13. 1 .5, 261 by ssvunnng that the granular bed is equivalent to a group of similar parallel darnels . The derival dependency up'Mn fractional void volume was (I-s=/e'. This factor is different by a (raaio . 0 - r)/q iron the factor derived by Burke for viscous flow. Fair and Hatdt 410) . Carman (4) . Inn and Surse (13), Fowler and Hertel (11) . and others (6. 13, 1:. 36) verified the Koteny factor experimentally, For a general correlation valid at all flow rotes, however, Carman recommended the plot of the dimensionless groups of Blake. Recently . Leva (24) anal horse (22) also adopted Blake's procedure in presenting the pressure drop data in filed beds . Lena, et al . (18) stated that the pressure drop was proportiorwl to (1 - .) /.' at lower dove rates and to (1 - s)/.' at higher flow rates.

Carman noted that at low fluid-flow rates the method of Blake leads to the Koreny eguatiou. hesxe to tux actor APR. (I- .)' p('

2. Viacom clergy kenos art pr .pxglknual to') t -c)'/ .' ae.l tIm kiotir energy I.to (I - .1/0. Since u a.Ml h of F.quatiat (4) represent the e..MTxkmls of viscous a,Ml kinetic energy losses . rcnprrdvtly. it is ,spoiled that a he pnpn,etbnal to (I - s)'/.' and h to O- .)/.' in order for the theory to be valid. .\ItMmch the above author. have curr,lat .,l tirade data suctt sfully single systems have nip been thoroughly examined at various frarti .n .al v, .kl volumes . One of tow aims of IIM• present work bas, been to inveslieatc Ow sinalr systems at various packing densities . A known amomn of solids was packed 6 t„ 20 different bulk densities each resulting in a different fractional void volume . For each packing the coefficients it and b of Equation (2) were determined from pressure drop and flow rate measurements (Fit. 1) . Firures 2 and 3 show typical plow of a against(, and b .globe (I-t)/e' obtained from Figure 1 . Saab plot, yield straight lines ach passing through the origin . The graphical representation is simple, yet most ective in tie investigation of the function of fractional void volume. A similar procedure has been adopted recently by Arthur, et at (1) it, testing the validity of the S.oaeny vuatias and by ErFun (0) in camrctiun with particle density determinations for porous solids, It is of in :crust also to note that the two extreme ranges of the Blake plot lead to the tern of the general equation proposed by Ergun and Oreins- The pro. parliorralities an he expressed in the formulae :

aneo"(=-~r) (7) :I, = b" .~' 1

(g )

where a" and b' are factors of proporti-

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(5) On the other hand, at high flow rates Bake t tttethod g iv ise t o th e e q uatio n es r fie. 2, la.ps.d.- .' is- wet hkw * .• oofBurke set Plummer for turbulent ansinl I.- - f .oakr .ol said -I.-, aq .. tint al .rod .(Q, lasnapis .wet dopes .ro_.b• . M Nf *d 1. at fig .. I by --*,W of asks 7- (6 )

p-t)' ir rig. 7. O.p..d. .r. at vista., ..orgy f.w .. . fc. .eel mid wotw.. tq,..riw. (7). 0 ... •, .w .brok.od hr akregw 4- through 7040 .lath, fags soh., liamak deWry as 1 .27 0./oe. Croe'u ..ri.o1 0- -0 It . %4. w 7.24 pose . Ink Dos in 740 . ..u M* laid 23' C

Chemical Engineering Progress

Februory, 1952


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alley . Their substitution into Equation (2) yields :

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(9) A rearrangement of Equation (9) leads to :

(10) Equation (10) makes it possible to group at data of Figure I on a single line by plotting AP s'

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LU (1 - .) " against G/(I- .) . This is demonstrated ht Figure 4 . Up to this point the aim has been to formulate the effect of fractional void volume in fixed beds, and the effect of orientation was not included . The orientation of the randomly Packed beds is not susceptible to exact mathematical formulation . This is especially true it the particles have odd shapes and are not negligible in size conpared with the diameter of the container . Furnas (12) has treated the subject at length and introduced the concept of "sarnwl packing" which was obtained by a slaunlard procedure. In tine present investigation, however, such a concept had to be abandoned, The problem was to pack a known amount of solids to various bulk densities . yet each packing had to be antforin and reproducible. This was accomplished by admitting gas below the supporting grid after the solids were pound in. The gas rate was sufficient to keep the bed in an expanded state and the use of a vibrator attached to she tube assured the uniformity of the packing . By varying the rate of upward gas flow, the bulk density could he varied from the tightest possible to tie loosest stable pack . ing, For crushed material the most tightly packed bed having a height of 30 cm. could easily be expanded by 6 to 7 con . When the desired pa,kiug density was at taincnl, the vibrator was ltxnlnulvcw1 anal the gas now rut off- The bed that was ready for pres . sure drop and flow rate measurements. Highly reproducible packings can be obtained by this method, and more important. the particles are believed to be oriented by the gas doming upward . This is evidenced by the existence of a theoretical relationship (7), verified experimentally, between the bed expansion and the flow rate . A further evidence for particle orientatio n was found in the fan that the most tightly • packed beds have been obtained by slowly reducing the rate of upward gas flow to an initially expanded bed while subjecting it to vibration . It will be evident on inspection of the form of Equation (9) that the estimation rat fractional void volume is important, particularly since it enters to second . and tlntrd-power terms aid is in many aces difficult sea measure directly. Whenever the particle density and the total weight of the granular material filling a given volume are known . a may be readily alculated . But the particle density of crushed porous materials is not readily known and its determination has presented a problem which was much discussed Fractional void volumes were usually calculated by the use of apparent specifu gravities which were determines by variant procedures . Use of such values for a in the pressurcdrop equations masticated the introduction of correction factors . This often caused the workers to doubt the validity of the factors describing the dependence of pressure dro p

,,' ;? Vol. 48, No. 2

upon . and to seek little correlations . However, this was believed to be unwarranted (g) sitter the determination of pressure drop through beds of porous panicles hinges upon the evaluation of the particle density. Therefore. a gas flow method was developed (8) for the determination of the particle density of porous granules . The method was ducked by the densities obtained for nonporous solids and the agreewin was good. Use of the particle densities of coke obtained by the method described, in the determination of fractional void volume sad hence in the promote drop equation, resulted in excellent agrexnwuu.

cases the concept of specific surface was believed to be not applicable by Burk : who suggested compensation by cmpirica factors in connection with the use e f the Blake plot . Determination of specific surface in~clves the mcasurerne" of the solid surface area as well as that of solid volume stint presents no problem for uniform geo :nctric shapes . For irregular solids, especially fur porous materials, however, surfs" area determination becomes involved. The surface of porous materials is necessari .y full of holes and projections. Different surface arms are usually defined in connection with porous materials, viz., total surface area

I

4. Sits, Shape and Smrface of the (including that of pores), external visibl e Particles T he effect o f the particle site . is-best surface area, Pt e area, Cie. and shape analysed in the light -of .A surface geometric sur face, as dis ti nct f rom ex . theoretical implications of the Blake plot .

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ternal visible surface, may be visual.ted as the surface of an impervious envelote surrounding the body in an aerodynamic sense . Irregularities and striae on the surface would not be taken into full accoui .t in a geometric surface area in contrast to external surface area. Whether the value of the total, external or geometric surface area is .lesircd will depend on the purpose for which it is to be used. Geometric surface APy ./L = 2"µS: U .(I --s)s/a' arm is believed (9) to be the relev .,nt one + (p/8)GU .S.(1 - .)/s' (11) in connection with the pressure crop in parked columns. This is made evident by where a and p are statistical constants, g, the close agreement between the nurface is the gravitational constant, and S. is the specific surface of solids . i.e.. surfs" of areas determined by gas-flaw methods and those by microscopic and light extinction the solids per out volume, of the solids. meshetls . inasmuch as the surface rough. Instead of specific surface. S., surface per ness affects both the geometric surface area unit packed volume . S . has been employed by some workers. Since the latter quantity' and the particle density, the deterntisation involves the fractional void volume, use of of its influence upon pressure drop ties in the evaluation of the effective values of specific surface has been preferred in the then quantities. present work. The relation between the two quantities is expressed by It has been customary to use a ch uracteristic dimension to represent the part cle site Sea (1- .)S« in pressure-drop atculations. The charocEquation Ill) involves the concept of terutie dimension generally used is the "mean hydraulic radius" in its theoretical diameter of a sphere having the specific development (7) . Its validity has been surface. S.. which is expressed by tested with spheres, cylinders, tablets, sot

The identity between the two extreme ranges of the Blake plot and the theoretical equations developed respectively by Kozeny and Burke for viscous . and turbulent-flow ranges has already been shown . Also, is has been pointed out that these two expressions cotnsinnad the following general equation developed by Ergun and Qrning (7) :

doles, round sand and crushed materials (glass. coke, coal, etc .) and found to be sotisneunry . The experiments have not been extended to inctale solids having holes and other special shapes. Few thos e

Snbatitutiat of Dr into Equation (11) yields : Ails. (I - .)' AU . + k 1 - . GU . . W ~so k (12)

where k. en 72 a and k. = 3/4 o, , Pinar torn of Equation (12) is :

(13 ) N ., = D p The left-hand side of Equation (13) is the ratio of pressure drop to the viscau energy term and will be designated by f .APD•

L U . (1^ .)

(13a )

/. = k.+ k . -I V-=~ (136)

B mtg . 4. A gsaersl plat for • single grins. petition to dgdarem #,*a* l odd "4e•. pats e tln gol.•~ .ap..» ~ arknj tqst w i p) • 'a straigh


According to Estwtiot (13) a linear relatio udnip exists between I. and A's ./ 1- e . Data of the present investigation mi those presented earlier have been treated accordsngly, std the coefficients Jr. and Its have been determined by the method of least squares. The values obtained are lit o ISO and ter at 1 .75 representing 64( experinicala. Data involved various-"l spheres.

sand, pulverized coke, and the ollowing gases : CO. N. CH . and Hs. Otwe the constants Jr. and A . were obtained it wa s

Page 91

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possible to construct the genera] equation . The results are shown on the top of Figure 5. To be able to include a wider range of data, a - logarithmic scale has been used which results in a curve for the straight line of Equation (13) . Data of Burke and Plummer and those of Morcom are also shown in Figure S . In all three cases the solid lines are identical and are drawn or . cording to the following equation : I. .— ISO+ 115

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1 Data shown in Figure 5 and some additional data obtained from the literature covering wider ranges of flow rate are included in Figure 6, together wilh the asymptotes of the resulting Curve on the logarithmic scale. Again the solid line represents Equation (13x) . A different form of Equation (121 is represented by :

APg . D, os :z k . 1a + k L GG . - rs N ..

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Chemical Engineering Progres s

(14) The left-hand side of Equation (14) is the ratio of total energy losses to the terns repeetertling kinetic energy losses and will be designated by / . ,_ PE D sot J . (14.) E 11150 1Ns. + 1 .75 (1db )

February, 1952


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1. is similar to the friction factor more commonly used and is identical with the dimensionless group of Blake . It will be noted that Burke and Plummer plotted essentially ft vs, ( ► - .)/M. . which per, according to Equation (14), should yield a straight inc lon an arithmetic scale. The authors apparently failed to recognize this fact . The best curve drawn through the expert usI pants on an arithmetic scale does not differ markedly from the line representing Equation (14b) . The scatter to be seen an the plot of Burke and Plummer was largely due to the systems involving mixtures and those for which the ratio of tube diameter to particle size was less than 10. 1 hew systems have been emitted in Figure 5, It has been customary, however, to plot f. against N.,/(1 - e) instead of the inverse of the last variable. This type of plot is the one suggested by Blake and adopted by Carman, Morse and others. Figure 7 shows I. plotted vs. N . ./(I- .) for the data already presented in Figure 5 . Figure I is a more comprehensive presentstion. The solid 1• act arc drawn according to &luation (lob) . A comparison of Figure 6 with 8 is analogous to that of 1. with ft. Both plots are capable of presenting the data. However, I. pus a big advantage over ft in that it is a linear functioc. of the modified Reynolds number , ,) . The curve of Figure 6 is a straight line on an arithmetic scale . On the other hard. I ., which has been used aln,ost exclusively, is an inverse function. A comparison of various empirical representations with Equation (I2) as to be seen in Figure 9. The foregoing treatment so far has been confined to studying the factors involved in the pressure loss in packed beds and to analyzing experimentally the theoretical developments presented earlier . It is only proper that the equations presented are also analyzed briefly from the standpoint of pure fluid dynamics . Fortunately, the equations lend themselves for such analyses . By definition :

1) ► to 6/S. (150 ) and S. = S,/AL(1 - .) (15b) where S, = total geometric surface area of the solids and A = cross-sectional area of the empty column . The total iorep exerted by the fluid on the solids = GPp,Ao . therefore the tractive force per unit solid surface area, usually referred to as the shear stress, e, is expressed by : is

r or .>Ag.A ./S, (15e) The ratio of the volume Occupied by the fluid in the bed, AL., to the surface area it sweeps, St, is the hydraulic radius, rs,

varied with the fractional void volume . Whether or not kt is a constant is to be decided on inspection of the lower end of Figure 6 and the upper end of Figure 8 where viscous energy losses are dominant However, the inherent inaccuracies involved in the meusurcments of specific surface, fractional void volume, eta, must be borne in mind In the present work, moreover, single systems were investigated at different fractional void volumes and no evidence of variance of its with . was found. This point is clearly supported by the proportionality of a to (I- .)s/ es as to be seen from Figures 2 and 3, and similar other graphical representations (1, 8, 9) . The factor ks(sa 3/4B) is subject to treatment similar ta that of kt (7, 8, 9),

Substitution of Equations (lSarr) into Equation (13a) give s f. s. 36 r (16) and into Equation (140) gives !ass6 P- (17 ) Similarly proper substitution will yield Na' as 6pnra (18) Therefore, Equations (13) and (14 ) respectively will become :

36 r=s •a 150 + 1 .75 6prs # at .0 (19 ) and 6Puts Its 130 s +1.75

Summary (20) It is seen that these transformations The laws of fluid flow through granemploying the absolute values of shear ular beds have several aspects of pracstress, fluid density, and velocity elimitical consequence . They generally find nate the fractional void volume. The use in correlating the rate of mass and terms involved in Equations (16.20) heat transfer to and from moving fluids are well known in the fields of hydro(24) . The extension of such relationand aerodynamics. Other forms of de- ships to packed columns will rtquire pendences upon . ascribed to a general formulation of the laws of fluid flow equation, as encountered in the litera- through granular beds . Empirics, corture, would not lead to complete elimi- relations are generally useful for the nation of the fractional void volume particular purpose for which tl..ey are upon transformation to these fundamen- made, but may not shard light for a tat variables. different purpose. For the sake of The theoretical significances of the clarity in the application and use of the constants let and lea have been omitted data obtained in packed columns, it in the foregoing treatment The former-* seemed desirable to develop expressions of these constants is discussed by Car- (Equation (12)) in a comprehensiv e form applicable to all typos of flow . I n man and Lea and Nurse (15) in connection with the Kozeny equation. As doing so the theoretical developments, a result of comparison of various sysas well as the empirical approaches, tems involving different fractional void' have been considered and the following volumes, Lea and Nurse (16) concluded conclusions have been drawn : that a(=let/72) was not a constant but 1. Total energy loss in fixed beds can

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rs or ALe/Sd (1Sd) The actual average velocity of the fluid in the bed is obtained from the ratio of the superficial fluid velocity to th e fractional voids, a to f1/.

. :952

Vol. 48, No. 2

2 3 4 6

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r1 . a a e.n pe .h.adv. s4a at pnwr4 dm0 in fund beds, Data dd rfe.e k or. ..pbno d . (15t ) This sep. at plus 4 idntnol with dear at sink. . a.ra atw is den rwu.rdt .9 sit aq ..t:o . (146) ,

Chemical Engineering progress

pogo 93


Use _ ouptrticial fluid velocity nice sacred at average pressure • a coefficient of viscous energy tern, in Equation (11 ) A = coefficient of kinetic energy tern, is Equation (I1 ) • = fractional void volume in bed p am absolute viscosity of fluid p = density of flui d = average shear stress, defined by Equation (lSc )

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1 . Arthur, J. R ., Linnet, J. W Raynor, E. J ., and Sington . E. P. E, Trans . Faraday Sat. . 46, 270 (1950) .

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Literature Cited

1

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Na, I-7 peg. 9. Coupwls.a at w eSa, .w*Uk•1 npr.mnta$ .ns rrta, e•w/fs . (12). be treated as the sum of viscous and kinetic energy losses. 2. Viscous energy losses per unit length are expressed by the first term of Equation (12) : ISO 0(1-e)s ssU„ es per and the kinetic energy losses by the second term :

. 3 . For any set of data the relative

amounts of viscous and kinetic energy losses can be obtained from either Equation (13) or (14) . 4 . A new form of friction factor, f,• representing the ratio of pressure drop to the viscous energy term has been given (Equation I3c) and should have advantages over the conventional type of friction (actor . 5. A linear equation .too been shown to represent the conventional type of friction factor, vie ., the ratio of pressure drop to energy term representing kinetic losses (Equation 146) . "

I

Acknowledgment Tlae author acknowledges the encouragement and advice of H . H . Lowry and J . C. Elgin, and the assistance rendered by Curtis W. Dewalt. Jr., in preparing this manuscript. Notatfofl n oo'd = coefficients in Equations (1), k (4), and (7), respectively A = cross-sectional area of the empty column

6.6" coefficients in Equations (1) and (8), respectively Ds a effective diameter of particles as defined by Equation (ISa ) . = friction factor, which repro )' . sends the ratio of pressure loss to viscous energy loss and which is linear with mass flow rate, defined by Equation (13a ) friction factor . identical with the dimensionless group of Blake, defined by Equation (1k) gravitational constan t 9. G m mass-flow rate of fluid. G=,v U At n coefficient of the viscous ear ergy term in Equation (12) ; k, = 15 0 kg - coefficient of the kinetic energy term in Equation (12) ; k2 - U S L = height of bed Nt,, = Reynolds number, Na, = D,G/p: . P s pressure loss, force units ra = hydraulic radius of packed bed, defined by Equation (lSd ) S = surface of sagida per unit vol . time of the bed St = total surface area of the solids in the bed S, = specific surface, surface of solids per unit volume of Solid s is actual velocity of fluid in the bed U superficial fluid velocity based on empty column croon section

2. Blake, F . E. Tram. Am. Inst. Cheap . Sages. . 14, 415 (1922) . 3. Burke, S. P ., and Plummer, W. B., l,sd. Env . Chen,., 20. 1196 (1928) . 4. Carman. P. C, Trans. last. Chsa, . Engrs. (London), IS, 150 (1937). 5. Chilton, T. H,, and Colbum . A. P ., lad . Exp. Chem ., 23, 913 (1931) . 6. Donal, J Wuurkrds a. Watsen,'irt,, 225 (1919) . 7. Ergun, S., and Orning, A. A. Ind, Eng . Chew., 41, 1179 (1549) . 8 . Ergun. S, Anal. Chem., 23, 151 (19$1) . 9. Ergun. S .. "Determfnatioa of Gm` metric Surface Area of Crushed Porous Solids." Not yet published.

10. Fair. G. 3f ., and Hatch, L P .. J . Aan. Naar Works Assoc., 2$, 1551 (1933) . It . Fowler, J. L, and Hertel. IC L, J. Ap Nied,Phy, ., It . 496 (1940) .

12. Furnas. C. C., U . S. Bar. Mines Bull ., 307 (1929) . Hatch, L. P., .13 J. Applied Vrrha,iee, 7 , 109 (1940) . 14 . Koseny, J ., Sit_l.cr. Abed. Wise. It'irn,^ blase.-nausea. Alan, 136 (AM. ]Ice), 271 (1927).

15. Lea, F. At., and Nurse, R. W., J. Soc. Clem . Ind ., 58, 277 (1939) .

16. Lea, F. It ., and Nurse, R . W., To-s. less. Chem . Loge . (London), 25, Supplement, pp . 47 (1947) . 17 . Lewis, W. IC .• Gilliland, E. R ., and Bauer, W . C.. led. Esp. Chen,., 41, 1104 (1949) .

18. Leas. It., and Grammer, H ., Chre, . Eng. Prop-,; 43, 549, 633, 713 (1947) . 19 . Lindquist, E., "?tvmier Ganges des Grander Barrages." Vol. V, pp. 8199, Stockholm (1933) . 20 . Marcum, A . R ., Trans. lest. Chum . Engr,. (London), 24, 311 (1946) . 21 . Morse, R. D, lad. R.O. Chew., 41 . 1117 (1949) . 22. Oman . A. O, and Watson K . 3d . . NaN. Pcaraleum New, 36, R79$ (1944) . 21 Reynolds. O, "Papers on Mechanical and Physical Subject., ." Cambridge -_ University Press (1900) ,

24, Sherwood . T. K., "Absorption and Extraction .' McGraw-Hilt Book Ca, New York N. Y. (1937) . 25. Stanton, T. E, and Pannell . J . R» 9 Tr (ans. )oy. So .. (Leaden), A214 , 19 R1914 . 26. Traxlee. R. N., and Baum. L A . K. Physic, 7, 9 (1936) . 27. Wilhelm, R. H, and Kwauk, al .. Chew. Exp. Progress, 44. 201 (1948) .

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Palo 94


February, 1957


V:


Fluid Flow Through Packed Columns, Ergun

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