PLATE-FIN-AND-TUBE CONDENSER PERFORMANCE PE RFORMANCE AND DESIGN DESIGN FOR REFRIGERANT REFRIGERANT R-410A AIR-CONDITIONER
A Thesis Presented to The Academic Faculty By Monifa Fela Wright
In Partia P artiall Fulfillmen of the Requirements for the Degree Master of Science in Mechanical Engineering
Georgia Institute of Technolog May 2000
PLATE-FIN-AND-TUBE CONDENSER PERFORMANCE PE RFORMANCE AND DESIGN DESIGN FOR REFRIGERANT REFRIGERANT R-410A AIR-CONDITIONER
Approved:
________________________________ Samuel V. Shelton
________________________________ James G. Hartley
________________________________ Prateen Desa
Date Approved____________________
ii
PLATE-FIN-AND-TUBE CONDENSER PERFORMANCE PE RFORMANCE AND DESIGN DESIGN FOR REFRIGERANT REFRIGERANT R-410A AIR-CONDITIONER
Approved:
________________________________ Samuel V. Shelton
________________________________ James G. Hartley
________________________________ Prateen Desa
Date Approved____________________
ii
TABLE OF CONTENTS
LIST OF TABLES
vi
LIST OF ILLUSTRATIONS
vii
NOMENCLATURE List of Symbols
xii xii
List of Symbols with Greek Letters SUMMARY
xxiii
CHAPTER I: INTRODUCTION Research Objectives
1 4
CHAPTER II: LITERATURE SURVEY Previous Studies on Variations of Heat Exchanger Geometric Parameters
5
Previous Work in R-22 Replacement Refrigerants
5 8
Two-Phase Flow Regime considerations in Condenser and Evaporator Design
13
Two-Phase Two-Phase Flow Heat Transfer Transfer Correlations Correlations
16
Two-Phase Flow Pressure Drop Correlations
19
CHAPTER III: AIR-CONDITIONING AIR-CONDITIONING SYSTEM AND COMPONENT COMPONENT MODELING Refrigeration Cycle System Component Models Compressor Condenser Condenser Fan Expansion Valve Evaporator Evaporator Fan
23 23 25 25 28 40 40 41 44
iii
Refrigerant Mass Inventory
45
CHAPTER IV: REFRIGERANT-SIDE HEAT TRANSFER TRANSFER COEFFIECIENT AND PRESSURE DROP MODELS 51 Single Phase Heat Transfer Coefficient 51 Condensation Heat Transfer Coefficient
56
Evaporative Heat Transfer Coefficient
61
Pressure Drop in the Straight Tubes
62
Pressure Drop In Tube Bends
70
CHAPTER V:
AIR-SIDE HEAT TRANSFER COEFFICIENT AND PRESSURE DROP MODELS 76 Heat Transfer Coefficient 76
Pressure Drop
81
CHAPTER VI: DESIGN AND OPTIMIZATION METHODOLOGY Figure of Merit (Coefficient of Performance)
89 89
System Design
94
Optimization Parameters Operating Parameters Geometric Parameters
94 95 96
Software Tools
97
CHAPTER VII: OPTIMIZATION OF OPERATING PARAMETERS Effects of Air Velocity, Ambient Temperature, and Sub-Cool
98 100
Effects on the Seasonal COP
109
Range of Optimum Operating Parameter
111
Effect of Operating Parameters Parameters on System Cost
111
CHAPTER VIII: OPTIMIZATION OF GEOMETRIC DESIGN PARAMETERS FOR FIXED CONDENSER COIL COST 112 Area Factor and Cost Facto 136 Varying Number of Rows of Condenser Tubes
113
Varying Condenser Tube Circuiting
115
Varying Fin Pitch
124
Varying Tube Diameter
137
iv
Operating Costs
145
CHAPTER IX: OPTIMIZATION OF GEOMETRIC DESIGN DESIGN PARAMETERS FOR FIXED CONDENSER FRONTAL AREA 152 Varying the Number of Rows of Condenser Tubes 153 Varying Fin Pitch
159
Varying Tube Diameter
163
Operating Costs
170
Varying the Base Configuration Frontal Area
179
CHAPTER X: CONCLUSIONS AND RECOMMENDATIONS RECOMMENDATIONS Conclusions List of Conclusions
185 185 188
Recommendations Optimization Parameters and Methodology Computational Methods Refrigerant-Side Heat Transfer and Pressure Drop Models Economic Analysis
191 191 193 196 196
APPENDIX A: AIR-CONDITIONING AIR-CONDITIONING SYSTEM: EES PROGRAM PROGRAM
197
REFERENCES
227
v
LIST OF TABLES
Table 2-1: List of Refrigerant R-22 Alternative Refrigerant Mixtures
12
Table 5-1: Coefficients for the Euler Number Inverse Power Series
84
Table 5-2: Staggered Array Geometry Factor
85
Table 5-3: Correction Factors for Individual Rows of Tubes
87
Table 6-1: Distribution of Cooling Load Hours, i.e. Distribution of Fractional Hours in Temperature Bins
91
Table 8-1: Material Costs (London Metals Exchange, 1999)
114
Table 8-2: Condenser Circuiting Configurations
124
Table 8-3: Refrigerant Pressure Drop Distributions at 82° F Ambient Temperature 128 Table 8-4: Seasonal COP and Area Factors for Varying Fin Pitch at Optimum Air Velocity and Sub-Cool for Fixed Condenser Material Cost 130 Table 8-5: Condenser Tube Dimensions (www.aaon.com. AAOP Heating and AirConditioning Products web site) 138 Table 8-6: Optimum Seasonal COP’s and Area Factors for Varying Tube Diameters
141
Table 9-1: Optimum Operating Conditions for Varying Number of Rows with Fixed Condenser Frontal Area
154
Table 9-2: Optimum Operating Conditions and Cost Factor for Varying Fin Pitch with Fixed Frontal Area 162 Table 9-3: Optimum Operating Conditions and Cost Factor For Varying Tube Diameters with Fixed Frontal Area 166
vi
LIST OF ILLUSTRATIONS
Figure 2-1: Typical Plate Fin-and-Tube Cross Flow Heat Exchange
5
Figure 2-2: Horizontal Two-Phase Flow Regime Patterns
14
Figure 3-1: The Actual Vapor-Compression Refrigeration Cycle
24
Figure 3-2: Typical Cross Flow Heat Exchanger (fins not displayed)
30
Figure 3-3: Hexagonal Fin Layout and Tube Array
37
Figure 4-1: Refrigerant-Side Single Nusselt Number vs. Reynolds Numbe
55
Figure 4-2: Condensation Heat Transfer Coefficient vs. Total Mass Flux Fo Refrigerant R-12
58
Figure 7-1: Effect of Operating Conditions on Evaporator Frontal Area
99
Figure 7-2: Effect of Air Velocity on COP for Various Ambient Temperatures and Optimum Degrees Sub-Cool 101 Figure 7-3: Effect of Air Velocity on Compressor and Condenser Fan Power 13° F Sub-cool at 95° F Ambient Temperature 103 Figure 7-4: Effect of Ambient Temperature on COP for Varying Degrees Sub-Cool at 95° F Ambient Temperature with an Air Velocity Over the Condenser of 8.5 ft/s 105 Figure 7-5: Effect of Ambient Temperature on the Evaporator Capacity for Varying Degrees Sub-Cool at 95° F Ambient Temperature with at Optimum Air Velocity
106
Figure 7-6: Evaporator Capacity vs. Ambient Temperature for Various Sub-Cool conditions at 95° F Ambient Temperature and Optimum Air Velocity 108
vii
Figure 7-7: Effect of Air Velocity on the Seasonal COP for Varying Sub-cool Conditions
110
Figure 8-1: Effect of Number of Rows on the Seasonal COP at Optimum Air Velocity and Varying Sub-Cool for Fixed Cost of Condenser Materials 116 Figure 8-2: Effect of Number of Rows on Compressor Power and Refrigerant Pressure Drop at Optimum Sub-Cool and Air Velocity for Fixed Condenser Material Cost at 82 ° F Ambient Temperature
118
Figure 8-3: Effect of Number of Rows of Tubes on Condenser Frontal Area fo Fixed Condenser Material Cost at Optimum Sub-Cool and Air Velocity 119 Figure 8-4: Effect of Number of Rows of Tubes on Condenser Fan Power and Airside Pressure Drop for Fixed Condenser Material Cost at 82° F Ambient Temperature at Optimum Sub-Cool and Air Velocity
120
Figure 8-5: Effect of Air Velocity on Seasonal COP for Varying Number of Rows at Optimum Sub-Cool for Fixed Condenser Material Cost 122 Figure 8-6: Effect of Number of Rows on the Optimum Air Velocity and Volumetric Flow Rate of Air Over the Condenser at Optimum SubCool for Fixed Condenser Material Cost 123 Figure 8-7: Seasonal COP vs. Varying Condenser Tube Circuiting at Optimum Sub-Cool and Air Velocity for Fixed Condenser Material Cost 126 Figure 8-8: Refrigerant-Side Pressure Drop for Various Circuiting at 82° F Ambient Temperature and at Optimum Sub-Cool and Air Velocity fo Fixed Condenser Material Cost 127 Figure 8-9: Seasonal COP vs. Air Velocity for Varying Fin Pitch at Fixed Condenser Material Cost and Optimum Sub-Cool
130
Figure 8-10: Effect of Fin Pitch on the Seasonal COP at Optimum Sub-Cool and Air Velocity Over the Condenser for Fixed Condenser Material Cost 131 Figure 8-11: Air-side Pressure Drop vs. Fin Pitch for Fixed Condenser Material Cost at Optimum Sub-Cool and Air Velocity at 95° F Ambient Temperature 133
viii
Figure 8-12: Power Requirements vs. Fin Pitch for Fixed Cost at Optimum SubCool and Air Velocity and 95 ° F Ambient Temperature 134 Figure 8-13: Effect of Fin Pitch on Condenser Frontal Area at Optimum Sub-Cool and Air Velocity for Fixed Condenser Material Cost 136 Figure 8-14: Optimum Seasonal COP for Varying Tube Diameter at Optimum SubCool and Air Velocity for Fixed Condenser Material Cost 138 Figure 8-15: Optimum Operating Parameters for Varying Tube Diameters at Fixed Condenser Material Cost 140 Figure 8-16: Condenser Tube Length Allocation for Varying Tube Diameters at Optimum Air Velocity and Sub-Cool and 82 ° F Ambient Temperature for Fixed Condenser Material Cost 141 Figure 8-17: Effect of Tube Diameter on Pressure Drop at Optimum Sub-Cool and Air Velocity at 82 ° F Ambient Temperature for Fixed Condenser Material Cost 143 Figure 8-18: Power Requirements for the Condenser Fan and the Compressor vs. Tube Diameter at Optimum Air Velocity and Sub-Cool for Fixed Condenser Material Cost and 82 ° F Ambient Temperature 144 Figure 8-19: Operating Costs vs. Area Factor For Various Geometric Parameter at Optimum Sub-Cool and Air Velocity with Fixed Condenser Material Cost 146 Figure 8-20: Seasonal COP at Optimum Sub-Cool and Air Velocity for Varying Condenser Tube Circuiting with Fixed Condenser Material Cost and 5/16” Tube Outer Diameter 149 Figure 8-21: Comparison of the Effect of the Number of Tubes per Circuit on Seasonal COP for 5/16” and 3/8” Outer Tube Diameters at Optimum Sub-Cool and Air Velocity with Fixed Condenser Material Cost 150 Figure 9-1: Effect of Air Velocity Over Condenser for Varying Numbers of Rows at Optimum Sub-Cool with Fixed Condenser Frontal Area 154 Figure 9-2: Effect of the Number of Rows of Tubes on the Seasonal COP at Optimum Sub-Cool and Air Velocity for Fixed Condenser Frontal Area 155
ix
Figure 9-3: Refrigerant-Side Pressure Drop vs. Number of Rows with Fixed Condenser Frontal Area for Optimum Sub-Cool and Air Velocity at 82° F Ambient Temperatur 157 Figure 9-4: Compressor and Condenser Fan Power for Varying Number of Rows with Optimum Sub-Cool and Air Velocity at 82° F Ambient Temperature for Fixed Condenser Frontal Area 158 Figure 9-5: Effect of Air Velocity on Seasonal COP for Varying Fin Pitch with Optimum Sub-Cool for Fixed Condenser Frontal Area
160
Figure 9-6: Effect of Fin Pitch on the Seasonal COP at Optimum Sub-Cool and Air Velocity for Fixed Condenser Frontal Area 161 Figure 9-7: Effect of Air Velocity For Varying Tube Diameter at Optimum SubCool for Fixed Condenser Frontal Area 164 Figure 9-8: Effect of Tube Diameter on the Seasonal COP for Fixed Condenser Frontal Area at Optimum Sub-Cool and Air Velocity 165 Figure 9-9: Refrigerant-Side Pressure vs. Tube Diameter for Fixed Frontal Area at 82° F Ambient Temperature, Optimum Sub-Cool and Air Velocity 168 Figure 9-10: Power Requirements for Varying Tube Diameters with Fixed Condenser Frontal Area at 82° F Ambient Temperature, Optimum Sub-Cool and Air Velocity 169 Figure 9-11: Air-Side Pressure Drop vs. Tube Diameter for Fixed Condenser Frontal Area at 82 ° F Ambient Temperature, Optimum Air Velocity and Sub-Cool 171 Figure 9-12: Operating Cost Factor vs. Cost Factor of Condenser Materials for Varying Geometric Parameters with Fixed Condenser Frontal Area and Optimum Air Velocity and Sub-Cool 172 Figure 9-13: Seasonal COP for Varying Condenser Tube Circuiting with Fixed Frontal Area and 5/16” Tube Outer Diameter at Optimum Sub-Cool and Air Velocity 175 Figure 9-14: Comparison of the Effect of the Number of Tubes per Circuit on th Seasonal COP for 5/16” and 3/8” Outer Tube Diameters with Fixed Frontal Area at Optimum Sub-Cool and Air Velocity 178
x
Figure 9-15: Operating Cost Factor vs. Condenser Material Cost Factor for Varying Tube Diameter and Tube circuiting at Optimum Air Velocity and Sub-Cool 180 Figure 9-16: Operating Cost Factor vs. Condenser Material Cost Factor for Varying Geometric Parameters and Various Fixed Frontal Areas at Optimum Air Velocity and Sub-Cool 182
xi
NOMENCLATURE
List of Symbols a
=
Ratio of the transverse tube spacing to the tube diameter
ast
=
Stanton Number coefficient in the Kays and London (1984) Correlation
ax
=
Axial acceleration due to gravity
A
=
Total heat transfer area
Ac
=
Minimum free-flow cross sectional area
Aci
=
Cross sectional area of the refrigerant-side of the tube
Afin
=
Total fin surface area
Afr,con
=
Frontal area of condenser
Amin
=
Minimum free-flow area
Ao
=
Total air-side heat transfer area including the fin and tube areas
AF
=
Area Factor
B
=
Buoyancy Modulus
Bθ
=
Two-phase flow refrigerant side pressure drop Coefficient for a tube bend o θ degrees
bst
=
Stanton Number coefficient in the Kays and London (1984) Correlation
b
=
Ratio of the tube spacing normal to the air flow, to the tube diameter
xii
C
=
Heat capacity
C1
=
Constant of the Hiller-Glicksman refrigerant-side pressure drop Correlation
C2
=
Constant of the Hiller-Glicksman refrigerant-side pressure drop Correlation
C3
=
Constant of the Hiller-Glicksman refrigerant-side pressure drop Correlation
cp
=
Specific heat at constant pressure
cp,eff
=
Effective specific heat at constant pressure
cp,l
=
Specific heat of fluid in the liquid phase
Cmin
=
Minimum heat capacity between that of the air and the refrigeran
Cmax
=
Maximum heat capacity between that of the air and the refrigerant
Cr
=
Ratio of the minimum heat capacity to the maximum heat capacity
Cz
=
Average row correction factor
cz
=
Individual row correction factor
CF
=
Cost factor
COP
=
Coefficient of Performance
COPseas
=
Seasonal Coefficient of Perfor mance
Cost
=
Cost of materials for the heat exchangers
Cost Al
=
Cost per pound of Aluminu
Cost Cu
=
Cost per pound of Copper
D
=
Tube diameter
Ddepc
=
Depth of condenser in the direction of air flow
Dh
=
Hydraulic diameter
xiii
d( )
=
Differential change in ( )
Eu
=
Euler number
Eucor
=
Corrected Euler number
f
=
Friction factor
f GO
=
Friction factor for fluid flowing as vapor onl
f LO
=
Friction factor for fluid flowing as liquid only
f fin
=
Fin friction factor
fri
=
Fraction of temperature bin hours
Fr
=
Froude number
G
=
Mass flux
Gmax
=
Mass flux of air through the minimum flow area
gcs
=
Units conversion constant
h
=
Specific enthalpy
h1
=
Specific enthalpy of refrigerant entering the compressor
h2
=
Actual specific enthalpy of refrigerant exiting the compressor
h2s
=
Ideal specific enthalpy of refrigerant exiting the compressor
h2a
=
Specific enthalpy of refrigerant exiting the superheated portion of the condenser
h2b
=
Specific enthalpy of refrigerant entering the sub-cooled portion of the condenser
h3
=
Specific enthalpy of refrigerant entering the expansion valve
h4
=
Specific enthalpy of refrigerant exiting the expansion valve
ha
=
Air-side heat transfer coefficien
xiv
hevap
=
Two-phase refrigerant-side evaporative heat transfer coefficien
hL
=
Liquid phase refrigerant side heat transfer coefficien
hr
=
Refrigerant-side heat transfer coefficient
hr,SP
=
Single phase refrigerant-side heat transfer coefficient
hTP
=
Two-phase refrigerant-side heat transfer coefficient
i
=
Temperature bin number
j
=
Colburn factor
JP
=
Parameter for the Colburn factor calculation
k
=
Thermal conductivity
k 1
=
Geometry factor for staggered tube array for the air-side pressure drop correlation
k l
=
Liquid phase thermal conductivity
k b,θ
=
Two-phase flow refrigerant side pressure drop Coefficient for a tube bend o θ degrees
L
=
Length
l
=
Integral variable evaporating tube length
Lcon,sa
=
Tube length of the saturated portion of the condenser tubes
Lcon,sc
=
Tube length of the sub-cooled portion of the condenser tubes
Lcon,sh
=
Tube length of the superheated portion of the condenser tubes
Levap,sat
=
Tube length of the saturated portion of the evaporator tubes
Levap,sh
=
Tube length of the superheated portion of the evaporator tubes
Lsat
=
Tube length of the saturated portion of the heat exchanger tubes
Ltot
=
Total tube length of the heat exchanger tubes
xv
m
=
mass
. m
=
mass flow rate
=
mass of flow rate of air f owing over the saturated portion of the condenser
=
total mass flow rate of air flowing over the condenser
mair
=
mass flow rate of air flowing over heat exchanger
mcon,sat
=
mass of refrigerant in the saturated portion of the condenser
mcon,sc
=
mass of refrigerant in the sub-cooled portion of the condenser
mcon,sh
=
mass of refrigerant in the superheated portion of the condenser
mes
=
extended surface geometric parameter
mevap,sat
=
mass of refrigerant in the saturated portion of the evaporator
mevap,sh
=
mass of refrigerant in the superheated portion of the evaporator
n
=
Blausius coefficien
NTU
=
Number of transfer units
NuD
=
Nusselt number based on the tube diameter
P
=
Pressure
pr
=
Reduced pressure
Prat
=
Ratio of the condenser saturation pressure to the evaporator saturation pressure
Pe
=
Perimeter
PD
=
Compressor piston displacemen
Pr . Q
=
Prandtl number
=
Rate of total heat trans erred between the refrigerant and the air
.
ma,sat .
ma,tot .
xvi
q
=
Amount of heat per unit mass transferred between the air and the refrigerant
Qave,seas
=
Average cooling load of the system over all cooling load hours
qcon,sat
=
Amount of heat per unit mass transferred between the air and the refrigerant in the saturated portion of the condenser
qcon,sc
=
Amount of heat per unit mass transferred between the air and the refrigerant in the sub-cooled portion of the condenser
qcon,sh
=
Amount of heat per unit mass transferred between the air and the refrigerant in the superheated portion of the condenser
qcst . Qe
=
Empirical constant for the Euler number correlation
=
Cooling capacity of the syste
. Qmax
=
Maximum possible amount of heat transferred between the refrigerant and the air
r
=
Outer radius of tube
rb
=
Radius of tube bend
Rb
=
Tube bend recovery length
rcst
=
Empirical constant for the Euler number correlati
Rcv,PD
=
Ratio of clearance volume to the piston displacemen
Re
=
Equivalent radius for a hexagonal fin
R” f ,r
=
Refrigerant-side heat exchanger fouling factor
R” f ,a
=
Air-side heat exchanger fouling factor
rr
=
Relative radius of tube bend
Rw
=
Tube wall thermal resistance
Re
=
Reynolds number
ReD
=
Reynolds number based on diameter
xvii
Rel
= Reynolds number based on transverse tube spacing
Rers
= Reynolds number based on row spacing
S
= Entropy
scst
= Empirical constant for the Euler number correlati
St
= Stanton Number
T
= Temperature
Tc,i
= Temperature of cold fluid entering the heat exchanger
tcst
= Empirical constant for the Euler number corr elation
Th,i
= Temperature of hot fluid entering the heat exchanger
Ti
= Representative bin temperature
Trat
= Ratio of the condenser saturation temperature to the evaporator saturation Temperature
U
= Overall heat transfer coefficient per unit area
u
= Empirical constant for the Euler number correlati
UA
= Overall heat transfer coefficien
UAhouse
= Overall “house” heat transfer coefficien
v
= Specific volume
v1
= Specific volume of refrigerant entering the compressor
v2
= Specific volume of refrigerant exiting the compressor
Va,con
= Velocity of the air flowing over the condenser
vl
= Specific volume of the fluid in the liquid phase
vm
= Mean specific volume of air flowing over the heat exchanger
vv
= Specific volume of the fluid in the vapor phase
xviii
Vol, Al,cond = Volume of the aluminum components of the condenser (fins) Vol, Al,eva = Volume of the aluminum components of the evaporator (fins) Vol, Cu,cond = Volume of the copper co ponents of the condenser (tubes) Vol, Cu,evap = Volume of the copper components of the evaporator (tubes) wa,com
= Actual compressor work per unit mass of refrigeran
Wave,seas
= Average electricity required by the system over all cooling load hours
. Wcom . Wf,con . Wf,evap
= Compressor power = Condenser fan power = Evaporator fan power
ws,com
= Isentropic compressor work per unit mass of refrigeran
x
= Vapor quality
xe
= Vapor quality at the exit of the heat exchanger
xi
= Vapor quality at the inlet of the heat exchanger
Xl
= Transverse tube spacing
Xt
= Tube spacing normal to air flow
X tt
= Lockhart-Martinelli Parameter
y
= Equivalent length of tube bend
z
= Number of rows of tubes
xix
List of Symbols with Greek Letters
α
= Local void fraction.
β
= Coefficient of the empirical relation for determining the equivalen circular radius for hexagonal fins
∆hlat
= Change in the latent enthalpy
∆hsens
= Change in the sensible enthalpy
∆htot
= Change in the total enthalpy
∆p
= Pressure drop
∆p a,con
= Pressure drop on the air-side of the condenser
∆p b
= Refrigerant-side pressure drop inside a tube bend
∆p b,LO
= Refrigerant-side pressure drop inside a tube bend with all fluid flowing as a liquid
∆p b,SP
= Single phase refrigerant-side pressure drop inside a tube bend
∆p b,TP
= Two-phase refrigerant-side pressure drop inside a tube bend
∆pf
= Friction component of the two-phase refrigerant-side pressure drop inside a straight tube
∆pfins
= Air-side pressure drop due to fins
∆pm
= Momentum component of the two-phase refrigerant-side pressure drop inside a straight tube
∆p S,SP
= Single phase refrigerant-side pressure drop inside a straight tube
∆p S,TP
= Two-phase refrigerant-side pressure drop inside a straight tube
∆p tot,ac
= Total air-side pressure drop
∆p tubes
= Air-side pressure drop due to tubes
xx
∆x
= Change in quality
ε
= Fin effectiveness
εpr
= Pipe roughness
φ
= Fin parameter that is a function of the equivalent circular radius of a hexagonal fin
Γ b2
= Physical property coefficient for the refrigerant-side pressure drop determination inside a tube bend
ηc
= Compressor thermal efficienc
ηf
= Fin efficienc
ηfan,con
= Condenser fan efficiency
ηs
= Surface efficiency
ηs,a
= Air-side surface efficienc
ηs,r
= Refrigerant-side surface efficienc
ηv
= Compressor volumetric efficiency
ϕ2b,LO
= Two-phase multiplier for the refrigerant side pressure drop inside tube bends
µ
= Viscosity
µl
= Viscosity of the fluid in the liquid phase
µm
= Viscosity of the fluid evaluated at the mean fluid temperature
µs
= Viscosity of the fluid evaluated at the temperature of the inner tube wall surface
µTP
= Two-phase fluid viscosity
µv
= Viscosity of the fluid in the vapor phase
xxi
π
= 3.14159…..
θ
= Angle of tube bend
ρ
= Density
ρl
= Density of the fluid in the liquid phase
ρv
= Density of the fluid in the vapor phase
σ
= Ratio of the minimum free-flow area to the frontal area of the hea exchanger
ψ
= Coefficient of the empirical relation for determining the equivalen circular radius for hexagonal fins
xxii
SUMMARY
Current residential air-conditioners and heat pumps use the hydrochlorofluorocarbon refrigerant, R-22, as the working fluid. In accordance with the Montreal Protocol, a production ban of all equipment utilizing R-22 will begin in 2005, and a total ban on the production of R-22 is also impending. A binary zeotropic mixture, R-410a, is a strong candidate for R-22 replacement due to its many favorable performance characteristics; e.g., non-flammability, high working pressures, and good cycle efficiency. Since R-410a has significantly higher working pressure and vapor densities than R22, current air cooled finned tube condenser designs are not appropriate. The optimum condenser and other high-pressure-side components are expected to employ smaller diameter tubes, which will affect other design parameters. At this time, there is limited information about condenser coil design and optimization using R-410a as the working fluid. Furthermore, the heat transfer and friction data are also limited. This work includes an examination of the available refrigerant-side two-phase flow heat transfer and pressure drop models for refrigerants. A model based on first principles is used to predict the performance of a unitary air-conditioning system with refrigerant R410a as the working fluid. The seasonal coefficient of performance of the airconditioning system is used as the figure of merit. The primary objective of this research was to provide guidelines for the design and optimization of the condenser coil for tw
xxiii
distinct criteria: (1) fixed condenser frontal area (size constraint), and (2) fixed condenser material cost (capital cost constraint). This study concludes that for both design criteria, the velocity of air flow over the condenser ranges between 7.5 ft/s and 8.5 ft/s while the optimum sub-cooling of the refrigerant exiting the condenser is approximately 15° F. It is also concluded that condensers employing tubes of smaller diameters yiel d the best system performance. Recommendations for further research into the modeling of the in-tube condensation o refrigerant R-410a are outlined. An exhaustive search optimization study could not be performed due to computational speed limitations, therefore more advanced optimization search techniques are also recommended for further study.
xxiv
CHAPTER I
INTRODUCTION
The decade of the 1990’s has been a challenging time for the Heating Ventilation Air Conditioning and Refrigeration (HVAC&R) industry worldwide. Due to their role in the destruction of the stratospheric ozone layer, provisions of the Montreal Protocol and its various amendments required the complete phase-out of chlorine-containing refrigerant such as chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs). These compounds have been used extensively as refrigerants in heat pumps, air conditioners and refrigeration systems (Ebisu and Torikoshi, 1998). CFCs, which are characterized by a high ozone-depletion potential (ODP), underwent a complete production phase-out in the United States in 1995. Because HCFC-22 (chlorodifluoromethane) has been readily available, inexpensive, and less harmful to the environment than CFCs, HCFC-22 has been widely used in the air-conditioning and heat pump industry, especially in residential unitary and central air-conditioning systems, for many years (Bivens et al., 1995). However, the 1992 revision of the Montreal Protocol stipulated the first producti ceiling for HCFCs starting in 1996 (Domanski and Didion, 1993). In the United States, regulations published by the Environmental Protection Agency (EPA) prohibit the
1
production of HCFC-22 after 2010 except for servicing equipment produced prior to 2010. The deadline is much earlier in some European countries (Gopalnarayanan and Rolotti, 1999). In addition, another international agreement, the Kyoto Protocol, has been initiated to reduce the emission of greenhouse gases (GHGs) in order to lower the potential risk o increased global warming. Representatives of more than 150 countries met in Kyoto, Japan in December of 1997. As a result of this agreement, the nations agreed to roll back emissions of carbon dioxide (CO2) and five other GHGs, including HFCs, to about 5.2% below 1990 levels by 2010. Individual emissions targets were adopted for most developed countries (Baxter et al., 1998). With CO2 emissions tied directly to energ use, the pressures for further HVAC&R equipment efficiency i mprovements will increase in the early decades of the next century. At the same time, pressures from internationa competition have continued unabated. The choices for short-term and long-term replacements for R-22 are being driven by environmental regulations, energy standard requirements, and the cost of implementation. The differences in R-22 phase-out dates for the different countries seem to significantl influence the choice of replacement refrigerants (Gopalnarayanan and Rolotti, 1999). However, several programs are underway for evaluating R-22 alternatives. One such industry program is the Alternative Refrigerants Program (AREP) initiated by the Air Conditioning & Refrigeration Institute (ARI). The objective of this program is to provide performance data on replacement refrigerants in compressors, air-conditioning syste components and/or systems by conducting tests with participating member companies.
2
Throughout the evaluation process, equipment manufacturers have made requests that the alternatives meet several requirements. In order to meet these customer needs, a family of alternatives has been developed for replacing R-22 (Bivens et al., 1995). Unfortunately, no single-component HFCs have been discovered that have thermodynamic properties close to that of R-22. Consequently, this has led to the introduction of binary or ternary refrigerant mixtures. Several alternatives, including binary and ternary blends o f HFCs, as well as propane, are being considered as potential R-22 replacement fluids (Gopalnarayanan and Rolotti, 1999). One very promising replacement, from the viewpoint of zero ODP and non-flammability, is the binary mixture, R-410a (Ebisu and Torikoshi, 1998). Note that R-410a is a near azeotropi mixture consisting of 50% (wt%) R-32 and 50% R-125. Besides the basic characteristics such as thermal properties and flammability, very little heat transfer and pressure drop data for R-410a is available; although Wijaya and Spatz (1995) have shown limited experimental data for heat transfer coefficients and pressure drops for R-410a inside a horizontal smooth tube. Yet, knowledge of the performance characteristics of air-cooled refrigerant heat exchangers with alternative refrigerants is of practical importance in designing air-cooled heat exchangers required in air-conditioning equipment. Therefore, more knowledge of the two-phase flow heat transfer and pressure drops that occur in refrigerant R-410a hea t exchangers is needed.
3
Research Objectives
The primary objective of this current work is to study the design and optimization o the operating conditions and the geometric design parameters for the air-cooled condenser coil of a vapor compression residential air-conditioning system wit refrigerant R-410a as the working fluid. The condenser and total system operating conditions are varied so that the system ’s coefficient of performance can be evaluated as a function of the heat exchanger design. Subsequently, it is also the intent of this stud that the optimization methodology detailed in this work provide guidelines to the coil designer for future design optimizations of this type. A secondary objective of this study is to investigate various two-phase flow heat transfer and pressure drop evaluation methods for refrigerant R-410a.
4
CHAPTER II
LITERATURE SURVEY
Previous Studies on Variations of Heat Exchanger Geometric Parameter
The heat exchanger of interest for this present study is of the plate-fin-and-tube configuration. A schematic of a typical plate-fin-and-tube heat exchanger is shown in Figure 2-1.
Air Cross Flow
Air Cross Flow
T= f(x,y)
Refrigerant Flow
Refrigerant Flow
Figure 2-1: Typical Plate Fin-and-Tube Cross Flow Heat Exchange
5
There have been several studies on heat exchangers of this type. Wang et al. (1999) conducted an experimental study on the air-side performance for two specific louver fin patterns and their plain plate fin counterparts. This study investigated the effects of fin pitch, longitudinal tube spacing and tube diameter on the air-side heat transfer performance and friction characteristics. This study found that for plain plate fin configurations ranging from 8 to 14 fins per inch, the effect of longitudinal tube pitch on the air-side was negligible for both the air-side heat transfer and pressure drop. However, the heat transfer performance increased with reduced fin pitch. Chi et al. (1998) conducted an experimental investigation of the heat transfer and friction characteristics of plate fin-and tube heat exchangers having 7 mm diameter tubes. In this study, 8 samples of commercially available plate-fin-and-tube heat exchangers were tested. It was found that the effect of varying fin pitch on the air-side heat transfer performance and friction characteristics was negligible for 4-row coils. However for 2row coils, the heat transfer performance increased with a decrease in fin pitch. This stud used a plate-fin-and tube heat exchanger configuration with louver fin surfaces, which are widely used in both automotive and residential air-conditioning systems. The transverse fin spacing ranged from 21 mm to 25.4 mm and longitudinal fin spacing ranged from 12.7 mm to 19.05 mm Wang et al. (1998) also collected experimental data on a plate-fin-and tube hea exchanger configuration. They examined the effect of the number of tube rows, fin pitch, tube spacing, and tube diameter on heat transfer and friction characteristics. This stud found that the effect of fin pitch on the air-side friction pressure drop was negligibly
6
small for air-side Reynolds numbers greater than 1000. It was also found that the hea transfer performance was independent of fin pitch for 4-row configurations. Furthermore, the results indicated that reducing the tube spacing and the tube diameter produced an increase in the air-side heat transfer coefficient. The fin surfaces utilized in this study were of the louver type, with transverse fin spacing ranging from 21 mm to 25.4 mm, and longitudinal fin spacing ranging from 12.7 mm to 19.05 mm. The longitudinal tube spacing investigated for this studied ranged from 15 mm to 19 mm and the tube diameters ranged from 7.94 mm to 9.52 mm. One of the earliest and most complete inv estigations of heat exchanger heat transfer and pressure drop characteristics was performed by Kays and London (1984). An extensive amount of experimental heat transfer and friction pressure d rop data were complied for several different plate-fin-and-tube heat exchanger configurations as part of this study. However, no optimization of the heat transfer surfaces and geometry was performed. Shepherd (1956) experimentally tested the effect of various geometric variations on 1-row plate fin-and-tube coils. He investigated the effects of varying the fin spacing, fin depth, tube spacing, and tube location on the heat transfer performance of the coil. The results of Shepherd’s study showed that as the fin pitch increased, the air-side hea transfer coefficient, for a given face velocity, increased only slightly. He also found tha as the fin depth and tube spacing increased, with all other variables constant, the air-side heat transfer coefficient decreased. Rich (1973) studied the effect of varying the fin spacing on the heat transfer and friction performance of multi-row heat exchanger coils.
7
Rich found that over the range from 3 to 14 fins per inch, the air-side heat transfer coefficient was independent of fin pitch. Neither Rich’s nor Shepherd’s investigations involved the optimization of the heat exchanger operating conditions and geometric parameters. All of the above studies provide valuable insight into the effects of varying different geometric parameters on the heat transfer and friction performance of plate-fin-tube heat exchangers. However none of the above works investigated the effects that varying these geometric parameters has on the optimization of a complete air-conditioning system
Previous Work in R-22 Replacement Refrigerants
Again, a major focus of this work is the study of the effect of the condenser plate-finand-tube heat exchanger design parameters on the performance of a refrigerant R-410a unitary air-conditioning system. However, as discussed in Chapter I, due to the impending ban of refrigerant R-22 production, there is a pressing need for studies on the performance characteristics of alternative refrigerants in air-conditioning and heat pump systems. Therefore a survey of the previous investigations on R-22 replacemen refrigerants in these systems is a very important part of this present study. There has been a substantial amount of work done in the area of air-conditioning and heat pump R-22 replacement refrigerants. Only some of the relevant studies are mentioned here. Radermacher and Jung (1991) conducted a simulation study of potential R-22 replacements in residential equipment. The coefficient of performance (COP) and
8
the seasonal performance factor (SPF) were calculated for binary and ternary sub stitutes for R-22. They found that for a ternary mixture of R-32/R-152a/R-124 with a weight concentration of 20 wt%/20 wt%/60 wt%, the COP was 13.7% larger and the compressor volumetric capacity was 23% smaller than the respective values for R-22. This stud found that in general, based on thermodynamic properties only, refrigerant mixtures have the potential to replace R-22 without a loss in efficiency. Efficiency gains are possible when counterflow heat exchangers are used and additional efficiency gains are possible when capacity modification is employed. Kondepudi (1993) performed experimental “drop-in” (unchanged system, same heat exchangers) testing of R-32/R-134a and R-32/R-152a blends in a two-ton split-system air conditioner. Five different refrigerant blends of R-32 with R-134a and R-152a were tested as “drop-in” refrigerants against a set of R-22 baseline tests for comparison. No hardware changes were made except for the use of a hand-operated expansion device, which allowed for a “drop-in” comparison of the refrigerant blends. Hence, other than the use of a different lubricant and a hand-operated expansion valve, no form of optimization was performed for the refrigerant blends. Parameters measured included capacity, efficiency, and seasonal efficiency. The steady state energy efficiency ratio (EER) and seasonal efficiency energy efficiency ratio (SEER) of all the R-32/R-134a and R-32/R-152a blends tested were within 2% of those for a system using R-22. The 40 wt%/60 wt% blend of R-32/R-134a performed the best in a non-optimized system. Fang and Nutter (1999) evaluated the effects of reversing valves on heat pump system performance with R-410a as the working fluid. A traditional reversing valve enables a
9
heat pump to operate in either the heating mode or cooling mode. It performs this function by switching the refrigerant flow path through the indoor and out door coils, thus changing the functions of the two heat exchangers. However, use of reversing valves causes increased pressure drops, refrigerant leakage from the high pressure side to the low pressure side, and undesired heat exchange. This study measured the overal effects of a reversing valve on a 3-ton heat pump system using R-410a and made comparisons to the same valve’s performance with R-22 as the working fluid. It was found that changing from r efrigerant R-22 to R-410a resulted in an increase in mass leakage, but did not significantly change the effect that the reversing valve had on the system COP. Domanski and Didion (1993) evaluated the performance of nine R-22 alternatives. The study was conducted using a semi-theoretical model of a residential heat pump with a pure cross-flow representation of heat transfer in the evaporator and condenser (Domanski and Mclinden, 1992). The models did not include transport properties since they carried the implicit assumption that transport properties (and the overall heat transfer coefficients) are the same for the fluids studied. Simulations were conducted for “dropin” performance, for performance in a modified system to assess the fluids’ potentials, and for performance in a modified system equipped with a liquid line/suction-line hea exchanger. The simulation results obtained from the “drop-in” evaluation predicted the performance of candidate replacement refrigerants tested in a system designed for the original refrigerant, with a possible modification of the expansion device. The “drop-in” model evaluations revealed significant differences in p erformance for high-pressure
10
fluids with respect to R-22 and indicated possible safety problems if those fluids were used in unmodified R-22 equipment. The simulation results obtained from the constantheat-exchanger-loading evaluation corresponded to a test in a system modified specifically for each refrigerant to obtain the same heat flux through the evaporator and condenser at the design rating point. This simulation constraint ensures that the evaporator pressures are not affected by the different volumetric capacities of the refrigerants studied. The results for the modified system performance showed tha capacity differences were larger for modified systems than for the “drop-in” evaluation. However, none of the candidate replacement refrigerants exceeded the COP of R-22 at any of the test conditions. Bivens et al. (1995) compared experimental performance tests with ternary and binary mixtures in a split system residential heat pump as well as a window air-conditioner. This study investigated refrigerants R-407c, a t ernary zeotropic mixture of 23 wt% R-32, 25 wt% R-125 and 52 wt% R-134a, and R-410b, a near azeotropic binary mixture composed of 45 wt% R-32 and 55 wt% R-125 as working fluids. The heat pump used for the evaluations was designed to operate with R-22 and was equipped with a fin-and-tube evaporator with 4 refrigerant flow parallel circuits, and a spined fin condenser with 5 circuits and 1 sub-cooling circuit. It was found that R-407c provided essentially the same cooling capacity as compared with R-22 with no equipment modification. R-410b provided a close match in cooling capacity using modified compressor and expansion devices. The energy efficiency ratio for R-407c versus R-22 during cooling ranged from 0.95 to 0.97. The energy efficiency ratio for R-410b versus R-22 during cooling ranged
11
from 1.01 to 1.04. Window air-conditioner tests were conducted with R-407c in three window air-conditioners ranging in size from 12 ,000 to 18,000 Btu/hr. The result demonstrated equivalent capacity and energy efficiency ranging from 0.96 to 0.98 compared with R-22. In summation, in the search for a replacement for refrigerant R-22 many refrigerants have been studied. As discussed throughout this work, many of those studied are refrigerant mixtures. A list of many of the refrigerant mixtures studied by the sources sited in this literature survey is shown in Table 2-1.
Table 2-1: List of Refrigerant R-22 Alternative Refrigerant Mixtures
Refrigerant
Weight Percent
R-410a
R-32/50%, R-125/50%
R-407b
R-32/45%, R-125/55%
R-407c
R-32/23%, R-125/25%, R-134a/52%
Radermacher and Jung (1991) Kondepudi (1993)
R-132/20%, R-R-152a/20%, R-124/60% R-32/40%, R-134a/60%
12
As a result of many of the studies discussed in this literature survey, refrigerant R-410a has emerged as the primary candidate to replace R-22 in many industrial and residential applications. There is at least one commercially available air-conditioning system using R-410a as the working fluid, which is made by Carrier. Therefore, as discussed in Chapter I, R-410a is the refrigerant of interest for this current study.
Two-Phase Flow Regime Considerations in Condenser and Evaporator Design
The prediction of flow patterns is a central issue in two-phase gas-liquid flow in hea exchangers. Design parameters such as pressure drop and heat and mass transfer are strongly dependent on the flow pattern. Hence, in order to accomplish a reliable design of gas-liquid systems such as pipelines, boilers and condensers, an a priori knowledge of the flow pattern is needed (Dvora et al., 1980). Figure 2-2 shows one version of the commonly recognized flow patterns for twophase flow inside horizontal tubes. Description of these patterns is highly subjective, of course, and there is some variation among researchers in the field concerning the characterization of the various patterns. However, the essential situation is this: For ordinary fluids under ordinary process con ditions, two forces control the behavior and distribution of the phases. These forces are gravity, always acting towards the center o the earth, and vapor shear forces, acting on the vapor-liquid interface in the direction o motion of the vapor. When gravity forces dominate (usually under condition s of low vapor and liquid flow rates), one obtains the stratified and wavy flow patterns shown
13
Figure 2-2: Horizontal Two-Phase Flow Regime Patterns
14
in Figure 2-2. When vapor shear forces dominate (usually at high vapor flow rates), one obtains the annular flow pattern (with or without entrained liquid in the core) shown on the diagram. When the flow rates are very high and the liquid mass fraction dominates, the dispersed bubble flow pattern is obtained, which is a shear-controlled flow of som importance in boiler design but of very limited interest in condensers. Intermediate flow rates correspond to patterns in which both gravitational and vapor shear forces are important (Bell, 1988). Although extensive r esearch on flow patterns has been conducted, most of this research has been concentrated on either horizontal or vertical flow. For horizontal flow the earliest and perhaps the most durable, and best known of pattern maps for two-phase gas-liquid flow was proposed by Baker (1954). Taitel and Dukler (1976) proposed a physical model capable of predicting flow regime transition in horizontal and near horizontal two-phase flow. There are several points that need to be emphasized concerning the use of any flow pattern map (Bell, 1988): 1. The definition of any two-phase flow pattern is highly subjective and differen observers may disagree upon exactly what they are looking at. Adding to this ambiguity are the various means of measuring two-phase flows and the resulting different criteria that are used to characterize two-phase flows.
2. The boundaries drawn on a map as lines should be viewed as very broad transition regions from one well defined flow pattern to another.
15
3. Few flow pattern maps are represented in non-dimensional form.
4. Most flow pattern maps are based on air-water flows. Hence it is assumed tha the ratio of the vapor to liquid mass flow does not change from one part of the conduit to another. Yet condensing and vaporizing flows are in a state of perpetual change form one quality to another. Even considering all of the above warnings, it is still better to use whatever limited information one can find and to use it with full recognition of its limitations than to totally ignore these considerations in the design of equipment (Bell, 1988).
Two-Phase Flow Heat Transfer Correlations
A very large number of techniques for predicting the heat-transfer coefficients during condensation and evaporation inside pipes have been proposed over the last 50 years or so. These range from very arbitrary correlations to highly sophisticated treatments of the mechanics of flow. While many of these have been valuable as practical design tools and have added to our understanding of the phenomena involved, there does not appear to be any general predictive technique which has been verified over a wide ran ge of parameters (Shah, 1979). Nusselt (1916) extended his ve rtical plate analysis to laminar film condensation inside a vertical tube with forced vapor flow. He assumed a constant condensate fil thickness, and that the condensing process in no way affected the vapor flow. He further
16
assumed that the shear at the edge of the condensate film is directly proportional to the pressure drop. This shear was expressed in terms of a constant friction factor and the vapor velocity. Consequently, Nusselt succeeded in obtaining a correlation for the hea transfer coefficient, which applies if the condensate is in laminar flow. However there are significant discrepancies between Nusselt’s theory and the experimental data when the condensate flow becomes turbulent or when the vapor velocity is very high (Soliman et al., 1968). Soliman et al. (1968) develop a model for two-phase flow heat transfer that includes the contribution of the gravity, momentum and frictional terms to the wall shear stress. In this work, a general correlation for the condensation heat transfer coefficient in the annular flow regime was developed. The major assumption used in the development o this correlation was that the major thermal resistance is in the laminar sublayer of the turbulent condensate film. Experimental data for several fluids (including steam, refrigerant R-22, and ethanol) was used to determine empirical coefficients and exponents. This correlation predicts the experimental data within ±25%. Yet another semi-empirical condensation heat transfer correlation for annular flow was developed by Akers et al. (1959). Correlations for both the local and average values of the condensation heat transfer coefficient were developed in the Akers study. The Akers correlation predicts the experimental heat transfer coefficients generated b Soliman et al. (1968), within ±35%. Traviss et al. (1973) applied the momentum and heat transfer analogy to an annular flow model using the von Karman universal velocity distribution to describe the liquid
17
film. Since the vapor core is very turbulent in this flow regime, radial temperature gradients were neglected, and the temperatures in the vapor core and at the liquid-vapor interface were assumed to be equal to the saturation temperature. Axial heat conduction and sub-cooling of the liquid film were also neglected. An order of magnitude analysis and non-dimensionalization of the heat transfer equa tions resulted in a simple formulation for the local heat transfer coefficient. The analysis was compared to experimental data for refrigerants R-12 and R-22 in a conden ser tube, and the results were used to substantiate a general equation for forced convection condensation. Since the heat transfer analysis assumed the existence of annular flow, the sensitivity of this analysis to deviations from the annular flow regime is important. When the mass flux of the refrigerant vapor exceeded 500,000 lbm/hr-ft2, there is appreciable entrainment of liquid in the upstream portion of the condenser tube. Since the analysis assumed tha annular film condensation exists and that all of the liquid is on the tube wall, analytical predictions are below the experimental data in the dispersed or misty flow regime. However, the entrainment of liquid is not very large because the main resistance to hea transfer occurs in the laminar sublayer, and liquid removed from the turbulent zone di not increase the heat transfer coefficient in direct relation to the amount of liquid removed. Yet, according to the experimental data collected and analyzed by Singh et al. (1996), the mean deviation for the Traviss correlation deviates by -%40 from the data. The above correlations were developed for o ne specific flow regime (annular flow). However, in many instances a correlation that is applicable to more than one flow regime is needed. Shah (1979) developed a very simple dimensionless correlation, which he
18
then verified by comparison with a wide variety of experimental data. Data analyzed included refrigerants, water, ethanol, and benzene, condensing in horizontal, vertical and inclined pipes and included diameters ranging from about 7 to 40mm. Very wide ranges of heat flux, mass flux, vapor velocities and pressures were covered. The 473 data points from 21 independent experimental studies were correlated with a mean deviation of abou 15%. From this study, Shah asserts that this semi-empirical correlation is recommended for use in all flow patterns and flow orientations. However, according to the experimental data collected and analyzed by Singh et al. (1996) the values for condensation heat transfer coefficients computed using the Shah correlation deviate by mean of –30% from the data. Again, a substantial amount of research has been performed in the development o two-phase flow heat transfer models. The models most applicable to this current stud are from the works of Akers et al. (1959), Traviss et al. (1976) and Shah (1979). A more detailed evalua tion of these models and their relevance to this current study is contained in Chapter IV.
Two-Phase-Flow Pressure Drop Correlations
Despite the importance of pressure drop in two-phase flow processes, and the consequent extensive research on the topic, there is still no satisfactory method for calculating two-phase pressure drop. The best current methods are cumbersome in structure, heavily dependent on empirically determined coefficients, and have
19
considerable uncertainty. Simpler forms or firmer theoretical bases for predictive methods can only be achieved with a narrowing of the ranges of applicability (Beattie and Whalley, 1982). Early two-phase flow studies emphasized the development of overall pressure drop correlations encompassing all types of flow regimes. Furthermore, most of the experimental data were obtained from relatively small and short pipes (Chen and Spedding, 1981). Hence, no satisfactory general correlation exists. For several years, experimental pressure drop data have been collected for horizontal gas-liquid systems, and many attempts have been made to develop, from the data, general procedures for predicting these quantities. Errors of about 20% to 40% can be expected in pressure-drop prediction, and even this range is optimistic if one attempts to use the various predictive schemes without applying a generous measure of experience and judgment. A major difficulty in developing a general correlation based on statistical evaluation of data is deciding on a method of properly weighing the fit in each flow regime. It is difficult to decide, for instance, whether a correlation giving a good fit with annular flow and a poor fit with stratified flow is a better correlation than one giving a fair fit for both kinds o flow (Russell et al., 1974). Lockhart and Martinelli (1949) developed one of the first general correlations. Although various other general correlations have since been proposed the original Lockhart-Martinelli approach is still in many respects the best. As discussed by Chen and Spedding (1981), this method continues to be one of the simplest procedures for calculating two-phase flow pressure drop. One of the biggest advantages of thi
20
procedure is that it can be used for all flow regimes. For this flexibility, however, relatively low accuracy must be accepted. Detailed checks with extensive data have shown that the correlation overpredicts the pressure drop for the stratified flow regime (Baker, 1954); it is quite reasonable for slug and plug flow (Dukler et al., 1964); and for annular flow, it underpredicts for small diameter pipes (Perry, 1963), but overpredicts for larger pipes (Baker, 1954). Souza et al. (1993) developed a correlation for two-phase frictional pressure drop inside smooth tubes for pure refrigerants using the Lockhart-Martinelli parameter, X tt (the square root of the ratio between the liquid only pressure drop and the vapor only pressure drop), the Froude number, Fr, and experimental data. The pressure drop due to acceleration was calculated using the Zivi (1964) equation for void fraction. A single tube evaporator test facility capable of measuring pressure drop and heat transfer coefficients inside horizontal tubes was utilized, and pressure drop data were collected. During the tests, the predominant flow pattern observed was annular flow. For lower mass fluxes and qualities, stratified-wavy, and semi-annular flow patterns were also observed. The resulting correlation of experimental data for refrigerants R-134a and R12 for turbulent two-phase flow predicted the pressure drop within
±10%.
Chisolm (1973,1983) has published important results on pressure drop and has improved several correlations that predicted the frictional pressure drop during two-phase flow for many different fluids. According to the data collected by Souza et al. (1993), Chisolm’s two-phase flow multipliers overpredicted the experimental data for low qualities and slightly underpredited those for high qualities. Overall, Chisolm’ s
21
correlation for friction pressure drop predicts the experimental values within
±30% with a
mean deviation of 14.7%. Jung and Radermacher (1989) developed a correlation for pressure drop during horizontal annular flow boiling of pure and mixed refrigerants. For this correlation, a two-phase multiplier based on total liquid flow was introduced for the total pressure drop (frictional and acceleration pressure drop) and was correlated as a function of the Lockhart and Martinelli parameter, Xtt. However, Jung and Radermacher’s correlation overpredicts the experimental data by an average of 29%. In summary, the general correlation procedures yield fair predictions of pressure drop for all flow regimes because they are based on a large amount of correlatable data. However, when these correlations are applied to systems other than those used in their development, or to flow over extended distances (fully established flow), predicted pressure drops can be in error by as much as a factor of 2. For more reliable predictions of pressure drop, correlations based on specific models for individual flow regimes are preferable, yet difficult to model analytically without concrete knowledge of the quality distribution throughout the tubes (Greslpvoch & Shrier, 1971).
22
CHAPTER III
AIR-CONDITIONING SYSTEM AND COMPONENT MODELING
Refrigeration Cycle
Heating, Ventilating, and Air-Conditioning (HVAC) systems that provide a cooling effect depend on a refrigeration cycle. Both the control and performance of HVAC systems are significantly affected by the performance of the refrigeration cycle. Therefore a basic understanding of the refrigeration cycle is needed in the design and optimization of HVAC systems. Of the three basic refrigeration cycles (vapor compression, absorption, and thermo-electric), the cycle typically used in the HVAC industry is the vapor compression cycle. Vapor compression refrigeration has many complex variations, but only the basic compression cycle will be discussed here. The working fluid for the system in this study is refrigerant R-410a. The vapor compression refrigeration cycle modeled for this study is shown in Figure 3-1. As the figure shows, low pressure, superheated refrigerant vapor from the evaporator enters the compressor (State 1) and leaves as high pressure, superheated vapor (State 2). This vapor enters the condenser where heat is rejected to outdoor air that is forced over the condenser coils. Next the refrigerant vapor is cooled to the saturation
23
T
S Condenser Sub-cooled
3
Saturated 2b
Superheated 2a
Expansion Valve
Compressor
Saturated 4
2
Superheated 4a
Evaporator
Figure 3-1: The Actual Vapor-Compression Refrigeration Cycle
24
1
temperature (State 2b), and then cooled to below the saturation point until only subcooled liquid is present (State 3). The high pressure liquid is then forced through the expansion valve into the evaporator (State 4). The refrigerant then absorbs heat from warm indoor air that is blown over the evaporator coils. The refrigerant is completel evaporated (State 4a) and heated above the saturation temperature before entering the compressor (State 1). The indoor air is cooled and dehumidified as it flows over the evaporator and returned to the living space.
System Component Models
Compressor
The purpose of the compressor is to increase the working pressure of the refrigerant. The compressor is the major energy-consuming component of the refrigeration system, and its performance and reliability are significant to the overall performance of the HVAC system. In general there are two categories of compressors: dynamic compressors and displacement compressors. Dynamic compressors convert angular momentum into pressure rise and transfer this pressure rise to the vapor (McQuiston and Parker, 1994). Positive displacement compressors increase the pressure of the vapor by reducing the volume. For this study scroll type positive displacement compressors, which dominate the residential air-conditioning industry, are utilized. The amount of specific work (work per unit mass of refrigerant) done by an ideal compressor can be expressed with the following:
25
ws ,com
= (h2 s − h1 )
(3-1)
where h is the refrigerant enthalpy. For a non-ideal compressor, the actual amount o work done depends on the efficiency,
wa, com
=
ws ,com
ηc
= (h2 − h1 )
(3-2)
where ηc is the compressor thermal efficiency. For a scroll type compressor, Klein and Reindl (1997) have determined that the thermal efficiency is related to a “pressure ratio” and a “temperature ratio” by the following relationship,
ηc
2 2 = −60.25 − 3.814 Prat − 0.281Prat + 111.3T rat − 50.31T rat + 3.061Prat T rat
26
(3-3)
where Prat is the “pressure ratio” and Trat is the “temperature ratio”, which are defined by the following relationships,
Prat
=
T rat =
Psat ,cond Psat ,evap
T sat ,cond T sat ,evap
(3-4)
(3-5)
The coefficients in this correlation are based on saturated temperatures and not on the actual temperatures at the inlet and outlet of the compressor. The volumetric efficiency is another important consideration in selecting and modeling compressors. The volumetric efficiency is the ratio of the mass of vapor that is compressed to the mass of vapor that could be compressed if the intake volume were equal to the compressor piston displacement. The volumetric efficiency is expressed as:
27
ηv
v = 1 − Rcv, pd 1 − 1 v 2
(3-6)
where ηv is the compressor volumetric efficiency, Rcv,pd is the ratio of clearance volume to the piston displacement, and v is the specific volume. The volumetric efficiency is .
also used to determine the mass flow rate of the refrigerant though the compressor, m, for a given compressor size by the following expression,
m=
η v PD v2
(3-7)
where PD is the Piston Displacement (Threlkeld, 1970).
Condenser
The condenser is a heat exchanger that rejects heat from the refrigerant to the outside air. Although there are many configurations of heat exchangers, finned-tube hea
28
exchangers are the type most commonly used for residential air conditioning applications. Refrigerant flows through the tubes, and a fan forces air between the fins and over the tubes. The heat exchangers used in this study are of the cross-flow, plate-fin-and-tube type. A schematic of this heat exchanger is shown in Figure 3-2. The plate fins are omitted from the schematic for simplicity. When the refrigerant exits the compressor, it enters the condenser as a superheated vapor and exits as a sub-cooled liquid. The condenser can be separated into three sections: superheated, saturated, and sub-cooled. The amount of heat per unit mass o refrigerant rejected from each section can be expressed as the difference between the refrigerant enthalpy at the inlet and at the outlet of each section:
qcon , sh
= h2 − h2 a ,
qcon , sat = h2 a
− h2b ,
(3-8)
(3-9)
and q con , sc
= h2b − h3 .
29
(3-10)
Horizontal Tube Spacing
Air Cross Flow Vertical Tube Spacing Height 1 Refrigerant Flow Parallel Circuit
Width
3 Tubes per Circuit row 1
row 2
row 3
Depth
Figure 3-2: Typical Cross Flow Heat Exchanger (fins not displayed)
30
The total heat rejected from the hot fluid, which in this case is the refrigerant, to the cold fluid, which is the air, is dependent on the heat exchanger effectiveness and the hea capacity of each fluid:
Q = ε C min (T h, i
− T c, i )
(3-11)
where ε is the heat exchanger effectiveness; Cmin is the smaller of the heat capacities o the hot and cold fluids, Ch and Cc respectively; Th,i is the inlet temperature of the hot fluid; and Tc,i is the inlet temperature of the cold fluid. The heat capacity C, is expressed as
C = mc p
(3-12)
.
where m is the mass flow rate of fluid and cp is the specific heat of the fluid. The hea capacity, C, is the extensive equivalent to the specific heat, and it determines the amoun of heat a substance absorbs or rejects when the temperature changes.
31
The amount of air flowing over each section of the condenser is proportional to the tube length, L, corresponding to each specific section. For example, the mass of air flowing over the saturated section of the condenser can be found by the following relation,
ma , sat ma , tot
= Lsat Ltot
(3-13)
The heat exchanger effectiveness discussed earlier in this chapter is th e ratio of the actua amount of heat transferred to the maximum possible amount of heat transferred,
ε =
Q Q max
(3-14)
The heat exchanger effectiveness is dependent on the temperature distribution within each fluid and on the paths of the fluids as the heat transfer takes place, i.e. parallel-flow, counter-flow, or cross-flow. In most typical condensers and evaporators, the refrigeran
32
mass flow flow is separated into a number of discrete tubes and does not mix between fluids. Furthermore, the plates of the heat exchanger exch anger prevent mixing of the air flowing over ov er the fins. Therefore, air at one end of the heat exchanger will not necessarily be the same temperature as the air at the the other end. For a cross flow flow heat exchanger exchan ger with both fluids unmixed, the eff ectiveness can be related to the number of transfer transfer units (NTU) with the followi following ng expression (Incropera & DeWitt, 1996):
1 0.22 0.78 ( ) NTU C NTU ( ) ( ) − − [ ] exp 1 , r C r
ε = 1 − exp
(3-15)
where Cr is the heat capacity ratio,
C r
=
33
C min C max
.
(3-16)
In the saturated portion of the condenser, the heat capacity on the refrigerant side approaches infinity and the heat capacity ratio, Cr goes to to zero. When Cr is zero, the effectiveness for any heat exchanger excha nger configurati configu ration on is expressed as
ε = 1 − exp (− NTU ).
(3-17)
The NTU is a function function of the overall heat transfer transfer coefficient, U, and is defined as
NTU =
UA C min
,
(3-18)
where A is the heat transfer area upon which the overall heat transfer coefficient, U, is based. The overall heat transfer coefficient accounts accounts for the total thermal resistance between the two fluids and is expressed as follows.
34
1 UA
=
"
1
η s ,a ha Aa
+
"
R f ,a
η s , a Aa
+ Rw +
R f , r
η s , r Ar
+
1
η s , r hr Ar
where R”f,(a or r) is the fouling factor, R w is the wall thermal resistance,
,
(3-19)
ηs(a or r) is the
surface efficiency, and h is is the heat heat transfer coefficient. There are no fins fins on the refrigerant side of the condensing tubes; therefore, the refrigerant side surface efficiency is 1. Neglecting the wall wall thermal resistance, Rw (this value is usually 3 orders o magnitude magnitude lower than the other resistances), and the fouling factors, fac tors, R” f,(a or r), the overall heat transfer coefficient reduces to:
−1
1 1 + UA = η s ,a ha Aa hr Ar
.
(3-20)
The methodology for for dete d etermining rmining the refrigerant and air-side heat transfer coefficients are discussed Chapter IV and Chapter V, respectively. To determine the overall surface efficiency for a finned tube heat exchanger, it is firs necessary to determine the efficiency of the fins as if they existed alone. For a plate-finplate-fin-
35
and-tube heat exchange exchangerr with multiple multiple rows of staggered tubes, the plates can be evenly divided into hexagonal hexagon al shaped fins as shown in Figure 3-3. Schmidt (1945) analyzed hexagonal fins and determined that they can be treated as circular fins by by replacing the outer oute r radius of the th e fin fin with an equivalent radius. radius. The empirical relation relation for for the equivalen radius is given by
Re r
= 1.27ψ (β − 0.3)1 / 2 ,
where r is the outside tube radiu radius. s. The coefficients
ψ =
(3-21)
ψ and β are defined as
X t
(3-22)
2r
and
1 2 X l β = X t
36
+
X t 2
4
1 / 2
,
(3-23)
Transverse Tube Spacing Xl
Air Flow
Xt Tube Spacing Normal to Air Flo
Figure 3-3: Hexagonal Fin Layout and Tube Array
37
where Xl is the tube spacing in the direction parallel to the direction of air flow, and X t is the tube spacing normal to the direction of air flow. Once the equivalent radius has been determined, the equations for standard circular fins can be used. For this study, the length of the fins is much greater than the fin thickness. Therefore, the standard extended surface parameter,
es
can be expressed as,
1 / 2
1 / 2 2ha h Pe = = kA kt c
mes
,
(3-24)
where ha is the air-side heat transfer coefficient, k is the thermal conduc tivity of the fin material, Pe is the fin perimeter,
c
is the fin cross sectional area, and t is the thickness o
the fin. For circular tubes, a parameter
φ can be defined as
Re − 1 R . 1 + 0.35 ln e r r
φ =
38
(3-25)
The fin efficiency,
ηf , for a circular fin is a function of
es,
Re, and f, and can be
expressed as
η f
The total surface efficiency of the fin,
ηs
=
tanh (mes Reφ ) . mes Reφ
(3-26)
ηs is therefore expressed as
=1−
A fin Ao
(1 − η f ),
(3-27)
where Afin is the total fin surface area, A o is the total air-side surface area of the tube and the fins.
39
Condenser Fan
Natural convection is not sufficient to attain the heat transfer rate required on the airside of the condenser used in a reasonably sized residential air-conditioning system. Therefore a fan must be employed to maintain the airflow at a sufficient rate of speed. Although much of the power consumed by the total system is due to the compressor, the condenser fan also requires a significant amount of power. The power required by the fan is directly related to the air-side pressure drop across the condenser and to the velocity of air across the condenser:
W f ,con
=
V a ,con ∆Pa ,con A fr ,con
(3-28)
η fan, con
where Va,con is the air velocity over the face of the condenser,
∆Pa,con is the air-side
pressure drop over the condenser, Afr,con is the frontal area of the condenser, a nd ηfan,con is the condenser fan efficiency. Calculations for the air-side pressure drop are discussed in Chapter V.
Expansion Valve
The expansion valve is used to control the refrigerant flow through the system Under normal operating conditions, the expansion valve opens and closes in order to 40
maintain a fixed amount of superheat in the exit of the evaporator. In this study, the superheat will be maintained at the typical 10° F. Because the expansion valve can only pass a limited volume of refrigerant, it cannot maintain the specified superheat at the evaporator exit if the refrigerant is not completely condensed into liquid. If incomplete condensation in the condenser occurs, the vapor refrigerant backs up behind the expansion valve and the pressure increases until the refrigerant is fully condensed. As a result, the expansion valve cannot regulate the refrigerant mass flow rate, and canno maintain a fixed superheat at the evaporator exit. The energy equation shows that the enthalpy is constant across the expansion valve.
h3
= h4
(3-29)
Evaporator
The purpose of the evaporator is to transfer heat from the room air in order to lower its temperature and humidity. Because the refrigerant enters the evaporator as a liquidvapor mixture, it is only divided into saturated and superheated sections. No sub-cooled section is necessary. The analysis of the thermodynamic parameters of the evaporator is
41
nearly identical to that of the condenser. However, the dehumidification process involving the evaporator results in some modifications of the analysis. To maintain the simplicity of the evaporator model, the evaporator coil is assumed to be dry, thus the airside heat transfer coefficient is not affected. However, because the air flowing over the evaporator is cooled to a temperature below the wet bulb temperature, some of the heat rejected by the air causes water to condense out of the air rather than simply lowering the temperature of the air. Therefore, the specific heat must be modified to account for this condensation. The total enthalpy change of the air is thus the sum of the enthalpy change due to the decrease in temperature (sensible heat), and the enthalpy change due to condensation (latent heat).
∆htot = ∆hsens + ∆hlat
(3-30)
If the specific heat for dry air is utilized in the model for the evaporator, the resulting exit temperatures will be too low for complete vaporization. Therefore, an effective specific heat that takes into account both the latent he at and the sensible heat must be utilized. Using an effective specific heat will result in a more accurate determination o
42
the evaporator exit temperature without the complications associated with using the standard equations for air-water mixtures. Since the evaporator is not the focus of this study, this approximation should not effect the condenser optimization methodology. Dividing (3-30) by the temperature change gives the following.
∆htot ∆hsens ∆hlat = + ∆T ∆T ∆T
(3-31)
The ratio of the sensible heat enthalpy change to the temperature change is by definition, the specific heat, cp. Therefore, after substituting cp into (3-31) and rearranging, the following expression is obtained:
c p , eff
= c p +
43
∆hlat ∆T
(3-32)
where cp is the specific heat ratio for dry air and cp,eff is the effective specific heat. To maintain indoor humidity, the latent heat accounts for approximately 25% of the tota enthalpy change of the air flowing over an evaporator. The effective specific heat can thus be expressed in terms of the specific heat for dry air only,
c p ,eff
0.25∆hlat ∆hsens = 1.33c p . = c p + ∆T 0.75∆htot
(3-33)
Evaporator Fan
Because the evaporator is not the primary focus of this study, introducing wet coils would present unwelcome complications in the overall analysis. In addition to affecting the heat transfer, wet coils also have an effect on the air-side pressure drop. Although there are correlations available for determining the pressure drop over wet coils, they are cumbersome to use and again, the evaporator is not the primary focus of this investigation. After the air flows over the evaporator, it enters a series of ducts that then return the air back inside the living space. The power required by the evaporator fan depends on the losses in these ducts and can vary from configuration to configuration. Therefore, the
44
default power requirement used by the Air-conditioning and Refrigeration Institute (ARI, 1989) of 365 Watts per 1000 ft3 /minute of air will be used.
Refrigerant Mass Inventory
The degrees of sub-cooling at the condenser exit are controlled by the syste operating conditions and the quantity of refrigerant mass in the system, as is discussed further in Chapter VI. The mass of refrigerant in the tubes connecting the components is neglected. Since the compressor contains only vapor, the mass of refrigerant in the compressor is also neglected. Therefore the total mass of the system includes the mass o refrigerant in the sub-cooled, saturated, and superheated portions of the condenser, and in the saturated and superheated portions of the evaporator. The following text outlines the procedure for finding the refrigerant mass in the saturated portion of the evaporator. The same procedure is also used to determine the mass of refrigerant in the saturated portion of the condenser, however the boundary conditions are different The mass of refrigerant can be expressed as
m=
Aci dl
∫
L
45
v
.
(3-34)
where, Aci is the cross sectional area of the refrigerant-side of the tube, and v is the specific volume, which at saturated conditions is a function of quality expressed as
v ( x ) = v l (1 − x ) + v v .
(3-35)
The boundary conditions for the saturated portion of the evaporator are
= 0) = xi
(3-36)
x(l = L ) = 1
(3-37)
x(l
and
46
where l is integral variable evaporating tube length and L is the total evaporating tube length. Using the boundary conditions and assuming the quality varies linearly with tube length, the following expression results
x(l ) =
1 − xi L
l + xi .
(3-38)
Substituting (3-38) into (3-35) yields an expression for the specific volume as a functi of length,
v(l ) = v l
1 − x i (v v ). + xi (v v − v l ) + l v − l L
For a uniform cross sectional area, substituting (3-39) into (3-34) yields
47
(3-39)
1 dl. = Aci ∫ 1 − xi (v − v ) l =0 v + x (v − v ) + l v l l i v l L l = L
msat ,evap
(3-40)
Integrating (3-40) yields the following expressi
= 1 − xi Aci L = ln v l + xi (v v − v l ) + l (v v − v l ) . 1 v v − x − L ( )( ) l =0 i l v l L
m sat ,evap
Substituting for l, the expression for the final mass in the saturated portion of the evaporator is expressed as:
48
(3-41)
m sat ,evap
=
Aci L sat ,evap
ln (1 − xi )(v v − v l ) x i (v v
. − v l ) + v l vv
(3-42)
The mass of refrigerant in the superheated portions of the condenser and evaporator are expressed simply as:
mcon,sh
= ρ v Aci Lcon,sh
(3-43)
mevap,sh
= ρ v Aci Levap, sh .
(3-44)
and
Finally, the mass of refrigerant in the sub-cooled section of the condenser is expressed as
49
mcon, sc
= ρ v Aci Lcon,sc .
50
(3-45)
CHAPTER IV
REFRIGERANT SIDE HEAT TRANSFER COEFFICIENT AND PRESSURE DROP MODELS
Single Phase Heat Transfer Coefficient
For a constant surface heat flux for single phase laminar flow, the Nusselt number can be approximated by the following expression.
Nu D
= 4.36
(4-1)
In the turbulent region, however, there are a number of expressions available for the Nusselt number. One of the more commonly used correlations for turbulent flow is the Dittus-Boelter equation. This correlation is valid for fully developed flow in circular
51
tubes with moderate temperature variations (Incropera & DeWitt, 1996). For refrigeran cooling in a condenser, the Dittus-Boelter equation is expressed as
Nu D
= 0.023 Re D0.8 Pr 0.3 .
(4-2)
This mathematical relation has been confirmed by experimental data for the following conditions: 0.7 ≤ Pr ≤ 160 ReD ≥ 10,000 L/D
≥ 10
In the sub-cooled portion of the condenser in this study, the temperature difference at the inlet and exit is usually less than 20° F, and the moderate temperature variation assumption is valid. However in the superheated portion of the condenser, the inlet and exit temperatures can differ by as much as 90° F. Therefore, the temperature difference between the air flowing over the tubes and the refrigerant flowing inside the tubes is large. This causes the temperature difference between the inner surface of the tubes and the refrigerant to also be large in the superheated portion of the condenser. Thus, under these conditions, the Dittus-Boelter equation is less accurate.
52
Yet another Nusselt number correlation for single phase turbulent flow has been developed by Sieder and Tate (1936). This correlation was developed for a large range of property variations based on the mean fluid temperature and the wall surface temperature, and is expressed as
0.14
Nu D
where all properties except for
= 0.027 Re
0.8 D
Pr
1 / 3
µ m µ s
,
(4-3)
µs are evaluated at the mean fluid temperature, and µs is
evaluated at the temperature of the inner tube wall surface. Again, since this model is developed for a large range of property variations, it is valid for larger temperature differences within the fluid flowing inside the tube. Kays and London (1984) have also developed a heat transfer correlation for single phase turbulent flow. This correlation was developed using empirical data taken from a variety of refrigerants in circular heat exchanger tubes under several thermodynamic conditions. Unlike most heat transfer correlations, Kays and London have developed the equations for the transition regio n between laminar and turbulent flow. The correlation is expressed as:
53
St Pr 2 / 3
= ast Rebst
(4-4)
where the coefficients a st and bst are as follows:
Laminar
Re < 3,500
ast = 1.10647,
bst = -0.78992
Transition 3,500 ≤ Re ≤ 6,000
ast = 3.5194 x 10-7,
bst = 1.03804
Turbulent 6,000 < Re
ast = 0.2243,
bst = -0.385
and the Stanton number, St is expressed as:
St =
Nu D
Re Pr
=
hr ,SP Gc p
(4-5)
where cp is the specific heat at constant pressure, and G is the total mass flux. The Nusselt numbers calculated using each of the correlations discussed above are plotted versus the Reynolds number in Figure 4-1. The difference between the wall temperature and the refrigerant is taken as 40° F. The calculations are performed using a
54
140 Laminar, Constant Heat Flux
120
Turbulent
Kays and Londo
Kays & Londo
Dittus Boelter
100
r e b m 80 u N t l e 60 s s u N
Sieder and Tate
Laminar
Transition
Sieder & Tate Dittus-Boelter
40 Kays & Londo
20
0 0
5000
10000
15000
20000
25000
Reynolds Number
Figure 4-1: Refrigerant-Side Single Nusselt Number vs. Reynolds Numbe
55
tube diameter of 0.2885 in, with refrigerant R-410a flowing as superheated vapor at a mean temperature of 140 °F and a pressure of 395 psia (conditions typically found in the superheated portion of the condenser for this study). In the turbulent region, the value o the Nusselt number calculated using the Kays and London correlation is on average about 70% higher than the Nusselt numbers calculated using both the Dittus-Boelter and the Sieder and Tate correlations. This is due to the fact that both the Sieder and Tate and Dittus-Boelter equations have assumed a smooth pipe. However the Kays and London correlation was developed with experimental data taken from actual heat exchangers which employ tubes with rougher surfaces. Because the Kays and London relation is based on experimental data taken directly from heat exchangers similar to those investigated in this work, and because the issue of the transition from laminar to turbulent flow has been addressed, this correlation is used.
Condensation Heat Transfer
As discussed in Chapter II, the hea transfer coefficient in two-phase flow is dependent on the flow regimes that are present. Annular flow is generally assumed to be the dominant flow pattern existing over most of the condensing length during bot horizontal and vertical condensing inside tubes (Soliman et al., 1968). Baker (1954) and Gouse (1964) have derived flow pattern maps from numerous data, and have verified the validity of this assumption. In most cases, annular flow is established soon after condensation begins, and continues to very low quality. For horizontal condensing,
56
gravity-induced stratification exists at low quality, but this usually occupies only a small portion of the overall condensing length (Soliman et al., 1968). Annular flow is a particularly important flow pattern since for a wide range of pressure and flow conditions, and it occurs over a major part of the mass quality range, from 0.1 up to unity (Collier & Thome, 1996). Therefore, heat transfer correlations developed for annular flow, in addition to a correlation developed for all flow regimes, are considered for use in this present study. Two-phase flow heat transfer correlations developed by Traviss et al. (1973), Akers e al. (1959), and Shah (1979) are evaluated for this current work. The correlations o Akers et al. and Traviss et al. were developed for annular flow, while the Shah correlation is proposed to be applicable to all flow regimes. Figure 4-2 shows the condensation hea transfer coefficients for refrigerant R-12 calculated from the correlations of Shah, Traviss et al., and Akers et al., versus the total mass flux. The figure also shows experimenta condensation heat transfer coefficients for refrigerant R-12 taken from experimental data collected by Eckels and Pate (1991). Using the parameters designated by the Baker (1954) flow regime map, it is determined that fo r the experimental conditions of Eckels and Pate, a slug flow pattern exists for mass fluxes between 100 and 250 kg/m2-s, and an annular flow regime exists for mass fluxes greater than 250 kg/
2
-s. As Figure 4-2
shows, the Traviss correlation overpredicts the experimental data for the entire range o mass fluxes shown. The Akers and Shah correlations slightl y underpredict the experimental values for relatively low mass fluxes and slightly overpredict the experimental data at higher mass fluxes (annular flow).
57
4000 3500
r e f s n ) s a r T 2 m t / a e W H ( t n n e o i i t i c a f s f n e e o d C n o C
Traviss, et al., correlation
3000 Shah correlation
2500
experimental data
2000 Eckels&Pate-experimental data
1500 Akers-correlation
Traviss, et al., -correlation
1000
Shah-correlation 500
Akers et al.,-correlation
0 0
100
200
300
400
500
600
700
2
Total Mass Flux (kg/m -s)
Figure 4-2: Condensation Heat Transfer Coefficient vs. Total Mass Flux Fo Refrigerant R-12
58
The fact that the Traviss correlation greatly overpredicts the experimental data for when the flow regime is annular is surprising since this correlation was developed for annular flow. The Akers correlation predicts the experimental data to within an average
±14.3% while the Shah correlation predicts the experimental data to within an average o ±14.7%.
Therefore the Shah and Akers correlations are in good agreement with each
other, and are both more accurate than the Traviss correlation for the conditions investigated. Using the parameters of the Baker (1954) flow regime map, and th e typical operating conditions of the condenser studied in this present work (mass fluxes approximately greater than or equal to 400 kg/ 2-s or 300,000 lbm/ft2-hr), it is determined that the dominant flow regime is indeed annular. However, this study also finds that for low qualities, stratified-wavy flow exists. As a result, the use of a general correlation that is valid for more than one flow regime is advantageous for the work of this investigation. Therefore, the correlations developed by Akers et al. and Traviss et al., are not used. Hence, the two-phase flow heat transfer correlation developed by Shah is used for this investigation. The two-phase flow heat transfer model developed by Shah is a simple correlation that has been verified over a large range of experimental data. In fact, experimental data from over 20 different researchers has been used in its development. The model has a mean deviation of about 15% and has been verified for many different fluids, tube sizes, and tube orientations.
59
For this model, at any given quality, the two-phase heat transfer coefficient is defined as:
hTP
0.04 3.8 x 0.76 (1 − x ) 0.8 = h L (1 − x ) + 0.38 p r
(4-6)
where hTP is the two-phase flow heat transfer coefficient, x is the quality, hL is the liquid only heat transfer coefficient, and p r is the reduced pressure. By integrating the expression (4-6) over the length of the tube, the mean two-phase flow heat transfer coefficient can be determined.
Le
3.8 x 0.76 (1 − x )0.04 0.8 hTPM = (1 − x ) + dL 0.38 ( Le − Li ) L∫ pr h L
(4-7)
i
If one assumes that the quality varies linearly with length, the mean two-phase flow hea transfer coefficient can be approximated b
60
− (1 − x )0.8 3.8 x1.76 0.04 x 2.76 x . + 0.37 − hTPM = ( xe − xi ) 1.8 1 . 76 2 . 76 pr x e
h L
(4-8)
i
This assumption of linearly varying quality typifies fixed heat transfer per unit length. For complete condensation, (x varying from 1 to 0), the mean two-phase heat transfer coefficient reduces to the following expression.
hTPM
2.09 = h L 0.55 + 0.38 p r
(4-9)
Evaporative Heat Transfer Coefficient
As discussed in Chapter III, the modeling of the evaporator is not the primary focus of this study. To this end, the correlations investigated to determine the evaporator hea transfer coefficient were limited. The expression for the average evaporative two-phase heat transfer coefficient is taken from Tong (1965). This relationship assumes a constan temperature difference between the wall and the fluid along the length of the pipe and is expressed as:
61
k hevap = (0.0186875) l D 0.2
0.4 µ l C p, l ρ 0.375 µ 0.075 G 0.8 x − x e i l v 0.325 0.325 (4-10) µ l k l ρ v µ l xe − xi
Pressure Drop in the Straight Tubes
The pressure drop in the straight-tube portions of the superheated and sub-cooled sections of the condenser (single phase vapor and liquid respectively) can be determined by applying the standard pressure drop relationship for pipe flow.
∆ p S ,SP =
fG 2 L
(4-11)
ρ
The friction factor, f , for circular pipes depends on the Reynolds number as shown in the following expressions:
62
f =
64 Re D
Laminar (Incropera & Dewitt, 1996)
(4-12)
and
1 f 1 / 2
ε / D 2.51 = −2 log10 pr + . Re D f 1 / 2 3.7
Turbulent (Colebrook, 1938)
(4-13)
where εpr is the pipe roughness, which for the drawn copper tubes utilized in this study, is assumed to be 0.000005 ft. For two-phase flow, determining the pressure drop is not as simple. As discussed in Chapter II, there is still no satisfactory, universal method for calculating the two-phase pressure drop, while taking into account flow regime considerations. Again using the parameters of the Baker (1954) flow regime map, and the typical operating conditions o the condenser studied in this present work (mass fluxes approximately greater than or equal to 400 kg/ 2-s or 300,000 lbm/ft2-hr), it is determined that the dominant flow regime is indeed annular. However for low qualities, stratified-wavy flow also exists. Therefore, only semi-empirical, general pressure dro p correlations are considered for use in this study. Although, various other general correlations have since been proposed, as discussed in Chapter II, the original Lockhart-Martinelli approach is still one of the
63
simplest, as discussed by Chen and Spedding (1981). Again, one of the bigges advantages of this procedure is that it can be used for all flow regimes. While the cost o this flexibility is decreased accuracy, as indicated in Chapter II, subsequent genera correlations do not appear to be substantially more accurate than the Lockhart-Martinell model. Therefore, the method of Lockhart and Martinelli is used to determine the twophase flow refrigerant-side pressure drop for the heat exchangers investigated in this study. The Lockhart-Martinelli method, or L and M method, is derived from the separated flow model of two-phase flow. This model considers the phases to be artificiall segregated into two streams; one of liquid and one of vapor (Collier and Thome, 1996). The separated flow model is based on the following assumptions:
1) constant but not necessarily equal velocities for the vapor and liquid phases, and 2) the attainment of thermodynamic equilibrium between the phases
Hiller and Glicksman (1976) detail the procedures for calculating the frictional momentum, and gravitational components of the two-phase flow pressure drop using the Lockhart-Martinelli model. Hiller and Glicksman expound on the method of LockhartMartinelli in the following manner. The total two-phase pressure drop is divided into frictional, gravitational, and momentum components as follows:
64
dP dz
dP dP dP = + + , dz f dz g dz m
(4-14)
Hiller and Glicksman then derive the following expression for the frictional component,
Gv2 − 0.2 µ dP = ρ v (0.09) v (1 + 2.85 X 0.523 )2 tt Gv D g cs D dz f
(4-15)
where gcs is a units conversion constant, and X tt is the Lockhart-Martinelli parameter which is expressed as:
0.875 ρ v 0.5 µ l 0.125 1 − x X tt = x ρ l µ v
65
(4-16)
The gravitational component is then expressed as:
Gv2 ρ l dP = ρ v 1 − B α 2 dz g D ρ Fr g cs v
(4-17)
where Fr is the Froude number based on the total flow (Traviss, 1973),
2
2
Fr
G ρ v = a x D
(4-18)
with ax defined as the axial acceleration due to gravity; B is the Buoyancy modulus;
B
=
ρ l
− ρ v ρ v
66
,
(4-19)
and α is the local void fraction
α =
1 2 / 3
ρ 1 − x v x ρ l
.
(4-20)
1+
Finally, Hiller and Glicksman give the momentum pressure drop component as:
1 / 3 2 / 3 ρ v ρ v ρ v dP = − G 2 dx 2 x + (1 − 2 x) + (1 − 2 x) ρ − 2(1 − x) ρ . ρ dz m g cs ρ v dz l l l
(4-21)
Unfortunately, it is difficult to predict the variation of the quality with length, dx/dz. However, as is the case with the condensation heat transfer coefficient, a linear profile is assumed for the work of this study. If the quality variation is divided in to small increments of ∆x, the resulting pressure drops over each small increment can be summed to yield the total pressure drop over the entire length. For horizontal tube flow, the gravitational pressure drop term is neglected. The pressure drop per unit length as a
67
function of the variation in quality for the frictional and momentum components are then integrated over the length of the tube, utilizing the aforementioned incremental procedure. The frictional pressure drop in the two-phase region then reduces to the following expression:
∆ p f = −C 2 0.357 x 2.8 + 2C 3 0.429 − 0.141 x − 0.288 x2 x 2.33 + C 32 (0.538 − 0.329 x ) x1.86 ] x x
(4-22) e i
where the constants C 3, C2, and C1 are determined by:
0.0523
µ C 3 = 2.85 l µ v
C 2
=
C 1
0.09µ vG1.8 1 .2 C 1g c ρ v D
=
xe − xi z e − z i
68
0.262
ρ v ρ l
.
(4-23)
(4-24)
(4-25)
The momentum pressure drop in the two-phase region then reduces to:
∆ pm
ρ ρ 1 / 3 ρ 2 / 3 v v 1 + v = − − ρ l ρ l x ρ v g c ρ l G2
(4-26) xe
ρ ρ 1 / 3 ρ 2 / 3 x . − 2 v − v − v ρ l ρ l ρ l x i
Hence, the total two-phase refrigerant pressure drop in the straight tube section is simply the sum of the momentum and frictional pressure drop components.
∆ p S ,TP = ∆ p m + ∆ p f
69
(4-27)
Pressure Drop In Tube Bends
The work of Chisolm (1983) is used to determine the pressure drop inside tube bends. For single phase flow, the pressure drop in tube bends is calculated simply by assigning an equivalent length to each bend based on the flow diameter and the bend radius. For two-phase flow in tube bends, the pressure drop is calculated for liquid-only flow, and correction factors are applied to determine the appro ximate two-phase flow pressure drop. Instead of predicting the two-phase pressure drop in inclined bends that are found in most heat exchangers, this method predicts the pressure drops for two-phase flow in horizontal bends. However, no accurate correlations are available for predicting the twophase flow pattern in an inclined bend. Furthermore, the pressure gradients due to elevation changes caused by the incline are negligible compared to friction pressure losses. Hence, the horizontal bend model developed by Chisolm is sufficient for this study. Since the bends are not finned and do not come into contact with air flow, the hea transfer in the bends is neglected. The first step in computing the pressure drop in tube a tube bend is to determine the equivalent length of the bend. The equivalent length, y, is a function of the relative radius, rr:
r r =
r b D
70
(4-28)
where rb is the radius of the bend, and D is the inner diameter of the tube. Most condensers utilize tubes with a relative radius between 1 and 3, which according to Chisolm’s model corresponds to an equivalent length of between 12 to 15 diameters for 90° bends. The equivalent length for a 180° return bend is approximately twice the equivalent length of a 90° bend. For this study, 180° return bends are assumed to have an equivalent length of 26 diameters. Chisolm approximates the single-phase pressure drop in a bend by simply substituting the equivalent length of the bend, y, for the straight pipe length in the standard pressure drop equation,
∆ pb ,SP =
fG
2
2 ρ
y D e
where ∆pb,SP is the single phase pressure drop in the bend.
71
(4-29)
For the two-phase flow pressure drop in bends, the calculations are more involved. Assuming homogeneous two-phase flow, the friction factor is determined by the same expressions that are used for single phase flow as shown in (4-12). However, Chisolm’s development uses a Reynolds number based on the two-phase flow viscosity.
Re =
GD
µ TP
(4-30)
The two-phase viscosity is a function of the quality and is determined by the following expression:
µ TP
= µ v x + (1 − x )µ l .
(4-31)
Chisolm defines a two-phase flow bend pressure drop coefficient for a 90° bend, k b,90°, which is expressed as:
72
k b ,90 o
y = f D e
(4-32)
Another coefficient for 90° bends, B90° is also defined, and is expressed by:
B 90o
= 1+
2.2 k b, 90o (2 + Rb / D )
(4-33)
where Rb is the bend recovery length. The B coefficient for bends that are not 90° is expressed as:
Bθ
= 1 + [B 90 − 1] o
k b ,90o k b ,θ
(4-34)
In the case of 180° bends, the bend pressure coefficient k b,180 °, is approximately twice the value of k b,90 °, so B180° can be calculated by the following expression.
73
B180 o
= 0.5 1 + B90
o
(4-35)
Chisolm defines a two-phase multiplier, ϕ2, for the pressure drop in a tube bend as:
ϕ b2,lo
= 1 + (Γ b2 − 1) Bθ x (2− n ) / 2 (1 − x )(2− n) / 2 + x (2−n )
(4-36)
where Γ b2 is the physical property coefficient for a tube bend and is determined by,
Γ = 2 b
ρ l ρ v
n
µ v , µ l
(4-37)
and n is the Blausius coefficient, which is calculated by the following expression.
74
f LO f GO n= µ v µ l ln
(4-38)
The friction factors f LO and f GO are determined using (4-12) by assuming all of the mass is flowing alone as either a liquid or a vapor. The two-phase pressure drop is then calculated as the product o f the liquid-onl single-phase pressure drop and the two-phase multiplier, ϕ2b,LO:
∆ pb ,TP = ∆ p b , LO ϕ b2, LO
The liquid-only bend pressure drop, ∆pb,LO is then determined by (4-28).
75
(4-39)
CHAPTER V
AIR-SIDE SIDE HEAT TRANSFER COEFFICIENT AND PRESSURE DROP MODELS
The works of McQuiston (McQuiston and Parker, 1994), Rich (1973), and Zukauskas and Ulinskas (1998), are used to evaluate the air-side heat transfer and pressure drop over finned tubes in air cross-flow. The following development of the work of McQuiston, Rich, and Zukauskas is taken from a thesis entitled “Optimization of a Finned-Tube Condenser for a Residential Air-Conditioner Using R-22” by Emma Saddler (2000). This development is detailed here in this study for completeness.
Heat Transfer Coefficient
The work of McQuiston (McQuiston and Parker, 1994) is used to evaluate the air-side convective heat transfer coefficient for a plate finned heat exchanger with multiple rows of staggered tubes. The model is developed for dry coils. The heat transfer coefficient is based on the Colburn j-factor, which is defined as:
76
j
= St Pr 2 / 3 .
(5-1)
Substituting the appropriate values for the Stanton number, St, gives the following relationship for the air-side convect ive heat transfer coefficient, ha,
ha
=
jc p Gmax
Pr 2 / 3
,
(5-2)
where cp is the specific heat, and G max is the mass flux of air through the minimum flow area which is expressed as:
Gmax
=
77
mair Amin
,
(5-3)
where Amin is the minimum air flow area. McQuiston (McQuiston and Parker, 1994) use a 4-row finned tube heat exchanger a s the baseline model, and define the Colburn j-factor for a 4-row finned-tube heat exchanger as:
j4
= 0.2675 JP + 1.325 × 10−6 ,
(5-4)
and the parameter JP is defined as:
−0.15
JP
− 0.4
= Re D
Ao At
,
(5-5)
where, Ao is the total air side heat transfer surface area (fin area plus tube area), and
t
the tube outside surface area. The Reynolds number, ReD in the above expression is based on the outside diameter of the tubes, Do, and the maximum mass flux, Gmax. The area ratio can be expressed as:
78
is
Ao At
=
4 X l X t σ , π Dh Ddepc
(5-6)
where Xl is the tube spacing parallel to the air flow (transverse), X t is the tube spacing normal to the air flow, D depc is the depth of the condenser in the direction of the air flow, Dh is the hydraulic diameter defined as:
Dh
=
4 Amin Ddepc Ao
,
(5-7)
and σ is the ratio of the minimum free-flow area to the frontal area,
σ =
Amin A fr
79
.
(5-8)
The j-factor for heat exchangers with four or fewer rows can then be found using the following correlation:
j z j 4
−
=
1 − 1280 z Re rs1.2 −1.2
1 − (1280 )(4 )Re rs
,
(5-9)
where z is the number of rows of tubes, and Re rs is the Reynolds number based on the row spacing, Xrs,
Re rs
=
80
Gmax X rs
µ
.
(5-10)
Pressure Drop
According to Rich (1973), the air-side pressure drop can be divided into two components, the pressure drop due to the tubes, ∆ptubes, and the pressure drop due to the fins, ∆pfin. The work of Rich is used to evaluate the air-side pressure drop due to the fins, which is expressed as
∆ pfin = f fin v m
2 G max Afin
2
Ac
,
where, f fin is the fin friction factor, v m is the mean specific volume,
(5-11)
fin
is the fin surface
area, and Ac is the minimum free-flow cross sectional area. In experimental tests, Rich found that the friction factor is dependent on the Reynolds number, but it is independent of the fin spacing for fin spacing between 3 and 14 fins per inch. In this range of fin spacing, Rich expresses the fin friction factor as:
f fin
= 1.7 Rel−0.5 ,
81
(5-12)
where the Reynolds number is based on the tube spacing parallel to the direction of the air flow (transverse tube spacing), Xl,
Rel
=
GX l
µ
.
(5-13)
To determine the pressure drop over the tubes, the relationships developed by Zukauskas and Ulinskas (1998) are used. The pressure drop over the banks of plain tubes is expressed as:
∆ p tubes = Eu
G
2
z ,
2 ρ
(5-14)
where z is the number of rows, and Eu is the Euler number. Rich expresses the Euler number as a function of the Reynolds number and the tube geometry. For staggered, equilateral triangle tube banks with several rows, Rich expresses the Euler number by a fourth order inverse power series by the following:
82
Eu = q cst +
r cst
Re D
+
s cst 2
Re D
+
t cst 3
Re D
+
u 4 Re D
(5-15)
where ReD is the Reynolds number based on the outer tube diameter. The coefficients qcst, rcst, scon, t cst, and u are dependent on the Reynolds number and the parameter “a”, which is defined as the ratio of the transverse tube spacing to the tube diameter. The coefficients for a range of Reynolds numbers and spacing to diameter ratios have been determined from experimental data by Zukauskas and Ulinskas (1998) and are expressed in Table 5-1. For non-equilateral triangle tube bank arrays, the staggered array geometry factor k 1 must be used as a correction factor to the coefficients in Table 5-1. The staggered arra geometry factor is dependent on the Reynolds number based o n: the outer tube diameter; the parameter “a”, which again is defined as the ratio of the transverse tube spacing to the tube diameter; and the parameter “b”, which is defined as the ratio of the tube spacing in the direction normal to the air flow and the tube diameter. The equations for k 1 are found in Table 5-2.
83
Table 5-1: Coefficients for the Euler Number Inverse Power Series a
Reynolds Number
qcst
rcst
scst
tcst
u
3 < Re D < 103
0.795
0.247 x 10 3
0.335 x 10 3
-0.155 x 10 4
0.241 x 10 4
103 < ReD < 2 x 106
0.245
0.339 x 10 4
-0.984 x 10 7
0.132 x 10 11
-0.599 x 10 13
3 < Re D < 103
0.683
0.111 x 10 3
-0.973 x 10 2
0.426 x 10 3
-0.574 x 10 3
103 < ReD < 2 x 106
0.203
0.248 x 10 4
-0.758 x 10 7
0.104 x 10 11
-0.482 x 10 13
7 < Re D < 102
0.713
0.448 x 10 2
-0.126 x 10 3
-0.582 x 10 3
0.000
102 < ReD < 104
0.343
0.303 x 10 3
-0.717 x 10 5
0.880 x 10 7
-0.380 x 10 9
104 < ReD < 2 x 106
0.162
0.181 x 10 4
-0.792 x 10 8
-0.165 x 10 13
0.872 x 10 16
102 < ReD < 5 x 103
0.330
0.989 x 10 2
-0.148 x 10 5
0.192 x 10 7
0.862 x 10 8
5 x 10 3 < ReD< 2 x 10 6
0.119
0.848 x 10 4
-0.507 x 10 8
0.251 x 10 12
-0.463 x 10 15
1.25
1.5
2.0
2.5
84
Table 5-2: Staggered Array Geometry Factor
ReD
a/b
102
1.25 < a/b < 3.5
k1
0.48
a k 1 = 0.93 b
(5-16)
−0.048
0.5 < a/b < 3.5 103
1.25 < a/b < 3.5
a k 1 = b
(5-17)
0.284
a k 1 = 0.951 b = 1.28 −
(5-18)
0.708 0.55 + (a / b ) (a / b )2
−
0.113
104
0.45 < a/b < 3.5
k 1
105
0.45 < a/b < 3.5
a a k 1 = 2.016 − 1.675 + 0.948 b b 3 4 a a − 0.234 + 0.021 b b
106
0.45 < a/b < 1.6
(a / b )3
(5-19)
2
85
(5-20)
If the tube bank has a small number of transverse rows, the average row correction factor, Cz, must be applied because the pressure drop over the first few rows will be different from the pressure drop over the subsequent rows. Cz is the average of the individual row correction factors, c z.
C z
=
1 z
z
∑c
z =1
z
(5-21)
The equations for the individual row correction factors are given in Table 5-3. Once the average row correction factor is found, the corrected Euler number can be determined as
Eu cor = k 1C z Eu.
86
(5-22)
Table 5-3: Correction Factors for Individual Rows of Tubes
ReD
z
cz
10
<3
c z
= 1.065 −
0.18 z − 0.297
(5-23)
102
<4
c z
= 1.798 −
3.497 z + 1.273
(5-24)
103
<3
c z
= 1.149 −
0.411 z − 0.412
104
<3
c z
= 0.924 −
> 105
<4
c z
= 0.62 −
(5-25)
0.269 z + 0.143
1.467 z + 0.667
(5-26)
(5-27)
For values of z greater than 4, cz = 1
87
The corrected Euler factor, Eucor can then be used in equation (5-14) to determine the pressure drop over the tubes. Since the relations in Table 5-1, Table 5-2, and Table 5-3, are given for discrete values of the “a” parameter and the Reynolds number, a linear interpolation is used to estimate the values of Eu, k 1, and cz. The total pressure drop over the heat exchanger is then simply the sum of the pressure drop over the tubes and the pressure drop over the fins:
∆ ptot ,ac = ∆ ptubes + ∆ pfin .
88
(5-28)
CHAPTER VI
DESIGN AND OPTIMIZATION METHODOLOGY
Figure of Merit (Coefficient of Performance)
In order to quantitatively evaluate the performance of any air-conditioning system, a figure of merit must be established. For an air-conditioning system utilizing a vapor compression refrigeration cycle, the efficiency is expressed in terms of the cooling coefficient of performance or the COP. The coefficient of performance is a dimensionless quantity. It is the ratio of the rate of cooling or refrigeration capacity (hea absorbed by the evaporator), to the electrical or mechanical power used to drive the system (compressor power, condenser fan power, and evaporator fan power). The COP is expressed as:
COP =
Qe W com
+ W f ,con + W f ,evap
89
.
(6-1)
In the United States, the performance of residential air-conditioning equipment is often given in dimensional terms, Btu/(W-hr), as an energy efficiency ratio or EER. Since 3.412 Btu = 1.0 W-hr, an EER rating of 10.0 would be equivalent to a COP o 10/3.412 or 2.93. The performance of an air-conditioning device over a summer is referred to as the seasonal COP, or COP seas, in dimensionless terms. The seasonal COP takes into account the effect of varying outside temperatures on the performance of the system. It is the ratio of the average cooling load for the system during its normal usage or “cooling load hours” to the average electricity required by the system over all cooling load hours. Cooling load hours are defined as hours when the temperature is above 65 ° F, which is when air-conditioning systems are typically operated. In warmer climates, there are more cooling load hours, per year than in cooler climates. In Atlanta, for example, the total cooling load hours are approximately 1300 hours per year, while in Detroit, MI the cooling hours are about 700 per year. The air-conditioning syste actually runs fewer hours than the cooling load hours since at ambient temperatures below 95° F, the system usually cycles on and off, as regulated by a thermostat. (The cycling inefficiencies that result from the system cycling o n and off are neglected in this study.) The distribution of temperature during these cooling hours is approximately the same for all major cities in the United States. Therefore, the Air-Conditioning Refrigeration Institute, ARI, has developed a temperature distribution model based on cooling load hours which is used throughout the United States. This is shown in Table 61 as the distribution of fractional hours in temperature “bins” (ARI, 1989). Table 6-1 shows for example that the outside temperature will be between 80° F and 84 ° F
90
Table 6-1: Distribution of Cooling Load Hours, i.e. Distribution of Fractional Hours in Temperature Bins
Bin #
Bin Temperature
Ti, Representative
fri, Fraction of Total
i
Range (°F)
Temperature for Bin ( °F)
Temperature Bin Hours
1
65-69
67
0.214
2
70-74
72
0.231
3
75-79
77
0.261
4
80-84
82
0.161
5
85-89
87
0.104
6
90-94
92
0.052
7
95-99
97
0.018
8
100-104
102
0.004
(temperature bin # 4) approximately 16.1% of the time that the ambient temperature is above 65° F. Again, the seasonal COP is therefore the ratio of the average cooling load for the system over all cooling load hours to the average electricity required by the system over all cooling load hours, and is expressed as:
COPseas
=
Qave,seas W ave, seas
91
.
(6-2)
The average cooling over all cooling load hours is calculated by summing the hourly “house” cooling load over all cooling load hours, and is expressed as:
8
Qave, seas
= ∑ UAhouse (Ti − 65 o F)fri , i =1
(6-3)
where UAhouse is the overall “house” heat transfer coefficient, i is the temperature bin number, Ti is the representative temperature bin, and fri is the fraction of total temperature bin hours (as shown in Table 6-1). The average electricity required by the system over all cooling load hours is expressed as:
UAhouse Ti − 65 o F fri = ∑ , COP i i =1 8
W ave, seas
where COPi is the COP at each representative temperature bin.
92
(6-4)
Since the overall “house” heat transfer coefficient, U
house,
is common to both
expressions, dividing (6-3) by (6-4) yields the following expression for the seasonal COP:
8
∑= (T − 65 F)fr o
i
COPseas
=
i 1
(Ti − 65 o F)fri ∑ COP i =1 i 8
(6-5)
i
.
The numerator of the above expression is a constant. Since the air-conditioning syste of this study is sized to deliver a specified amount of cooling at 95 °F ambient temperature, the indoor temperature will rise when the ambient temperature is greater than 95 °F. As a result, the temperature difference of (Ti - 65°F) is limited to a maximu of 30 °F for this study. In dimensional terms, the seasonal COP can be given as the seasonal energ efficiency ratio, or SEER, and is expressed in Btu/W-hr. This is the efficiency rating tha is required by the United States Department of Energy to be placed on a yellow sticker on all air-conditioning systems sold in the United States.
93
System Design
The primary focus of this study is the optimization of the condenser configuration. However, some assumptions about the parameters of the complete air-conditioning system must be made. Air-conditioning systems are characterized by their cooling capacity at 95° F ambient temperature. The most common residential air-conditioning systems sold in the United States have a cooling capacity rating o f 30,000 Btu/hr (2 1/2 tons). Hence, for the air-conditioning system modeled in this study, the cooling capacity at 95° F is fixed to 30,000 Btu/hr. It is also customary in most residential airconditioning applications to employ an evaporator that has a 45 ° F saturation temperature. At this temperature, humidity control is maintained by removing sufficient water vapor from the cooled air. Therefore the evaporator saturation temperature is fixed at 45° F in this study. As discussed in Chapter III, the evaporator fan power and the volume flow rate of air over the evaporator, are fixed to 365 Watts per 1000 ft 3 /minute o air flow respectively (equates to a c onstant fan power of 1245 Btu/hr).
Optimization Parameters
When designing and optimizing the condenser to yield the maximum seasonal COP of the air conditioning system there are a large number of parameters that can be varied. For this investigation, these optimization parameters have been divided into two categories: operating parameters for the system, and geome tric design parameters specific to the condenser coil.
94
As part of the optimization process, comparisons are made between the seasona performances of air-conditioning systems with condensers of various geometric configurations (tube diameters, fin spacing, etc.). However, it is not possible to make valid comparisons between different heat exchanger configurations without first optimizing the operating parameters at each configuration to yield the maximum seasonal COP. For example, it is erroneous to compare the performance of a system with a 3-row condenser coil configuration in which the operating parameters have been optimized to system with a 2-row condenser coil configuration in which the operating parameters have not been optimized. No valid conclusions can be made about which configuration yields the best performance unless the operating parameters are re-optimized for each new geometric configuration tested. Therefore, in this study, the performance of each configuration at its optimum operating conditions will be determined and compared.
Operating Parameters
The operating parameters of the system studied are the refrigerant charge, the ambien temperature, the level of superheat exiting the evaporator, the amount of sub-cool exiting the condenser, and the velocity of the air flowing over the condenser. For this study, the level of superheat exiting the evaporator is fixed at a constant value of 10° F, which is typically used in most residential air-conditioning systems, and required by the compressor manufacturers to prevent liquid from returning to the compressor. For this study, the air velocity over the condenser and the sub-cool in the condenser are specified at 95° F. The resultant compressor piston displacement and mass o 95
refrigerant in the system (refrigerant charge) that yield 30,000 Btu/hr of cooling capacity at 95° F are determined. The mass inventory at 95° F dictates the sub-cool at other ambient temperatures. Hence, the air velocity over the condenser and the sub-cool in the condenser at 95° F are the two operating parameters that are optimized for each condenser geometric configuration investig ated during this study. The method for calculating the mass of refrigerant in the system (mass inventory) is detailed in Chapter III.
Geometric Parameters
There are a large number of condenser coil geometric design parameters that can be varied in order to optimize the seasonal pe rformance of an air-conditioning system. These parameters include the tube diameter, the tube spacing, the number of refrigerant parallel flow circuits, the number of tubes per refrigerant parallel flow circuit, and the fin spacing or pitch. For this study, the tube diameter, the number of refrigerant parallel flow circuits, the number of tubes per refrigerant flow circuit, and the fin spacing will be optimized. In all cases, the vertical and horizontal tube spacing are specified as 1 in. and 0.625 in., respectively. These values are typical of those found in condenser coils for unitary air-conditioning systems. In Chapter III, Figure 3-2 shows a schematic of a typical finned-tube condenser coil. In this figure, geometric parameters such as the tube spacing, number of tube refrigerant flow circuits, number of tubes per refrigerant flow circuit, and the number of rows of tubes are detailed.
96
Software Tools
For this study, all modeling and simulations are performed using Engineering Equation Solver (EES). EES is a software package developed by Dr. Sanford Klein of
the University of Wisconsin. EES incorporates the programming structures of C and FORTRAN with a built-in iterator, thermodynamic and transport property relations,
graphical capabilities, numerical integration, and many other useful mathematica functions. By grouping equations that are to be solved simultaneously, EES is able to rapidly solve large numbers of transcendent al equations. EES can also be used to perform parametric studies. Most important for this study, EES has the ability to seamlessly incorporate fluid property calls. Thermodynamic transport properties for steam, air, and many different refrigerants are built into EES.
97
CHAPTER VII
OPTIMIZATION OF OPERATING PARAMETERS
The performance of air conditioning systems is highly dependent on specific operating conditions and parameters. As detailed in Chapter VI, without optimizing th e operating conditions, it is not possible to determine the condenser configuration that yields the optimum seasonal COP. Again, the operating parameters investigated for this study are the air velocity over the condenser and the refrigerant charge measured by the sub-cool in the condenser at 95° F ambient temperature. To determine the effects of the various operating parameters on the seasonal COP, a typical evaporator and condenser coil pair is arbitrarily selected for the “base configuration”. All of the charact eristics o the condenser are specified, and all but the frontal area of the evaporator are specified. The dimensions of the heat exchangers are shown in Table 7-1. Figure 7-1 shows the effect that the operating parameters of refrigerant charge (given here in terms of the degrees of sub-cool at 95 °F ambient temperature) has on the frontal area of the evaporator for the given design conditions and a fixed cond enser geometry.
98
Table 7-1: Base Case Condenser and Evaporator Characteristics
Dimension
Condenser
Evaporator
Tube Spacing (in x in) Tube inner diameter (in)
1.25 x 1.083
1.00 x 0.625
0.349
0.349
Tube outer diameter (in)
0.375
0.375
Height (ft
2.5
1.5
Finned width (ft)
3
N/A
Fin pitch (fin/in)
12
12
Number of rows
3
4
Number of circuits
12
9
Number of tubes per circuit
2
2
3.4
) 3.2 t f ( a e r 3 A l a t n2.8 o r F r o2.6 t a r o p a 2.4 v E
2
Tsubcool=5 F Tsubcool=10 F Tsubcool=15 F Tsubcool = 20 F
2.2
2 5
6
7
8
9
10
11
12
13
14
15
Air Velocity Over Condenser (ft/s)
Figure 7-1: Effect of Operating Conditions on Evaporator Frontal Area
99
As the figure shows, the necessary finned frontal area of the evaporator is virtually independent of this operating parameter. The compressor piston displacement is calculated such that at each design condition, the system will deliver an evaporator capacity of 30,000 Btu/hr at 95° F ambient temperature. The mass inventory at 95° F ambient temperature dictates the sub-cool at other ambient temperatures. Therefore, the air velocity over the condenser and the sub-cool (refrigerant charge) are the operating parameters optimized for each condenser geometric configuration investigated in this study.
Effects of Air Velocity, Ambient Temperature, and Sub-Cool
For a fixed amount of sub-cool at 95 ° F ambient temperature, there is an air velocity that yields the maximum COP. Figure 7-2 shows the effect of air velocity on the COP for various ambient temperatures at optimum degrees sub-cool. As the figure shows, the COP has an optimum with respect to the air velocity for any ambient temperature. For ambient temperatures ranging from
° F to 97° F, the maximum seasonal COP occurs at
an air velocity between 8.0 ft/s and 9.0 ft/s for this sub-cool condition (15 ° F). For each ambient temperature, in this range o f velocities the COP is relatively insensitive to the air velocity and varies by less than 1%. For example, at 77° F sub-cool, the maximum COP is 4.31 and it occurs at an air velocity of 8.5 ft/s. Because the COP varies so little with air velocity in the optimum range, it is difficult to determine the exact optimum velocity for each sub-cool within an accuracy of more than ± 0.1 ft/s. However in actual practice,
100
4.90
4.40
P O3.90 C
Locus of Optimums
Tambient = 67 F Tambient = 77 F Tambient = 97 F
3.40
2.90 4
6
8
10
12
14
16
Air Velocity Over Condenser (ft/s)
Figure 7-2: Effect of Air Velocity on COP for Various Ambient Temperatures and Optimum Degrees Sub-Cool
101
the air speed cannot be specified to such high tolerances. Hence, the accuracy which is indicated in this investigation is sufficient. The above observations of the insensitivity of the COP to air velocity near the optimum range may initially be counter intuitive. Since the condenser fan power increases proportionally with the cube of the velocity, one does not expect the COP to become insensitive to changes in the velocity. However, in this range of velocities, as the condenser fan power requirement is increasing, the required compressor power is decreasing by approximately the same amount. This phenomenon is demonstrated in Figure 7-3, which shows the effect of the air velocity on the compressor power and the condenser fan power at 95° F ambient temperatures and optimum sub-cool. As the air velocity over the condenser increases, the condensing temperature decreases, and the inlet enthalpy to the evaporator also increases. This causes a reduction of the mass flow rate of refrigerant required to maintain the evaporator cooling capacity. Hence, the amount of compressor work is decreased. The condensing temperature of the refrigeran can never be lower than the inlet air temperature. Thus, there is a minimum power requirement for the compressor. As the air velocity increases beyond the optimal recommended range, the power required for the condenser fan begins to grow rapidly. At this point, the decrease in the compressor power requirement will not compensate for this increase in the condenser fan requirement, thus resulting in lower values of the seasonal COP.
102
9000 8000 7000 6000
) r h / u5000 t B ( r e4000 w o P
Total Power Compressor Power Condenser Fan Power
3000 2000 1000 0 5
6
7
8
9
10
11
12
13
14
15
Air Velocity Over Condenser (ft/s)
Figure 7-3: Effect of Air Velocity on Compressor Compressor and Condenser Fan Power 13° F Sub-cool at 95° F Ambient Temperature
103
Figure 7-2 also shows that as the ambient temperature decreases, the COP increases. This phenomenon is also also displayed displayed in Figure 7-4. This figure shows shows how the COP varies with the ambient temperature for various various sub-cool sub-cool conditio condit ions. ns. This phenomenon can be explained by an analysis of the effects of ambient temperature on the th e condensing temperature and pressure, the compressor power, and the evaporator cooling capacity. As the ambient temperature t emperature decreases, the saturation pressure in the condenser also decreases. Therefore, the pressure pressure rise rise in the compressor decreases. As a result, the compressor requires requires less power, and hence, the COP increases. Furthermore, as the ambient temperature decreases, decreases, the condensing temperature decreases. Thus, the enthalpy of the refrig refrigerant erant enteri e ntering ng the evaporator is reduced. The decrease in the enthalpy of the refrigerant entering the evaporator that is produced by the decrease in the ambient temperature causes the the evaporator cooling capacit ca pacity y to increase. This decrease in the enthalpy of the refrigerant entering the evaporator also causes a reduction of the mass flow rate of refrigerant refrigerant required to maintain maintain the evaporator cooling capacity. Hence, the amount of compressor compressor work is decreased. Therefore, the th e ultimate result of decreasing the ambient temperature is an increase in the COP of the t he system. Figure 7-5 shows how evaporator capacity capacity varie variess with ambient temperature. te mperature. For the reasons mentioned mentioned above, abo ve, the figure shows that as the ambient temperature decreases, the evaporator capacity increases. Unfortunately, this this trend is the opposite opposite of the trend in in the residential cooling requirements, which which increase with ambient temperature.
104
5.50 5.00 4.50 Locus of Optimums
4.00 3.50
P O C
3.00 Tambient=67 F
2.50
Tambient=82 F
2.00
Seasonal Tambient=97 F
1.50
Tambient=102 Tambient=102 F
1.00 0
5
10
15
20
25
Degrees of Sub-Cool at 95 F Ambient Temperature (F)
Figure 7-4: Effect of Ambient Temperature on COP for for Varying Degrees Sub-Cool at 95° F Ambient Temperature Temperature with an Air Velocity Over the Condenser of 8.5 ft/s
105
5.00
4.50
4.00
P O3.50 C Tsubcool at 95 F =10 Tsubcool at 95 F =15
3.00
Tsubcool at 95 F =5 Tsubcool at 95 F = 20
2.50
2.00 60
65
70
75
80
85
90
95
100
105
Ambient Temperature (F)
Figure 7-5: Effect of Ambient Temperature Temperature on the Evaporator Capacity for Varying Degrees Sub-Cool at 95° F Ambient Temperature with at Optimum Air Velocity
106
Figure 7-6 shows the effect that th at the refrigerant refrigerant charge (sub-cool at 95 ° F ambient temperature) has on the COP at various various ambient temperatures te mperatures at optimum opt imum air velocity over the condenser. cond enser. According to the figure, as the ambient temperature temperature decreases the optimum sub-cool at 95° F increases. increases. As discussed in Chapter VI, the sub-cool sub-cool is specified at 95° F. The resultant mass of of refrigerant in the system system (refrigerant charge) that yields 30,000 Btu/hr of cooling capacity at a t 95° F is determined, and the mass inventory at 95° F dictates the sub-cool at other ambient ambient temperatures. As noted not ed earlier, earlier, as the ambient temperature decreases, the condensing temperature also decreases, and the enthalpy of the th e refri refrigerant gerant entering ente ring the evaporator is reduced. As a result, the inle quality is also lower and more of the refrigerant in the evaporator exists in the liquid state. The total mass of refrigerant in the entire system system is constant. Hence, as the ambient temperature decreases, the mass of refrigerant in the evaporator evapor ator increases and the mass of refrigerant in in the condenser decreases. When the mass of refrigerant in the condenser cond enser decreases, the volume volume of the condenser cond enser that contains low density density refrigerant vapor increases and the volume of refrigerant in the condenser that contains higher higher density sub-cooled liquid decreases, causing an overall decrease in the mass of the syste (refrigerant charge). Thus it is possible for the mass of refrigerant in the condenser to drop to very low low levels such that complete condensati c ondensation on does not occur. In these instances where the refrigerant is not completely condensed when it exits the condenser
107
33500 33000
) r h 32500 / u t 32000 B ( y t i 31500 c a p 31000 a C r o 30500 t a r o 30000 p a v E29500
Tsubcool at 95 F = 20 Tsubcool at 95 F = 15 Tsubcool at 95 F = 10
29000 28500 65
70
75
80
85
90
95
10
10
Ambient Temperature (F)
Figure 7-6: Evaporator Capacity vs. Ambient Temperature for Various Sub-Cool conditions at 95° F Ambient Temperature and Optimum Air Velocity
108
and enters the expansion valve, the valve goes to its wide open position, and a fixed superheat cannot be maintained. This reduces the COP of the system. As a result, more sub-cool at 95 °F is needed to maintain some sub-cool a t the lower ambient temperatures (i.e. as the ambient temperature decreases, the degrees of sub-cool also decrease).
Effects on the Seasonal COP
Figure 7-4 also shows that the seasonal COP is nearly identical to the COP that exist at 82° F ambient temperature for a vast range of sub-cool conditions. This is due to the relatively large seasonal weighting assigned to the 82 ° F ambient temperature. Ambien temperatures at 82 ° F and below constitute more than 82% of the seasonal COP weightings. Thus the performance of the system at these ambient temperatures greatly influence the seasonal performance of the system. Figure 7-7 shows the effect of air velocity on the seasonal COP at varying sub-cool conditions. As the figure shows, the COP varies quadratically with the air velocity for any sub-cool condition. For sub-cools ranging from 5 ° F to 30 ° F, the maximum seasonal COP occurs at an air velocity between 7.5 ft/s and 10.0 ft/s. This figure also shows that the maximum seasonal COP occurs at a sub-cool between 10° F and 15 ° F, while the minimum seasonal COP occurs at a sub-cool of 20° F.
109
4.05 4.00 3.95 3.90
P3.85 O C3.80 l a n o3.75 s a e S3.70 Tsub-cool at 95 F = 15
3.65
Tsub-cool at 95 F = 10 Tsub-cool at 95 F = 5
3.60
Tsub-cool at 95 F = 20
3.55 3.50 4
6
8
10
12
14
16
Air Velocity Over Condenser (ft/s)
Figure 7-7: Effect of Air Velocity on the Seasonal COP for Varying Sub-cool Conditions
110
Range of Optimum Operating Parameter
Based on the results discussed in this chapter, it is clear that there is a range of operating parameters that yield the optimum performance for the base configuration system. It is determined that systems with between 10 ° F and 16° F degrees sub-cool in the condenser and air flowing over the condenser with velocities ranging from 6 ft/s and 12 ft/s will yield the optimum seasonal COP for the base configuration investigated in this study.
Effect of Operating Parameters on System Cost
As the sub-cool and the air velocity over the condenser are varied for a fixed condenser geometric configuration, the cost of the entire system is affected. This is because the size and cost of the condenser fan are also assumed to vary with changes in the operating conditi ons. This varying condenser fan and compressor equipment cos analysis is beyond the scope of this study, however the variation of these costs is not expected to be large. Therefore, only the condenser cost of materials is considered in this study. However, the designer should be aware of the effects of these factors on syste costs.
111
CHAPTER VIII
OPTIMIZATION OF GEOMETRIC DESIGN PARAMETERS FOR FIXED CONDENSER COIL COST
The two most pertinent constraints on condenser design are its costs and space requirements (frontal area). It is not possible to maintain a fixed condenser frontal area and a fixed condenser cost while varying only one geometric design parameter. Yet, it i very difficult to isolate the effects of individual geometric design parameters while simultaneously varying more than one. The condenser frontal area is the dominant geometric design variable, since it determines the volume of the entire system. Hence, for this study, two distinct investigations of the condenser geometric design effects are considered: (1) effects of geometric design changes with fixed condenser cost, and (2) effects of geometric design changes with fixed condenser frontal area. Each, geometric design parameter is isolated and varied while the others are maintained at the values of the base configuration. After an analysis of these results, the geometric parameters having the greatest effect on the COP are varied simultaneously in the appropriate combinations to yield a more nearly absolute optimum configuration. In this chapter, the
112
cost of the condenser is fixed while the condenser frontal area is allowed to vary for each of the configurations investigated.
Area Factor and Cost Facto
In order to compare the frontal area of each condenser configuration investigated, an area factor, defined as the ratio of the frontal are a of the test configuration to that of the base configuration (detailed in Chapter VII) is given by the following. .
AF =
Frontal Area Frontal Areabase
(8-1)
To compare the relative cost of each condenser and evaporator configuration a cost factor, defined as the ratio of the cost of the test configuration to that of the base configuration (detailed in Chapter VII) is given by the following.
CF =
Cost Cost base
113
(8-2)
The cost of the heat exchanger is determined primarily by the cost of materials. Hence the cost of each heat exchanger configuration is defined as:
Cost = (VolCu ,cond + VolCu , evap ) ρ CuCost Cu
+ (Vol Al ,cond + Vol Al ,evap ) ρ Al Cost Al
(8-3)
where Vol is the volume of the component, ρx is the density of the x material, and Cost x is the cost per lbm of the x material. The costs of the heat exchanger materials per lbm are summarized in Table 8-1.
Table 8-1: Material Costs (London Metals Exchange, 1999)
Material
Cost ($/lbm)
Copper Aluminum
0.8 0.7
The material cost of the base condenser configuration is $26.00. The optimum compressor piston displacement, and thus the compressor size, will change with each condenser configuration. For the vast majority of reasonable operating conditions, the
114
compressor piston displacement varies 3% from the optimum configuration to the base case configuration. Therefore, the cost of the compressor will not be considered for this investigation.
Varying Number of Rows of Condenser Tubes
The number of rows of condenser tubing, which dictates the condenser coil depth, is the first geometric design parameter studied. For this investigation, the height of the condenser remained constant while the width of the condenser was free to vary. The number of tubes per circuit, the fin spacing, the tube diameter, and the tube spacing were fixed to the values of the base configuration. Figure 8-1 shows the effect of the number of rows of condenser tubing on the optimum seasonal COP at the optimum air velocity over the condenser and varying degrees sub-cool at
° F ambient temperature.
One would expect that a heat exchanger with only one long row of tubes and no tube bends, providing the largest heat exchanger frontal area possible, would yield the best performance. This prediction is verified by Figure 8-1, which shows that as the number of rows of tubes decreases, the seasonal COP increases. This is because decreasing the number of rows of tubing also decreases the number of tube bends. Hence the frictional losses in the tubes and the required compressor work are also reduced, increasing the seasonal COP. The difference between the temperature of the refrigerant flowing inside the condenser tubes and the temperature of the air flowing over the condenser tubes is
115
4.30 4.25 4.20 4.15
P O4.10 C l a n 4.05 o s a e 4.00 S 3.95 15 degreees sub-cool at 95 F
3.90 10 degreees sub-cool at 95 F
3.85
20 degreees sub-cool at 95 F
3.80 0
1
2
3
4
5
Number of Rows of Condenser Tubes
Figure 8-1: Effect of Number of Rows on the Seasonal COP at Optimum Air Velocity and Varying Sub-Cool for Fixed Cost of Condenser Materials
116
also maximized by using only one row of tubes. This also results in a decrease in compressor power, further contributing to increasing the COP. This reduction in compressor power and refrigerant-side pressure drop is shown in Figure 8-2. While decreasing the number of rows produces an increase in the seasonal COP, it also causes an increase in the frontal area of the condenser. Figure 8-3 shows the effec of the number of rows of tubes on the frontal area. As the number of rows is decreased from 4 to 1, where the seasonal COP is the maximum, the frontal area of the condenser nearly quadruples from approximately 5.9 ft2 to 23.2 ft2. A condenser that has a frontal area of 23.2 ft2 is generally not feasible in most residential air-conditioning applications. Therefore, when determining the number of rows of tubes, one must make a tradeoff between space constraints and optimum performance when the c ost of the configuration is fixed. Although the main cause of the increased seasonal COP with decreased number of tube rows (decreased coil depth) is the decrease in comp ressor power, there is also a decrease in condenser fan power with a decreased number of tube rows. Figure 8-4 shows the effect of the number of tube rows on the condenser fan power and the air-side pressure drop. The figure shows that the air-side pressure drop also decreases as the number of tube rows decreases. In fact, the decrease in the condenser fan power is due to the reduction in air-side pressure drop, which results from a decrease in the depth of the air passage produced by using fewer tube rows.
117
6600
20
6500
5800
19 R e f r i g 18 e r a n 17 t S i d e 16 P r e s 15 s u r e D 14 r o p ( 13 p s i a ) 12
5700
11
Compressor Power
) r 6400 h / u t 6300 B ( r e w6200 o P r o6100 s s e r p6000 m o C5900
Refrigerant Side Pressure Drop
0
1
2
3
4
5
Number of Rows of Condenser Tubes
Figure 8-2: Effect of Number of Rows on Compressor Power and Refrigerant Pressure Drop at Optimum Sub-Cool and Air Velocity for Fixed Condense Material Cost at 82 ° F Ambient Temperature
118
25
20
) f ( a e r A15 l a t n o r F r e s 10 n e d n o C
2 t
5
0 0
1
2
3
4
5
Number of Rows of Condenser Tubes
Figure 8-3: Effect of Number of Rows of Tubes on Condenser Frontal Area fo Fixed Condenser Material Cost at Optimum Sub-Cool and Air Velocity
119
350
0.007
300
0.006
) r h / u250 t B ( r e w200 o P n a F150 r e s n e d100 n o C
0.005
0.004
0.003
0.002 Condenser Fan Power
50
A i r S i d e P r e s s u r e D r o p ( p s i a )
0.001
Air Side Pressure Drop
0
0 0
1
2
3
4
5
Number of Rows of Condenser Tubes
Figure 8-4: Effect of Number of Rows of Tubes on Condenser Fan Power and Airside Pressure Drop for Fixed Condenser Material Cost at 82° F Ambient Temperature at Optimum Sub-Cool and Air Velocity
120
The number of rows of condenser tubes also effects the optimum operating parameters such as the air velocity over the condenser. Figure 8-5 displays the effect o air velocity over the condenser on the optimum seasonal COP for varying number o rows. The figure shows that as the number of rows increases, the optimum air velocit increases. For example, for a condenser configuration utilizing only 1 row of tubes, the optimum seasonal COP occurs at an air velocity of approximately 7.0 ft/s. However for a deeper condenser configuration utilizing 4 rows of tubes, the optimum seasonal COP occurs at an air velocity of approximately 9.0 ft/s. The increase in optimal air velocity coupled with the increase in air-side pressure drop shown in Figure 8-4 causes the fan power to more than double as the number of rows is increased from 1 to 4. Figure 8-6 shows the effect of the number of rows on the optimum air velocity and optimum volumetric flow rate of air over the condenser. This figure shows that while the optimum air velocity increases as the number of rows increases, the optimum volumetric flow rate of air over the condenser decreases.
121
4.35 4.30 4.25 4.20
P4.15 O C4.10 l a n o 4.05 s a e S4.00
Locus of Optimums
3.95 1 row
3.90
2 rows 3 rows
3.85
4 rows
3.80 5
6
7
8
9
10
11
12
13
14
Air Velocity Over Condenser (ft/s)
Figure 8-5: Effect of Air Velocity on Seasonal COP for Varying Number of Rows at Optimum Sub-Cool for Fixed Condenser Material Cost
122
10000
) e i n t a m / R 3 t w f ( o l r F e r s i n A e d m n u o m C i t r p e v O O
10
9000
9
8000
8
7000
7
6000
6
5000
5
4000
4
3000
3
2000
Optimum Air Flow Rate
2
1000
Optimum Air Velocity Over Condenser
1
0
O p t i m u m A i r V e l o ( f c t / i s t ) y O v e r C o n d e n s e r
0 0
1
2
3
4
5
Number of Rows of Condenser Tubes
Figure 8-6: Effect of Number of Rows on the Optimum Air Velocity and Volumetric Flow Rate of Air Over the Condenser at Optimum Sub-Cool for Fixed Condenser Material Cost
123
Varying Condenser Tube Circuiting
Another condenser geometric design parameter that has an effect on system performance is the tube circuiting. Varying the number of condenser tubes per circui does not affect either the cost factor or the configuration or the frontal area of the condenser. For this investigation, the number of rows, the tube diameter, the tube spacing, and fin spacing were fixed to the values used for the base configuration. While varying the number of tubes per circuit, the number of circuits was also varied in order to maintain a nearly constant height to width ratio of approximately 0.83. The refrigeran flow circuit configurations investigated for this study are summarized in Table 8-2. Each configuration was tested for air velocities ranging from 6 ft/s to 13 ft/s and sub-cools ranging from
° F to 20° F at 95° F ambient temperature.
The maximum seasonal COP
for every configuration tested occurs within this selected range of operating conditions.
Table 8-2: Condenser Circuiting Configurations
Number of
Condenser Width
Circuits
(ft)
2
12
3.0
3
8
3.0
4
6
3.0
5
5
2.9
Tubes/Circuit
124
Figure 8-7 shows the effect of the number of tubes per circuit on the optimum seasonal COP based on the optimum operating conditions for each configuration. The figure shows that the maximum seasonal COP occurs when the refrigerant flow is divided among 3 tubes. However, the seasonal COP for the optimal configuration is only approximately 2.0 % greater than that of the base configuration (2 tubes per circuit), 0.2 % greater than a configuration utilizing 4 tubes per circuit, and 0.6 % greater than a configuration with five tubes per circuit. Hence, in the range of optimum operating conditions, the seasonal COP is relatively insensitive to variation s in the number of tubes per circuit. The improved seasonal COP that occurs when the tubes per circuit increases from 2 to 3 results from the decrease in refrigerant pressure drop which tends to reduce the required compressor power. The decrease in pressure drop occurs because as the number of tubes per circuit increases, the mass flow of refrigerant through each individual tube decreases. This decrease in the amount of mass flowing in each tube leads to a decrease in the pressure drop through each tube. Figure 8-8 shows how the refrigerant-side pressure drop varies with changes in the number of tubes per circuit at an ambient temperature of 82° F for the optimum operating conditi ons for each configuration. As the figure shows, the refrigerant-side pressure drop does indeed decrease with an increased number of tubes per circuit. However, increasing the number of tubes per circuit also causes the refrigerant-side heat transfer coefficient to decrease, which has a negative effect on the seasonal COP. Therefore, two competing effects are at work. At a certain point, the decrease in the refrigerant-side heat transfer coefficient that results fro
125
4.16 4.15 4.14
P4.13 O C l a 4.12 n o s a e S4.11 4.10 4.09 4.08 1
2
3
4
5
6
Number of Condenser Tubes per Circuit
Figure 8-7: Seasonal COP vs. Varying Condenser Tube Circuiting at Optimum Sub-Cool and Air Velocity for Fixed Condenser Material Cost
126
18 16
) a i s p 14 ( p o r D12 e r u s 10 s e r P e 8 d i S - 6 t n a r e g 4 i r f e R
Total Straight Pipe Bends
2 0 1
2
3
4
5
6
Number of Condenser Tubes per Circuit
Figure 8-8: Refrigerant-Side Pressure Drop for Various Circuiting at 82° F Ambient Temperature and at Optimum Sub-Cool and Air Velocity for Fixed Condenser Material Cost
127
increasing the number of tubes per circuit has a larger effect than the resulting decrease in the refrigerant-side pressure drop. As a result, the seasonal COP begins to decrease as the number of tubes per circuit increases. According to Figure 8-7, this point occurs when the number of tubes per circuit is increased from 3 to 4. While the total refrigerant-side pressure drop decreases with an increase in the number of tubes per circuit, the percentage of the total pressure drop due to tube bends actually increases considerably. This is because the actual number of bends is increased by increasing the number of parallel flow passages. The refrigerant-side pressure drop distribution between the straight tube and the tube bends at an ambient temperature of 82° F for various condenser tube circuit configurations is shown in Table 8-3.
Table 8-3: Refrigerant Pressure Drop Distributions at 82° F Ambient Temperature
Tubes per Circuit
Bend Pressure Drop (psia)
Straight Pipe Pressure Drop (psia)
2
5.40
11.2
16.6
32.5 %
3
1.70
3.32
5.02
33.9 %
4
0.77
1.31
2.08
37.0 %
5
0.41
0.44
0.85
48.2 %
128
Total Pressure Drop (psia)
% of Total Pressure Drop Due To Bends
Varying Fin Pitch
The condenser fin pitch is another geometric design parameter considered for this study. To investigate the effect of condenser tube fin pitch on system performance for a fixed heat exchanger cost factor, the tube size, tube spacing, circuiting, and number o rows were fixed to the values of the base configuration. For this study, the syste performance was calculated for fin pitches ranging from 8 fins per inch to 14 fins per inch (fpi). With the cost of the condenser materials fixed, varying the fin pitch involves a compromise between purchasing more aluminum fins versus purchasing more copper tubing. Figure 8-9 shows the variation of seasonal COP with air velocity for various fin pitch values at optimal sub-cool conditions for each configuration (15° sub-cool at 95° F for every case). As the figure shows, the optimum velocity for every fin pitch configuration occurs between 8 ft/s and 9 ft/s. Thus, according to these results, the fin spacing has very little affect on the seasonal COP or the optimal operating conditions. The optimu seasonal COPs and area factors for varying fin pitch at fixed heat exchanger cost ar shown in Table 8-4. Figure 8-10 shows a graphical demonstration of the effect of the fin pitch on the optimum seasonal COP that is documented in Table 8-4. Table 8-4 and Figure 8-10 also show that as the number of fins per inch increases from 8 fins per inch to 10 fins per inch, the seasonal COP increases slightly from 4.10 to 4.11. However, as the fin pitch increases from 10 fins per inch to 14 fins per inch, the seasonal COP decreases steadily from 4.11 to 4.08. It might be expected that increasing
129
4.15 4.10 4.05 4.00
P O C3.95 l a n o s 3.90 a e S
8 fins per inch
3.85
10 fins per inch
3.80
12 fins per inch 14 fins per inch
3.75 3.70 4
5
6
7
8
9
10
11
12
13
14
15
16
Air Velocity Over Condenser (ft/s)
Figure 8-9: Seasonal COP vs. Air Velocity for Varying Fin Pitch at Fixed Condenser Material Cost and Optimum Sub-Cool
Table 8-4: Seasonal COP and Area Factors for Varying Fin Pitch at Optimum Air Velocity and Sub-Cool for Fixed Condenser Material Cost
Fin Pitch (fpi)
Optimum Seasonal COP
Area Factor
8
4.10
1.30
10
4.11
1.15
12
4.09
1.00
14
4.08
0.93
130
4.12
4.11
4.10
P O C l a n 4.09 o s a e S 4.08
4.07
4.06 6
8
10
12
14
16
Fin Pitch (fins/inch)
Figure 8-10: Effect of Fin Pitch on the Seasonal COP at Optimum Sub-Cool and Air Velocity Over the Condenser for Fixed Condenser Material Cost
131
the fins per inch should decrease the power requirements of the compressor and thus increase the seasonal COP. However, as the fin spacing becomes smaller, the air-side pressure drop also increases thus increasing the required power for the condenser fan. This phenomenon is displayed in Figure 8-11, which shows the air-side pressure drop versus the fin pitch at optimal operating conditions. At a certain plateau, the fin spacing becomes too small and produces a pressure drop so large that the resultant increase in condenser fan power is more than the decrease in the compressor power requirement. Hence, the seasonal COP is lower. Figure 8-12 shows how the power requirements of the condenser fan and the compressor vary with the fin pitch at the maximum seasonal COP. The figure shows that, as expected, the condenser fan power requirement increases with increasing fin pitch. Figure 8-12 also shows that as the fin pitch increases from eight fins per inch to ten fins per inch, the compressor power decreases from 7440 Btu/hr to approximatel 7400 Btu/hr at the maximum seasonal COP of each configuration. However, this figure appears to contradict the theoretical prediction of decreased compressor power with increased fin pitch since as the fin pitch increases from ten fins per inch to fourteen fins per inch, the compressor power increases from approximately 7400 Btu/hr to 7420 Btu/hr. While this increase is a very small percentage of the total power requirement, it is still surprising given the theoretical prediction. One possible explanation for this result can be found in an analysis of the optimal operating conditions yielding the maximu seasonal COP. The optimum seasonal COP occurs at a slightly different air velocity for each configuration. The compressor power requirement steadily decreases with increased
132
0.0050
0.0045
) a i s p ( p0.0040 o r D e r 0.0035 u s s e r P e 0.0030 d i s r i A 0.0025
0.0020 7
8
9
10
11
12
13
14
15
Fin Pitch (fins/inch)
Figure 8-11: Air-side Pressure Drop vs. Fin Pitch for Fixed Condenser Material Cost at Optimum Sub-Cool and Air Velocity at 95° F Ambient Temperature
133
7500
350
7450
7250
300 C o n d e n 250 s e r F a n 200 P o w e r 150 ( B t u / h r ) 100
7200
50
) r h / u t B ( 7400 r e w o P7350 r o s s e r 7300 p m o C
Compressor Power Condenser Fan Power
7
8
9
10
11
12
13
14
15
Fin Pitch (fins/inch)
Figure 8-12: Power Requirements vs. Fin Pitch for Fixed Cost at Optimum SubCool and Air Velocity and 95 ° F Ambient Temperature
134
air velocity over the condenser while conversely the condenser fan work steadily increases with increased air velocity. The maximum COP for each fin pitch configuration occurs where the combined power requirement for the condenser fan and the compressor is at a minimum. For fixed cost of condenser materials, as the number o fins per inch increases, the air-side heat transfer area increases thus causing a decrease in the compressor power and an increase in the seasonal COP. However increasing the fin pitch also reduces the refrigerant-side heat transfer area, since for fixed cost, the fronta area decreases with increasing fin pitch. Therefore, two competing effects are at work. When the fin pitch is increased from 8 to 10, the effect of the increase in the air-side hea transfer area is larger than the effect of the decrease in the refrigerant-side heat transfer area. Therefore, the compressor power is decreased, producing an increase in the seasonal COP. However when the fin pitch is further increased from 10 to 12, the effec of the reduction in the refrigerant-side heat transfer area is larger than the effect of the increase in the air-side heat transfer area. Hence, the compressor power begins to increase, thus causing the seasonal COP to decrease. While the fin pitch has very little effect on the seasonal COP it does affect another important aspect of heat exchanger design, the frontal area. Figure 8-13 shows the effec of the fin pitch on the condenser frontal area. The figure shows that as the fin pitch is increased, the frontal area decreases. This is due to the fixed material cost constrain requiring less tubing with increasing fin pitch. This trend can also be seen in Table 8-4, which shows that increasing the fin pitch causes a decrease in the area factor. Thus, if the
135
10.00 9.50
) f ( 9.00 a e r A 8.50 l a t n o r 8.00 F r e s 7.50 n e d n o 7.00 C
2 t
6.50 6.00 6
8
10
12
14
16
Fin Pitch (fins/inch)
Figure 8-13: Effect of Fin Pitch on Condenser Frontal Area at Optimum Sub-Cool and Air Velocity for Fixed Condenser Material Cost
136
designer’s primary goal is for a more compact heat exchanger, a larger fin pitch should be utilized. Again, the fin pitch has very little effect on the optimal operating conditions and the seasonal COP; thus using the maximum fin pitch would create a compact he a exchanger without significantly sacrificing performance.
Varying Tube Diameter
Yet another geometric design parameter studied in this work is the condenser tube diameter. The tube sizes considered for this study are taken from the AAON Heating and Refrigeration Products specifications (www.aaon.co . AAON Heating and AirConditioning Products web site). The dimensions of the tubes investigated are summarized in Table 8-5. For this investigation, the number of rows, number of tubes per circuit and number of fins per inch were all maintained at the values used in the base configuration. Figure 8-14 shows how the optimum seasonal COP is affected by the tube diameter. For all sub-cool conditions in the recommended range of 10° F to 20 ° F at 95° F ambient temperature, utilizing tubes of 5/8” outer diameter yields unreasonably low condensing temperatures inside the tubes for the resultant frontal area at the given fixed hea exchanger cost.
137
Table 8-5: Condenser Tube Dimensions (www.aaon.com. AAOP Heating and AirConditioning Products web site)
Outside Diameter (in.)
Inside Diameter (in.)
Wall Thickness (in.)
0.3125
0.3005
0.0120
0.3750
0.3630
0.0120
0.5000
0.4840
0.0160
0.6250
0.6170
0.0180
4.12
4.08
4.04
P O C l a 4.00 n o s a e S 3.96
3.92
3.88 1/4
5/16
3/8
7/16
1/2
9/16
Outer Tube Diameter (in)
Figure 8-14: Optimum Seasonal COP for Varying Tube Diameter at Optimum SubCool and Air Velocity for Fixed Condenser Material Cost
138
Tubes of this size also greatly deteriorate the system performance, and thus are not considered in this discussion of the effect of tube size at fixed heat exchanger cost. As Figure 8-14 shows, the optimum seasonal COP occurs with a tube diameter of 3/8”. The optimum seasonal COP of 4.09 is exactly equal to the optimum value for the base configuration. While varying the tube circuiting and the fin spacing has very little effect on the optimal operating conditions, varying the tube diameter does indeed have a significan effect. Figure 8-15 shows the effect of the tube diameter on the optimum air velocity over the condenser and the optimum sub-cool conditions. As the figure shows, the optimum air velocity increases continuously with tube diameter. However the optimu sub-cool has a distinct minimum which exists at a tube size of 3/8”. The optimu seasonal COP, area factor, and operating con ditions for each tube size in vestigated are shown in Table 8-6. The decreasing frontal area with increasing tube diameter is a result of the fixed condenser material cost constraint. As both Table 8-6 and Figure 8-15 demonstrate, the optimum air velocity varies with changes in tube diameter. The length of condenser tubing allocated to the superheated, saturated, and subcooled portions of the condenser is also affected by the tube diameter, as shown in Figure 8-16. The figure shows that as the tube diameter increases from 5/16” to 5/8”, the condenser allocation for the superheated and the saturated portions of the condenser tube increases steadily while that o f the sub-cooled portion decreases steadily. The portion o
139
16
11
15
10 O p t i m u m 9 A i r V e l o c i 8 t y ( f t / s )
) F ( l o o 14 C b u S m u 13 m i t p O
Subcool Air Velocity
12
7
11
6 9/16
1/4
5/16
3/8
7/16
1/2
Outer Tube Diameter (in)
Figure 8-15: Optimum Operating Parameters for Varying Tube Diameters at Fixed Condenser Material Cost
140
Table 8-6: Optimum Seasonal COPs and Area Factors for Varying Tube Diameters
Outer Tube Diameter (in)
Optimum Seasonal COP
Optimum Air Velocity (ft/s)
Optimum Degrees Sub-cool (F)
Area Factor
5/16”
3.91
8.0
15
1.13
3/8”
4.09
8.5
15
1.00
1/2”
3.99
10.0
15
0.82
0.8 0.7 0.6
n o i t a 0.5 c o l l A r 0.4 e s n e d 0.3 n o C
Saturated Subcooled Superheated
0.2 0.1 0 1/4
5/16
3/8
7/16
1/2
9/16
Tube Outer Diameter (in)
Figure 8-16: Condenser Tube Length Allocation for Varying Tube Diameters at Optimum Air Velocity and Sub-Cool and 82 ° F Ambient Temperature for Fixed Condenser Material Cost
141
the condenser allocated to the sub-cooled and superhe ated portions is nearly identical a the optimum tube diameter of 3/8”. The amount of tube length allocated to each portion of the condenser will have an effect on t he refrigerant pressure drop, which in turn affects the compressor power required. Figure 8-17 shows the effect of the tube diameter on the refrigerant-side pressure drop at optimum operating conditions and 82° F ambient temperature. As the figure shows, the refrigerant-side pressure drop decreases as the tube diameter increases. Figure 8-18 shows the effect of the tube diameter on the power required for the condenser fan and the compressor for the optimum seasonal COP at each tube diameter. As the figure shows, the compressor power required is a minimum at the optimum tube diameter 3/8”. Again, the optimum seasonal COP occurs where the total power required by the condenser fan and th e compressor is at a minimum. Just as with the fin spacing, the minimum power required varies as the tube diameter varies. While the total power required at the optimum steadily decreases with increased tube diameter, the required compressor power reaches a minimum at a tube diameter of 3/8” and then increases when the tube diameter increases to 1/2”.
142
45
a 40 i s p ( p35 o r D 30 e r u s s 25 e r P e 20 d i S t n15 a r e g10 i r f e R 5
Total saturated superheated subcooled
0 1/4
5/16
3/8
7/16
1/2
9/16
Tube Outer Diameter (in)
Figure 8-17: Effect of Tube Diameter on Pressure Drop at Optimum Sub-Cool and Air Velocity at 82 ° F Ambient Temperature for Fixed Condenser Material Cost
143
8000
800
7900
700
r e w 7800 o P r o s 7700 s e ) r r p h m / u 7600 o t C B ( & 7500 r e w o P 7400 l a t o T 7300
600 500 400 300 Total Power
200
Compressor Power Condenser Fan Power
7200
C o n d e n s e r F a n P o w e r ( B t u / h r )
100 0
1/4
5/16
3/8
7/16
1/2
9/16
Tube Outer Diameter (in)
Figure 8-18: Power Requirements for the Condenser Fan and the Compressor vs. Tube Diameter at Optimum Air Velocity and Sub-Cool for Fixed Condenser Material Cost and 82 ° F Ambient Temperature
144
Operating Costs
The operating costs for the air-conditioning system are inversely proportional to the seasonal COP (1/COP
∝ operating cost).
In this study, an operating cost factor is defined
as: 1/COP = operating cost factor. Figure 8-19 shows how the operating cost factor varies with the area factor for all of the geometric parameters investigated for this study. According to the figure, decreasing the number of rows from the base configu ration value of 3 rows to 1 row produces the largest decrease in operating costs. In fact, the lowes operating cost occurs when 1 row of tubing is used. However, the frontal area for this configuration is more than 3 times that of the base configuration. The frontal area for the 2-row configuration is 50% greater than that of the base configuration. Hence configurations utilizing 2 and 3 rows of tubing generally may not be feasible at a fixed condenser material cost factor when space constraints are of concern. In this space constrained situation, the configuration using 3 rows of tubing will yield the best performance and lowest operating cost with the most reasonable frontal area. Once configurations of 1, 2, and 4 rows of tubes are eliminated, Figure 8-19 shows that the two geometric parameters having the most significant effect on the operating cos of the complete air-conditioning system are the tube diameter and the number of tubes per circuit. The figure shows that when only 2 tubes per circuit are used, as in the base configuration, the optimum tube diameter is 3/8” with fixed heat exchanger cost. However, the figure also shows that for a tube diameter of 3/8”, and 3 rows of tubing, the lowest operating cost occurs for a condenser configuration utilizing 3 tubes per circuit. The initial investigations outlined throughout this chapter did not test the effect of tube
145
0.258 5/16"
Base configuration: 12 Fins Per Inch (FPI)
0.254
) r o t c0.250 a F t s o C0.246 g n i t a r 0.242 e p O ( P O0.238 C / 1
2 Tubes per Circuit (TPC)
Tube Diameter
1/2"
3 Rows
4 rows
3/8" Diameter
14 FPI Base Case
10 FPI
Tube Diameter
8 FPI
5 TPC
Fin Pitch
4 TPC 3 TPC
Number of Rows Tubes per Circui 2 rows
0.234 1 row
0.230 0
0.5
1
1.5
2
2.5
3
3.5
Area Factor
Figure 8-19: Operating Costs vs. Area Factor For Various Geometric Parameter at Optimum Sub-Cool and Air Velocity with Fixed Condenser Material Cost
146
diameter on the system performance when tube circuiting other than the base configuration of 2 tubes per circuit is utilized. From an analysis of the figure, it is obvious that an examination of this effect is warranted. While Figure 8-19 shows that for fixed heat exchanger cost, a configuration with a 5/16” tube diameter yields the highest operating cost and the worst performance, this tube diameter was only investigated for the base case configuration of 2 tubes per circuit. When the number of tubes per circuit is increased, the amount of mass of refrigerant flowing through each individual tube is decreased. Therefore, tubes of smaller diameter can be utilized without degrading system performance. Employing a smaller diameter tube does not greatly increase the frontal area with fixed condenser material cost since the resultant area factor is only 1.13. Furthermore, increasing the number of tubes per circuit has no affect on the frontal area. As a result of the above analysis, the effect of the number of tubes per circuit on the system performance was investigated, for a configuration utilizing a tube diameter of 5/16”, 3 rows of tubes, and 12 fins per inch. Although Figure 8-10 shows that the optimum fin pitch is 10 fins per inch for the base case configuration, Figure 8-19 shows that the fin pitch has virtually no effect on the optimum system operating cost and system performance. In fact, both Figure 8-10 and Figure 8-19 show that there is very littl difference in the optimum seasonal COP (minimum operating cost) for a range of 8 fins per inch to 12 fins per inch. Therefore since there is very little difference in the operating cost for the varying fin pitch, a configuration employing 12 fins per inch was used in this supplemental investigation of the effect of tubes circuiting with tubes of 3/8 ” and 5/16”
147
diameter. A configuration of 12 fins per inch will yield an area factor of unity while stil providing a near optimum seasonal CO P (lowest operating cost). Figure 8-20 shows the effect of the number of tubes per circuit on the optimum seasonal COP for a condenser configuration with a tube diameter of 5/16” with a fixed cost factor. The figure shows that for a tube diameter of 5/16”, as the number of tubes per circuit increases from 2 to 4, the optimum seasonal COP increases by approximately 8% from approximately 3.91 to 4.22. As the number of tubes per circuit increases from 4 to 5, the optimum seasonal COP increases from 4.22 to a maximum of 4.23. The optimum seasonal COP then decreases to 4.21 when the number of tubes per circuit increases from 5 to 6. The explanations for this trend are the same as for the trends discussed earlier in this chapter under the section entitled “Varying Condenser Tube Circuiting”. As discussed in that section, the improved seasonal COP that occurs when the tubes per circuit increases from 2 to 5 results from the decrease in the refrigeran pressure drop having a larger effect on increasing the COP than the decrease in the refrigerant-side heat transfer has on decreasing the COP. Figure 8-21 shows the optimum seasonal COP versus the number of tubes per circuit for the both 3/8” tube diameter configuration (base configuration) and the 5/16” tube diameter configuration. As the figure shows, the optimum seasonal COPs achieved for condensers using a 5/16” diameter tube are higher than those with a 3/8” diameter tube. For a condenser with a tube diameter of 3/8”, the optimum seasonal COP occurs when 3 tubes per circuit is used. However, when the diameter is decreased to 5/16”, the optimu
148
4.25 4.20 4.15 P4.10 O C l a n4.05 o s a e S4.00
5/16" Tube Diameter 3 rows of tubes 12 fins per inch
3.95 3.90 3.85 1
2
3
4
5
6
7
Number of Condenser Tubes per Circuit
Figure 8-20: Seasonal COP at Optimum Sub-Cool and Air Velocity for Varying Condenser Tube Circuiting with Fixed Condenser Material Cost and 5/16” Tube Outer Diameter
149
4.25 4.20 4.15
P4.10 O C l a 4.05 n o s a e S4.00
5/16" Outer Tube Diameter 3/8" Outer Tube Diameter
3.95
Fixed Material Cost 3 rows of tubes 12 fins per inch
3.90 3.85 1
2
3
4
5
6
7
Number of Condenser Tubes per Circuit
Figure 8-21: Comparison of the Effect of the Number of Tubes per Circuit on Seasonal COP for 5/16” and 3/8” Outer Tube Diameters at Optimum Sub-Cool and Air Velocity with Fixed Condenser Material Cost
150
seasonal COP occurs when 5 tubes per circuit circuit are used. When the number of tubes per circuit is the value used for the base configuration (2 tubes per circuit), a tube diameter o f 3/8” yields a slightly higher optimum seasonal seasonal COP than a 5/16” diameter tube. Conversely, when the tubes per circuit are increased, configurations with a tube diameter of 5/16” yield the highest highest seasonal season al COP. The optimum optimum seasonal COP C OP for the 5/16” tube diameter configurat con figuratio ion n is 4.23, which is approximately approximately 2% greater than the optimum op timum seasonal COP COP for the t he 3/8” diameter tube configuration that has a value of 4.15. Therefore, when the cost factor of the heat he at exchanger configuration is fixed, a condenser with an outer oute r tube diameter diameter of 5/16”, 5 tubes per circuit, circuit, 3 rows of tubes, and 12 fins per inch yields the highest seasonal COP (lowest operating cost) with the most reasonable frontal area.
151
CHAPTER IX
OPTIMIZATION OF GEOMETRIC DESIGN PARAMETERS FOR FIXED F IXED CONDENSER FRONTAL AREA
The effects of of varying the number of rows, r ows, the number of tubes per circuit, the tube diameter, and the fin pitch while keeping the heat exchanger costs constant wer presented in the previous chapter. While producing changes in perfo perform rmance, ance, varying these parameters (with the exception of the tubes per circuit) also produces changes in in the frontal area area of the condenser since it is allowed allowed to vary freely. freely. However, as discussed earlier, the residential air-conditioning system system designer encounters space constraints tha prevent the use of a heat exchanger with a large large frontal area. In this chapter, the effects of varying the number of rows, rows, the tube diameter, and the fin pitch pitc h for fixed frontal frontal area with variable cost will be inves tigated.
152
Varying the Number of Rows of Condenser Tubes
Varying the depth of the coil by changing the number of rows of condenser tubing with fixed frontal area is the first geometric design parameter investigation investigation considered considered for this part of the study. The number of o f tubes per circuit, circuit, fin spacing, tube diameter, diameter, frontal fronta l area, and tube spacing are all fixed to the the values of the base configuration. configuration. Figure 9-1 shows the effect of the air velocity on the seasonal COP for varying numbers of rows rows with optimum sub-cool at 95° F ambient ambient temperature. According to the figure, for much much of the range of o f air velocities shown, the optimum seasonal COP occurs for configurations utilizing 3 rows of tubes. The figure also also shows that as the number of rows decreases, the optimum opti mum air velocity velocity increases. This trend trend is summarized in Table 9-1, 9-1, which shows shows the optimum opti mum operating conditi conditions ons for for each row configuration. Figure 9-2 shows the effect effect o the number of rows rows on the t he seasonal COP at optimum operating conditions. conditions. This figure reinforces the trends observed in Figure 9-1, and again shows that th e maximum maximum seasonal COP occurs when 3 rows of tubes are employed. As shown in the Table 9-1, the maximum maximum seasonal COP occurs occu rs when 3 rows of tubes are utilized with 15° F sub-cool at
velocity of 8.5 ° F ambient temperature and an air velocity
ft/s. The seasonal COP while showing a majo majorr increase when the number number of rows rows i increased from 2 rows to 3 rows, actually shows a slight decrease when the numbe r o rows is further increased from from 3 to 4. Continuing Continuing to increase increase the number of rows of tubes also further increases the heat transfer area. Hence, intuitively intuitively one might assume assume that the seasonal COP would also continue to increase. However as both Figure Figure 9-1 and Table 91 have shown this is not the case.
153
4.15 4.10 4.05 4.00
P O3.95 C l a n 3.90 o s a 3.85 e S
Locus of Optimums
3.80 3 rows
3.75
4 rows 2 rows
3.70 3.65 5
6
7
8
9
10
11
12
13
14
Air Velocity Over Condenser (ft/s)
Figure 9-1: Effect of Air Velocity Over Condenser for for Varying Numbers of Rows at Optimum Sub-Cool with Fixed Condenser Frontal Area
Table 9-1: Optimum Operating Conditions for for Varying Number of Rows with Fixed Condenser Frontal Area Area
Number of Rows
Seasonal COP
Cost Factor
Air Velocity (ft/s)
Degrees Sub-coo at 95° F (° F)
2
3 .9 8
0 .7 5
1 1. 0
13
3
4 .0 9
1 .0 0
8. 5
15
4
4 .0 7
1 .3 2
7. 0
13
154
4.10
4.08
4.06
P O C4.04 l a n o s 4.02 a e S 4.00
3.98
3.96 1
2
3
4
5
Number of Rows of Tubes
Figure 9-2: Effect of the Number of Rows of Tubes on the Seasonal COP at Optimum Sub-Cool and Air Velocity for Fixed Condenser Frontal Fro ntal Area
155
As the number of rows of tubes increases, the depth of the condenser increases and both the refrigerant-side and air-side heat transfer areas increase. However, increasing the number of rows also increases the refrigerant flow path, as well as the air flow path (deeper coil), thus increasing both the refrigerant-side and air-side pressure drops. The increase in the refrigerant-side pressure drop with increasing number of rows is shown in Figure 9-3. Therefore, two competing effects are at work. As the number of rows is increased from 2 to 3, the increase in the overall heat transfer area has a larger effect on the seasonal COP than the resultant increase in the in the pressure drop, hence the seasonal COP increases. Figure 9-4 displays the compressor and condenser fan power versus the number of rows, and shows that the compressor power decreases when the number of rows is increased from 2 to 3. Again, this is because the increase in the overall heat transfer area has a larger effect on the seasonal COP th an the increase in pressure drop. However, when the number of rows is increased from 3 to 4, the resultant increase in the pressure drop has a larger effect on the seasonal COP than the increase in the overall heat transfer area, thus the seasonal COP decreases. Figure 9-4 shows that as the number of rows is increased from 3 to 4, the compressor power actually increases, thus confirming the aforementioned trend.
156
25
) a i s p20 ( p o r D e r 15 u s s e r P e d i 10 S t n a r e g i r 5 f e R
Total Straight Tube Bends
0 1
2
3
4
5
Number of Rows of Tubes
Figure 9-3: Refrigerant-Side Pressure Drop vs. Number of Rows with Fixed Condenser Frontal Area for Optimum Sub-Cool and Air Velocity at 82° F Ambient Temperature
157
6000
350
5975
325
) r h / 5950 u t B ( r e 5925 w o P r o 5900 s s e r p m5875 o C
300
275
250
225 Compressor Power
5850
C o n d e n s e r F a n P o w e r ( B t u / h r )
200
Condenser Fan Power
5825
175 1
2
3
4
5
Number of Rows of Condenser Tubes
Figure 9-4: Compressor and Condenser Fan Power for Varying Number of Rows with Optimum Sub-Cool and Air Velocity at 82° F Ambient Temperature for Fixed Condenser Frontal Area
158
Varying Fin Pitch
The next geometric design parameter varied while fixing the condenser frontal area is the fin pitch. The frontal area, tube diameter, number of rows, number of tubes per circuit and the tube spacing are all fixed to the values of the base configuration. Figure 95 shows the effect of air velocity on the seasonal COP for varying fin pitch with optimu sub-cool at 95° F ambient temperature. As the figure shows, varying the fin pitch has a small affect on the optimum seasonal COP when keeping the frontal area of the condenser fixed (optimums range from 4.00 to 4.10). According to the figure, the recommended range of operation is between air velocities of 8.0 ft/s and 11.0 ft/s. The optimum air velocity increases from 8.0 ft/s to 10.5 ft/s as the fin pitch decreases from 14 fins per inch to 8 fins per inch. Figure 9-6 shows the effect of the fin pitch on the seasonal COP at optimum air velocity and sub-cool. The figure shows that the optimum seasonal COP increases as the number of fins per inch increases. However, the increase in the optimum seasonal COP is only approximately 2.5 % when the number of fins per inch increases form 8 to 14. Thus the fin pitch has only a small on the optimum seasonal COP when the frontal area of the condenser is fixed. Varying the fin pitch also has very little affect on the optimum sub-cool conditions. However, unlike the case in the previous chapter where the hea exchanger cost is fixed, varying the fin pitch does have a significant affect on the optimum air velocity over the condenser when the frontal area of the condenser is fixed.
159
4.20
4.10
4.00 Locus of Optimums
P O3.90 C l a n o s 3.80 a e S
14 fins per inch
3.70
12 fins per inch 10 fins per inch
3.60
8 fins per inch
3.50 4
6
8
10
12
14
16
Air Velocity Over Condenser (ft/s)
Figure 9-5: Effect of Air Velocity on Seasonal COP for Varying Fin Pitch with Optimum Sub-Cool for Fixed Condenser Frontal Area
160
4.12
4.10
4.08
P O C4.06 l a n o s 4.04 a e S 4.02
4.00
3.98 6
8
10
12
14
16
Fin Pitch (fins/inch)
Figure 9-6: Effect of Fin Pitch on the Seasonal COP at Optimum Sub-Cool and Air Velocity for Fixed Condenser Frontal Area
161
As is demonstrated in Figure 9-5, at a fin pitch of 8 fins per inch the optimum air velocity is approximately 10.5 ft/s. Yet when the fin itch increases to 14 fins per inch, the optimum air velocity decreases to 8.0 ft/s. Using condenser designs with more fins per inch yields better performance. The maximum variation in the optimum seasonal COP as the fin pitch is varied from 8 fins per inch to 14 fins per inch is approximately 2.0 %. For this improvement in the seasona COP, the cost of this configuration increases by approximately 41% as shown in Table 92. This table shows the cost factor, optimum operating conditions, and the optimum seasonal COP for varying fin pitch with fixed condenser frontal area.
Table 9-2: Optimum Operating Conditions and Cost Factor for Varying Fin Pitch with Fixed Frontal Area
8
Optimum Sub-cool a 95° F ambient Temperature (95° F) 15
10
15
9.5
4.05
0.89
12
15
8.5
4.09
1.00
14
15
8.0
4.10
1.10
Fin Pitch
Optimum Air Velocity (ft/s)
Optimum Seasonal COP
Cost Factor
10.5
4.00
0.78
162
As the fin pitch increases, the airside pressure drop over the fins also increases. When the frontal area of the condenser is fixed, the increased pressure drop due to increasing fin pitch is transferred directly to the fan power, causing it to increase as well. However, the compressor fan power required decreases by approximately the same amount as the fan power increases. Thus, the phenomenon of increased airside pressure drop resulting fro increased fin pitch does not cause the seasonal COP to decrease.
Varying Tube Diameter
The final geometric parameter varied with fixed conde nser frontal area is the tube diameter. The frontal area, the number of rows, the fin pitch, the tube spacing and the number of tubes per circuit are all maintained at the values utilized for the base configuration. Figure 9-7 shows the effect of the air velocity on the seasonal COP for various tube diameters at optimum sub-cool. According to the figure, the absolute maximum seasonal COP is 4.11 and occurs at a tube diameter of 1/2”. Conversely, in the previous chapter is was found that for fixed heat exchanger cost and variable frontal area, the maximum seasonal COP is 4.09 and occurs for a tube diameter of 3/8”. Figure 9-8 shows how the seasonal COP varies with the tube diameter at optimum operating conditions. Figure 9-8 only reinforces the trends displayed in Figure 9-7. The seasonal COP increases by approximately 5.4 % from 3.88 to 4.09 as the tube diameter is increased from 5/16” to 3/8”. The seasonal COP th en increases by only 0.5% from 4.09
163
4.20
4.10
4.00
P O C l a 3.90 n o s a e S
Locus of Optimums
3.80
1/2" tube diameter 3/8" tube diameter
3.70
5/8" tube diameter 5/16" tube diameter
3.60 5
6
7
8
9
10
11
12
13
14
Air Velocity Over Condenser (ft/s)
Figure 9-7: Effect of Air Velocity For Varying Tube Diameter at Optimum SubCool for Fixed Condenser Frontal Area
164
4.15
4.10
4.05
P O C l a n 4.00 o s a e S 3.95
3.90
3.85
1/4
5/16
3/8
7/16
1/2
9/16
5/8
11/16
Outer Tube Diameter (in)
Figure 9-8: Effect of Tube Diameter on the Seasonal COP for Fixed Condenser Frontal Area at Optimum Sub-Cool and Air Velocity
165
to 4.11 when the tube diameter increases from 3/8” to 1/2”. When the diameter is further increased from 1/2” to 5/8”, the optimum seasonal COP decreases by only 2.8 % fro 4.11 to 4.00. These results, along with the optimum operating conditions and cost factors for varying tube diameters, are shown in Table 9-3.
Table 9-3: Optimum Operating Conditions and Cost Factor For Varying Tube Diameters with Fixed Frontal Area
Tube Diameter (in) 5/16”
Optimum Sub-coo at 95° F ambient Temperature (95° F) 15
Optimum Air Velocity (ft/s) 8.5
3/8”
15
1/2” 5/8”
Optimum Seasonal COP
Cost Factor
3.88
0.92
8.5
4.09
1.00
15
8.5
4.11
1.20
15
8.0
4.00
1.59
Increasing the tube diameter has a large impact on many physical phenomena in the system. Increasing the tube diameter causes a decrease in the refrigerant-side pressure drop and an increase in the refrigerant-side heat transfer area. Both of these phenomena have a positive impact on the seasonal COP. However, increasing the tube diameter also reduces the minimum air flow area, producing an increase in the air drag. As a result, the
166
air-side pressure drop increases, and the condenser fan power also steadily increases. These phenomena have a negative impact on the COP. Hence, there are competing negative phenomena and positive phenomena at work. According to Figure 9-8, when the tube diameter is increased from 5/16” to 1/2”, the increase in the COP that results from the reduction in the refrigerant-side pressure drop and the increase in the refrigerant-side heat transfer area is larger than the reduction in the COP that results from the increased air-side pressure drop and increased condenser fan power. Thus, as Figure 9-8 shows, the seasonal COP increases when the diameter is increased from 5/16” to 1/2”. Figure 9-9 shows the refrigerant-side pressure drop versus tube diameter at optimum sub-cool and air velocity. According to Figure 9-9, when the tube diameter is increased from 5/16” to 1/2”, the refrigerant-side pressure drop decreases significantly. Figure 9-10 shows the power requirements of the compressor and the condenser fan versus the tube diameter at optimum sub-cool and air velocity. The figure shows that the reduction in the refrigerant-side pressure drop is indeed large enough to produce a decrease in the compressor power as the tube diameter is increased from 5/16” to 1/2”, while the condenser fan power increases steadily. Conversely, when the tube diameter is further increased from 1/2” to 5/8”, the reduction in the COP that results from the increased air-side pressure drop and increased condenser fan power is larger than the increase in the COP that results from the decrease in the refrigerant-side pressure drop and the increase in the refrigerant-side heat transfer area. Therefore, when the diameter is increased from 1/2” to 5/8”, the seasonal COP
167
40
) a 35 i s p ( p 30 o r D e r 25 u s s e r P 20 e d i S 15 t n a r e 10 g i r f e R 5
Total Straight Tube Bends
0 1/4
5/16
3/8
7/16
1/2
9/16
5/8
11/16
Outer Tube D iameter (in)
Figure 9-9: Refrigerant-Side Pressure vs. Tube Diameter for Fixed Frontal Area at 82° F Ambient Temperature, Optimum Sub-Cool and Air Velocity
168
6700
1000
6600
900
Total Power (Fixed Area)
) r h / 6500 u t B ( 6400 r e w o6300 P r o6200 s s e r p6100 m o C6000 & l a 5900 t o T
Compressor Power (Fixed Area) Condenser Fan Power (Fixed Area)
5800
C 800 o n d 700 e n s e r 600 F a n 500 P o w e 400 r ( B t 300 u / h r ) 200
100
5700 1/4
5/16
3/8
7/16
1/2
9/16
5/8
0 11/16
Tube Outer Diameter (in)
Figure 9-10: Power Requirements for Varying Tube Diameters with Fixed Condenser Frontal Area at 82° F Ambient Temperature, Optimum Sub-Cool and Air Velocity
169
decreases, as shown in Figure 9-8. The increase in the air-side pressure drop tha accompanies an increase in the tube diameter is displayed in Figure 9-11. Figure 9-10 shows that when the tube diameter is increased from 1/2” to 5/8”, the compressor power actually increases. Moreover, Table 9-3 shows that when the tube diameter is increased from 1/2” to 5//8”, the optimum air velocity decreases in an effort to reduce the increase in the fan power that results from the increased drag . The reduction in the optimum airvelocity results in a decrease in the effective temperature difference between the refrigerant and the air. Therefore, the reduction in the minimum air flow area coupled with the decrease in the effective refrigerant-to-air temperature difference produces a decrease in the air-side heat transfer coefficient. Hence, the negative effects on the seasonal COP become even larger.
Operating Costs
As discussed in the previous chapter, the operating cost of the air-conditioning system is inversely proportional to the seasonal COP (1/COP
∝ operating cost).
In this study, an
operating cost factor is defined as: 1/COP = operating cost factor. Figure 9-12 shows how the operating cost factor, varies with the condenser material cost factor with fixed frontal area for all of the geometric parameters investigated for this study. According to the figure, when the frontal area of the condenser is fixed, the lowest operating cost i achieved when a configuration utilizing 3 rows of tubes, with tube of diameter 3/8”, a fin
170
0.008 Total (Fixed Area)
0.007
) a i s 0.006 p ( p o r 0.005 D e r u 0.004 s s e r P0.003 e d i S r 0.002 i A
Fins (Fixed Area) Tubes (Fixed Area)
0.001 0 1/4
5/16
3/8
7/16
1/2
9/16
5/8
11/16
Tube Outer Diameter (in)
Figure 9-11: Air-Side Pressure Drop vs. Tube Diameter for Fixed Condenser Frontal Area at 82 ° F Ambient Temperature, Optimum Air Velocity and Sub-Cool
171
0.260 Base Configuration 12 Fins Per Inch (FPI)
0.258
Tube Diameter
5/16"
Number of Rows 2 Tubes per Circuit (TPC)
) r 0.256 o t c a F0.254 t s o0.252 C g n i t 0.250 a r e p0.248 O ( P0.246 O C / 1 0.244
Fin Pitch
Tube Diameter
3 Rows
Tubes per Circuit
3/8" Diameter Tube 2 rows
5/8"
8 FPI
10 FPI 4 rows
Base Configuration 14 FPI 5 TPC
0.242
1/2"
4 TPC
3 TPC
0.240 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Condenser Material Cost Factor
Figure 9-12: Operating Cost Factor vs. Cost Factor of Condenser Materials for Varying Geometric Parameters with Fixed Condenser Frontal Area and Optimum Air Velocity and Sub-Cool
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pitch of 12 fins per inch, and employing 3 tubes per circuit. This configuration has a cost factor of unity, and thus cost the same as that of the base configuration. Figure 9-12 also shows that, unlike in Chapter VIII where the cost factor of the hea exchangers is fixed, when the frontal area is fixed the lowest operating cost occurs when 3 rows of tubes are used. Increasing the number of rows to 4 actually increases both the coil material cost and the operating cost factor. Although there is a relatively significant 2.3% decrease in operating cost when the fin pitch decreases from 8 fins per inch to 12 fins per inch, there is only a 0.2% decrease in the operating cost when the fin pitch is further decreased to 14 fins per inch. A configuration using 14 fins per inch yields lower operating costs than do those employing fewer fins per inch. However, the material cos factor of this configuration is 1.1, which is 10% greater than the base case configuration (12 fins per inch). Therefore, when the frontal area of the condenser is fixed, it is recommended that a fin pitch of 12 fins per inch be employed. It can also be discerned from Figure 9-12 that the tube diameter and the number o tubes per circuit have a significant effect on the operating cost of the complete airconditioning system. Figure 9-12 shows that when only 2 tubes per circuit are used, as in the base configuration, the optimum tube diameter is 1/2” with fixed heat exchanger cost. However, the figure also shows that for a tube diameter of 3/8” and 3 rows of tubes, the lowest operating cost occurs for a condenser configuration utilizing 3 tubes per circuit. The initial investigations outlined throughout this chapter did not examine the effect of tube diameter on the system performance when tube circuiting other than the base
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configuration of 2 tubes per circuit is utilized. From an analysis of Figure 9-12, it is obvious that an examination of this effect is warranted. While Figure 9-12 shows that for fixed heat exchanger cost, a configuration with a 5/16” tube diameter yields the highest operating cost, and the worst performance. However, this tube diameter was only tested for the base case configuration of 2 tubes per circuit. This low performance is related to the higher refrigerant-side pressure drop tha results when a tube diameter this small is employed with only 2 tubes per refrigerant flow circuits. Increasing the number of tubes per circuit should relieve the detrimental effec of the higher refrigerant-side pressure drop. When the number of tubes per circuit is increased, the amount of mass of refrigerant flowing through each individual tube is decreased. Therefore, tubes of smaller diameter can be utilized without degrading syste performance. Employing a 5/16” diameter tube with the frontal area of the condenser fixed actually reduces the cost factor to 0.92. Furthermore, increasing the number o tubes per circuit has no effect on the frontal area. Therefore, configurations with smaller diameter tubes and a greater number of tubes per circuit do not increase the cost o materials for the total system when the frontal area of the heat exchangers is fixed. As a result of the above analysis, the effect of the number of tubes per circuit on t he system performance will be investigated, for a configuration utilizing a tube diameter of 5/16”, 3 rows of tubes, and 12 fins per inch. Figure 9-13 shows the effect of the number of tubes per circuit on the seasonal COP at optimum operating conditions for a heat exchanger configuration with a tube diameter
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4.20
4.15
4.10
P O C4.05 l a n o s 4.00 a e S
5/16" Tube Diameter 3 rows of tubes 12 fins per inch
3.95
3.90
3.85 1
2
3
4
5
6
7
Number of Condenser Tubes per Circuit
Figure 9-13: Seasonal COP for Varying Condenser Tube Circuiting with Fixed Frontal Area and 5/16” Tube Outer Diameter at Optimum Sub-Cool and Ai Velocity
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of 5/16” with a fixed cost factor. The figure shows that for a tube diameter of 5/16”, as the number of tubes per circuit increases from 2 to 4, the seasonal COP increases by approximately 7.2% from approximately 3.88 to 4.16. As the number of tubes per circuit increases from 4 to 5, the optimum seasonal COP increases from 4.16 to a maximum o 4.17. The optimum seasonal COP then decreases to 4.14 when the number of tubes per circuit increases from 5 to 6. The explanations for the aforementioned trends in the optimum seasonal COP with varying number of tubes per circuit a re the same as for the trends discussed earlier in Chapter VIII under the section entitled “Varying Condenser Tube Circuiting ”. As discussed in that section, the improved seasonal COP that occurs when the tubes per circuit increases from 2 to 5 results from the decrease in refrigerant pressure drop which reduces the required compressor power. The decrease in pressure drop occurs because, as the number of tubes per circuit increases, the amount of mass of refrigerant through each individual tube decreases. This decrease in the amount of mass flowing in each tube leads to a decrease in the refrigerant-side pressure drop through each tube, which has a positive effect on the seasonal COP. However increasing the number of tubes per circuit also decreases the refrigerant-side heat transfer coefficient, which has a negative effect on the seasonal COP. For the 5/16” diameter tube configuration, when the number of tubes per circuit is increased from 2 to 5, the positive effect of the reduced refrigerant-side pressure drop has a larger impact on the seasonal COP than the negative effect of the decreased refrigerant-side heat transfer coefficient. Thus the seasonal COP increases. However, when the tubes per circuit is increased from 5 to 6 for the 5/16” tube
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configuration, the decreased refrigerant-side heat transfer coefficient has a larger effec on the seasonal COP than the decreased refrigerant-side pressure drop, and the seasonal COP decreases. Hence, there is a certain plateau at which the number of tubes per circuit cannot increase without causing a decrease in system performance. For a condenser with a tube diameter of 3/8”, this occurs when 3 tubes per circuit are used. However when the tube diameter is decreased to 5/16”, this plateau occurs at a configuration utilizing 5 tubes per circuit. Figure 9-14 shows the optimum seasonal COP versu s the number of tubes per circui for the both 3/8” tube diameter configuration (base configuration) and the 5/16” tube diameter configuration with fixed condenser frontal area. As the figure shows, the values of the optimum seasonal COP achieved for condensers using a 5/16” diameter tube are slightly higher than those with a 3/8” diameter tube. For a condenser with a tube diameter of 3/8”, the optimum seasonal COP is 4.15 and occurs when 3 tubes per circuit is used. However, when the diameter is decreased to 5/16”, the optimum seasonal COP is 4.17 and occurs when 5 tubes per circuit are used. Thus, the optimum seasonal COP obtained when using tubes of 5/16” diameter is 0.5 % higher than the optimum obtained using tubes of 3/8” diameter. When the number of tubes per circuit is th e value used for the base configuration (2 tubes per circuit), a condenser using tubes of diameter of 3/8” yields a much higher optimum seasonal COP, COP = 4.09, than a condenser using tubes of diameter 5/16”, COP = 3.88. Conversely, when the number of tubes per circuit is increased,
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4.20
4.15
4.10
P O C4.05 l a n o s 4.00 a e S
5/16" Outer Tube Diameter 3/8" Outer Tube Diameter
3.95
Fixed Frontal Area
3.90
3 rows of tubes 12 fins per inch
3.85 1
2
3
4
5
6
7
Number of Condenser Tubes per Circuit
Figure 9-14: Comparison of the Effect of the Number of Tubes per Circuit on th Seasonal COP for 5/16” and 3/8” Outer Tube Diameters with Fixed Frontal Area at Optimum Sub-Cool and Air Velocity
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configurations utilizing tubes of 5/16” diameter yield the highest seasonal COP. Furthermore, the cost factor of a configuration utilizing tubes of diameter 5/16” and 5 tubes per circuit, 0.92, is 8.0% lower than the 1.0 cost factor o btained when condenser tubes of 3/8” diameter are employed. Figure 9-15 shows the operating cost versus the material cost factor for varying tube circuiting and tube diameter. Only the tube circuiting of the base configuration, 2 tubes per circuit, is utilized for tube diameters o 1/2” and 5/8”. As shown in Figure 9-15, condensers with tube diameters of 1/2” and 5/8” have not only significantly higher material cost factors but also higher operating cost than condensers employing tubes of 5/16” and 1/2” diameter. Therefore, when the frontal area of the heat exchanger is fixed to the area of the base configuration (7.5 ft2), a condenser with an outer tube diameter of 5/16”, 5 tubes per circuit, 3 rows of tubes, and 12 fins per inch yields the highest seasonal COP (lowest operating cost) of all configurations investigated in this study, and has the most reasonable heat exchanger material co st (cost factor lower than the base configuration).
Varying the Base Configuration Frontal Area
As discussed in Chapter VIII, the frontal area of the base heat condenser configuration, 7.5 f 2, has been selected as a value typically found in most residential airconditioning systems rated at 30,000 Btu/hr. In many instances, there are space constraints and/or material cost constraints imposed on the heat exchanger designer that
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0.260 3/8" Tube Diameter 2 tpc
r 0.256 o t c a F t 0.252 s o C g n i t 0.248 a r e p O ( 0.244 P O C / 1
5/16" Tube Diameter 1/2" Tube Diameter 5/8" Tube Diameter
2 tpc
2 tpc
6 tpc 4 tpc
0.240
2 tpc
5 tpc 4 tpc 3 tpc
5 tpc
0.236 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Material Cost Factor
Figure 9-15: Operating Cost Factor vs. Condenser Material Cost Factor for Varying Tube Diameter and Tube circuiting at Optimum Air Velocity and Sub-Cool
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restrict the size of the condenser and consequently sacrificing performance. In these situations, the frontal area of the condenser may have to be even smaller and/or cheaper than that of the base configuration used in this study. However, there are also examples in which space and material cost constraints are not stringent, and a larger and/or more expensive condenser can be employed to produce a lower operating cost (or higher seasonal COP). Yet, as shown in Figure 9-13, the cost of materials can be increased or decreased in a number of ways including: increasing the number of rows, increasing the fin pitch, increasing the tube diameter, or by simply increasing the frontal area. Hence, two hypothetical questions arise from this: (1) If the material cost of the condenser must be reduced by a specified amount, what geometric parameter or dimension should be reduced to ensure that only a minimum increase in the operating cost results? (2) If the cost of materials is allowed to increase by a specified amount, what geometric parameter or dimension should be increased in order to produ ce the maximum decrease in the operating cost? As discussed earlier, Figure 9-15 shows that condenser configurations employing tube diameters of 1/2” and 5/16” do not yield the best system performance. Therefore in addressing the two hypothetical questions posed above, tube diameters of this size are not studied. Figure 9-16 shows the operating cost factor versus the material cost factor for varying fin pitch and varying numbers of rows for the base configuration. This figure also shows the operating cost of the condenser configuration utilizing 5/16” diameter tubes, 5 tubes per circuit, 3 rows of tubes, and 12 fins per inch for 3 condenser frontal areas: (1) frontal area equal to the base configuration, (2) frontal area 20% lower than the
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0.256 Fin Pitch
0.252
) r o t c 0.248 a F t s o 0.244 C g n i t a r 0.240 e p O ( P0.236 O C / 1
20% smaller frontal area
Number of Rows
2 rows 8 FPI
5/16" diameter, 5 tubes per circuit
10 FPI
4 rows 14 FP
5/16" diameter
Base Configuration
Base case frontal area Base Configuration 12 Fins PerInch (FPI) 20% greater frontal area
2 Tubes per Circuit (TPC)
0.232
3 Rows 3/8" Diameter Tube
0.228 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Material Cost Factor
Figure 9-16: Operating Cost Factor vs. Condenser Material Cost Factor for Varying Geometric Parameters and Various Fixed Frontal Areas at Optimum Air Velocity and Sub-Cool
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base configuration, and frontal area 20% greater than the base configuration. An investigation of the slopes of the curves in this figure is needed to discern the best methods to vary the frontal area in order to achieve reductions in the material cost or the operating cost. In question (1), the material cost of the condenser is to be reduced by a specified amount. The three methods considered for reducing the material cost are: reducing the number of rows, reducing the fin pitch, and reducing the frontal area. According to Figure 9-16, decreasing the fin pitch from the base configuration value of 12 fins per inch to 8 fins per inch produces a smaller increase in operating cost than decreasing either the number of rows or decreasing the frontal area. The slope of the line of row variation is smaller than the slopes for frontal area variation and fin pitch variation in the d irection of decreasing material cost. In question (2), the material cost of the condenser is allowed to increase by a specified amount in order to reduce the operating cost. Again, the three methods considered for increasing the material cost are: increasing the number of rows, increasing the fin pitch, and increasing the frontal area. According to Figure 9-16, increasing the frontal area, produces the largest reduction in the operating cost. The slope of the line o frontal area variation is negative in the direction of increased material cost. The slope of the line of fin pitch variation is also negative in the direction of increased mat erial cost. However, increasing the fin pitch produces only a slight decrease in the operating cost. Conversely, increasing the number of rows actually increases the operating cost for the
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base configuration detailed in the figure. Therefore, increasing the material cost in this manner is a “lose-lose” proposition in that no reduction in the operating cost results. While the above analysis attempts to address the two hypothetical questions posed in regards to methods of increasing and decreasing material cost, the questions have not been universally answered by the work of this study. As indicated in the figure, the frontal area was varied only for the configuration optimized with fixed frontal area configuration (5/16” diameter tubes, 5 tubes per circuit, 3 rows of tubes, and 12 fins per inch). For the reasons detailed in the section of this chapter entitled “Operating Costs”, the number of rows and the fin pitch were varied only for the base configuration (3/8” diameter tubes, 2 tubes per circuit, 3 rows of tubes, and 12 fins per inch). Therefore in the above discussion it is assumed that the slopes of the lines of varying fin pitch and varying number of rows will be the same regardless of tube d iameter and tube circuiting in order to address the hypothetical questions posed in this study.
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CHAPTER X
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
Refrigerant R-410a is one of the primary candidates to replace refrigerant R-22 in residential heat pump and air-conditioning applications. As a result of this current study, many conclusions can be drawn regarding the design of a fin-and-tube condenser coil for a unitary air-conditioning system with refrigerant R-410a as the working fluid. A computational model that determines the seasonal COP of an air-conditioning system for various operating conditions and geometric configurations of the condenser is also used. In addition, a methodology is detailed for optimizing the condenser design using the seasonal COP of the system as the figure of merit. While the primary objectives of this work are not to perform detailed economic analyses, the system operating cost factor and the capital cost factor for the heat exchanger materials are both considered when detailing the selection of the best design. Design guidelines taking into account space constraints have also been given. It is concluded that selecting the final optimum configuration depends on the constraints imposed upon the heat exchanger designer. If the space constraints are stringent, then the base condenser configuration for the system
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investigated with the frontal area of the condenser fixed is the opt imum (5/16” tube diameter, 5 tubes per circuit, 3 rows of tubes, 12 fins per inch, 7.5 ft2 frontal area). However, if the space constraints are not stringent, and a higher seasonal COP is the primary goal, then the condenser configuration for the system optimized with the cost o heat exchanger materials fixed may be preferred (5/16” tube diameter, 5 tubes per circuit, 3 rows of tubes, 12 fins per inch, 8.5 ft2 frontal area). Hence, more information about the space and economic constraints imposed on the designer is required before the bes condenser configuration of those investigated in this study can be selected. As discussed in previous chapters, due to the impending ban of refrigerant R-22 production there is a pressing need for studies on air-conditioning systems that utilize alternative refrigerants. Therefore, in this current study comparisons are made between the condenser configurations and seasonal performance of air-conditioning systems designed using refrigerant R-410a as the working fluid (this current study) to systems designed using refrigerant R-22 as the working fluid. A thesis entitled “Optimization of Finned-Tube Condenser for a Residential Air-Conditioner Using R-22” by Emma Saddler (Saddler, 2000), details the design methodology for an air-conditioning system with refrigerant R-22 as the working fluid. The base configuration condenser, as well as the component and property models used in Saddler’s study are similar to those used in this current work. Likewise, the geometric and operating parameters varied in Saddler’s optimization are also similar to those of this current study. According to Saddler’s results, the R-22 air-conditioning system designed with the frontal area of the condenser fixed has a maximum seasonal COP of 4.18, 13 degrees of
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sub-cool in the condenser, and an air velocity of 8.3 ft/s over the condenser with the following geometric parameters: a frontal area of 7.5 ft2, 4 rows of tubes, 6 tubes per circuit, tubes that are 5/16” in diameter, and 12 fins per inch. The major differences between the geometric and operating parameters of this system and those of the R-410a system designed with the fixed condenser frontal area constraint are the number of rows, and the tube circuiting. The maximum seasonal COP for the R-22 system designed with the fixed condenser frontal area constraint is approximately 0.2% greater than the maximum seasonal COP for the comparable R-410a system. With a fixed heat exchanger cost constraint identical to that used in this current study, Saddler’s results show that the R-22 air-conditioning system has a maximum seasonal COP of 4.22. For this maximum seasonal COP design, the R-22 system has 10 degrees of sub-cool in the condenser, and an air velocity of 8.3 ft/s over the condenser with the following geometric parameters: a frontal area of 10.6 ft2, 3 rows of tubes, 6 tubes per circuit, 5/16” tube diameter, and 8 fins per inch. The major differences between the geometric and operating parameters of this system an d those of the R-410a system optimized with the fixed heat exchanger cost constraint are the tube circuiting, and the fin pitch. The maximum seasonal COP for the R-22 system designed with the fixed cost constraint is approximately 0.2% lower than the maximum seasonal COP for the comparable R-410a system. Because the seasonal COP of the R-22 systems and the R-410a systems optimized with both the fixed material cost and fixed frontal area constraints are nearly identical (vary within ± 0.3%), the estimated operating costs of both systems are also roughl
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equivalent. In addition, for both the R-22 and R-410a air-conditioning systems, the best performing condenser configurations investigated utilize the smallest tube diameter examined in both studies (tube diameter = 5/16”). It is expected that the best performing condenser configurations investigated for the R-410a air-conditioning system would require fewer tubes per circuit than the best condenser configurations investigated for the R-22 air-conditioning system. This is because the working pressure and the vapor phase density for R-410a are much higher than for R-22. Based on the results of both this current work and Saddler’s thesis, this expected trend has been confirmed. The results of this study confirm the viability of refrigerant R-410a as a replacemen for refrigerant R-22 in vapor compression air-conditioning systems similar to those investigated in this work. The R-410a systems have seasonal performance and operating costs equivalent to those of the R-22 systems designed with the same frontal area and material cost constraints. Therefore environmental safety is achieved without sacrificing cost and performance.
List of Conclusions
The specific conclusions drawn from this study are as follows:
•
Condenser design for air-conditioning systems must be based on seasonal performance. The United States Department of Energy regulations require a seasonal
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performance rating, which incorporates the system’s performance at ambien temperatures ranging from
°F to 102 °F, weighted with the cooling load
distribution factors. Whether this rating is consistent with actual practice is questionable. However, the United States Department of Energy regulations require all residential air-conditioning systems to be labeled with this rating.
•
The seasonal performance of an air-conditioning system can be closely approximated by calculating the system’s performance at 82 °F ambient temperature.
•
Condenser tubes of smaller diameter enhance performance.
•
When packaging and space constraints are not present, the condenser configuration with the largest frontal area possible yields the best system performance.
•
When typical volume and space constraints are imposed, cond ensers employing 3 rows of tubes yield the best performance. Contrary to intuition, increasing the number of rows to 4 actually increases the material cost of the coil and decreases the system performance when space constraints are imposed.
•
For all geometric configurations investigated, a refrigerant charge producing between 10 and 15 degrees sub-cool at 95 °F ambient temperature produces the optimu performance.
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•
For all geometric configurations investigated, the optimum velocity of air flow over the condenser coil ranges from roughly 6 ft/s and 12 ft/s.
•
As the ambient temperature decreases, the sub-cool at 95 °F ambient temperature that is needed to produce the highest COP increases.
•
If the material cost of the condenser must be reduced, decreasing the fin pitch from the base configuration value of 12 fins per inch to 8 fins per inch produces a smaller increase in operating cost than decreasing either the number of rows or the frontal area.
•
If the cost of materials is allowed to increase by a specified amount, increasing the frontal area produces the largest reduction in the operating cost. However, increasing the number of rows or the fin pitch actually increases the operating cost for the base configuration detailed in the figure. Therefore, increasing the material cost in this manner is a “lose-lose” proposition, in that no reduction in the operating cost results.
•
All parameters that do not affect material cost of the condenser, such as the operating parameters and the tube circuiting, should be optimized for every geometric configuration investigated before the performance of different systems is compared.
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Recommendations
Optimization Parameters and Methodology
Again, a principal goal of this study was to provide heat exchanger designers with guidelines for optimizing a condenser with the alternative refrigerant R-410a as the working fluid using the seasonal COP of the air-conditioning system as the figure of merit. Perhaps the most salient lesson learned during this study is the significant effec that the operating conditions have on the system performance, and subsequently the optimization process. The operating parameters examined in this study include the subcool in the condenser and the velocity of airflow over the condenser. It is of the utmost importance that heat exchanger designers be aware that it is not possible to make valid comparisons between heat exchangers of different geometric configurations without first optimizing the operating parameters at each configuration to yield the maximum seasonal COP. Therefore, in all future studies of this kind, it is recommended that the operating parameters continue to be optimized at each geometric configuration in a manner similar to the method detailed in this study. Varying the sub-cool in the condenser and the air velocity over the condenser does not significantly alter the frontal area or the material cost of the heat exchanger. During this study, it has also been determined that varying the number of tubes per refrigeran flow parallel circuits also does not alter the cost of materials or the frontal area of the hea exchanger. However, as discussed in Chapter VIII and Chapter IX, the refrigerant flow tube circuiting does have a major effect on the optimum seasonal COP, and hence, the optimum design. Therefore, for future optimization studies of this kind, it is
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recommended that in addition to the operating conditions, the condenser tube circuiting should also be optimized at each geometric configuration investigated. For example, in order make a valid comparison between a system using a condenser with 2 rows of tubes to one using 3 rows of tubes, the optimum air velocity, the optimum degrees sub-cool in the condenser, and the optimum tube circuiting arrangement should be determined for both systems. The spacing of the tubes in the condenser during this investigation is the standard recommended for condensers by most heat exchanger manufacturers. However, it is possible that this spacing is not the optimum spacing. The tube spacing affects the efficiency of the fins. The closer the tube spacing, the higher the fin efficiency, and hence a higher air-side heat transfer coefficient is produced. As a result, it is recommended that the tube spacing be varied and optimized for future studies of this kind. Due to the limitations of the air-side pressure drop and heat transfer models, condensers utilizing tubes of diameter smaller than 5/16” have not been investigated in this study. As stated previously, for the air-conditioning systems investigated in this study, the optimum condenser configurations utilize the smallest tube diameter investigated, 5/16”. It is therefore recommended that condensers with tubes of 1/4” outer diameter be included in future optimization studies, since it is possible that even better performance can be achieved. As a result, air-side pressure drop and heat transfer models that are valid for tubes of smaller outer diameter must be used.
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Computational Methods
For this study, all modeling computations were performed using Engineering Equation Solver (EES) operating on a 250 MHz Intel Pentium II processor. The
optimization parameters analyzed in this study included the sub-cool in the condenser, the air velocity over the condenser, the number of rows of tubes, the refrigerant tube circuiting, the fin pitch, and the tube diameter. A breakdown of the computational time involved to determine the effects of these various parameters on the system performance is as follows:
•
For this study, in order to calculate the seasonal COP at one condenser geometri configuration and with the operating parameters specified (1 “run”), 5 minutes o computational ti me was needed: 5 minutes/ ”run”
•
Determining the optimum air velocity at one sub-cool condition at one geometric configuration required a minimum of 12 “runs”: 12 “runs” / velocity
•
Determining the optimum sub-cool at one condenser geometric configurati required 12 “runs”: 12 “runs” / sub-coo
Therefore calculating the seasonal COP for one condenser geometric configuration required:
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(12 “runs” /velocity) x (12 “runs” /sub-cool) x (5 minutes/ “run”) = 720 minutes (12 hours) of run time to determine the optimum sub-cool and air velocity for one geometric configuration of the condenser.
An exhaustive search over the range of geometric design parameters requires:
•
Investigating fin pitch varying from 8 to 14 fins/inch: 4 “runs” /fin pitch
•
Investigating tube diameter varying from 5/16” to 5/8”: 4 “runs” /tube diameter
•
Investigating tube circuiting varying from 2 to 6 tubes per circuit: 5 “runs” /tube circuiting
•
Investigating the number of tube rows varying from 1 to 4: 4 “runs” /number of rows
•
Design constraints of fixed frontal area and fixed material cost: 2 “runs” /design constraint
Therefore the total computational time required for an exhaustive optimization search scheme is:
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(5 minutes/ “run”) x (12 “runs” /velocity) x (12 “runs” /sub-cool) x (4 “runs” /fin pitch) x (4 “runs” /tube diameter) x (5 “runs” /tube circuiting) x (4 “runs” /number of rows) x (2 “runs” /design constraints) = 460,800 minutes or 7,680 hours of computational time.
Hence, the total computational time involved is 7,680 hours, or more than 10 and 1/2 months. The EES model developed to calculate the system performance for this stud involves more than 2000 equations. Of these 2000 equations, 1000 must be solved through iteration. The solution of these 100 simultaneous equations is heavily dependen on the “guess values” for each variable. For varying geometric configurations and operating conditions, the guess values must be continuously adjusted in order to ensure the convergence of the solution. Therefore, the researcher is required to be in attendance for all computations, since in nearly all instances, the guess values must be adjusted for every “run”. Therefore, the actual total time for this exhaustive search is considerably longer than the 7,680 hours that have been calculated. Hence, for future studies of this kind, a more powerful and concise method for finding the optimum values of each parameter should be developed. For example, entropy minimization techniques tha quantify the tradeoff between pressure drop irreversibilities and heat transfer irreversiblilities might be useful in finding a universal optimization relation for the tube circuiting. More advanced search techniques will allow further investigation into the coupling and interactions of the geometric parameters for a larger number of configurations.
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Refrigerant Side Heat Transfer and Pressure Drop Models
Several echniques for predicting the heat transfer coefficients and pressure drops during condensation and evaporation inside tubes have been evaluated during this study Many of the current methods are cumbersome in structure, heavily dependent on empirically determined coefficients, and have considerable uncertainty. In this work, general correlations based on statistical evaluation of data, and proposed to be valid for all flow regimes, were used to calculate the condensing heat transfer coefficients and pressure drop. While it was determined that the dominant flow regime for the conditions of this present study is the annular flow regime, at low qualities, stratified-wavy flow also exists. Furthermore it was assumed that the quality varies linearly with length. It is recommended that this assumption be studied further, and that correlations based on specific models for individual flow regimes should be used.
Economic Analysis
Again, the goal of this study is not to conduct a detailed economic analysis for residential air-conditioning systems. Moreover, the cost of the compressor and condenser fan units are excluded from the cost analysis (material cost factor) for this investigation. However, in determining the optimum heat exchanger configuration, a tradeoff must be made between the capital cost and the operating cost (using the reciprocal of the seasonal COP as an operating cost factor). It is recommended that a detailed economic analysis be performed that includes both the capital cost and the operating cost of each component o the system.
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APPENDIX A
AIR-CONDITIONING SYSTEM: EES PROGRAM
{I. Refrigerant-Side Procedures and Functions A. Pressure Drop 1. singledp 2. twophasedp 3. tpbenddrop B. Heat transfer Coefficients 1. h_bar_single 2. h_bar_c 3. h_bar_e II. Air -Side A. Heat Transfer coefficients 1. ha B. Pressure Drop 1. GetEuler III. Heat Exchanger Procedures and Functions A. Surf_eff B. Exch_size C. Exch_size_un_un D. sat_size E. Tubing IV. Compressor Procedure A. Compeff }
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PROCEDURE singledp(m_r, nr,D,L,f,rho:delP) {Purpose-to determine the single phase pressure drop for flow in tubes} {velocity of refrigerant through tube, ft/hr} vel=m_r/((pi*D^2/4)*rho*nr) "[ft/hr]" delP=(f*(L/D)*(rho*Vel^2)/2)*convert(lbm/ft-hr2,psi) end PROCEDURE twophasedp(xi,xf,T1, T2, D, m_dot, nr,L:DP) {Purpose- to determine the two phase pressure drop f or flow in tubes Inputs D- equivalent diameter of flow passage , ft E- surface roughness, ft G- mass flow per unit area lbm/hr-ft^2 mu_v- viscosity of vapor phase, lbm/hr-ft mu_l- viscosity of liquid phase, lbm/hr-ft rhov- density of liquid phase rhol- density of vapor phase ReV- Reynold's number of vapor phase ReL- Reynold's number of liquid phase Dztp- length of two phase region xf- final quality xi- initial quality
v- exit specific volume of vapor phase, ft^3/lb nr- number of flow passages L- length of tube Output DeltaP- pressure drop over two phase region }
Tav=(T1+T2)/2 G=(m_dot/(D^2*pi/4))/nr mu_v=viscosity(R410A, T=Tav, x=1) mu_l=viscosity(R410A, T=Tav, x=0) rhov=density(R410A, T=Tav, x=1) rhol=density(R410A, T=Tav, x=0) {Momentum component of 2 phase pressure drop} DpM=((xf^2-xi^2)*(1+rhov/rhol-(rhov/rhol)^.333-(rhov/rhol)^(2/3))-(xfxi)*(2*rhov/rhol-(rhov/rhol)^(1/3)-(rhov/rhoL)^(2/3))*G^2/(rhov)*convert(lbm/hr^2ft,psi)) C1=(xf-xi)/L "[1/ft]" C2=.09*mu_v^.2*G^1.8/(C1*rhov*D^1.2*32.2*convert(ft/s^2,ft/hr^2))*convert(lbf/ft^2 , psia)
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C3=2.85*(mu_l/mu_v)^(.0523)*(rhov/rhoL)^.262 {Friction component of 2 phase pressure drop}
DPf2=2*c3*(.429*(xf^2.33-xi^2.33)-.141*(xf^3.33-xi^3.33)-.0287*(xf^4.33-xi^4.33)) DPf3=C3^2*(.538*(xf^1.86-xi^1.86)-.329*(xf^2.86-xi^2.86)) DPf=c2*(.357*(xf^2.8-xi^.28)+DPf2+DPf3) DP=(DpM+DPf end Procedure singlebenddrop(tpc, D_i, m_dot_r,P, T1, T2, L, Width, f:DP) {Pressure Drop in bends for single phase regions} T=(T1+T2)/2 G=m_dot_r/(tpc*D_i^2*pi/4) equiv_L=13*2 rho=density(R410A, T=T, P=P) grav=32.2*convert(1/s^2,1/hr^2) ncirc=trunc(L/width) DP=f*G^2*equiv_L/(2*grav*rho)*convert(lbf/ft^2, psia)*ncirc end PROCEDURE tpbenddrop(nr,D_i_1,m_dot_r, h_f, T_c, L_c, L_22a, L_2a2b,width:DP) {Pressure Drop In bends for two-phase regions} {for 180 degree bends} equiv_L=13*2 R_b=h_f/2 "[ft]" z=R_b/D_i_1 G=m_dot_r/(nr*D_i_1^2*pi/4) e=.000005 DP=0 num_circuit_2a2b=trunc(l_2a2b/Width) num_circuit_22a=trunc(L_22a/width) L_o=L_22a-Width*num_circuit_22a L=width-L_o mu_v=viscosity(R410A, T=T_c, x=1) mu_l=viscosity(R410A, T=T_c, x=0) grav=32.2*convert(1/s^2,1/hr^2) Rho_l=density(R410A, T=T_c, x=0) rho_v=density(R410A, T=T_c, x=1) Re_l=G*D_i_1/mu_l A_l=(2.457*ln(1/((7/Re_l)^0.9+.27*e/d_i_1)))^16 B_l=(37530/Re_l)^16 lambda_l=8*((8/Re_l)^12+(1/((A_l+B_l)^(3/2))))^(1/12)
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Re_v=G*D_i_1/mu_v A_v=(2.457*ln(1/((7/Re_v)^0.9+.27*e/d_i_1)))^16 B_v=(37530/Re_v)^16 lambda_v=8*((8/Re_v)^12+(1/((A_v+B_v)^(3/2))))^(1/12) n=ln(lambda_l/lambda_v)/ln(mu_l/mu_v) i=0 repea i=i+1 x=-L/L_2a2b+1 If x<=0 then goto 10 mu_TP=mu_v*x+mu_l*(1-x) Re_tp=G*D_i_1/mu_tp A_tp=(2.457*ln(1/((7/Re_tp)^0.9+.27*e/d_i_1)))^16 B_tp=(37530/Re_tp)^16 lambda_tp=8*((8/Re_tp)^12+(1/((A_tp+B_tp)^(3/2))))^(1/12) DELTAp_b_lo=lambda_l*G^2*equiv_L/(2*grav*rho_l)*convert(lbf/ft^2, psia) {k_b for 90 degree bend} k_b=lambda_tp*equiv_L/2 GAMMA_B=rho_l/ ho_v*(mu_v/mu_l)^n B=1+2.2/(k_b*(2+R_b/D_i_1)) {B for 90 degree bend} {B for 180 degree bend} B=.5*(1+B) phi_b_lo=1+(GAMMA_b-1)*(B*x^((2-n)/2)*(1-x)^((2-n)/2)+x^(2-n)) DELTAp_b=DELTAp_b_lo*phi_b_lo DP=DP+DELTAp_b L=L+width until i>=num_circuit_2a2b-1 10:DP=Dp end
Procedure h_bar_single22ash(D, m_dot_r, T1, T2, P:Re,h_bar, rho) {single phase heat transfer coefficient in the superheated portion of the condenser} Area=(D/2)^2*pi G=m_dot_r/Area Tav=(T1+T2)/2 rho=density(R410A, T=Tav,P=P) c_p=specheat(R410A, T=Tav, P=P) mu=viscosity(R410A, T=Tav, P=P) Pr=prandtl(R410A, T=Tav, P=P) If Re<3500 then a=1.10647 b=-.078992 endIF if (Re>3500) and (Re<6000) then a=3.5194e-7
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b=1.03804 ENDIF if Re>6000 then a=.2243 b=-.385 endif St=a*Re^b/(Pr^(2/3)) h_bar=St*G*C_p end Procedure h_bar_single4a1sh(D, m_dot_r, T1, T2, P:Re,h_bar, rho) {Single phase refrigerant heat transfer coefficient for the superheated portion of th evaporator} Area=(D/2)^2*pi G=m_dot_r/Area Tav=(T1+T2)/2 rho=density(R410A, T=Tav,P=P) c_p=specheat(R410A, T=Tav, P=P) mu=viscosity(R410A, T=Tav, P=P) Re=m_dot_r*D/(Area*mu) Pr=prandtl(R410A, T=Tav, P=P) If Re<3500 then a=1.10647 b=-.078992 endIF if (Re>3500) and (Re<6000) then a=3.5194e-7 b=1.03804 ENDIF if Re>6000 then a=.2243 b=-.385 endif St=a*Re^b/(Pr^(2/3)) h_bar=St*G*C_p end
Procedure h_bar_single2b3sc(D, m_dot_r, T1, T2, P:Re,h_bar, rho) {Single refrigerant heat transfer coefficient for the sub-cooled portion of the condenser} Area=(D/2)^2*pi G=m_dot_r/Area Tav=(T1+T2)/2 rho=density(R410A, T=Tav,P=P)
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c_p=specheat(R410A, T=Tav, P=P) mu=viscosity(R410A, T=Tav, P=P) Re=m_dot_r*D/(Area*mu) Pr=prandtl(R410A, T=Tav, P=P) If Re<3500 then a=1.10647 b=-.078992 endIF if (Re>3500) and (Re<6000) then a=3.5194e-7 b=1.03804 ENDIF if Re>6000 then a=.2243 b=-.385 endif St=a*Re^b/(Pr^(2/3)) h_bar=St*G*C_p end
FUNCTION h_bar_c(T, P,D, m_dot_r,nr) {Shah-Correlation: Two-phase refrigerant heat transfer coefficient in the condenser } G=m_dot_r/(D^2*nr/4) mu_l=viscosity(R410A, T=T, x=0) mu_g=viscosity(R410A, T=T, x=1) rho_l=density(R410A, T=T, x=0) rho_g=density(R410A, T=T, x=1) Pr_l=prandtl(R410A, T=T-1, P=P) k_l=conductivi ty(R410A, T=T, x=0) P_r=P/p_crit(R410A) Re_l=G*D/mu_l h_l=0.023*Re_l^.8*Pr_l^.4*k_l h_bar_c=h_l*(.55+2.09/(P_r^.38)) end Function h_bar_e(Te, Pe,De, m_r, x_in) {Purpose to evaluate the evaporation two phase heat transfer coefficient for forced convection flow inside tubes} x_i:=x_in "Prandtl # of liquid phase in evaporator" Pr_L=prandtl(R410A, T=Te-1, P=Pe) kl=conductivity(R410A, T=Te, x=0) "conductivity of liq. phase" mu_v=viscosity(R410A, T=Te, x=1) "viscosity of vap. phase"
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mu_l=viscosity(R410A, T=Te, x=0) rho_l=density(R410A, T=Te, x=0) rho_v=density(R410A, T=Te, x=1) x_e:=1 g:=m_r/(pi*De^2/4) h_bar_ave_e1 := 0.023 * 0.325 * 2.5 * kl * (g / mu_l) ^ 0.8 * De ^ (-0.2) * Pr_L ^ 1.4 h_bar_ave_e2 := (rho_L / rho_V) ^ 0.375 * (mu_v / mu_l) ^ 0.075 * (x_e - x_i) / (x_e ^.325 - (x_i ^ 0.325)) h_bar_e := h_bar_ave_e1 * h_bar_ave_e2 End FUNCTION ha(hf, eta,t,L, ma, mu, D_o, Ao,At, Cp, Pr, n) {Returns air-side heat transfer coefficient based on McQuiston Method} {h_bar_a- external heat transfer coefficient (btu/hr-ft^2-R)} "[ft^2]" A_min=(hf/2)*(1/eta-t) "[lbm/hr-ft^2]" Gmax=ma*(1/eta-t)/(A_min*L) Re_D=Gmax*D_o/m Re_L=Gmax*hf/mu dum1=(Ao/(At)) JP=Re_D^(-.4)*(Ao/(At/(1-t*eta)))^(-.15) j4=.2675*JP+1.325*10^(-3) jn=(1-n*1280*Re_L^(-1.2))*j4/(1-4*1280*Re_L^(-1.2)) ha=jn*Cp*Gmax/(Pr^(2/3))*convert(1/s,1/hr) end FUNCTION geteuler(Re, h_f, dep_f, D, nrow) {finds Euler number for staggered banks of tubes for a fin-and-tube cross flow hea exchanger} {Modify Euler number to account for non- equilateral geometry find correction factor k1 to account for a/b ratio, use k1 with other relationships to correct Euler # for row spacing}
a=dep_f/D b=h_f/D Check1=1 Check2=1 Check3=1 spacerat=a/b Eu=0 k1=0 If (spacerat>.5) and (spacerat<1.2) and (re>=1000) and (Re<10000) then {this relationship is stated for Re=1000, not the range 1000
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k1=(k2-k1)/(10000-1000)*(Re-1000)+k1 endIF if (spacerat>1.25) and (spacerat<3.5) and (Re>1000) and (Re<10000) then k1=.951*spacerat^.284 k2=1.28-.708/spacerat+.55/(spacerat^2)-0.113/(spacerat^3) k1=(k2-k1)/(10000-1000)*(Re-1000)+k1 endIF If (spacerat>.45) and (spacerat<3.5) and (Re>=10000) and (Re<100000) then {stated for Re=10000} k1=1.28-.708/spacerat+.55/(spacerat^2)-0.113/(spacerat^3) k2=2.016-1.675*spacerat+.948*spacerat^2-.234*spacerat^3+.021*spacerat^4 k1=(k2-k1)/(100000-10000)*(Re-10000)+k1 endif If ((spacerat>.45) and (spacerat<3.5) and (Re>=100000)) o r ((spacerat>.45) and (spacerat<1.6) and (Re>=1000000)) then {stated for Re=100000} k1=2.016-1.675*spacerat+.948*spacerat^2-.234*spacerat^3+.021*spacerat^4 endIF if (spacerat>1.25) and (spacerat<3.5) and (Re>100) and (Re<1000) then k1=.93*spacerat^.48 k2=spacerat^(-.048) k1=(k2-k1)/(1000-100)*(Re-100)+k1 endIF if (spacerat=1.155) then k1=1 endif If k1=0 then check1=0 If (a>=1.25) and (a<1.5) and (Re>3) and (re<1000) then a=1.25} Eu1:=(.795+247/re+335/(re^2)-1550/Re^3+2410/Re^4) eu2:=(.683+1.11e2/re-97.3/Re^2+426/re^3-574/re^4) Eu=(Eu2-Eu1)/(1.5-1.25)*(a-1.25)+Eu1 endif If (a>=1.25) and (a<1.5) and (Re>1000) and (Re<2e6) then Eu1:=(.245+3390/Re-9.84e6/Re^2+1.32e10/re^3-5.99e12/Re^4) Eu2:=(.203+2480/re-7.58e6/re^2+1.04e10/re^3-4.82e12/re^4) Eu=(Eu2-Eu1)/(1.5-1.25)*(a-1.25)+Eu1 endif If (a>=1.5) and (a<2) and (Re>3) and (Re<100) then eu1:=(.683+1.11e2/re-97.3/Re^2+426/re^3-574/re^4) Eu2:=(.713+44.8/Re-126/Re^2-582/Re^3) Eu=(Eu2-Eu1)/(2-1.5)*(a-1.5)+Eu1
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{Stated for
endif If (a>=1.5) and (a<2) and (Re>100) and (Re<1000) then eu1:=(.683+1.11e2/re-97.3/Re^2+426/re^3-574/re^4) Eu2:=(.343+303/re-7.17e4/re^2+8.8e6/re^3-3.8e8/Re^4) Eu=(Eu2-Eu1)/(2-1.5)*(a-1.5)+Eu1 endif If (a>=1.5) and (a<2) and (Re>1000) (Re>100 0) and (Re<10000) then then Eu1:=(.203+2480/re-7.58e6/re^2+1.04e10/re^3-4.82e12/re^4) Eu2:=(.343+303/re-7.17e4/re^2+8.8e6/re^3-3.8e8/Re^4) Eu=(Eu2-Eu1)/(2-1.5)*(a-1.5)+Eu1 endif If (a>=1.5) and (a<2) (a <2) and (Re>10000) and (Re<200000) then then Eu1:=(.203+2480/re-7.58e6/re^2+1.04e10/re^3-4.82e12/re^4) Eu2=(.162+1810/Re+7.92e7/re^2-1.65e12/Re^3+8.72e15/re^4) Eu=(Eu2-Eu1)/(2-1.5)*(a-1.5)+Eu1 endif If (a>=2) and (a<2.5) and (Re>7) and (Re<100) then Eu1:=(.713+44.8/Re-126/Re^2-582/Re^3) Eu2:=(.33+98.9/re-1.48e4/Re^2+1.92e6/re^3-8.62e7/re^4) Eu=(Eu2-Eu1)/(2.5-2)*(a-2)+Eu1 endif If (a>=2) and (a<2.5) and (Re>100) and (Re<5000) then Eu1:=(.343+303/re-7.17e4/re^2+8.8e6/re^3-3.8e8/Re^4) Eu2:=(.33+98.9/re-1.48e4/Re^2+1.92e6/re^3-8.62e7/re^4) Eu=(Eu2-Eu1)/(2.5-2)*(a-2)+Eu1 endif If (a>=2) and (a<2.5) and (Re>5000) (Re>500 0) and (Re<10000) then then Eu1:=(.343+303/re-7.17e4/re^2+8.8e6/re^3-3.8e8/Re^4) Eu2:=(.119+498/Re-5.07e8/Re^2+2.51e11/Re^3-4.62e14/re^4) Eu=(Eu2-Eu1)/(2.5-2)*(a-2)+Eu1 endif If (a>=2) and (a<2.5) (a<2.5 ) and (Re>10000) and (Re<2000000) (Re<2000000) then Eu1:=(.162+1810/Re+7.92e7/re^2-1.65e12/Re^3+8.72e15/re^4) Eu2:=(.119+4980/Re-5.07e7/Re^2+2.51e11/Re^3-4.62e14/re^4) Eu=(Eu2-Eu1)/(2.5-2)*(a-2)+Eu1 endif If (a>=2.5) and (Re>100) and (Re<5000) then Eu:=(.33+98.9/re-1.48e4/Re^2+1.92e6/re^3-8.62e7/re^4) endif
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If (a>=2.5) and (Re>5000) and (Re<2000000) then Eu:=(.119+4980/Re-5.07e7/Re^2+2.51e11/Re^3-4.63e14/re^4) endif If Eu=0 then Check2=0 {Modify for less than 4 rows}
z=1 C=0 c_z=0 if nrow<10 then repea If z>=3 then c_z=1 else IF Re>=10 THEN c_z1=1.065-(.180/(z-.297)) c_z2=1.798-(3.497/(z+1.273)) c_z=(c_z2-c_z1)/(100-10)*(Re-10)+c_z1 endif IF Re>=100 THEN c_z1=1.798-(3.497/(z+1.273)) c_z2=1.149-(.411/(z-.412)) c_z=(c_z2-c_z1)/(1000-100)*(Re-100)+c_z1 endif IF Re>=1000 THEN c_z1=1.149-(.411/(z-.412)) c_z2=.924+(.269/(z+.143)) c_z=(c_z2-c_z1)/(10000-1000)*(Re-1000)+c_z1 endif IF Re>=10000 THEN c_z1=.924+(.269/(z+.143)) c_z2=.62+(1.467/(z+.667)) c_z=(c_z2-c_z1)/(100000-10000)*(Re-10000)+c_z1 endif IF Re>=100000 THEN c_z=.62+(1.467/(z+.667)) endif endif z=z+1 C=C+c_z until z>nrow C=C/nrow
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If C=0 then Check3=0 endif Eu=Eu*C*k1 geteuler=Eu end Procedure surf_eff(D_o_1, h_bar_a,h_f, h _bar_a,h_f, d_f,t, d_f,t, Af,Ao:fin_eff, Af,Ao:fin_eff,surf surfeff) eff) {finds the tube surface efficiencey and fin efficiency } h_f=h_f*convert(in,ft) d_f=d_f*convert(in,ft) "[ft]" {outside radius of tube} r_t=D_o_1/2 "[ft]" M=h_f/2 L=.5*sqrt(d_f^2+M^2) "[ft]" psi=M/r_t BETA=L/M "[ft]" R_e=R_t*1.27*psi*(BETA-.3)^.5 for pure k=237*convert(W/m-K, BTU/hr-ft-R) "[BTU/hr-ft-R]" {conductivity for Aluminum, Incropera & Dewit Dewitt} t} m_eff=sqrt(2*h_bar_a/(k*t)) "[1/ft]" phi=(R_e/R_t-1)*(1+.35*ln(R_e/r_t)) fin_eff=tanh(m_eff*r_t*phi)/(m_eff*r_t*phi) surfeff = 1 - Af/Ao*(1-fin_eff) end
Procedure sat_size(Cunmixed, E:UA) {Finds the UA of the saturated portions of the heat exchangers} Cr:=0 NTU:=-ln(1-E) UA:=NTU*Cunmixed end Procedure exch_size_un_un(Cair, Cfridge,UA:E) {Finds the UA of the sub-cooled and/or superheated sections of the heat exchangers} Cmin=min(Cair, Cfridge) Cmax=max(Cair, Cfridge) Cr=Cmin/Cmax NTU=UA/Cmin E=1-exp((1/Cr)*NTU^.22*(exp(-Cr*(NTU^.78))-1)) end Procedure tubing(Type:D_i,D_o) {Returns the inner and outer diameter of copper tubes based on AAON product specifications
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Type 1 2 3 4 }
Stan St and dard size(in) 5/16 3/8 1/2 5/8
if type=1 then D_i=.2885 D_o=.3125 endIF if type=2 then D_i=.3490 D_o=.375 endIF if type=3 then D_i=.4680 D_o=.5000 endIF if type=4 then D_i=.5810 D_o=.6250 endIF D_i=D_i/12 D_o=D_o/12 end
Function Function compeff(P_o, P_i, T_o, T_i) {computes efficiency of scroll compressor based on condensing and evaporating Temperature and pressure} Pr=P_o/P_i Tr=(T_o+459)/(T_i+459) compeff=-60.25-3.614*Pr-.0281*Pr^2+111.3*Tr-50.31*Tr^2+3.061*Tr*Pr end Function fri(Tac) {Sets the ambient temperature weight fractions in order to compute the seasonal COP} fri=0 If (Tac>65) and (Tac<69) then fri =.214 endif If Tac=72 then fri =.231 endif
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If Tac =77 then fri =.216 endif If Tac = 82 then fri =.161 endif If Tac=87 then fri =.104 endif If Tac=92 then fri =.052 endif If Tac=97 then fri =.018 endif If Tac=102 then fri =.004 END
Module At95(Tsc, V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c:PD, m_sys, A_e, A_c, Tc_ave,width_e,width_c,W_dot_fc,W_dot_com,DELTAP_tot_ac,CF_e,CF_c,DELTAP_ ResideBEND_total,DELTAP_Residecondnser_total,L_22a,L_2a2b,L_2b3) {This model returns the compressor piston displacement, amount of sub-cool, e vaporator frontal area, condenser frontal area and mass of refrigerant in the system in order to provide an evaporator capacity of 30,000 Btu/hr at 95 F ambient temperature} {System Constraints} {variable refrigeration cycle parameters} {Design Conditions @ Tac1=95 F} "[F]" T4a=45 "[Btu/hr]" Q_dot_e=30000
x4a=1 x2a=1 "[F]"" {r "[F] {refr efrige igeran rantt superh superheat eat in evapo evaporat rator or from from states states 4a-1, 4a-1, F} F} Tsh=10 Tc_ave=(T2a+T2b)/2 "[F]" "[ft ft]]" {roughn hneess fo forr draw awn n tubing (White) e),, ft} e=.000005 "[lbm/ "[l bm/hr] hr]"" {ma {mass ss flow flow rate rate per per tube} tube} m_r_t=m_dot_r/tpc_c
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{Air flow over Condenser}
Tac1=95 "[F]" {Air inlet T into Condenser} V_ac=V_dot_ac*convert(1/min,1/sec)/A_c {Air velocity over condenser} "[ft/s]" {viscosity of air flowing over the mu_ac=viscosity(AIR, T=Tac1)*convert(1/hr,1/s) condenser} “[lbm/ft-s]" rho_ac1=density(AIR, T=Tac1, P=Pac1) "{density of air flowing over the condenser} [lbm/ft^3]" m_dot_ac=m_ac*convert(1/hr,1/s) "{mass flow rate of air flowing over the condenser} [bm/s]" m_ac=V_dot_ac*convert(1/min,1/hr)*rho_ac1 "{mass of air flowing over the condenser} [lbm/hr]" h_bar_ac=ha(h_c, eta_c,t_c, width_c,m_dot_ac, mu_ac, D_o_c, A_o_c,A_t_c, C_p_air, Pr_ac, nrow_c) {air-side heat transfer coefficient over the condenser} "[Btu/hr-ft^2-R]" c_p_air=specheat(AIR, T=Tac1) {specific heat at constant pressure of air flowing over the condenser} "[Btu/lbm-R]" Pr_ac=prandtl(AIR, T=Tac1) {Prandtl number of air flowing over the condenser} {Air Flow over Evaporator} "[F]" {Air inlet T into Evaporator} Tae1=80 "[cfm]" {air flow rate over evaporator in cfm V_dot_ae=30000*400/12000 assuming 400 cfm/ton at design Q_e of 30,000 BTU/hr} V_ae=V_dot_ae*convert(1/min,1/sec)/A_e "{Velocity of air flow over evaporator} [ft/sec]" rho_ae1=density(AIR, T=Tae1, P=14.7) {Density of air flow over the evaporator} "[lbm/ft^3]" m_ae=V_dot_ae*convert(1/min,1/hr)*rho_ae1 {mass of air flowing over the evaporator} "[lbm/hr]" {mass flow rate of air flowing over the m_dot_ae=m_ae*convert(1/hr,1/s) evaporator} "[lbm/s]" mu_ae=viscosity(AIR, T=Tae1)*convert(1/hr,1/sec) {Viscosity of are flowing over the evaporatr} " [lbm/ft-s]"
h_bar_ae=ha(h_e, eta_e,t_e, width_e,m_dot_ae, mu_ae, D_o_e, A_o_e, A_t_e, C_p_air, Pr_ac, nrow_e) {heat transfer coefficient of air flowing over the evaporator} "[Btu/hr ft^2-R]" W_dot_fe=365*V_dot_ae*convert(W, BTU/hr)/1000 {Compressor}
nc=compeff(P2a,P4,tc_ave,T4)
{compressor efficiency thermal efficiency}
{specific heat ratio of Cp/Cv} gamma_R410A=1.16 {Percent} Clearance=.05 "[%]" v1=volume(R410A, P=P1,T=T1)"[ft^3/lbm]"
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v2=volume(R410A, P=P2,T=T2)"[ft^3/lbm]" {Compressor volumetric efficiency, Klein} nv=1-R*(v1/v2-1) R=.025 {ratio of clearance volume to displacement} PD=m_dot_r*v1/nv "[ft^3/hr]" {compressor piston displacement} {condenser Characteristics} {Variable Condenser characteristics} spac_rat=h_f_c/d_f_c {tube spacing ratio-horizontal to vertical tube spacing} "[ft]" {condenser depth, ft} Dep_c=d_fft_c*nrow_c width_c=3 {base configuration width} "[ft]" L_c=Width_c*nrow_c*ncircuit_c "[ft]" {Total length of condenser} "[ft]" {height of condenser, ft} H_c=h_fft_c*tpc_c*ncircuit_c V_c=Width_c*h_c*dep_c "[ft^3]" {Volume of Condenser} "[ft^2]" {frontal area of condenser, ft^2} A_c=Width_c*H_c CALL Surf_eff(D_o_c, h_bar_ac,h_f_c, d_f_c,t_c, A_f_c,A_o_c:phi_f,phi_c) {calls the fin efficiency and tube surface efficiency for the condenser} Call tubing(TubeType_c:D_i_c,D_o_c) {calls the tube diameter based on the 4 tube types for the condenser} A_i_c=L_c*D_i_c*pi*tpc_c {the condenser refrigerant-side inner tube heat transfer area} "[ft^2]" "[ft^2]" A_t_c=D_o_c*pi*L_c*(1-t_c*eta_c)*tpc_c A_f_c=2*L_c*eta_c*tpc_c*(h_fft_c*d_fft_c-pi*(D_o_c/2)^2) {the total fin heat transfer area} "[ft^2]" A_o_c=A_t_c+A_f_c {the total heat transfer area -air-side and refrigerant} "[ft^2]" A_flow_c=Width_c*(1-eta_c*t_c)*(H_c-D_o_c*ncircuit_c*tpc_c)"{the total refrigerant flow area} [ft^2]" {the condenser refrigerant-side inner tube heat A_i_c=A_i_22a+A_i_2a2b+A_i_2b3 transfer area} "[ft^2]"
{evaporator Characteristics} {Variable Evaporator characteristics} "[in]" {tube vertical spacing on centers, in} h_f_e=1 "[ft]" {thickness of fins, ft} t_e=.006/12 eta_e=12*12 "[1/ft]" {evaporator fin pitch, fins/ft} "[in]" {evaporator fin depth per tube, in} d_f_e=.625 {evaporator depth, ft} Dep_e=d_f_e*nrow_e*convert(in,ft) "[ft]" {number of tubes per refrigerant flow parallel circuit} tpc_e=2 {number of rows of tubing} nrow_e=4 ncircuit_e=9 L_e=Width_e*nrow_e*ncircuit_e {evaporator tube length} "[ft]" H_e=h_f_e*tpc_e*ncircuit_e*convert(in,ft "[ft]" {height of evaporator ft} TubeType_e=2
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V_e=width_e*h_e*dep_e "[ft^3]" {Volume of evaporator} "[ft^2]" {frontal area of evaporator ft^2} A_e=Width_e*H_e CALL Surf_eff(D_o_e, h_bar_ae,h_f_e, d_f_e,t_e, A_f_e,A_o_e:phi_f_e,phi_e) {calls the fin efficiency and tube surface efficiency for the evaporator} Call tubing(TubeType_e:D_i_e,D_o_e) {calls the tube diameter based on the 4 tub types for the evaporator} d_fft_e=d_f_e*convert(in,ft) "[ft]" "[ft]" h_fft_e=h_f_e*convert(in,ft "[ft^2]" {the evaporator refrigerant-side inner tube A_i_e=L_e*D_i_e*pi*tpc_e heat transfer area} A_t_e=D_o_e*pi*L_e*(1-t_e*eta_e)*tpc_ {the total refrigerant side tube heat transfer area for the evaporator} "[ft^2]" A_f_e=2*h_fft_e*tpc_e*d_fft_e*eta_e*L_e-2*pi*(D_o_e/2)^2*eta_e*L_e*tpc_e {the "[ft^2]" total fin heat transfer area for the evaporator} A_o_e=A_t_e+A_f_e {the total heat transfer area -air-side and refrigerant} "[ft^2]" A_flow_e=width_e*(1-eta_e*t_e)*(H_e-D_o_e*ncircuit_e*tpc_e) { the total refrigerant flow area for the evaporator}"[ft^2]"
{*********************************************************** Begin Cycle Analysis -analyzes the vaporcompression refrigeration cycle ************************************************************}
{Compressor Equations} "[Btu/lbm]" h1=enthalpy(R410A, T=T1, P=P1) "[Btu/lbm-R]" s1=entropy(R410A, T=T1, P=P1) "[Btu/lbm-R]" s1=s2s h2s=enthalpy(R410A, P=P2, s=s2s) "[btu/lbm]" wcs= h2s-h1 "[btu/lbm]" "[Btu/lbm]" wc=wcs/nc "[btu/lbm]" h2=h1+wc T2=temperature(R410A, P=P2, h=h2) "[F]" {Condenser Equations} {pressure of refrigerant exiting the P2a=P2-DELTAP_22a-DELTAP_b_22a superheated portion of the condenser} "[psia]" P2b=P2a-DELTAP_2a2b-DELTAP_b_2a2b {pressure of refrigerant exiting the saturated portion of the condenser} "[psia]" P3=P2b-DELTAP_2b3-DELTAP_b_2b3 {pressure of refrigerant exiting the subcooled portion of the condenser} "[psia]"
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{Superheated portion of condenser} T2a=temperature(R410A, P=P2a, x=x2a) {Temperature of refrigerant exiting the superheated portion of the condenser} "[F]" {Enthalpy of refrigerant exiting th h2a=enthalpy(R410A, T=T2a, x=x2a) superheated portion of the condenser} "[Btu/lbm]" Q_22a=m_dot_r*(h2-h2a) "[Btu/hr]" "[Btu/hr]" Q_22a=E_22a*min(C_22a,C_a22a)*(T2-Tac1) C_a22a=m_ac*specheat(AIR, T=Tac1)*L_22a/L_c "[Btu/hr-R]" "[Btu/hr-R]" C_22a=m_dot_r*specheat(R410A, T=T2, P=P2) Call exch_size_un_un(C_a22a, C_22a,UA_22a:E_22a) UA_22a=U_o_22a*A_o_22a {Superheated UA} "[Btu/hr-R]" U_o_22a=(1/(phi_c*h_bar_ac)+A_o_22a/(h_bar_22a*A_i_22a))^(-1) "[Btu/hr-ft^2-R]" Call h_bar_single(D_i_c, m_r_t, T2, T2a, P2:Re_22a, h_bar_22a, rho_22a) A_t_22a=D_o_c*pi*L_22a*(1-t_c*eta_c)*tpc_c "[ft^2]" A_f_22a=2*h_f_c*tpc_c*convert(in, ft)*d_f_c*convert(in,ft)*eta_c*L_22a2*pi*(D_o_c/2)^2*eta_c*L_22a*tpc_c "[ft^2]" "[ft^2]" A_o_22a=A_t_22a+A_f_22a "[ft^2]" A_i_22a=L_22a*tpc_c*pi*D_i_c "[Btu/hr]" Q_22a=(A_i_22a/A_i_c)*m_ac*(hac22a-hac1) Q_2a2b=(A_i_2a2b/A_i_c)*m_ac*(hac2a2b-hac1) "[Btu/hr]" "[Btu/hr]" Q_2b3=(A_i_2b3/A_i_c)*m_ac*(hac2b3-hac1) "[F]" Tac22a=temperature(AIR, h=hac22a) Tac2a2b=temperature(AIR,h=hac2a2b) "[F]" "[F]" Tac2b3=temperature(AIR,h=hac2b3) "[Btu/lbm]" hac1=enthalpy(AIR, T=Tac1) Call SingleDP(m_dot_r, tpc_c,D_i_c,L_22a,f_22a,rho_22a:DELTAP_22a) call singlebenddrop(tpc_c, D_i_c, m_dot_r,P1, T2, T2a, L_22a, Width_c, f_22a:DELTAP_b_22a) 1/f_22a^0.5=-2*log10((e/(D_i_c*3.7))+2.51/(Re_22a*f_22a^0.5)) {Saturated portion of condenser} T2b=temperature(R410A, P=P2b, x=.1) "[F]" CALL TwophaseDp(x2a, x2b,T2a, T2b, D_i_c, m_dot_r, tpc_c,L_2a2b:DELTAP_2a2b) CALL tpbenddrop(tpc_c,D_i_c,m_dot_r, h_f_c, T_c, L_c, L_22a, L_2a2b, Width_c:DELTAP_b_2a2b) x2b=0 h2b=enthalpy(R410A, T=T2b, x=x2b) "[Btu/lbm]" Q_2a2b=m_dot_r*(h2a-h2b)"[Btu/hr]" Q_2a2b=E_2a2b*C_a2a2b*(T2a-Tac1) "[Btu/hr-R]" C_a2a2b=m_ac*specheat(AIR, T=Tac1)*L_2a2b/L_c
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Call sat_size(C_a2a2b, E_2a2b:UA_2a2b) U_o_2a2b=(1/(phi_c*h_bar_ac)+A_o_2a2b/(h_bar_2a2b*A_i_2a2b))^(-1) "[Btu/hr ft^2-R]" UA_2a2b=U_o_2a2b*A_o_2a2b"[Btu/hr-R]" h_bar_2a2b= h_bar_c(T2a, P2a,D_i_c, m_dot_r,tpc_c) "[Btu/hr-ft^2-R]" A_t_2a2b=D_o_c*pi*L_2a2b*(1-t_c*eta_c)*tpc_c "[ft^2]" A_f_2a2b=2*tpc_c*L_2a2b*eta_c*(h_fft_c*D_fft_c-pi*(D_o_c/2)^2) "[ft^2]" A_o_2a2b=A_t_2a2b+A_f_2a2b"[ft^2]" A_i_2a2b=L_2a2b*tpc_c*pi*D_i_c "[ft^2]" {Sub-cooled portion of Condenser} "[Btu/lbm]" h3=enthalpy(R410A, T=T3, P=P3) T3=T2b-Tsc "[F]" Q_2b3=m_dot_r*(h2b-h3) "[Btu/hr]" "[Btu/hr]" Q_2b3=E_2b3*min(C_2b3, C_a2b3)*(T2b-Tac1) C_a2b3=m_ac*specheat(AIR, T=Tac1)*L_2b3/L_c "[Btu/hr-R]" "[Btu/hr-R]"{assume Cp for C_2b3=m_dot_r*specheat(R410A, T=T3, P=P3) R410A constant over Tsc} {CALL Exch_size(C_2b3, C_a2b3, E_2b3:UA_2b3)} Call exch_size_un_un(C_a2b3, C_2b3,UA_2b3:E_2b3) UA_2b3=U_o_2b3*A_o_2b3 "[Btu/hr-R]" U_o_2b3=(1/(phi_c*h_bar_ac)+A_o_2b3/(h_bar_2b3*A_i_2b3))^(-1) "[Btu/hr-ft^2-R]" Call h_bar_single(D_i_c, m_r_t, T2b, T3, P2b:Re_2b3,h_bar_2b3, rho_2b3) A_t_2b3=D_o_c*pi*L_2b3*(1-t_c*eta_c)*tpc_c "[ft^2]" A_f_2b3=2*h_f_c*tpc_c*convert(in, ft)*d_f_c*convert(in,ft)*eta_c*L_2b32*pi*(D_o_c/2)^2*eta_c*L_2b3*tpc_c "[ft^2]" A_o_2b3=A_t_2b3+A_f_2b3 "[ft^2]" "[ft^2]" A_i_2b3=L_2b3*tpc_c*pi*D_i_c "[ft/hr]" {velocity of refrigerant vel_2b3=m_r_t/((pi*D_i_c^2/4)*rho_2b3) through tube, ft/hr} Call SingleDP(m_dot_r, tpc_c,D_i_c,L_2b3,f_2b3,rho_2b3:DELTAP_2b3) call singlebenddrop(tpc_c, D_i_c, m_dot_r,P2b, T2b, T3, L_2b3, Width_c, f_2b3:DELTAP_b_2b3) 1/f_2b3^0.5=-2*log10((e/(D_i_c*3.7))+2.51/(Re_2b3*f_2b3^0.5))
{Total Refrigerant Side Pressure Drop for condenser} DELTAP_Residecondnser_total= DELTAP_22a+DELTAP_b_22a+DELTAP_2a2b+ DELTAP_b_2a2b+DELTAP_2b3+DELTAP_b_2b3 "[psia]" {Total Refrigerant Side Pressure Drop Due To Bends in the Condenser}
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DELTAP_ResideBEND_total=DELTAP_b_22a+DELTAP_b_2a2b+DELTAP_b_2b3 "[psia]" {Valve Equation} h4=h3 "[Btu/lbm]" {Evaporator Equations} "[psia]" P4=P4a "[psia]" P4=P1 Q_dot_e=Q_44a+Q_4a1"[Btu/hr]" A_i_e=A_i_44a+A_i_4a1 "[ft^2]" {A_o_e=A_i_e*D_o_1/D_i_c} T4=T4a"[F]" P4=pressure(R410A, T=T4, h=h4) "[psia]" {m_ae=V_dot_ae*convert(1/min,1/hr)/volume(AIR, T=Tac1, P=14.7)} x4=quality(R410A, T=T4, h=h4) {saturated portion of evaporator} Cp_44a_cor=specheat(AIR, T=Tae1)*1.33 "[Btu/lbm-F]" "[Btu/lbm]" h4a=enthalpy(R410A, T=T4a, x=x4a) "[Btu/hr]" Q_44a=m_dot_r*(h4a-h4) Q_44a=E_44a*C_a44a*(Tae1-T4) Q_44a=(A_i_44a/A_i_e)*C_a44a*(-Tae44a+Tae1) "[Btu/hr]" C_a44a=m_ae*Cp_44a_cor*A_i_44a/A_i_e call sat_size(C_a44a,E_44a:UA_44a) UA_44a=U_i_44a*A_i_44a "[Btu/hr-R]" "[Btu/hr-ft^2-R]" U_i_44a=(C1+1/h_bar_44a)^(-1) h_bar_44a=h_bar_e(T4, P4,D_i_c, m_dot_r, x4) "[Btu/hr-ft^2-R]" {CALL TwophaseDp(x4a,x4, T4, T4a, D_i_e, m_dot_r, tpc_e,L_44a:DELTAP_44a) CALL tpbenddrop(tpc_c,D_i_c,m_dot_r, h_f_c, T_c, L_c, L_22a, L_2a2b, Width_c:DELTAP_b_2a2b)} "[ft^2]" A_i_44a=L_44a*tpc_e*pi*D_i_e {superheated portion of evaporator} "[F]" T1=T4a+Tsh Q_4a1=m_dot_r*(h1-h4a) "[btu/hr]" Q_4a1=E_4a1*min(C_4a1, C_a4a1)*(Tae1-T4a) C_a4a1=m_ae*specheat(AIR, T=Tae1)*A_i_4a1/A_i_e C_4a1=m_dot_r*specheat(R410A, T=T1, P=P1) call exch_size_un_un(C_4a1,C_a4a1, UA_4a1:E_4a1) "[Btu/hr-R]" UA_4a1=U_i_4a1*A_i_4a1 "[Btu/hr-ft^2-R]" U_i_4a1=(C1+1/h_bar_4a1)^(-1) Call h_bar_single(D_i_e, m_dot_r, T4a, T1, P4a:Re_4a1,h_bar_4a1, rho_4a1)
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{C1=D_i_c/(D_o_1*h_bar_ae)} {constant represents air side term for evaporator U} C1=1/(Area_rat*h_bar_ae) "[hr-ft^2-R/BTU]" Area_rat=A_o_e/A_i_e {Call SingleDP(m_dot_r, tpc_e,D_i_e,L_4a1,f_4a1,rho_4a1:DELTAP_4a1) call singlebenddrop(tpc_e, D_i_e, m_dot_r,P4a, T4a, T1, L_4a1, Width_e, f_4a1:DELTAP_b_4a1)} 1/f_4a1^0.5=-2*log10((e/(D_i_e*3.7))+2.51/(Re_4a1*f_4a1^0.5)) "[ft^2]" A_i_4a1=L_4a1*tpc_e*pi*D_i_e {COP} W_dot_com=wc*m_dot_r "[Btu/hr]" Q_c=Q_22a+Q_2a2b+Q_2b3 "[Btu/hr]" COP=Q_dot_e/(W_dot_com+W_dot_fc+W_dot_fe) {Mass balances} Vol_22a=L_22a*D_i_c^2*pi*tpc_c/4 "[ft^3]" Vol_2a2b=L_2a2b*D_i_c^2*pi*tpc_c/4 "[ft^3]" Vol_2b3=L_2b3*D_i_c^2*pi*tpc_c/4 "[ft^3]" "[ft^3]" Vol_44a=A_i_44a*D_i_c/4 Vol_4a1=A_i_4a1*D_i_c/4 "[ft^3]" m_22a=rho_22a*Vol_22a "[lbm]" vfg2a2b=volume(R410A, T=T2a, x=1)-volume(R410A, T=T2a, x=0) "[ft^3/lbm]" m_2a2b=-(Vol_2a2b/vfg2a2b)*ln(volume(R410A, T=T2a, x=0)/volume(R410A, T=T2a, x=1)) "[lbm]" "[lbm]" m_2b3=rho_2b3*Vol_2b3 m_c=m_22a+M_2a2b+m_2b3 "[lbm]" "[lbm]" m_4a1=rho_4a1*Vol_4a1 vfg44a=volume(R410A, T=T4a, x=1)-volume(R410A, T=T4a, x=0) "[ft^3/lbm]" m_44a=(Vol_44a/(x4*vfg44a))*ln(volume(R410A, T=T4, x=1)/(volume(R410A, T=T4, x=0)+x4*vfg44a)) "[lbm] check this equation" m_sys=m_4a1+m_44a+m_c "[lbm]" m_e=m_4a1+m_44a {Air Side Pressure Drop} "fan efficiency" E_fc=.65 W_dot_fc=V_ac*DELTAP_tot_ac*convert(psia,lbf/ft^2)*A_c/E_fc*convert(ftlbf/s,btu/hr) "[Btu/hr]" d_fft_c=d_f_c*convert(in,ft) "[ft]" "[ft]" h_fft_c=h_f_c*convert(in,ft
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{Flow rate} G_max_ac=m_ac/A_flow_c "[lbm/ft^2 hr]" "[psia]" Pac2=P_atm "[psia]" P_atm=14.7 grav=32.2*convert(1/s^2,1/hr^2)"[lbm-ft/hr^2-lbf]" Re_D_c=G_max_ac*D_o_c/(mu_ac*convert(1/s,1/hr)) {Pressure Drop Calculation} DELTAP_tot_ac=Pac1-Pac2
"[psia]"
Eu_c=GetEuler(Re_d_c, d_f_c, h_f_c, D_o_c, nrow_c) DELTAP_tubes=Eu_c*G_max_ac^2*n row_c/(2*rho_ac1)*convert(lbm-ft/ft2-hr2, psia) "[psia]" DELTAP_tubes_inH2O=DELTAP_tubes*convert(psia, inH2O) "[inH2O]" DELTAP_tot_ac=DELTAP_tubes+DELTAP_fin DELTAP_fin=(f_f*G_max_ac^2*A_f_c/(2*A_flow_c*grav*rho_ac1))*convert(1/ft^2,1/ in^2) "[psia]" DELTAP_fin_inH2O=DELTAP_fin*convert(psia,inh2o) "[inh2O]" f_f=1.7*Re_L_ac^(-.5) Re_L_ac=G_max_ac*h_fft_c/mu_ac*convert(1/hr, 1/s) {Cost Factors for metals Fins made from pure aluminum Tubes made from pure copper} {copper is about $0.8/lb on the London Metals Cf_cu=.8 "[1/lbm]" Exchange} "[1/lbm]" {aluminum is about $0.7/lb} Cf_al=.7 "[lbm/ft^3]"{Incropera and DeWitt} rho_al=2702*convert(kg/m^3,lbm/ft^3) "[lbm/ft^3]" rho_cu=8933*convert(kg/m^3, lbm/ft^3) V_cu=L_c*pi*(D_o_c^2/4-D_i_c^2/4)*tpc_c+L_e*pi*(D_o_e^2/4-D_i_e^2/4)*tpc_e "[ft^3]" "[ft^3]" V_al=A_f_c*t_c/2+A_f_e*t_e/2 CF=(rho_al*V_al*Cf_al+rho_cu*V_cu*Cf_cu)/CF_base_total CF_base_total=35.88 {CF=1} CF_e=(rho_al*A_f_e*t_e/2*Cf_al+rho_cu*L_e*pi*(D_o_e^2/4D_i_e^2/4)*tpc_e*Cf_cu)/CF_base_e CF_c=(rho_al*A_f_c*t_c/2*Cf_al+rho_cu*L_c*pi*(D_o_c^2/4D_i_c^2/4)*tpc_c*Cf_cu)/CF_base_c End
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Module WithSubcool(Tac1, PD, A_e, A_c, m_sys, V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c: Den_COPseas_i, Tsc) {This module returns the seasonal COP of the system and the sub-cool in the condenser for the various ambient temperatures for a system whose compressor has been sized for a system capacity of 30,000 Btu/hr at 95 F ambient temperature}
x4a=1 x2a=1 "[F]" {refrigerant superheat in evaporator from states 4a-1, F} Tsh=10 Tc_ave=(T2a+T2b)/2 "[F]" "[ft]" {roughness for drawn tubing (White), ft} e=.000005 "[lbm/hr]" {mass flow rate per tube} m_r_t=m_dot_r/tpc_c {Air flow over Condenser} V_ac=V_dot_ac*convert(1/min,1/sec)/A_c {Air velocity over condenser} "[ft/s]" {viscosity of air flowing over the mu_ac=viscosity(AIR, T=Tac1)*convert(1/hr,1/s) condenser} “[lbm/ft-s]" rho_ac1=density(AIR, T=Tac1, P=Pac1) "{density of air flowing over the condenser} [lbm/ft^3]" m_dot_ac=m_ac*convert(1/hr,1/s) "{mass flow rate of air flowing over the condenser} [bm/s]" m_ac=V_dot_ac*convert(1/min,1/hr)*rho_ac1 "{mass of air flowing over the condenser} [lbm/hr]" h_bar_ac=ha(h_c, eta_c,t_c, width_c,m_dot_ac, mu_ac, D_o_c, A_o_c,A_t_c, C_p_air, Pr_ac, nrow_c) {air-side heat transfer coefficient over the condenser} "[Btu/hr-ft^2-R]" c_p_air=specheat(AIR, T=Tac1) {specific heat at constant pressure of air flowing over the condenser} "[Btu/lbm-R]" Pr_ac=prandtl(AIR, T=Tac1) {Prandtl number of air flowing over the condenser} {Air Flow over Evaporator} "[F]" {Air inlet T into Evaporator} Tae1=80 "[cfm]" {air flow rate over evaporator in cfm V_dot_ae=30000*400/12000 assuming 400 cfm/ton at design Q_e of 30,000 BTU/hr} V_ae=V_dot_ae*convert(1/min,1/sec)/A_e "{Velocity of air flow over evaporator} [ft/sec]" rho_ae1=density(AIR, T=Tae1, P=14.7) {Density of air flow over the evaporator} "[lbm/ft^3]" m_ae=V_dot_ae*convert(1/min,1/hr)*rho_ae1 {mass of air flowing over the evaporator} "[lbm/hr]" m_dot_ae=m_ae*convert(1/hr,1/s) {mass flow rate of air flowing over the evaporator} "[lbm/s]"
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mu_ae=viscosity(AIR, T=Tae1)*convert(1/hr,1/sec) {Viscosity of are flowing over the evaporatr} " [lbm/ft-s]" h_bar_ae=ha(h_e, eta_e,t_e, width_e,m_dot_ae, mu_ae, D_o_e, A_o_e, A_t_e, C_p_air, Pr_ac, nrow_e) {heat transfer coefficient of air flowing over the evaporator} "[Btu/hr ft^2-R]" W_dot_fe=365*V_dot_ae*convert(W, BTU/hr)/1000 {Compressor} nc=compeff(P2a,P4,tc_ave,T4)
{compressor efficiency thermal efficiency}
{specific heat ratio of Cp/Cv} gamma_R410A=1.16 Clearance=.05 "[%]" {Percent} v1=volume(R410A, P=P1,T=T1)"[ft^3/lbm]" v2=volume(R410A, P=P2,T=T2)"[ft^3/lbm]" nv=1-R*(v1/v2-1) {Compressor volumetric efficiency, Klein} {ratio of clearance volume to displacement} R=.025 PD=m_dot_r*v1/nv "[ft^3/hr]" {compressor piston displacement} {condenser Characteristics} {Variable Condenser characteristics} spac_rat=h_f_c/d_f_c {tube spacing ratio-horizontal to vertical tube spacing} Dep_c=d_fft_c*nrow_c "[ft]" {condenser depth, ft} L_c=Width_c*nrow_c*ncircuit_c "[ft]" {Total length of condenser} "[ft]" {height of condenser, ft} H_c=h_fft_c*tpc_c*ncircuit_c V_c=Width_c*h_c*dep_c "[ft^3]" {Volume of Condenser} "[ft^2]" {frontal area of condenser, ft^2} A_c=Width_c*H_c CALL Surf_eff(D_o_c, h_bar_ac,h_f_c, d_f_c,t_c, A_f_c,A_o_c:phi_f,phi_c) {calls the fin efficiency and tube surface efficiency for the condenser} Call tubing(TubeType_c:D_i_c,D_o_c) {calls the tube diameter based on the 4 tube types for the condenser} A_i_c=L_c*D_i_c*pi*tpc_c {the condenser refrigerant-side inner tube heat transfer area} "[ft^2]" "[ft^2]" A_t_c=D_o_c*pi*L_c*(1-t_c*eta_c)*tpc_c A_f_c=2*L_c*eta_c*tpc_c*(h_fft_c*d_fft_c-pi*(D_o_c/2)^2) {the total fin heat transfer area} "[ft^2]" A_o_c=A_t_c+A_f_c {the total heat transfer area -air-side and refrigerant} "[ft^2]" A_flow_c=Width_c*(1-eta_c*t_c)*(H_c-D_o_c*ncircuit_c*tpc_c)"{the total refrigerant flow area} [ft^2]" {the condenser refrigerant-side inner tube heat A_i_c=A_i_22a+A_i_2a2b+A_i_2b3 transfer area} "[ft^2]"
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{evaporator Characteristics} {Variable Evaporator charcteristics} h_f_e=1 "[in]" {tube vertical spacing on centers, in} "[ft]" {thickness of fins, ft} t_e=.006/12 "[1/ft]" {evaporator fin pitch, fins/ft} eta_e=12*12 d_f_e=.625 "[in]" {evaporator fin depth per tube, in} Dep_e=d_f_e*nrow_e*convert(in,ft) "[ft]" {evaporator depth, ft} {number of tubes per refrigerant flow parallel circuit} tpc_e=2 {number of rows of tubing} nrow_e=4 ncircuit_e=9 L_e=Width_e*nrow_e*ncircuit_e {evaporator tube length} "[ft]" {height of evaporator ft} H_e=h_f_e*tpc_e*ncircuit_e*convert(in,ft "[ft]" TubeType_e=2 V_e=width_e*h_e*dep_e "[ft^3]" {Volume of evaporator} "[ft^2]" {frontal area of evaporator ft^2} A_e=Width_e*H_e CALL Surf_eff(D_o_e, h_bar_ae,h_f_e, d_f_e,t_e, A_f_e,A_o_e:phi_f_e,phi_e) {calls the fin efficiency and tube surface efficiency for the evaporator} Call tubing(TubeType_e:D_i_e,D_o_e) {calls the tube diameter based on the 4 tub types for the evaporator} "[ft]" d_fft_e=d_f_e*convert(in,ft) "[ft]" h_fft_e=h_f_e*convert(in,ft "[ft^2]" {the evaporator refrigerant-side inner tube A_i_e=L_e*D_i_e*pi*tpc_e heat transfer area} A_t_e=D_o_e*pi*L_e*(1-t_e*eta_e)*tpc_ {the total refrigerant side tube heat transfer area for the evaporator} "[ft^2]" A_f_e=2*h_fft_e*tpc_e*d_fft_e*eta_e*L_e-2*pi*(D_o_e/2)^2*eta_e*L_e*tpc_e {the "[ft^2]" total fin heat transfer area for the evaporator} A_o_e=A_t_e+A_f_e {the total heat transfer area -air-side and refrigerant} "[ft^2]" A_flow_e=width_e*(1-eta_e*t_e)*(H_e-D_o_e*ncircuit_e*tpc_e) { the total refrigerant flow area for the evaporator}"[ft^2]"
{*********************************************************** Begin Cycle Analysis -analyzes the vaporcompression refrigeration cycle ************************************************************}
{Compressor Equations} "[Btu/lbm]" h1=enthalpy(R410A, T=T1, P=P1) s1=entropy(R410A, T=T1, P=P1) "[Btu/lbm-R]" "[Btu/lbm-R]" s1=s2s
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h2s=enthalpy(R410A, P=P2, s=s2s) "[btu/lbm]" "[btu/lbm]" wcs= h2s-h1 wc=wcs/nc "[Btu/lbm]" "[btu/lbm]" h2=h1+wc T2=temperature(R410A, P=P2, h=h2) "[F]" {Condenser Equations} P2a=P2-DELTAP_22a-DELTAP_b_22a { pressure of refrigerant exiting the superheated portion of the condenser} "[psia]" P2b=P2a-DELTAP_2a2b-DELTAP_b_2a2b {pressure of refrigerant exiting the saturated portion of the condenser} "[psia]" {pressure of refrigerant exiting the subP3=P2b-DELTAP_2b3-DELTAP_b_2b3 cooled portion of the condenser} "[psia]" {Superheated portion of condenser} T2a=temperature(R410A, P=P2a, x=x2a) {Temperature of refrigerant exiting the superheated portion of the condenser} "[F]" {Enthalpy of refrigerant exiting th h2a=enthalpy(R410A, T=T2a, x=x2a) superheated portion of the condenser} "[Btu/lbm]" "[Btu/hr]" Q_22a=m_dot_r*(h2-h2a) "[Btu/hr]" Q_22a=E_22a*min(C_22a,C_a22a)*(T2-Tac1) C_a22a=m_ac*specheat(AIR, T=Tac1)*L_22a/L_c "[Btu/hr-R]" C_22a=m_dot_r*specheat(R410A, T=T2, P=P2) "[Btu/hr-R]" Call exch_size_un_un(C_a22a, C_22a,UA_22a:E_22a) UA_22a=U_o_22a*A_o_22a "[Btu/hr-R]" U_o_22a=(1/(phi_c*h_bar_ac)+A_o_22a/(h_bar_22a*A_i_22a))^(-1) "[Btu/hr-ft^2-R]" Call h_bar_single(D_i_c, m_r_t, T2, T2a, P2:Re_22a, h_bar_22a, rho_22a) A_t_22a=D_o_c*pi*L_22a*(1-t_c*eta_c)*tpc_c "[ft^2]" A_f_22a=2*h_f_c*tpc_c*convert(in, ft)*d_f_c*convert(in,ft)*eta_c*L_22a2*pi*(D_o_c/2)^2*eta_c*L_22a*tpc_c "[ft^2]" "[ft^2]" A_o_22a=A_t_22a+A_f_22a A_i_22a=L_22a*tpc_c*pi*D_i_c "[ft^2]" "[Btu/hr]" Q_22a=(A_i_22a/A_i_c)*m_ac*(hac22a-hac1) Q_2a2b=(A_i_2a2b/A_i_c)*m_ac*(hac2a2b-hac1) "[Btu/hr]" "[Btu/hr]" Q_2b3=(A_i_2b3/A_i_c)*m_ac*(hac2b3-hac1) "[F]" Tac22a=temperature(AIR, h=hac22a) "[F]" Tac2a2b=temperature(AIR,h=hac2a2b) "[F]" Tac2b3=temperature(AIR,h=hac2b3) "[Btu/lbm]" hac1=enthalpy(AIR, T=Tac1) Call SingleDP(m_dot_r, tpc_c,D_i_c,L_22a,f_22a,rho_22a:DELTAP_22a)
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call singlebenddrop(tpc_c, D_i_c, m_dot_r,P1, T2, T2a, L_22a, Width_c, f_22a:DELTAP_b_22a) 1/f_22a^0.5=-2*log10((e/(D_i_c*3.7))+2.51/(Re_22a*f_22a^0.5)) {Saturated portion of condenser} T2b=temperature(R410A, P=P2b, x=.1) "[F]" CALL TwophaseDp(x2a, x2b,T2a, T2b, D_i_c, m_dot_r, tpc_c,L_2a2b:DELTAP_2a2b) CALL tpbenddrop(tpc_c,D_i_c,m_dot_r, h_f_c, T_c, L_c, L_22a, L_2a2b, Width_c:DELTAP_b_2a2b) x2b=0 h2b=enthalpy(R410A, T=T2b, x=x2b) "[Btu/lbm]" Q_2a2b=m_dot_r*(h2a-h2b)"[Btu/hr]" Q_2a2b=E_2a2b*C_a2a2b*(T2a-Tac1) "[Btu/hr-R]" C_a2a2b=m_ac*specheat(AIR, T=Tac1)*L_2a2b/L_c Call sat_size(C_a2a2b, E_2a2b:UA_2a2b) U_o_2a2b=(1/(phi_c*h_bar_ac)+A_o_2a2b/(h_bar_2a2b*A_i_2a2b))^(-1) "[Btu/hr ft^2-R]" UA_2a2b=U_o_2a2b*A_o_2a2b"[Btu/hr-R]" h_bar_2a2b= h_bar_c(T2a, P2a,D_i_c, m_dot_r,tpc_c) "[Btu/hr-ft^2-R]" "[ft^2]" A_t_2a2b=D_o_c*pi*L_2a2b*(1-t_c*eta_c)*tpc_c A_f_2a2b=2*tpc_c*L_2a2b*eta_c*(h_fft_c*D_fft_c-pi*(D_o_c/2)^2) "[ft^2]" A_o_2a2b=A_t_2a2b+A_f_2a2b"[ft^2]" A_i_2a2b=L_2a2b*tpc_c*pi*D_i_c "[ft^2]" {Sub-cooled portion of Condenser} h3=enthalpy(R410A, T=T3, P=P3) "[Btu/lbm]" "[F]" T3=T2b-Tsc Q_2b3=m_dot_r*(h2b-h3) "[Btu/hr]" "[Btu/hr]" Q_2b3=E_2b3*min(C_2b3, C_a2b3)*(T2b-Tac1) C_a2b3=m_ac*specheat(AIR, T=Tac1)*L_2b3/L_c "[Btu/hr-R]" "[Btu/hr-R]"{assume Cp for C_2b3=m_dot_r*specheat(R410A, T=T3, P=P3) R410A constant over Tsc} {CALL Exch_size(C_2b3, C_a2b3, E_2b3:UA_2b3)} Call exch_size_un_un(C_a2b3, C_2b3,UA_2b3:E_2b3) UA_2b3=U_o_2b3*A_o_2b3 "[Btu/hr-R]" U_o_2b3=(1/(phi_c*h_bar_ac)+A_o_2b3/(h_bar_2b3*A_i_2b3))^(-1) "[Btu/hr-ft^2-R]" Call h_bar_single(D_i_c, m_r_t, T2b, T3, P2b:Re_2b3,h_bar_2b3, rho_2b3) A_t_2b3=D_o_c*pi*L_2b3*(1-t_c*eta_c)*tpc_c "[ft^2]" A_f_2b3=2*h_f_c*tpc_c*convert(in, ft)*d_f_c*convert(in,ft)*eta_c*L_2b32*pi*(D_o_c/2)^2*eta_c*L_2b3*tpc_c "[ft^2]" A_o_2b3=A_t_2b3+A_f_2b3 "[ft^2]" A_i_2b3=L_2b3*tpc_c*pi*D_i_c "[ft^2]"
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"[ft/hr]" {velocity of refrigerant vel_2b3=m_r_t/((pi*D_i_c^2/4)*rho_2b3) through tube, ft/hr} Call SingleDP(m_dot_r, tpc_c,D_i_c,L_2b3,f_2b3,rho_2b3:DELTAP_2b3) call singlebenddrop(tpc_c, D_i_c, m_dot_r,P2b, T2b, T3, L_2b3, Width_c, f_2b3:DELTAP_b_2b3) 1/f_2b3^0.5=-2*log10((e/(D_i_c*3.7))+2.51/(Re_2b3*f_2b3^0.5))
{Total Refrigerant Side Pressure Drop for condenser} DELTAP_Residecondnser_total= DELTAP_22a+DELTAP_b_22a+DELTAP_2a2b+ DELTAP_b_2a2b+DELTAP_2b3+DELTAP_b_2b3 "[psia]" {Total Refrigerant Side Pressure Drop Due To Bends in the Condenser} DELTAP_ResideBEND_total=DELTAP_b_22a+DELTAP_b_2a2b+DELTAP_b_2b3 "[psia]" {Valve Equation} h4=h3 "[Btu/lbm]" {Evaporator Equations} {Neglect Pressure drop across evaporator} "[psia]" P4=P4a P4=P1 "[psia]" Q_dot_e=Q_44a+Q_4a1"[Btu/hr]" A_i_e=A_i_44a+A_i_4a1 "[ft^2]" {A_o_e=A_i_e*D_o_1/D_i_c} T4=T4a"[F]" P4=pressure(R410A, T=T4, h=h4) "[psia]" {m_ae=V_dot_ae*convert(1/min,1/hr)/volume(AIR, T=Tac1, P=14.7)} x4=quality(R410A, T=T4, h=h4) {saturated portion of evaporator} Cp_44a_cor=specheat(AIR, T=Tae1)*1.33 "[Btu/lbm-F]" "[Btu/lbm]" h4a=enthalpy(R410A, T=T4a, x=x4a) "[Btu/hr]" Q_44a=m_dot_r*(h4a-h4) Q_44a=E_44a*C_a44a*(Tae1-T4) "[Btu/hr]" Q_44a=(A_i_44a/A_i_e)*C_a44a*(-Tae44a+Tae1) C_a44a=m_ae*Cp_44a_cor*A_i_44a/A_i_e call sat_size(C_a44a,E_44a:UA_44a) UA_44a=U_i_44a*A_i_44a "[Btu/hr-R]" "[Btu/hr-ft^2-R]" U_i_44a=(C1+1/h_bar_44a)^(-1) h_bar_44a=h_bar_e(T4, P4,D_i_c, m_dot_r, x4) "[Btu/hr-ft^2-R]" {CALL TwophaseDp(x4a,x4, T4, T4a, D_i_e, m_dot_r, tpc_e,L_44a:DELTAP_44a)
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CALL tpbenddrop(tpc_c,D_i_c,m_dot_r, h_f_c, T_c, L_c, L_22a, L_2a2b, Width_c:DELTAP_b_2a2b)} A_i_44a=L_44a*tpc_e*pi*D_i_e "[ft^2]" {superheated portion of evaporator} T1=T4a+Tsh "[F]" Q_4a1=m_dot_r*(h1-h4a) "[btu/hr]" Q_4a1=E_4a1*min(C_4a1, C_a4a1)*(Tae1-T4a) C_a4a1=m_ae*specheat(AIR, T=Tae1)*A_i_4a1/A_i_e C_4a1=m_dot_r*specheat(R410A, T=T1, P=P1) call exch_size_un_un(C_4a1,C_a4a1, UA_4a1:E_4a1) "[Btu/hr-R]" UA_4a1=U_i_4a1*A_i_4a1 U_i_4a1=(C1+1/h_bar_4a1)^(-1) "[Btu/hr-ft^2-R]" Call h_bar_single(D_i_e, m_dot_r, T4a, T1, P4a:Re_4a1,h_bar_4a1, rho_4a1) {C1=D_i_c/(D_o_1*h_bar_ae)} {constant represents air side term for evaporator U} "[hr-ft^2-R/BTU]" C1=1/(Area_rat*h_bar_ae) Area_rat=A_o_e/A_i_e {Call SingleDP(m_dot_r, tpc_e,D_i_e,L_4a1,f_4a1,rho_4a1:DELTAP_4a1) call singlebenddrop(tpc_e, D_i_e, m_dot_r,P4a, T4a, T1, L_4a1, Width_e, f_4a1:DELTAP_b_4a1)} 1/f_4a1^0.5=-2*log10((e/(D_i_e*3.7))+2.51/(Re_4a1*f_4a1^0.5)) "[ft^2]" A_i_4a1=L_4a1*tpc_e*pi*D_i_e {COP} W_dot_com=wc*m_dot_r "[Btu/hr]" Q_c=Q_22a+Q_2a2b+Q_2b3 "[Btu/hr]" COP=Q_dot_e/(W_dot_com+W_dot_fc+W_dot_fe) {Mass balances} Vol_22a=L_22a*D_i_c^2*pi*tpc_c/4 "[ft^3]" Vol_2a2b=L_2a2b*D_i_c^2*pi*tpc_c/4 "[ft^3]" Vol_2b3=L_2b3*D_i_c^2*pi*tpc_c/4 "[ft^3]" Vol_44a=A_i_44a*D_i_c/4 "[ft^3]" "[ft^3]" Vol_4a1=A_i_4a1*D_i_c/4 m_22a=rho_22a*Vol_22a "[lbm]" vfg2a2b=volume(R410A, T=T2a, x=1)-volume(R410A, T=T2a, x=0) "[ft^3/lbm]" m_2a2b=-(Vol_2a2b/vfg2a2b)*ln(volume(R410A, T=T2a, x=0)/volume(R410A, T=T2a, x=1)) "[lbm]" m_2b3=rho_2b3*Vol_2b3 "[lbm]" m_c=m_22a+M_2a2b+m_2b3 "[lbm]"
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"[lbm]" m_4a1=rho_4a1*Vol_4a1 vfg44a=volume(R410A, T=T4a, x=1)-volume(R410A, T=T4a, x=0) "[ft^3/lbm]" m_44a=(Vol_44a/(x4*vfg44a))*ln(volume(R410A, T=T4, x=1)/(volume(R410A, T=T4, x=0)+x4*vfg44a)) "[lbm] check this equation" m_sys=m_4a1+m_44a+m_c "[lbm]" m_e=m_4a1+m_44a {Air Side Pressure Drop} "fan efficiency" E_fc=.65 W_dot_fc=V_ac*DELTAP_tot_ac*convert(psia,lbf/ft^2)*A_c/E_fc*convert(ftlbf/s,btu/hr) "[Btu/hr]" d_fft_c=d_f_c*convert(in,ft) "[ft]" "[ft]" h_fft_c=h_f_c*convert(in,ft {Flow rate} "[lbm/ft^2 hr]" G_max_ac=m_ac/A_flow_c "[psia]" Pac2=P_atm "[psia]" P_atm=14.7 grav=32.2*convert(1/s^2,1/hr^2)"[lbm-ft/hr^2-lbf]" Re_D_c=G_max_ac*D_o_c/(mu_ac*convert(1/s,1/hr)) {Pressure Drop Calculation} DELTAP_tot_ac=Pac1-Pac2
"[psia]"
Eu_c=GetEuler(Re_d_c, d_f_c, h_f_c, D_o_c, nrow_c) DELTAP_tubes=Eu_c*G_max_ac^2*n row_c/(2*rho_ac1)*convert(lbm-ft/ft2-hr2, psia) "[psia]" DELTAP_tubes_inH2O=DELTAP_tubes*convert(psia, inH2O) "[inH2O]" DELTAP_tot_ac=DELTAP_tubes+DELTAP_fin DELTAP_fin=(f_f*G_max_ac^2*A_f_c/(2*A_flow_c*grav*rho_ac1))*convert(1/ft^2,1/ in^2) "[psia]" DELTAP_fin_inH2O=DELTAP_fin*convert(psia,inh2o) "[inh2O]" f_f=1.7*Re_L_ac^(-.5) Re_L_ac=G_max_ac*h_fft_c/mu_ac*convert(1/hr, 1/s) Den_COPseas_i = (Tac1-67)*fri(Tac1)/COP End
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Call At95(Tsc, V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c:PD, m_sys, A_e, A_c, Tc_95, width_e, width_c, W_dot_fc, W_dot_com, DELTAP_tot_ac95,CF_e,CF_c,DP_Rbend_95,DP_Rtotal_95,L_22a95,L_2a2b_95,L_2b 3_95) Call WithSubcool(67,PD,A_e,A_c, m_sys,V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c: Den_COPseas_67,Tsc[1]) Call WithSubcool(72,PD,A_e,A_c, m_sys,V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c: Den_COPseas_72,Tsc[1]) Call WithSubcool(77,PD,A_e,A_c, m_sys,V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c: Den_COPseas_77,Tsc[1]) Call WithSubcool(82,PD,A_e,A_c, m_sys,V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c: Den_COPseas_82,Tsc[1]) Call WithSubcool(87,PD,A_e,A_c, m_sys,V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c: Den_COPseas_87,Tsc[1]) Call WithSubcool(92,PD,A_e,A_c, m_sys,V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c: Den_COPseas_92,Tsc[1]) Call WithSubcool(97,PD,A_e,A_c, m_sys,V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c: Den_COPseas_97,Tsc[1]) Call WithSubcool(102,PD,A_e,A_c, m_sys,V_ac, h_f_c, t_c, eta_c, d_f_c, tpc_c, nrow_c, Tubetype_c, ncircuit_c: Den_COPseas_102,Tsc[1]) COPseas = 11.27/(Den_COPseas_67 + Den_COPseas_72 + Den_COPseas_77 + Den_COPseas_82 + Den_COPseas_87 + Den_COPseas_92 + Den_COPseas_97 + Den_COPseas_102) Tsc=15 “[F]” {Sub-cool in the condenser} V_ac=8 “[ft/s]” {Air velocity over condenser, ft/s} h_f_c=1.25 "[in]" {tube vertical spacing on centers, in} "[ft]" {thickness of fins, ft} t_c=.006/12 "[1/ft]" {condenser fin pitch, fins/ft} eta_c=12*convert(1/in,1/ft) "[in]" {condenser fin depth per tube, in} d_f_c=1.083 {number of rows per refrigerant flow parallel circuit} tpc_c=2 {number of columns of tubing} nrow_c=3 {Indicates tube diameter for standard copper pipe} TubeType_c=2 ncircuit_c=12 {indicates number of refrigerant flow parallel circuits}
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REFERENCES
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