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Overall conductance and heat transfer area minimization of refrigerators and heat pumps with finite heat reservoirs
Energy Conversion and Management 48 (2007) 803–808 www.elsevier.com/locate/enconman
Overall conductance and heat transfer area minimization of refrigerators and heat pumps with finite heat reservoirs J. Sarkar, Souvik Bhattacharyya
*
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721 302, India Received 4 March 2005; received in revised form 28 August 2006; accepted 10 September 2006 Available online 27 October 2006
Abstract In the present study, the overall conductance and the overall heat transfer area per unit capacity of refrigeration and heat pump systems have been minimized analytically considering both internal and external irreversibilities with variable temperature (finite capacity) heat reservoirs. Hot and cold side refrigerant temperatures, conductance and heat transfer area ratios have been optimized to attain this goal. The results have been verified with the more elaborate numerical optimization results obtained for ammonia based vapour compression refrigeration and heat pump systems working with variable temperature reservoirs. It is observed that the analytical results for optimum refrigerant temperatures, minimum overall conductance and heat transfer area deviate marginally from the numerically optimized results (within 1%), if one assumes a constant heat rejection temperature. The deviation of minimum overall conductance and heat transfer area is more (about 20%), if one considers both the desuperheating and condensation regions separately. However, in the absence of complex and elaborate numerical models, the simple analytical results obtained here can be used as reasonably accurate preliminary guidelines for optimization of refrigeration and heat pump systems. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Conductance; Heat transfer area; Finite heat reservoir; Ammonia plant; Numerical verification
1. Introduction Minimization of the heat exchanger area of the evaporator and condenser is important for the design engineer to reduce the size, weight and cost of refrigeration, air conditioning and heat pump systems. This is all the more significant because specifications of other components such as the compressor, expansion devices and accessories are somewhat fixed for given capacity and outlet conditions. Hence, it is necessary to optimize the operating conditions of the systems to get the desired output for minimum heat exchanger area. Although a large body of work has been reported on maximization of COP, minimization of work input and maximization of cooling or heating capacities for finite or infinite reservoirs with fixed conductance and
*
Corresponding author. Tel.: +91 3222 282904; fax: +91 3222 255303. E-mail address: [email protected] (S. Bhattacharyya).
0196-8904/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2006.09.001
heat transfer area and with linear or nonlinear heat transfer correlations within the last two decades [1–3], literature on total area minimization is limited. Bejan [4] first reported minimization of total heat exchanger inventory for an endoreversible cycle with constant temperature heat reservoirs. Most of the other authors focused their attention on the optimization of conductance and heat exchanger area ratio [5–12]. Numerical validation of the analytical results is valuable to the design engineer. Klein and Reindl [11] numerically optimized the conductance ratio based on COP, cooling load and entropy generation. Chen et al. [12] have compared the optimized area ratio with the numerical result of actual refrigeration and air conditioning plants. However, it may be noted that the reported literature, cited above, do not present sufficient work on total area minimization, which is important to reduce the size, weight and cost of refrigeration, air conditioning and heat pump systems. The present study is an effort toward filling that gap in information.
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J. Sarkar, S. Bhattacharyya / Energy Conversion and Management 48 (2007) 803–808
Nomenclature A C COPc COPh h NTU s T TCI TCO
heat transfer area, m2 secondary fluid heat capacity, W K1 cooling coefficient of performance heating coefficient of performance specific enthalpy, kJ kg1 number of transfer units specific entropy, kJ kg1 K1 temperature, K secondary fluid inlet temperature in condenser, K secondary fluid outlet temperature in condenser, K
In this study, minimization of the overall heat conductance and heat transfer area of irreversible refrigerators and heat pumps with finite capacity (variable temperature) heat reservoirs has been performed analytically. In this way, the hot and cold side refrigerant temperatures and the conductance and area ratios have been optimized. Finally, the theoretical results have been verified with results obtained from an elaborate numerical simulation of an ammonia based refrigeration and heat pump system with variable temperature heat reservoirs. 2. Irreversible refrigerator and heat pump with finite heat reservoirs The model of an irreversible Carnot refrigerator or heat pump cycle (1–2–3–4–1), as shown in Fig. 1, incorporates two types of irreversibilities: (i) irreversibility due to heat transfer between internal and external fluids and (ii) irreversibilities associated with compression and expansion processes. Here, a linear heat transfer law has been employed, which is common in practical applications. So, the heat released to the high temperature
TEI TEO THC TLC U
secondary fluid inlet temperature in evaporator, K secondary fluid outlet temperature in evaporator, K hot working fluid temperature, K cold side working fluid temperature, K overall heat transfer coefficient, W m2 K1
Subscripts H high temperature side (condenser) L low temperature side (evaporator)
reservoir (heating capacity) by the working fluid can be written as QH ¼ C H ðT CO T CI Þ ¼ AH U H
ðT HC T CI Þ ðT HC T CO Þ : ln½ðT HC T CI Þ=ðT HC T CO Þ ð1Þ
Since the working fluid heat capacity is much larger than the secondary fluid heat capacity, the heat exchanger effectiveness eH and QH can be written as [10]: eH ¼ 1 expðNTUH Þ; QH ¼ C H eH ðT HC T CI Þ:
NTUH ¼ AH U H =C H
ð2Þ ð3Þ
From a similar analysis, the effectiveness eL and the heat absorbed by the working fluid from the cold side heat reservoir (cooling capacity, QL) can be expressed as eL ¼ 1 expðNTUL Þ; where NTUL ¼ AL U L =C L ; QL ¼ C L ðT EI T EO Þ ¼ C L eL ðT EI T LC Þ:
ð4Þ ð5Þ
Heat transfer between the two fluids in both heat exchangers requires that the temperatures satisfy the following: T HC > T CO > T CI
and
T EI > T EO > T LC :
ð6Þ
Following the second law of thermodynamics applied to the working fluid undergoing an irreversible Carnot refrigeration or heat pump cycle for steady state condition: THC
3 2
Temperature
QH TCO TCI TEI
TLC
TEO
QL
4
1
Specific entropy
Fig. 1. Irreversible Carnot refrigeration/heat pump cycle on the T–s plane.
QH T HC ðs2 s3 Þ ¼ QL T LC ðs1 s4 Þ
or
QH T HC ¼/ ; QL T LC
ð7Þ
where / is the irreversibility parameter, which is the ratio of entropy loss and entropy gain by the working fluid with the hot and cold reservoir, respectively (/ P 1). The total conductance (UA) and total area (A) for both the heat exchangers are taken as UA ¼ U L AL þ U H AH ; A ¼ AL þ AH :
ð8Þ ð9Þ
These two, the total conductance and total area, are the objective functions of this study, and they have to be minimized for optimized system performance.
J. Sarkar, S. Bhattacharyya / Energy Conversion and Management 48 (2007) 803–808
2.1. Total conductance minimization for a refrigerator Combining Eqs. (2)–(5), (7) and (8), the total conductance per unit cooling load can be expressed as 1 1 /ðT HC =T LC Þ 1 UA ¼ C H ln 1 þ C L ln 1 : C H ðT HC T CI Þ C L ðT EI T LC Þ
ð10Þ
The above equation has two degrees of freedom. Most of the studies reported earlier have considered either the ratio (THC/TLC) or the COP as the constant parameter, which can be shown to yield identical results. For the present analysis, the cooling COP (COPc) is assumed to be constant and the following relation can be derived: T LC COPc ¼ /X c ¼/ T HC COPc þ 1
ðsayÞ:
ð11Þ
Combining Eqs. (10) and (11) and equating the derivative of UA with respect to TLC to zero, the following expression is obtained for the optimal cold side working fluid temperature (for / > 1): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T LC;opt ¼ B B2 4ð/ 1ÞD =½2ð/ 1Þ; ð12Þ where B = 2/(TEI XcTCI) (//CL + //CH) and D ¼ /T 2EI ð/X c T CI Þ2 /T EI =C L /2 X c T CI =C H Sample calculations show that the negative sign (in Eq. (12)) satisfies the conditions of heat transfer (Eq. (6)). Hence, the hot and cold side optimal working fluid temperatures are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T LC;opt ¼ B B2 4ð/ 1ÞD =½2ð/ 1Þ; T HC;opt ¼ T LC;opt =ð/X c Þ:
ð13Þ
Substituting the working fluid temperatures in Eq. (10) for the optimum values (from Eq. (13)), the minimum overall conductance for unit cooling capacity and for a specified COP can be obtained.
and where E = 2(TEI/Xh TCI) (1/CL + 1/CH) 2 F ¼ ðT EI =X h Þ =/ T 2CI T CI =C H T EI =ð/X h C L Þ. Similar to the previous case, adopting the negative sign to satisfy the heat transfer condition (Eq. (6)), the hot side and cold side optimal working fluid temperatures for a heat pump can be derived as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T HC;opt ¼ E E2 þ 4ð/ 1ÞF =½2ð/ 1Þ; T LC;opt ¼ uX h T HC;opt :
ð16Þ
Substituting the optimum working fluid temperatures in Eq. (14), one can obtain the minimum overall conductance for unit heating capacity and a particular value of COP. In Eqs. (10) and (14), the 1st and 2nd terms represent the conductances for the hot and cold side heat exchangers, respectively. Thus, replacing the working fluid temperatures by their optimum values, one can find the conductances of the hot and cold side heat exchanger individually and, hence, finally, the optimum conductance ratio can be obtained as well. 2.3. Total area minimization for refrigerator Combining Eqs. (2)–(5), (7) and (8), the total heat exchanger area required for unit cooling output can be shown to be given by 1 CH /T HC =T LC A¼ ln 1 UH C H ðT HC T CI Þ 1 CL 1 þ ln 1 : ð17Þ C L ðT L T LC Þ UL Adopting the same technique as before to resolve the problem of two degrees of freedom, after replacing THC by Eq. (11) in the above equation and equating the derivative of A with respect to TLC to zero, the following expression for the optimal cold side working fluid temperature is obtained: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T LC;opt ¼ G G2 4ð1 d=/ÞH =½2ð1 d=/Þ for d 6¼ /;
2.2. Total conductance minimization for heat pump
ð18Þ
Combining Eqs. (2)–(5), (7) and (8), the total conductance per unit heating load is given by, 1 1 UA ¼ C H ln 1 C H ðT HC T CI Þ 1 ðT LC =/T HC þ C L ln 1 : ð14Þ C L ðT EI T LC Þ Performing an analysis similar to that for the refrigerator, the optimum hot side working fluid temperature can be obtained as (for / > 1) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T HC;opt ¼ E E2 þ 4ð/ 1ÞF =½2ð/ 1Þ; X h ¼ ðCOPh 1Þ=COPh ;
805
ð15Þ
where G = 2(TEI dXcTCI) (1/CL + d/CH) and H ¼ T 2EI d/ðX c T CI Þ2 T EI =C L d/X c T CI =C H and d is defined as d = UH/UL. Sample calculations show that the negative sign in Eq. (18) will satisfy the heat transfer condition, Eq. (6). Hence, the cold and hot side optimal working fluid temperatures can be derived to yield qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T LC;opt ¼ G G2 4ð1 d=/ÞH =½2ð1 d=/Þ; T HC;opt ¼ T LC;opt =/X c
ð19Þ
For d = /, the optimal working fluid temperatures will simply become T LC;opt ¼ H =G
and
T HC;opt ¼ T LC;opt =/X c :
ð20Þ
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J. Sarkar, S. Bhattacharyya / Energy Conversion and Management 48 (2007) 803–808
Substituting the optimal working fluid temperatures, Eq. (17) will yield the minimum overall heat transfer area per unit cooling load. 2.4. Total area minimization for heat pump Combining Eqs. ((2)–(5), (7) and (9)), the total heat exchanger area required for unit heating output is given by, 1 1 CH 1 CL T LC =ð/T HC Þ A¼ ln 1 þ ln 1 : UH UL C H ðT HC T CI Þ C L ðT L T LC Þ ð21Þ
As was shown for the refrigerator, the optimum hot side working fluid temperature for minimum total area per unit heating load is obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T HC;opt ¼ J J 2 4ð/ dÞK =½2ð/ dÞ for d 6¼ /;
against results obtained from an elaborate numerical simulation of ammonia based refrigeration and heat pump plants. The NH3 based refrigeration/heat pump with variable temperature heat reservoirs is shown in Fig. 2 (cycle 1–2 0 –3 0 –4–1). As illustrated in the diagram, the main difference with the irreversible Carnot cycle is in the heat rejection process in the condenser. The cold side refrigerant temperature (TLC) is the same as the evaporation temperature and the hot side refrigerant temperature (THC) is taken as the thermodynamic average temperature of the refrigerant in the condenser, which, mathematically, is the ratio of the enthalpy and entropy differences: T HC ¼ ðh20 h30 Þ=ðs20 s30 Þ: 0
The compression (1–2 ) and expansion (3 –4) processes are assumed to be adiabatic and irreversible. The evaporation and condensation processes are considered to be isobaric. The cooling and heating capacity can be written as
ð22Þ where J = 2(TEI/Xh dTCI) (1/CL + d/CH) and K = (TEI/Xh)2// dTCI2 dTCI/CH TEI/(/XhCL) Similar to the previous cases, adopting the negative sign to satisfy the heat transfer condition, Eq. (6), the hot side and cold side optimal working fluid temperatures for the heat pump can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T HC;opt ¼ J J 2 4ð/ dÞK =½2ð/ dÞ;
ð25Þ
0
QL ¼ AL U L
ðT EI T LC Þ ðT EO T LC Þ ¼ m_ r ðh1 h4 Þ ln½ðT EI T LC Þ=ðT EO T LC Þ ð26Þ
QH ¼ AH U H
ðT HC T CO Þ ðT HC T CI Þ ¼ m_ r ðh20 h30 Þ ln½ðT HC T CO Þ=ðT HC T CI Þ ð27Þ
For d = /, the optimal working fluid temperatures will simply become
where m_ r is the mass flow rate of refrigerant. However, a better approximation of the heat rejection process is to divide the condenser into saturated and superheated zones and employ the LMTD equation separately, where the secondary fluid temperature corresponding to that of saturated refrigerant vapour (Trv) is given by
T HC;opt ¼ K=J
T SV ¼ T CI þ
and
ð23Þ
T LC;opt ¼ /X h T HC;opt :
ð24Þ
Substituting the optimal working fluid temperatures in Eq. (21) from Eq. (23), the minimum overall heat transfer area per unit heating load and fixed COP can be obtained. Similar to the case of conductance analysed before, replacing the working fluid temperatures by their optimum values separately in Eqs. (17) and (21), one can obtain the minimum hot and cold side heat exchanger areas individually. Subsequently, the optimum heat exchanger area ratio for both the refrigeration and heat pump systems for minimum total area can be obtained as well. 3. Numerical verification The theoretical analyses show that it is possible to design refrigeration and heat pump systems for minimum overall conductance and heat exchanger area for certain coefficients of performance and cooling or heating output employing finite heat reservoirs. The hot and cold side refrigerant temperatures/pressures and individual areas can be set at optimum values accordingly. The results obtained from the above analyses have been verified
ðhrv h3 Þ ðT CO T CI Þ ðh2 h3 Þ
ð28Þ
The isentropic efficiency is assumed to be 0.8 for all cases. For unit cooling or heating load and a particular value of COP, QL and QH are fixed. Assuming the refrigerant outlet in the evaporator as saturated vapor, the simulation code,
2'
Temperature
T LC;opt ¼ /X h T HC;opt
QH
3'
rv sv
TCO
TCI TEI
TLC
TEO
QL
4
1
Specific entropy
Fig. 2. Ammonia based refrigeration/heat pump cycle on the T–s plane.
J. Sarkar, S. Bhattacharyya / Energy Conversion and Management 48 (2007) 803–808
0.22 c
0.2
= 1. 6
= 1. 4
c
= 1. 6
Open symbol - Numerical Filled symbol - Analytical
0.18
UA (1/K)
c
0.16 c
= 1.2
0.14 c
0.12
= 1.4
0.1
c
= 1. 2
0.08 240
245
250
255
260
Evaporation temperature (K) Fig. 3. Variation of total conductance per unit cooling capacity.
0.085
0.075
UA (1/K)
which is based on the fundamental thermodynamic equations of the refrigeration/heat pump cycle, solves for the other state points of the cycle and m_ r for a certain evaporator temperature. The thermophysical properties of ammonia have been estimated employing the Haar and Gallagher correlations [13] through independently developed exclusive property subroutines. Using the LMTD Eqs. (26) and (27), individual conductances and heat exchanger areas and, finally, the total values have been obtained. Following a variation of the evaporation temperature, the minimum total conductance and area have been found. The inlet temperatures of the secondary fluids are taken as 280 K and 320 K for the evaporator and condenser, respectively, in this analysis. The variation in overall conductance with TLC for a refrigerator of 1 kW cooling capacity is shown in Fig. 3 for CL = CH = 0.05 kW/K. It is common knowledge that a lower temperature difference between the working fluid and the secondary fluid yields a higher COP, but it also requires a higher conductance for the same capacity. So, higher COP demands higher conductance as shown in Fig. 3. The results show that for a constant refrigerant temperature (THC) in the LMTD equation throughout the condensation process (solid lines in Fig. 3), the deviations are smaller (both the cold side and hot side working fluid temperature deviate below 1%, whereas the overall conductance deviates less than 0.5%) because of closely similar assumptions made in the two forms of modeling. As the assumption of isothermality is partially matched with that in the numerical model of the condenser, the individual optimum value of conductance of the condenser deviates slightly more than that of the evaporator. On the other hand, bifurcation of the condensation process into saturated and superheated zones (dotted lines in Fig. 3) results in significantly larger deviation (optimum TLC deviates below 2%, whereas the overall conductance deviates 20– 30%) as this is not compatible with the analytical model. Hence, the concept of THC has been used for the remaining results as it matches the analytical model better.
807
h
Open symbol - Numerical Filled symbol - Analytical
= 2.6
0.065
= 2.4
h
0.055 h
0.045
= 2.2
0.035 350
360
370
380
390
400
THC(K)
Fig. 4. Variation of total conductance per unit heating capacity.
The variation of overall conductance with THC for a heat pump is shown in Fig. 4 for QH = 1 kW and CL = CH = 0.04 kW/K. Following the same arguments presented earlier, a higher COP calls for a higher conductance as shown in Fig. 4. The results show that both the cold side and hot side optimum working fluid temperatures from the numerical study deviate below 1% when compared with that from the analytical model, whereas the overall conductance deviates within 0.5%. Similar to the above case, the theoretically obtained optimum value of the individual conductance of the condenser deviates more than that of the evaporator when compared against the numerical results. The variations of total heat transfer area per unit cooling capacity with TLC for a refrigeration plant are shown in Fig. 5 with a set of input parameters given by: UL = 0.5 kW/m2 K, UH = 0.5 kW/m2K. Because of the same reason discussed earlier, a higher COP requires a higher heat transfer area as is evident from Fig. 5. The difference of the analytically obtained cold side and hot side optimum working fluid temperatures and minimum area from the numerical predictions are 0.6–0.9%, 0.7–1% and 0.3– 0.5%, respectively, whereas the individual condenser area differs slightly more than that of the evaporator. The variations of total area with THC for the heat pump are shown in Fig. 6 for QH = 1 kW, CL = CH = 0.05 kW/K and overall heat transfer coefficients of 0.5 kW/m2 K for both the evaporator and condenser. Similar to the above case, a higher COP requires a higher heat transfer area. The analytical results compare well with the numerical results with the differences within the range of 0.7–1% for both the hot and cold side refrigerant temperatures and within 0.3–0.5% for total area, whereas the individual area of the condenser shows slightly higher differences due to the same reason discussed earlier. The minimum heat transfer area per unit capacity depends on the individual values of UL and UH. It is observed that the heat transfer area will be minimum at a particular ratio (d) of UH and UL, keeping their sum constant. By replacing TLC,opt and THC,opt in Eq. (17) or
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J. Sarkar, S. Bhattacharyya / Energy Conversion and Management 48 (2007) 803–808
0. 35
c = 1.8
0. 3
2
Tot al area ( m /kW)
Open symbol - Numerical Filled symbol - Analytical
0. 25
c = 1.5
0. 2
c = 1.2
0. 15 232
237
242
247
252
257
Evaporation temperature (K) Fig. 5. Variation of total heat exchanger area per unit cooling capacity.
0.19
Open symbol - Numerical Filled symbol - Analytical
h = 2.8
0.15
2
Total area (m /kW)
0.17
0.13
as the working fluid. Accurate property subroutines, developed exclusively for this study, are employed to assist in the simulation. Numerical results have been obtained assuming a single equivalent heat rejection temperature in the condenser and also considering desuperheating and condensation regions separately. The results show that the hot and cold side refrigerant temperatures and the conductance and area ratio between evaporator and condenser can be optimized to obtain minimum overall conductance and heat exchanger area. The deviation of analytical results from those yielded by the numerical model is reasonably small (below 1% for all hot and cold side optimum refrigerant temperatures, and minimum total heat transfer area) if a constant refrigerant temperature (THC) is assumed throughout the condensation process because it closely resembles the analytical model. The deviations between analytical and numerical results are larger when condensation and desuperheating regions are treated separately (maximum deviation is of the order of 20%). The deviation between the analytical results and actual plant data would be higher than this due to the presence of other irreversibilities such as pressure drop and heat interaction with the surroundings, which have not been incorporated in the present analyses. However, the simple analytical expressions developed here will help in guiding the design engineer in the preliminary optimization of actual plants in the absence of more elaborate numerical models.
h = 2.5
0.11
References h = 2.2
0.09 0.07 350
360
370
380
390
THC (K) Fig. 6. Variation of total heat exchanger area per unit heating capacity.
(21), taking UH + UL= constant and considering oAodmin ¼ 0, it is possible to obtain an optimum d. The numerical results show that the optimum value of overall heat transfer coefficient ratio (UH/UL) varies between 1.08 and 1.12. In an actual evaporator or condenser design, UH and UL depend on the fluid properties, flow and geometry of the heat exchangers, and the heat transfer area also depends on geometry. Hence, both the concepts of conductance and heat transfer area minimizations are required in actual design practice. 4. Conclusions The overall conductance and the overall heat transfer area per unit capacity have been minimized analytically for refrigerator and heat pump systems working with variable temperature (finite) heat reservoirs. The analytical results are verified against numerical results obtained from an elaborate simulation of the systems based on ammonia
[1] Bejan A. Entropy generation minimization: the new thermodynamics of finite-size devices and finite-time processes. J Appl Phys 1996;79:1191–218. [2] Chen L, Wu C, Sun F. Finite-time thermodynamic optimization or entropy generation minimization of energy systems. J Non-Eqm Therm 1999;24:327–59. [3] Durmayaz A, Sogut OS, Sahin B, Yavuz H. Optimization of thermal system based on finite-time thermodynamics and thermoeconomics. Prog Energ Combust Sci 2004;30:175–217. [4] Bejan A. Power and refrigeration plants for minimum heat exchanger inventory. J Energ Res Techol 1993;115:148–50. [5] Wu C, Chen L, Sun F. Optimization of steady flow heat pumps. Energ Convers Mgmt 1998;39:767–73. [6] Chen L, Wu C, Sun F. Influence of internal heat leaks on the performance of refrigerators. Energ Convers Mgmt 1998;39:45–50. [7] Chen J, Wu C. Design considerations of primary performance parameters for irreversible refrigeration cycles. Int J Amb Energ 1995;16:17–22. [8] Chen L, Sun F, Wu C, Kiang RL. A generalized model of a real refrigerator and its performance. Appl Therm Engg 1997;17:401–12. [9] Al-Din MMS. Optimization of totally irreversible refrigerators and heat pumps. Energ Convers Mgmt 1999;40:423–36. [10] Al-Din MMS. Performance analysis of heat pumps and refrigerators with variable resorvoir temperatures. Energ Convers Mgmt 2001;42:201–16. [11] Klein SA, Reindl DT. The relationship of optimum heat exchanger allocation and minimum entropy generation rate for refrigeration cycles. J Energ Res Techol 1998;20:172–8. [12] Chen L, Sun F, Wu C. Optimal allocation of heat exchanger area for refrigeration and air-conditioning plants. Appl Energ 2004;77:339–54. [13] Haar L, Gallagher JS. Thermodynamic properties of ammonia. J Phys Chem Ref Data 1978;7:635–792.
Overall conductance and heat transfer area minimization of refrigerators and heat pumps with finite heat reservoirs J. Sarkar, Souvik Bhattacharyya
*
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721 302, India Received 4 March 2005; received in revised form 28 August 2006; accepted 10 September 2006 Available online 27 October 2006
Abstract In the present study, the overall conductance and the overall heat transfer area per unit capacity of refrigeration and heat pump systems have been minimized analytically considering both internal and external irreversibilities with variable temperature (finite capacity) heat reservoirs. Hot and cold side refrigerant temperatures, conductance and heat transfer area ratios have been optimized to attain this goal. The results have been verified with the more elaborate numerical optimization results obtained for ammonia based vapour compression refrigeration and heat pump systems working with variable temperature reservoirs. It is observed that the analytical results for optimum refrigerant temperatures, minimum overall conductance and heat transfer area deviate marginally from the numerically optimized results (within 1%), if one assumes a constant heat rejection temperature. The deviation of minimum overall conductance and heat transfer area is more (about 20%), if one considers both the desuperheating and condensation regions separately. However, in the absence of complex and elaborate numerical models, the simple analytical results obtained here can be used as reasonably accurate preliminary guidelines for optimization of refrigeration and heat pump systems. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Conductance; Heat transfer area; Finite heat reservoir; Ammonia plant; Numerical verification
1. Introduction Minimization of the heat exchanger area of the evaporator and condenser is important for the design engineer to reduce the size, weight and cost of refrigeration, air conditioning and heat pump systems. This is all the more significant because specifications of other components such as the compressor, expansion devices and accessories are somewhat fixed for given capacity and outlet conditions. Hence, it is necessary to optimize the operating conditions of the systems to get the desired output for minimum heat exchanger area. Although a large body of work has been reported on maximization of COP, minimization of work input and maximization of cooling or heating capacities for finite or infinite reservoirs with fixed conductance and
*
Corresponding author. Tel.: +91 3222 282904; fax: +91 3222 255303. E-mail address: [email protected] (S. Bhattacharyya).
0196-8904/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2006.09.001
heat transfer area and with linear or nonlinear heat transfer correlations within the last two decades [1–3], literature on total area minimization is limited. Bejan [4] first reported minimization of total heat exchanger inventory for an endoreversible cycle with constant temperature heat reservoirs. Most of the other authors focused their attention on the optimization of conductance and heat exchanger area ratio [5–12]. Numerical validation of the analytical results is valuable to the design engineer. Klein and Reindl [11] numerically optimized the conductance ratio based on COP, cooling load and entropy generation. Chen et al. [12] have compared the optimized area ratio with the numerical result of actual refrigeration and air conditioning plants. However, it may be noted that the reported literature, cited above, do not present sufficient work on total area minimization, which is important to reduce the size, weight and cost of refrigeration, air conditioning and heat pump systems. The present study is an effort toward filling that gap in information.
804
J. Sarkar, S. Bhattacharyya / Energy Conversion and Management 48 (2007) 803–808
Nomenclature A C COPc COPh h NTU s T TCI TCO
heat transfer area, m2 secondary fluid heat capacity, W K1 cooling coefficient of performance heating coefficient of performance specific enthalpy, kJ kg1 number of transfer units specific entropy, kJ kg1 K1 temperature, K secondary fluid inlet temperature in condenser, K secondary fluid outlet temperature in condenser, K
In this study, minimization of the overall heat conductance and heat transfer area of irreversible refrigerators and heat pumps with finite capacity (variable temperature) heat reservoirs has been performed analytically. In this way, the hot and cold side refrigerant temperatures and the conductance and area ratios have been optimized. Finally, the theoretical results have been verified with results obtained from an elaborate numerical simulation of an ammonia based refrigeration and heat pump system with variable temperature heat reservoirs. 2. Irreversible refrigerator and heat pump with finite heat reservoirs The model of an irreversible Carnot refrigerator or heat pump cycle (1–2–3–4–1), as shown in Fig. 1, incorporates two types of irreversibilities: (i) irreversibility due to heat transfer between internal and external fluids and (ii) irreversibilities associated with compression and expansion processes. Here, a linear heat transfer law has been employed, which is common in practical applications. So, the heat released to the high temperature
TEI TEO THC TLC U
secondary fluid inlet temperature in evaporator, K secondary fluid outlet temperature in evaporator, K hot working fluid temperature, K cold side working fluid temperature, K overall heat transfer coefficient, W m2 K1
Subscripts H high temperature side (condenser) L low temperature side (evaporator)
reservoir (heating capacity) by the working fluid can be written as QH ¼ C H ðT CO T CI Þ ¼ AH U H
ðT HC T CI Þ ðT HC T CO Þ : ln½ðT HC T CI Þ=ðT HC T CO Þ ð1Þ
Since the working fluid heat capacity is much larger than the secondary fluid heat capacity, the heat exchanger effectiveness eH and QH can be written as [10]: eH ¼ 1 expðNTUH Þ; QH ¼ C H eH ðT HC T CI Þ:
NTUH ¼ AH U H =C H
ð2Þ ð3Þ
From a similar analysis, the effectiveness eL and the heat absorbed by the working fluid from the cold side heat reservoir (cooling capacity, QL) can be expressed as eL ¼ 1 expðNTUL Þ; where NTUL ¼ AL U L =C L ; QL ¼ C L ðT EI T EO Þ ¼ C L eL ðT EI T LC Þ:
ð4Þ ð5Þ
Heat transfer between the two fluids in both heat exchangers requires that the temperatures satisfy the following: T HC > T CO > T CI
and
T EI > T EO > T LC :
ð6Þ
Following the second law of thermodynamics applied to the working fluid undergoing an irreversible Carnot refrigeration or heat pump cycle for steady state condition: THC
3 2
Temperature
QH TCO TCI TEI
TLC
TEO
QL
4
1
Specific entropy
Fig. 1. Irreversible Carnot refrigeration/heat pump cycle on the T–s plane.
QH T HC ðs2 s3 Þ ¼ QL T LC ðs1 s4 Þ
or
QH T HC ¼/ ; QL T LC
ð7Þ
where / is the irreversibility parameter, which is the ratio of entropy loss and entropy gain by the working fluid with the hot and cold reservoir, respectively (/ P 1). The total conductance (UA) and total area (A) for both the heat exchangers are taken as UA ¼ U L AL þ U H AH ; A ¼ AL þ AH :
ð8Þ ð9Þ
These two, the total conductance and total area, are the objective functions of this study, and they have to be minimized for optimized system performance.
J. Sarkar, S. Bhattacharyya / Energy Conversion and Management 48 (2007) 803–808
2.1. Total conductance minimization for a refrigerator Combining Eqs. (2)–(5), (7) and (8), the total conductance per unit cooling load can be expressed as 1 1 /ðT HC =T LC Þ 1 UA ¼ C H ln 1 þ C L ln 1 : C H ðT HC T CI Þ C L ðT EI T LC Þ
ð10Þ
The above equation has two degrees of freedom. Most of the studies reported earlier have considered either the ratio (THC/TLC) or the COP as the constant parameter, which can be shown to yield identical results. For the present analysis, the cooling COP (COPc) is assumed to be constant and the following relation can be derived: T LC COPc ¼ /X c ¼/ T HC COPc þ 1
ðsayÞ:
ð11Þ
Combining Eqs. (10) and (11) and equating the derivative of UA with respect to TLC to zero, the following expression is obtained for the optimal cold side working fluid temperature (for / > 1): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T LC;opt ¼ B B2 4ð/ 1ÞD =½2ð/ 1Þ; ð12Þ where B = 2/(TEI XcTCI) (//CL + //CH) and D ¼ /T 2EI ð/X c T CI Þ2 /T EI =C L /2 X c T CI =C H Sample calculations show that the negative sign (in Eq. (12)) satisfies the conditions of heat transfer (Eq. (6)). Hence, the hot and cold side optimal working fluid temperatures are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T LC;opt ¼ B B2 4ð/ 1ÞD =½2ð/ 1Þ; T HC;opt ¼ T LC;opt =ð/X c Þ:
ð13Þ
Substituting the working fluid temperatures in Eq. (10) for the optimum values (from Eq. (13)), the minimum overall conductance for unit cooling capacity and for a specified COP can be obtained.
and where E = 2(TEI/Xh TCI) (1/CL + 1/CH) 2 F ¼ ðT EI =X h Þ =/ T 2CI T CI =C H T EI =ð/X h C L Þ. Similar to the previous case, adopting the negative sign to satisfy the heat transfer condition (Eq. (6)), the hot side and cold side optimal working fluid temperatures for a heat pump can be derived as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T HC;opt ¼ E E2 þ 4ð/ 1ÞF =½2ð/ 1Þ; T LC;opt ¼ uX h T HC;opt :
ð16Þ
Substituting the optimum working fluid temperatures in Eq. (14), one can obtain the minimum overall conductance for unit heating capacity and a particular value of COP. In Eqs. (10) and (14), the 1st and 2nd terms represent the conductances for the hot and cold side heat exchangers, respectively. Thus, replacing the working fluid temperatures by their optimum values, one can find the conductances of the hot and cold side heat exchanger individually and, hence, finally, the optimum conductance ratio can be obtained as well. 2.3. Total area minimization for refrigerator Combining Eqs. (2)–(5), (7) and (8), the total heat exchanger area required for unit cooling output can be shown to be given by 1 CH /T HC =T LC A¼ ln 1 UH C H ðT HC T CI Þ 1 CL 1 þ ln 1 : ð17Þ C L ðT L T LC Þ UL Adopting the same technique as before to resolve the problem of two degrees of freedom, after replacing THC by Eq. (11) in the above equation and equating the derivative of A with respect to TLC to zero, the following expression for the optimal cold side working fluid temperature is obtained: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T LC;opt ¼ G G2 4ð1 d=/ÞH =½2ð1 d=/Þ for d 6¼ /;
2.2. Total conductance minimization for heat pump
ð18Þ
Combining Eqs. (2)–(5), (7) and (8), the total conductance per unit heating load is given by, 1 1 UA ¼ C H ln 1 C H ðT HC T CI Þ 1 ðT LC =/T HC þ C L ln 1 : ð14Þ C L ðT EI T LC Þ Performing an analysis similar to that for the refrigerator, the optimum hot side working fluid temperature can be obtained as (for / > 1) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T HC;opt ¼ E E2 þ 4ð/ 1ÞF =½2ð/ 1Þ; X h ¼ ðCOPh 1Þ=COPh ;
805
ð15Þ
where G = 2(TEI dXcTCI) (1/CL + d/CH) and H ¼ T 2EI d/ðX c T CI Þ2 T EI =C L d/X c T CI =C H and d is defined as d = UH/UL. Sample calculations show that the negative sign in Eq. (18) will satisfy the heat transfer condition, Eq. (6). Hence, the cold and hot side optimal working fluid temperatures can be derived to yield qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T LC;opt ¼ G G2 4ð1 d=/ÞH =½2ð1 d=/Þ; T HC;opt ¼ T LC;opt =/X c
ð19Þ
For d = /, the optimal working fluid temperatures will simply become T LC;opt ¼ H =G
and
T HC;opt ¼ T LC;opt =/X c :
ð20Þ
806
J. Sarkar, S. Bhattacharyya / Energy Conversion and Management 48 (2007) 803–808
Substituting the optimal working fluid temperatures, Eq. (17) will yield the minimum overall heat transfer area per unit cooling load. 2.4. Total area minimization for heat pump Combining Eqs. ((2)–(5), (7) and (9)), the total heat exchanger area required for unit heating output is given by, 1 1 CH 1 CL T LC =ð/T HC Þ A¼ ln 1 þ ln 1 : UH UL C H ðT HC T CI Þ C L ðT L T LC Þ ð21Þ
As was shown for the refrigerator, the optimum hot side working fluid temperature for minimum total area per unit heating load is obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T HC;opt ¼ J J 2 4ð/ dÞK =½2ð/ dÞ for d 6¼ /;
against results obtained from an elaborate numerical simulation of ammonia based refrigeration and heat pump plants. The NH3 based refrigeration/heat pump with variable temperature heat reservoirs is shown in Fig. 2 (cycle 1–2 0 –3 0 –4–1). As illustrated in the diagram, the main difference with the irreversible Carnot cycle is in the heat rejection process in the condenser. The cold side refrigerant temperature (TLC) is the same as the evaporation temperature and the hot side refrigerant temperature (THC) is taken as the thermodynamic average temperature of the refrigerant in the condenser, which, mathematically, is the ratio of the enthalpy and entropy differences: T HC ¼ ðh20 h30 Þ=ðs20 s30 Þ: 0
The compression (1–2 ) and expansion (3 –4) processes are assumed to be adiabatic and irreversible. The evaporation and condensation processes are considered to be isobaric. The cooling and heating capacity can be written as
ð22Þ where J = 2(TEI/Xh dTCI) (1/CL + d/CH) and K = (TEI/Xh)2// dTCI2 dTCI/CH TEI/(/XhCL) Similar to the previous cases, adopting the negative sign to satisfy the heat transfer condition, Eq. (6), the hot side and cold side optimal working fluid temperatures for the heat pump can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T HC;opt ¼ J J 2 4ð/ dÞK =½2ð/ dÞ;
ð25Þ
0
QL ¼ AL U L
ðT EI T LC Þ ðT EO T LC Þ ¼ m_ r ðh1 h4 Þ ln½ðT EI T LC Þ=ðT EO T LC Þ ð26Þ
QH ¼ AH U H
ðT HC T CO Þ ðT HC T CI Þ ¼ m_ r ðh20 h30 Þ ln½ðT HC T CO Þ=ðT HC T CI Þ ð27Þ
For d = /, the optimal working fluid temperatures will simply become
where m_ r is the mass flow rate of refrigerant. However, a better approximation of the heat rejection process is to divide the condenser into saturated and superheated zones and employ the LMTD equation separately, where the secondary fluid temperature corresponding to that of saturated refrigerant vapour (Trv) is given by
T HC;opt ¼ K=J
T SV ¼ T CI þ
and
ð23Þ
T LC;opt ¼ /X h T HC;opt :
ð24Þ
Substituting the optimal working fluid temperatures in Eq. (21) from Eq. (23), the minimum overall heat transfer area per unit heating load and fixed COP can be obtained. Similar to the case of conductance analysed before, replacing the working fluid temperatures by their optimum values separately in Eqs. (17) and (21), one can obtain the minimum hot and cold side heat exchanger areas individually. Subsequently, the optimum heat exchanger area ratio for both the refrigeration and heat pump systems for minimum total area can be obtained as well. 3. Numerical verification The theoretical analyses show that it is possible to design refrigeration and heat pump systems for minimum overall conductance and heat exchanger area for certain coefficients of performance and cooling or heating output employing finite heat reservoirs. The hot and cold side refrigerant temperatures/pressures and individual areas can be set at optimum values accordingly. The results obtained from the above analyses have been verified
ðhrv h3 Þ ðT CO T CI Þ ðh2 h3 Þ
ð28Þ
The isentropic efficiency is assumed to be 0.8 for all cases. For unit cooling or heating load and a particular value of COP, QL and QH are fixed. Assuming the refrigerant outlet in the evaporator as saturated vapor, the simulation code,
2'
Temperature
T LC;opt ¼ /X h T HC;opt
QH
3'
rv sv
TCO
TCI TEI
TLC
TEO
QL
4
1
Specific entropy
Fig. 2. Ammonia based refrigeration/heat pump cycle on the T–s plane.
J. Sarkar, S. Bhattacharyya / Energy Conversion and Management 48 (2007) 803–808
0.22 c
0.2
= 1. 6
= 1. 4
c
= 1. 6
Open symbol - Numerical Filled symbol - Analytical
0.18
UA (1/K)
c
0.16 c
= 1.2
0.14 c
0.12
= 1.4
0.1
c
= 1. 2
0.08 240
245
250
255
260
Evaporation temperature (K) Fig. 3. Variation of total conductance per unit cooling capacity.
0.085
0.075
UA (1/K)
which is based on the fundamental thermodynamic equations of the refrigeration/heat pump cycle, solves for the other state points of the cycle and m_ r for a certain evaporator temperature. The thermophysical properties of ammonia have been estimated employing the Haar and Gallagher correlations [13] through independently developed exclusive property subroutines. Using the LMTD Eqs. (26) and (27), individual conductances and heat exchanger areas and, finally, the total values have been obtained. Following a variation of the evaporation temperature, the minimum total conductance and area have been found. The inlet temperatures of the secondary fluids are taken as 280 K and 320 K for the evaporator and condenser, respectively, in this analysis. The variation in overall conductance with TLC for a refrigerator of 1 kW cooling capacity is shown in Fig. 3 for CL = CH = 0.05 kW/K. It is common knowledge that a lower temperature difference between the working fluid and the secondary fluid yields a higher COP, but it also requires a higher conductance for the same capacity. So, higher COP demands higher conductance as shown in Fig. 3. The results show that for a constant refrigerant temperature (THC) in the LMTD equation throughout the condensation process (solid lines in Fig. 3), the deviations are smaller (both the cold side and hot side working fluid temperature deviate below 1%, whereas the overall conductance deviates less than 0.5%) because of closely similar assumptions made in the two forms of modeling. As the assumption of isothermality is partially matched with that in the numerical model of the condenser, the individual optimum value of conductance of the condenser deviates slightly more than that of the evaporator. On the other hand, bifurcation of the condensation process into saturated and superheated zones (dotted lines in Fig. 3) results in significantly larger deviation (optimum TLC deviates below 2%, whereas the overall conductance deviates 20– 30%) as this is not compatible with the analytical model. Hence, the concept of THC has been used for the remaining results as it matches the analytical model better.
807
h
Open symbol - Numerical Filled symbol - Analytical
= 2.6
0.065
= 2.4
h
0.055 h
0.045
= 2.2
0.035 350
360
370
380
390
400
THC(K)
Fig. 4. Variation of total conductance per unit heating capacity.
The variation of overall conductance with THC for a heat pump is shown in Fig. 4 for QH = 1 kW and CL = CH = 0.04 kW/K. Following the same arguments presented earlier, a higher COP calls for a higher conductance as shown in Fig. 4. The results show that both the cold side and hot side optimum working fluid temperatures from the numerical study deviate below 1% when compared with that from the analytical model, whereas the overall conductance deviates within 0.5%. Similar to the above case, the theoretically obtained optimum value of the individual conductance of the condenser deviates more than that of the evaporator when compared against the numerical results. The variations of total heat transfer area per unit cooling capacity with TLC for a refrigeration plant are shown in Fig. 5 with a set of input parameters given by: UL = 0.5 kW/m2 K, UH = 0.5 kW/m2K. Because of the same reason discussed earlier, a higher COP requires a higher heat transfer area as is evident from Fig. 5. The difference of the analytically obtained cold side and hot side optimum working fluid temperatures and minimum area from the numerical predictions are 0.6–0.9%, 0.7–1% and 0.3– 0.5%, respectively, whereas the individual condenser area differs slightly more than that of the evaporator. The variations of total area with THC for the heat pump are shown in Fig. 6 for QH = 1 kW, CL = CH = 0.05 kW/K and overall heat transfer coefficients of 0.5 kW/m2 K for both the evaporator and condenser. Similar to the above case, a higher COP requires a higher heat transfer area. The analytical results compare well with the numerical results with the differences within the range of 0.7–1% for both the hot and cold side refrigerant temperatures and within 0.3–0.5% for total area, whereas the individual area of the condenser shows slightly higher differences due to the same reason discussed earlier. The minimum heat transfer area per unit capacity depends on the individual values of UL and UH. It is observed that the heat transfer area will be minimum at a particular ratio (d) of UH and UL, keeping their sum constant. By replacing TLC,opt and THC,opt in Eq. (17) or
808
J. Sarkar, S. Bhattacharyya / Energy Conversion and Management 48 (2007) 803–808
0. 35
c = 1.8
0. 3
2
Tot al area ( m /kW)
Open symbol - Numerical Filled symbol - Analytical
0. 25
c = 1.5
0. 2
c = 1.2
0. 15 232
237
242
247
252
257
Evaporation temperature (K) Fig. 5. Variation of total heat exchanger area per unit cooling capacity.
0.19
Open symbol - Numerical Filled symbol - Analytical
h = 2.8
0.15
2
Total area (m /kW)
0.17
0.13
as the working fluid. Accurate property subroutines, developed exclusively for this study, are employed to assist in the simulation. Numerical results have been obtained assuming a single equivalent heat rejection temperature in the condenser and also considering desuperheating and condensation regions separately. The results show that the hot and cold side refrigerant temperatures and the conductance and area ratio between evaporator and condenser can be optimized to obtain minimum overall conductance and heat exchanger area. The deviation of analytical results from those yielded by the numerical model is reasonably small (below 1% for all hot and cold side optimum refrigerant temperatures, and minimum total heat transfer area) if a constant refrigerant temperature (THC) is assumed throughout the condensation process because it closely resembles the analytical model. The deviations between analytical and numerical results are larger when condensation and desuperheating regions are treated separately (maximum deviation is of the order of 20%). The deviation between the analytical results and actual plant data would be higher than this due to the presence of other irreversibilities such as pressure drop and heat interaction with the surroundings, which have not been incorporated in the present analyses. However, the simple analytical expressions developed here will help in guiding the design engineer in the preliminary optimization of actual plants in the absence of more elaborate numerical models.
h = 2.5
0.11
References h = 2.2
0.09 0.07 350
360
370
380
390
THC (K) Fig. 6. Variation of total heat exchanger area per unit heating capacity.
(21), taking UH + UL= constant and considering oAodmin ¼ 0, it is possible to obtain an optimum d. The numerical results show that the optimum value of overall heat transfer coefficient ratio (UH/UL) varies between 1.08 and 1.12. In an actual evaporator or condenser design, UH and UL depend on the fluid properties, flow and geometry of the heat exchangers, and the heat transfer area also depends on geometry. Hence, both the concepts of conductance and heat transfer area minimizations are required in actual design practice. 4. Conclusions The overall conductance and the overall heat transfer area per unit capacity have been minimized analytically for refrigerator and heat pump systems working with variable temperature (finite) heat reservoirs. The analytical results are verified against numerical results obtained from an elaborate simulation of the systems based on ammonia
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