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Design and rating of an integrated mechanical-subcooling vapor-compression refrigeration system
Energy Conversion & Management 41 (2000) 1201±1222
www.elsevier.com/locate/enconman
Design and rating of an integrated mechanical-subcooling vapor-compression refrigeration system Jameel-ur-Rehman Khan, Syed M. Zubair* Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Abstract Subcooling of the refrigerant at the exit of the condenser in a simple vapor-compression refrigeration system allows the refrigerant to enter the main cycle evaporator with low quality. Thus, allowing the refrigerant to absorb more heat in the evaporator; thereby improving the coecient of performance (COP) of the system. In an integrated mechanical-subcooling vapor-compression refrigeration system, the subcooling is performed by utilizing a small integrated vapor-compression refrigeration cycle, known as the subcooler cycle. This subcooler cycle is coupled to the main cycle at the exit of the condenser and it utilizes the main cycle condenser for rejecting the heat. In this paper, thermodynamic models of an integrated mechanical subcooling system are developed to simulate the actual performance of the subcooling system, particularly with respect to the subcooler saturation temperature in addition to heat exchanger areas. It is demonstrated that the performance of the overall cycle is improved over the corresponding simple cycle. This improvement is found to be related to the refrigerant saturation temperature of the subcooler. The model is also used for predicting an optimum distribution of the total heat exchanger area between the evaporator and condenser. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Integrated mechanical-subcooling system; Finite-time thermodynamic model; Characteristic performance curves
* Corresponding author: Tel.: +966-3-860-2540; fax: +966-3-860-2949. E-mail address: [email protected] (S.M. Zubair). 0196-8904/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 9 9 ) 0 0 1 6 9 - 7
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Nomenclature
Tcond T in cond Tevap T in evap T out, sub T sub _ W _s W Z e
out, sub dierence between T main cond and T _ p (kW Kÿ1) capacitance rate for the external ¯uids C_ mc coecient of performance ratio of condenser eectiveness-capacitance product to the sum of the eectiveness-capacitance rate products for the condenser and evaporator of the cycle. subcooler heat rejection factor. speci®c enthalpy of refrigerant at state point j (kJ kgÿ1) constant thermal inventory (kW Kÿ1) mass ¯ow rate (kg sÿ1) rate of heat rejection in the condenser (kW) rate of heat leak from the hot refrigerant (kW) rate of heat absorbed in the evaporator (kW) rate of heat leak from the ambient to cold refrigerant (kW) rate of heat leak from the compressor shell to ambient (kW) internal entropy generation, primarily due to non-isentropic compression and expansion (kW Kÿ1) refrigerant temperature in the condenser (K) condenser coolant inlet temperature (K) refrigerant temperature in the evaporator (K) evaporator coolant inlet temperature (K) temperature of the main cycle refrigerant after leaving the subcooler (K) subcooler saturation temperature (K) electrical power input to the compressor (kW) power required for isentropic compression (kW) eciency heat exchanger eectiveness
Subscripts 1, 2, 3, . . . comp cond evap gen ref total W
state points compressor condenser evaporator generation refrigerant total heat loss from compressor shell
AMTS C_ COP fh frejection hj k m_ Q_ cond Q_ loss cond Q_ evap Q_ loss evap Q_ loss W S_igen
Superscripts in inlet loss heat loss
J-u-R. Khan, S.M. Zubair / Energy Conversion & Management 41 (2000) 1201±1222
main sub out
1203
main cycle subcooler cycle outlet
1. Introduction The performance of a simple vapor-compression refrigeration system can be signi®cantly improved by further cooling the liquid refrigerant leaving the condenser coil. This subcooling of the liquid refrigerant can be accomplished by adding a mechanical-subcooling loop in a conventional vapor compression cycle. The subcooling system can either be a dedicated mechanical-subcooling system or an integrated mechanical-subcooling system. In a dedicated mechanical-subcooling system, there are two condensers, one for each of the main cycle and the subcooler cycle, whereas for an integrated mechanical-subcooling system, there is only one condenser serving both the main cycle and the subcooler cycle. Thornton et al. [1] and Couvillion et al. [2] have investigated a dedicated mechanical-subcooling system, whereas Bahel and Zubair [3], Zubair [4] and Zubair et al. [5] have investigated an integrated mechanicalsubcooling system from the thermodynamic standpoint. In either of these studies, they found that the system performance improved when operating in situations where the dierence between the condensing and evaporating temperatures is large. The development of refrigeration system models that simulates the actual working of the mechanical-subcooling system, has been the goal of many researchers (Thornton et al. [1]; Couvillion et al. [2]; Bahel and Zubair [3]; Zubair [4]; and Zubair et al. [5]). The most simple model is an ideal temperature-dependent model, which represents an ideal mechanicalsubcooling system without any irreversible losses. Thornton et al. [1] have developed an ideal temperature-dependent thermodynamic model for predicting the performance of mechanicalsubcooling system with irreversibilities only in the subcooler. A temperature-dependent model with irreversibilities in all the heat exchangers of the system can be considered as an improved temperature-dependent model. However, the temperature-dependent models do not take into account the actual properties of the refrigerants that are used in the systems. The propertydependent thermodynamic models takes into account the refrigerant properties and are developed by considering the mass and energy balance across each component of the system. It should be noted that the ®nite-time thermodynamic model, accounts for the irreversibilities existing due to the ®nite temperature dierence in the heat exchangers as well as the losses due to non-isentropic compression and expansion in the compressors and expansion valves of the system, respectively. Zubair [4] and Zubair et al. [5] have investigated the second-law analysis of an integrated mechanical-subcooling system using an ideal refrigerant cycle model. They showed that the irreversible losses in the expansion device (the major source of irreversibility) can be signi®cantly reduced by operating the system at the optimum subcooling conditions. This optimum condition was found to occur at a subcooler saturation temperature about halfway between the condensation and evaporation temperatures. The objective of this paper is to investigate temperature-dependent and property-dependent thermodynamic models of integrated mechanical-subcooling systems.
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2. Cycle description The major components of an integrated mechanical subcooling vapor-compression refrigeration system includes two reciprocating compressors, two expansion valves, condenser, evaporator, receiver and a subcooler. The system consists of two simple cycles coupled to each other via a subcooler as shown in Fig. 1, while its pressure enthalpy diagram is shown in Fig. 2. The bigger cycle is known as the main cycle and the smaller cycle is known as the subcooler cycle. The two cycles have a common condenser, and the components of the two cycles are connected in a closed loop through a piping system that has heat transfer with the surroundings. The ®gure shows that the main-cycle refrigerant leaves the main-cycle evaporator at state 1 as a low pressure, low temperature, saturated vapor and enters the main cycle compressor at state 2. The refrigerant, from state 1 to 2 takes heat from the surroundings in
Fig. 1. Schematic of an integrated mechanical-subcooling refrigeration system.
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Fig. 2. Pressure±enthalpy diagram of an integrated mechanical-subcooling refrigeration system.
the suction line. At state 3, it leaves the compressor as a high temperature, high pressure, superheated vapor. The refrigerant, from state 3 to 4 rejects heat to the surroundings in the discharge line. At state 4, it mixes with the subcooler cycle refrigerant coming from the subcooler cycle compressor and attains state 13, and the mixture enters the condenser. The mixture after leaving the condenser is collected in the receiver. Some of this liquid refrigerant mixture is extracted from the receiver and is expanded in the expansion valve of the subcooler cycle and is then passed through the subcooler. The remaining liquid refrigerant in the receiver enters the subcooler, where it is cooled below the saturated liquid state at a constant pressure to state 6 by the subcooler cycle refrigerant. It enters the main cycle expansion valve and at state 7 it leaves the expansion valve as a low quality vapor and enters the evaporator. In the evaporator, it is evaporated at a constant pressure to the saturated vapor state. The subcooler-cycle refrigerant after cooling the main-cycle refrigerant in the subcooler, leaves as a low-pressure, low-temperature, saturated vapor at state 9 and enters the subcooler cycle-compressor at state 10. The refrigerant from state 9 to 10 takes heat from the surroundings. At state 11, it leaves the compressor as a superheated vapor where it is mixed with the main cycle refrigerant coming from the main-cycle compressor, and attains state 13, as shown in Fig. 1. 3. Analysis of the cycle Referring to the main cycle loop in Fig. 1, from the ®rst law of thermodynamics and the fact that the change in internal energy is zero for a cyclic process, we can write loss, main loss, main loss, sub loss, sub main sub _ _ _ _ _ _ _ W Qcond Qcond QW QW Qcond W 1 main loss, main loss, sub ÿ Q_ : Q_ Q_ evap
evap
evap
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Since entropy is a state function, it's net change over the cycle is zero. Therefore, the contributions to the entropy generation are from heat transfer (at the condenser, evaporator and subcooler) and the entropy generation due to non-isentropic compression and expansion in the compressors and the expansion valves, respectively. In addition to various irreversible losses, there is also a contribution to the entropy generation due to mixing of the refrigerants after exiting their respective compressors, as shown in Fig. 1. main loss, main out, sub _ _ Qevap Qevap ÿ ÿ main main Q_ cond T T13 total _ _ p ref ln _ p ref ln ÿ mc ÿ ÿ mc Sigen Tcond Tcond T3 T main evap sub loss, sub Q_ evap Q_ evap ÿ sub T13 _ p ref ln ÿ mc ÿ : 2 T sub T11 evap The heat transfer to and from the cycle occur by convection to ¯owing ¯uid streams with ®nite mass-¯ow rates and speci®c heats. Therefore, the rate of heat transfer to the cycle at a low temperature in the main-cycle evaporator can be written as ÿ main main main main Q_ evap eC_ evap T in, 3 ÿ T evap evap , while the rate of heat transfer between the refrigeration cycle and the sink in the main cycle condenser is ÿ ÿ 4 Q_ cond eC_ cond Tcond ÿ T in cond : De®ning the COPtotal as refrigerating eect over the net work input, we have COPtotal
main Q_ evap
_ main W _ sub W
:
5
_ sub from Eq. (1) into the above equation and _ main and W By substituting the expression of W expressing refrigeration temperatures in terms of the more readily available coolant temperatures, one obtains an analytical formula for the COP as a function of cooling capacity, coolant temperatures, heat exchanger characteristics, heat leak terms, internal cycle losses and the exit temperature of the main-cycle refrigerant from the subcooler, as loss, main loss, main loss, sub loss, sub _ _ _ _ QW QW Qcond Qcond 1 X1 ÿ1 main COPtotal X2 Q_ evap
ÿ
loss, main loss, sub Q_ evap Q_ evap main Q_ evap
where in the above equation we have,
6
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0 " # _ main Q_ loss, main out, sub ÿ ÿ Q ÿ T T evap evap total i3 main B _ ! mc _ p ref ln X1 eC_ cond T in cond BSigen main B Tcond T3 Q_ evap @ in T evap ÿ main eC_ evap
sub loss, sub Q_ evap Q_ evap
1
sub ÿ T13 C _ p ref ln mc C T11 C A
T sub evap
1207
7
8 0 # " main loss, main > _ _ > ÿ Q Q T out, sub Ti3 > main ÿ evap evap main < total B ! mc _ p ref ln X2 Q_ evap eC_ cond ÿBS_igen main B Tcond T3 Q_ evap > @ in > T evap ÿ main > : eC_ evap
sub loss, sub Q_ evap Q_ evap
T sub evap
19 > > ÿ sub T13 C > = _ p ref ln mc C T11 C A> > > ;
8
Eq. (6) gives the COPtotal of the system for one set of input data. The required input data can be measured experimentally from an actual operating system or calculated from the thermodynamic model, which is discussed in the latter part of the paper. Table 1 shows the required input data obtained from the thermodynamic model. The value of COPtotal obtained from Eq. (6) using the input data given in Table 1 was found to be 4.13 which is exactly equal to the value obtained from the thermodynamic model. Therefore, the analytical expression given in Eq. (6) and the thermodynamic model are correct and hence they can be used for design and performance evaluation purpose. It should be noted that the heat leak terms are neglected in the above comparison because they do not contribute much towards the overall system performance as reported by Gordon et al. [6]. Fig. 3 shows the characteristic performance curve (1.0/COP vs. 1:0=Q_ evap of the system. The plots are drawn using the input data given in Table 1. The design point is also shown in the ®gure, where for Fig. 3(a) the design point is (COP = 2.763). Note that the design point is located away from the minimum point because of the losses mainly due to non-isentropic compression in the compressors of the system. However, the design point is at the minimum point (COP = 4.1297) when the eciency of the compressors is assumed to be 100%, as main and a shown in Fig. 3(b). These plots are hypothetical curves drawn for a constant T in, evap main _ range of main cycle evaporator capacities Qevap : The shapes of the curves are similar to the one obtained for a simple cycle [6]. It shows that the COP increases with evaporator capacity up to the minimum point, due to irreversibilities such as ¯uid friction (some times referred to as non-isentropic losses) in the compressors and expansion valves. While at evaporator capacities greater than the minimum point, COP decreases signi®cantly because of the losses
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Table 1 Input data for Eq. (6), obtained from the thermodynamic model System operating parameters
For Fig. 3(a)
For Fig. 3(b)
Cooling capacity of the main cycle evaporator, Q_ main evap main Heat leak into the suction line of the main cycle, Q_ loss, evap main Heat leak from the discharge line of the main cycle, Q_ loss, cond loss, main Heat leak from the compressor shell of the main compressor, Q_ W _ main Eectiveness and capacitance rate product for the main evaporator, eC evap main _ Eectiveness-capacitance rate product for the main condenser, eCcond total Internal entropy generation due to non isentropic compression and expansion, S_igen in, main Coolant temperature at the inlet to the main evaporator, T evap Coolant temperature at the inlet to the condenser, T in cond Isentropic eciency of the main cycle compressor, Z main comp Isentropic eciency of the subcooler cycle compressor, Z sub comp Temperature of the main cycle refrigerant after exiting the subcooler, T out, sub Refrigerant temperature at the main condenser, Tcond Subcooler saturation temperature, T sub evap _ p main Mass ¯ow rate-speci®c heat product of the main cycle re®gerant, mc ref _ p sub Mass ¯ow rate-speci®c heat product of the subcoler cycle re®gerant, mc ref Cooling capacity of the subcooler, Q_ sub evap loss, sub Heat leak into the suction line of the subcooler cycle, Q_ evap sub Heat leak from the discharge line of the subcooler cycle, Q_ loss, cond sub Heat leak from the compressor shell of the subcooler compressor, Q_ loss, W Temperature at state 13, T13 Temperature at state 3, T3 Temperature at state 11, T11
30.0 kW 0.0 W 0.0 W 0.0 W 5.0 kW Kÿ1 5.5 kW Kÿ1 13.958 W Kÿ1 0.08C 40.08C 65% 70% 34.6538C 46.6538C 30.5078C 309.46 W Kÿ1 182.33 W Kÿ1 3713 W 0.0 W 0.0 W 0.0 W 54.7178C 57.96998C 49.1968C
30.0 kW 0.0 W 0.0 W 0.0 W 5.0 kW Kÿ1 5.5 kW Kÿ1 3.517 W Kÿ1 0.08C 40.08C 100% 100% 34.0398C 46.0398C 30.0298C 306.85 W Kÿ1 195.85 W Kÿ1 3682.20 W 0.0 W 0.0 W 0.0 W 52.10388C 54.35298C 48.588C
due to the ®nite rate of heat transfer in the heat exchangers of the system. It should be noted that the performance of the system in the light of various losses of the system is similar to the performance of a simple cycle discussed by Gordon and Choon [7].
4. Performance evaluation of the system Fig. 1 shows that the main-cycle refrigerant and the subcooler-cycle refrigerant mixed at state 13, after leaving their respective compressors. The mixture then rejects heat in the condenser at a constant temperature Tcond : It can be deduced that each refrigerant cycle uses a fraction of the total condenser area for rejecting it's heat. To investigate the fraction of area utilized by each refrigerant, a factor de®ned as the subcooler heat-rejection factor is de®ned as ÿ sub eC_ 9 frejection ÿ main cond ÿ sub eC_ eC_ cond
cond
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Fig. 3. Performance curves of an integrated mechanical-subcooling vapor-compression refrigeration system, for an input data given in Table 1.
_ main eC _ sub is the total thermal inventory of the condenser eC _ cond : The individual where eC cond cond areas can be expressed in terms of the total area as follows ÿ ÿ sub eC_ cond frejection eC_ cond ,
10
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ÿ main ÿ ÿ eC_ cond 1:0 ÿ frejection eC_ cond :
11
The performance of a mechanical-subcooling system is controlled mostly by the subcooler temperature T sub evap , as reported by Thornton et al. [1]. Mechanical-subcooling allows the main cycle refrigerant to enter the evaporator with a low quality (state 7), see Fig. 2, compared with a typical simple vapor-compression cycle (state 5'). The low quality of the refrigerant at the evaporator inlet corresponds to an increase in the refrigeration capacity per unit mass of refrigerant circulated in the main cycle. However, it should be noted that the increase in refrigeration capacity is not without cost since there are additional components in the system. Neglecting losses to the environment, an energy balance on the subcooler indicates that the amount of subcooling provided to the main cycle must equal the heat addition to the subcooling cycle evaporator. The heat addition to the subcooler must be rejected in the subcooling cycle condenser at the cost of the work consumed in the subcooling cycle compressor. It should be emphasized that there is an ``optimum'' temperature for the subcooler at which the COPtotal of the cycle is maximized. This ``optimum'' temperature can be investigated by developing a temperature-dependent model of the system. Similar to Thornton et al. [1], the following assumptions were made in the development of the mechanicalsubcooling model: . Both the main and subcooling cycle condensers reject heat at the sink temperature, T in cond : , an intermediate temperature . The subcooling cycle heat addition occurs at T sub evap sub in T in evap RT evap RT cond ). . There is no thermal energy loss to the environment. . The main cycle compressor work is not in¯uenced by the amount of subcooling provided to the main cycle. . The exit states of the main cycle condenser and evaporator are unaected by the amount of subcooling performed. . Isentropic expansion and compression are assumed for both the main and subcooling cycles. If no subcooling is provided to the main cycle, the COP of the main cycle can be written as, main
COP
main Q_ evap
_ main W
T main evap Tcond ÿ T main evap
main T in, ÿ DTm evap in, main DTm T in ÿ T evap cond
_ main and Q_ cond de®ned earlier, we can write using the de®nitions of W # main " Q_ evap T in in cond Tcond T cond ÿ main : main T in, ÿ DTm evap eC_
12
13
cond
The subcooling cycle operates between the temperatures T in cond and Tcond : Therefore, the COP of the subcooling cycle may be expressed as sub
COP
sub Q_ evap sub
_ W
T sub evap
T sub cond
ÿ
T sub evap
ÿ
Tcond ÿ DTs : ÿ Tcond DTs
T in cond
14
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The overall cycle COP may be expressed as the total refrigeration capacity divided by the total power. The capacity of the overall cycle is simply the capacity of the main cycle without subcooling Q_ main evap ), plus the increment in capacity of the main cycle due to the subcooling performed Q_ sub evap ). The total power is simply the sum of the compressor work for both the subcooling and the main cycles. With these de®nitions, the overall cycle COP may be expressed as COPtotal
main sub Q_ evap Q_ evap
15
_ sub _ main W W
After manipulations of the above equations we can express the overall COPtotal of the system in terms of the subcooler cycle temperature T sub evap and other variables as 1 X Tcond ÿ T sub evap , 16 COPtotal in main T cond ÿT in, DTm X Tcond ÿT sub evap DTTs evap in, main in T ÿ DTTs ÿ DT T m
evap
cond
where in the above equation, we have subcooler parameter given by ÿ X
main
esub ref , main, nosub Q_ evap
_ p mc
17
and various temperatures as " # 1 1 main _ DTm Qevap ÿ main ÿ , ÿ 1:0 ÿ frejection eC_ cond eC_ evap " DTs Q
Tcond
sub
1
main ÿ _ p ref esub mc
# 1 ÿ , frejection eC_ cond
# " _ main in Q T evap cond , T in ÿ cond ÿ in, main 1:0 ÿ frejection eC_ cond T evap ÿ DTm
DTTs DTs T in cond ÿ Tcond :
18
19
20
21
It should be noted that Eq. (16) considers the irreversible losses in all the heat exchangers. _ main , eC _ main to be very large, it gives losses in the _ sub , eC However, when we consider eC cond cond evap subcooler only. Substituting the above values in Eqs. (18)±(21), we get DTm 0:0,
22
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J-u-R. Khan, S.M. Zubair / Energy Conversion & Management 41 (2000) 1201±1222 sub Q_ evap sub DTs ÿ main sub Tcond ÿ T evap , _ p ref eevap mc
23
in T main cond T cond ,
24
DTTs DTs :
25
and
Substituting the above values in Eq. (16) we get sub 1 X T in ÿ T cond evap COPtotal in 2 in, main in T cond ÿT evap T cond ÿT sub evap X in, main sub T T
26
evap
evap
which is exactly the same equation as that discussed by Thornton et al. [1]. To show the predicted improvement in performance of a mechanical-subcooling refrigeration system with respect to the single stage cycle, the curves are plotted in a normalized form de®ned as COPN
COPtotal , COPmain
27
similarly, a reduced subcooler saturation temperature y is de®ned as y
main T sub evap ÿ T evap
Tcond ÿ T main evap
:
28
Fig. 4(a) shows the plot of normalized coecient of performance (COPN) and the reduced subcooler saturation temperature y). The plots are obtained by using Eq. (27) for the main _ main _ main 0:08C, T in following set of input data T in, evap cond 40:08C, Qevap 30:0 kW, eCevap _ cond 5500:0 W Kÿ1. In these plots, y varies between T main and Tcond : 5000:0 W Kÿ1, and eC evap It can be seen from the plots that maximum COPN occurs at approximately y 0:5, and it almost remains constant for all the values of the subcooler cycle heat exchanger parameter X. main For y lower than 0.5, the dierence between T sub evap and T evap is less and hence maximum amount of subcooling is being performed; in this case, the thermal lift of the subcooling cycle is close to that of the main cycle. Therefore, the subcooling cycle compressor consumes more power, and the advantage of using an integrated mechanical-subcooling is destroyed and the COPN of the system reduces. For y greater than 0.5, the dierence between T sub evap and Tcond is less and hence a relatively low amount of subcooling is being performed and the overall cycle behaves very close to a simple cycle, thus the value of COPN is lower than the maximum possible value. Between these two extremes, there is an optimum T sub evap at which COPN is maximum, as shown in the ®gure. Fig. 4(b) shows the variation of the subcooler heat rejection factor frejection with the reduced subcooler temperature y: It can be seen from the plots that at high values of y, the amount of
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Fig. 4. Variations of COPN (a) and frejection (b) with y for a completely irreversible temperature-dependent integrated mechanical-subcooling system.
subcooling performed is lower. Hence, the amount of heat to be rejected by the subcooler cycle refrigerant is also low. Therefore, it uses a small fraction of the condenser area, however, as the value of y reduces, the amount of subcooling performed is high; thereby requiring more subcooler compressor work. Hence, the amount of heat to be rejected by the subcooler cycle refrigerant is relatively high. Thus, it uses a large fraction of the condenser area. Also, for a given value of y, the heat rejection factor increases with X since subcooling increases with the subcooler heat exchanger parameter X.
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Fig. 5(a) shows the eect of condenser coolant inlet temperature T in cond on the subcooler heat rejection factor frejection for constant values of other system parameters. The ®gure shows that for high values of T in cond and particularly at low values of y, the heat rejection factor frejection increases, since the amount of subcooling increases; thereby requiring more subcooler compressor work. Thus, an increase in heat rejection factor for the subcooling cycle is expected in with T in cond : The ®gure also shows that at high values of y, for all the values of T cond , the amount of subcooling performed is low. Therefore, the amount of heat to be rejected by the
in, main Fig. 5. Variations of frejection with y for various T in (b). cond (a) and T evap
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subcooler cycle refrigerant is relatively low. Thus, it uses a small fraction of the condenser area. main on the subcooler Fig. 5(b) shows the eect of evaporator coolant inlet temperature T in, evap heat rejection factor frejection for constant values of other system parameters. Similar to Fig. 5(a), main and y, the heat rejection factor frejection this ®gure also shows that for low values of T in, evap increases, since the amount of subcooling increases; thereby requiring more subcooler compressor work and hence an increase in heat rejection factor for the subcooling cycle is expected. The ®gure also shows that at high values of y and almost for all the values of main , the amount of subcooling performed is lower and hence the amount of heat to be T in, evap rejected by the subcooler cycle refrigerant is relatively low. Therefore, it uses a small fraction of the condenser area, however, as described earlier that as the value of y reduces, the amount of subcooling performed is high and hence the amount of heat to be rejected by the subcooler cycle refrigerant is relatively high. Thus, it uses a large fraction of the condenser area.
5. A ®nite time thermodynamic model of the system As mentioned earlier, the heat transfer to and from the cycle occur by convection to ¯owing ¯uid streams having ®nite mass ¯ow rates and speci®c heats. Therefore, the rate of heat transfer to the cycle in the main cycle evaporator can be expressed in terms of temperature, mass ¯ow rates, and change in speci®c enthalpy of the refrigerant, given by ÿ main main main main _ main 29 ÿ T Q_ evap eC_ evap T in, evap evap m ref h1 ÿ h7 : The compressor operation is described in terms of an isentropic eciency Zmain comp , so that the power requirement for the main cycle compressor is given by _ main W main main s _ m_ ref h3 ÿ h2 main : W Zcomp
30
In the subcooler, the rate of heat transfer between the main cycle refrigerant and the subcooler cycle refrigerant is given by ÿ main ÿ sub _ p Tcond ÿ T out, sub m_ main Q_ evap mc 31 ref h5 ÿ h6 : loss, main of heat, leaks into the suction line of the main cycle Assuming that an amount of Q_ evap compressor, which can be expressed as loss, main m_ main Q_ evap ref h2 ÿ h1 :
32
loss, main loss, main Similarly, assuming that an amount of Q_ cond Q_ W of heat leaks from the discharge line and compressor shell given by loss, main loss, main Q_ cond Q_ W m_ main ref h3 ÿ h4 :
33
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For the subcooler cycle loop, the rate of heat transfer between the subcooler cycle refrigerant and the main cycle refrigerant in the subcooler can be written in terms of temperatures, the mass ¯ow rates, and speci®c enthalpy of the refrigerant, expressed as, ÿ main sub _ sub h9 ÿ h8 : _ p ref esub Tcond ÿ T sub Q_ evap mc 34 evap m The work input to the subcooler cycle compressor is given by _ sub W sub sub _ W m_ ref h11 ÿ h10 subs : Zcomp
35
loss, sub of heat leaks into the suction line of the subcooler cycle, Assuming that an amount of Q_ evap which can be expressed as sub Q_ loss m_ sub ref h10 ÿ h9 :
36
loss, sub loss, sub of heat leaks from the discharge line Similarly, assuming that an amount of Q_ cond Q_ W and the compressor shell. This is given by loss, sub loss, sub m_ sub Q_ cond Q_ W ref h11 ÿ h12 :
37
Also, the rate of heat transfer between the refrigeration cycle and sink in the main cycle condenser is expressed as ÿ ÿ in, main _ _ Qcond eC cond Tcond ÿ T cond m_ main m_ sub h13 ÿ h5 : 38 The mixing of the main cycle refrigerant and subcooler cycle refrigerant after leaving their respective compressors is given by m_ main h4 ÿ h13 m_ sub h13 ÿ h12 : From the above equation, we get ÿ m_ main h4 m_ sub h12 : h13 ÿ main m_ m_ sub
39
40
The above equations have been solved numerically by using the thermodynamic property data for several dierent refrigerants. The ¯ow chart representing the method of solving these equations is shown in Fig. 6, wherein the terms SATPRP and TRIAL represent the subroutines for calculating the refrigerant saturated and vapor properties, respectively. These subroutines need any two independent intensive properties of the refrigerant to ®nd other properties at a given state. For this purpose, a computer program, originally developed by Kartsounes and Erth [8] and modi®ed by Fisher and Rice [9] and Khan and Zubair [10], have been used. The program gives the COPtotal and all other parameters of the system for the main _ main , eC _ main , T in , T in, main , AMTS and esub : following set of input data: Q_ evap , eC cond evap cond evap evap Fig. 7(a) shows the plot of COPN vs. the reduced subcooler saturation temperature y and
J-u-R. Khan, S.M. Zubair / Energy Conversion & Management 41 (2000) 1201±1222
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Fig. 6. Flow chart for a thermodynamic model of an integrated mechanical-subcooling system.
the heat exchanger thermal inventory parameter k, de®ned as ÿ ÿ k eC_ evap eC_ cond :
41
The results shown in the ®gure are obtained from the above described thermodynamic model main main sub sub _ p main for R-134a when: T in, 0:08C, T in ref e evap cond 40:08C, Zcomp 0:65, Zcomp 0:7, mc main ÿ1 _ 0:1 kW K and Qevap 30:0 kW. Fig. 7(b) shows the corresponding variation of the system operating temperatures for k 12:5 kW Kÿ1. The ®gure shows that as the value of heat exchanger thermal inventory parameter k increases, the value of COPN also increases, because as k increases the irreversible losses due to heat transfer decreases in the heat exchangers of the system, thus increasing the system performance. We note that the property-dependent model
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main Fig. 7. Variations of COPN (a) and operating temperatures (b) with y for the following input data: Q_ evap 30:0 ÿ1 main sub main sub in, main in _ p ref e 0:1 kW K , Zcomp 0:65, Zcomp 0:70, T evap 273:0 K, T cond 313:0 K. kW, mc
predicts the same relationship between COPN and the reduced subcooler saturation temperature y, as that discussed earlier by the temperature-dependent model (see Fig. 4(a)). It should be noted that the heat leak terms are neglected in the above analysis because they do not contribute much towards the overall system performance (see Ref. [6]). Referring to Fig. 7(b), we note that the condenser temperature more-or-less remains constant, however, one would expect a variation in Tcond with the reduced subcooler saturation temperature y: The calculations indicate a somewhat constant temperature varying between 319.42 and 319.33 K,
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1219
Fig. 8. Variation of COPtotal with y for an ideal temperature-dependent and property-dependent mechanicalsubcooling system: irreversible losses only in the subcooler.
which is mainly due to the fact that the variation in Q_ cond is relatively small and moreover the _ cond 6:5 kW Kÿ1) in the calculation of Tcond is high. value of eC Fig. 8 shows the plot of COPtotal vs. the reduced subcooler saturation temperature y and the subcooler parameter X of the property-dependent and temperature-dependent ideal mechanicalsubcooling vapor-compression refrigeration system with irreversible losses only in the subcooler and using R-134a as the refrigerant. The plots are drawn for the following set of input data: main _ main 0:08C, T in T in, evap cond 40:08C, and Qevap 30:0 kW. The ®gure shows that the shapes of the curves for the two models are similar and the dierence in the COPtotal values is mainly due to non-isentropic expansion in the expansion valves of the property-dependent model. The maximum value of COPtotal shown in the ®gure is for X = 0.0066, and is equal to 7.29. It should be noted that this value is much larger than the Carnot cycle limit of 6.825, which is main ÿ0:08C and T in the ideal COP for T in, evap cond 40:08C. This can be explained by the fact that the temperature dependent model will not show the eect of mass ¯ow rates on the system performance. Therefore, the COPtotal is somewhere between the Carnot limit of the main and subcooler cycle. It is thus important to note that the maximum possible value of COPtotal is the Carnot cycle COP of the subcooler cycle, which is given by COPtotal
T sub evap
sub T in cond ÿ T evap
:
42
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In the absence of the subcooler cycle, the maximum possible COPtotal is the Carnot cycle COP of the main cycle only, which is given by COPtotal
main T in, evap
in, main T in cond ÿ T evap
:
43
This is also established from Eq. (16), which reduces to Eq. (42) when irreversible losses in all the heat exchangers including that of the subcooler are removed.
main Fig. 9. Variations of COPtotal (a) and operating temperatures (b) with fh for the following input data: Q_ evap 30:0 sub in, main kW, Zmain 273:0 K, T in comp 0:65, Zcomp 0:70, T evap cond 313:0 K.
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1221
6. Optimum distribution of the heat exchanger area A set of sample plots, shown in Fig. 9(a), is drawn between COPtotal and the dimensionless heat exchanger parameter fh ), de®ned as ÿ eC_ cond ÿ fh ÿ 44 eC_ cond eC_ evap using the output from the model; for the following set of input data and using R-134a as the main ÿ1 main 273:0 K, T in refrigerant: Q_ evap 30:0 kW, T in, evap cond 313:0 K, k 10:0 kW K , sub _ p main Zmain ref comp 0:65, Zcomp 0:7: The plots are drawn for dierent values of the parameter mc sub and for each value of the dimensionless heat exchanger parameter fh ); the global e optimum value of the subcooler saturation temperature T sub evap is calculated as shown in Fig. 7(a). For better explanation of the plots, the variation of system temperatures are shown sub _ p main 0:2 kW Kÿ1. The ®gure shows that COPtotal is maximum at in Fig. 9(b) for mc ref e about fh 0:55, for all the curves. The dierence between T in cond and Tcond is high at low values of the parameter fh ; therefore, the heat transfer in the condenser takes place at high temperature dierence resulting in high irreversible losses due to ®nite rate of heat transfer. main and These irreversible losses decrease as fh increases. However, the dierence between T in, evap main T evap increases as fh increases and the heat transfer in the evaporator takes place at a high temperature dierence, resulting in an increase in irreversible losses due to the ®nite rate of heat transfer. At fh 0:55, the temperature dierence in the evaporator and condenser is minimum, hence COPtotal is maximum at this point. It should also be noted that the heat transfer in the subcooler takes place through the temperature dierence between Tcond and T sub evap , and the ®gure shows that this temperature dierence is approximately constant for all values of fh :
7. Conclusions Thermodynamic models of an integrated mechanical-subcooling vapor-compression refrigeration system have been developed to study the design and performance evaluation of the system. The models developed include both the temperature and property-dependent models. The performance evaluation of the systems has been investigated by considering the temperature model. It is demonstrated that the optimum subcooler saturation temperature at which the COP of the system is maximum, is not a function of the subcooler heat exchanger parameter X. This optimum temperature is found to be about halfway between the condensation and evaporation temperatures (i.e., y 0:50 for almost all values of the parameter X. The property-dependent model showed the same trends as that predicted by the temperature model, that is, the existence of an optimum subcooling saturation temperature. The optimum distribution of total heat exchanger areas between the condenser and evaporator has also been investigated. It is found that COP of the system is maximum when a large fraction of the total area is allocated to the condenser than to the evaporator of the cycle. It is also demonstrated
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that this optimum distribution occurs when the irreversibilities due to the ®nite rate of heat transfer in the heat exchangers of the system are minimum. Acknowledgements The authors acknowledge the support provided by King Fahd University of Petroleum and Minerals for this research project. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Thornton JW, Klein SA, Mitchell JW. International Journal of Refrigeration 1994;17(8):508±15. Couvillion RJ, Larson MW, Somerville MH. ASHRAE Transactions 1988;94(2):641±59. Bahel V, Zubair SM. Heating Piping Air Conditioning 1988;60(2):105±7. Zubair SM. Energy 1994;19(6):707±15. Zubair SM, Yaqub M, Khan SH. International Journal of Refrigeration 1996;19(8):506±16. Gordon JM, Chua HT, NG, KC. International Journal of Heat and Mass Transfer 1996;39(11):2195±204. Gordon JM, Choon NG. International Journal of Heat and Mass Transfer 1995;38(5):807±18. Kartsounes GT, Erth RA. ASHRAE Transactions 1971;77(2):88±103. Fisher SK, Rice CK. The Oak Ridge heat pump models: Part 1. A steady-state computer design model for airto-air heat Pumps, ORNL/CON-80/RI, Oak Ridge National Laboratory, Oak Ridge: TN, 1983. [10] Khan SH, Zubair SM. Energy 1993;18(7):717±26.
www.elsevier.com/locate/enconman
Design and rating of an integrated mechanical-subcooling vapor-compression refrigeration system Jameel-ur-Rehman Khan, Syed M. Zubair* Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Abstract Subcooling of the refrigerant at the exit of the condenser in a simple vapor-compression refrigeration system allows the refrigerant to enter the main cycle evaporator with low quality. Thus, allowing the refrigerant to absorb more heat in the evaporator; thereby improving the coecient of performance (COP) of the system. In an integrated mechanical-subcooling vapor-compression refrigeration system, the subcooling is performed by utilizing a small integrated vapor-compression refrigeration cycle, known as the subcooler cycle. This subcooler cycle is coupled to the main cycle at the exit of the condenser and it utilizes the main cycle condenser for rejecting the heat. In this paper, thermodynamic models of an integrated mechanical subcooling system are developed to simulate the actual performance of the subcooling system, particularly with respect to the subcooler saturation temperature in addition to heat exchanger areas. It is demonstrated that the performance of the overall cycle is improved over the corresponding simple cycle. This improvement is found to be related to the refrigerant saturation temperature of the subcooler. The model is also used for predicting an optimum distribution of the total heat exchanger area between the evaporator and condenser. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Integrated mechanical-subcooling system; Finite-time thermodynamic model; Characteristic performance curves
* Corresponding author: Tel.: +966-3-860-2540; fax: +966-3-860-2949. E-mail address: [email protected] (S.M. Zubair). 0196-8904/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 9 9 ) 0 0 1 6 9 - 7
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Nomenclature
Tcond T in cond Tevap T in evap T out, sub T sub _ W _s W Z e
out, sub dierence between T main cond and T _ p (kW Kÿ1) capacitance rate for the external ¯uids C_ mc coecient of performance ratio of condenser eectiveness-capacitance product to the sum of the eectiveness-capacitance rate products for the condenser and evaporator of the cycle. subcooler heat rejection factor. speci®c enthalpy of refrigerant at state point j (kJ kgÿ1) constant thermal inventory (kW Kÿ1) mass ¯ow rate (kg sÿ1) rate of heat rejection in the condenser (kW) rate of heat leak from the hot refrigerant (kW) rate of heat absorbed in the evaporator (kW) rate of heat leak from the ambient to cold refrigerant (kW) rate of heat leak from the compressor shell to ambient (kW) internal entropy generation, primarily due to non-isentropic compression and expansion (kW Kÿ1) refrigerant temperature in the condenser (K) condenser coolant inlet temperature (K) refrigerant temperature in the evaporator (K) evaporator coolant inlet temperature (K) temperature of the main cycle refrigerant after leaving the subcooler (K) subcooler saturation temperature (K) electrical power input to the compressor (kW) power required for isentropic compression (kW) eciency heat exchanger eectiveness
Subscripts 1, 2, 3, . . . comp cond evap gen ref total W
state points compressor condenser evaporator generation refrigerant total heat loss from compressor shell
AMTS C_ COP fh frejection hj k m_ Q_ cond Q_ loss cond Q_ evap Q_ loss evap Q_ loss W S_igen
Superscripts in inlet loss heat loss
J-u-R. Khan, S.M. Zubair / Energy Conversion & Management 41 (2000) 1201±1222
main sub out
1203
main cycle subcooler cycle outlet
1. Introduction The performance of a simple vapor-compression refrigeration system can be signi®cantly improved by further cooling the liquid refrigerant leaving the condenser coil. This subcooling of the liquid refrigerant can be accomplished by adding a mechanical-subcooling loop in a conventional vapor compression cycle. The subcooling system can either be a dedicated mechanical-subcooling system or an integrated mechanical-subcooling system. In a dedicated mechanical-subcooling system, there are two condensers, one for each of the main cycle and the subcooler cycle, whereas for an integrated mechanical-subcooling system, there is only one condenser serving both the main cycle and the subcooler cycle. Thornton et al. [1] and Couvillion et al. [2] have investigated a dedicated mechanical-subcooling system, whereas Bahel and Zubair [3], Zubair [4] and Zubair et al. [5] have investigated an integrated mechanicalsubcooling system from the thermodynamic standpoint. In either of these studies, they found that the system performance improved when operating in situations where the dierence between the condensing and evaporating temperatures is large. The development of refrigeration system models that simulates the actual working of the mechanical-subcooling system, has been the goal of many researchers (Thornton et al. [1]; Couvillion et al. [2]; Bahel and Zubair [3]; Zubair [4]; and Zubair et al. [5]). The most simple model is an ideal temperature-dependent model, which represents an ideal mechanicalsubcooling system without any irreversible losses. Thornton et al. [1] have developed an ideal temperature-dependent thermodynamic model for predicting the performance of mechanicalsubcooling system with irreversibilities only in the subcooler. A temperature-dependent model with irreversibilities in all the heat exchangers of the system can be considered as an improved temperature-dependent model. However, the temperature-dependent models do not take into account the actual properties of the refrigerants that are used in the systems. The propertydependent thermodynamic models takes into account the refrigerant properties and are developed by considering the mass and energy balance across each component of the system. It should be noted that the ®nite-time thermodynamic model, accounts for the irreversibilities existing due to the ®nite temperature dierence in the heat exchangers as well as the losses due to non-isentropic compression and expansion in the compressors and expansion valves of the system, respectively. Zubair [4] and Zubair et al. [5] have investigated the second-law analysis of an integrated mechanical-subcooling system using an ideal refrigerant cycle model. They showed that the irreversible losses in the expansion device (the major source of irreversibility) can be signi®cantly reduced by operating the system at the optimum subcooling conditions. This optimum condition was found to occur at a subcooler saturation temperature about halfway between the condensation and evaporation temperatures. The objective of this paper is to investigate temperature-dependent and property-dependent thermodynamic models of integrated mechanical-subcooling systems.
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2. Cycle description The major components of an integrated mechanical subcooling vapor-compression refrigeration system includes two reciprocating compressors, two expansion valves, condenser, evaporator, receiver and a subcooler. The system consists of two simple cycles coupled to each other via a subcooler as shown in Fig. 1, while its pressure enthalpy diagram is shown in Fig. 2. The bigger cycle is known as the main cycle and the smaller cycle is known as the subcooler cycle. The two cycles have a common condenser, and the components of the two cycles are connected in a closed loop through a piping system that has heat transfer with the surroundings. The ®gure shows that the main-cycle refrigerant leaves the main-cycle evaporator at state 1 as a low pressure, low temperature, saturated vapor and enters the main cycle compressor at state 2. The refrigerant, from state 1 to 2 takes heat from the surroundings in
Fig. 1. Schematic of an integrated mechanical-subcooling refrigeration system.
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1205
Fig. 2. Pressure±enthalpy diagram of an integrated mechanical-subcooling refrigeration system.
the suction line. At state 3, it leaves the compressor as a high temperature, high pressure, superheated vapor. The refrigerant, from state 3 to 4 rejects heat to the surroundings in the discharge line. At state 4, it mixes with the subcooler cycle refrigerant coming from the subcooler cycle compressor and attains state 13, and the mixture enters the condenser. The mixture after leaving the condenser is collected in the receiver. Some of this liquid refrigerant mixture is extracted from the receiver and is expanded in the expansion valve of the subcooler cycle and is then passed through the subcooler. The remaining liquid refrigerant in the receiver enters the subcooler, where it is cooled below the saturated liquid state at a constant pressure to state 6 by the subcooler cycle refrigerant. It enters the main cycle expansion valve and at state 7 it leaves the expansion valve as a low quality vapor and enters the evaporator. In the evaporator, it is evaporated at a constant pressure to the saturated vapor state. The subcooler-cycle refrigerant after cooling the main-cycle refrigerant in the subcooler, leaves as a low-pressure, low-temperature, saturated vapor at state 9 and enters the subcooler cycle-compressor at state 10. The refrigerant from state 9 to 10 takes heat from the surroundings. At state 11, it leaves the compressor as a superheated vapor where it is mixed with the main cycle refrigerant coming from the main-cycle compressor, and attains state 13, as shown in Fig. 1. 3. Analysis of the cycle Referring to the main cycle loop in Fig. 1, from the ®rst law of thermodynamics and the fact that the change in internal energy is zero for a cyclic process, we can write loss, main loss, main loss, sub loss, sub main sub _ _ _ _ _ _ _ W Qcond Qcond QW QW Qcond W 1 main loss, main loss, sub ÿ Q_ : Q_ Q_ evap
evap
evap
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Since entropy is a state function, it's net change over the cycle is zero. Therefore, the contributions to the entropy generation are from heat transfer (at the condenser, evaporator and subcooler) and the entropy generation due to non-isentropic compression and expansion in the compressors and the expansion valves, respectively. In addition to various irreversible losses, there is also a contribution to the entropy generation due to mixing of the refrigerants after exiting their respective compressors, as shown in Fig. 1. main loss, main out, sub _ _ Qevap Qevap ÿ ÿ main main Q_ cond T T13 total _ _ p ref ln _ p ref ln ÿ mc ÿ ÿ mc Sigen Tcond Tcond T3 T main evap sub loss, sub Q_ evap Q_ evap ÿ sub T13 _ p ref ln ÿ mc ÿ : 2 T sub T11 evap The heat transfer to and from the cycle occur by convection to ¯owing ¯uid streams with ®nite mass-¯ow rates and speci®c heats. Therefore, the rate of heat transfer to the cycle at a low temperature in the main-cycle evaporator can be written as ÿ main main main main Q_ evap eC_ evap T in, 3 ÿ T evap evap , while the rate of heat transfer between the refrigeration cycle and the sink in the main cycle condenser is ÿ ÿ 4 Q_ cond eC_ cond Tcond ÿ T in cond : De®ning the COPtotal as refrigerating eect over the net work input, we have COPtotal
main Q_ evap
_ main W _ sub W
:
5
_ sub from Eq. (1) into the above equation and _ main and W By substituting the expression of W expressing refrigeration temperatures in terms of the more readily available coolant temperatures, one obtains an analytical formula for the COP as a function of cooling capacity, coolant temperatures, heat exchanger characteristics, heat leak terms, internal cycle losses and the exit temperature of the main-cycle refrigerant from the subcooler, as loss, main loss, main loss, sub loss, sub _ _ _ _ QW QW Qcond Qcond 1 X1 ÿ1 main COPtotal X2 Q_ evap
ÿ
loss, main loss, sub Q_ evap Q_ evap main Q_ evap
where in the above equation we have,
6
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0 " # _ main Q_ loss, main out, sub ÿ ÿ Q ÿ T T evap evap total i3 main B _ ! mc _ p ref ln X1 eC_ cond T in cond BSigen main B Tcond T3 Q_ evap @ in T evap ÿ main eC_ evap
sub loss, sub Q_ evap Q_ evap
1
sub ÿ T13 C _ p ref ln mc C T11 C A
T sub evap
1207
7
8 0 # " main loss, main > _ _ > ÿ Q Q T out, sub Ti3 > main ÿ evap evap main < total B ! mc _ p ref ln X2 Q_ evap eC_ cond ÿBS_igen main B Tcond T3 Q_ evap > @ in > T evap ÿ main > : eC_ evap
sub loss, sub Q_ evap Q_ evap
T sub evap
19 > > ÿ sub T13 C > = _ p ref ln mc C T11 C A> > > ;
8
Eq. (6) gives the COPtotal of the system for one set of input data. The required input data can be measured experimentally from an actual operating system or calculated from the thermodynamic model, which is discussed in the latter part of the paper. Table 1 shows the required input data obtained from the thermodynamic model. The value of COPtotal obtained from Eq. (6) using the input data given in Table 1 was found to be 4.13 which is exactly equal to the value obtained from the thermodynamic model. Therefore, the analytical expression given in Eq. (6) and the thermodynamic model are correct and hence they can be used for design and performance evaluation purpose. It should be noted that the heat leak terms are neglected in the above comparison because they do not contribute much towards the overall system performance as reported by Gordon et al. [6]. Fig. 3 shows the characteristic performance curve (1.0/COP vs. 1:0=Q_ evap of the system. The plots are drawn using the input data given in Table 1. The design point is also shown in the ®gure, where for Fig. 3(a) the design point is (COP = 2.763). Note that the design point is located away from the minimum point because of the losses mainly due to non-isentropic compression in the compressors of the system. However, the design point is at the minimum point (COP = 4.1297) when the eciency of the compressors is assumed to be 100%, as main and a shown in Fig. 3(b). These plots are hypothetical curves drawn for a constant T in, evap main _ range of main cycle evaporator capacities Qevap : The shapes of the curves are similar to the one obtained for a simple cycle [6]. It shows that the COP increases with evaporator capacity up to the minimum point, due to irreversibilities such as ¯uid friction (some times referred to as non-isentropic losses) in the compressors and expansion valves. While at evaporator capacities greater than the minimum point, COP decreases signi®cantly because of the losses
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Table 1 Input data for Eq. (6), obtained from the thermodynamic model System operating parameters
For Fig. 3(a)
For Fig. 3(b)
Cooling capacity of the main cycle evaporator, Q_ main evap main Heat leak into the suction line of the main cycle, Q_ loss, evap main Heat leak from the discharge line of the main cycle, Q_ loss, cond loss, main Heat leak from the compressor shell of the main compressor, Q_ W _ main Eectiveness and capacitance rate product for the main evaporator, eC evap main _ Eectiveness-capacitance rate product for the main condenser, eCcond total Internal entropy generation due to non isentropic compression and expansion, S_igen in, main Coolant temperature at the inlet to the main evaporator, T evap Coolant temperature at the inlet to the condenser, T in cond Isentropic eciency of the main cycle compressor, Z main comp Isentropic eciency of the subcooler cycle compressor, Z sub comp Temperature of the main cycle refrigerant after exiting the subcooler, T out, sub Refrigerant temperature at the main condenser, Tcond Subcooler saturation temperature, T sub evap _ p main Mass ¯ow rate-speci®c heat product of the main cycle re®gerant, mc ref _ p sub Mass ¯ow rate-speci®c heat product of the subcoler cycle re®gerant, mc ref Cooling capacity of the subcooler, Q_ sub evap loss, sub Heat leak into the suction line of the subcooler cycle, Q_ evap sub Heat leak from the discharge line of the subcooler cycle, Q_ loss, cond sub Heat leak from the compressor shell of the subcooler compressor, Q_ loss, W Temperature at state 13, T13 Temperature at state 3, T3 Temperature at state 11, T11
30.0 kW 0.0 W 0.0 W 0.0 W 5.0 kW Kÿ1 5.5 kW Kÿ1 13.958 W Kÿ1 0.08C 40.08C 65% 70% 34.6538C 46.6538C 30.5078C 309.46 W Kÿ1 182.33 W Kÿ1 3713 W 0.0 W 0.0 W 0.0 W 54.7178C 57.96998C 49.1968C
30.0 kW 0.0 W 0.0 W 0.0 W 5.0 kW Kÿ1 5.5 kW Kÿ1 3.517 W Kÿ1 0.08C 40.08C 100% 100% 34.0398C 46.0398C 30.0298C 306.85 W Kÿ1 195.85 W Kÿ1 3682.20 W 0.0 W 0.0 W 0.0 W 52.10388C 54.35298C 48.588C
due to the ®nite rate of heat transfer in the heat exchangers of the system. It should be noted that the performance of the system in the light of various losses of the system is similar to the performance of a simple cycle discussed by Gordon and Choon [7].
4. Performance evaluation of the system Fig. 1 shows that the main-cycle refrigerant and the subcooler-cycle refrigerant mixed at state 13, after leaving their respective compressors. The mixture then rejects heat in the condenser at a constant temperature Tcond : It can be deduced that each refrigerant cycle uses a fraction of the total condenser area for rejecting it's heat. To investigate the fraction of area utilized by each refrigerant, a factor de®ned as the subcooler heat-rejection factor is de®ned as ÿ sub eC_ 9 frejection ÿ main cond ÿ sub eC_ eC_ cond
cond
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Fig. 3. Performance curves of an integrated mechanical-subcooling vapor-compression refrigeration system, for an input data given in Table 1.
_ main eC _ sub is the total thermal inventory of the condenser eC _ cond : The individual where eC cond cond areas can be expressed in terms of the total area as follows ÿ ÿ sub eC_ cond frejection eC_ cond ,
10
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ÿ main ÿ ÿ eC_ cond 1:0 ÿ frejection eC_ cond :
11
The performance of a mechanical-subcooling system is controlled mostly by the subcooler temperature T sub evap , as reported by Thornton et al. [1]. Mechanical-subcooling allows the main cycle refrigerant to enter the evaporator with a low quality (state 7), see Fig. 2, compared with a typical simple vapor-compression cycle (state 5'). The low quality of the refrigerant at the evaporator inlet corresponds to an increase in the refrigeration capacity per unit mass of refrigerant circulated in the main cycle. However, it should be noted that the increase in refrigeration capacity is not without cost since there are additional components in the system. Neglecting losses to the environment, an energy balance on the subcooler indicates that the amount of subcooling provided to the main cycle must equal the heat addition to the subcooling cycle evaporator. The heat addition to the subcooler must be rejected in the subcooling cycle condenser at the cost of the work consumed in the subcooling cycle compressor. It should be emphasized that there is an ``optimum'' temperature for the subcooler at which the COPtotal of the cycle is maximized. This ``optimum'' temperature can be investigated by developing a temperature-dependent model of the system. Similar to Thornton et al. [1], the following assumptions were made in the development of the mechanicalsubcooling model: . Both the main and subcooling cycle condensers reject heat at the sink temperature, T in cond : , an intermediate temperature . The subcooling cycle heat addition occurs at T sub evap sub in T in evap RT evap RT cond ). . There is no thermal energy loss to the environment. . The main cycle compressor work is not in¯uenced by the amount of subcooling provided to the main cycle. . The exit states of the main cycle condenser and evaporator are unaected by the amount of subcooling performed. . Isentropic expansion and compression are assumed for both the main and subcooling cycles. If no subcooling is provided to the main cycle, the COP of the main cycle can be written as, main
COP
main Q_ evap
_ main W
T main evap Tcond ÿ T main evap
main T in, ÿ DTm evap in, main DTm T in ÿ T evap cond
_ main and Q_ cond de®ned earlier, we can write using the de®nitions of W # main " Q_ evap T in in cond Tcond T cond ÿ main : main T in, ÿ DTm evap eC_
12
13
cond
The subcooling cycle operates between the temperatures T in cond and Tcond : Therefore, the COP of the subcooling cycle may be expressed as sub
COP
sub Q_ evap sub
_ W
T sub evap
T sub cond
ÿ
T sub evap
ÿ
Tcond ÿ DTs : ÿ Tcond DTs
T in cond
14
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1211
The overall cycle COP may be expressed as the total refrigeration capacity divided by the total power. The capacity of the overall cycle is simply the capacity of the main cycle without subcooling Q_ main evap ), plus the increment in capacity of the main cycle due to the subcooling performed Q_ sub evap ). The total power is simply the sum of the compressor work for both the subcooling and the main cycles. With these de®nitions, the overall cycle COP may be expressed as COPtotal
main sub Q_ evap Q_ evap
15
_ sub _ main W W
After manipulations of the above equations we can express the overall COPtotal of the system in terms of the subcooler cycle temperature T sub evap and other variables as 1 X Tcond ÿ T sub evap , 16 COPtotal in main T cond ÿT in, DTm X Tcond ÿT sub evap DTTs evap in, main in T ÿ DTTs ÿ DT T m
evap
cond
where in the above equation, we have subcooler parameter given by ÿ X
main
esub ref , main, nosub Q_ evap
_ p mc
17
and various temperatures as " # 1 1 main _ DTm Qevap ÿ main ÿ , ÿ 1:0 ÿ frejection eC_ cond eC_ evap " DTs Q
Tcond
sub
1
main ÿ _ p ref esub mc
# 1 ÿ , frejection eC_ cond
# " _ main in Q T evap cond , T in ÿ cond ÿ in, main 1:0 ÿ frejection eC_ cond T evap ÿ DTm
DTTs DTs T in cond ÿ Tcond :
18
19
20
21
It should be noted that Eq. (16) considers the irreversible losses in all the heat exchangers. _ main , eC _ main to be very large, it gives losses in the _ sub , eC However, when we consider eC cond cond evap subcooler only. Substituting the above values in Eqs. (18)±(21), we get DTm 0:0,
22
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J-u-R. Khan, S.M. Zubair / Energy Conversion & Management 41 (2000) 1201±1222 sub Q_ evap sub DTs ÿ main sub Tcond ÿ T evap , _ p ref eevap mc
23
in T main cond T cond ,
24
DTTs DTs :
25
and
Substituting the above values in Eq. (16) we get sub 1 X T in ÿ T cond evap COPtotal in 2 in, main in T cond ÿT evap T cond ÿT sub evap X in, main sub T T
26
evap
evap
which is exactly the same equation as that discussed by Thornton et al. [1]. To show the predicted improvement in performance of a mechanical-subcooling refrigeration system with respect to the single stage cycle, the curves are plotted in a normalized form de®ned as COPN
COPtotal , COPmain
27
similarly, a reduced subcooler saturation temperature y is de®ned as y
main T sub evap ÿ T evap
Tcond ÿ T main evap
:
28
Fig. 4(a) shows the plot of normalized coecient of performance (COPN) and the reduced subcooler saturation temperature y). The plots are obtained by using Eq. (27) for the main _ main _ main 0:08C, T in following set of input data T in, evap cond 40:08C, Qevap 30:0 kW, eCevap _ cond 5500:0 W Kÿ1. In these plots, y varies between T main and Tcond : 5000:0 W Kÿ1, and eC evap It can be seen from the plots that maximum COPN occurs at approximately y 0:5, and it almost remains constant for all the values of the subcooler cycle heat exchanger parameter X. main For y lower than 0.5, the dierence between T sub evap and T evap is less and hence maximum amount of subcooling is being performed; in this case, the thermal lift of the subcooling cycle is close to that of the main cycle. Therefore, the subcooling cycle compressor consumes more power, and the advantage of using an integrated mechanical-subcooling is destroyed and the COPN of the system reduces. For y greater than 0.5, the dierence between T sub evap and Tcond is less and hence a relatively low amount of subcooling is being performed and the overall cycle behaves very close to a simple cycle, thus the value of COPN is lower than the maximum possible value. Between these two extremes, there is an optimum T sub evap at which COPN is maximum, as shown in the ®gure. Fig. 4(b) shows the variation of the subcooler heat rejection factor frejection with the reduced subcooler temperature y: It can be seen from the plots that at high values of y, the amount of
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Fig. 4. Variations of COPN (a) and frejection (b) with y for a completely irreversible temperature-dependent integrated mechanical-subcooling system.
subcooling performed is lower. Hence, the amount of heat to be rejected by the subcooler cycle refrigerant is also low. Therefore, it uses a small fraction of the condenser area, however, as the value of y reduces, the amount of subcooling performed is high; thereby requiring more subcooler compressor work. Hence, the amount of heat to be rejected by the subcooler cycle refrigerant is relatively high. Thus, it uses a large fraction of the condenser area. Also, for a given value of y, the heat rejection factor increases with X since subcooling increases with the subcooler heat exchanger parameter X.
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Fig. 5(a) shows the eect of condenser coolant inlet temperature T in cond on the subcooler heat rejection factor frejection for constant values of other system parameters. The ®gure shows that for high values of T in cond and particularly at low values of y, the heat rejection factor frejection increases, since the amount of subcooling increases; thereby requiring more subcooler compressor work. Thus, an increase in heat rejection factor for the subcooling cycle is expected in with T in cond : The ®gure also shows that at high values of y, for all the values of T cond , the amount of subcooling performed is low. Therefore, the amount of heat to be rejected by the
in, main Fig. 5. Variations of frejection with y for various T in (b). cond (a) and T evap
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subcooler cycle refrigerant is relatively low. Thus, it uses a small fraction of the condenser area. main on the subcooler Fig. 5(b) shows the eect of evaporator coolant inlet temperature T in, evap heat rejection factor frejection for constant values of other system parameters. Similar to Fig. 5(a), main and y, the heat rejection factor frejection this ®gure also shows that for low values of T in, evap increases, since the amount of subcooling increases; thereby requiring more subcooler compressor work and hence an increase in heat rejection factor for the subcooling cycle is expected. The ®gure also shows that at high values of y and almost for all the values of main , the amount of subcooling performed is lower and hence the amount of heat to be T in, evap rejected by the subcooler cycle refrigerant is relatively low. Therefore, it uses a small fraction of the condenser area, however, as described earlier that as the value of y reduces, the amount of subcooling performed is high and hence the amount of heat to be rejected by the subcooler cycle refrigerant is relatively high. Thus, it uses a large fraction of the condenser area.
5. A ®nite time thermodynamic model of the system As mentioned earlier, the heat transfer to and from the cycle occur by convection to ¯owing ¯uid streams having ®nite mass ¯ow rates and speci®c heats. Therefore, the rate of heat transfer to the cycle in the main cycle evaporator can be expressed in terms of temperature, mass ¯ow rates, and change in speci®c enthalpy of the refrigerant, given by ÿ main main main main _ main 29 ÿ T Q_ evap eC_ evap T in, evap evap m ref h1 ÿ h7 : The compressor operation is described in terms of an isentropic eciency Zmain comp , so that the power requirement for the main cycle compressor is given by _ main W main main s _ m_ ref h3 ÿ h2 main : W Zcomp
30
In the subcooler, the rate of heat transfer between the main cycle refrigerant and the subcooler cycle refrigerant is given by ÿ main ÿ sub _ p Tcond ÿ T out, sub m_ main Q_ evap mc 31 ref h5 ÿ h6 : loss, main of heat, leaks into the suction line of the main cycle Assuming that an amount of Q_ evap compressor, which can be expressed as loss, main m_ main Q_ evap ref h2 ÿ h1 :
32
loss, main loss, main Similarly, assuming that an amount of Q_ cond Q_ W of heat leaks from the discharge line and compressor shell given by loss, main loss, main Q_ cond Q_ W m_ main ref h3 ÿ h4 :
33
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For the subcooler cycle loop, the rate of heat transfer between the subcooler cycle refrigerant and the main cycle refrigerant in the subcooler can be written in terms of temperatures, the mass ¯ow rates, and speci®c enthalpy of the refrigerant, expressed as, ÿ main sub _ sub h9 ÿ h8 : _ p ref esub Tcond ÿ T sub Q_ evap mc 34 evap m The work input to the subcooler cycle compressor is given by _ sub W sub sub _ W m_ ref h11 ÿ h10 subs : Zcomp
35
loss, sub of heat leaks into the suction line of the subcooler cycle, Assuming that an amount of Q_ evap which can be expressed as sub Q_ loss m_ sub ref h10 ÿ h9 :
36
loss, sub loss, sub of heat leaks from the discharge line Similarly, assuming that an amount of Q_ cond Q_ W and the compressor shell. This is given by loss, sub loss, sub m_ sub Q_ cond Q_ W ref h11 ÿ h12 :
37
Also, the rate of heat transfer between the refrigeration cycle and sink in the main cycle condenser is expressed as ÿ ÿ in, main _ _ Qcond eC cond Tcond ÿ T cond m_ main m_ sub h13 ÿ h5 : 38 The mixing of the main cycle refrigerant and subcooler cycle refrigerant after leaving their respective compressors is given by m_ main h4 ÿ h13 m_ sub h13 ÿ h12 : From the above equation, we get ÿ m_ main h4 m_ sub h12 : h13 ÿ main m_ m_ sub
39
40
The above equations have been solved numerically by using the thermodynamic property data for several dierent refrigerants. The ¯ow chart representing the method of solving these equations is shown in Fig. 6, wherein the terms SATPRP and TRIAL represent the subroutines for calculating the refrigerant saturated and vapor properties, respectively. These subroutines need any two independent intensive properties of the refrigerant to ®nd other properties at a given state. For this purpose, a computer program, originally developed by Kartsounes and Erth [8] and modi®ed by Fisher and Rice [9] and Khan and Zubair [10], have been used. The program gives the COPtotal and all other parameters of the system for the main _ main , eC _ main , T in , T in, main , AMTS and esub : following set of input data: Q_ evap , eC cond evap cond evap evap Fig. 7(a) shows the plot of COPN vs. the reduced subcooler saturation temperature y and
J-u-R. Khan, S.M. Zubair / Energy Conversion & Management 41 (2000) 1201±1222
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Fig. 6. Flow chart for a thermodynamic model of an integrated mechanical-subcooling system.
the heat exchanger thermal inventory parameter k, de®ned as ÿ ÿ k eC_ evap eC_ cond :
41
The results shown in the ®gure are obtained from the above described thermodynamic model main main sub sub _ p main for R-134a when: T in, 0:08C, T in ref e evap cond 40:08C, Zcomp 0:65, Zcomp 0:7, mc main ÿ1 _ 0:1 kW K and Qevap 30:0 kW. Fig. 7(b) shows the corresponding variation of the system operating temperatures for k 12:5 kW Kÿ1. The ®gure shows that as the value of heat exchanger thermal inventory parameter k increases, the value of COPN also increases, because as k increases the irreversible losses due to heat transfer decreases in the heat exchangers of the system, thus increasing the system performance. We note that the property-dependent model
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main Fig. 7. Variations of COPN (a) and operating temperatures (b) with y for the following input data: Q_ evap 30:0 ÿ1 main sub main sub in, main in _ p ref e 0:1 kW K , Zcomp 0:65, Zcomp 0:70, T evap 273:0 K, T cond 313:0 K. kW, mc
predicts the same relationship between COPN and the reduced subcooler saturation temperature y, as that discussed earlier by the temperature-dependent model (see Fig. 4(a)). It should be noted that the heat leak terms are neglected in the above analysis because they do not contribute much towards the overall system performance (see Ref. [6]). Referring to Fig. 7(b), we note that the condenser temperature more-or-less remains constant, however, one would expect a variation in Tcond with the reduced subcooler saturation temperature y: The calculations indicate a somewhat constant temperature varying between 319.42 and 319.33 K,
J-u-R. Khan, S.M. Zubair / Energy Conversion & Management 41 (2000) 1201±1222
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Fig. 8. Variation of COPtotal with y for an ideal temperature-dependent and property-dependent mechanicalsubcooling system: irreversible losses only in the subcooler.
which is mainly due to the fact that the variation in Q_ cond is relatively small and moreover the _ cond 6:5 kW Kÿ1) in the calculation of Tcond is high. value of eC Fig. 8 shows the plot of COPtotal vs. the reduced subcooler saturation temperature y and the subcooler parameter X of the property-dependent and temperature-dependent ideal mechanicalsubcooling vapor-compression refrigeration system with irreversible losses only in the subcooler and using R-134a as the refrigerant. The plots are drawn for the following set of input data: main _ main 0:08C, T in T in, evap cond 40:08C, and Qevap 30:0 kW. The ®gure shows that the shapes of the curves for the two models are similar and the dierence in the COPtotal values is mainly due to non-isentropic expansion in the expansion valves of the property-dependent model. The maximum value of COPtotal shown in the ®gure is for X = 0.0066, and is equal to 7.29. It should be noted that this value is much larger than the Carnot cycle limit of 6.825, which is main ÿ0:08C and T in the ideal COP for T in, evap cond 40:08C. This can be explained by the fact that the temperature dependent model will not show the eect of mass ¯ow rates on the system performance. Therefore, the COPtotal is somewhere between the Carnot limit of the main and subcooler cycle. It is thus important to note that the maximum possible value of COPtotal is the Carnot cycle COP of the subcooler cycle, which is given by COPtotal
T sub evap
sub T in cond ÿ T evap
:
42
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In the absence of the subcooler cycle, the maximum possible COPtotal is the Carnot cycle COP of the main cycle only, which is given by COPtotal
main T in, evap
in, main T in cond ÿ T evap
:
43
This is also established from Eq. (16), which reduces to Eq. (42) when irreversible losses in all the heat exchangers including that of the subcooler are removed.
main Fig. 9. Variations of COPtotal (a) and operating temperatures (b) with fh for the following input data: Q_ evap 30:0 sub in, main kW, Zmain 273:0 K, T in comp 0:65, Zcomp 0:70, T evap cond 313:0 K.
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6. Optimum distribution of the heat exchanger area A set of sample plots, shown in Fig. 9(a), is drawn between COPtotal and the dimensionless heat exchanger parameter fh ), de®ned as ÿ eC_ cond ÿ fh ÿ 44 eC_ cond eC_ evap using the output from the model; for the following set of input data and using R-134a as the main ÿ1 main 273:0 K, T in refrigerant: Q_ evap 30:0 kW, T in, evap cond 313:0 K, k 10:0 kW K , sub _ p main Zmain ref comp 0:65, Zcomp 0:7: The plots are drawn for dierent values of the parameter mc sub and for each value of the dimensionless heat exchanger parameter fh ); the global e optimum value of the subcooler saturation temperature T sub evap is calculated as shown in Fig. 7(a). For better explanation of the plots, the variation of system temperatures are shown sub _ p main 0:2 kW Kÿ1. The ®gure shows that COPtotal is maximum at in Fig. 9(b) for mc ref e about fh 0:55, for all the curves. The dierence between T in cond and Tcond is high at low values of the parameter fh ; therefore, the heat transfer in the condenser takes place at high temperature dierence resulting in high irreversible losses due to ®nite rate of heat transfer. main and These irreversible losses decrease as fh increases. However, the dierence between T in, evap main T evap increases as fh increases and the heat transfer in the evaporator takes place at a high temperature dierence, resulting in an increase in irreversible losses due to the ®nite rate of heat transfer. At fh 0:55, the temperature dierence in the evaporator and condenser is minimum, hence COPtotal is maximum at this point. It should also be noted that the heat transfer in the subcooler takes place through the temperature dierence between Tcond and T sub evap , and the ®gure shows that this temperature dierence is approximately constant for all values of fh :
7. Conclusions Thermodynamic models of an integrated mechanical-subcooling vapor-compression refrigeration system have been developed to study the design and performance evaluation of the system. The models developed include both the temperature and property-dependent models. The performance evaluation of the systems has been investigated by considering the temperature model. It is demonstrated that the optimum subcooler saturation temperature at which the COP of the system is maximum, is not a function of the subcooler heat exchanger parameter X. This optimum temperature is found to be about halfway between the condensation and evaporation temperatures (i.e., y 0:50 for almost all values of the parameter X. The property-dependent model showed the same trends as that predicted by the temperature model, that is, the existence of an optimum subcooling saturation temperature. The optimum distribution of total heat exchanger areas between the condenser and evaporator has also been investigated. It is found that COP of the system is maximum when a large fraction of the total area is allocated to the condenser than to the evaporator of the cycle. It is also demonstrated
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that this optimum distribution occurs when the irreversibilities due to the ®nite rate of heat transfer in the heat exchangers of the system are minimum. Acknowledgements The authors acknowledge the support provided by King Fahd University of Petroleum and Minerals for this research project. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Thornton JW, Klein SA, Mitchell JW. International Journal of Refrigeration 1994;17(8):508±15. Couvillion RJ, Larson MW, Somerville MH. ASHRAE Transactions 1988;94(2):641±59. Bahel V, Zubair SM. Heating Piping Air Conditioning 1988;60(2):105±7. Zubair SM. Energy 1994;19(6):707±15. Zubair SM, Yaqub M, Khan SH. International Journal of Refrigeration 1996;19(8):506±16. Gordon JM, Chua HT, NG, KC. International Journal of Heat and Mass Transfer 1996;39(11):2195±204. Gordon JM, Choon NG. International Journal of Heat and Mass Transfer 1995;38(5):807±18. Kartsounes GT, Erth RA. ASHRAE Transactions 1971;77(2):88±103. Fisher SK, Rice CK. The Oak Ridge heat pump models: Part 1. A steady-state computer design model for airto-air heat Pumps, ORNL/CON-80/RI, Oak Ridge National Laboratory, Oak Ridge: TN, 1983. [10] Khan SH, Zubair SM. Energy 1993;18(7):717±26.