Entransy and entropy analyses of heat pump systems

Article Engeerg Thermophysics December 2013 Vol.58 No.36: 46964702 doi: 10.1007/s11434-013-6096-4 Entransy and entropy analyses of heat pump systems CHENG XueTao & LIANG XGang * Key Laboratory for Thermal Science and Power Engeerg of Mistry of Education, Department of Engeerg Mechanics, Tsghua University, Beijg 100084, Cha Received March 27, 2013; accepted May 6, 2013 In this paper, heat pump systems are analyzed with entransy crease and entropy generation. The extremum entransy crease prciple is developed. hen the equivalent temperatures of the high and low temperature heat sources are fixed, the theoretical analyses and numerical results both show that the maximum COP leads to the maximum entransy crease rate for fixed put power, while it leads to the mimum entransy crease rate for fixed heat flow absorbed from the low temperature heat source. The mimum entropy generation prciple shows that the mimum entropy generation rate always leads to the maximum COP for fixed put power or fixed heat flow absorbed from the low temperature heat source when the equivalent thermodynamic forces of the high and low temperature heat sources are given. Further discussions show that only the entransy crease rate always creases with creasg heat flow rate to the high temperature heat source for the discussed cases. entransy loss, entransy crease,entropy generation, heat pump, optimization Citation: Cheng X T, Liang X G. Entransy and entropy analyses of heat pump systems. Ch Sci Bull, 2013, 58: 46964702, doi: 10.1007/s11434-013-6096-4 *Correspondg author (email: liangxg@tsghua.edu.cn) Heat pump systems are common dustrial equipments and have many applications [1 15]. For stance, the heatg and air conditiong system, the heat pump is applied to drivg the heat from the environment to the room [15]. The optimization design of heat pump systems has received more and more attention because it can improve the system performance and crease the energy utilization efficiency [6 12]. There are different optimization objectives for heat pump systems, such as the thermo-economic performance [6] and the thermodynamic performance optimization [4,5,7 12]. For stance, uoil et al. [6] analyzed the thermo-economic performance of heat pump systems. Chen et al. [4] optimized the piston speed ratios to get the maximum COP for the irreversible Carnot refrigerator and heat pump usg the fite time thermodynamics. In this paper, we focus on the thermodynamic performance optimization. As the thermodynamic processes heat pump systems are maly composed of heat transfer processes and thermodynamic cycles, the analyses of the heat transfer processes and thermodynamic cycles are very important for the optimization designs. In the past decades, some optimization theories have already been developed and applied to heat transfer and thermodynamic cycles [15 18]. Practical heat transfer processes are irreversible from the thermodynamic viewpot, and entropy generation will be produced. Many researchers applied the entropy generation mimization method to analyzg and optimizg heat transfer processes [18 20]. However, the entropy generation paradox tells us that the effectiveness of heat exchangers ε does not always decrease when the entropy generation number creases [18]. hen Bejan [18] analyzed a balanced counter flow heat exchanger, he explaed the entropy generation paradox as followg: when ε 0, the heat exchanger would disappear as an engeerg component, and such case does not exist. The entropy generation paradox could not be removed with this explanation because the effectiveness can be a value the range of [0, 0.5] which the entropy generation number and the effectiveness still crease at the same time. In addition, Shah and Skiepko [21] also noticed that ε may be the maximum, an termediate value or the mimum when the entropy generation reaches The Author(s) 2013. This article is published with open access at Sprgerlk.com csb.scicha.com www.sprger.com/scp

Cheng X T, et al. Ch Sci Bull December (2013) Vol.58 No.36 4697 the maximum value. Cheng et al. [22,23] analyzed the entropy generation of the heat exchangers and heat exchanger networks with two streams and found that the entropy generation does not decrease monotonically with the crease of the heat transfer rate and effectiveness. Guo et al. [16] developed the concept of entransy, which describes the heat transfer ability. Entransy dissipation always exists durg practical heat transfer processes [16,24]. Guo et al. [16] derived the extremum entransy dissipation prciple and the mimum thermal resistance prciple, which have been applied to the optimizations of conductive heat transfer [16,25 32], convective heat transfer [33], radiative heat transfer [34,35], heat exchangers and heat exchanger networks [17,22,23,36,37]. In the analyses of heat exchangers with the entransy theory, there is no paradox like the entropy generation paradox [17,22,36]. For thermodynamic cycles, more entropy generation means that more ability to do work is lost [38,39]. Hence, the entropy generation mimization method has been widely applied to the analyses and optimizations of thermodynamic processes because it can decrease the loss of the ability to do work [38 42]. For stance, Myat et al. [42] showed that the entropy generation mimization leads to the largest COP when they analyzed an absorption chiller. However, there are also some different viewpots for the applicability of the entropy generation mimization to the optimization of thermodynamic cycles [43,44]. For stance, Kle and Redl [43] analyzed the refrigeration system and found that the entropy generation mimization does not always lead to the best system performance unless the refrigeration capacity is given. The entransy theory is also used to analyze thermodynamic cycles [15,44 50]. u [45] defed the conversion entransy by which the thermodynamic processes with work were analyzed. In the recent vestigations of thermodynamic cycles, Cheng et al. [15,44,47] defed a new concept, entransy loss rate, that is the difference between the entransy flow rate to the system and that of the system. It is shown that the maximum entransy loss rate leads to the maximum put work for the discussed systems [15,44, 47 50]. The above troduction shows that the applicability of the entropy generation mimization to the analyses and optimizations of heat transfer processes and thermodynamic cycles is limited. For the concept of entransy, it has been applied to the analyses and optimizations of heat transfer and thermodynamic cycles, but there are not many reports. For the heat pump system, Chen et al. [4] applied the concept of entropy to its optimization. However, Kle and Redl [43] found that the mimum entropy generation rate does not always lead to the best system performance. So, the applicability of the entropy generation mimization to the heat pump systems needs further discussion. On the other hand, there are few reports on the applicability of the entransy theory to the analyses and optimization designs of heat pump systems. Therefore, it is also necessary for us to discuss the applicability of the entransy theory to heat pump systems. 1 Extremum entransy crease prciple and mimum entropy generation prciple for heat pump system As shown Figure 1, the heat pump system is maly composed of four parts, which are the cooler, the compressor, the cold storage (the heat source with low temperature) and the expander, respectively. The workg fluid absorbs heat flow from the cold storage durg process 4 1. Then, it is compressed durg process 1 2 by the compressor. In the next process, the workg fluid releases heat flow durg process 2 3 the cooler (the heat source with high temperature). Fally, the workg fluid gets beck to the itial state when the expansion process 3 4 fishes the expander. The heat the cold storage is pumped to the cooler when a cycle fishes, and the put power is P. For the heat pump system, the energy conversation gives The system COP is P. (1) COP P 1 P. (2) Eq. (2) shows that larger leads to larger COP with fixed P, while smaller P leads to larger COP with fixed. For the heat pump system, assume that there are n low temperature heat sources and m high temperature heat sources. The temperature of the ith low temperature heat source is T, while that of the jth high temperature heat source is T -j. The thermodynamic processes of the workg fluid are shown Figure 2, which can be divided to two parts. One is the heat transfer processes between the workg fluid and the heat sources, while the other is the thermodynamic cycle. Durg heat transfer, the entransy theory gives [16] Figure 1 Sketch of a heat pump.

4698 Cheng X T, et al. Ch Sci Bull December (2013) Vol.58 No.36 n i 1, (9) m j 1. (10) n m loss dis1 dis2 1 i j1 -j -j -j The equivalent temperature of the low temperature heat sources can be defed as n i1 T T, (11) while that of the high temperature heat sources can be defed as m j 1 -j -j T T. (12) Figure 2 Thermodynamic process of the heat pump system. Then, eq. (8) can be changed to G T T. (13) loss G n T q T d A, (3) dis1 i1 f A dis2 f j 1 -j -j A G q T m d A T, (4) where G dis1 is the entransy dissipation rate durg the heat transfer process between the low temperature heat sources and the workg fluid, G dis2 is that durg the heat transfer process between the high temperature heat sources and the workg fluid, is the heat transfer rate between the ith low temperature heat source and the workg fluid, -j is that between the jth high temperature heat source and the workg fluid, T f is the temperature of the workg fluid, q and q are the heat fluxes absorbed and released by the workg fluid, and A and A are the correspondg heat transfer areas. For the thermodynamic cycle, we have [15,44] G T T P G fδ fδ, (5) where G is the heat entransy flow rate, δ is the heat flow absorbed by the workg fluid, δp is the power put, and G is the work entransy flow rate. There is G q T da q T da. (6) So, we have A A f f A G q T da q T da. (7) f f A Considerg the defition of entransy loss rate and eqs. (3), (4) and (7), we obta G G G G T T. (8) For Figure 1, we have The entransy loss rate is negative for the heat pump systems because is smaller than and T is lower than T. Therefore, the system entransy does not decrease, but creases. The power put to the system, δp eq. (5), is negative, and the work entransy flow is also negative. This means that the work entransy flow gets to the system, which makes the system entransy crease. Therefore, the entransy crease rate can be defed as G G T T. (14) c loss Accordg to eq. (3), we have G P T T T T P T. (15) c hen T and T are given, the maximum G c leads to the maximum and the maximum COP (see eq.(2)) for fixed P, and the mimum G c leads to the mimum P and the maximum COP for fixed. This is the extremum entransy crease prciple of the heat pump system. On the other hand, the entropy balance equation gives [51] ds ds δs, (16) f where ds f is the entropy flow, ds is the entropy change, and δs g is the entropy generation. As the thermodynamic processes are steady, ds is zero. Therefore, the entropy generation is δs ds. (17) g Therefore, the entropy generation rate of the system is where system, and f m -j n -g S S f- S f- j1 i1 T- j T, (18) S f- is the entropy flow rate that gets of the g S f- is that gets to the system. e can de-

Cheng X T, et al. Ch Sci Bull December (2013) Vol.58 No.36 4699 fe the equivalent thermodynamic forces of the low and the high temperature heat sources as n - i n n - i H i 1 i 1 i1, (19) T T H. (20) m -j m m - 1 1 -j j j j1 T- j T- j Then, eq. (18) can be changed to S H H. (21) g Considerg eq. (1), we have S H P H H H P H.(22) g As H is smaller than H, the term the last bracket is negative. Therefore, when H and H are given, the mimum S g leads to the maximum with fixed P, while it leads to the mimum P with fixed. Considerg eq. (2), we can see that the maximum COP leads to the mimum S g with either fixed P or fixed. This is the mimum entropy generation prciple of the heat pump system. As above, we get different optimization prciples for heat pump systems. Eq. (15) shows that the maximum COP sometimes leads to the maximum entransy crease rate, and sometimes leads to the mimum entransy crease rate. On the other hand, the mimum entropy generation rate always leads to the maximum COP for fixed equivalent thermodynamic forces of heat sources. Therefore, when COP is the optimization objective of heat pump systems, the mimum entropy generation prciple is convenient, though extremum entransy crease prciple is also applicable. 2 Optimization examples and discussions 2.1 Numerical examples of the heat pump system with reversed Brayton cycle Let us discuss a heat pump system composed of the reversed Brayton cycle. For the workg fluid, its thermodynamic processes are shown Figure 3. The reversed Brayton cycle works between the low and high temperature heat sources with constant temperatures T and T, respectively. The temperatures of the workg fluid at the state pots are T 1, T 2, T 3 and T 4, respectively. The workg fluid absorbs heat flow from the low temperature stream under constant pressure, then its temperature creases to T 1. The next process is an isentropic process and the temperature of the workg fluid creases to T 2. Then, the workg fluid releases heat flow to the high temperature heat source under constant pressure, and its temperature decreases to T 3. Fally, the workg fluid is expanded and gets back to the itial state. Durg the whole thermodynamic processes, the put mechanical power is P. For the system Figure 3, it is assumed that U U U const, (23) where U is the thermal conductance of the heat exchanger between the cold storage and the workg fluid, and U is that of the heat exchanger between the cooler and the workg fluid. Then, the distribution of U is to be optimized to crease the COP. The heat transfer rates the heat exchangers are [15] 1 exp Cf T T4 U C f, (24) 1 exp Cf T2 T U C f, (25) where C f is the heat capacity flow rate of the workg fluid. The energy conservation gives f 1 4 C T T, (26) f 2 3 C T T. (27) In the Brayton cycle, there is [52] T2 T1 T3 T4. (28) hen T, T, and C f are fixed, the values of, T 1, T 2, T 3 and T 4 can be calculated with eqs. (23) (28) for every distribution of U. Then, the put power can be obtaed from eq. (1), and the correspondg COP can be calculated from eq. (2). On the other hand, when T, T, P and C f are fixed,,, T 1, T 2, T 3 and T 4 can also be calculated with eqs. (1), (23) (28) for every distribution of U. Then, the correspondg COP can also be obtaed with eq. (2). Accordg to eqs. (14) and (21), the entransy crease rate and entropy generation rate could also be calculated. Let us discuss some numerical examples below. Let U=10 /K, T =270 K, T =300 K, C f =2 /K, and =100. The variations of the COP, the entransy crease rate and the entropy generation rate with U can be seen Figure 4. The mimum entransy crease rate and the mimum entropy generation rate both lead to the maximum Figure 3 T-S diagram of the workg fluid reversed Brayton cycle.

4700 Cheng X T, et al. Ch Sci Bull December (2013) Vol.58 No.36 Figure 4 Variations of the COP, the entransy crease rate and the entropy generation rate with U when is fixed. COP of the system when T, T, C f and are fixed. On the other hand, if the fixed parameter is not, but the put power P, we assume that P =100, and the values of U, T, T and C f are the same as those of the first case. The variations of the COP, the entransy crease rate and the entropy generation rate with U are shown Figure 5. It can be seen that the maximum entransy crease rate and the mimum entropy generation rate both lead to the maximum COP of the system. Therefore, both the extremum entransy crease prciple and the mimum entropy generation prciple can be applied to optimizg the heat pump system. The thermodynamic forces of the high and low temperature heat sources are given when their temperatures are fixed. Accordg to eqs. (15) and (22), the preconditions of the prciples are both satisfied. This is the reason why the prciples are effective optimizg the system. 2.2 Discussions For the heat pump system with the reversed Brayton cycle discussed above, we can make a discussion which the heat flow rate released to the high temperature heat source is the optimization objective. hen the heat flow rate pumped from the low temperature heat source, the temperatures of the heat sources, T and T, are fixed, it can be seen that the entransy crease rate, the entropy generation rate and the heat flow rate to the high temperature heat source all decrease with decreasg put power from eqs. (1), (14) and (21). It means that both the maximum (not the mimum) entropy generation rate and the maximum entransy crease rate lead to the maximum heat flow rate to the high temperature heat source. The variations of, P, G c and S g with U are shown Figure 6. It can be seen that the variation tendencies of, G c and S g are all same as that of P. The results verify the analyses above. Furthermore, when the put power is fixed, eqs. (1), (15) and (22) show that both the entransy crease rate and the heat flow rate to the high temperature heat source crease with creasg, and the entropy generation rate decreases. It means that both larger entransy crease rate and smaller entropy generation rate lead to larger heat flow rate to the heat source high temperature. For the cases discussed above, larger entransy crease rate always leads to larger heat flow rate to the high temperature heat source, while smaller entropy generation rate does not always. Therefore, if the optimization objective is the heat flow rate to the heat source with high temperature, the concept of entransy crease rate is more convenient. This is the difference between the concepts of entransy crease and entropy generation. Last, let us analyze the simple heat pump system with reversed Carnot cycle as shown Figure 7. The workg fluid absorbs heat flow at temperature T L from the low temperature heat source whose temperature is T, and releases heat flow at temperature T H to the high temperature heat source whose temperature is T. The put power is P. Accordg to the Carnot theorem, we have TH TL. (29) Therefore, combg eqs. (1), (14) and (21) leads to Figure 5 Variations of the COP, the entransy crease rate and the entropy generation rate with U when P is fixed. Figure 6 Variations of the entransy crease rate, the entropy generation rate, the heat flow rate to the high temperature heat source and the put power with U when is fixed.

Cheng X T, et al. Ch Sci Bull December (2013) Vol.58 No.36 4701 For given T H and T L, it can also be seen that both and G c crease with creasg or P. However, the entropy generation rate is always zero. For the reversed Carnot cycle, the entropy generation rate does not relate to the released heat flow rate because entropy generation is the measure of the irreversible degree of thermodynamic process. It does not directly relate to heat or work for the reversible processes. The entransy crease rate always is always related to heat or power, which can be seen from their expressions. 3 Conclusions Figure 7 Sketch of the heat pump system with reversed Carnot cycle. 1 TH TL P TL T H, (30) TH G c T T T T TL S g T T 1, H H P T T TL TL TH 1 T T TLT T TH 1 T H P 1. TT L T T L (31) (32) hen the temperatures T, T, T H and T L are fixed, it can be seen that, G c and S g all crease with creasg or P. Hence, larger entransy crease rate is always compatible with larger heat flow rate to the high temperature heat source, while smaller entropy generation rate is not always. hen heat is removed from a low temperature to a high temperature, the total entransy creases. The power of the heat pump becomes heat to the high temperature heat source and contributes to the entransy crease. The entropy generation only comes from the heat transfer between heat sources and workg fluid because the Carnot cycle is reversible. So, drivg more heat from the low temperature heat source the high temperature one will crease entropy generation. Let us just only look at the reversed Carnot cycle. Assume that the absorbed heat flow rate is, the released heat flow rate is, the put power is P, and the high and low workg temperatures of the workg fluid are T H and T L, respectively. In this case, eq. (29) is still tenable. The entransy crease rate is 2 2 G T T T 1 P T T. (33) c L H L H L This paper discusses the optimization of heat pump systems by the concepts of entransy crease and entropy generation and proposes the extremum entransy crease prciple. The maximum COP leads to the maximum entransy crease rate for fixed put power and to the mimum entransy crease rate for fixed heat flow rate absorbed from the low temperature heat source when the equivalent temperatures of the high and low temperature heat sources are given. On the other hand, the mimum entropy generation rate always leads to the maximum COP for fixed put power or fixed heat flow rate absorbed from the low temperature heat source with given equivalent thermodynamic forces of the high and low temperature heat sources. These different prciples are applied to the analyses of the heat pump system with reversed Brayton cycle. hen the optimization objective is the heat flow rate to the high temperature heat source, it is shown that larger entransy crease rate always leads to larger heat flow rate released to the high temperature heat source, while smaller entropy generation rate does not always. The difference between the concepts of entransy crease and entropy generation mechanisms and the mechanisms of the prciples are discussed. This work was supported by the National Natural Science Foundation of Cha (51376101) and the Tsghua University Initiative Scientific Research Program. 1 David B, Ramousse J, Luo L. Optimization of thermoelectric heat pumps by operatg condition management and heat exchanger design. Energy Convers Manage, 2012, 60: 125 133 2 Zhang L, Hihara E, Saikawa M. Combation of air-source heat pumps with liquid desiccant dehumidification of air. Energy Convers Manage, 2012, 57: 107 116 3 Fernández-Seara J, Piñeiro C, Dopazo J A, et al. Experimental analysis of a direct expansion solar assisted heatpump with tegral storage tank for domestic water heatg under zero solar radiation conditions. Energy Convers Manage, 2012, 59: 1 8 4 Chen L G, Feng H, Sun F R. Optimal piston speed ratios for irreversible Carnot refrigerator and heat pump usg fite time thermodynamics, fite speed thermodynamics and the direct method. J Energy Inst, 2011, 84: 105 112 5 Chen L G, Dg Z, Sun F R. Model of a total momentum filtered energy selective electron heat pump affected by heat leakage and its performance characteristics. Energy, 2011, 26: 4011 4018 6 uoil S, Declaye S, Tchanche B F, et al. Thermo-economic opti-

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