Energy Conversion and Management

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1 Energy Conversion and Management 58 (202) Contents lists available at SciVerse ScienceDirect Energy Conversion and Management journal homepage: Computation of effectiveness of two-stream heat exchanger networks based on concepts of entropy generation, entransy dissipation and entransy-dissipation-based thermal resistance Xuetao Cheng, Xingang Liang Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 00084, China article info abstract Article history: Received 6 December 20 Accepted 28 January 202 Available online 22 February 202 Keywords: Two-stream heat exchanger networks Effectiveness Entransy dissipation Entropy generation Thermal resistance The two-stream heat exchanger networks (THENs) are widely used in industry. The effectiveness of the THENs is analyzed in this paper. The general expressions for the entransy dissipation, the entransy-dissipation-based thermal resistance and the entropy generation for a generalized THEN are developed. It is found that the expressions are independent of the specific constitution of the THENs. Only the entransydissipation-based thermal resistance always decreases monotonously with the increase in effectiveness, while the entransy dissipation and the entropy generation do not. Therefore, the entransy-dissipationbased thermal resistance is most applicable for the optimization of the THENs. Ó 202 Elsevier Ltd. All rights reserved.. Introduction Corresponding author. Fax: address: liangxg@tsinghua.edu.cn (X.G. Liang). The worldwide appeals for energy conservation draw great attention on the effective energy utilization and exchange. Improvement on heat transfer performance is of great significance to the reduction of the energy consumption as nearly 80% of the total energy consumption is related to heat transfer []. The two-stream heat exchanger network (THEN) is one of typical euipments that has wide applications in thermal engineering, such as room heating and air conditioning [2], ground-coupled heat pump [3], and loop units in power plants [4]. Therefore, optimization of heat transport in the THENs is an important topic. For the optimization of heat transport, many theories have been developed during the recent years, such as the thermodynamic optimization [5] and the entransy theory [6]. In the optimization of the distribution of the limited high conducting materials in the Volume-to-Point conduction problem, Bejan [7] developed the constructal theory, in which the basic, imagined structure of the high conductivity material is given manually, and then the aspect ratio of the structure is optimized through theoretical derivation and numerical simulation to decrease the highest temperature of the heated domain. The constructal theory has been extended to optimize heat convection [8 ]. Furthermore, Bejan [7,2] related the optimized heat transfer to the minimum production of entropy generation, and extended this idea into other heat transfer systems, such as the optimization of the parameters in heat exchangers. A lot of research work has been reported on heat transfer optimization based on the minimum entropy generation principle [3 6]. As the objective of this kind of optimization is to minimize the entropy generation, it is called thermodynamic optimization. However, it is observed in some cases that the construct beyond the first order does not contribute to the performance improvement [7 20]. In the second law analysis of different complexity levels of the tree network of conducting paths, Ghodoossi [7,8] found that the heat flow performance does not essentially improve if the internal complexity of the heat generating area increases. It is reported that the hierarchical constructal designs from simple towards more complex tree-shaped internal structures does not improve the heat flow performance in general, it improves the performance only up to the first order construct. In heat convection with prescribed flow rate, Escher et al. [20] found that the parallel channel network provides more than fivefold increase in performance coefficient as compared to the bifurcating tree-like network, with fourfold increase in heat removal rate at a specific pressure gradient across the network. An entropy generation paradox has been observed when the minimum entropy generation principle was used to analyze heat exchangers [2]. The heat exchanger effectiveness does not always increase with decrease in entropy generation number. Shah and Skiepko [2] have analyzed the relationship between the effectiveness and entropy generation for heat exchangers and found that the heat exchanger effectiveness can be maximum, intermediate or minimum at the maximum entropy generation. Therefore, the /$ - see front matter Ó 202 Elsevier Ltd. All rights reserved. doi:0.06/j.enconman

2 64 X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (202) Nomenclature A area (m 2 ) C heat capacity flow rate (W/K) C ratio between the smaller and bigger heat capacity flow rate of the streams c specific capacity (J/(kg K)) G dis entransy dissipation (W K) N s revised entropy generation number n normal vector heat transfer rate (W) heat flux (W/m 2) R thermal resistance (K/W) S g entropy generation (W/K) T temperature (K) DT temperature difference (K) u velocity (m/s) V volume (m 3 ) Greek symbols e effectiveness k thermal conductivity (W/m/K) density (kg/m 3 ) Subscripts c cold stream h hot stream i inlet of streams max the maximum value min the minimum value o outlet of streams minimum entropy generation principle is not always applicable to heat exchanger optimization. During the last decade, the concept of entransy has been developed based on the analogy between heat conduction and electrical conduction, in which heat flux corresponds to electrical current, thermal resistance to electrical resistance, temperature to electric voltage, and heat capacity to capacitance [6]. Therefore, entransy is actually the potential energy of the heat in a body, corresponding to the electrical energy in a capacitor [6]. Based on this concept, Guo et al. [6], Han et al. [22] and Cheng et al. [23,24] proved that the entransy would always decrease during the thermal euilibrium processes. Furthermore, Cheng et al. [24] found that the entransy of an isolated system that does not involve any work always decreases during heat transfer. The decrease of entransy is called entransy dissipation which could be used to describe the irreversibility of heat transfer [6,22 25]. It is found that the maximum entransy dissipation corresponds to the maximum heat transfer rate when the heat transfer temperature difference is prescribed, while the minimum entransy dissipation corresponds to the minimum heat transfer temperature difference when the heat transfer rate is prescribed. Furthermore, Guo et al. [6] defined an euivalent thermal resistance based on the entransy dissipation and heat flux. Then, the extreme entransy dissipation principle is euivalent to the minimum thermal resistance principle [6]. These principles were used in heat conduction [6,26,27], heat convection [6,25,28], thermal radiation [29,30] and the design of heat exchangers [,3 33]. The entransy theory has been found to be more applicable to the optimization of heat transfer processes [,6,29,32] than the thermodynamic optimization. For instance, the Volume-to-Point problem in heat conduction has been analyzed based on the entransy theory and the minimum entropy generation principle [6]. It has been found that the distribution of the high conductivity material optimized by the entransy theory would lead to lower average temperature than that optimized by the minimum entropy generation. For heat exchangers optimization, it has been shown that the entransy-dissipation-based thermal resistance decreases monotonously with the increase in effectiveness [,32,33]. For some typical THENs, the entransy theory has also been applied to analyze the heat transfer processes. Chen et al. [34] analyzed a THEN connected by a hydronic fluid based on the minimum entransy-dissipation-based thermal resistance, and found that minimizing the thermal resistance results in the maximum heat transfer rate in the THEN. ian et al. [35] analyzed three constitutions of THENs and found that the maximum heat transfer rate between two streams corresponds to the minimum entransy-dissipationbased thermal resistance. In this paper, we apply the entransy theory to investigate generalized THENs and makes comparisons between entransy dissipation, the entransy-dissipation-based thermal resistance and the entropy generation. 2. Analysis of THENs A THEN may consist of heat exchangers, distributors, mixers, and medial fluids. The three THENs analyzed by ian et al. [35] is shown in Figs. 3, where the heat capacity flow rates of the hot stream and the cold stream are C h and C c, the inlet and outlet temperatures of the hot stream are T h i and T h o, while those of the cold stream are T c i and T c o, respectively. In Fig., a medial fluid works between the hot and cold streams. It absorbs heat from the hot stream with inlet temperature T m through heat exchanger, and releases heat to the cold stream with inlet temperature T m2 through heat exchanger 2. In Fig. 2, there is no medial fluid. The hot and cold streams are distributed into two heat exchangers. In Fig. 3, the THEN has a medial fluid, two distributors and two mixers. The medial fluid absorbs heat from the hot stream through heat exchanger 3. Then, the medial fluid is distributed into two heat exchangers to release heat to the cold stream. Based on the three THENs, a generalized THEN is proposed as shown in Fig. 4, where there may be many heat exchangers, distributors, mixers, and medial fluids. A scheme like this is analyzed in this paper. During a heat transfer process, the temperature of the hot stream decreases from T h i to T h o, while that of the cold stream increases from T c i to T c o. It is assumed that there is no heat exchange between the THEN and the environment, the fluids are incompressible, and the influence of viscous dissipation on heat exchange, entropy generation and entransy could be ignored. With these assumptions, the effectiveness of the generalized THEN could be defined, and the Fig.. THEN with a medial fluid [35].

3 X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (202) Fig. 2. THEN with mixers [35]. where C min ¼ minðc h ; C c Þ: Similar to the definition of effectiveness of a heat exchanger [36], the effectiveness of the THEN could be defined as e ¼ ¼ C hðt h i T h o Þ max C min ðt h i T c i Þ ¼ C cðt c o T c i Þ C min ðt h i T c i Þ : ð5þ It is evident that higher effectiveness means higher heat transfer rate at prescribed inlet temperatures of the two streams or smaller difference between the inlet temperatures of the two streams at a prescribed heat transfer rate. Therefore, the effectiveness defined in E. (5) is a measure of the performance of the THEN of Fig. 4. For the generalized THEN, the expressions of the entransy dissipation, the entransy-dissipation-based thermal resistance, and the entropy generation are G dis ¼ 2 C h T 2 h i T2 h o þ 2 C c T 2 c i T2 c o : ð6þ ð4þ R ¼ G dis = 2 ¼ 2 C h T 2 h i T2 h o þ 2 C c T 2 c i T2 c o = 2 : ð7þ S g ¼ C h ln T h o T h i þ C c ln T c o T c i ; ð8þ Fig. 3. THEN with a medial fluid and mixers [35]. respectively. It could be found that the expressions are the same as those reported in Ref. [34]. The detailed derivation of the expressions is in the Appendix A at the end of this paper. Combinations of Es. (), (6) (8) yield G dis ¼ h i 2 C h T 2 h i ðt h i =C h Þ 2 þ h i 2 C c T 2 c i ðt c i þ =C c Þ 2 ¼ðT h i T c i Þ 2 2 ð=c h þ =C c Þ: ð9þ R ¼ G dis = 2 ¼ðT h i T c i Þ= 2 ð=c h þ =C c Þ: ð0þ S g ¼ C h ln þ C c ln þ : ðþ C h T h i C c T c i expressions of the entransy dissipation, the entransy-dissipationbased thermal resistance, and the entropy generation can be derived. In the case that there is no heat source in the THEN and no heat exchange between the THEN and the environment, the energy conservation gives ¼ C h ðt h i T h o Þ¼C c ðt c o T c i Þ; where is the heat transfer rate between the hot and cold streams in the THEN. The temperature of the hot stream would never be lower than T c i, while that of the cold stream would never be higher than T h i. The maximum temperature change of the two streams is jdt max j¼t h i T c i : Fig. 4. Sketch of a generalized THEN. Therefore, the maximum possible heat transfer rate is max ¼ C min jdt max j¼c min ðt h i T c i Þ; ðþ ð2þ ð3þ For the THEN in Fig. 4, there are two kinds of optimization problems: the inlet temperatures of the two streams are prescribed or the heat transfer rate is prescribed. For both kinds of problems, the objective is to maximize the effectiveness of the exchanger. When the inlet temperatures of the two streams are prescribed, from E. (5), ¼ e max ¼ ec min ðt h i T c i Þ: Substituting E. (2) into E. (9) gives ð2þ G dis ¼ ec min ðt h i T c i Þ 2 ½ 2 ec minðt h i T c i ÞŠ 2 ð=c h þ =C c Þ ¼ C min ðt h i T c i Þ 2 e 2 e2 ð þ C Þ ; ð3þ where C ¼ minðc h; C c Þ maxðc h ; C c Þ : ð4þ It can be seen that the entransy dissipation is a uadratic function of the effectiveness and reaches its maximum value when e ¼ þ C < : ð5þ

4 66 X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (202) As the value range of the effectiveness is [0,], the maximum entransy dissipation does not correspond to the maximum effectiveness of the THEN. The entropy generation is given by S g ¼ C h ln C minðt h i T c i Þ C h T h i e þc c ln þ C minðt h i T c i Þ C c T c i Letting the derivative of E. (6) eual zero, we get e : ¼ C C min ðt h i T c i Þ h C h T h i C min ðt H i T c i Þe þ C C min ðt h i T c i Þ c C c T c i þ C min ðt h i T c i Þe ¼ 0: Therefore, C e c C h ¼ C min ðc c þ C h Þ ¼ minðc h ; C c ÞmaxðC h ; C c Þ C min ½minðC h ; C c ÞþmaxðC h ; C c ÞŠ ¼ þ C < : It could be proved that ð7þ 2 S g < 0: The entropy generation reaches its maximum value when E. (8) is satisfied. However, keeping in mind that the value range of the effectiveness is [0, ], the entropy generation would not always decrease with the increase of the effectiveness. The entropy generation paradox thus exists in the THEN as well. For the entransy-dissipation-based thermal resistance, we have R ¼ ec min 2 C h þ C c ¼ C min e 2 ð þ C Þ : ð20þ This relationship is the same as that of the two-stream heat exchangers reported in Ref. [32]. The thermal resistance would decrease monotonously with the increase in effectiveness and hence it can describe the performance of the THEN. Let us consider a numerical example. Let C h = 2 W/K, C c = 3 W/K, T h i = 350 K, and T c i = 300 K. The variations of the entransy dissipation, the thermal resistance, and the entropy generation with the effectiveness are calculated from Es. (3), (6), and (20), respectively and the results are shown in Fig. 5. It is seen that the entransy dissipation and the entropy generation both get to their maximum value when the effectiveness is 0.6. For the entransy dissipation, the extreme entransy dissipation principle tells us that the maximum entransy dissipation corresponds to the maximum heat transfer rate with the prescribed euivalent heat transfer temperature difference, while the minimum entransy dissipation corresponds to the minimum heat transfer temperature difference when the heat transfer rate is prescribed [6]. In the present case when the inlet temperatures of the streams are given, neither the euivalent heat transfer temperature difference nor the heat transfer rate is prescribed and conseuently, the extreme entransy dissipation does not correspond to the best performance of the THEN. For the entropy generation, when the effectiveness is smaller than 0.6, the entropy generation does not decrease, but increase with the increase in effectiveness. This is a paradox of entropy generation in the THEN. Entropy generation is not directly related to heat transfer rate but is related to the loss of the ability of doing work during an irreversible process. Less entropy generation means less loss of the ability of doing work. However, in THENs, what we are concerned of is not the loss of the ability of doing work, but the effectiveness or the heat transfer rate. Therefore, the minimum entropy generation principle is not always applicable to the design optimization of heat exchangers. For the thermal resistance, it decreases monotonously with the increase in effectiveness. Therefore, it is the appropriate parameter that could be used to optimize the performance of the THEN. Fig. 6 demonstrates the variation of the thermal resistance with the effectiveness at different values of C c. The minimum thermal resistance always corresponds to the maximum effectiveness, and there is no paradox similar to the entropy generation. Thus, the minimum principle of entransy-dissipation-based thermal resistance is a more general rule in heat transfer optimization than the extreme entransy dissipation principle and the minimum entropy generation principle. When the heat transfer rate is prescribed, E. (2) tells us that higher effectiveness means smaller difference between the inlet temperatures of the two streams. Substituting E. (2) into E. (9), we get G dis ¼ 2 C min e 2 ð þ C Þ : ð2þ The entransy dissipation decreases monotonously with the increase in effectiveness. The entransy dissipation extreme principle holds good in this case. The entropy generation is S g ¼ C h ln þ C c ln þ : ð22þ C h T h i ½T h i =ðec min ÞŠC c for prescribed T h i,oris S g ¼ C h ln þ C c ln þ : ð23þ C h ½=ðeC min ÞþT c i Š T c i C c for prescribed T c i. The entropy generation also decreases monotonously with the increase in effectiveness. The minimum entropy Fig. 5. Variation of entransy dissipation, thermal resistance, and entropy generation with effectiveness of THEN when the inlet temperatures of the streams are prescribed. Fig. 6. Variation of thermal resistance with effectiveness of THEN at different heat capacity flow rates of cold stream.

5 X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (202) Fig. 7. Variation of entransy dissipation, thermal resistance, and entropy generation with effectiveness of THEN when the heat transfer rate is prescribed. Fig. 8. Variation of effectiveness, entransy dissipation, thermal resistance and entropy generation with thermal conductance of heat exchanger 2 of Fig.. generation principle also holds good in this case. The thermal resistance, E. (20), is tenable for all the cases. The following numerical example accords with above discussion. Let C h = 2 W/K, C c = 3 W/K, = 0 W, and T c i = 300 K. The results, shown in Fig. 7, indicate that the entransy dissipation, the thermal resistance, and the entropy generation are all monotonically related to the performance of the THEN in such a case. When the heat transfer rate is prescribed, the extreme entransy dissipation principle is applicable, and the minimum entransy dissipation corresponds to the best performance of the heat transfer process [6]. This is the reason why the entransy dissipation decreases monotonously with the increase of the effectiveness. On the other hand, as the optimization objective of the extreme entransy dissipation principle is to decrease the heat transfer temperature difference, the entropy generation is also decreased due to the decrease of the heat transfer temperature difference. Cheng et al. [37] drew a similar conclusion when they investigated the Volume-to-Point problem. When the outlet boundaries are isothermal (there could be more than one outlet of heat but their temperatures must be the same), the optimization results of the extreme entransy dissipation principle and the minimum entropy generation principle are the same. This is the reason why the entropy generation could also represent the performance of the THEN when the heat transfer rate of the streams is prescribed. From the above discussions of both kinds of problems, it could be concluded that only the thermal resistance could represent the change of the effectiveness of the THEN without limitations. Neither the entransy dissipation nor the entropy generation would always change monotonously with the increase in effectiveness, so their application is conditional. Let us consider a numerical example for the THEN sketched in Fig.. Let T h i = 350 K, T c i = 300 K, C h = 2 W/K, C c = 3 W/K, the heat capacity flow rate of the medial fluid is 2.5 W/K, and the thermal conductance of heat exchanger, F, is 0 W/K. The variations of the effectiveness, the entransy dissipation, the thermal resistance, and the entropy generation with the thermal conductance of heat exchanger 2, F 2, are shown in Fig. 8. It can be seen that the effectiveness increases monotonously with the increase of F 2, while the thermal resistance decreases monotonously. However, the entransy dissipation and the entropy generation both increase first and then decrease. This example also indicates that the application of entransy dissipation and the entropy generation is conditional to the performance of the THEN, while the thermal resistance is not. To improve the application of entropy generation optimization to heat exchangers, Hesselgreaves [38] developed a revised entropy generation number for two-stream heat exchanger analysis based on the work of Witte and Shamsundar [39]. For any two-stream heat exchanger, the expression of the revised entropy generation number is defined as [38] N s ¼ T L is g ; ð24þ It was found that the effectiveness varies monotonically with N s. There is no paradox between N s and the effectiveness. As the expressions of the effectiveness and entropy generation for twostreams heat exchanger networks are the same as those of twostream heat exchangers, the revised entropy generation number can also be used to describe the performance of two-stream heat exchanger networks for given heat capacity flow rates. Compared with the revised entropy generation number, the entransy-dissipation-based thermal resistance has a direct relation with the heat transfer rate in the heat exchanger and its physical meaning is clear for heat transfer. Let s consider the case in which both /C h and /C c keep constant. The resistance of E. (0) can be rewritten as R ¼ G dis = 2 ¼ T h i T c i 2 ð=c h þ =C c Þ =: ð25þ where ð=c h þ =C c Þ on the right is constant for constant /C h and /C c. Hence, R decrease with the increase of, indicating that lower thermal resistance corresponds to larger heat transfer rate. On the other hand, we can also rewrite the revised entropy generation number from E. (), N s ¼ S gt c i ¼ T c i C h ln C h T h i þ C c ln þ T c i C c : ð26þ The revised entropy generation number is constant for given inlet temperature in such case. Raising C h and C c proportionally will make become larger but the revised entropy generation number remains constant. The revised entropy generation number has no direct relation with the heat transfer rate. It is not related to the difference of the stream temperatures but the difference of the reciprocal of the log-mean temperatures as indicated by E. (7) in Ref. [38]. The target of minimizing N s is not to reduce the heat transfer difference or increase heat transfer rate, but to reduce the difference of thermodynamics potential that is the reciprocal of absolute temperature. On the other hand, thermal resistance is closely connected with the stream temperature difference and heat flow which are the most important parameters in heat transfer optimizations. The different objectives of the thermal resistance and the entropy optimizations have been shown clearly in the example of the Volume-to-Point problem [40] in which the uniform heat source in

6 68 X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (202) the suare domain is specified. The entransy dissipation is euivalent to thermal resistance and the entropy generation is euivalent to N s because the total heat flow rate is prescribed in this example. It was found that the minimum thermal resistance principle results in smaller average and maximum temperatures in the domain but a larger difference of thermodynamic potential (reciprocal of temperature) than the minimum entropy generation does. The entropy generation is reduced by decreasing the difference of thermodynamic potential, not by temperature difference. 3. Conclusions The general expressions for entransy dissipation, entransydissipation-based thermal resistance and entropy generation for a generalized THEN are developed. The relationships between effectiveness and entransy dissipation, entransy-dissipationbased thermal resistance and entropy generation in THENs are derived and analyzed. When the inlet temperatures of the two streams of the THEN are prescribed, it is found that the entransy-dissipation-based thermal resistance decreases monotonously with the increase in effectiveness, while the entransy dissipation and entropy generation do not. When the heat transfer rate of the THEN is prescribed, the entransy dissipation, the entransy-dissipation-based thermal resistance and the entropy generation all decrease monotonously with the increase in effectiveness. The overall conclusion is that the thermal resistance is most suitable to optimize the effectiveness and describe the performance of the THEN. Acknowledgements The present work is supported by the Natural Science Foundation of China (Grant No ) and the Tsinghua University Initiative Scientific Research Program. Appendix A The heat transfer process in the generalized THEN in Fig. 4 is composed of three parts. The first part is the heat conduction through the solid structure, the second part is the heat convection between the medial fluids and the solid channel surface, and the third part is the heat convection between the solid channel surface and the hot and cold streams. For the heat conduction, the governing euation is r ¼ 0; ða:þ where is the heat flux vector. Multiplying E. (A.) by the temperature T leads to [6] r ðtþ rt ¼ 0: ða:2þ Applying the Gauss divergence theorem to the integration of E. (A.2) over the whole heat conduction region yields [6] ½rðTÞ rtšdv ¼ V ðtþn da A rtdv ¼ 0; V ða:3þ where V is the volume of the heat conduction region, A is the surface area of the region, and n is the normal vector of the surface of the region. According to the Fourier s law, the entransy dissipation in the heat conduction region is [6] G dis ¼ kðrtþ 2 dv ¼ V ðtþn da ; A ða:4þ where k is the thermal conductivity. As there is no heat exchange between the THEN and the environment, the heat flux only exists at the interface between the solid and the two streams or between the solid and the medial fluids. Therefore, we get G dis ¼ ðtþn f d ðtþn m d ; ða:5þ where A f s is the area of the interface with the normal vector n f between the solid and the two streams, and is the area of the interface with the normal vector n m between the solid and the medial fluids. For the heat convection region of the medial fluids, the governing euation is cðu rtþ ¼ r þ U; ða:6þ where is the density of the fluid, c is the specific heat capacity, u is the velocity of the fluid, and U is the viscous dissipation function. Multiplying E. (A.6) by the temperature T gives [6] 2 c½u rðt2 ÞŠ ¼ r ðtþþ rt þ UT; ða:7þ Apply the Fourier s law to E. (A.7), and integrate it over the region [6] G dis 2 ¼ kðrtþ 2 dv 2 V 2 ¼ V 2 c½u rðt2 ÞŠdV r ðtþdv þ UTdV; ða:8þ V V For the incompressible medial fluid, r u ¼ 0: ða:9þ Based on the Gauss divergence theorem and E. (A.9), the entransy dissipation of the heat convection region of the medial fluids could be expressed as G dis 2 ¼ A 2 2 ct2 u n 2 da 2 ðtþn 2 da 2 þ ðutþdv 2 ; ða:0þ V 2 where V 2 is the volume of the heat convection region of the medial fluids, A 2 is the area of the surface of the region, and n 2 is the normal vector of the surface. As there are no outlets or inlets for the medial fluids, A 2 is, hence, u n 2 ¼ 0: ða:þ Substituting E. (A.) in E. (A.0) and ignoring the influence of viscous dissipation on the entransy of the region, we get G dis 2 ¼ ðtþn 2 d : ða:2þ For the heat convection between the solid surface and the two streams, E. (A.7) is applicable. Accordingly, G dis 3 ¼ A 3 2 ct2 u n 3 da 3 ðtþn 3 da 3 þ ðutþdv 3 ; A 3 V 2 ða:3þ where V 3 is the volume of the heat convection region of the two streams, n 3 is the normal vector of the surface of the region, and A 3 is the area of the surface of the region which is the sum of the area A f s and the area of the inlet and the outlet of the two streams. Ignoring the influence of the viscous dissipation on the entransy of the region and the heat conduction at the inlet and the outlet of the two streams, E. (A.3) reduces to G dis 3 ¼ A 3 2 ct2 u n 3 da 3 ðtþn 3 d : ða:4þ Assuming that the velocity and the temperature of the fluids at the inlet and the outlet are uniform and the velocity of the fluids in the normal direction at the interface is zero, we get

7 G dis 3 ¼ 2 h c hu h i A h i T 2 h i 2 h c hu h o A h o T 2 h o þ 2 c c cu c i A c i T 2 c i 2 c c cu c o A c o T 2 c o ðtþn 3 d ¼ 2 C h T 2 h i T2 h o þ 2 C c T 2 c i T2 c o ðtþn 3 d ; ða:5þ where the subscripts h, c, i and o stand for hot stream, cold stream, inlet and the outlet of the streams, respectively. Combination of Es. (A.5), (A.2), and (A.5) yields the entransy dissipation of the THEN, G dis ¼ G dis þ G dis 2 þ G dis 3 ¼ ðtþðn f þ n 3 Þd ðtþðn m þ n 2 Þd X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (202) S g 3 ¼ S out S in þ T n 3d : ða:24þ where S in is the entropy that the fluids bring into the region, while S out is that the fluids take away from the region. For any stream with inlet temperature T i and outlet temperature T o, the entropy change is DS fluid ¼ f 0 d f T To ¼ C f dt ¼ C f ln T o : T i T T i ða:25þ where f is the change of the internal energy, C f is the heat capacity flow rate of the stream. Hence, we could get S g 3 ¼ DS fluid-h þ DS fluid-l þ T n 3d ¼ C h ln T h o þ C c ln T c o þ T h i T c i T n 3d : ða:26þ where DS fluid-h and DS fluid-l are the entropy changes of the hot and the cold streams, respectively. Combining Es. (A.7), (A.8), (A.22), (A.23), and (A.26), we get þ 2 C hðt 2 h i T2 h o Þþ 2 C cðt 2 c i T2 c o Þ: ða:6þ At the interface between the solid surface and the fluids, n m þ n 2 ¼ 0; ða:7þ S g ¼ S g þ S g 2 þ S g 3 ¼ C h ln T h o T h i þ C c ln T c o T c i : References ða:27þ n f þ n 3 ¼ 0: Then, E. (A.6) could be reduced to G dis ¼ 2 C hðt 2 h i T2 h o Þþ 2 C cðt 2 c i T2 c o Þ: ða:8þ ða:9þ ian et al. [35] defined the entransy-dissipation-based thermal resistance for the THENs shown in Figs. 3. With their definition, the thermal resistance of the generalized THEN shown in Fig. 4 could be defined as R ¼ G dis = 2 ¼ 2 C hðt 2 h i T2 h o Þþ 2 C cðt 2 c i T2 c o Þ = 2 : ða:20þ Let us now proceed to the estimation of the entropy generation of heat exchanger. For any thermodynamic process, the entropy euation is ds ¼ ds f þ ds g ; ða:2þ where ds is the entropy change, ds f is the entropy flux, and ds g is the entropy generation rate of the heat transfer processes. At steady state, ds is zero. For the heat conduction region, considering only the heat flux at the interface between the solid surface and the two streams or between the solid surface and the medial fluids, we get S g ¼ ds g ¼ ds f ¼ r dv ¼ V V V T A T n da ¼ T n f d þ T n md : ða:22þ For the heat convection region of the medial fluids, the entropy generation is S g 2 ¼ T n 2d : ða:23þ The heat convection region of the two streams is composed of the two parts, the entropy flux at the interface and the entropy flux at the fluid inlet and fluid outlet. Therefore, the entropy generation is [] ian XD, Li X. Analysis of entransy dissipation in heat exchangers. Int J Thermal Sci 20;50: [2] Kreider JF. Heating and cooling of buildings: design for efficiency. New York: McGraw-Hill; 994. 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