Application of entransy to optimization design of parallel thermal

SCIENCE CHINA Technological Sciences • RESEARCH PAPER •

April 2011 Vol.54 No.4: 964–971 doi: 10.1007/s11431-011-4294-3

Application of entransy to optimization design of parallel thermal network of thermal control system in spacecraft CHENG XueTao, XU XiangHua* & LIANG XinGang Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China Received October 20, 2010; accepted December 30, 2010; published online January 25, 2011

For distribution optimization of the flow rate of cold fluid and heat transfer area in the parallel thermal network of the thermal control system in spacecraft, a physical and mathematical model is set up, analyzed and discussed with the entransy theory. It is found that the optimization objective of this problem and the optimization direction of the extremum entransy dissipation principle are consistent in theory. For a two-branch thermal network system, the distributions of the flow rate of the cold fluid and the heat transfer area are optimized by calculating the extremum entransy dissipation with the Newton method. The influential factors of the optimized distributions are also analyzed and discussed. The results show that the main influence factors are the heat transfer rate of the branches and the total heat transfer area. The total flow rate of the cold fluid has a threshold, beyond which further increasing its value brings very little influence on the optimization results. Moreover, the difference between the extremum entransy dissipation principle and the minimum entropy generation principle is also discussed when they are used to analyze the problem in this paper, and the extremum entransy dissipation principle is found to be more suitable. In addition, the Newton method is mathematically efficient to solve the problem, which could accomplish the optimized distribution in a very short time for a ten-branch thermal network system. thermal control system, parallel thermal network, optimization design, extremum entransy dissipation principle Citation:

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Cheng X T, Xu X H, Liang X G. Application of entransy to optimization design of parallel thermal network of thermal control system in spacecraft. Sci China Tech Sci, 2011, 54: 964971, doi: 10.1007/s11431-011-4294-3

Introduction

It is very important to release the heat produced in the equipment on a spacecraft to keep the equipment working at a normal temperature. Therefore, in the spacecraft, there are often some fluid loops, such as the gas recycling thermal control loop and the liquid recycling thermal control loop, which have high ability to control the temperature of the equipment [1]. The equipment that releases heat is often in parallel connection in the thermal control loop [2]. Equipment could also be connected in series. Compared with the

*Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2011

thermal network in series, the parallel configuration has higher heat transfer efficiency, smaller heat transfer area and lower pressure drop. Therefore, the weight of parallel thermal network may be less. Moreover, the equipment in parallel connection is more independent from each other. It is easier to control their temperatures separately. Hence, the parallel thermal network is widely used in the thermal control system of spacecraft, such as the International Space Station [3]. For the parallel thermal network, there are mainly two kinds of optimization problems. First, the weight of the network should be decreased as much as possible to reduce the launching cost of the spacecraft. Mark and David [4] pointed out that lightweight design should be the optimizatech.scichina.com

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tion objective of the thermal control system in 1985. Second, the heat transfer processes in the thermal control system should be optimized under some restrained conditions because most equipment in a spacecraft requires appropriate temperature and an increasing temperature would decrease the working performance. For the lightweight design problem, Zhang et al. [2] analyzed a parallel thermal network in spacecraft using the nonlinear mathematical programming method. Based on the precondition that the equipment works at certain temperatures and could release their heat, they discussed the relationship between the system weight and the cold fluid distribution. The nonlinear mathematical programming method may not be very suitable for a complex parallel thermal network because of its large amount of calculations. Therefore, Zhang et al. mainly discussed a two-branch system in their research. To decrease the calculation amount, Cheng et al. [5, 6] optimized some complex parallel thermal network more efficiently by using the variable substitution method and Lagrange multiplier method and derived correlations to calculate the optimized flow rate distributions based on their calculation results. It is also very common and important to optimize the heat transfer processes in thermal control system of spacecraft to decrease the working temperatures of the equipment under some restrained conditions, such as limited weight of the thermal control system. The present work analyzes and discusses this problem. In heat transfer optimization, Guo et al. [7] introduced the concept of entransy by analogy between heat and electrical conduction. It was also called the heat transport potential capacity at the beginning [8, 9]. For a system that is at its thermal equilibrium state, its entransy is 1 G  UT , 2

(1)

where U is the internal energy and T is the temperature of the system. Entransy describes the ability of a body to release heat. Cheng et al. [10] proved that the entransy would always decrease in the heat transfer process of an isolated system, which means that heat transfer would always accompany entransy dissipation. Based on this new quantity, Guo et al. [7] derived the extremum entransy dissipation principle, defined an equivalent thermal resistance of the system based on the entransy dissipation and heat flux, and further developed the minimum thermal resistance principle. These principles gradually found their applications in heat transfer, such as in heat conduction [8, 9, 11–16], in heat convection [17–19], in thermal radiation [20, 21], and in the design of heat exchangers [22–26], as well as in the heat transfer processes of phase change [27]. These principles could optimize the heat transfer processes, and increase the heat transfer efficiency and the energy utilization efficiency. For the heat transfer optimization in thermal control system in a spacecraft, Cheng et al. [28] discussed the homog-

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enization of temperature field for the thermal radiator in space based on the entransy theory. They found that the performance of the thermal radiator could increase due to the decrease of its thermal conduction resistance and thermal radiation resistance defined based on entransy dissipation. However, there are a few reports on the application of entransy in the optimization design of thermal control system of spacecraft. Therefore, it is worth making further investigation of entransy theory on the optimization of the parallel thermal network. The research may expand the application field of entransy theory, and introduce a new view of point for this kind of optimization problems.

2 Optimization problem for parallel thermal network dispersion A parallel thermal network could be simplified as shown in Figure 1. In the thermal network, the cold fluid, whose flow rate is q and temperature is Tin, is guided into the branches by the distributor, then it absorbs a heat, Qi, from the ith branch, and flows into the main outlet pipe finally. For this parallel thermal network, it is necessary to optimize the heat transfer processes to decrease the heat transfer temperature and increase the performance of the equipments. The distribution of the flow rate of the cold fluid should be optimized under the limitation of the cost and the weigh of the thermal control system. In heat transfer, the limitation can be interpreted as a limitation on the total heat transfer area of all the branches. Therefore, the optimization is the distribution of the given heat transfer area. The total flow rate of the cold fluid, q, and the total heat transfer area, A, are given. Assume that the flow rate into the ith branch is qi, and the heat transfer area of the ith branch is Ai. The distribution coefficients of the flow rate and the area are defined as xi  qi q ,

(2)

ai  Ai A .

(3)

The heat, Qi, transferred in different branches may not be the same. For the branches where Qi is bigger, there may be more high-power electronic devices. Therefore, the temperature increase may have greater effect on the performance

Figure 1

The parallel thermal network.

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of the equipments. Hence, the branches where the task of releasing heat is heavier should be considered more during the optimization. With this consideration, the heat transfer temperature of the equipments weighted by heat flux could be defined as n

n

T   QiTi

n

Q  QT

i 1

i 1

i

i 1

i i

Q,

(4)

where Ti is the temperature of the ith branch when Qi is transferred into the fluid, and Q is the sum of the heat of the branches. Then, the optimization in the parallel thermal network could be expressed as min T ;  n n  s.t. x 1, ai  1, xi   0,1 , ai   0,1 .    i  i 1 i 1 

(5)

3 Model and theoretical analysis The extremum entransy dissipation principle tells us that the minimum entransy dissipation leads to the minimum heat transfer temperature difference when the heat fluxes of the boundaries are prescribed [7]. For the optimization problem shown in eq. (5), the heat flux of each branch is known, therefore the total heat transfer rate is given. Would the optimization results of eq. (5) correspond to the minimum heat transfer temperature difference, or be consistent with those of the extremum entransy dissipation principle? The entransy dissipation of the heat transfer process would be analyzed. In the thermal network system shown in Figure 1, there are two kinds of entransy flux inlet. One is due to the inlet of the fluid, while the other is due to the heat transfer in the equipment in the branches. The total entransy flux that flows into the system is n 1 Gin   QiTi  cqTin2 , 2 i 1

(6)

where c is the specific heat capacity of the fluid. The entransy flux that flows out of the system from the outlet of the fluid is Gout

1 2  cqTout , 2

(7)

where Tout is the outlet temperature of the fluid. Based on eqs. (6) and (7), the entransy dissipation rate of the system is n 1 1 2 . Gdis  Gin  Gout   QiTi  cqTin2  cqTout 2 2 i 1

(8)

It could be rewritten as n

Gdis   QiTi  i 1

Tout  Tin cq Tout  Tin   . 2

(9)

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In the heat transfer process, as all the heat released from the equipment is absorbed by the fluid, there are Q  cq Tout  Tin  ,

(10)

Tout  Q cq  Tin .

(11)

Substituting eqs. (10) and (11) into eq. (9) produces n

Gdis   QiTi  Q 2 2cq  QTin .

(12)

i 1

Considering the definition of the weighted heat transfer temperature of the equipment, there is Gdis  Q T   Q 2cq  Tin   ,

(13)

where Q, c, q and Tin are all known. In eq. (5), the optimization objective is to get the minimum temperature T. This objective is consistent with getting the minimum entransy dissipation rate. In eq. (13), the term in the parentheses is the sum of the inlet temperature and half of the total temperature increase of the fluid. This part could be treated as the equivalent average temperature of the fluid. Therefore, the part in the square bracket is the difference of the heat transfer temperature of the equipment and the equivalent average temperature of the fluid. This part could be taken as the equivalent heat transfer temperature difference, T. Then, there is Gdis  QT .

(14)

It is obvious that the optimization expressed in eq. (5) is equivalent to the one whose objective is to make the heat transfer temperature difference minimum with the prescribed total heat transfer rate. Therefore, the optimization results of the problem expressed in eq. (5) is consistent with the extremum entransy dissipation principle. Then eq. (5) could be replaced by min Gdis ,  n n  s.t.  xi  1,  ai  1, xi   0,1 , ai   0,1 . i 1 i 1 

(15)

The entransy dissipation rate is calculated in the following way. By ignoring the heat conduction resistance in the branches in Figure 1, the heat transfer coefficient between the fluid and the equipment could be assumed as [2, 5, 6] ki  k0 qi  k0 q xi ,

(16)

where k0 is a constant and the value of  is between 0.5 and 0.8 (the values 0.5 and 0.8 correspond to the laminar flow and turbulence flow of the fluid, respectively). Based on eq. (16), the energy balance equation of the ith branch could be expressed as Qi  ki Ai Ti  k0 q Axi ai Ti ,

(17)

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where Ti is the logarithmic mean temperature difference of the ith branch, whose expression is Ti 

Touti  Tin

ln Ti  Tin  Ti  Touti  

(18)

.

Another energy balance equation of the ith branch is Qi  cqi Touti  Tin  .

(19)

Combination of eqs. (17)–(19) gives the temperature of the ith branch





1

Ti  ri xi1 1  exp sxi 1ai   Tin ,

(20)

ri  Qi cq , s  k0 q A cq .

(21)

where

Therefore, based on eq. (12), the entransy dissipation rate of the thermal network could be determined by n







1

Gdis   Qi ri xi1 1  exp sxi 1ai   Tin i 1

 (22)

 Q 2 2cq  QTin .

To find the extreme value of eq. (22) with the limiting condition of eq. (15), a functional is established as  n   n    Gdis  1   xi  1  2   ai  1 ,  i 1   i 1 

(23)

where 1 and 2 are the Lagrange multipliers. The values of xi and ai that lead to an extremum value of eq. (23) must satisfy         0. xi ai 1 2

(24)

Equation (24) is a group of nonlinear equations. It could be treated as F(Y) = 0, where Y is a vector composed of the distribution coefficient and the Lagrange multipliers, that is [x1 x2  xn a1 a2  an 1 2]T. It could be solved by Newton method [29]. The iterative formula is 1

Y j 1  Y j   F (Y j )  F (Y j ),

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In general, the heat generated in each equipment cabinet of the spacecraft is less than 1,000 W, so we assume that Q1 = 800 W and Q2 = 500 W. The cold fluid is water and its specific heat capacity is 4,200 J/kg/K. Considering the system works in manned spacecraft and the inlet temperature of the fluid is not too low. Let Tin be 290 K. Suppose the total flow rate of the cold fluid, q, is 0.1 kg/s, the total heat transfer area, A, is 0.5 m2, and k0 is 104. The exponent, , is determined by the flow state of the fluid in the thermal network. The flow state is affected both by flow rate of the fluid and the heat transfer area of the branches. In this paper, we simply assume  is 0.8 at first, and the influence of different values of  is then discussed. By using the Newton method expressed in eq. (25), we could obtain the optimized distributions for the two-branch thermal network described above. The results of the flow rate distribution are x1 = 0.40 and x2 = 0.60, while those of the area distribution are a1 = 0.43 and a2 = 0.57. The minimum entransy dissipation rate is 4.03×103 WK and the minimum heat transfer temperature of the thermal network is 294.64 K. As verification, the heat transfer temperature and the entransy dissipation rate of the thermal network are calculated for some other distributions of the flow rate and the heat transfer area. The results are shown in Figures 2 and 3. Both the heat transfer temperature and the entransy dissipation rate reach their minimum values when x1 = 0.40 and a1 = 0.43. It could also be found that the entransy dissipation rate monotonously corresponds to the heat transfer temperature, and their change tendencies are consistent. In heat transfer optimization, there is thermodynamics optimization principle except the entransy optimization theory. Bejan [30, 31] developed a constructal theory to optimize the distribution of the limited high conducting materials for the volume-to-point problem, and related the optimal heat transfer to the minimum production of entropy generation. Bejan [32, 33] thought that the thermodynamics performance of a system might be the best when its entropy

(25)

where Yj is the result when the iteration number is j, and [F′(Yj)]1 is the inverse matrix of the derivative matrix of F(Y) at point Yj.

4

Numerical examples and discussion

4.1 Entransy optimization and thermodynamics optimization Let us discuss a simple two-branch thermal network system.

Figure 2 The heat transfer temperature of the two-branch thermal network under different distributions of the flow rate and the heat transfer area.

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Figure 3 The entransy dissipation rates of the two-branch thermal network under different distributions of the flow rate and the heat transfer area.

generation reached the minimum value. Many researchers (Poulikakos and Bejan [34], Erek and Dincer [35], Chen et al. [36] and Zhou et al. [37]) applied the entropy generation to optimize heat transfer problems. The entropy generation optimization could also be used to analyze the system shown in Figure 1. According to the entropy equation of the open system, there is [38] dS  dSf  Sg ,

(26)

where dS is the entropy change, dSf is the entropy flux, and δSg is the entropy generation rate of the system. As the system is steady, dS equals zero. Considering that the entropy flux is composed of the inlet and outlet entropy flux and the entropy flux of heat transfer, there is Sg  dSf  

Q  d(qsw ), T

(27)

where Q is the heat transported, q is the flow rate of the fluid, and sw is the entropy of the fluid at the inlet and the outlet per unit mass. When eq. (27) is applied to the system shown in Figure 1, it could be found that the state of the fluid at the inlet and the outlet would not change with the distributions of the flow rate and the heat transfer area, because the inlet temperature of the fluid and the total heat transfer rate are both given. Therefore, sw is certain. Then, there is Qi  const. i 1 Ti n

Sg  

(28)

Based on eq. (28), the entropy generation rate of the system could be calculated under different distributions of the flow rate and the heat transfer area. It is found that the optimized distributions of the minimum entropy generation principle and the extremum entransy dissipation principle are consis-

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tent in the two-branch system described above. This is because the optimization objectives of the two principles are consistent with each other. When the exchanged heat is prescribed, the optimization objective of the extremum entransy dissipation principle is to decrease the heat transfer temperature difference and the entransy dissipation rate. At the same time, the entropy generation rate would decrease due to the decrease of the heat transfer temperature difference. Cheng et al. [39] drew a similar conclusion when they researched 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 extremum entransy dissipation principle and the minimum entropy generation principle are the same. Their optimization objectives are consistent. When the optimization objectives are different, the difference between the two principles would be apparent. For the system shown in Figure 1, there is an inverse problem. When the temperatures of the equipment, Ti, are given, the problem is how to distribute the given total flow rate of the cold fluid and the given total heat transfer area to increase the heat transfer ability of the system and make the fluid absorb more heat from the system. For the heat transfer process between the equipments and the fluid in any branch, the entropy generation rate is  1 1 Sg  i  Q    ,  Tw Ti 

(29)

where Tw is the temperature of the fluid. It could be found that the minimum entropy generation corresponds to the minimum heat transfer. At the same time, the entransy dissipation rate of the same process is Gdis  i  Q Ti  Tw  .

(30)

The maximum entransy dissipation rate corresponds to the maximum heat transfer. In this problem, the optimization objective is to increase the heat transfer ability of the system and the heat transfer rate. If we use the minimum entropy generation principle to optimize this problem, the heat transfer rate of the system would not have the maximum value, but decrease instead. However, we could reach the expected objective when we use the extremum entransy dissipation principle. Therefore, the minimum entropy generation principle is not applicable to this kind of problems. Comparison between the two optimization principles indicates that entropy generation is a physical quantity that is used to analyze the irreversibility of heat-work conversion. Less entropy generation means less loss of the ability of doing work. However, in heat transfer, many processes are not related to heat-work conversion. In other words, what we concern in heat transfer is not the loss of the ability of doing work, but the efficiency or rate of heat transfer. Therefore, the minimum entropy generation principle is not always applicable to heat transfer optimization [39]. For

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instance, Bejan [33] noted there is an entropy generation paradox in analyzing heat exchangers. The heat exchanger effectiveness does not always increase with decreasing entropy generation. Instead, it becomes smaller under some conditions. Shah et al. [40] analyzed the relationship between the heat exchanger effectiveness and entropy generation for 18 kinds of heat exchangers. They found that the heat exchanger effectiveness could be maximum, intermediate or minimum at the maximum entropy generation. However, the extremum entransy dissipation principle was developed based on the nature of heat transfer, and there is no such paradox in application [23]. The physical natures of the two principles are different although they are consistent with each other sometimes. The extremum entransy dissipation principle is more applicable to heat transfer optimization. 4.2

The influencing factors of the optimization

In our model, the value of  is not fixed because we could not simply determine the flow state of the fluid in the heat transfer branches. In the following, we change the value of , and discuss its influence on the optimization distributions. The results shown in Figure 4 show that the flow rate and the heat transfer distribution coefficients of the first branch, x1 and a1, increase slightly with the increase of . However, the increase range is less than 4% and its influence is negligible. Therefore, the results in Figures 2 and 3 are universal. For the two-branch parallel thermal network, we could investigate the influence on the optimization results from the heat transfer rates of the branches, the total flow rate of the fluid and the total heat transfer area. First, we change the heat transfer rate of the first branch and keep other parameters to observe its influences shown in Figure 5. Second, the heat transfer rate of the first branch keeps at 500 W, and the total flow rate of the cold fluid is changed. The influence of the total flow rate of the cold fluid is depicted in Figure 6. Finally, the influence of the total heat transfer area is shown in Figure 7 by keeping the total flow rate of the cold fluid at 0.1 kg/s.

Figure 4

The influence of the exponent  on the optimization results.

Figure 5 The influence of the heat transfer rate of the first branch on the optimization results.

Figure 6 The influence of the total flow rate of the cold fluid on the optimization results.

Figure 7 results.

The influence of the total heat transfer area on the optimization

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From Figures 5 to 7, it could be found that the heat transfer rates of the branches have the most influential effect on the optimization distributions. With the increase of the heat transfer rate of one branch, the optimized distributions of the flow rate and the heat transfer area for this branch both increase. Furthermore, the total heat transfer area is also a main influential factor of the optimization results. The distributed flow rate of the cold fluid decreases but the distributed heat transfer area increases for the branch whose heat transfer rate is smaller when the total heat transfer area increases. At the same time, the flow rate of the branch whose heat transfer rate is bigger increases. This is because the increase of the flow rate for the branch whose heat transfer rate is bigger could increase the heat transfer coefficient of this branch greatly and decrease the heat transfer temperature of this branch. Therefore, the total heat transfer rate of the system decreases. When the total flow rate of the cold fluid is small (smaller than 0.1 kg/s in the two-branch system), its increase could improve the heat transfer coefficients of the branches, and then affect the optimization results. However, when the total flow rate of the cold fluid is bigger, its increase would bring less improvement to the heat transfer coefficients, and then the influence on the optimization results is smaller. The above discussion demonstrates that the main influence factors of the optimized flow rate and heat transfer area in parallel thermal network are the heat transfer rate of the branches and the total heat transfer area. For the total flow rate of the cold fluid, there is a threshold, beyond which further increasing its value brings very little influence on the optimization results. 4.3

The optimization example of a complex system

A two-branch thermal network system is optimized with the Newton method expressed in eq. (25). For a complex thermal network system with more branches, the optimization results could also be realized quickly by using the Newton method. Table 1 shows the optimized distributions of a Table 1 The optimization distributions of a ten-branch system Number

Q (W)

T (K)

x

a

1

500

296.52

0.083

0.087

2

600

296.30

0.096

0.097

3

700

296.12

0.110

0.108

4

600

296.30

0.096

0.097

5

800

295.98

0.123

0.117

6

400

296.81

0.069

0.075

7

800

295.98

0.123

0.117

8

500

296.52

0.083

0.087

9

750

296.05

0.116

0.112

10

650

296.20

0.103

0.103

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complex parallel thermal network with 10 branches. In the thermal network, the total flow rate of the cold fluid is 0.5 kg/s, the total heat transfer area is 1.5 m2, and  is 0.8. The computing time is less than one second with a computer whose memory is 2 G, and the main frequency of the two CPUs is 2.7 GHz.

5

Conclusions

In the parallel thermal network of a spacecraft, the distribution of the flow rate of the cold fluid and the heat transfer area should be optimized to decrease the heat transfer temperature of the network when the heat transfer rates of the branches are prescribed. For this problem, a physical and mathematical model is set up. With the model, the problem is analyzed and discussed. Some conclusions could be drawn from the investigation. The parallel thermal network could be optimized with the extremum entransy dissipation principle. The present work proves that the optimization objective of the minimum heat transfer temperature and the optimization direction of the extremum entransy dissipation principle are consistent in theory. When the minimum entropy generation principle is used to optimize this problem, the optimization results are consistent with those of the extremum entransy dissipation principle. However, when we analyze the inverse problem in which the temperatures of the equipment are given and the maximum heat transfer rate is the optimization objective, the minimum entropy generation principle is not suitable, and the extremum entransy dissipation principle is appropriate. This is because the optimization objective of the minimum entropy generation principle is to decrease the loss of the ability of doing work, while that of the extremum entransy dissipation principle is to increase the heat transfer rate. Therefore, the entransy theory is more suitable in heat transfer optimization. For a two-branch thermal network system, the distributions of the flow rate and the heat transfer area are optimized by calculating the extremum entransy dissipation with the Newton method. The influential factors of the optimized distributions are discussed. It is found that the heat transfer rates of the branches and the total heat transfer area are the main influential factors. For the total flow rate of the cold fluid, it has a threshold value, beyond which it has very little influence on the optimized distributions. The Newton method is used to optimize a ten-branch complex thermal network system. This method is efficient to optimize the problem in this paper, and its computing time is very short. This work was supported by Tsinghua University Initiative Scientific Research Program.

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Hou Z Q, Hu J G. Thermal Control Technology in Spacecraft (in Chinese). Beijing: Chinese Scientific and Technological Press, 2007 Zhang X R, Ren J X, Liang X G, et al. Optimization of the thermal network in parallel connect ion of manned spacecraft (in Chinese). J Tsinghua Univ (Sci & Tech), 2002, 42(4): 462–465 Patel V, Barido R, Johnson B, et al. Development of the internal thermal control system (ITCS) for International Space Station (ISS). 31st International Conference on Environmental Systems Orlando, Florida July 9-12, 2001 (SAE2001-01-2332) Mark T H, David W P. Preliminary design of the space station internal thermal control system. SAE Paper, 1985, 96(6): 712–723 Cheng X T, Xu X H, Ren J X, et al. Optimization of the liquid cooling network system in parallel connection with Lagrange multipliers method (in Chinese). J Tsinghua Univ (Sci & Tech), 2008, 48(8): 1359–1361, 1367 Cheng X T, Xu X H, Ren J X, et al. Optimization of the thermal network in parallel connection (in Chinese). J Eng Thermophys, 2008, 29(12): 2125–2127 Guo Z Y, Zhu H Y, Liang X G. Entransy – A physical rate describing heat transfer ability. Int J Heat Mass Transf, 2007, 50(13): 2545– 2556 Guo Z Y, Cheng X G, Xia Z Z. Least dissipation principle of heat transport potential capacity and its application in heat conduction optimization. Chin Sci Bull, 2003, 48(4): 406–410 Cheng X G, Li Z X, Guo Z Y. Heat conduction optimization based on least dissipation principle of heat transport potential capacity (in Chinese). J Eng Thermophys, 2003, 24(1): 94–96 Cheng X T, Liang X G, Guo Z Y. Entransy decrease principle of heat transfer in an isolated system. Chin Sci Bull, (in press) Zhu H Y, Chen Z J, Guo Z Y. Thermal electric analogy experimental research for the extremum principle of entransy dissipation (in Chinese). Prog Nat Sci, 2007, 17(12): 1692–1698 Chen L G, Wei S H, Sun F R. Constructal entransy dissipation minimization for ‘volume-point’ heat conduction. J Phys D-Appl Phys, 2008, 41: 195506 Chen L G, Wei S H, Sun F R. Constructal entransy dissipation minimization of an electromagnet. J Appl Phys, 2009(105): 094906 Xie Z H, Chen L G, Sun F R. Constructal optimization for geometry of cavity by taking entransy dissipation minimization as objective. Sci China Ser E-Tech Sci, 2009, 52(12): 3413–3504 Wei S H, Chen L G, Sun F R. “Volume-Point” heat conduction constructal optimization with entransy dissipation minimization objective based on rectangular element. Sci China Ser E-Tech Sci, 2008, 51(8): 1283–1295 Cheng X T, Xu X H, Liang X G. Homogenization of temperature field and temperature gradient field. Sci China Ser E-Tech Sci, 2009, 52(10): 2937–2942 Chen Q, Ren J X. Generalized thermal resistance for convective heat transfer and its relation to entransy dissipation. Chin Sci Bull, 2008, 53(23): 3753–3761 Wu J, Cheng X G, Meng J A, et al. Potential capacity dissipation extremum and entropy generation minimization in laminar convective heat transfer (in Chinese). J Eng Thermophys, 2006, 27(1): 100–102 Chen Q, Wang M R, Pan N, et al. Optimization principles for convective heat transfer. Energy, 2009, 34(9): 1199–1206 Wu J, Liang X G. Application of entransy dissipation extremum

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28

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37

38 39

40

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principle in radiative heat transfer optimization. Sci China Ser ETech Sci, 2008(51): 1306–1314 Cheng X T, Liang X G. Entransy flux of thermal radiation and its application to enclosures with opaque surfaces. Int J Heat Mass Transf 2011, 54(1-3): 269–278 Liu X B, Meng J A, Guo Z Y. Thermal resistance analysis for the heat exchangers based on entransy dissipation. Prog Nat Sci, 2008, 18(10): 1186–1190 Liu X B, Guo Z Y. A novel method for heat exchanger analysis. Acta Phys Sin, 2009, 58(7): 4766–4771 Liu X B, Meng J A, Guo Z Y. Entropy generation extremum and entransy dissipation extremum for heat exchanger optimization. Chin Sci Bull, 2009, 54(6): 943–947 Xia S J, Chen L G, Sun F R. Optimization for entransy dissipation minimization in heat exchanger. Chin Sci Bull, 2009, 54(19): 3587– 3595 Guo J F, Chen L, Xu M T. Entransy dissipation number and its application to heat exchanger performance evaluation. Chin Sci Bull, 2009, 54(15): 2708–2713 Xia S J, Chen L G, Sun F R. Entransy dissipation minimization for liquid-solid phase change processes. Sci China Ser E-Tech Sci, 2010, (53)4: 960–968 Cheng X T, Xu X H, Liang X G. Homogenization of temperature field for thermal radiator in space (in Chinese). J Eng Thermophys, 2010, 31(6): 1031–1033 Cai D Y, Bai F S. Modern Scientific Calculation (in Chinese). Beijing: Science Press, 2000 Bejan A. Constructal-theory network of conducting paths for cooling a heat generating volume. Int J Heat Mass Transf, 1997, 40(4): 779– 816 Bejan A. Entropy Generation through Heat and Fluid Flow. New York: John Wiley & Sons, 1982 Bejan A. A study of entropy generation in fundamental convective heat transfer. ASME J Heat Transf, 1979, 101: 718–725 Bejan A. Second-law analysis in heat transfer and thermal design. Adv Heat Transf, 1982, 15: 1–58 Poulikakos D, Bejan A. Fin geometry for minimum entropy generation in forced convection. ASME J Heat Transf, 1982, 104: 616–623 Erek A, Dincer I. An approach to entropy analysis of a latent heat storage module. Int J Therm Sci, 2008, 47(8): 1077–1085 Chen L, Wu C, Sun F. Finite time thermodynamic optimization or entropy generation minimization of energy systems. J Non-Equilib Thermodyn, 1999, 24(4): 327–359 Zhou S B, Chen L G, Sun F R. Constructal optimization for a solid-gas reactor based on triangular element. Sci China Ser E-Tech Sci, 2008, 51(9): 1554–1562 Zeng D L, Ao Y, Zhang X M, et al. Engineering Thermodynamics (in Chinese). Beijing: High Eduction Press, 2002. 83–88 Cheng X G, Meng J A, Guo Z Y. Potential capacity dissipation minimization and entropy generation minimization in heat conduction optimization (in Chinese). J Eng Thermophys, 2005, 26(6): 1034–1036 Shah R K, Skiepko T. Entropy generation extrema and their relationship with heat exchanger effectiveness—Number of transfer unit behavior for complex flow arrangements. J Heat Transf, 2004, 126(6): 994–1002