Heat exchangers are very important and widely used in thermal engineering. For the analyses and design of heat exchangers, Guo et al.^[1,2] proposed a parameter, the uniformity factor of temperature difference field. For a two-stream heat exchanger, its definition is^[1–3] 1 where T₁ and T₂ are the local temperatures of the streams, is the local temperature difference, A is the heat transfer area, and [0,1]. When the number of heat transfer units (NTU) and the ratio of the heat capacity flow rates are fixed, larger uniformity factor of temperature difference field leads to larger heat exchanger effectiveness. Especially, = const when f = 1. This is the uniformity principle of temperature difference field, which has been verified with many cases.^[1–8] For instance, Guo et al.^[1] showed the validity of the principle with theoretical analyses and experiments in two-stream heat exchangers. Gao et al.^[5] proved the principle directly in multi-pressure condenser. Fu and Zhang^[8] gave a rigorous proof for the principle with deductive method.

The principle finds many applications in engineering because it is very convenient for designing and optimizing heat exchangers. For instance, if some local temperature differences in one heat exchanger are obviously higher than other places, the principle tells us that our technical design should decrease the local temperature differences to increase the heat exchanger effectiveness. However, it should be pointed out that the principle is not from the viewpoint of the irreversibility of heat transfer, but is intuitive and phenomenological. Therefore, the physical basis of the principle should be investigated. In the entransy theory,^[9] it has been proved that the concept of entransy dissipation can describe the irreversibility of heat transfer.^[10,11] With this theory, Song et al.^[6] analyzed one-dimensional two-stream and three-stream heat exchangers and proved that the principle holds when the heat transfer rate or the entransy dissipation rate is fixed. In the analyses of two-stream heat exchangers, Cheng et al.^[4] set up the mathematical relationship between the uniformity factor of temperature difference field and the heat exchanger effectiveness with the help of the entransy theory, and found that larger uniformity factor of temperature difference field always corresponds to smaller entransy-dissipation-based thermal resistance. Therefore, larger uniformity factor of temperature difference field means better heat transfer, and the entransy theory can be the physical basis of the intuitive and phenomenological principle.

Furthermore, it is obvious that the application scope of the principle is limited because it can only be used to analyze and optimize heat exchangers. In this paper, we try to extend the application scope of the principle, and analyze the applicability of the principle to the heat transfer processes in the thermal management system of spacecrafts. The cooling problem of thermal components, the one-stream heat exchanger networks, the volume-to-point heat conduction optimization for cooling electronic devices, and the radiative heat transfer optimization are analyzed. The physical basis of the principle in the new application cases is also discussed.

In the thermal management system of spacecrafts, the cooling of thermal components may be simplified to the steady heat transfer system shown in Fig. 1. There are n thermal components from which heat is released to the environment at temperature T₀. For the i-th thermal component, the released heat flow rate is Q_i, and the thermal conductance between the thermal component and the environment is K_i. First, we can assume that the released heat flow rates are fixed, and the limiting condition is 2 where K is the total thermal conductance. In this case, the distribution of the total thermal conductance is optimized to obtain the minimum average temperature of the thermal components weighted by released heat flow rate 3 Considering the energy conservation, we have 4 Therefore 5 To find the minimum value of Eq. (5) with the limitation condition of Eq. (2), a function can be set up 6 where is the Lagrange multiplier. The variations that lead to the minimum average temperature should satisfy 7 Then, we can obtain that 8 Substituting Eq. (8) into Eq. (2) gives 9 The distribution of the total thermal conductance is obtained 10

Fig. 1.

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	Fig. 1. A thermal system with n thermal components.

With Eq. (10), the heat transfer temperature difference of the i-th thermal components can also be obtained 11 When the heat transfer of the system is optimized, it can be seen that the heat transfer temperature differences of thermal components equal each other.

As below, a numerical example is presented. Assume that there are three thermal components in the system, the heat transfer rates Q₁ = 300 W, Q₂ = 150 W, and Q₃ = 50 W, the total thermal conductance K = 20 W/K and the temperature T₀ = 300 K. For the system, the uniformity factor of temperature difference field can be defined as 12 In this case, the variations of the average temperature of the thermal components weighted by released heat flow rate and the uniformity factor of temperature difference field with the distribution of the total thermal conductance are calculated and shown in Fig. 2. The results show that the uniformity factor of temperature difference field and the average temperature have opposite variation tendencies. Especially, when the thermal conductances K₁ =12 W/K, K₂ = 6 W/K, and K₃ = 2 W/K, the average temperature reaches the minimum value, 325 K, and the uniformity factor of temperature difference field reaches the maximum value, 1. Therefore, the theoretical analyses have been verified by the numerical results.

Fig. 2.

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	Fig. 2. Variations of the average temperature of the thermal components weighted by released heat flow rate and the uniformity factor of temperature difference field with the distribution of the total thermal conductance. (a) Variation of the uniformity factor of temperature difference field. (b) Variation of the average temperature of the thermal components.

Second, we can assume that the thermal conductance of each thermal component is given, and the limiting condition is 13 where Q is the total released heat flow rate. In this case, the distribution of the total heat flow rate can also be optimized to obtain the minimum value of Eq. (5). We can also set up a function for this problem 14 where is the Lagrange multiplier. The optimized distribution of the total released heat flow rate should satisfy 15 Then, we can obtain that 16 Substituting Eq. (16) into Eq. (13) gives 17 So, we have 18

With Eq. (18), we can also have 19 The heat transfer temperature differences of thermal components also equal each other when the optimal heat transfer is obtained.

As below, a numerical example can also be presented. We can also assume that there are three thermal components, the thermal conductance K₁ = 6 W/K, K₂ = 3 W/K, and K₃ = 1 W/K, the total heat transfer rate Q = 600 W, and the temperature T₀ = 300 K. In this case, the variations of the average temperature of the thermal components weighted by released heat flow rate and the uniformity factor of temperature difference field with the distribution of the total released heat flow rate are calculated and shown in Fig. 3. It is very clear that the uniformity factor of temperature difference field and the average temperature have opposite variation tendencies. When the heat transfer rates Q₁ = 360 W, Q₂ = 180 W, and Q₃ = 60 W, the results show that the average temperature reaches the minimum value, 360 K, and the uniformity factor of temperature difference field reaches the maximum value, 1. The theoretical analyses have also been verified by the numerical results.

Fig. 3.

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	Fig. 3. Variations of the average temperature of the thermal components weighted by released heat flow rate and the uniformity factor of temperature difference field with the distribution of the total released heat flow rate. (a) Variation of the uniformity factor of temperature difference field. (b) Variation of the average temperature of the thermal components.

It can be seen that the theoretical and numerical optimization results of the cases above all lead to uniform heat transfer temperature difference fields. Therefore, the uniformity principle of temperature difference field holds in these heat transfer optimization problems.

Heat exchanger networks are widely used in the thermal management system of spacecrafts. As shown in Fig. 4, we can analyze a simple one-stream series-wound heat exchanger network. In the network, the n thermal components are cooled by the cold stream with mass flow rate m and inlet temperature T_in. The specific heat capacity and the outlet temperature of the stream are c and T_out, and the released heat flow rate and the corresponding heat transfer area of the i-th thermal component are Q_i and A_i, respectively. We assume that the limiting conditions of the system are Eq. (13) and 20 where A is the total heat transfer area. In this case, the distributions of the total heat transfer area and the total released heat flow rate are optimized at the same time to obtain the minimum value of Eq. (3).

Fig. 4.

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	Fig. 4. A one-stream series-wound heat exchanger network.[13]

For the i-th thermal component, we can assume that^[12,13] 21 where k_i is the heat transfer coefficient, k is a constant, the value of α is between 0.5 and 0.8 (0.5 and 0.8 correspond to the laminar flow and the turbulent flow of the stream, respectively). Therefore, we have 22 where is the inlet temperature of the stream for the i-th thermal component 23 Then, we can obtain that 24

Based on the entransy theory,^[9] we have proved that the minimization of Eq. (3) is equivalent to the entransy dissipation minimization.^[13] Therefore, we can find the optimal distributions of the total heat transfer area and the total released heat flow rate that lead to the entransy dissipation minimization below. For the i-th thermal component, the entransy dissipation rate is 25 where and are the inlet entransy flow rate and outlet entransy flow rate in the heat transfer process of the i-th thermal component, and is the outlet temperature of the stream. The entransy dissipation rate of the system is 26 To find the minimum value of Eq. (26) with the limiting condition expressed by Eqs. (13) and (20), a function can be set up here 27 where and are the Lagrange multipliers. The optimal distribution of the total heat transfer area and the total released heat flow rate should satisfy 28 So, the optimal distribution of the total released heat flow rate gives 29 The entransy dissipation rate of the i-th thermal component can be rewritten as 30 where is the equivalent heat transfer temperature difference of the thermal component. Considering Eqs. (25) and (29), we can obtain that 31 From the derivation above, it can be seen that equation (31) also holds when the heat transfer areas of the thermal components are fixed and only the distribution of the total released heat flow rate is optimized. If the one-stream series-wound heat exchange network is treated as a heat exchanger, the temperature difference field is uniform. Therefore, the uniformity principle of temperature difference field is also tenable.

Here, a numerical example can also be presented. Assume that there are three thermal components, the heat capacity flow rate cm = 100 W/K, the heat transfer coefficient k, the mass flow rate m and the heat capacity c satisfy , the heat transfer areas A₁ = 0.5 m², A₂ = 0.4 m², and A₃ = 0.1 m², the inlet temperature of the stream T_in = 300 K, and the total heat transfer rate Q = 1000 W. The values of the parameters above are given according to the relevant values in the thermal control systems of spacecrafts.^[12] In this case, the uniformity factor of temperature difference field can be defined as 32 The variations of average temperature of the thermal components weighted by released heat flow rate and the uniformity factor of temperature difference field with the distribution of the total released heat flow rate are calculated and shown in Fig. 5. When the distribution of the total released heat flow changes, the average temperature of the thermal components and the uniformity factor of temperature difference field also have opposite variation tendencies. Obviously, the numerical results can verify the theoretical analyses above.

Fig. 5.

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Fig. 5. Variations of the average temperature of the thermal components weighted by released heat flow rate and the uniformity factor of temperature difference field with the distribution of the total released heat flow rate for the one-stream heat exchanger network. (a) Variation of the uniformity factor of temperature difference field. (b) Variation of the average temperature of the thermal components.

In the thermal management system of spacecrafts, there are many electronic devices that should be cooled, and the cooling problem may be simplified to be the volume-to-point heat conduction problem. As shown in Fig. 6, there is an internal heat source in the two-dimensional region, and the generated heat can only be transferred to the environment through a “point” boundary with temperature T₀ on one boundary. In order to decrease the average temperature of the region weighted by heat flow, a certain amount of high thermal conductivity material can be inserted in the region. As the amount of the high thermal conductivity material is fixed, we can find an optimal distribution of the limited high thermal conductivity material to minimize the average temperature of the heated region 33 where V is the volume, q_L is the local inner heat source, and T_L is the local temperature. It can be seen the average temperature weighted by heat flow equals the volume average temperature when the inner heat source is distributed uniformly.

Fig. 6.

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	Fig. 6. Volume-to-point heat conduction problem.[9,14]

The limiting condition of the problem is 34 35 where K_c is a constant, k_c is the local thermal conductivity, and Q is the total heat transfer rate.

With the entransy theory,^[9] we can find that the entransy balance of the system is 36 where G_dis is the entransy dissipation rate, and G_in and G_out are the entransy flow rates into and out of the system, respectively. As the temperature T₀ and the generated heat in the system are fixed, it can be seen that the minimization of Eq. (36) is equivalent to the entransy dissipation minimization.

In heat conduction, the entransy dissipation rate can be expressed as^[9,14] 37 Therefore, to find the minimum value of Eq. (37) with the limiting condition expressed by Eq. (35), we can set up a function 38 where is the Lagrange multiplier. The optimal distribution of the thermal conductivity should satisfy 39 Therefore, we have^[9,14] 40 Then, with Eqs. (34) and (39), we can obtain that 41 Obviously, there is 42 It can be seen that the temperature gradient field is uniform. Obviously, the local temperature gradient is the expression of the temperature difference in the micro-body. From this point of view, the uniformity principle of temperature difference field also holds here. As the volume-to-point heat conduction problem has been analyzed many times, we do not give any numerical example here.

The optimization analyses of radiative heat transfer processes also widely exist in the thermal management system of spacecrafts. Assume that there are n thermal components on a flat plate, and the thermal components release heat flow into the environment at temperature T₀ only through radiative heat transfer. For the i-th thermal component, the temperature is T_i, the released heat flow rate is Q_i, the heat transfer area is A_i, and the thermal emissivity is , respectively. In radiative heat transfer, the driving potential is ,^[15] where σ is the Stefan-Boltzmann constant. Therefore, we can find the optimal design that leads to the minimum equivalent thermal potential weighted by released heat flow rate 43 where U_i is the thermal potential of the i-th thermal component.

First, we assume that the limiting condition is 44 In this case, we have 45 where U₀ is the thermal potential of the environment, which equals . So, 46 and 47 To find the minimum value of Eq. (47) with the limiting condition expressed by Eq. (44), we can set up a function 48 The optimal distribution of E_i should satisfy 49 where is the Lagrange multiplier. Then, we can obtain that 50 51 Considering the definition of thermal potential, we can see that the temperatures and the heat transfer temperature differences of the thermal components are all the same. Obviously, the uniformity principle of temperature difference field is also tenable here.

Furthermore, we can analyze a case in which the values of E_i are fixed, and the limiting condition is expressed by Eq. (13). In this case, we can also set up a function 52 where is the Lagrange multiplier. The optimal distribution of the total released heat flow rate should satisfy 53 Therefore, we can obtain that 54 55 In this case, the thermal potentials, the temperatures, and the heat transfer temperature differences of the thermal components are also distributed uniformly. Hence, the uniformity principle of temperature difference field also holds here.

For the two optimization problems discussed above, it can be seen that the mathematical expressions are very similar to those in Section 2, and the variation tendencies of the variables are also similar to each other. Therefore, we do not present numerical examples for the optimization problems of radiative heat transfer, either.

In this paper, it is found that the uniformity principle of temperature difference field holds not only for heat exchanger optimization, but also for some other heat transfer optimization problems in the thermal management system of spacecrafts, including the cooling of thermal components, the optimization of a one-stream series-wound heat exchanger network, the volume-to-point heat conduction optimization problem, and the radiative heat transfer optimization. Numerical examples are also presented, and the results have verified the theoretical analyses.

The uniformity principle of temperature difference field is intuitive and phenomenological, and reflects the characteristic of the distribution of potential in the heat transfer optimization cases with specific constraints and design objectives. However, it is shown that the uniformity principle of temperature difference field is consistent with the entransy theory. Therefore, the entransy theory can work as the physical basis of the principle and provide the corresponding explanations of the physical mechanisms.