Improvement of Constructal Optimization for “Volume-Point” Heat Conduction Based on Uniformity Principle of Temperature Difference Fields

Abstract

The uniformity principle of temperature difference fields (TDFs) is applied in this study to improve the constructal optimization for “volume-point” heat conduction based on entransy dissipation rate (EDR) minimization without the premise of an optimal last-order construct, and the constructal optimization algorithm based on EDR minimization is simplified in this paper. The results further prove that the uniformity principle of TDF is consistent with entransy theory. The constructal optimization of “volume-point” heat conduction based on EDR minimization is conducted not only to lower the average temperature but also to obtain a more uniform TDF distribution. Through comparing the optimal results based on EDR minimization without the premise of an optimal last-order construct with those based on maximum temperature difference (MTD) minimization, some criteria and formulas for designing conductivity paths based on EDR minimization and MTD minimization are proposed, and the idea and method of improving constructal optimization via the variational principle are proposed.

1. Introduction

Due to the limitations of the efficiency of electronic devices, nearly 80% of the electric power dissipation of electronic devices becomes waste heat. When the temperature of electronic devices increases by 10 °C, the reliability decreases by 50% [1]. The thermal failures of electronic equipment caused by high temperatures account for more than 55% of the failures of electronic equipment, and the heat dissipation of electronic devices with a high heat flux has become a bottleneck restricting their development [2]. There is an urgent need to improve and develop a series of heat transfer control and enhancement methods.

Constructing an efficient heat transfer channel between a device and an external cooler and efficiently transferring the dissipative waste heat generated by the device to the external radiator is the key to the thermal management of electronic equipment [3]. The main methods include filling the heating body with high-thermal-conductivity materials [4], constructing cooling medium channels [5,6], burying or affixing heat pipes [7,8], etc. A lot of studies have been carried out on the layout and optimization design of high-heat-transfer channels. The main methods are topology optimization [9,10], bionic optimization [11,12], optimization designs based on fractal theory [13,14], and constructal theory [15,16,17,18,19,20].

Constructal theory was put forward by Bejan [16], which can be expressed by the constructal law [17] as follows: “For a finite-size system to persist in time (to live) its configuration must change such that it provides easier access to its currents”. The statement of constructal law can be further simplified as follows [18]: “The structures of matters come from their optimal performance”. Compared with topology optimization, bionic optimization, and fractal theory, constructal theory combines the formation of matter’s structures with the least-action principle and can be used as a guideline in engineering practice, which has the advantage of explaining the formation mechanism of matter’s structures. Constructal theory has been widely used in various fields of nature, engineering, and society [17,18,19,20]. The idea of constructal law was first used in the design of high-conductivity paths in a rectangular uniform heat generating area, and tree-like network high-conductivity paths were constructed through assembling so that the maximum thermal resistance of the heat-generating area was minimized [15]. Based on this, a lot of follow-up studies have been carried out. For the cross-section of high-conductivity paths, Ledezma et al. [21] proposed the continuous-variable cross-section high-conductivity path model to minimize thermal resistance, which was extended to the discrete-variable cross-section high-conductivity path model [22] and the continuous-and-discrete-variable cross-section high conductivity path model [23]. The rectangular-element model was extended to the triangular-element model [24], the trapezoidal-element model [25], the disk-shape model [26], the leaf-shape model [27], etc. The tree-like network was extended to the fork-shaped network [28], X-shaped network [29], Y-shaped network [30], phi- and psi-shaped networks [31], and the tree-like network without the premise of an optimal last-order construct [32,33,34]. The problems of heat conduction were extended to heat convection [35,36] and heat radiation [37], heat transfer in porous media [38], and heat transfer at the micro- and nanoscales [39]. The problems of uniform heat generation were extended to the problems of nonuniform heat generation [40]. The optimization objective was extended from the maximum temperature difference (MTD) minimization problem to the entransy dissipation rate (EDR) minimization [23,33,34], the entropy generation minimization [41,42], and various multi-objective optimization problems [43,44].

The uniformity principle of temperature difference fields (TDFs) was firstly proposed for the optimization of heat exchangers [45], and it was a phenomenological principle. It can be expressed as [46]: “The more uniform the TDF, the higher the effectiveness of heat exchanger for the fixed N_tu and C_r”. Cheng et al. [47] and Fu et al. [48] proved the correctness and validity of this principle. Guo et al. [49] put forward the entransy theory and proved by strict mathematics that the most uniform temperature gradient in an area corresponds to the lowest average temperature in the area. Cheng et al. [50] proved that the uniformity principle of TDFs was also effective in other heat transfer optimization problems, and it was consistent with entransy theory. From the studies mentioned above, constructal optimizations based on entransy theory can also be carried out using the uniformity principle of TDFs. Constructal optimizations based on EDR minimization have been widely discussed in the literature [18,23,31,33]. However, whether these studies have obtained a more uniform temperature difference field has not been discussed further. In this paper, the uniformity principle of TDFs is applied to improve the “volume-point” heat conduction optimization without the premise of an optimal last-order construct proposed in refs. [32,33,34], and the optimization algorithm based on the EDR minimization is simplified. The consistency between the uniformity principle of TDFs and entransy theory is further verified by constructal theory. Some design criteria and formulas for high-conductivity path optimization are proposed, and some new ideas and methods to improve constructal optimization methods are proposed.

2. Constructal Heat Conduction Optimization Based on the Uniformity Principle of TDFs

2.1. Element Area

As sketched in Figure 1, a rectangular heating area ($H_0$ × $L_0$) penetrated by a high-conductivity path is considered in this study, in which the heat is generated uniformly at a constant rate $q^{‴}$. The area of the rectangle $A_0$ is fixed, while the width–length ratio $H_0/L_0$ can be varied. The heat conductivity rate of the heating body is $k_0$. The heat fluxes converge into the high-conductivity path with a width of $D_0\left(x\right)$ (the thermal conductivity of the path is $k_p$, $k_p>>k_0$), and it flows out from the $M_0$ Point, except for the rest of the boundaries of $A_0$, which are adiabatic. It is assumed that $k_0/k_p<<{\varphi }_0<<1$ (${\varphi }_0=A_{p,0}/A_0$), $A_{p,0}$ is the area occupied by the high conductivity path, and the heat flux generated in the $k_0$ material can be simplified to first conduct along the y-axis and then flow out through the high-conductivity path $D_0\left(x\right)$. [21]. Obviously, the heat flux in the high-conductivity path increases along the x-axis direction, and the width of the high-conductivity path should be a continuous-variable cross-section.

Considering the symmetry of the area, the case of y > 0 is analyzed, and the case of y < 0 can be obtained by replacing y with –y.

The temperature gradient in the $k_0$ material is [21]:

$$\left(\frac{dT}{dy}\right)=-\frac{q^{‴}}{k_0}\left(\frac{H_0}{2}-y\right)$$

(1)

The temperature gradient in the $k_p$ material is [21]:

$$\left(\frac{dT}{dx}\right)=-\frac{q^{‴}{H}_{0}x}{k_p{D}_0\left(x\right)}$$

(2)

According to the uniformity principle of TDFs, the heat transfer performance is the best when the temperature gradient is uniform along the heat flux direction. In other words, the heat transfer performance is the best when the temperature gradient is constant. No variable can be used to optimize the temperature gradient in Equation (1), but a constant temperature gradient can be obtained by varying the width of the high-conductivity path, $D_0\left(x\right)$. If the temperature gradient in Equation (2) is constant and equal to C, the width of the high-conductivity path with a uniform temperature gradient is:

$$D_0\left(x\right)=-\frac{q^{‴}{H}_{0}x}{k_{p}C}$$

(3)

Substituting $A_{p,0}={\displaystyle {\int }_0^{L_0}{D}_0\left(x\right)}dx$ and ${\varphi }_0=A_{p,0}/A_0$ into Equation (3) yields:

$$D_0\left(x\right)=2{\varphi }_0\frac{H_0}{L_0}x$$

(4)

Substituting Equation (4) into Equation (2) yields:

$$\left(\frac{dT}{dx}\right)=-\frac{q^{‴}{L}_0}{2k_p{\varphi }_0}$$

(5)

The width of the high-conductivity path, $D_0\left(x\right)$, increases linearly along the x-axis. The TDF distribution in the element according to the uniformity principle of TDFs can be obtained from Equations (1) and (5), that is:

$$T\left(x,y\right)-T_{M0}=\frac{q^{‴}}{2k_0}\left(H_{0}y-y^2\right)+\frac{q^{‴}{L}_0}{2k_p{\varphi }_0}\left(L_0-x\right)$$

(6)

Equation (6) is consistent with the TDF for EDR minimization in ref. [34], and the EDR of the element ${\dot{E}}_{h\varphi ,A_0}$ can be calculated as:

$$\begin{array}{l}{\dot{E}}_{h,A_0}={\displaystyle \underset{A_0}{\int }\dot{q}\cdot \nabla Tds}=2{\displaystyle {\int }_0^{H_0/2}{\displaystyle {\int }_0^{L_0}{q}^{‴}\left[\frac{q^{‴}}{2k_0}\left(H_{0}y-y^2\right)+\frac{q^{‴}{L}_0}{2k_p{\varphi }_0}\left(L_0-x\right)\right]}}dxdy\\ =\frac{{q^{‴}}^2{A}_0^2}{4}\left(\frac{1}{3k_0}\frac{H_0}{L_0}+\frac{1}{k_p{\varphi }_0}\frac{L_0}{H_0}\right)\end{array}$$

(7)

${\dot{E}}_{h\varphi ,A_0}$ can be minimized with respect to $H_0/L_0$, and the optimal width–length ratio, ${\left(H_0/L_0\right)}_{opt}$, the minimum, EDR ${\dot{E}}_{h\varphi ,A_0,m}$, and the corresponding dimensionless minimum average temperature difference, $\mathsf{\Delta }{\tilde{\overline{T}}}_{0,\min }$, are also consistent with ref. [23].

$${\left(\frac{H_0}{L_0}\right)}_{opt}=\sqrt{\frac{3k_0}{k_p{\varphi }_0}}$$

(8)

$${\dot{E}}_{h\varphi ,A_0,m}=\frac{\sqrt{3}{A}_0^2{q^{‴}}^2}{6\sqrt{k_0{k}_p{\varphi }_0}}$$

(9)

$$\mathsf{\Delta }{\tilde{\overline{T}}}_{0,\min }=\frac{\mathsf{\Delta }{\overline{T}}_{0,\min }}{q^{″}{A}_0/k_0}=\frac{\sqrt{3}}{6}{\left(\frac{k_0}{k_p{\varphi }_0}\right)}^{1/2}$$

(10)

The dimensionless minimum MTD, $\mathsf{\Delta }{\tilde{\mathrm{T}}}_{0,\min }$, is:

$$\mathsf{\Delta }{\tilde{\mathrm{T}}}_{0,\min }=\left(\frac{T_{\max 0}-T_{M0}}{q^{‴}{A}_0/k_0}\right)=\frac{1}{2}{\left(\frac{k_0}{k_p{\varphi }_0}\right)}^{1/2}$$

(11)

Equation (11) is consistent with the results of the MTD minimization with a fixed-cross-section high-conductivity path in ref. [15], but it is less than the results of the MTD minimization with a variable-cross-section high-conductivity path in ref. [21] which is: $\mathsf{\Delta }{\tilde{\mathrm{T}}}_{0,\min }=\frac{1}{3}\sqrt{\frac{2}{\tilde{k}{\varphi }_0}}$.

2.2. First-Order Construct

The first-order construct is obtained by assembling $2n_1$($n_1>>1$) elements along a new conductivity path, $D_1$, as sketched in Figure 2. The area of the first-order construct is $A_1$($A_1=H_1\times L_1$), and the boundaries are adiabatic, except for $M_0$.

According to the conclusions in refs. [32,33,34], assembling the next-order construct by an optimal last-order construct does not achieve better performance. $H_0/L_0$ and $D_0(x)$ are re-used as optimization variables in the optimization of the first-order construct. According to the uniformity principle of TDFs, the formula of $D_0(x)$ is unchanged, as per Equation (4), and the EDR of the element ${\dot{E}}_{h\varphi ,A_0}$ is the same as Equation (7). In the high-conductivity path, $D_1$, the heat flux increases discretely along the x-axis with the increase in the inflow nodes of the element. The high-conductivity path is divided into $n_1$ segments by $n_1$ nodes, and the width of each segment along the x-axis is defined as $D_{1,1},\begin{array}{cccc}{D}_{1,2},& \cdots ,& D_{1,n1-1},& D_{1,n1}\end{array}$. The formula of the temperature gradient in the high-conductivity path $D_1$ is:

$${\left(\frac{dT}{dx}\right)}_{D1,i}=\frac{i2q^{‴}{A}_0}{k_p{D}_{1,i}},\begin{array}{cc}& i=1,\cdots ,n_1\end{array}$$

(12)

According to the uniformity principle of TDFs, the temperature gradients of each segment in the high-conductivity path, $D_1$, are equal:

$${\left(\frac{dT}{dx}\right)}_{D_{1,i-1}}={\left(\frac{dT}{dx}\right)}_{D_{1,i}}\begin{array}{cc}& i=2,\cdots n_1\end{array}$$

(13)

The width ratio of each segment is:

$$D_{1,i-1}:D_{1,i}=i-1:i\begin{array}{cc}& i=2,\cdots ,n_1\end{array}$$

(14)

The EDR, ${\dot{E}}_{h\varphi ,D_1}$, and the MTD, $\mathsf{\Delta }{T}_{\max ,D_1}$, along the high-conductivity path, D₁, can be calculated with Equations (12) and (14).

$${\dot{E}}_{h\varphi ,D_1}=\frac{{\left(2q^{‴}{A}_0\right)}^2{H}_0}{k_p{D}_{1,1}}\cdot \frac{n_1^2}{2}$$

(15)

$$\mathsf{\Delta } T_{\max ,D_1}=\frac{q^{‴}{A}_0{H}_0\left(2n_1-1\right)}{k_p{D}_{1,1}}$$

(16)

The EDR, ${\dot{E}}_{h\varphi ,A_1}$, and the MTD, $\mathsf{\Delta }{T}_{\max ,A_1}$, of the first-order construct area, A₁, are:

$${\dot{E}}_{h\varphi ,A_1}=\frac{{q^{‴}}^2}{2}\frac{A_1^2}{4n_1}\left(\frac{1}{3k_0}\frac{H_0}{L_0}+\frac{1}{k_p{\varphi }_0}\frac{L_0}{H_0}\right)+\frac{{q^{‴}}^2{H}_0}{k_p{D}_{1,1}}\frac{A_1^2}{2}$$

(17)

$$\mathsf{\Delta }{T}_{\max ,A_1}=\frac{q^{‴}{A}_1}{4n_1}\left(\frac{1}{4k_0}\frac{H_0}{L_0}+\frac{1}{k_p{\varphi }_0}\frac{L_0}{H_0}\right)+\frac{q^{‴}{A}_1{H}_0\left(2n_1-1\right)}{2n_1{k}_p{D}_{1,1}}$$

(18)

Supposing that the area of the high-conductivity material in the first-order construct is $A_{p,1}$ and the proportion is ${\varphi }_1$(${\varphi }_1=A_{p,1}/A_1$), the proportion of $D_1$ is:

$${\varphi }_{D_1}={\varphi }_1-{\varphi }_0=\frac{D_{1,1}\cdot H_0+D_{1,2}\cdot H_0+\cdots +D_{1,n1-1}\cdot H_0+D_{1,n1}\cdot \frac{H_0}{2}}{A_1}=\frac{D_{1,1}{n}_1}{4L_0}$$

(19)

By solving Equation (19), the width $D_{1,1}$ is obtained:

$$D_{1,1}=\frac{4L_0\left({\varphi }_1-{\varphi }_0\right)}{n_1}$$

(20)

Substituting Equation (20) into Equations (17) and (18), respectively, yields:

$${\dot{E}}_{h\varphi ,A_1}=\frac{{q^{‴}}^2{A}_1^2}{8}\left[a\left(\frac{1}{3k_0{n}_1}+\frac{n_1}{k_p\left({\varphi }_1-{\varphi }_0\right)}\right)+\frac{1}{a}\frac{1}{n_1{k}_p{\varphi }_0}\right]$$

(21)

$$\mathsf{\Delta }{T}_{\max ,A_1}=\frac{q^{‴}{A}_1}{4}\left[\frac{1}{a}\frac{1}{k_p{\varphi }_0{n}_1}+a\frac{k_p\left({\varphi }_1-{\varphi }_0\right)+4k_0{n}_1^2-2k_0{n}_1}{4k_p{k}_0\left({\varphi }_1-{\varphi }_0\right)n_1}\right]$$

(22)

where $a=H_0/L_0$. Optimizing Equations (21) and (22) with respect to a yields:

$$a_{h,opt}=\sqrt{\frac{3k_0\left({\varphi }_1-{\varphi }_0\right)}{3n_1^2{k}_0{\varphi }_0+k_p{\varphi }_0\left({\varphi }_1-{\varphi }_0\right)}}$$

(23)

$${\dot{E}}_{h\varphi ,A_1,\min }=\frac{{q^{‴}}^2{A}_1^2}{4k_0}\sqrt{\frac{1}{{\tilde{k}}^2{\varphi }_0\left({\varphi }_1-{\varphi }_0\right)}+\frac{1}{3\tilde{k}{\varphi }_0{n}_1^2}}$$

(24)

$$a_{T,opt}=\sqrt{\frac{4k_0\left({\varphi }_1-{\varphi }_0\right)}{\left(4n_1^2-2n_1\right)k_0{\varphi }_0+k_p{\varphi }_0\left({\varphi }_1-{\varphi }_0\right)}}$$

(25)

$$\mathsf{\Delta }{T}_{\max ,A_1,\min }=\frac{q^{‴}{A}_1}{2k_0}\left(\sqrt{\frac{1}{4\tilde{k}{\varphi }_0{n}_1}+\frac{\left(1-1/\left(2n_1\right)\right)}{{\tilde{k}}^2{\varphi }_0\left({\varphi }_1-{\varphi }_0\right)}}\right)$$

(26)

According to Equations (24) and (26), ${\dot{E}}_{h\varphi ,A_1}$ and $\mathsf{\Delta }{T}_{\max ,A_1}$ can be can be optimized for a second time by $n_1$. When $n_{1,opt}\to \infty$, ${\dot{E}}_{h\varphi ,A_1}$ and $\mathsf{\Delta }{T}_{\max ,A_1}$ achieve their second minimum values:

$${\dot{E}}_{h\varphi ,A_1,mm}=\frac{{q^{‴}}^2{A}_1^2}{4k_0}\sqrt{\frac{1}{{\tilde{k}}^2{\varphi }_0\left({\varphi }_1-{\varphi }_0\right)}}$$

(27)

$$\mathsf{\Delta }{T}_{\max ,A_1,mm}=\frac{q^{‴}{A}_1}{2k_0}\sqrt{\frac{1}{{\tilde{k}}^2{\varphi }_0\left({\varphi }_1-{\varphi }_0\right)}}$$

(28)

$n_{1,opt}\to \infty$ indicates that the width of high-conductivity path, $D_1$, increases linearly along the x-axis, and $D_1\left(x\right)$ can be expressed as:

$$D_1\left(x\right)=2\left({\varphi }_1-{\varphi }_0\right)\frac{H_1}{L_1}x$$

(29)

Substituting $n_{1,opt}\to \infty$ into Equations (23) and (25), respectively, yields $H_0/L_0\to 0$. According to Equations (27) and (28), ${\dot{E}}_{h\varphi ,A_1}$ and $\mathsf{\Delta }{T}_{\max ,A_1}$ can be can be optimized for a third time by ${\varphi }_0$. When ${\varphi }_0={\varphi }_1/2$, ${\dot{E}}_{h\varphi ,A_2}$ and $\mathsf{\Delta }{T}_{\max ,A_2}$ achieve their third minimum values:

$${\dot{E}}_{h\varphi A_1,mmm}=\frac{{q^{‴}}^2{A}_1^2}{2k_p{\varphi }_1}$$

(30)

$$\mathsf{\Delta }{T}_{\max ,A_1,mmm}=\frac{q^{‴}{A}_1}{k_p{\varphi }_1}$$

(31)

The optimal width–length ratio, ${\left(H_1/L_1\right)}_{opt}$, the dimensionless minimum average temperature difference, $\mathsf{\Delta }{\tilde{\overline{T}}}_{1,\min }$, and the dimensionless minimum MTD, $\mathsf{\Delta }{\tilde{\mathrm{T}}}_{1,\min }$, are:

$${\left(\frac{H_1}{L_1}\right)}_{opt}=2$$

(32)

$$\mathsf{\Delta }{\tilde{\overline{T}}}_{1,\min }=\frac{1}{2\tilde{k}{\varphi }_1}$$

(33)

$$\mathsf{\Delta }{\tilde{\mathrm{T}}}_{1,\min }=\frac{1}{\tilde{k}{\varphi }_1}$$

(34)

$\mathsf{\Delta }{\tilde{\overline{T}}}_{1,\min }$ is consistent with the results of the EDR minimization with a variable-cross-section path in ref. [33]. $\mathsf{\Delta }{\tilde{\mathrm{T}}}_{1,\min }$ is consistent with the results of the MTD minimization with a fixed-cross-section path in ref. [32], but it is less than the results of the MTD minimization with variable-cross-section high-conductivity path in ref. [33], which is $\mathsf{\Delta }{\tilde{\mathrm{T}}}_{1,\min }=8/\left(9\tilde{k}{\varphi }_1\right)$.

2.3. Second-Order Construct

The second-order construct is obtained by assembling $2n_2$($n_2>>1$) first-order constructs along a new conductivity path, $D_2$, as sketched in Figure 3. The area of the second-order construct is $A_2$($A_2=H_2\times L_2$), and the boundaries are adiabatic, except for $M_0$.

The EDR of the second-order construct, ${\dot{E}}_{h\varphi ,A_2}$, consists of the total EDR of the elements and the EDR of the high-conductivity paths.

$$\begin{array}{l}{\dot{E}}_{h\varphi ,A_2}={\dot{E}}_{h\varphi ,4n_1{n}_2{A}_0}+{\dot{E}}_{h\varphi ,D_1}+{\dot{E}}_{h\varphi ,D_2}\\ =n_2{n}_1{q^{‴}}^2{A}_0^2\left(\frac{1}{3k_0}\frac{H_0}{L_0}+\frac{1}{k_p{\varphi }_0}\frac{L_0}{H_0}\right)+2n_2\frac{{\left(2q^{‴}{A}_0\right)}^2{H}_0}{k_p{D}_{1,1}}\frac{n_1^2}{2}+\frac{{\left(2q^{‴}{A}_1\right)}^2{H}_1}{k_p{D}_{2,1}}\frac{n_2^2}{2}\end{array}$$

(35)

The MTD of the second-order construct $\mathsf{\Delta }{T}_{\max ,A2}$ is:

$$\begin{array}{l}{\dot{E}}_{h\varphi ,A_2}={\dot{E}}_{h\varphi ,4n_1{n}_2{A}_0}+{\dot{E}}_{h\varphi ,D_1}+{\dot{E}}_{h\varphi ,D_2}\\ =n_2{n}_1{q^{‴}}^2{A}_0^2\left(\frac{1}{3k_0}\frac{H_0}{L_0}+\frac{1}{k_p{\varphi }_0}\frac{L_0}{H_0}\right)+2n_2\frac{{\left(2q^{‴}{A}_0\right)}^2{H}_0}{k_p{D}_{1,1}}\frac{n_1^2}{2}+\frac{{\left(2q^{‴}{A}_1\right)}^2{H}_1}{k_p{D}_{2,1}}\frac{n_2^2}{2}\end{array}$$

(36)

Supposing that the area of the high-conductivity material in the second-order construct is $A_{p,2}$ and the proportion is ${\varphi }_2$(${\varphi }_2=A_{p,2}/A_2$), $D_{2,1}$ can be obtained according to the same calculation method of Equation (19).

$$D_{2,1}=\frac{4n_1{H}_0\left({\varphi }_2-{\varphi }_1\right)}{n_2}$$

(37)

Substituting Equations (20) and (37) into Equations (35) and (36), respectively, yields:

$${\dot{E}}_{h\varphi ,A_2}=\frac{{q^{‴}}^2{A}_2^2}{16k_0}\left[\frac{1}{n_2}\left(\frac{a}{3n_1}+\frac{1}{\tilde{k}{\varphi }_0{n}_{1}a}+\frac{n_{1}a}{\tilde{k}\left({\varphi }_1-{\varphi }_0\right)}\right)+n_2\frac{4}{\tilde{k}\left({\varphi }_2-{\varphi }_1\right)n_{1}a}\right]$$

(38)

$$\mathsf{\Delta }{T}_{\max ,A_2}=\frac{q^{‴}{A}_2}{k_0}\left[\frac{1}{n_2}\left(\frac{a}{32n_1}+\frac{1}{8n_{1}a\tilde{k}{\varphi }_0}+\frac{a\left(2n_1-1\right)}{16\tilde{k}\left({\varphi }_1-{\varphi }_0\right)}\right)+n_2\frac{1}{2n_{1}a\tilde{k}\left({\varphi }_2-{\varphi }_1\right)}-\frac{1}{4n_{1}a\tilde{k}\left({\varphi }_2-{\varphi }_1\right)}\right]$$

(39)

Optimizing Equations (38) and (39) with respect to $n_2$ yields:

$$n_{2,h,opt}=\frac{1}{2}\sqrt{\left({\varphi }_2-{\varphi }_1\right)\left(\frac{\tilde{k}{a}^2}{3}+\frac{1}{{\varphi }_0}+\frac{n_1^2{a}^2}{{\varphi }_1-{\varphi }_0}\right)}$$

(40)

$${\dot{E}}_{h\varphi ,A_2,\min }=\frac{{q^{‴}}^2{A}_2^2}{4k_0}\sqrt{\frac{1}{3n_1^2\tilde{k}\left({\varphi }_2-{\varphi }_1\right)}+\frac{1}{{\tilde{k}}^2{\varphi }_0\left({\varphi }_2-{\varphi }_1\right)n_1^2{a}^2}+\frac{1}{{\tilde{k}}^2\left({\varphi }_2-{\varphi }_1\right)\left({\varphi }_1-{\varphi }_0\right)}}$$

(41)

$$n_{2,T,opt}=\sqrt{\frac{a^2\tilde{k}\left({\varphi }_2-{\varphi }_1\right)}{16}+\frac{\left({\varphi }_2-{\varphi }_1\right)}{4{\varphi }_0}+\frac{\left(2n_1^2-n_1\right)a^2\left({\varphi }_2-{\varphi }_1\right)}{8\left({\varphi }_1-{\varphi }_0\right)}}$$

(42)

$$\mathsf{\Delta }{T}_{\max ,{\mathrm{A}}_2,\min }=\frac{q^{‴}{A}_2}{k_0}\left[\begin{array}{l}2\sqrt{\frac{1}{64n_1^2\tilde{k}\left({\varphi }_2-{\varphi }_1\right)}+\frac{1}{16n_1^2{a}^2\tilde{k}\left({\varphi }_2-{\varphi }_1\right){\varphi }_0}+\frac{1-1/\left(2n_1\right)}{16{\tilde{k}}^2\left({\varphi }_2-{\varphi }_1\right)\left({\varphi }_1-{\varphi }_0\right)}}\\ -\frac{1}{4n_{1}a\tilde{k}\left({\varphi }_2-{\varphi }_1\right)}\end{array}\right]$$

(43)

According to Equations (41) and (43), ${\dot{E}}_{h\varphi ,A_2}$ and $\mathsf{\Delta }{T}_{\max ,A_2}$ can be can be optimized for a second time by $n_1,n_{1}a$ and $n_{2,opt}$. When $n_1,n_{1}a\to \infty$ and $n_{2,opt}\to \infty$, ${\dot{E}}_{h\varphi ,A_2}$ and $\mathsf{\Delta }{T}_{\max ,A_2}$ achieve their second minimum values:

$${\dot{E}}_{h\varphi ,A_2,mm}=\frac{{q^{‴}}^2{A}_2^2}{4k_0}\sqrt{\frac{1}{{\tilde{k}}^2\left({\varphi }_2-{\varphi }_1\right)\left({\varphi }_1-{\varphi }_0\right)}}$$

(44)

$$\mathsf{\Delta }{T}_{\max ,A_2,mm}=\frac{q^{‴}{A}_2}{2k_0}\sqrt{\frac{1}{{\tilde{k}}^2\left({\varphi }_2-{\varphi }_1\right)\left({\varphi }_1-{\varphi }_0\right)}}$$

(45)

According to Equations (44) and (45), ${\dot{E}}_{h\varphi ,A_2}$ and $\mathsf{\Delta }{T}_{\max ,A_2}$ can be optimized for a third time by ${\varphi }_0$ and ${\varphi }_1$. When ${\varphi }_0=0,{\varphi }_1={\varphi }_2/2$, ${\dot{E}}_{h\varphi ,A_2}$ and $\mathsf{\Delta }{T}_{\max ,A_2}$ achieve their third minimum values. The second-order construct then degenerates into the first-order construct.

$${\dot{E}}_{h\varphi ,A_2,mmm}=\frac{{q^{‴}}^2{A}_2^2}{2k_p{\varphi }_2}$$

(46)

$$\mathsf{\Delta }{T}_{\max ,A_2,mmm}=\frac{q^{‴}{A}_2}{k_p\varphi }$$

(47)

The optimal width of the high-conductivity path, $D_2\left(x\right)$, the optimal width–length ratio, ${\left(H_2/L_2\right)}_{opt}$, the dimensionless minimum average temperature difference, $\mathsf{\Delta }{\tilde{\overline{T}}}_{2,\min }$, and the dimensionless minimum MTD, $\mathsf{\Delta }{\tilde{\mathrm{T}}}_{2,\min }$, are the same as with the first-order construct.

3. Comparison and Analyses

The optimization results obtained in this paper are listed in

Table 1. The results of variable-cross-section high-conductivity path without the premise of an optimal last-order construct based on the uniformity principle of TDFs.

Construct Order i	${\mathit{n}}_{\mathit{i},\mathit{o}\mathit{p}\mathit{t}}$	${\left({\mathit{H}}_{\mathit{i}}/{\mathit{L}}_{\mathit{i}}\right)}_{\mathit{o}\mathit{p}\mathit{t}}$	${\left({\mathit{\varphi }}_{\mathit{i}-1}/{\mathit{\varphi }}_{\mathit{i}}\right)}_{\mathit{o}\mathit{p}\mathit{t}}$	${\mathit{D}}_{\mathit{i},\mathit{o}\mathit{p}\mathit{t}}\left(\mathit{x}\right)$	$\mathit{\Delta }{\widetilde{\overline{\mathit{T}}}}_{\mathit{i},\mathbf{min}}$	$\mathit{\Delta }{\tilde{\mathit{T}}}_{\mathit{i},\mathbf{min}}$
Element area		$\sqrt{\frac{3}{\tilde{k}{\varphi }_0}}$		$2{\varphi }_0\frac{H_0}{L_0}x$	$\frac{\sqrt{3}}{6}{\left(\frac{1}{\tilde{k}{\varphi }_0}\right)}^{1/2}$	$\frac{1}{2}{\left(\frac{1}{\tilde{k}{\varphi }_0}\right)}^{1/2}$
First-order construct	$\infty$	2	1/2	$2\left({\varphi }_1-{\varphi }_0\right)\frac{H_1}{L_1}x$	$\frac{1}{2\tilde{k}{\varphi }_1}$	$\frac{1}{\tilde{k}{\varphi }_1}$
Second- and higher-order constructs	Degenerate into first-order construct

, and they were consistent with the results based on EDR minimization presented in ref. [33]. The comparison further verified the consistency between the uniformity principle of TDFs and entransy theory. The “volume-point” heat conduction optimization based on EDR minimization was conducted to not only to lower the average temperature but also to obtain a more uniform TDF. The optimization process showed that the “volume-point” heat conduction optimization algorithm based on EDR minimization could be simplified by using the uniformity principle of TDFs.

The results of the variable-cross-section high-conductivity path without the premise of an optimal last-order construct based on MTD minimization presented in ref. [33] are listed in

Table 2. The results of variable-cross-section high-conductivity path without the premise of an optimal last-order construct based on MTD minimization [33].

Construct Order i	${\mathit{n}}_{\mathit{i},\mathit{o}\mathit{p}\mathit{t}}$	${\left({\mathit{H}}_{\mathit{i}}/{\mathit{L}}_{\mathit{i}}\right)}_{\mathit{o}\mathit{p}\mathit{t}}$	${\left({\mathit{\varphi }}_{\mathit{i}-1}/{\mathit{\varphi }}_{\mathit{i}}\right)}_{\mathit{o}\mathit{p}\mathit{t}}$	${\mathit{D}}_{\mathit{i},\mathit{o}\mathit{p}\mathit{t}}\left(\mathit{x}\right)$	$\mathit{\Delta }{\widetilde{\overline{\mathit{T}}}}_{\mathit{i},\mathbf{min}}$	$\mathit{\Delta }{\tilde{\mathit{T}}}_{\mathit{i},\mathbf{min}}$
Element area		$\frac{4}{3}\sqrt{\frac{2}{\tilde{k}{\varphi }_0}}$		$\frac{3{\varphi }_0{H}_0}{2}{\left(\frac{x}{L_0}\right)}^{1/2}$	$\frac{19\sqrt{2}}{90}{\left(\frac{1}{\tilde{k}{\varphi }_0}\right)}^{1/2}$	$\frac{\sqrt{2}}{3}{\left(\frac{1}{\tilde{k}{\varphi }_0}\right)}^{1/2}$
First-order construct	$\infty$	2	1/2	$\frac{3\left({\varphi }_1-{\varphi }_0\right)H_1}{2}{\left(\frac{x}{L_1}\right)}^{1/2}$	$\frac{31}{60\tilde{k}{\varphi }_1}$	$\frac{8}{9}\frac{1}{\tilde{k}{\varphi }_1}$
Second- and higher-order construct	Degenerate into first-order construct

, in which the dimensionless minimum average temperature difference, $\mathsf{\Delta }{\tilde{\overline{T}}}_{i,\min }$, is derived from the optimal construct. The results of the fixed-cross-section high-conductivity paths without the premise of an optimal last-order construct based on EDR minimization and MTD minimization presented in ref. [34] are listed in

Table 3. The results of fixed-cross-section high-conductivity path without the premise of an optimal last-order construct [34].

Construct Order i	${\mathit{n}}_{\mathit{i},\mathit{o}\mathit{p}\mathit{t}}$	${\left({\mathit{H}}_{\mathit{i}}/{\mathit{L}}_{\mathit{i}}\right)}_{\mathit{o}\mathit{p}\mathit{t}}$	${\left({\mathit{\varphi }}_{\mathit{i}-1}/{\mathit{\varphi }}_{\mathit{i}}\right)}_{\mathit{o}\mathit{p}\mathit{t}}$	${\mathit{D}}_{\mathit{i},\mathit{o}\mathit{p}\mathit{t}}\left(\mathit{x}\right)$	$\mathit{\Delta }{\widetilde{\overline{\mathit{T}}}}_{\mathit{i},\mathbf{min}}$	$\mathit{\Delta }{\tilde{\mathit{T}}}_{\mathit{i},\mathbf{min}}$
Element area		$\frac{2}{\sqrt{\tilde{k}{\varphi }_0}}$		$H_0{\varphi }_0$	$\frac{1}{3}{\left(\frac{1}{\tilde{k}{\varphi }_0}\right)}^{1/2}$	$\frac{1}{2}{\left(\frac{1}{\tilde{k}{\varphi }_0}\right)}^{1/2}$
First-order construct	$\infty$	2	1/2	$H_0\left({\varphi }_1-{\varphi }_0\right)$	$\frac{2}{3\tilde{k}{\varphi }_1}$	$\frac{1}{\tilde{k}{\varphi }_1}$
Second- and higher-order construct	Degenerate into first-order construct

. It is worth noting that with a fixed-cross-section high-conductivity path and without the premise of an optimal last-order construct, the results based on EDR minimization are the same as those based on MTD minimization. By comparing Table 1, Table 2 and Table 3, the following conclusions can be drawn: A lower minimum average temperature difference, $\mathsf{\Delta }{\tilde{\overline{T}}}_{i,\min }$, or a minimum MTD, $\mathsf{\Delta }{\tilde{\mathrm{T}}}_{i,\min }$, can be obtained by optimizing the width of the high-conductivity path $D_i\left(x\right)$; the optimal width of the high-conductivity path, $D_{i,opt}\left(x\right)$,based on MTD minimization is different from that based on EDR minimization. The width of the high-conductivity path, $D_i\left(x\right)$, is an important optimization variable, and it is necessary to further analyze its influence on the two different optimization objectives.

A one-dimensional heat conduction path is sketched in Figure 4. The heat flux, $q^{‴}\left(x\right)$, is transferred through a high-conductivity path, the conductivity rate of which is $k_p$. The length of the high-conductivity path is L, and its width, $D\left(x\right)$, is a variable cross-section. The total area of the high-conductivity path, $A_p$, is a constant. It is assumed that the upper and lower boundaries of the high-conductivity path are not adiabatic and that heat flux varies along the heat conduction direction.

The temperature difference gradient along the heat conduction path is:

$$\frac{dT}{dx}=-\frac{q^{‴}\left(x\right)}{k_{p}D\left(x\right)}$$

(48)

The MTD and EDR can be calculated as follows:

$$\mathsf{\Delta }{T}_{\max }={\displaystyle {\int }_0^L\frac{q^{‴}\left(x\right)}{k_{p}D\left(x\right)}dx}$$

(49)

$${\dot{E}}_{h\varphi }={\displaystyle {\int }_0^L\frac{{q^{‴}}^2\left(x\right)}{k_{p}D\left(x\right)}dx}$$

(50)

Taking Equations (49) and (50) as the objective functions, respectively, and considering the constraint that $A_p={\displaystyle {\int }_0^{L}D\left(x\right)}dx$, the Lagrange functions, ${\mathsf{\Phi }}_T$ and ${\mathsf{\Phi }}_h$, can be set up, respectively:

$${\mathsf{\Phi }}_T={\displaystyle {\int }_0^L\left(\frac{q^{‴}\left(x\right)}{k_{p}D\left(x\right)}+{\lambda }_{T}D\left(x\right)\right)}dx$$

(51)

$${\mathsf{\Phi }}_h={\displaystyle {\int }_0^L\left(\frac{{q^{‴}}^2\left(x\right)}{k_{p}D\left(x\right)}+{\lambda }_{h}D\left(x\right)\right)}dx$$

(52)

where ${\lambda }_T$ and ${\lambda }_h$ are the Lagrange multipliers. By minimizing ${\mathsf{\Phi }}_T$ and ${\mathsf{\Phi }}_h$, respectively, the optimal widths of the high-conductivity path for MTD minimization and EDR minimization are derived as follows:

$$D{\left(x\right)}_{T,opt}=\frac{\sqrt{q^{‴}\left(x\right)}{A}_p}{{\displaystyle {\int }_0^L\sqrt{q^{‴}\left(x\right)}dx}}$$

(53)

$$D{\left(x\right)}_{h,opt}=\frac{q^{‴}\left(x\right)A_p}{{\displaystyle {\int }_0^L{q}^{‴}\left(x\right)dx}}$$

(54)

The corresponding optimal temperature gradient expressions are:

$${\left(\frac{dT}{dx}\right)}_{T,opt}=\frac{-\sqrt{q^{‴}\left(x\right)}{\displaystyle {\int }_0^L\sqrt{q^{‴}\left(x\right)}dx}}{k_p{A}_p}$$

(55)

$${\left(\frac{dT}{dx}\right)}_{h,opt}=\frac{-{\displaystyle {\int }_0^L{q}^{‴}\left(x\right)dx}}{k_p{A}_p}$$

(56)

The following optimal design criteria for the high-conductivity path can be derived from Equations (53)–(56).

(1) The width of the high-conductivity path for the minimum MTD $D{\left(x\right)}_{T,opt}$ is proportional to $\sqrt{q^{‴}\left(x\right)}$, and the corresponding optimal temperature gradient ${\left(dT/dx\right)}_{T,opt}$ is also proportional to $\sqrt{q^{‴}\left(x\right)}$. The design of the high-conductivity path for the minimum MTD can be simplified using Equations (53) and (55).

(2) The width of the high-conductivity path for the minimum EDR $D{\left(x\right)}_{h,opt}$ is proportional to $q^{‴}\left(x\right)$, but the corresponding optimal temperature gradient ${\left(dT/dx\right)}_{h,opt}$ is a constant. No matter how the heat flux $q^{‴}\left(x\right)$ varies, the most uniform TDF can be derived based on EDR minimization. The design of the high-conductivity path for the minimum EDR can be simplified using Equations (54) and (56).

(3) When the upper and lower boundaries of the high-conductivity path are adiabatic, the heat flux $q^{‴}\left(x\right)$ is a constant. The results of the MTD minimization are consistent with that of the EDR minimization, which leads to a more uniform temperature difference field. The optimal width of the high-conductivity path is $D{\left(x\right)}_{opt}=A_p/L$ and the corresponding optimal temperature gradient is ${\left(dT/dx\right)}_{opt}=-\left(q^{‴}L\right)/\left(k_p{A}_p\right)$.

4. Conclusions

The constructal optimization for “volume-point” heat conduction was improved based on the uniformity principle of TDFs in this paper. Without the premise of an optimal last-order construct, the optimization results were the same as those of the EDR minimization, which further verified the consistency between the uniformity principle of TDFs and entransy theory. The “volume-point” heat conduction optimization based on EDR minimization was carried out not only to lower the average temperature but also to obtain a more uniform TDF. The constructal optimization algorithm based on EDR minimization could be simplified by the uniformity principle of TDFs. Comparisons between the constructal optimization based on EDR minimization and the MTD minimization with fixed- or variable-cross-section paths were made, and it was pointed out that the variation in the width of the high-conductivity path was an important reason for the difference between the constructs of the MTD minimization and those of the EDR minimization. Some criteria and formulas for the design of high-conductivity paths are proposed: The width of a high-conductivity path for a minimum MTD $D{\left(x\right)}_{T,opt}$ is proportional to $\sqrt{q^{‴}\left(x\right)}$, and the corresponding optimal temperature gradient ${\left(dT/dx\right)}_{T,opt}$ is also proportional to $\sqrt{q^{‴}\left(x\right)}$; the width of a high-conductivity path for a minimum EDR $D{\left(x\right)}_{h,opt}$ is proportional to $q^{‴}\left(x\right)$, but the corresponding optimal temperature gradient ${\left(dT/dx\right)}_{h,opt}$ is kept constant; when the heat flux $q^{‴}\left(x\right)$ in the high-conductivity path is a constant, the results of the minimum MTD are consistent with those of the minimum EDR.

Some new ideas and methods for improving other constructal optimizations by the variational principle were also provided in this paper. Some criteria for optimization variables can be derived by the Euler–Lagrange equation when the optimization objective and constraints are given, and the criteria can be used to simplify the optimization process.