# 多层介质传热的计算模拟 Simulation of Heat Transfer in Multi-Layered Medium

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The temperature changes with time and location for heat transfer with steady heat source flowing through a multi-layered dielectric material. In this paper, the temperature distribution of steady heat source flowing through a multi-layered medium is studied. By using heat conduction equation, we derive the heat-stable temperature distribution and the temperature regularity of the innermost material changes as time. Our method is applicable to the composite problem of various thermal insulation materials, and can be used to determine the temperature distribution of the multi-layered medium material.

1. 引言

2. 模型的建立

2.1. 热传递温度分布

Figure 1. The schematic of one dimensional multi-layered media

$\frac{\partial u\left(x,t\right)}{\partial t}-{a}^{2}\frac{{\partial }^{2}u\left(x,t\right)}{\partial {x}^{2}}=0$ , (1)

$u\left(x=0,t\right)={T}_{1}$ , $u\left(d,t=\infty \right)={T}_{5}$ . (2)

${X}_{k}\left(x\right)={p}_{k}\mathrm{cos}\left({\omega }_{k}x\right)+{q}_{k}\mathrm{sin}\left({\omega }_{k}x\right)$ , (3)

${T}_{k}\left(t\right)={n}_{k}{\text{e}}^{-{a}^{2}{\omega }_{k}^{2}t}$ ( $k=0,1,2,\cdots$ ) (4)

$u\left(x,t\right)=\underset{k}{\sum }{X}_{k}\left(x\right){T}_{k}\left(t\right)$ (5)

${Q}_{i}=A\frac{{\lambda }_{i}}{{d}_{i}}\left({T}_{i}-{T}_{i+1}\right)$ .

$\frac{\Delta {T}_{1,2}}{{R}_{1}}=\frac{\Delta {T}_{2,3}}{{R}_{2}}=\frac{\Delta {T}_{3,4}}{{R}_{3}}=\frac{\Delta {T}_{4,5}}{{R}_{4}}\equiv Q$ , (6)

2.2. 温度稳定前多层材料最内侧温度随时间的变化

$\begin{array}{l}\rho =\frac{{\rho }_{1}{v}_{1}+{\rho }_{2}{v}_{2}}{{v}_{1}+{v}_{2}},\\ c=\frac{{c}_{1}{v}_{1}+{c}_{2}{v}_{2}}{{v}_{1}+{v}_{2}},\\ \lambda =\frac{{\lambda }_{2}}{\left(\frac{{\lambda }_{2}}{{\lambda }_{1}}-1\right){v}_{1}+1}\end{array}$ (7)

$u\left(x=d,t\right)={C}_{1}{\text{e}}^{-{a}^{2}{\omega }^{2}t}+{C}_{0}$ , (8)

3. 计算模拟结果与讨论

Table 1. The parameter for related materials

$\rho =370.07,\text{\hspace{0.17em}}c=1622.68,\text{\hspace{0.17em}}\lambda =0.053,\text{\hspace{0.17em}}d=15.2$

$u\left(x=15.2,t\right)=-11.08{\text{e}}^{-0.0038t}+48.08$ .

Figure 2. Comparison of fitted curve and experimental value

I、II层接触面：

$u\left(d=0.6,t\right)=-37.302{\text{e}}^{-5.4419t}+74.302$ .

II、III层接触面：

$u\left(d=6.6,t\right)=-35.754{\text{e}}^{-0.0385t}+72.754$ .

III、IV层接触面：

$u\left(d=10.2,t\right)=-28.117{\text{e}}^{-0.0088t}+65.117$ .

Figure 3. The curve for temperature varies as time on three interfaces between layers

4. 实验探究

Table 2. The parameter for related materials

$\rho =1185.5,\text{\hspace{0.17em}}c=1210.1,\text{\hspace{0.17em}}\lambda =0.097,\text{\hspace{0.17em}}d=11.04$

$u\left(d=11.04,t\right)=-23.3{\text{e}}^{-0.0055t}+50$

Figure 4. Comparison of fitted curve and experimental value

5. 总结

NOTES

*通讯作者。

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