# 基于分层差分方程的热防护服热量分布规律的模拟Simulation of Heat Distribution Law of Thermal Protective Clothing Based on Lay-ered Difference Equation

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In order to study the heat distribution of heat-resistant garments in high-temperature environ-ments and further to reduce the production cost of heat-resistant garments under high-temperature environment, and shorten the development cycle, in this paper, we first establish a model of high-temperature protective clothing-air-skin for different material layers. The heat conduction equation is used to study the heat distribution of different layers, where the thermal diffusivity in the heat transfer equation depends on the parameter values of the dielectric material. We propose two models to solve the boundary condition problem in partial differential equations. The concept of thermal resistance is introduced in Model I. According to the Crank-Nicolson implicit scheme for solving partial differential equations in numerical analysis, the temperature-space-time correlation results in different material layers can be obtained. In Model II, four different material layers are considered as a whole. Thus, the classical heat differential equations are used to solve the partial differentials of the heat conduction equations for the four layers. Numerical solution of the equation shows that the methods are workable.

1. 引言

2. 分层热传导模型

Figure 1. Schematic diagram of thermal conduction model

(1)

3. 模型的求解方法

3.1. 基于热阻法求解边界条件以及Crank-Nicolson隐式格式求解偏微分方程

${T}_{i}\left(x,t\right)=\left({T}_{i}\left(1,1\right),\cdots ,{T}_{i}\left(Ax-2,1\right),{T}_{i}\left(1,2\right),\cdots {T}_{i}\left(Ax-2,2\right),\cdots {T}_{i}\left(1,Bt-1\right),\cdots ,{T}_{i}\left(Ax-2,Bt-1\right)\right)$ (2)

Figure 2. Implicit schematic diagram of Crank-Nicolson

$\begin{array}{l}-\frac{1}{2}r{T}_{i}\left(x-1,t+1\right)+\left(1+r\right){T}_{i}\left(x,t+1\right)-\frac{1}{2}r{T}_{i}\left(x+1,t+1\right)\\ =\frac{1}{2}r{T}_{i}\left(x-1,t\right)+\left(1-r\right){T}_{i}\left(x,t\right)+\frac{1}{2}r{T}_{i}\left(x+1,t\right)\end{array}$ (3)

$\begin{array}{l}\left(\begin{array}{cccc}1+2r& -r& \cdots & 0\\ -r& 1+2r& \ddots & ⋮\\ 0& \ddots & \ddots & -r\\ 0& 0& -r& 1+2r\end{array}\right)\cdot \left(\begin{array}{c}{T}_{i}\left(1,t+1\right)\\ {T}_{i}\left(2,t+1\right)\\ ⋮\\ {T}_{i}\left(Ax-2,t+1\right)\end{array}\right)\\ =\left(\begin{array}{cccc}1+2r& -r& \cdots & 0\\ -r& 1+2r& \ddots & ⋮\\ 0& \ddots & \ddots & -r\\ 0& 0& -r& 1+2r\end{array}\right)\cdot \left(\begin{array}{c}{T}_{i}\left(1,t\right)\\ {T}_{i}\left(2,t\right)\\ ⋮\\ {T}_{i}\left(Ax-2,t\right)\end{array}\right)\\ +\left(\begin{array}{c}0.5r{T}_{i}\left(0,t\right)+0.5r{T}_{i}\left(0,t+1\right)\\ 0\\ ⋮\\ 0.5r{T}_{i}\left(Ax-2,t\right)+0.5r{T}_{i}\left(Ax-2,t+1\right)\end{array}\right)\end{array}$ (4)

Figure 3. Temperature-time-space Distribution of Layer I-IV Based on the Implicit Method of Solving Partial Differential Equation of CN

Figure 4. The change of temperature at the last moment (5400 s) with the position based on the method of solving partial differential equation with CN implicit scheme

3.2. 基于整体化法以及古典显式差分法求解偏微分方程

$\frac{{T}_{i}\left(x,t+1\right)-{T}_{i}\left(x,t\right)}{k}-\frac{{T}_{i}\left(x+1,t\right)-2{T}_{i}\left(x,t\right)+{T}_{i}\left(x-1,t\right)}{{h}^{2}}=0$

${T}_{i}\left(x,t+1\right)=r{T}_{i}\left(x-1,t\right)+\left(1-2r\right){T}_{i}\left(x,t\right)+r{T}_{i}\left(x+1,t\right)$ (5)

Figure 5. Temperature-time-space distribution of layer I-IV based on classical explicit partial differential equation

Figure 6. On the basis of the classical explicit partial differential equation solution method, the temperature change of the boundary layer I-IV with time

4. 模型检验

Figure 7. The calculation and measurement of skin temperature and its error based on the method of solving implicit partial differential equation of CN

5. 结语

NOTES

*通讯作者。

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