﻿ 载人潜水器水下传热问题近似计算方法
 舰船科学技术  2020, Vol. 42 Issue (1): 83-87, 94 PDF

An approximate method of computing heat transfer problem of manned submersible
ZHAO Yuan-hui, XU Meng, WANG Bo, ZHANG Yi-chi
State Key Laboratory of Deep-sea Manned Vehicles, China ship Scientific Research Center, Wuxi 214082, China
Abstract: The article provides a heat transfer model for solving manned submersibles heat exchange problem. The factors including thermal insulation、heat capacity、air conditioning are firstly take into account in this model. With some assumptions, the problem can be simplified to 1D unsteady heat transfer, A finite difference method is used to numerically solve the problem and is carried out using Matlab codes. Calculating different thermal condition for a submersible based on Matlab program. The results are compared with the simulation results of AMESim, which will verify the proposed heat transfer model and calculation method. The method will provide a more convenient approach to help research on thermal comfort of manned submersible.
Key words: submersible     thermal insulation     finite difference     thermal comfort
0 引　言

1 潜器传热的数学描述 1.1 应用背景

1.2 基本假设

1）本文的计算仅考虑厚度方向的热传递，忽略长度和宽度的影响，也即一维热传递问题；

2）由于舱壁厚度、隔热层厚度与潜水器的直径、长度相比是一个小量，因而采用平壁传热模型，而非圆柱或球形传热，整个计算在笛卡尔坐标系中完成；

3） 所有物体材料均匀且它们的热物理属性与温度无关；

4）在隔热层与舱壁接触处，隔热层表面温度与舱内壁温度相等；

5）舱内空气仅通过对流方式与隔热层进行换热；

6）与外界海水直接接触的金属耐压舱壁外表面，其温度与海水温度时刻保持一致。

1.3 建立数学模型

 图 1 隔热层的传热模型 Fig. 1 Heat transfer model of insulation layer

1）控制方程

 $\frac{\partial }{{\partial x}}\left( {{k_i}\frac{{\partial T}}{{\partial x}}} \right) = {\rho _i}{c_i}\frac{{\partial T}}{{\partial t}}\begin{array}{*{20}{c}} {} \end{array}\left( {0 < x < {L_i}} \right)\text{，}$
 $\frac{\partial }{{\partial x}}\left( {{k_h}\frac{{\partial T}}{{\partial x}}} \right) = {\rho _h}{c_h}\frac{{\partial T}}{{\partial t}}\begin{array}{*{20}{c}} {} \end{array}\left( {{L_i} < x < ({L_i} + {L_h})} \right)\text{。}$

 ${C_g}{V_g}{\rho _g} \cdot \frac{{{T_g}\left( {{t_0} + \Delta t} \right) - {T_g}\left( {{t_0}} \right)}}{{\Delta t}} = {Q_1} - {Q_2} - {Q_3}\text{。}$

 ${Q_2}\left( t \right) = {C_g}{V_{ac}}{\rho _g}\left( {{T_g}\left( t \right) - {T_{ac}}} \right)\text{，}$
 ${Q_3}\left( t \right) = {h_i}\left( {{T_g}\left( t \right) - T\left( {0,t} \right)} \right)\text{。}$

2）初始条件

$\tau = 0$ 时，潜水器处于水面状态，准备开始下潜。此时，隔热层与舱壁处于均匀温度，其数值与水面海水温度一致：

 $T\left( {x,0} \right) = {T_{sea}}\text{，}$

 ${T_g}\left( 0 \right) = 27\text{。}$

3）边界条件

 ${\left. { - {k_i}\frac{{\partial T}}{{\partial x}}} \right|_{x = 0}} = {h_i}\left( {{T_g} - T\left( {0,t} \right)} \right)\text{。}$

x= ${L_i}$ 时，隔热层与舱壁接触的温度相等，当x= ${L_i} + {L_h}$ 时，舱壁与海水接触进行对流热交换，但根据假设6所述，由于舱壁对流导热计算过程中的毕渥数较大，因此，导热热阻远大于对流换热热阻，可取壁面的温度与海水的温度相同，即表示为：

 $T\left( {{L_i} + {L_h},t} \right) = {T_{sea}}\text{。}$

2 传热问题的数值解法

 图 2 计算域节点划分图 Fig. 2 Computational domain node sketch

 图 3 时间空间区域离散图 Fig. 3 Time-space region scatter graph
 $\frac{{T_i^m - T_i^{m - 1}}}{{\Delta t}} = \alpha \frac{{T_{i - 1}^m - 2 \cdot T_i^m + T_{i + 1}^m}}{{\Delta {x^2}}}\text{。}$

 $\frac{1}{{\Delta t}}T_i^{m - 1} = - \frac{\alpha }{{\Delta {x^2}}}T_{i - 1}^m + \left( {\frac{{2\alpha }}{{\Delta {x^2}}} + \frac{1}{{\Delta t}}} \right)T_i^m - \frac{\alpha }{{\Delta {x^2}}}T_{i + 1}^m\text{，}$

 ${ A} \cdot{ B} ={ C}\text{，}$
 ${{ A} = {\left[\!\! {\begin{array}{*{20}{c}} {{b_1}}\!\!\!&\!\!\!{{c_1}}\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\!0 \\ { - \dfrac{\alpha }{{\Delta {x^2}}}}\!\!\!&\!\!\!{\dfrac{1}{{\Delta t}} + \dfrac{{2\alpha }}{{\Delta {x^2}}}}\!\!\!&\!\!\!{ - \dfrac{\alpha }{{\Delta {x^2}}}}\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\!0 \\ 0\!\!\!&\!\!\!{ - \dfrac{\alpha }{{\Delta {x^2}}}}\!\!\!&\!\!\!{\dfrac{{2\alpha }}{{\Delta {x^2}}} + \dfrac{1}{{\Delta t}}}\!\!\!&\!\!\!{ - \dfrac{\alpha }{{\Delta {x^2}}}}\!\!\!&\!\!\!0\!\!\!&\!\!\!0 \\ 0\!\!\!&\!\!\!0\!\!\!&\!\!\! \ddots \!\!\!&\!\!\! \ddots \!\!\!&\!\!\! \ddots \!\!\!&\!\!\!0 \\ 0\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\!{ - \dfrac{\alpha }{{\Delta {x^2}}}}\!\!\!&\!\!\!{\dfrac{{2\alpha }}{{\Delta {x^2}}} + \dfrac{1}{{\Delta t}}}\!\!\!&\!\!\!{ - \dfrac{\alpha }{{\Delta {x^2}}}} \\ 0\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\!0\!\!\!&\!\!\!1 \end{array}}\!\! \right]_{N \times N}}}\text{，}$
 ${ B}= {\left[ {T_1^m,T_2^m,T_3^m \ldots T_{N - 1}^m,T_N^m} \right]^{\rm{T}}}\text{，}$
 ${ C} = {\left[ {{d_1},\frac{1}{{\Delta t}}T_2^{m - 1},\frac{1}{{\Delta t}}T_3^{m - 1} \ldots \frac{1}{{\Delta t}}T_{N - 1}^{m - 1},{d_N}} \right]^{\rm{T}}}\text{。}$

 $h\left( {T_g^{m + 1} - T_1^{m + 1}} \right) + k\frac{{T_2^{m + 1} - T_1^{m + 1}}}{{\Delta x}} = \rho \frac{{\Delta x}}{2}c\frac{{T_1^{m + 1} - T_1^m}}{{\Delta t}}\text{。}$

 $\begin{gathered} \left( {\frac{1}{{\Delta t}} + 2\frac{\alpha }{{\Delta {x^2}}} + 2\frac{h}{{\rho c\Delta x}}} \right)T_1^m - 2\frac{\alpha }{{\Delta {x^2}}}T_2^m= \\ 2\frac{h}{{\rho c\Delta x}} \cdot T_g^m + \frac{1}{{\Delta t}}T_1^{m - 1} \text{，} \end{gathered}$

 $\begin{gathered} \frac{1}{{\Delta t}}T_j^m = \left( {\frac{1}{{\Delta t}} + \frac{{2{k_i} + 2{k_h}}}{{\Delta {x^2}\left( {{c_i}{\rho _i} + {c_h}{\rho _h}} \right)}}} \right)T_j^{m + 1} - \\ \frac{{2{k_i}}}{{\Delta {x^2}\left( {{c_i}{\rho _i} + {c_h}{\rho _h}} \right)}}T_{j - 1}^{m + 1} - \frac{{2{k_h}}}{{\Delta {x^2}\left( {{c_i}{\rho _i} + {c_h}{\rho _h}} \right)}}T_{j + 1}^{m + 1} \text{。} \end{gathered}$

 ${ A}=\scriptsize{\left[\!\!\!\!\begin{array}{*{20}{c}} {\dfrac{1}{{\Delta t}} + 2\dfrac{{{k_i}}}{{{\rho _i}{c_i}\Delta {x^2}}} + 2\dfrac{{{h_i}}}{{{\rho _i}{c_i}\Delta x}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! { - 2\dfrac{{{k_i}}}{{{\rho _i}{c_i}\Delta {x^2}}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \\ { - \dfrac{{{k_i}}}{{{\rho _i}{c_i}\Delta {x^2}}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! {\dfrac{1}{{\Delta t}} + 2\dfrac{{{k_i}}}{{{\rho _i}{c_i}\Delta {x^2}}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! { - \dfrac{{{k_i}}}{{{\rho _i}{c_i}\Delta {x^2}}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! { - \dfrac{{{k_i}}}{{{\rho _i}{c_i}\Delta {x^2}}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! {\dfrac{1}{{\Delta t}} + 2\dfrac{{{k_i}}}{{{\rho _i}{c_i}\Delta {x^2}}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! { - \dfrac{{{k_i}}}{{{\rho _i}{c_i}\Delta {x^2}}}} \\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! \ddots \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! \ddots \\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! { - \dfrac{{2{k_i}}}{{\Delta {x^2}\left( {{c_i}{\rho _i} + {c_h}{\rho _h}} \right)}}} \\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0\\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0\\ \ddots \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \\ {\dfrac{1}{{\Delta t}} + \dfrac{{2{k_i} + 2{k_h}}}{{\Delta {x^2}\left( {{c_i}{\rho _i} + {c_h}{\rho _h}} \right)}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! { - \dfrac{{2{k_h}}}{{\Delta {x^2}\left( {{c_i}{\rho _i} + {c_h}{\rho _h}} \right)}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \\ \ddots \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! \ddots \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0\\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! { - \dfrac{{{k_h}}}{{{\rho _h}{c_h}\Delta {x^2}}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! {\dfrac{1}{{\Delta t}} + 2\dfrac{{{k_h}}}{{{\rho _h}{c_h}\Delta {x^2}}}} \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! { - \dfrac{{{k_h}}}{{{\rho _h}{c_h}\Delta {x^2}}}}\\ 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 0 \!\!\!\!\!\!\!\!\!&\!\!\!\!\!\!\!\!\! 1 \end{array}\!\! \right]}\text{。}$
3 算例及验证

 ${T_{sea}} = 16.16{e^{ - 0.007478x}} + 14.21{e^{ - 0.001098x}}\text{，}\begin{array}{*{20}{c}} {}&{0 \leqslant x \leqslant 1000} \text{。} \end{array}$

 图 4 海水温度随时间变化关系 Fig. 4 Temperature of seawater varies with time
3.1 无隔热层情况

 图 5 无隔热层下潜舱内气温变化 Fig. 5 Temperature varies with time without thermal insulation

3.2 有隔热层不开启空调

 图 6 敷设隔热层下潜舱内气温变化 Fig. 6 Temperature varies with time with thermal insulation

2种软件的计算结果吻合也较好，仅在最高气温值上存在较小差异，Matlab计算最高温度为48 ℃，AMESim计算最高温度为47 ℃，相差仅为1 ℃。从计算结果可以看出，敷设隔热层后，由于隔热层的保温效应，舱内气温显著上升，且气温的变化速率相对无隔热层时更加平缓。舱内的最高温度在1.5 h到达最大47 ℃，最后的平衡温度约为32 ℃。

3.3 有隔热层开启空调

 图 7 敷设隔热层开空调下潜舱内气温变化 Fig. 7 Temperature varies with time with thermal insulation and air conditioning
4 结　语

本文提出了一种用于具有热源、隔热层、空调等多因素综合影响下的水下载人潜水器热交换计算模型，并利用有限差分方法给出了该模型的数值计算过程。尽管给定了部分假设，但通过与主流传热计算软件的计算结果对比发现该计算模型及方法仍具有较高准确性。后期，通过利用该计算方法可为水下载人潜器在舱室热舒适性上的工程研究及优化，重点包括隔热层材料及厚度的选取、空调参数选配、舱室环境温度预估等问题提供更便捷计算方法。

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