Heat Transfer|L3_One Dementional Steady Heat Conduction_1

原创于 2020-03-06 11:43:29 发布 · 208 阅读

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本文深入探讨了一维稳态导热问题在不同几何形状下的数学模型，包括平面壁、圆柱壁和球壳的导热方程。详细介绍了各坐标系下导热方程的形式，并给出了平面壁、管道壁和壳体的热阻和热流量计算公式。

CATALOG

1D steady conduction problem without heat souce:

1D steady conduction problem without heat souce:

The totle gavorning function

$\frac{1}{r^{n}} \frac{\partial}{\partial r}\left(\lambda r^{n} \frac{\partial T}{\partial r}\right)+\dot{\Phi}=\rho C_{p} \frac{\partial T}{\partial t}$
$\mathrm{n}=0 \quad \text{for a plane wall}\\ \mathrm{n}=1 \quad \text{for a cylinder}\\ \mathrm{n}=2 \quad \text{for a sphere}$

Cartesian Coordinate
$\frac{\partial}{\partial x}\left(\lambda \frac{\partial T}{\partial x}\right)+\dot{\Phi}=\rho C_{p} \frac{\partial T}{\partial t}$
Cylindrical Coordinate
$\frac{1}{r} \frac{\partial}{\partial r}\left(\lambda r \frac{\partial T}{\partial r}\right)+\dot{\Phi}=\rho C_{p} \frac{\partial T}{\partial t} \quad\text { in the } r \text { direction }$
Spherical Coordinate
$\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(\lambda r^{2} \frac{\partial T}{\partial r}\right)+\dot{\Phi}=\rho C_{p} \frac{\partial T}{\partial t} \quad\text { in the } r \text { direction }$

a) plane walls

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$R=\frac{\delta_{1}}{\lambda_{1}}+\frac{\delta_{2}}{\lambda_{2}}+\frac{\delta_{3}}{\lambda_{3}}$

$\phi=\frac{T_{w 1}-T_{w 4}}{A(\frac{\delta_{1}}{\lambda_{1}}+\frac{\delta_{2}}{\lambda_{2}}+\frac{\delta_{3}}{\lambda_{3}})}$

b) pipe walls

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$\begin{aligned} R_{t} &=R_{1}+R_{2}+R_{3} \\ &=\frac{\ln \left(r_{2} / r_{1}\right)}{2 \pi L \lambda_{1}}+\frac{\ln \left(r_{3} / r_{2}\right)}{2 \pi L \lambda_{2}}+\frac{\ln \left(r_{4} / r_{3}\right)}{2 \pi L \lambda_{3}} \end{aligned}$

$\Phi=\frac{2 \pi L\left(T_{1}-T_{4}\right)}{\frac{\ln \left(r_{2} / r_{1}\right)}{\lambda_{1}}+\frac{\ln \left(r_{3} / r_{2}\right)}{\lambda_{2}}+\frac{\ln \left(r_{4} / r_{3}\right)}{\lambda_{3}}}$