流体传热方程
Equation of Energy in terms of $\pmb q$
控制方程表达通式：
1.直角坐标系($ x, y, z $)
| 直角坐标系Cartesian coordinates ($\textit { x,y,z }$): | NO. |
|---|---|
| $\rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_x \dfrac {\partial T}{\partial x} +v_y \dfrac {\partial T}{\partial y} + v_z \dfrac {\partial T}{\partial z}\right) = - \left [\dfrac {\partial q_x}{\partial x} +\dfrac {\partial q_y}{\partial y}+\dfrac {\partial q_z}{\partial z}\right] -\left(\dfrac {\partial ln \rho}{\partial ln T}\right)_p \dfrac {Dp}{Dt}-\pmb \tau : \nabla \pmb v $ | 1-1 |
2.圆柱坐标系（$r,\theta, z$)
| 圆柱坐标系Cylindrical coordinates coordinates ($\textit {r, $\theta$, z }$): | NO. |
|---|---|
| $\rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_r \dfrac {\partial T}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial T}{\partial \theta} + v_z \dfrac {\partial T}{\partial z}\right) = - \left [\dfrac {1}{r} \dfrac {\partial}{\partial r} (r q_r) + \dfrac {1}{r} \dfrac {\partial q_\theta}{\partial \theta}+\dfrac {\partial q_z}{\partial z}\right] -\left(\dfrac {\partial ln \rho}{\partial ln T}\right)_p \dfrac {Dp}{Dt}-\pmb \tau : \nabla \pmb v $ | 2-1 |
3.球坐标系（$r, \theta, \phi $)
| 球坐标系Spherical coordinates（$\textit {r, $\theta$, $\phi$ }$): | NO. |
|---|---|
| $\rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_r \dfrac {\partial T}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial T}{\partial \theta} + \dfrac {v_\phi}{rsin\theta} \dfrac {\partial T}{\partial \phi}\right) = - \left [\dfrac {1}{r^2} \dfrac {\partial}{\partial r} (r^2 q_r) + \dfrac {1}{r sin\theta} \dfrac {\partial}{\partial \theta}(q_\theta sin\theta)+ \dfrac {1}{r sin\theta} \dfrac {\partial q_\phi}{\partial \phi}\right] -\left(\dfrac {\partial ln \rho}{\partial ln T}\right)_p \dfrac {Dp}{Dt}-\pmb \tau : \nabla \pmb v $ | 3-1 |
注：黏度耗散项$（-\pmb \tau : \nabla \pmb v）$很小，可以被忽略，除非速度的梯度非常大。另外对于恒定密度的流体$\dfrac {\partial ln \rho}{\partial ln T}$项等于零。
Equation of Energy for Newtonian Fluids with Constant $\rho$ and $k$
控制方程表达通式：
1.直角坐标系($ x, y, z $)
| 直角坐标系Cartesian coordinates ($\textit { x,y,z }$): | NO. |
|---|---|
| $\rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_x \dfrac {\partial T}{\partial x} +v_y \dfrac {\partial T}{\partial y} + v_z \dfrac {\partial T}{\partial z}\right) = k \left [\dfrac {\partial^2 T}{\partial x^2} +\dfrac {\partial ^2 T}{\partial y^2}+\dfrac {\partial ^2 T}{\partial z^2}\right] +\mu \pmb \Phi_v $ | 1-1 |
2.圆柱坐标系（$r,\theta, z$)
| 圆柱坐标系Cylindrical coordinates coordinates ($\textit {r, $\theta$, z }$): | NO. |
|---|---|
| $\rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_r \dfrac {\partial T}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial T}{\partial \theta} + v_z \dfrac {\partial T}{\partial z}\right) = k\left [\dfrac {1}{r} \dfrac {\partial}{\partial r} \left(r \dfrac{\partial T}{\partial r}\right) + \dfrac {1}{r^2} \dfrac {\partial^2 T}{\partial \theta^2}+\dfrac {\partial^2 T}{\partial z^2}\right] +\mu \pmb \Phi_v $ | 2-1 |
3.球坐标系（$r, \theta, \phi $)
| 球坐标系Spherical coordinates（$\textit {r, $\theta$, $\phi$ }$): | NO. |
|---|---|
| $\rho \hat {C}_p \left (\dfrac {\partial T}{\partial t}+ v_r \dfrac {\partial T}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial T}{\partial \theta} + \dfrac {v_\phi}{rsin\theta} \dfrac {\partial T}{\partial \phi}\right) =k \left [\dfrac {1}{r^2} \dfrac {\partial}{\partial r} \left(r^2 \dfrac {\partial T}{\partial r} \right) + \dfrac {1}{r^2 sin\theta} \dfrac {\partial}{\partial \theta} \left(sin\theta \dfrac {\partial T}{\partial \theta}\right)+ \dfrac {1}{r^2 sin^2\theta} \dfrac {\partial^2 T}{\partial \phi^2}\right] +\mu \pmb \Phi_v $ | 3-1 |
参考文献
- R. Byron Bird, Warren E. stewart, Edwin N. Lightfoot. Transport phenomena:Revised second edition John Wiely &Sons, Inc.