流体传质方程
Equation of Continuity for Species $ \alpha$ in terms of $ \pmb j_\alpha$
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 \left (\dfrac {\partial w_\alpha}{\partial t}+ v_x \dfrac {\partial w_\alpha}{\partial x} +v_y \dfrac {\partial w_\alpha}{\partial y} + v_z \dfrac {\partial w_\alpha}{\partial z}\right) = - \left [\dfrac {\partial j_{\alpha x}}{\partial x} +\dfrac {\partial j_{\alpha y}}{\partial y}+\dfrac {\partial j_{\alpha z}}{\partial z}\right] + r_\alpha $ | 1-1 |
2.圆柱坐标系（$r,\theta, z$)
| 圆柱坐标系Cylindrical coordinates coordinates ($\textit {r, $\theta$, z }$): | NO. |
|---|---|
| $\rho \left (\dfrac {\partial w_\alpha}{\partial t}+ v_r \dfrac {\partial w_\alpha}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial w_\alpha}{\partial \theta} + v_z \dfrac {\partial w_\alpha}{\partial z}\right) = - \left [\dfrac {1}{r} \dfrac {\partial}{\partial r} (r j_{\alpha r}) + \dfrac {1}{r} \dfrac {\partial j_{\alpha \theta}}{\partial \theta}+\dfrac {\partial j_{\alpha z}}{\partial z}\right] + r_\alpha$ | 2-1 |
3.球坐标系（$r, \theta, \phi $)
| 球坐标系Spherical coordinates（$\textit {r, $\theta$, $\phi$ }$): | NO. |
|---|---|
| $\rho \left (\dfrac {\partial w_\alpha}{\partial t}+ v_r \dfrac {\partial w_\alpha}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial w_\alpha}{\partial \theta} + \dfrac {v_\phi}{rsin\theta} \dfrac {\partial w_\alpha}{\partial \phi}\right) = \left [\dfrac {1}{r^2} \dfrac {\partial}{\partial r} (r^2 j_{\alpha r}) + \dfrac {1}{r sin\theta} \dfrac {\partial}{\partial \theta}(j_{\alpha \theta} sin\theta)+ \dfrac {1}{rsin\theta} \dfrac {\partial j_{\alpha \phi}}{\partial \phi}\right] + r_\alpha $ | 3-1 |
注：如果密度被浓度代替。都要做相应的替换
| 质量浓度 | $\rho$ | $w_\alpha$ | $\pmb j_\alpha$ | $\pmb v$ | $r_\alpha $ |
|---|---|---|---|---|---|
| 摩尔浓度 | $c$ | $x_\alpha$ | $\pmb J_\alpha^*$ | $\pmb v^*$ | $R_\alpha-x_\alpha \sum\limits_{\beta=1}^{N}R_\beta $ |
Equation of Continuity for Species A with Constant $\rho \mathscr{D}_{AB}$ in terms of $w_A$
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 \left (\dfrac {\partial w_A}{\partial t}+ v_x \dfrac {\partial w_A}{\partial x} +v_y \dfrac {\partial w_A}{\partial y} + v_z \dfrac {\partial w_A}{\partial z}\right) = \rho \mathscr{D}_{AB} \left [\dfrac {\partial^2 w_A}{\partial x^2} +\dfrac {\partial ^2 w_A}{\partial y^2}+\dfrac {\partial ^2 w_A}{\partial z^2}\right] +r_A $ | 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 w_A}{\partial t}+ v_r \dfrac {\partial w_A}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial w_A}{\partial \theta} + v_z \dfrac {\partial w_A}{\partial z}\right) = \rho \mathscr{D}_{AB}\left [\dfrac {1}{r} \dfrac {\partial}{\partial r} \left(r \dfrac{\partial w_A}{\partial r}\right) + \dfrac {1}{r^2} \dfrac {\partial^2 w_A}{\partial \theta^2}+\dfrac {\partial^2 w_A}{\partial z^2}\right] +r_A $ | 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 w_A}{\partial t}+ v_r \dfrac {\partial w_A}{\partial r} +\dfrac {v_\theta}{r} \dfrac {\partial w_A}{\partial \theta} + \dfrac {v_\phi}{rsin\theta} \dfrac {\partial w_A}{\partial \phi}\right) =\rho \mathscr{D}_{AB} \left [\dfrac {1}{r^2} \dfrac {\partial}{\partial r} \left(r^2 \dfrac {\partial w_A}{\partial r} \right) + \dfrac {1}{r^2 sin\theta} \dfrac {\partial}{\partial \theta} \left(sin\theta \dfrac {\partial w_A}{\partial \theta}\right)+ \dfrac {1}{r^2 sin^2\theta} \dfrac {\partial^2 w_A}{\partial \phi^2}\right] +r_A $ | 3-1 |
参考文献
- R. Byron Bird, Warren E. stewart, Edwin N. Lightfoot. Transport phenomena:Revised second edition John Wiely &Sons, Inc.