Chemical Engineering Fluid Mechanics, 3rd Edition

Appendix E: Microscopic Conservation Equations in Rectangular, Cylindrical, and Spherical Coordinates

E.1  CONTINUITY EQUATION

Rectangular coordinates (x, y, z):

$\frac{\partial \text{ρ}}{\partial t}+\frac{\partial }{\partial x}\left({\text{ρv}}_x\right)+\frac{\partial }{\partial y}\left({\text{ρv}}_y\right)+\frac{\partial }{\partial z}\left({\text{ρv}}_z\right)=0$

Cylindrical coordinates (r, θ, z):

$\frac{\partial \text{ρ}}{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}\left(\text{ρ}r{\text{v}}_r\right)+\frac{1}{r}\frac{\partial }{\partial \text{θ}}\left({\text{ρv}}_{\text{θ}}\right)+\frac{\partial }{\partial z}\left({\text{ρv}}_z\right)=0$

Spherical coordinates (r, θ, ϕ):

$\frac{\partial \text{ρ}}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(\text{ρ}{r}^2{\text{v}}_r\right)+\frac{1}{r\sin \text{θ}}\frac{\partial }{\partial \text{θ}}\left({\text{ρv}}_{\text{θ}}\sin \text{θ}\right)+\frac{1}{r\sin \text{θ}}\frac{\partial }{\partial \varphi }\left({\text{ρv}}_{\varphi }\right)=0$

E.2  MOMENTUM EQUATION IN RECTANGULAR COORDINATES

x-component:

$\begin{array}{l}\text{ρ}\left(\frac{\partial {\text{v}}_x}{\partial t}+{\text{v}}_x\frac{\partial {\text{v}}_x}{\partial x}+{\text{v}}_y\frac{\partial {\text{v}}_x}{\partial y}+{\text{v}}_z\frac{\partial {\text{v}}_x}{\partial z}\right)\\ =-\frac{\partial P}{\partial x}+\left(\frac{\partial {\text{τ}}_{xx}}{\partial x}+\frac{\partial {\text{τ}}_{yx}}{\partial y}+\frac{\partial {\text{τ}}_{zx}}{\partial z}\right)+\text{ρ}{g}_x\end{array}$

y-component:

$\begin{array}{l}\text{ρ}\left(\frac{\partial {\text{v}}_y}{\partial t}+{\text{v}}_x\frac{\partial {\text{v}}_y}{\partial x}+{\text{v}}_y\frac{\partial {\text{v}}_y}{\partial y}+{\text{v}}_z\frac{\partial {\text{v}}_y}{\partial z}\right)\\ =-\frac{\partial P}{\partial y}+\left(\frac{\partial {\text{τ}}_{xy}}{\partial x}+\frac{\partial {\text{τ}}_{yy}}{\partial y}+\frac{\partial {\text{τ}}_{zy}}{\partial z}\right)+\text{ρ}{g}_y\end{array}$

z-component:

$\begin{array}{l}\text{ρ}\left(\frac{\partial {\text{v}}_z}{\partial t}+{\text{v}}_x\frac{\partial {\text{v}}_z}{\partial x}+{\text{v}}_y\frac{\partial {\text{v}}_z}{\partial y}+{\text{v}}_z\frac{\partial {\text{v}}_z}{\partial z}\right)\\ =-\frac{\partial P}{\partial y}+\left(\frac{\partial {\text{τ}}_{xz}}{\partial x}+\frac{\partial {\text{τ}}_{yz}}{\partial y}+\frac{\partial {\text{τ}}_{zz}}{\partial z}\right)+\text{ρ}{g}_z\end{array}$

E.3  MOMENTUM EQUATION IN CYLINDRICAL COORDINATES

r-component:

$\begin{array}{l}\text{ρ}(\frac{\partial {\text{v}}_r}{\partial t}+{\text{v}}_r\frac{\partial {\text{v}}_r}{\partial r}+\frac{{\text{v}}_{\text{θ}}}{r}\frac{\partial {\text{v}}_r}{\partial }\end{array}$