# 质量通量

## 定義

${\displaystyle j_{m}=\lim \limits _{A\rightarrow 0}{\frac {I_{m}}{A}}}$

${\displaystyle I_{m}=\lim \limits _{\Delta t\rightarrow 0}{\frac {\Delta m}{\Delta t}}={\frac {dm}{dt}}}$

${\displaystyle m=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} _{m}\cdot \mathbf {\hat {n}} {\rm {d}}A{\rm {d}}t}$

${\displaystyle \mathbf {j} _{m}\cdot \mathbf {\hat {n}} =j_{m}\cos \theta }$

## 流體方程

### 替代方程

${\displaystyle \mathbf {j} _{\rm {m}}=\rho \mathbf {u} }$

• ρ 為密度，
• u 為流體的流速

### 混合流體的質量通量及莫耳通量

#### 質量通量

${\displaystyle \mathbf {j} _{{\rm {m}},\,i}=\rho _{i}\mathbf {u} _{i}}$

${\displaystyle \mathbf {j} _{{\rm {m}},\,i}=\rho \left(\mathbf {u} _{i}-\langle \mathbf {u} \rangle \right)}$

${\displaystyle \langle \mathbf {u} \rangle ={\frac {1}{\rho }}\sum _{i}\rho _{i}\mathbf {u} _{i}={\frac {1}{\rho }}\sum _{i}\mathbf {j} _{{\rm {m}},\,i}}$

• ρ為混合物的平均密度
• ρi為成份i的密度
• u i為成份i的速度

#### 莫耳通量

${\displaystyle \mathbf {j} _{\rm {n}}=n\mathbf {u} }$

${\displaystyle \mathbf {j} _{{\rm {n}},\,i}=n_{i}\mathbf {u} _{i}}$

${\displaystyle \mathbf {j} _{{\rm {n}},\,i}=n\left(\mathbf {u} _{i}-\langle \mathbf {u} \rangle \right)}$

${\displaystyle \langle \mathbf {u} \rangle ={\frac {1}{n}}\sum _{i}n_{i}\mathbf {u} _{i}={\frac {1}{n}}\sum _{i}\mathbf {j} _{{\rm {m}},\,i}}$

## 用法

${\displaystyle \nabla \cdot \mathbf {j} _{\rm {m}}+{\frac {\partial \rho }{\partial t}}=0}$

${\displaystyle \nabla \cdot \mathbf {j} _{\rm {n}}=-\nabla \cdot D\nabla c}$

## 參考資料

1. ^ Vectors, Tensors, and the basic Equations of Fluid Mechanics, R. Aris, Dover Publications, 1989, ISBN(10) 0-486-66110-5