Diffusion mass transfer in a stationary medium

The governing equations for diffusion mass transfer are [23]

$$\boldsymbol{ j}_A = - \rho \boldsymbol{D}_{AB}\nabla m_A$$ (62)

and

$$\nabla \cdot \boldsymbol{ j}_A + \dot{n_A} = \frac{\partial \rho_A}{\partial t},$$ (63)

where

$$m_A = \frac{\rho_A}{\rho_A + \rho_B}$$ (64)

and

$$\rho = \rho_A + \rho_B.$$ (65)

In these equations $\boldsymbol{j}_A$ is the mass flux of species A, $\boldsymbol{D}_{AB}$ is the mass diffusivity, $m_A$ is the mass fraction of species A and $\rho_A$ is the density of species A. Furthermore, $\dot{n_A}$ is the rate of increase of the mass of species A per unit volume of the mixture. Another way of formulating this is:

$$\boldsymbol{ J}_A^* = - C \boldsymbol{D}_{AB}\nabla x_A$$ (66)

and

$$\nabla \cdot \boldsymbol{ J}_A^* + \dot{N_A} = \frac{\partial C_A}{\partial t}.$$ (67)

where

$$x_A = \frac{C_A}{C_A + C_B}$$ (68)

and

$$C = C_A + C_B.$$ (69)

Here, $\boldsymbol{J}_A^*$ is the molar flux of species A, $\boldsymbol{D}_{AB}$ is the mass diffusivity, $x_A$ is the mole fraction of species A and $C_A$ is the molar concentration of species A. Furthermore, $\dot{N_A}$ is the rate of increase of the molar concentration of species A.

The resulting equation now reads

$$\nabla \cdot (- \rho \boldsymbol{ D}_{AB} \cdot \nabla m_A)+ \frac{\partial \rho_A}{\partial t} = \dot{n_A} .$$ (70)

or

$$\nabla \cdot (- C \boldsymbol{ D}_{AB} \cdot \nabla x_A)+ \frac{\partial C_A}{\partial t} = \dot{N_A} .$$ (71)

If $C$ and $\rho$ are constant, these equations reduce to:

$$\nabla \cdot (- \boldsymbol{ D}_{AB} \cdot \nabla \rho_A)+ \frac{\partial \rho_A}{\partial t} = \dot{n_A} .$$ (72)

or

$$\nabla \cdot (- \boldsymbol{ D}_{AB} \cdot \nabla C_A)+ \frac{\partial C_A}{\partial t} = \dot{N_A} .$$ (73)

Accordingly, by comparison with the heat equation, the correspondence in Table  (8) arises.

Table 8: Correspondence between the heat equation and mass diffusion equation.

heat	mass diffusion
T	$\rho$	$C_A$
$\boldsymbol{ q}$	$\boldsymbol{j}_A$	$\boldsymbol{J}_A^*$
$q_n$	${j_A}_n$	${J_{A^*}}_n$
$\boldsymbol{\kappa}$	$\boldsymbol{D}_{AB}$	$\boldsymbol{D}_{AB}$
$\rho h$	$\dot{n_A}$	$\dot{N_A}$
$\rho c$	$1$	$1$