Diffusion mass transfer in a stationary medium

The governing equations for diffusion mass transfer are [35]

\boldsymbol{ j}_A = - \rho \boldsymbol{D}_{AB}\nabla m_A

(367)

and

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

(368)

where

m_A = \frac{\rho_A}{\rho_A + \rho_B}

(369)

and

\rho = \rho_A + \rho_B.

(370)

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

(371)

and

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

(372)

where

x_A = \frac{C_A}{C_A + C_B}

(373)

and

C = C_A + C_B.

(374)

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} .

(375)

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

(376)

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} .

(377)

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

(378)

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

**Table:** 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