Diffusion mass transfer in a stationary medium next up previous contents
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Diffusion mass transfer in a stationary medium

The governing equations for diffusion mass transfer are [29]

$\displaystyle \boldsymbol{ j}_A = - \rho \boldsymbol{D}_{AB}\nabla m_A$ (113)

and

$\displaystyle \nabla \cdot \boldsymbol{ j}_A + \dot{n_A} = \frac{\partial \rho_A}{\partial t},$ (114)

where

$\displaystyle m_A = \frac{\rho_A}{\rho_A + \rho_B}$ (115)

and

$\displaystyle \rho = \rho_A + \rho_B.$ (116)

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:

$\displaystyle \boldsymbol{ J}_A^* = - C \boldsymbol{D}_{AB}\nabla x_A$ (117)

and

$\displaystyle \nabla \cdot \boldsymbol{ J}_A^* + \dot{N_A} = \frac{\partial C_A}{\partial t}.$ (118)

where

$\displaystyle x_A = \frac{C_A}{C_A + C_B}$ (119)

and

$\displaystyle C = C_A + C_B.$ (120)

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

$\displaystyle \nabla \cdot (- \rho \boldsymbol{ D}_{AB} \cdot \nabla m_A)+ \frac{\partial \rho_A}{\partial t} = \dot{n_A} .$ (121)

or

$\displaystyle \nabla \cdot (- C \boldsymbol{ D}_{AB} \cdot \nabla x_A)+ \frac{\partial C_A}{\partial t} = \dot{N_A} .$ (122)

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

$\displaystyle \nabla \cdot (- \boldsymbol{ D}_{AB} \cdot \nabla \rho_A)+ \frac{\partial \rho_A}{\partial t} = \dot{n_A} .$ (123)

or

$\displaystyle \nabla \cdot (- \boldsymbol{ D}_{AB} \cdot \nabla C_A)+ \frac{\partial C_A}{\partial t} = \dot{N_A} .$ (124)

Accordingly, by comparison with the heat equation, the correspondence in Table  (15) arises.


Table 15: 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$


next up previous contents
Next: Aerodynamic Networks Up: Types of analysis Previous: Stationary groundwater flow   Contents
guido dhondt 2012-10-06