Fick's law

Fick's law relates the molecular (diffusive) flux to its mass fraction or mole fraction gradient in a multicomponent mixture.

Fick's law on mass basis

In one dimension, Fick's law is expressed as:

$j_{A,z}=-\rho {{𝒟}}_A\frac{d{w}_A}{dz}$

(1)

where

• $j_{A,x}$ is the mass diffusion flux, which is expressed as the mass of $A$ per unit area per unit time.
• $\rho$ is the mass density of the mixture.
• ${{𝒟}}_A$ is the diffusion coefficient.
• $w_A$ is the mass fraction of A in the mixture.
• $z$ is position, the dimension of which is length.

In two or more dimensions we use $\mathrm{\nabla }$, the gradient operator, which is independent of the chosen coordinate system. The generalized form of Fick's equation is

${\mathbf{j}}_{\mathbf{A}}=-\rho {{𝒟}}_A\mathrm{\nabla }{w}_A$

(2)

where ${\mathbf{j}}_{\mathbf{A}}$ denotes the diffusion flux vector.

Fick's law on a mole basis

Fick's law can also be written on a mole basis:

$j_{\mathbf{A}}^{\mathbf{*}}=-c{{𝒟}}_A\mathrm{\nabla }{x}_A$

(3)

where

• $j_{\mathbf{A}}^{\mathbf{*}}$ is the molar diffusion flux, which is expressed as the moles of $A$ per unit area per unit time.
• $c$ is the molar concentration of the mixture.
• $x_A$ is the mole concentration of A.

Relationship between the convective flux, molecular flux, and species velocities

Mass basis

The convective flux of $A$ on a mass basis is

${\mathrm{\Phi }}_A={\rho }_A\mathbf{v}+{\mathbf{j}}_A$

(4)

which represents the transport of species $A$ due to both bulk flow and diffusion. Here, $\mathbf{v}$ is the mass-average velocity. In a multicomponent fluid with $N$ distinct species, the mass-average velocity is

$\mathbf{v}=\sum _{i=1,N}{w}_i{\mathbf{v}}_i=\frac{1}{\rho }\sum _{i=1,N}{\mathrm{\Phi }}_i$

(5)

where

• $w_i$ is the weight fraction of species $i$.
• ${\mathbf{v}}_i$ is the velocity of species $i$.
• $\rho$ is the total density of the mixture.

Hence, the molecular flux of species $A$ is related to the difference between the velocity of $A$ and the mass-average velocity of the fluid.

${\mathbf{j}}_{A_{}}={\rho }_A\left({\mathbf{v}}_A-\mathbf{v}\right)$

(6)

Mole basis

The convective flux of $A$ on a mole basis is

${\mathrm{\Phi }}_A^{*}=c_A{\mathbf{v}}^{*}+{\mathbf{j}}_A^{*}$

(7)

which represents the transport of species $A$ due to both bulk flow and diffusion. Here, ${\mathbf{v}}^{*}$ is the mole-average velocity. In a multicomponent fluid with $N$ distinct species, the mole-average velocity is

${\mathbf{v}}^{*}=\sum _{i=1,N}{x}_i{\mathbf{v}}_i=\frac{1}{c}\sum _{i=1,N}{\mathrm{\Phi }}_i^{*}$

(8)

where

• $x_i$ is the mole fraction of species $i$.
• ${\mathbf{v}}_i$ is the velocity of species $i$.
• $c$ is the total mole concentration of the mixture.

Hence, the molecular flux of species $A$ is related to the difference between the velocity of $A$ and the mole-average velocity of the fluid.

${\mathbf{j}}_A^{*}=c_A\left({\mathbf{v}}_A-{\mathbf{v}}^{*}\right)$

(9)