Fick's law

From Chemepedia

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:

            (1)

where

  • is the mass diffusion flux, which is expressed as the mass of per unit area per unit time.
  • is the mass density of the mixture.
  • is the diffusion coefficient.
  • is the mass fraction of A in the mixture.
  • is position, the dimension of which is length.

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

            (2)

where denotes the diffusion flux vector.

Fick's law on a mole basis

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

            (3)

where

  • is the molar diffusion flux, which is expressed as the moles of per unit area per unit time.
  • is the molar concentration of the mixture.
  • is the mole concentration of A.

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

Mass basis

The convective flux of on a mass basis is

            (4)

which represents the transport of species due to both bulk flow and diffusion. Here, is the mass-average velocity. In a multicomponent fluid with distinct species, the mass-average velocity is

            (5)

where

  • is the weight fraction of species .
  • is the velocity of species .
  • is the total density of the mixture.

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

            (6)

Mole basis

The convective flux of on a mole basis is

            (7)

which represents the transport of species due to both bulk flow and diffusion. Here, is the mole-average velocity. In a multicomponent fluid with distinct species, the mole-average velocity is

            (8)

where

  • is the mole fraction of species .
  • is the velocity of species .
  • is the total mole concentration of the mixture.

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

            (9)