9.1. Diffusion Fundamentals

The diffusion process is governed by the first Fick's law:

(9–1)

It is also governed by the mass balance equation:

(9–2)

where:

{J} = diffusion flux vector
Dxx, Dyy, and Dzz = the diffusivity coefficients in the element's X, Y, and Z directions, respectively (input as DXX, DYY, DZZ on MP command)
C = C(x,y,z,t) = concentration (CONC degree of freedom)
{v} = transport velocity vector (input as VELO on BF command for PLANE238, SOLID239, and SOLID240 only)
G = diffusing substance generation rate per unit volume (input as DGEN on BF or BFE commands)
= divergence operator
t = time

Substituting Equation 9–1 into Equation 9–2 produces the second Fick’s law augmented with the transport term:

(9–3)

The applicable boundary conditions and loads are:

  1. Specified concentration (input as CONC on D, DA, DK, DL commands). Initial concentration can also be applied using the CONC label with the IC command.

  2. Specified diffusion flux acting on a surface (input as DFLUX on SF or SFE commands):

    (9–4)

    where:

    {n} = unit outward normal vector
    {r} = diffusion flow vector

  3. Diffusing substance generation rate G (input as DGEN on BF or BFE commands).

  4. Transport velocity vector {v} (input as VELO on BF command for PLANE238, SOLID239, and SOLID240 only).

  5. Diffusion flow rate (input as RATE on F and FK commands).