The capability to perform a coupled thermal-diffusion analysis exists in the following elements:

These elements support the effects of thermomigration (the transport of particles due to a temperature gradient) and temperature-dependent saturated concentration when used in coupled-field analyses having thermal and diffusion DOFs.

Constitutive Equations

In thermal-diffusion analyses, the diffusion flux {J} is coupled to temperature as follows:

(10–78)

where:

= diffusivity matrix

Dxx, Dyy, and Dzz = diffusivity coefficients in the element's X, Y, and Z directions, respectively (input as DXX, DYY, DZZ on MP command)

C = concentration; where

= normalized concentration (input/output as CONC)

= saturated concentration (input as MP,CSAT)

T = absolute temperature = Tc + Toff

Tc = current temperature (input/output as TEMP on D or BF commands)

Toff = offset temperature from absolute zero to zero (input on TOFFST command)

Q/k = heat transport constant (input as C3 on TBDATA command with TB,MIGR)

Q = particle heat of transport

k = Boltzmann constant

If saturated concentration is a function of temperature C_sat = C_sat(T), Equation 10–78 takes the form:

(10–79)

where the derivative of C_sat with respect to the temperature is numerically evaluated by the program.

For more information on Equation 10–78 and the related material constant input, see Migration Model in the Material Reference.

For more information on diffusion analysis, see Diffusion.

Derivation of Thermal-Diffusion Matrices

Applying the variational principle with respect to thermal and diffusion DOFs to the equations of heat flow Equation 6–1 and mass continuity equations Equation 9–7 coupled by the constitutive equation Equation 10–79, we obtain the following finite element matrix equation for the electric-diffusion analysis:

(10–80)

where:

{T} = nodal temperature vector (input/output as TEMP)

{C} = nodal concentration vector (input/output as CONC)

[Ct] = element specific heat matrix (defined by Equation 6–22)

[Cd] = element diffusion damping matrix (defined by Equation 9–9)

[Kt] = element thermal conductivity matrix (defined by Equation 6–22)

[Kd] = element diffusion conductivity matrix (defined by Equation 9–9)

= element transport conductivity matrix

= nonlinear part of the element transport conductivity matrix

= thermal gradient (output as TG)

= element thermal-diffusion conductivity matrix produced by

= nonlinear part of the element diffusion conductivity matrix associated with thermomigration

nonlinear part of the element diffusion conductivity matrix produced by

= element thermal-diffusion damping matrix produced by

{Q} = sum of the nodal heat generation and convection loads (input/output as HEAT)

{R} = nodal diffusion flow rate vector (input/output as RATE)

{N} = element shape functions

The finite element equation Equation 10–80 is unsymmetric. If a symmetric equation is desirable, the thermomigration and temperature-dependent C_sat coupling can be applied as a load vector by setting KEYOPT(2) = 1 for the coupled-field elements (PLANE223, SOLID226, SOLID227).