The capability to model thermoelectric effects exists in the following elements:

These elements support the Joule heating effect (irreversible), and the Seebeck, Peltier, and Thomson effects (reversible).

In addition to the above, the following elements support a basic thermoelectric analysis that takes into consideration Joule heating effect only:

Constitutive Equations of Thermoelectricity

The coupled thermoelectric constitutive equations (Landau and Lifshitz([355])) are:

(10–108)

(10–109)

Substituting [Π] with T[α] to further demonstrate the coupling between the above two equations,

(10–110)

(10–111)

where:

[Π] = Peltier coefficient matrix = T[α]

T = absolute temperature

{q} = heat flux vector (output as TF)

{J} = electric current density (output as JC for elements that support conduction current calculation)

{E} = electric field (output as EF)

α xx, α yy, α zz = Seebeck coefficients (input as SBKX, SBKY, SBKZ on MP command)

Kxx, Kyy, Kzz = thermal conductivities (input as KXX, KYY, KZZ on MP command)

ρxx, ρyy, ρzz = resistivity coefficients (input as RSVX, RSVY, RSVZ on MP command)

Note that the Thomson effect is associated with the temperature dependencies of the Seebeck coefficients (MPDATA,SBKX also SBKY, SBKZ).

Derivation of Thermoelectric Matrices

After the application of the variational principle to the equations of heat flow (Equation 6–1) and of continuity of electric charge (Equation 5–5) coupled by Equation 10–108 and Equation 10–109, the finite element equation of thermoelectricity becomes (Antonova and Looman([91])):

(10–112)

where:

[Kt] = element thermal conductivity matrix (defined by Equation 6–28)

[Ct] = element specific heat matrix (defined by Equation 6–28)

{Q} = sum of the element heat generation load and element convection surface heat flow vectors (defined by Equation 6–28)

[Kv] = element electrical conductivity coefficient matrix (defined by Equation 5–115)

[Cv] = element dielectric permittivity coefficient matrix (defined by Equation 5–115)

= Peltier heat contribution to the element thermal conductivity matrix

= element Seebeck coefficient coupling matrix

= element Peltier heat coupling matrix

= element Joule heat load vector

{N} = element shape functions

{I} = vector of nodal current load

The finite element equation Equation 10–112 is unsymmetric when Seebeck coefficients are defined by issuing MP,SBKX (also RSVY, RSVZ). To make it symmetric, use KEYOPT(2) = 1 to apply Seebeck current and Peltier heat coupling as load vectors.

When thermal conductivity, electrical resistivity, and Seebeck coefficients are temperature dependent (MPTEMP/MPDATA), the stiffness matrix [K] in Equation 10–112 includes terms associated with the derivatives with respect to temperature of these material properties. These terms that also make Equation 10–112 unsymmetric can be turned off by setting KEYOPT(2) = 1.