The capability to model thermoelectric effects exists in the following elements:
| PLANE223 - 2-D 8-Node Coupled-Field Solid |
| SOLID226 - 3-D 20-Node Coupled-Field Solid |
| SOLID227 - 3-D 10-Node Coupled-Field Solid |
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:
| SOLID5 - 3-D 8-Node Coupled-Field Solid |
| LINK68 - 3-D 2-Node Coupled Thermal-Electric Line |
| SOLID98 - 3-D 10-Node Coupled-Field Solid |
| SHELL157 - 3-D 4-Node Thermal-Electric Shell |
Constitutive Equations of Thermoelectricity
The coupled thermoelectric constitutive equations (Landau and Lifshitz([359])) are:
 | (10–66) |
 | (10–67) |
Substituting [Π] with T[α] to further demonstrate the coupling between the above two equations,
 | (10–68) |
 | (10–69) |
where:
| [Π] = Peltier coefficient matrix = T[α] |
| T = absolute temperature |
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| {q} = heat flux vector (output as TF) |
| {J} = electric current density (output as JC for elements that support conduction current calculation) |
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| {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 10–66 and Equation 10–67, the finite element equation of thermoelectricity becomes (Antonova and Looman([90])):
 | (10–70) |
where:
| [Kt] = element thermal conductivity matrix (defined by Equation 6–22) |
| [Ct] = element specific heat matrix (defined by Equation 6–22) |
| {Q} = sum of the element heat generation load and element convection surface heat flow vectors (defined by Equation 6–22) |
| [Kv] = element electrical conductivity coefficient matrix (defined by Equation 5–115) |
| [Cv] = element dielectric permittivity coefficient matrix (defined by Equation 5–115) |
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| {N} = element shape functions |
| {I} = vector of nodal current load |