Joule Heating and Thermoelectricity

Joule (Ohmic) heating and thermoelectricity describe the relationship between electric currents and temperature changes in conducting materials.

Joule Heating

Electric currents flowing in resistive materials generate heat. Simcenter STAR-CCM+ accounts for this effect by adding a source term to the energy equation.

Joule’s Law gives the heat source per unit volume due to an electric current density $J$ :

Figure 1. EQUATION_DISPLAY

Q = J \cdot E

(4357)

where $E$ is the electric field. For harmonic fields (see Harmonic Time dependence), Simcenter STAR-CCM+ calculates a cycle-averaged heat source per unit volume, as:

Figure 2. EQUATION_DISPLAY

Q = \frac{1}{2} Re (\hat{J} \cdot {\hat{E}}^{*})

(4358)

where ${\hat{E}}^{*}$ is the complex conjugate of $\hat{E}$ (see Eqn. (4269)).

The total power that is dissipated from a volume of conducting material is:

Figure 3. EQUATION_DISPLAY

P = \int_{V} J \cdot E d V

(4359)

When modeling Joule heating, Simcenter STAR-CCM+ adds Eqn. (4359) as a source term to the energy equation (see Eqn. (1657) for fluids and Eqn. (1660) for solids).

On wall boundaries and contact interfaces with an electric resistance (see Eqn. (4279) and Eqn. (4281)), the heat source per unit area is:

Figure 4. EQUATION_DISPLAY

P = J_{n}^{2} R^{User}

(4360)

or, for harmonic fields (see Harmonic Time dependence):

Figure 5. EQUATION_DISPLAY

P = \frac{1}{2} [{Re ({\hat{J}}_{n})}^{2} + {I m ({\hat{J}}_{n})}^{2}] R^{User}

(4361)

The heat source per unit area contributes to the face neighboring cell center source term of the energy equation. On contact interfaces, the contribution to the cell center source term of the energy equation is weighted by the effective thermal conductivities of the neighboring physics continua.

In Simcenter STAR-CCM+, Joule (Ohmic) heating for Eulerian multiphase uses the effective conductivity $k_{i}^{e f f}$ to obtain the heat distribution that is proportionately allocated to each Eulerian phase, as:

Figure 6. EQUATION_DISPLAY

Q = k_{i}^{e f f} J \cdot E

(4362)

and:

Figure 7. EQUATION_DISPLAY

k_{i}^{e f f} = \frac{α_{i} ‖ k_{i} ‖}{\sum_{j} α_{j} ‖ k_{j} ‖}

(4363)

where $α$ is the phase volume fraction.

Thermoelectricity

Thermoelectricity refers to the combination of three effects—the Seebeck effect, the Peltier effect, and the Thomson effect. These effects describe the relationship between temperature gradients and electric voltage in solid conductors. Unlike Joule heating, thermoelectricity is a thermodynamically reversible process.

The Seebeck effect is the generation of electromotive force due to a temperature gradient:

Figure 8. EQUATION_DISPLAY

E_{e m f} = - α \nabla T

(4364)

where $α$ is the Seebeck coefficient of the material. This electromotive force contributes to the electric current density, modifying Ohm's law (see Eqn. (4228)):

Figure 9. EQUATION_DISPLAY

J = σ (E - α \nabla T)

(4365)

Simcenter STAR-CCM+ includes the additional term $- α \nabla T$ in Eqn. (4234) and, therefore, in the electric potential equation (Eqn. (4242)).

Typically, the Seebeck coefficient of a material is a function of temperature. Therefore, a temperature gradient can lead to a gradient in the Seebeck coefficient. This effect is known as the Thomson effect.

The Peltier effect describes the relationship between the electric current density and the heat flux under isothermal conditions:

Figure 10. EQUATION_DISPLAY

q = Π J

(4366)

where $q$ is the isothermal heat flux and $Π$ is the Peltier coefficient of the material, which is related to the Seebeck coefficient through the Kelvin relation:

Figure 11. EQUATION_DISPLAY

Π = α T

(4367)

When modeling thermoelectricity, Simcenter STAR-CCM+ adds the term:

Figure 12. EQUATION_DISPLAY

- \oint_{A} Π J \cdot d a

(4368)

to the right-hand side of the energy equation for the solid conductor (Eqn. (1660)).