Theory for the Thermoelectric Effect Interface
The Thermoelectric Effect Interface implements thermoelectric effect, which is the direct conversion of temperature differences to electric voltage or vice versa. Devices such as thermoelectric coolers for electronic cooling or portable refrigerators rely on this effect. While Joule heating (resistive heating) is an irreversible phenomenon, the thermoelectric effect is, in principle, reversible.
Historically, the thermoelectric effect is known by three different names, reflecting its discovery in experiments by Seebeck, Peltier, and Thomson. The Seebeck effect is the conversion of temperature differences into electricity, the Peltier effect is the conversion of electricity to temperature differences, and the Thomson effect is heat produced by the product of current density and temperature gradients. These effects are thermodynamically related by the Thomson relations:
where P is the Peltier coefficient (SI unit: V), S is the Seebeck coefficient (SI unit: V/K), T is the temperature (SI unit: K), and μTh is the Thomson coefficient (SI unit: V/K). These relations show that all coefficients can be considered different descriptions of one and the same quantity. The COMSOL formulation primarily uses the Seebeck coefficient. The Peltier coefficient is also used as an intermediate variable, but the Thomson coefficient is not used.
When simulating the thermoelectric effect, the following fluxes are the quantities of interest:
•
Conductive heat flux q, defined by
(4-121)
•
Electric current density J, defined by
(4-122)
Thermoelectric efficiency is measured by the figure of merit Z (SI unit: 1/K), defined as:
where σ is the electrical conductivity and k the thermal conductivity.
Some other quantities of relevance are the electric field E and the Joule heat source Q:
From these definitions, conservation of heat energy and electrical current in an immobile solid reads:
where ρ is the density, Cp the heat capacity, and Qj is the current source.
How the Seebeck, Peltier, and Thomson Effects are Included in the General Formulation
The general formulation of thermoelectric effect redefines the heat flux and the electric current according to Equation 4-121 and Equation 4-122, respectively. This formulation does not necessarily correspond to the formulation used when only a particular aspect of thermoelectric effect is considered: Seebeck, Peltier, or Thomson. This paragraph describes how these separated effects can be recognized in the general formulation.
Seebeck Effect
The Seebeck effect is described as the conversion of temperature gradient into electric current. The contribution of the Seebeck effect is defined as a current contribution
This formulation corresponds directly to Equation 4-122 used in the general formulation.
Peltier Effect
The Peltier effect is described as the conversion of t electric current in heat source or sink. It is defined as an heat source contribution
This contribution is obtained by developing the divergence of q term in the heat equation when q is defined following Equation 4-121.
Thomson Effect
The Thomson effect defines the heat source induced by a current in presence of a temperature gradient in thermoelectric material. The heat source is defined by
This contribution is obtained again by developing the divergence of the q term in the heat equation when q is defined following Equation 4-121. This time consider the term −TJ ⋅ ∇S. Assuming that S is function of T, then: