Thermodynamics of Thermoelectric Phenomena and Applications

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Seebeck actually misinterpreted the observed effect as a thermomagnetic coupling. The correct interpretation of the effect was given later by Oersted. Nevertheless, some authors consider Alessandro Volta to be at the origin of the discovery of this thermoelectric effect [172,173]. The introduction of additional effects due to a magnetic field into the Onsager relations is possible, see e.g. the Nernst effect and Ettinghausen effect [21,22,35,117].

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The transport parameters are often used without subscripts: σ ≡ σT is the isothermal conductivity and κ ≡ κJ the thermal conductivity under zero current.

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Strictly speaking, a small interaction level is needed in order to reach thermalization of the gas.

The chemical potential is defined as the mean Gibbs free energy per particle. In TE materials, this holds for the electrochemical potential.

Normally e is the elementary charge. Here e is negative for electrons, but positive if holes are considered.

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The off-diagonal terms of the kinetic matrix are symmetric, only if the correct thermodynamic potentials of the system has been chosen. In the case of a Fermi gas the correct potentials are μe/T and 1/T, see Section 2.2 for an example and 5.1 for a counter-example.

This time reversal symmetry is broken under the application of Coriolis or magnetic force.

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The reader will take note of the naming conventions with respect to fluxes and flows: Q ˙ = ∂ Q ∂ t is the heat transfer rate or heat flow in units of 1 W. For the sake of simplicity the dot is sometimes omitted. Often used is the heat flux which is the heat transfer rate per cross-sectional area J Q = 1 A c Q ˙ = 1 A c ∂ Q ∂ t in units of 1 W/m2.

With the restriction to one type of semiconductor (p-type, the index can be omitted from the material coefficients in the following calculations, i.e., αp = α or Kp = K.

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The Joule effect induced by the circulation of the electrical current has been often considered to be spread over the hot and cold source in equal contributions 1 2 R in I 2 . Obviously, this is valid only under CPM condition, see Equations (43), but sufficient for most design.

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We follow here Sherman’s notation and use ϕ instead of C.O.P. in appropriate formulae.

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A more general definition of u seems possible when writing the relative current density in terms of fluctuating currents which are indeed 3D. In this context, particle and heat flow should be considered in an anisotropic medium where the material parameters are tensors, see e.g., [174].

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Power output is defined here according to thermodynamic rules: quantities input to the system are positive.

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The Carnot engine exchanges energy in a reversible form, that means after an infinite time. Therefore, the Carnot engine transforms energy efficiently but with zero power.

Replace in [116] Equation (9), right hand side, the ± sign by a minus sign valid for both the pump up and the pump down situation.

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In [116] and [153] reduced efficiencies η r ( g ) ≡ η r , η r ( c ) ≡ φ r are introduced for both TEG and TEC, respectively.

Seifert, W., Pluschke, V., Goupil, C., Zabrocki, K., Müller, E., and Snyder, G. (2011). Maximum performance in self-compatible thermoelectric elements. J. Mater. Res., in press.

The relative current density is defined by Equation (75); the 1D variants are u ( x ) = J κ T ′ ( x ) espectively u ( T ) = − J κ x ′ ( T ) , whereby the temperature gradient decides the sign of u. Note that T(x) peaks in the interior of the TE element only far above the optimum current for maximum efficiency and maximum coefficient of performance. Then u has a pole.

Note that Sherman et al. used the function ϵ(T) instead of optimal reduced efficiency.

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