A Quantitative Model for the Thermocouple Effect Using Statistical and Quantum Mechanics

Abstract

This paper employs statistical and quantum mechanics to develop a model for the mechanism underlying the Seebeck effect. The conventional view of the equilibrium criterion for valence electrons in a material is that the Fermi Energy should be constant throughout the system. However, this criterion is an approximation and it is shown to be inadequate for thermocouple systems. An improved equilibrium criterion is developed by applying statistical and quantum mechanics to determine the total flow of electrons across an arbitrary boundary within a system. Dynamic equilibrium is then considered to be the situation where the Fermi Energy either side of the boundary is such that the flow of electrons in each direction is the same. This equilibrium criterion is then applied to the conditions along the thermocouple wires and at the junctions in order to generate a model for the Seebeck effect. The equations involved for calculating the electronic structure of a material cannot be solved analytically, so a solution is achieved using numeric models employing CASTEP code running on a Sun Beowulf cluster and iterative algorithms written in the Excel VBA language on a PC. The model is used to calculate the EMF versus temperature function for the gold versus platinum thermocouple, which is then compared with established experimental data.

Fully Screen Printed Carbon Black-Only Thermocouple and the Corresponding Seebeck Coefficients

“…However, carbon black does not have one uniquely defined Seebeck coefficient. The reason for this lies in the Fermi level of carbon black [4], which is affected by the density of states [5]. Expressed in classical material parameters, the Seebeck coefficient is influenced by charge carrier transport properties.…”

Towards an ab initio calculation of elemental thermocouple output

“…The quantities τ (E) and v(E), and their energy dependence, are not practically calculable, so the assumption is made that they do not vary with energy over the range selected by the Fermi occupation function (3). Then most of the terms in (6) cancel, and the thermopower dV /dT can be expressed in terms of the electron density of states (see also [7]):…”

“…As the temperature increases above absolute zero, this means the value of E F must change in order to ensure that the number of valence electrons is conserved. The value of E F can be determined by balancing the number of electrons able to take part in conduction (determined from the density of states) [7] with the number of electrons in a unit volume calculated from the density as yielded by the lattice parameter calculated by ABINIT: recalling that N (E) denotes the electron density of states, and that P is the (energy dependent) Fermi function of (3). N A is Avogadro's constant, ρ is the zero temperature metal density (determined from the lattice parameter calculated by ABINIT), A is the atomic mass, and there are L valence electrons per atom (10, 11, and 10 for Pt, Au, and Pd respectively).…”