A method of calculating (usually for the purpose of maximizing) the power-conversion efficiency of a segmented thermoelectric generator is based on equations derived from the fundamental equations of thermoelectricity. Because it is directly traceable to first principles, the method provides physical explanations in addition to predictions of phenomena involved in segmentation. In comparison with the finite-element method used heretofore to predict (without being able to explain) the behavior of a segmented thermoelectric generator, this method is much simpler to implement in practice: in particular, the efficiency of a segmented thermoelectric generator can be estimated by evaluating equations using only handheld calculator with this method. In addition, the method provides for determination of cascading ratios. The concept of cascading is illustrated in the figure and the definition of the “cascading ratio” is defined in the figure caption.
An important aspect of the method is its approach to the issue of compatibility among segments, in combination with introduction of the concept of compatibility within a segment. Prior approaches involved the use of only averaged material properties. Two materials in direct contact could be examined for compatibility with each other, but there was no general framework for analysis of compatibility. The present method establishes such a framework.
The mathematical derivation of the method begins with the definition of reduced efficiency of a thermoelectric generator as the ratio between (1) its thermal-to-electric power-conversion efficiency and (2) its Carnot efficiency (the maximum efficiency theoretically attainable, given its hot- and cold-side temperatures). The derivation involves calculation of the reduced efficiency of a model thermoelectric generator for which the hot-side temperature is only infinitesimally greater than the cold-side temperature. The derivation includes consideration of the ratio (u) between the electric current and heat-conduction power and leads to the concept of compatibility factor (s) for a given thermoelectric material, defined as the value of u that maximizes the reduced efficiency of the aforementioned model thermoelectric generator. It turns out that s depends only on the absolute temperature (T) and on intrinsic properties of the material that may vary with the temperature. The equation for s is
where Z is the traditional thermoelectric figure of merit, defined as Z = α2/ρκ; α is the Seebeck coefficient; ρ is the electrical resistivity; and κ is the thermal conductivity.
For maximum efficiency, u should be equal to s, both within a single material, and throughout a segmented thermoelectric-generator leg as a whole. It is in this sense that s serves as a basis for assessing both compatibility among segments and compatibility within a segment (self-compatibility). Given that u remains relatively constant throughout the thermoelectric element, the degree to which s varies with temperature along a given segment or differs among adjacent segments thus serves as a measure of incompatibility that one strives to minimize.
The compatibility factor can further be used as a quantitative guide for deciding whether a thermoelectric material is better suited for segmentation or cascading. Cascading enables the use of a material that may be suitable for a given temperature stage but is incompatible for segmentation with one or more other materials in the same temperature stage. The simplest option that may be available in a given case is to choose the numbers of unicouples in both temperature stages such that each stage is operating at its optimal u value. Such a choice is embodied in the following expression for the cascading ratio:
g'/g = u'/u
where g is the number of unicouples per stage and the prime mark distinguishes one stage from the other.