Performance optimization of a thermoelectric generator element with linear, spatial material profiles in a one-dimension
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lfgang Seifert and Steffen Trimper Institute of Physics, University Halle-Wittenberg, D-06099 Halle, Germany (Received 15 December 2010; accepted 15 March 2011)
Graded and segmented thermoelectric elements are studied in order to improve the performance of thermogenerators that are exposed to a large temperature difference. The linear thermodynamics of irreversible processes is extended by assuming spatially dependent material parameters like the Seebeck coefficient, the electrical and thermal conductivities. For the particular case in which these transport coefficients exhibit a constant gradient, we present an analytical solution of the onedimensional thermal energy balance in terms of Bessel functions. Given linear spatial material profiles, we discuss the optimization of performance parameters like the electrical power Pel and the efficiency g of a graded thermogenerator element of fixed length and fixed boundary temperatures. The results are compared with the constant properties model, i.e., physically and chemically homogeneous material, as a suitable reference for performance evaluation.
I. INTRODUCTION
Although the direct conversion of heat flow into electrical power based on the occurrence of thermovoltage from a temperature difference known as the Seebeck effect, and likewise the inverse Peltier effect, which is the transport of heat directly linked to a current flow, have been discussed for a long time, there has been great progress in understanding these phenomena from a more microscopic point of view.1,2 More recently the thermoelectric (TE) spin transfer in textured magnets3 and the TE effect of Dirac fermions in graphene has been studied numerically.4 Despite the effort in microscopic modeling, the application of linear thermodynamics of irreversible processes within a continuum approach is still under debate, especially for graded and segmented TE elements. They are characterized by locally nonconstant transport coefficients like the Seebeck coefficient, the electrical and thermal conductivities (see, e.g., Refs. 5 and 6). One aim of this study is to improve the performance of thermogenerators (TEGs) that are exposed to a large temperature difference. In 1909 Altenkirch7 had already calculated the efficiency of a thermopile concerning the material properties that are needed to build practical devices. He provided the first evidence7,8 that good TE materials should have a large Seebeck coefficient S, a high electrical conductivity r (low electrical resistivity) to minimize Joule heat, and a low thermal conductivity j to retain heat a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2011.91 J. Mater. Res., Vol. 26, No. 15, Aug 14, 2011
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at the junctions and maintain a large temperature gradient. Later these qualitative aspects were embodied in the material figure of merit z 5 S2r/j.9 Note that there are correlations between the material coefficients like the Wiedemann–Franz law (connecting the electronic part of therma
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