Performance coefficients of thermoelectric cascade modules and modules with segmented branches

The paper presents calculations of the performance coefficient for thermoelectric modules with segmented branches and cascade coolers in a wide range of temperature differences. The objects of calculation are a single-stage thermoelectric module with two-section branches and a two-stage thermoelectric cooler. The calculation of thermoelectric modules is carried out for the maximum performance coefficient mode. In the case of a single-stage module, the operation of one branch is considered. For a two-stage module, the number of branches in the first and second stages is the same. The length of the branches in sections and stages is the same. The calculation does not take into account the temperature losses at heat transfers and the Joule heat released during switching. The temperature dependences of thermoelectric parameters are not taken into account in analytical expressions, and are taken into account numerically (by the method of successive approximations) when calculating modules. The calculation results showed that a two-stage cooler is always energetically more advantageous than a cooler with two-section branches, and the greater the operating temperature difference of the module, the greater the difference in their maximum coefficients of performance. The advantage of a cascade cooler is due to the fact that each stage operates in the maximum coefficient of performance mode, and for a segmented branch it is impossible to ensure the maximum coefficient of performance of each section. The calculation results are confirmed by the results of measurements of the energy parameters of real thermoelectric modules in two operating modes ΔT = 77 and 55 K. Single-stage and two-stage thermoelectric modules are designed so that their coefficients of performance are maximum in these operating modes. For the mode ΔT = 77 K, the coefficient of performance of the two-stage module exceeds the coefficient of performance of the single-stage module by five times. When the temperature difference decreases to 55 K, the two-stage module remains a more energy-efficient solution. These results are important to consider for the competent design of thermoelectric modules.

1. Nolas G.S., Sharp J., Goldsmid H.J. Thermoelectrics: Basic principles and new materials developments. Springer Berlin; 2001. 293 p. https://doi.org/10.1007/978-3-662-04569-5

2. Semenyouk V.A., Nechiporuk O.L. Maximum temperature reduction in composite semiconductor thermocouples. Izvestiya vuzov: Energetika. 1976; (2): 105—110. (In Russ.)

3. Bian Zh., Shakouri A. Beating the maximum cooling limit with graded thermoelectric materials. Applied Physics Letters. 2006; 89(21): 212101. https://doi.org/10.1063/1.2396895

4. Ioffe A.F. Semiconductor thermoelements and thermoelectric cooling. London: Infosearch ltd.; 1957. 254 p.

5. Burnshtein A.I. The physical basis of the calculation of semiconductor thermoelectric devices. Moscow: Gosudarstvennoe izdatelstvo fiziki-tekhnicheskoi literatury; 1962. 135 p. (In Russ.)

6. Stil'bans L.S. Physics of semiconductors. Moscow: Sovetskoe radio; 1967. 452 p. (In Russ.)

7. Buist R.J. The extrinsic Thomson effect (ETE). In: Proceed. of XIV Inter. conf. on thermoelectrics. St. Petersburg, Russia. June 27–30, 1995. St. Petersburg: A.F. Ioffe Physical-Technical Institute; 1995. P. 301—304.

8. Lukishker E.M., Vainer A.L., Somkin M.N., Vodolagin V.Yu. Thermoelectric coolers. Moscow: Radio i svyaz; 1983. 176 p. (In Russ.)

9. Drabkin I. Optimization of thermoelectric cooling and generator batteries. LAP LAMBERT Academic Publishing; 2017. 60 p.

10. Harman T.C. Special techniques for measurement of thermoelectric properties. Journal of Applired Physics. 1958; 29(9): 1373—1374. https://doi.org/10.1063/1.1723445

11. Babin V.P., Gorodetskiy S.M. Thermoelectric modules quality testing by a manufacturer. In: Proceed. of XIV Inter. conf. on thermoelectrics. St. Petersburg, Russia. June 27–30, 1995. St. Petersburg: A.F. Ioffe Physical-Technical Institute; 1995. P. 338—339.

12. Buist R.J. Methodology for testing thermoelectric materials and devices. In: Rowe D.M. (ed.). CRC Handbook of Thermoelectrics. London: CRC Press LLC; 1995. P. 189—209.

13. Vainer A.L., Kolomoets N.V., Lukishker E.M., Rzevskiy V.M. Towards the theory of a composite thermoelectric element. Fizika i tekhnika poluprovodnikov. 1977; 11(3): 546—552. (In Russ.)

14. Semeniouk V.A., Svechnikova T.E., Ivanova L.D. Single stage thermoelectric coolers with temperature difference of 80 K. In: Proceed. of XIII Inter. conf. on thermoelectrics. Kansas City, MO, 1994. New York: AIP Press; 1994. P. 485—489.

15. Goldsmid H.J. Conversion efficiency and figure-of-merit. In: Rowe D.M. (ed.). CRC Handbook of Thermoelectrics. London: CRC Press; 1995. P. 19—25.

16. Vainer A.L. Cascade thermoelectric cold sources. Moscow: Sovetskoe radio; 1976. 137 p. (In Russ.)

17. Anatychuk L.I., Vikhor L.N., Malyshko V.V, Nekhai V.A. Cooling modules of segmented thermoelements. Journal of Thermoelectricity. 2006; (3): 64—71. (In Russ.)

18. Ball C.A., Jesser W.A., Maddux J.R. The distributed Peltier effect and its influence on cooling devices. In: Proceed. of XIV Inter. conf. on thermoelectrics. St. Petersburg, Russia. June 27–30, 1995. St. Petersburg: A.F. Ioffe Physical-Technical Institute; 1995. P. 305—309.

19. Drabkin I.A. Optimization of segmented cooling leg. Semiconductors. 2017; 51(7): 913—915. https://doi.org/10.1134/S1063782617070077

20. Marlow R., Burke E. Module design and fabrication. In: Rowe D.M. (ed.). CRC Handbook of Thermoelectrics; 1995. P. 597—607.

21. Mahan G., Sales B., Sharp J. Thermoelectric materials: New approaches to an old problem. Physics Today. 1997; 50(3): 42—47. https://doi.org/10.1063/1.881752

22. Galperin V.L. A Generalized procedure to study multistage thermoelectric cooler operated in the regime of maximum energy efficiency. In: Proceed. of XIV Inter. conf. on thermoelectrics. St. Petersburg, Russia. June 27–30, 1995. St. Petersburg: A.F. Ioffe Physical-Technical Institute; 1995. P. 442—445.

23. Lavrentev M.G., Drabkin I.A., Ershova L.B., Volkov M.P. Improved extruded thermoelectric materials. Journal of Electronic Materials. 2020; 49(7): 2937—2942. https://doi.org/10.1007/s11664-020-07988-0

24. Drabkin I.A. Coefficient of the perfomance of a segmented thermoelectric cooling leg. Semiconductors. 2019; 53(5): 678—681. https://doi.org/10.1134/S1063782619050051

25. Drabkin I.A., Ershova L.B. Comparison of different approaches to optimization of single-cascade thermoelectric modules. In: Proceed. of the 10th Inter. seminar "Thermoelectrics and Their Application". St. Petersburg, November 2006. St. Petersburg: FTI; 2006. P. 386—390. (In Russ.) 26. Drabkin I.A., Dashevskiy Z.M. The main energy ratios for the refrigeration leg taking into account the temperature dependences of thermoelectric parameters. In: Proceed. of the VII Inter. seminar on thermoelectricity. St. Petersburg; 2000. P. 292—297. (In Russ.)

26. Semeniouk V.A., Berzverkbov D.B. Modeling and minimization of intercascade thermal resistance in multi-stage thermoelectric cooler, XVI ICT '97. In: Proceed. ICT'97. 16th Inter. conf. on thermoelectrics (Cat. No. 97TH8291). Dresden, Germany, 26–29 August 1997. IEEE; 1997. P. 701—704. https://doi.org/10.1109/ICT.1997.667627