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Thermoelectric Thomson Relations Revisited for a Linear Energy Converter

  • Saul Gonzalez-Hernandez and Luis-Antonio Arias-Hernandez EMAIL logo
Published/Copyright: June 4, 2019
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Journal of Non-Equilibrium Thermodynamics
From the journal Journal of Non-Equilibrium Thermodynamics Volume 44 Issue 3

Abstract

In this paper we revisit the classic thermocouple model, as a Linear Irreversible Thermodynamic (LIT) energy converter. In this model we have two types of phenomenological coefficients: the first comes from some microscopic models, such as the coefficient associated with the electric conductivity, and the second comes from experimental facts, such as the coefficient associated with the Seebeck power. We show that in the last case, these coefficients can be related to the thermodynamic operation modes of the energy converter. These relations between the experimental phenomenological coefficients and the regimes of performance allow us to propose a first and a second Thomson-type relation, which give us 12 new relations between the Seebeck power, the Peltier heat and the Thomson heat. With this purpose we develop the idea of non-isothermal linear energy converters operated either “directly” (like a heat engine) or “inversely” (like a refrigerator). We analyze the energetics associated to these converters operating under steady states corresponding to different modes of performance, all of them satisfying the fundamental Onsager symmetry relations.

Keywords: first and second Thomson relations; non-isothermal energy converters; performance of energy conversion systems

Funding source: Comisión de Operación y Fomento de Actividades Académicas, Instituto Politécnico Nacional

Award Identifier / Grant number: 1750

Award Identifier / Grant number: 20171231

Award Identifier / Grant number: 5406

Funding source: Consejo Nacional de Ciencia y Tecnología

Award Identifier / Grant number: 16051

Funding source: Consejo Nacional de Ciencia y Tecnología

Award Identifier / Grant number: 275589

Funding statement: This work was supported in part by EDI-SIP-COFAA-IPN (Grant Numbers: 1750, 20171231, 5406), SNI-CONACYT (Grant Number: 16051) and PNPC-CONACYT (Grant Number: 275589), MÉXICO.

Acknowledgment

We thank Marco Antonio Ramirez Moreno and Fernando Angulo-Brown for stimulating discussions, useful suggestions and invaluable help in the preparation of the manuscript.

Appendix

The efficient power Pη is defined as Pη=P×η, which can be written in terms of generalized fluxes and forces as follows:

(65)Pη=TcJD1XD12JD2=ηc2xx+q2qx+1L22X2

or

(66)Pη∗=ηcxx+q2qx+1.

Finding the maximum of eq. (65) with respect to x, it is possible to obtain the maximum efficient power force ratio xMPη (see Figure 5), i. e.,

(67)xMPηq=461−q2+q24−1+q22q;

we derive the energetics of the D-LEC from eq. (67).

References

[1] J. W. Stucki, The optimal efficiency and the economic degrees of coupling of oxidative phosphorylation, Eur. J. Biochem. 109 (1980), 269–283.10.1111/j.1432-1033.1980.tb04792.xSearch in Google Scholar PubMed

[2] S. R. Caplan and A. Essig, Bioenergetics and Linear Nonequilibrium Thermodynamics: The Steady State, 1st ed., Harvad University Press, Cambridge, MA, 1983.10.4159/harvard.9780674732063Search in Google Scholar

[3] M. Santillan, L. A. Arias-Hernandez and F. Angulo-Brown, Some optimization criteria for biological systems in linear irreversible thermodynamics, Il Nuovo Cim. Soc. Ital. Fis. D 19 (1999), 99–109.Search in Google Scholar

[4] L. A. Arias-Hernandez, F. Angulo-Brown and R. T. Paez-Hernandez, First-order irreversible thermodynamic approach to a simple energy converter, Phys. Rev. E 77 (2008), 011123.10.1103/PhysRevE.77.011123Search in Google Scholar PubMed

[5] L. Onsager, Reciprocal relations in irreversible processes. I, Phys. Rev. 37 (1931), 405–426.10.1103/PhysRev.37.405Search in Google Scholar

[6] L. Onsager, Reciprocal relations in irreversible processes. II, Phys. Rev. 38 (1931), 2265–2279.10.1103/PhysRev.38.2265Search in Google Scholar

[7] H. T. Odum and R. C. Pinkerton, Time’s speed regulator, Am. Sci. 43 (1955), 331–343.Search in Google Scholar

[8] G. Valencia-Ortega and L. A. Arias-Hernandez, Thermodynamic optimization of an electric circuit as a non-steady energy converter, J. Non-Equilib. Thermodyn. 42 (2017), 187–199.10.1515/jnet-2016-0037Search in Google Scholar

[9] F. Angulo-Brown, An ecological optimization criterion for finite-time heat engines, J. Appl. Phys. 69 (1991), 7465.10.1063/1.347562Search in Google Scholar

[10] K. H. Hoffmann, J. M. Burzler and S. Schubert, Endoreversible thermodynamics, J. Non-Equilib. Thermodyn. 22 (1997), 311–355.Search in Google Scholar

[11] L. A. Arias-Hernandez and F. Angulo-Brown, A general property of endoreversible thermal engines, J. Appl. Phys. 81 (1997), 2973–2979.10.1063/1.364090Search in Google Scholar

[12] A. Calvo-Hernández, A. Medina, J. M. M. Roco, J. A. White and S. Velasco, Unified optimization criterion for energy converters, Phys. Rev. E 63 (2001), 037102–1.10.1103/PhysRevE.63.037102Search in Google Scholar PubMed

[13] C. Van den Broeck, Thermodynamic efficiency at maximum power, Phys. Rev. Lett. 95 (2005), 190602.10.1103/PhysRevLett.95.190602Search in Google Scholar PubMed

[14] B. Jiménez de Cisneros, L. A. Arias-Hernandez and A. Calvo Hernández, Linear irreversible thermodynamics and coefficient of performance, Phys. Rev. E 73 (2006), 057103.10.1103/PhysRevE.73.057103Search in Google Scholar PubMed

[15] B. Jiménez de Cisneros and A. Calvo Hernández, Coupled heat devices in linear irreversible thermodynamics, Phys. Rev. E 77 (2008), 041127.10.1103/PhysRevE.77.041127Search in Google Scholar PubMed

[16] R. T. Páez-Hernández, P. Portillo-Díaz, Ladino-Luna, D. A. Ramírez-Rojas and J. C. Pacheco-Paez, An analytical study of the endoreversible Curzon–Ahlborn cycle for a non-linear heat transfer law, J. Non-Equilib. Thermodyn. 41 (2016), 19–27.10.1515/jnet-2015-0032Search in Google Scholar

[17] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, Wiley, 1985.Search in Google Scholar

[18] T. J. Seebeck, Ueber den Magnetismus der galvenischen Kette, Royal Prussian Academy of Science, 1821.Search in Google Scholar

[19] T. J. Seebeck, Ueber die magnetische Polarisation der Metalle und Erze durch Temperatur-Differenz, Ann. Phys. 82 (1826), 1–20.10.1002/andp.18260820102Search in Google Scholar

[20] J. C. A. Peltier, Nouvelles experiences sur la caloricite des courants électrique, Ann. Chim. Phys. 56 (1834), 371–386.Search in Google Scholar

[21] M. Tribus, Thermostatistics and Thermodynamics, D. Van Nostrand Company, Inc., 1961.Search in Google Scholar

[22] H. B. Callen, The application of Onsager’s reciprocal relations to thermoelectric, thermomagnetic, and galvanomagnetic effects, Phys. Rev. 73 (1948), 1349–1358.10.1103/PhysRev.73.1349Search in Google Scholar

[23] S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover Publications, 1984.Search in Google Scholar

[24] L. García Colín-Scherer and P. Goldstein Menache, La física de los procesos irreversibles, El Colegio Nacional, México, 2003.Search in Google Scholar

[25] C. Goupil, W. Seifert, K. Zabrocki, E. Müller and G. J. Snyder, Thermodynamics of thermoelectric phenomena and applications, Entropy 13 (2011), 1481–1517.10.3390/e13081481Search in Google Scholar

[26] F. Angulo-Brown and L. A. Arias-Hernandez, Reply to comment on: A general property of endoreversible thermal engines, J. Appl. Phys. 89 (2001), 1520–1521.10.1063/1.1335619Search in Google Scholar

[27] L. A. Arias-Hernandez, G. Ares de Parga and F. Angulo-Brown, On some nonendoreversible engine models with nonlinear heat transfer laws, Open Syst. Inf. Dyn. 10 (2003), 351–375.10.1023/B:OPSY.0000009556.27759.11Search in Google Scholar

[28] K. Wagner and K. H. Hoffmann, Endoreversible modeling of a PEM fuel cell, J. Non-Equilib. Thermodyn. 40 (2015), 283–294.10.1515/jnet-2015-0061Search in Google Scholar

[29] W. Köhler and K. I. Morozov, The Soret effect in liquid mixtures – A review, J. Non-Equilib. Thermodyn. 41 (2016), 151–197.10.1515/jnet-2016-0024Search in Google Scholar

[30] C. De Tomás, J. M. M. Roco, A. Calvo-Hernández, Y. Wang and Z. C. Tu, Low-dissipation heat devices: Unified trade-off optimization and bounds, Phys. Rev. E 87 (2013), 012105–1.10.1103/PhysRevE.87.012105Search in Google Scholar PubMed

[31] L. Partido Tornez, Aplicación del criterio omega y ecológico generalizados a diferentes convertidores de energía, Tesis de Maestría, ESFM-IPN, México, 2006.Search in Google Scholar

[32] International Energy Agency, Energy Technology Perspectives 2012, OECD/IEA, Paris, 2012.Search in Google Scholar

[33] D. Ürge-Vorsatz, L. F. Cabeza, S. Serrano, C. Barreneche and K. Petrichenko, Heating and cooling energy trends and drivers in buildings, Renew. Sustain. Rev. 41 (2015), 85–98.10.1016/j.rser.2014.08.039Search in Google Scholar

[34] M. Santamouris, Cooling the buildings – past, present and future, Energy Build. 128 (2016), 617–638.10.1016/j.enbuild.2016.07.034Search in Google Scholar

[35] S. Serrano, D. Ürge-Vorsatz, C. Barreneche, A. Palacios and L. F. Cabeza, Heating and cooling energy trends and drivers in Europe, Energy 119 (2017), 425–434.10.1016/j.energy.2016.12.080Search in Google Scholar

[36] W. Yourgrau, A. van der Merwe and G. Raw, Treatise on Irreversible and Statistical Thermophysics: An Introduction to Nonclassical Thermodynamics, Dover Publications Inc., N. Y., 1982.Search in Google Scholar

Received: 2017-12-20
Revised: 2019-04-10
Accepted: 2019-05-10
Published Online: 2019-06-04
Published in Print: 2019-07-26

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/jnet-2017-0068
Keywords for this article
first and second Thomson relations; non-isothermal energy converters; performance of energy conversion systems

Articles in the same Issue

  1. Frontmatter
  2. Special Issue Articles of the International Workshop on Non-Equilibrium Thermodynamics IWNET2018
  3. Editorial
  4. Entropy and Entropy Production in Multiscale Dynamics
  5. Micro-Inertia Effects in Material Flow
  6. Kluitenberg–Verhás Rheology of Solids in the GENERIC Framework
  7. A Rigorous Scattering Approach to Quasifree Fermionic Systems out of Equilibrium
  8. Regular Research Articles
  9. Temperature changes accompanying signal propagation in axons
  10. Development of Hybrid ANFIS–CFD Model for Design and Optimization of Membrane Separation of Benzoic Acid
  11. On the Combined Use of Friction Matrices and Dissipation Potentials in Thermodynamic Modeling
  12. Focalization of Heat Waves in an Inhomogeneous System
  13. Thermoelectric Thomson Relations Revisited for a Linear Energy Converter
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