Research Article
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An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators

Year 2021, Volume: 24 Issue: 2, 184 - 192, 26.05.2021
https://doi.org/10.5541/ijot.877687

Abstract

We introduce a reduced-order endoreversible model of a Vuilleumier refrigerator for waste heat recovery. Based on the Vuilleumier cycle, in this refrigerator a working gas is alternately displaced between three subsystems that are in thermal contact with external heat reservoirs. Regarding refrigeration performance, very crucial components of the Vuilleumier machine are its two regenerators. For obtaining a sufficiently accurate model of the Vuilleumier machine, it is hence essential to incorporate a proper description of the regenerators. This can be achieved by using one-dimensional continuum models, e.g. with a finite volume approach, which brings about a large number of degrees of freedom and significant numerical effort. As opposed to that, the model presented in this paper utilizes a novel modeling ansatz for the regenerators that reduces the number of degrees of freedom per regenerator to three. It leads to a considerable reduction in numerical effort and computation time and is hence predestined for applications like design and control optimizations. For an exemplary set of design parameters and operational conditions, we validate the model against a detailed finite volume model of the regenerators in order to work out limitations and perspectives.

Supporting Institution

German Federal Ministry of Education and Research

Project Number

FKZ01LY1706B

Thanks

The authors thank the German Federal Ministry of Education and Research for supporting this work carried out within the framework of “KMU-innovativ”, support code FKZ01LY1706B.

References

  • R. Stirling, “Inventions for diminishing the consumption of fuel and in particular an engine capable of being applied to the moving of machinery on a principle entirely new,” British Patent 4081, 1816.
  • G.T. Reader, “Stirling regenerators,” Heat Transf. Eng., 15, 19-25, 1994.
  • H. Hausen, “Über die Theorie des Wärmeaustausches in Regeneratoren,” Zeitschr. f. angew. Math. und Mech., 9, 173-200, 1929.
  • M. Tanaka, I. Yamashita, F. Chisaka, “Flow and heat transfer characteristics of the Stirling engine regenerator in an oscillating flow,” JSME Int. J., 33, 283-289, 1990.
  • A.J. Willmott, “The development of thermal regenerator theory 1931 – the present,” J. Inst. Energy, 66, 54-70, 1993.
  • A.J. Organ, “Transient thermal performance of the Stirling engine wire regenerator,” Proc. R. Soc. Lond. A, 444, 53-72, 1994.
  • K. Matsumoto, M. Shiino, “Thermal regenerator analysis: analytical solution for effectiveness and entropy production in regenerative process,” Cryogenics, 29, 888-894, 1989.
  • J.A. Wills, T. Bello-Ochende, “Exergy analysis and optimization of an alpha type Stirling engine using the implicit filtering algorithm,” Front. Mech. Eng., 3, 21, 2017.
  • H.D. Kuehl, S. Schulz, “A 2nd order regenerator model including flow dispersion and bypass losses,” in IECEC 96: 31st Intersociety Energy Conversion Engineering Conference, Washington, DC, USA, pp. 1343-1348, 1996.
  • T.J. Lambertson, “Performance factors of a periodic-flow heat exchanger,” Trans. Am. Soc. Mech. Eng., 159, 586-592, 1958.
  • A.J. Willmott, “Digital computer simulation of a thermal regenerator,” Int. J. Heat Mass Transf., 7, 1291-1302, 1964.
  • Urieli, I. (1977). A computer simulation of Stirling cycle machines (Doctoral dissertation), University of Witwatersrand, Johannesburg, South Africa.
  • N. Andersson, L.-E. Eriksson, M. Nilsson, “Numerical simulation of Stirling engines using an unsteady quasi-one-dimensional approach,” J. Fluids Eng., 137, 051104, 2015.
  • R. Vuilleumier, “Method and apparatus for inducing heat changes,” U.S. Patent 1 275 507, 1918.
  • R. Paul, A. Khodja, and K.H. Hoffmann, “Nodal modeling of a Vuilleumier refrigerator for waste heat recovery on refrigerator trucks,” in ECOS 2019: Proceedings of the 32nd International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems, Wroclaw, Poland, pp. 97-108, 2019.
  • K.H. Hoffmann, J.M. Burzler, S. Schubert, “Endoreversible thermodynamics,” J. Non-Equilib. Thermodyn., 22, 311-355, 1997.
  • K.H. Hoffmann, J.M. Burzler, A. Fischer, M. Schaller, S. Schubert, “Optimal process paths for endoreversible systems,” J. Non-Equilib. Thermodyn., 28, 233-268, 2003.
  • K.H. Hoffmann, “An introduction to endoreversible thermodynamics,” Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat., 86, 1-19, 2008.
  • B. Andresen, R.S. Berry, A. Nitzan, P. Salamon, “Thermodynamics in finite time. I. The step-Carnot cycle,” Phys. Rev. A, 15, 2086-2093, 1977.
  • P. Salamon, B. Andresen, R.S. Berry, “Thermodynamics in finite time. II. Potentials for finite-time processes,” Phys. Rev. A, 15, 2094-2102, 1977.
  • B. Andresen, P. Salamon, R.S. Berry, “Thermodynamics in finite time: Extremals for imperfect heat engines,” J. Chem. Phys., 66, 1571-1577, 1977.
  • B. Andresen, P. Salamon, R.S. Berry, “Thermodynamics in finite time,” Phys. Today, 37, 62-70, 1984.
  • A. De Vos, “Reflections on the power delivered by endoreversible engines,” J. Phys. D Appl. Phys., 20, 232-236, 1987.
  • J. Chen, Z. Yan, “Optimal performance of an endoreversible-combined refrigeration cycle,” J. Appl. Phys., 63, 4795-4798, 1988.
  • V. Bǎdescu, “On the theoretical maximum efficiency of solar-radiation utilization,” Energy, 14, 571-573, 1989.
  • A. De Vos, “Is a solar cell an edoreversible engine?,” Sol. Cells, 31, 181-196, 1991.
  • Wagner, K. (2014). An extension to endoreversible thermodynamics for multi-extensity fluxes and chemical reaction processes (Doctoral dissertation), Chemnitz University of Technology, Chemnitz, Germany.
  • K. Wagner, K.H. Hoffmann, “Chemical reactions in endoreversible thermodynamics,” Eur. J. Phys., 37, 015101, 2016.
  • K.Schwalbe, K.H. Hoffmann, “Novikov engine with fluctuating heat bath temperature,” J. Non-Equilib. Thermodyn., 43, 141-150, 2018.
  • A. Tsirlin, I.A. Sukin, A. Balunov, K. Schwalbe, “The rule of temperature coefficients for selection of optimal separation sequence for multicomponent mixtures in thermal systems,” J. Non-Equilib. Thermodyn., 42, 359-369, 2017.
  • F. Marsik, B. Weigand, M. Thomas, O. Tucek, P. Novotny, “On the efficiency of electrochemical devices from the perspective of endoreversible thermo¬dynamics,” J. Non-Equilib. Thermodyn., 44, 425-437, 2019.
  • A. Fischer, K.H. Hoffmann, “Can a quantitative simulation of an Otto engine be accurately rendered by a simple Novikov model with heat leak?,” J. Non-Equilib. Thermodyn., 29, 9-28, 2004.
  • Z. Ding, L. Chen, F. Sun, “Finite time exergoeconomic performance for six endoreversible heat engine cycles: Unified description,” Appl. Math. Mod., 35, 728-736, 2011.
  • R.T. Paéz-Hernández, J.C. Chimal-Eguía, N. Sánchez-Salas, D. Ladino-Luna, “General properties for an Agrowal thermal engine,” J. Non-Equilib. Thermodyn., 43, 131-139, 2018.
  • R. Masser, K.H. Hoffmann, “Dissipative endoreversible engine with given efficiency,” Entropy, 21, 1117, 2019.
  • E. Açıkkalp, H. Yamık, “Modeling and optimization of maximum available work for irreversible gas power cycles with temperature dependent specific heat,” J. Non-Equilib. Thermodyn., 40, 25-39, 2015.
  • R. Masser, K.H. Hoffmann, “Endoreversible modeling of a hydraulic recuperation system,” Entropy, 22, 383, 2020.
  • A. De Vos, “Endoreversible models for the thermodynamics of computing,” Entropy, 22, 660, 2020.
  • W. Muschik, K.H. Hoffmann, “Modeling, simulation, and reconstruction of 2-reservoir heat-to-power processes in finite-time thermodynamics,” Entropy, 22, 997, 2020.
  • W. Muschik, K.H. Hoffmann, “Endoreversible thermodynamics: A tool for simulating and comparing processes of discrete systems,” J. Non-Equilib. Thermodyn., 31, 293-317, 2006.
  • S.C. Kaushik, S. Kumar, “Finite time thermodynamic analysis of endoreversible Stirling heat engine with regenerative losses,” Energy, 25, 989-1003, 2000.
  • I. Tlili, “Finite time thermodynamic evaluation of endoreversible Stirling heat engine at maximum power conditions,” Renew. Sust. Energ. Rev., 16, 2234-2241, 2012.
  • A. Sharma, S. K. Shukla, A. K. Rai, “Finite time thermodynamic analysis and optimization of solar-dish Stirling heat engine with regenerative losses,” Therm. Sci., 15, 995-1009, 2011.
  • S. Bhattacharyya, D. A. Blank, “Design considerations for a power optimized regenerative endoreversible Stirling cycle,” Int. J. Energy Res., 24, 539-547, 2000.
  • D. A.Blank, G. W. Davis, C. Wu, “Power optimization of an endoreversible stirling cycle with regeneration,” Energy, 19, 125-133, 1994.
  • R. Masser, A. Khodja, M. Scheunert, K. Schwalbe, A. Fischer, R. Paul, K.H. Hoffmann, “Optimized piston motion for an alpha-type Stirling engine,” Entropy, 22, 700, 2020.
  • M. Scheunert, R. Masser, A. Khodja, R. Paul, K. Schwalbe, A. Fischer, K.H. Hoffmann, “Power-optimized sinusoidal piston motion and its performance gain for an alpha-type Stirling engine with limited regeneration,” Energies, 13, 4564, 2020.
  • F. Wu, L. Chen, C. Wu, F. Sun, “Optimum performance of irreversible Stirling engine with imperfect regeneration,” Energy Convers. Mgmt., 39, 727-732, 1998.
  • Z.M. Ding, L.G. Chen, F.R. Sun, “Performance optimization of a linear phenomenological law system Stirling engine,” J. Energy Inst., 88, 36-42, 2015.
  • Paul, R. (2020). Optimal control of Stirling engines (Doctoral dissertation), Chemnitz University of Technology, Chemnitz, Germany.
  • T.J. Lu, H.A. Stone, M.F. Ashby, “Heat transfer in open-cell metal foams,” Acta mater., 49, 3619-3635, 1998.
Year 2021, Volume: 24 Issue: 2, 184 - 192, 26.05.2021
https://doi.org/10.5541/ijot.877687

Abstract

Project Number

FKZ01LY1706B

References

  • R. Stirling, “Inventions for diminishing the consumption of fuel and in particular an engine capable of being applied to the moving of machinery on a principle entirely new,” British Patent 4081, 1816.
  • G.T. Reader, “Stirling regenerators,” Heat Transf. Eng., 15, 19-25, 1994.
  • H. Hausen, “Über die Theorie des Wärmeaustausches in Regeneratoren,” Zeitschr. f. angew. Math. und Mech., 9, 173-200, 1929.
  • M. Tanaka, I. Yamashita, F. Chisaka, “Flow and heat transfer characteristics of the Stirling engine regenerator in an oscillating flow,” JSME Int. J., 33, 283-289, 1990.
  • A.J. Willmott, “The development of thermal regenerator theory 1931 – the present,” J. Inst. Energy, 66, 54-70, 1993.
  • A.J. Organ, “Transient thermal performance of the Stirling engine wire regenerator,” Proc. R. Soc. Lond. A, 444, 53-72, 1994.
  • K. Matsumoto, M. Shiino, “Thermal regenerator analysis: analytical solution for effectiveness and entropy production in regenerative process,” Cryogenics, 29, 888-894, 1989.
  • J.A. Wills, T. Bello-Ochende, “Exergy analysis and optimization of an alpha type Stirling engine using the implicit filtering algorithm,” Front. Mech. Eng., 3, 21, 2017.
  • H.D. Kuehl, S. Schulz, “A 2nd order regenerator model including flow dispersion and bypass losses,” in IECEC 96: 31st Intersociety Energy Conversion Engineering Conference, Washington, DC, USA, pp. 1343-1348, 1996.
  • T.J. Lambertson, “Performance factors of a periodic-flow heat exchanger,” Trans. Am. Soc. Mech. Eng., 159, 586-592, 1958.
  • A.J. Willmott, “Digital computer simulation of a thermal regenerator,” Int. J. Heat Mass Transf., 7, 1291-1302, 1964.
  • Urieli, I. (1977). A computer simulation of Stirling cycle machines (Doctoral dissertation), University of Witwatersrand, Johannesburg, South Africa.
  • N. Andersson, L.-E. Eriksson, M. Nilsson, “Numerical simulation of Stirling engines using an unsteady quasi-one-dimensional approach,” J. Fluids Eng., 137, 051104, 2015.
  • R. Vuilleumier, “Method and apparatus for inducing heat changes,” U.S. Patent 1 275 507, 1918.
  • R. Paul, A. Khodja, and K.H. Hoffmann, “Nodal modeling of a Vuilleumier refrigerator for waste heat recovery on refrigerator trucks,” in ECOS 2019: Proceedings of the 32nd International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems, Wroclaw, Poland, pp. 97-108, 2019.
  • K.H. Hoffmann, J.M. Burzler, S. Schubert, “Endoreversible thermodynamics,” J. Non-Equilib. Thermodyn., 22, 311-355, 1997.
  • K.H. Hoffmann, J.M. Burzler, A. Fischer, M. Schaller, S. Schubert, “Optimal process paths for endoreversible systems,” J. Non-Equilib. Thermodyn., 28, 233-268, 2003.
  • K.H. Hoffmann, “An introduction to endoreversible thermodynamics,” Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat., 86, 1-19, 2008.
  • B. Andresen, R.S. Berry, A. Nitzan, P. Salamon, “Thermodynamics in finite time. I. The step-Carnot cycle,” Phys. Rev. A, 15, 2086-2093, 1977.
  • P. Salamon, B. Andresen, R.S. Berry, “Thermodynamics in finite time. II. Potentials for finite-time processes,” Phys. Rev. A, 15, 2094-2102, 1977.
  • B. Andresen, P. Salamon, R.S. Berry, “Thermodynamics in finite time: Extremals for imperfect heat engines,” J. Chem. Phys., 66, 1571-1577, 1977.
  • B. Andresen, P. Salamon, R.S. Berry, “Thermodynamics in finite time,” Phys. Today, 37, 62-70, 1984.
  • A. De Vos, “Reflections on the power delivered by endoreversible engines,” J. Phys. D Appl. Phys., 20, 232-236, 1987.
  • J. Chen, Z. Yan, “Optimal performance of an endoreversible-combined refrigeration cycle,” J. Appl. Phys., 63, 4795-4798, 1988.
  • V. Bǎdescu, “On the theoretical maximum efficiency of solar-radiation utilization,” Energy, 14, 571-573, 1989.
  • A. De Vos, “Is a solar cell an edoreversible engine?,” Sol. Cells, 31, 181-196, 1991.
  • Wagner, K. (2014). An extension to endoreversible thermodynamics for multi-extensity fluxes and chemical reaction processes (Doctoral dissertation), Chemnitz University of Technology, Chemnitz, Germany.
  • K. Wagner, K.H. Hoffmann, “Chemical reactions in endoreversible thermodynamics,” Eur. J. Phys., 37, 015101, 2016.
  • K.Schwalbe, K.H. Hoffmann, “Novikov engine with fluctuating heat bath temperature,” J. Non-Equilib. Thermodyn., 43, 141-150, 2018.
  • A. Tsirlin, I.A. Sukin, A. Balunov, K. Schwalbe, “The rule of temperature coefficients for selection of optimal separation sequence for multicomponent mixtures in thermal systems,” J. Non-Equilib. Thermodyn., 42, 359-369, 2017.
  • F. Marsik, B. Weigand, M. Thomas, O. Tucek, P. Novotny, “On the efficiency of electrochemical devices from the perspective of endoreversible thermo¬dynamics,” J. Non-Equilib. Thermodyn., 44, 425-437, 2019.
  • A. Fischer, K.H. Hoffmann, “Can a quantitative simulation of an Otto engine be accurately rendered by a simple Novikov model with heat leak?,” J. Non-Equilib. Thermodyn., 29, 9-28, 2004.
  • Z. Ding, L. Chen, F. Sun, “Finite time exergoeconomic performance for six endoreversible heat engine cycles: Unified description,” Appl. Math. Mod., 35, 728-736, 2011.
  • R.T. Paéz-Hernández, J.C. Chimal-Eguía, N. Sánchez-Salas, D. Ladino-Luna, “General properties for an Agrowal thermal engine,” J. Non-Equilib. Thermodyn., 43, 131-139, 2018.
  • R. Masser, K.H. Hoffmann, “Dissipative endoreversible engine with given efficiency,” Entropy, 21, 1117, 2019.
  • E. Açıkkalp, H. Yamık, “Modeling and optimization of maximum available work for irreversible gas power cycles with temperature dependent specific heat,” J. Non-Equilib. Thermodyn., 40, 25-39, 2015.
  • R. Masser, K.H. Hoffmann, “Endoreversible modeling of a hydraulic recuperation system,” Entropy, 22, 383, 2020.
  • A. De Vos, “Endoreversible models for the thermodynamics of computing,” Entropy, 22, 660, 2020.
  • W. Muschik, K.H. Hoffmann, “Modeling, simulation, and reconstruction of 2-reservoir heat-to-power processes in finite-time thermodynamics,” Entropy, 22, 997, 2020.
  • W. Muschik, K.H. Hoffmann, “Endoreversible thermodynamics: A tool for simulating and comparing processes of discrete systems,” J. Non-Equilib. Thermodyn., 31, 293-317, 2006.
  • S.C. Kaushik, S. Kumar, “Finite time thermodynamic analysis of endoreversible Stirling heat engine with regenerative losses,” Energy, 25, 989-1003, 2000.
  • I. Tlili, “Finite time thermodynamic evaluation of endoreversible Stirling heat engine at maximum power conditions,” Renew. Sust. Energ. Rev., 16, 2234-2241, 2012.
  • A. Sharma, S. K. Shukla, A. K. Rai, “Finite time thermodynamic analysis and optimization of solar-dish Stirling heat engine with regenerative losses,” Therm. Sci., 15, 995-1009, 2011.
  • S. Bhattacharyya, D. A. Blank, “Design considerations for a power optimized regenerative endoreversible Stirling cycle,” Int. J. Energy Res., 24, 539-547, 2000.
  • D. A.Blank, G. W. Davis, C. Wu, “Power optimization of an endoreversible stirling cycle with regeneration,” Energy, 19, 125-133, 1994.
  • R. Masser, A. Khodja, M. Scheunert, K. Schwalbe, A. Fischer, R. Paul, K.H. Hoffmann, “Optimized piston motion for an alpha-type Stirling engine,” Entropy, 22, 700, 2020.
  • M. Scheunert, R. Masser, A. Khodja, R. Paul, K. Schwalbe, A. Fischer, K.H. Hoffmann, “Power-optimized sinusoidal piston motion and its performance gain for an alpha-type Stirling engine with limited regeneration,” Energies, 13, 4564, 2020.
  • F. Wu, L. Chen, C. Wu, F. Sun, “Optimum performance of irreversible Stirling engine with imperfect regeneration,” Energy Convers. Mgmt., 39, 727-732, 1998.
  • Z.M. Ding, L.G. Chen, F.R. Sun, “Performance optimization of a linear phenomenological law system Stirling engine,” J. Energy Inst., 88, 36-42, 2015.
  • Paul, R. (2020). Optimal control of Stirling engines (Doctoral dissertation), Chemnitz University of Technology, Chemnitz, Germany.
  • T.J. Lu, H.A. Stone, M.F. Ashby, “Heat transfer in open-cell metal foams,” Acta mater., 49, 3619-3635, 1998.
There are 51 citations in total.

Details

Primary Language English
Subjects Thermodynamics and Statistical Physics, Energy Systems Engineering (Other)
Journal Section Regular Original Research Article
Authors

Raphael Paul

Abdellah Khodja This is me

Karl Heinz Hoffmann This is me

Project Number FKZ01LY1706B
Publication Date May 26, 2021
Published in Issue Year 2021 Volume: 24 Issue: 2

Cite

APA Paul, R., Khodja, A., & Hoffmann, K. H. (2021). An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators. International Journal of Thermodynamics, 24(2), 184-192. https://doi.org/10.5541/ijot.877687
AMA Paul R, Khodja A, Hoffmann KH. An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators. International Journal of Thermodynamics. May 2021;24(2):184-192. doi:10.5541/ijot.877687
Chicago Paul, Raphael, Abdellah Khodja, and Karl Heinz Hoffmann. “An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators”. International Journal of Thermodynamics 24, no. 2 (May 2021): 184-92. https://doi.org/10.5541/ijot.877687.
EndNote Paul R, Khodja A, Hoffmann KH (May 1, 2021) An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators. International Journal of Thermodynamics 24 2 184–192.
IEEE R. Paul, A. Khodja, and K. H. Hoffmann, “An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators”, International Journal of Thermodynamics, vol. 24, no. 2, pp. 184–192, 2021, doi: 10.5541/ijot.877687.
ISNAD Paul, Raphael et al. “An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators”. International Journal of Thermodynamics 24/2 (May 2021), 184-192. https://doi.org/10.5541/ijot.877687.
JAMA Paul R, Khodja A, Hoffmann KH. An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators. International Journal of Thermodynamics. 2021;24:184–192.
MLA Paul, Raphael et al. “An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators”. International Journal of Thermodynamics, vol. 24, no. 2, 2021, pp. 184-92, doi:10.5541/ijot.877687.
Vancouver Paul R, Khodja A, Hoffmann KH. An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators. International Journal of Thermodynamics. 2021;24(2):184-92.