Modified Cattaneo-Vernotte equation for heat transfer in solids

Victor L. Mironov

Article ID: 8050
Vol 7, Issue 2, 2024

VIEWS - 366 (Abstract)

Abstract


We propose a modified relation between heat flux and temperature gradient, which leads to a second-order equation describing the evolution of temperature in solids with finite rate of propagation. A comparison of the temperature field spreading in the framework of Fourier, Cattaneo-Vernotte (CV) and modified Cattaneo-Vernotte (MCV) equations is discussed. The comparative analysis of MCV and Fourier solutions is carried out on the example of simple one-dimensional problem of a plate cooling.


Keywords


Non-Fourier thermal conductivity; modified Cattaneo-Vernotte equation; microscale heat transfer

Full Text:

PDF


References


1. Carslow HS, Jaeger JC. Conduction of heat in solids. Oxford University Press, Oxford, UK, 1959.

2. Christov CI, Jordan PM. Heat Conduction Paradox Involving Second-Sound Propagation in Moving Media. Physical Review Letters. 2005; 94(15). doi: 10.1103/physrevlett.94.154301

3. Chandrasekharaiah DS. Hyperbolic Thermoelasticity: A Review of Recent Literature. Applied Mechanics Reviews. 1998; 51(12): 705-729. doi: 10.1115/1.3098984

4. Joseph DD, Preziosi L. Heat waves. Reviews of Modern Physics. 1989; 61(1): 41-73. doi: 10.1103/revmodphys.61.41

5. Green AE, Naghdi PM. A re-examination of the basic postulates of thermomechanics. In: Proceedings of the Royal Society of London Series A: Mathematical and Physical Sciences. 1991; 432(1885): 171-194. doi: 10.1098/rspa.1991.0012

6. Cimmelli VA, Kosinski W, Saxton K. Modified Fourier law: Comparison of two approaches, Archives of Mechanics. 1992; 44, 409-415.

7. Ignaczak J. Soliton-like solutions in a nonlinear dynamic coupled thermoelasticity. Journal of Thermal Stresses. 1990; 13(1): 73-98. doi: 10.1080/01495739008927025

8. Tzou DY. A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales. Journal of Heat Transfer. 1995; 117(1): 8-16. doi: 10.1115/1.2822329

9. Christov CI. On frame indifferent formulation of the Maxwell–Cattaneo model of finite-speed heat conduction. Mechanics Research Communications. 2009; 36(4): 481-486. doi: 10.1016/j.mechrescom.2008.11.003

10. Oldroyd JG. On the formulation of rheological equations of state. In: Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences. 1950; 200(1063): 523-541. doi: 10.1098/rspa.1950.0035

11. Khan U, Ahmad S, Hayyat A, et al. On the Cattaneo–Christov Heat Flux Model and OHAM Analysis for Three Different Types of Nanofluids. Applied Sciences. 2020; 10(3): 886. doi: 10.3390/app10030886

12. Jafarimoghaddam A, Turkyilmazoglu M, Pop I. Threshold for the generalized Non-Fourier heat flux model: Universal closed form analytic solution. International Communications in Heat and Mass Transfer. 2021; 123: 105204. doi: 10.1016/j.icheatmasstransfer.2021.105204

13. Turkyilmazoglu M. Heat Transfer Enhancement Feature of the Non-Fourier Cattaneo–Christov Heat Flux Model. Journal of Heat Transfer. 2021; 143(9). doi: 10.1115/1.4051671

14. Joseph DD, Preziosi L. Addendum to the paper “Heat waves” [Rev. Mod. Phys. 61, 41 (1989)]. Reviews of Modern Physics. 1990; 62(2): 375-391. doi: 10.1103/revmodphys.62.375

15. Sobolev SL. On hyperbolic heat-mass transfer equation. International Journal of Heat and Mass Transfer. 2018; 122: 629-630. doi: 10.1016/j.ijheatmasstransfer.2018.02.022

16. Cattaneo C. On the Calculation of Certain Potentials and Their Intervention in the Solving of Particular Harmonic Problems (Italian). In: Proceedings of the Mathematical and Physical Seminar of the University of Modena and Reggio Emilia, 1948; 3, 29-45.

17. Cattaneo C. A form of heat equation which eliminates the paradox of instantaneous propagation. Comptes Rendus de l'Academie des Sciences. 1958; 247, 431-433.

18. Vernotte P. Paradoxes in the continuous theory of the heat equation. Comptes Rendus de l'Academie des Sciences. 1958; 246, 3154-3155.

19. Vernotte P. The true heat equation. Comptes Rendus de l'Academie des Sciences. 1958; 247, 2103.

20. O¨zis¸ik MN, Tzou DY. On the Wave Theory in Heat Conduction. Journal of Heat Transfer. 1994; 116(3): 526-535. doi: 10.1115/1.2910903

21. Mandrusiak GD. Analysis of Non-Fourier Conduction Waves from a Reciprocating Heat Source. Journal of Thermophysics and Heat Transfer. 1997; 11(1): 82-89. doi: 10.2514/2.6204

22. Xu M, Wang L. Thermal oscillation and resonance in dual-phase-lagging heat conduction, International Journal of Heat and Mass Transfer. 2002; 45(5), 1055-1061.

23. Barletta A, Zanchini E. Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady periodic electric field, International Journal of Heat and Mass Transfer. 1996; 39 (6), 1307-1315.

24. Özisik MN, Vick B. Propagation and reflection of thermal waves in a finite medium, International Journal of Heat and Mass Transfer. 1984; 27 (10), 1845-1854.

25. Tzou DY. Shock wave formation around a moving heat source in a solid with finite speed of heat propagation, International Journal of Heat and Mass Transfer. 1989; 32 (10), 1979-1987.

26. Carey GF, Tsai M. Hyperbolic heat transfer with reflection. Numerical Heat Transfer. 1982; 5(3): 309-327. doi: 10.1080/10407788208913451

27. Kheibari AK, Jafari M, Nazari MB. Propagation of heat wave in composite cylinder using Cattaneo- Vernotte theory. International Journal of Heat and Mass Transfer. 2020; 160: 120208. doi: 10.1016/j.ijheatmasstransfer.2020.120208

28. van der Merwe AJ, van Rensburg NFJ, Sieberhagen RH. Comparing the dual phase lag, Cattaneo-Vernotte and Fourier heat conduction models using modal analysis. Applied Mathematics and Computation. 2021; 396: 125934. doi: 10.1016/j.amc.2020.125934

29. Nosko O. Perfect thermal contact of hyperbolic conduction semispaces with an interfacial heat source. International Journal of Heat and Mass Transfer. 2021; 164: 120541. doi: 10.1016/j.ijheatmasstransfer.2020.120541

30. Yuvaraj R, Senthil Kumar D. Numerical simulation of thermal wave propagation and collision in thin film using finite element solution. Journal of Thermal Analysis and Calorimetry. 2020; 142(6): 2351-2369. doi: 10.1007/s10973-020-09346-y

31. Kovács R, Rogolino P. Numerical treatment of nonlinear Fourier and Maxwell-Cattaneo-Vernotte heat transport equations. International Journal of Heat and Mass Transfer. 2020; 150: 119281. doi: 10.1016/j.ijheatmasstransfer.2019.119281

32. Auriault JL. The paradox of fourier heat equation: A theoretical refutation. International Journal of Engineering Science. 2017; 118: 82-88. doi: 10.1016/j.ijengsci.2017.06.006

33. Maillet D. A review of the models using the Cattaneo and Vernotte hyperbolic heat equation and their experimental validation. International Journal of Thermal Sciences. 2019; 139: 424-432. doi: 10.1016/j.ijthermalsci.2019.02.021

34. Mehrer H. Diffusion in Solids. Springer Berlin Heidelberg; 2007. doi: 10.1007/978-3-540-71488-0

35. Litvinenko YE, Schlickeiser R. The telegraph equation for cosmic-ray transport with weak adiabatic focusing. Astronomy & Astrophysics. 2013; 554: A59. doi: 10.1051/0004-6361/201321327

36. Tautz RC, Lerche I. Application of the three-dimensional telegraph equation to cosmic-ray transport. Research in Astronomy and Astrophysics. 2016; 16(10): 162. doi: 10.1088/1674-4527/16/10/162

37. Zhang D, Ostoja-Starzewski M. Telegraph equation: two types of harmonic waves, a discontinuity wave, and a spectral finite element. Acta Mechanica. 2019; 230(5): 1725-1743. doi: 10.1007/s00707-018-2356-3

38. Sommerfeld A. Introduction to Partial Differential Equations. Partial Differential Equations in Physics. 1949; 32-62. doi: 10.1016/b978-0-12-654658-3.50006-9




DOI: https://doi.org/10.24294/tse.v7i2.8050

Refbacks

  • There are currently no refbacks.


Copyright (c) 2024 Victor L. Mironov

License URL: https://creativecommons.org/licenses/by/4.0/

This site is licensed under a Creative Commons Attribution 4.0 International License.