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Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source

Published online by Cambridge University Press:  17 June 2016

M. R. Talaee  and
V. Sarafrazi
M. R. Talaee*
Affiliation:
Rolling StockSchool of Railway EngineeringIran University of Science and TechnologyTehran, Iran
V. Sarafrazi
Affiliation:
Rolling StockSchool of Railway EngineeringIran University of Science and TechnologyTehran, Iran
*
*Corresponding author ([email protected])
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Abstract

This paper is devoted to the analytical solution of three-dimensional hyperbolic heat conduction equation in a finite solid medium with rectangular cross-section under time dependent and non-uniform internal heat source. The closed form solution of both Fourier and non-Fourier profiles are introduced with Eigen function expansion method. The solution is applied for simple simulation of absorption of a continues laser in biological tissue. The results show the depth of laser absorption in tissue and considerable difference between the Fourier and Non-Fourier temperature profiles. In addition the solution can be applied as a verification branch for other numerical solutions.

Keywords

Hyperbolic heat conductionSource termLaser interaction of biological tissueEigen function expansion method
Type
Research Article
Information
Journal of Mechanics , Volume 33 , Issue 1 , February 2017 , pp. 65 - 75
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2017 

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References

1. Cattaneo, C., “A Form of Heat Conduction Equation Which Eliminates the Paradox of Instantaneous Propagation,” CompteRendus, 247, pp. 431–433 (1958).Google Scholar
2. Vernotte, P., “Les Paradoxes De La Théorie Continue De Léquation De La Chaleur,” Comptes Rendus Hebdomadaires Des Seances De Academie Des Sciences, 246, pp. 3154–3155 (1958).Google Scholar
3. Talaee, M. R., Sarafrazi, V. and Bakhshandeh, S., “Exact Analytical Hyperbolic Temperature Profile in a Three-Dimensional Media Under Pulsed Surface Heat Flux,” Journal of Mechanics, doi:10.1017/jmech.2015.97 (2015).Google Scholar
4. Lewandowska, M. and Malinowski, L., “An Analytical Solution of the Hyperbolic Heat Conduction Equation for the Case of a Finite Medium Symmetrically Heated on Both Sides,” International Communications in Heat and Mass Transfer, 33, pp. 61–69 (2006).Google Scholar
5. Lewandowska, M. and Malinowski, L., “Hyperbolic Heat Conduction in the Semi-Infinite Body with the Heat Source Which Capacity Linearly Depends on Temperature,” Heat and Mass Transfer, 33, pp. 389–393 (1998).Google Scholar
6. Al-Khairy, R. T. and Al-Ofey, Z. M., “Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium Under the Effect of Time-Dependent Laser Heat Source,” Journal of Applied Mathematics, doi:10.1155/2009/604695 (2009).Google Scholar
7. Yoo, J., Lim, C. H. and Kim, C., “Numerical Study of Thermal Wave Propagation in a Disc Slab Irradiated by Ultrashort Laser Pulses,” Journal of the Korean Physical Society, 47, pp. 63–67 (2005).Google Scholar
8. Zhang, D., Li, L., Li, Z., Guan, L. and Tan, X., “Non-Fourier Conduction Model with Thermal Source Term of Ultra Short High Power Pulsed Laser Ablation and Temperature Evolvement Before Melting,” Physica B: Condensed Matter, 364, pp. 285–293 (2005).Google Scholar
9. Torii, S. and Yang, W. J., “Heat Transfer Mechanisms in Thin Film with Laser Heat Source,” International Journal of Heat and Mass Transfer, 48, pp. 537–544 (2005).Google Scholar
10. Ahmadikia, H., et al. “Analytical Solution of Non-Fourier and Fourier Bioheat Transfer Analysis During Laser Irradiation of Skin Tissue,” Journal of Mechanical Science and Technology, 26, pp. 1937–1947 (2012).Google Scholar
11. Trujillo, M., Rivera, M. J. and Lopez Molina, J. A., “Analytical Thermal-Optic Model for Laser Heating of Biological Tissue Using the Hyperbolic Heat Transfer Equation,” Mathematical Medicine and Biology, 26, pp. 187–200 (2009).Google Scholar
12. Zubair, S. M. and Chaudhry, M. A., “Heat Conduction in a Semi-Infinite Solid Due to Time Dependent Laser Source,” International Journal of Heat and Mass Transfer, 39, pp. 3067–3074 (1996).Google Scholar
13. Talaee, M. R. and Atefi, G., “Non-Fourier Heat Conduction in a Finite Hollow Cylinder with Periodic Surface Heat Flux,” Archive of Applied Mechanics, 81, pp. 1793–1806 (2011).Google Scholar
14. Asmar, Nakhle H., Partial Differential Equations with Fourier Series and Boundary Value Problems, 2nd Edition, Pearson Prentice Hall, New Jersey (2004).Google Scholar
15. Simmons, G. Finlay, and Robertson, John S.. Differential Equations: With Applications and Historical Notes, McGraw-Hill, New York (1972).Google Scholar
16. Banerjee, A., Ogale, A. A., Das, C., Mitra, K. and Subramanian, C., “Temperature Distribution in Different Materials Due to Short Pulse Laser Irradiation,” Heat Transfer Engineering, 26, pp. 41–49 (2005).CrossRefGoogle Scholar
17. Ooi, E.-H. and Ang, W.-T., “A Boundary Element Model of the Human Eye Undergoing Laser Thermokeratoplasty,” Computers in Biology and Medicine, 38, pp. 727–737 (2008).Google Scholar