A new approximate-analytical method to solve non-Fourier heat conduction problems

Mohammad J. Noroozi, Seyfolah Saedodin, Davood D. Ganji

Abstract


In this paper, the effect of laser as a heat source on a thin film was investigated. The non-Fourier heat conduction modelof Cattaneo-Vernotte was used for thermal analysis of the problem. The thermal conductivity was assumed temperaturedependent, which resulted in a non-linear equation and by assuming the role of laser as a heat source, a non-homogeneous
equation was obtained. The obtained equations were solved using approximate-analytical method of variational iteration method (VIM). It was concluded that the non-linear analysis is important in non-Fourier heat conduction problems. Significant differences were observed between the Fourier and non-Fourier solutions which stresses the importance of non-Fourier solutions in the similar problems.


Keywords


Non-Fourier; C-V Model; Variational Iteration Method; Laser Heating; Thin Film

Full Text:

PDF

References


Adomian, G. (1983). Stochastic systems, Academic Press, New York.

Bargmann, S. & Favata, A. (2014). Continuum mechanical modeling

of laser-pulsed heating in polycrystals: A multi-physics problem of

coupling diffusion, mechanics, and thermal waves. ZAMM - Journal

of Applied Mathematics and Mechanics / Zeitschrift fr Angewandte

Mathematik und Mechanik, 94(6):487498.

Bayat, M.,, Pakar, I. & Bayat, Mahdi. (2015). Nonlinear vibration

of mechanical systems by means of Homotopy perturbation method.

Kuwait Journal of Science, 42(3):6485.

Bergman, T.L. & Incropera, F.P. (2011). Introduction to heat transfer,

John Wiley & Sons, 6th Edition.

Blackwell, B.F. (1990). Temperature profile in semi-infinite body with

exponential source and convective boundary condition. Journal of Heat

Transfer, 112(3):567571.

Catteneo, C. (1958). A form of heat conduction equation, which

eliminates the paradox of instantaneous propagation. Compte rendus,

:431433.

Dogan, N. (2013). Numerical solution of chaotic Genesio system with

multi-step Laplace Adomian decomposition method. Kuwait Journal of

Science, 40(1):109121.

Elsayed, A.F. (2013). Comparison between variational iteration method

and homotopy perturbation method for thermal diffusion and diffusion

thermo effects of thixotropic fluid through biological tissues with laser

radiation existence. Applied Mathematical Modelling, 37(6):3660

Fong, E. & Lam, T.T. (2014). Asymmetrical collision of thermal waves

in thin films: An analytical solution. International Journal of Thermal

Sciences, 77:5565.

Garg, M. & Manohar, P.(2013). Analytical solution of the reactiondiffusion

equation with space-time fractional derivatives by means

of the generalized differential transform method. Kuwait Journal of

Science, 40(1):2334.

He, J. (1998a). An approximate solution technique depending on an

artificial parameter: A special example. Communications in Nonlinear

Science and Numerical Simulation, 3(2):9297.

He, J., 1. (1998b). Newton-like iteration method for solving algebraic

equations. Communications in Nonlinear Science and Numerical

Simulation, 3(2):106109.

He, J. (1999). Variational iteration method a kind of non-linear

analytical technique : some examples. International Journal of Nonlinear

Mechanics, 34(4):699708.

Lewandowska, M. (2001). Hyperbolic heat conduction in the semiinfinite

body with a time-dependent laser heat source. Heat and Mass

Transfer, 37(4):333342.

Liao, S.J. (1992). The proposed homotopy analysis technique for the

solution of nonlinear problems. Tong University.

Liu, C.-F., Kong, S.-S. & Yuan, S.J. (2013). Reconstructive schemes

for variational iteration method within Yang-Laplace transform with

application to fractal heat conduction problem. Thermal Science,

(3):715721.

Malekzadeh, P. & Rahideh, H. (2007). IDQ two-dimensional nonlinear

transient heat transfer analysis of variable section annular fins. Energy

Conversion and Management, 48(1):269276.

Malvandi, A. & Ganji, D.D. (2013). A general mathematical

expression of amperometric enzyme kinetics using Hes variational iteration method with Pad approximation. Journal of Electroanalytical

Chemistry, 711(0):3237.

Mishra, S.C. & Sahai, H. (2014). Analysis of non-Fourier conduction

and volumetric radiation in a concentric spherical shell using lattice

Boltzmann method and finite volume method. Heat and Mass Transfer,

:5166.

Odibat, Z.M. (2010). A study on the convergence of variational iteration

method. Mathematical and Computer Modelling, 51(9-10):11811192.

zi?ik, M.N. & Tzou, D.Y. (1994). On the wave theory in heat

conduction. Journal of Heat Transfer, 116(3):526535.

Peshkov, V. (1944). Second sound in Helium II. Journal of Physics,

USSR, 3, p.381.

Rahbari, I., Mortazavi, F. & Rahimian, M.H. (2014). High order

numerical simulation of non-Fourier heat conduction: An application of

numerical Laplace transform inversion. International Communications

in Heat and Mass Transfer, 51:5158.

Ramos, J.I. (2008). On the variational iteration method and other

iterative techniques for nonlinear differential equations. Applied

Mathematics and Computation, 199(1):3969.

Saadati, R., Dehghan, M., Vaezpour, S.M. & Saravi, M. (2009). The

convergence of Hes variational iteration method for solving integral

equations. Computers & Mathematics with Applications, 58(11-

:21672171.

Saedodin, S., Yaghoobi, H. & Torabi, M. (2011). Application of the

variational iteration method to nonlinear non-Fourier conduction heat

transfer equation with variable coefficient., 40(6):513523.

Saha Ray, S. & Gupta, A.K. (2014). Comparative analysis of variational

iteration method and Haar wavelet method for the numerical solutions

of BurgersHuxley and Huxley equations. Journal of Mathematical

Chemistry, 52(4):10661080.

Salkuyeh, D.K. (2008). Convergence of the variational iteration

method for solving linear systems of ODEs with constant coefficients.

Computers & Mathematics with Applications, 56(8):20272033.

Samaee, S.S., Yazdanpanah, O., Ganji, D.D. & Mofidi A.A. (2014).

Analytical solution for a suspension bridge by applying HPM and VIM.

International Journal of Computer Mathematics, 92(4):120.

Samaee, S.S., Yazdanpanah, O. & Ganji, D.D. (2015). New approaches

to identification of the Lagrange multiplier in the variational iteration

method. Journal of the Brazilian Society of Mechanical Sciences and

Engineering, 37(3):937944.

Sasmal, A. & Mishra, S.C. (2014). Analysis of non-Fourier conduction

and radiation in a differentially heated 2-D square cavity. International

Journal of Heat and Mass Transfer, 79:116125.

Shirmohammadi, R. (2011). Thermal response of microparticles due

to laser pulse heating. Nanoscale and Microscale Thermophysical

Engineering, 15(3):151164.

Tang, D.W. & Araki, N. (1999). Wavy, wavelike, diffusive thermal

responses of finite rigid slabs to high-speed heating of laser-pulses.

International Journal of Heat and Mass Transfer, 42(5):855860.

Tatari, M. & Dehghan, M. (2007). On the convergence of He s

variational iteration method. Journal of Computational and Applied

Mathematics, 207(424):121128.

Torabi, M., Yaghoobi, H. & Boubaker, K. (2013). Thermal analysis

of non-linear convective radiative hyperbolic lumped systems with

simultaneous variation of temperature-dependent specific heat and

surface emissivity by MsDTM and BPES. International Journal of

Thermophysics, 34(1):122138.

Torabi, M., Yaghoobi, H. & Saedodin, S. (2011). Assessment of

homotopy perturbation method in non-linear convective-radiative nonfourier

conduction. Thermal Science, 15(2):263274.

Torii, S. & Yang W. (2005). Heat transfer mechanisms in thin film

with laser heat source. International Journal of Heat and Mass Transfer,

:(3-4)537544.

Tung, M.M., Trujillo, M., Molina, J.A., Lpez, M.J. & Berjano,

E. J. (2009). Modeling the heating of biological tissue based on

the hyperbolic heat transfer equation. Mathematical and Computer

Modelling, 50(5-6):665672.

Vernotte, P. 1961). Some possible complications in the phenomenon of

thermal conduction. Compte Rendus, 247:21902191.

Wang, X. & Xu, X. (2002). Thermoelastic wave in metal induced by

ultrafast laser pulseS. Journal of Thermal Stresses, 25(5):457473.

Wu, G.-C. & Baleanu, D. (2013). Variational iteration method for the

Burgers flow with fractional derivativesNew Lagrange multipliers.

Applied Mathematical Modelling, 37(9):61836190.

Yang, S., Xiao, A. & Su, H. (2010). Convergence of the variational

iteration method for solving multi-order fractional differential equations.

Computers & Mathematics with Applications, 60(10):28712879.

Zhao, W.T., Wu, J.H. & Chen, Z. (2014). Analysis of non-Fourier

heat conduction in a solid sphere under arbitrary surface temperature

change. Archive of Applied Mechanics, 84(4):505518.

Zhou, J.K. (1986). Differential transform and its applications for

electrical circuIts. Huarjung University Press.

Zubair, S.M. & Chaudhry, M.A. (1996). Heat conduction in a semiinfinite

solid due to time-dependent laser source. International Journal

of Heat and Mass Transfer, 39(14):30673074.


Refbacks

  • There are currently no refbacks.