Hyperbolic Heat Conduction Equation

The lagging response in time between the heat flux vector and the temperature gradient observed by Bertman and Sandiford (1970) requires a new look into the macroscopic relationship between the heat flux and the temperature gradient. The following constitutive relationship is proposed for heat flux  

The above equation indicates that the temperature gradient established at time t results in a heat flux vector at a later time t + t. 

The constitutive relationship proposed in equation (8.11) can be expanded using Taylor s expansion as 

When T is small, second-order term can be neglected. Thus, we have 

This is same as the Cattaneo type of model originally developed for gases. This assumes flnite speed of heat propagation contrary to the infinite propagation speed assumption of Fourier s model (zero relaxation time). 

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This paper deals with thermal wave behavior during frmisient heat conduction in a film (solid plate) subjected to a laser heat source with various time characteristics from botii side surfaces. Emphasis is placed on the effect of the time characteristics of the laser heat source (constant, pulsed and periodic) on tiiermal wave propagation. Analytical solutions are obtained by memis of a numerical technique based on MacCormack s predictor-corrector scheme to solve the non-Fourier, hyperbolic heat conduction equation. 

Heat waves have been theoretically studied in a very thin film subjected to a laser heat source and a sudden symmetric temperature change at two side walls. The non-Fourier, hyperbolic heat conduction equation is solved using a numerical technique based on MacCormak s predictor-corrector scheme. Results have been obtained for ftie propagation process, magnitude and shape of thermal waves and the range of film ftiickness Mid duration time wiftiin which heat propagates as wave. 

Baumeister, K.J. and Hamill, T.D. (1969) Hyperbolic heat conduction equation - a solution for the semiinfinite body problem. Journal of Heat Transfer, Vol. 91, pp. 543-548. 

Vick, B., and Ozisik, M.N. (1983) Growth and Decay of a Thermal Pulse Predicted by the Hyperbolic Heat Conduction Equation, Journal of Heat Transfer, Vol. 105, pp. 902-907. 

Kinetic theory is introduced and developed as the initial step toward understanding microscopic transport phenomena. It is used to develop relations for the thermal conductivity which are compared to experimental measurements for a variety of solids. Next, it is shown that if the time- or length scale of the phenomena are on the order of those for scattering, kinetic theory cannot be used but instead Boltzmann transport theory should be used. It was shown that the Boltzmann transport equation (BTE) is fundamental since it forms the basis for a vast variety of transport laws such as the Fourier law of heat conduction, Ohm s law of electrical conduction, and hyperbolic heat conduction equation. In addition, for an ensemble of particles for which the particle number is conserved, such as in molecules, electrons, holes, and so forth, the BTE forms the basis for mass, momentum, and energy conservation equa-... 

Baumeister and Hamill [32] solved the hyperbolic heat conduction equation in a semi-infinite medium subjected to a step change in temperature at one of its ends using the method of Laplace transform. The space-integrated expression for the temperature in the Laplace domain had the inversion readily available within the tables. This expression was differentiated using Leibniz s rule, and the resulting temperature distribution was given for x > X as... 

The application of Fourier s conduction equation and hyperbolic heat conduction equation to the transient heating of semi-infinite solid is discussed in the following section. 

The application of hyperbolic heat conduction equation is shown here using a pulse heating example of semi-infinite solid. The wall at x = 0 is impulsively stepped to a temperature (Figure 8.4). 

The density of state D e) is the number of states of particle between energy e and e + de). Fourier s law, Ohm s law, Pick s law, hyperbolic heat conduction equation, and mass, momentum, and energy equation can be derived from the BTE. 

The above equation is the hyperbolic heat conduction equation. 

The above phenomena me physically miomalous and can be remedied through the introduction of a hyperbolic equation based on a relaxation model for heat conduction, which accounts for a finite thermal propagation speed. Recently, considerable interest has been generated toward the hyperbolic heat conduction (HHC) equation and its potential applications in engineering and technology. A comprehensive survey of the relevant literature is available in reference [6]. Some researchers dealt with wave characteristics and finite propagation speed in transient heat transfer conduction [3], [7], [8], [9] and [10]. Several analytical and numerical solutions of the HHC equation have been presented in the literature. 

Soles CL, Yee AF (2000) A discussion of the molecular mechanisms of moisture transput in epoxy resins. J Polym Sci B Polym Phys 38(5) 792-802 Soles CL, Chang FT, Gidley DW, Yee AF (2000) Contributions of the nanovoid structtue to the kinetics of moisture transport in epoxy resins. J Polym Sci B Polym Phys 38(5) 776-791 Suh D, Ku M, Nam J, Kim B, Yoon S (2001) Equilibrium water uptake of epoxy/carbon fiber composites in hygrothermal environmental conditions. J Compos Mater 35(3) 264—278 Taitel Y (1972) On the parabolic, hyperbolic and discrete formulation of the heat conduction equation, hit J Heat Mass Transf 15(2) 369-371... 

The hyperbolic heat conduction governing equation for the ID problem is... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear parabolic partial differential equation. In this chapter, we describe how one can arrive at the analytical solutions for linear first order hyperbolic partial differential equations and parabolic partial differential equations in finite domains using the Laplace transform technique. 

Temperature Distribution in a Hollow Sphere. Derive Eq. (4.2-14) for the steady-state conduction of heat in a hollow sphere. Also, derive an equation which shows that the temperature varies hyperbolically with the radius r. 

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