Hyperbolic heat equation

Local thermodynamic equilibrium in space and time is inherently assumed in the kinetic theory formulation. The length scale that is characteristic of this volume is i whereas the timescale is xr. When either L i, ir or t x, xr or both, the kinetic theory breaks down because local thermodynamic equilibrium cannot be defined within the system. A more fundamental theory is required. The Boltzmann transport equation is a result of such a theory. Its generality is impressive since macroscopic transport behavior such as the Fourier law, Ohm s law, Fick s law, and the hyperbolic heat equation can be derived from this in the macroscale limit. In addition, transport equations such as equation of radiative transfer as well as the set of conservation equations of mass, momentum, and energy can all be derived from the Boltzmann transport equation (BTE). Some of the derivations are shown here. 

Bright, T.J., Zhang, Z.M. Common misperceptions of the hyperbolic heat equation. J. Ther-mophys. Heat Transfer 23(3), 601-607 (2009)... 

A brief review of the different classes of second-order partial differential equations is appropriate. Parabolic PDFs are typified by the heat equation duldx = Dd uldy, hyperbolic equations by the wave equation d uldx = c d utdy, and elliptical equations by Laplace s equation d utdx + d uldy = 0. x and y are spatial coordinates, and u is representative of forces, stresses, or similar quantities. The diffusion coefficient D in the heat equation must have units of length, and the "sound speed" in the wave equation is dimensionless. 

The description of phenomena in a continuous medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of wave propagation are described by a class of partial differential equations called hyperbolic, and these are essentially different in their properties from other classes such as those that describe equilibrium ( elhptic ) or diffusion and heat transfer ( para-bohc ). Prototypes are ... 

In estimating the value of Ed by means of the transcendental equations (28), the circumstance utilized is that the variation of em for a given change in Tm is much less than the variation of exp(em) (31). Until now, only particular solutions have been available for the hyperbolic and linear heating schedules and for the first-order and second-order desorptions. They can be found for example in the fundamental papers by Redhead (31) and Carter (32) or in the review by Contour and Proud homme (106), and therefore will not be repeated here. Recently, a universal procedure for the... 

Hence, we arrive at the conclusion that only in the limit a - 0 the Hookean body is the ideal energy-elastic one (r = 0) and the uniform deformation of a real system is accompanied by thermal effects. Equation (19) shows also that the dependence of the parameter q (as well as to) on strain is a hyperbolic one and a, the phenomenological coefficient of thermal expansion in the unstrained state, is determined solely by the heat to work and the internal energy to work ratios. From Eqs. (17) and (18), we derive the internal energy of Hookean body... 

Assuming the validity of Adam-Gibbs equation for relaxation dynamics and the hyperbolic temperature dependence of heat capacity, the strength parameter is found to be inversely proportional to the change in heat capacity [see Eq. (2.10)] at the glass transition temperature [48,105]. 

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. 

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. 

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. 

Using the transfer function concept, Koppel (1967) derived the optimal control policy for a heat exchanger system described by hyperbolic partial differential equations using the lumped system approach. Koppel and Shih (1968) also presented a feedback interior control for a class of hyperbolic differential equations with distributed control. In an earlier paper Koppel e/ al. (1968) discussed the necessary conditions for the system with linear hyperbolic partial differential equations having a control which is independent of spatial coordinates. The optimal feedback-feedforward control law for linear hyperbolic systems, whose dynamical response to input variations is characterized by an initial pure time delay, was derived by Denn... 

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-... 

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