Damped-Wave Conduction and Relaxation in a Finite Sphere and Cylinder

S IX reasons were given by Sharma [1] to seek a generalized Fourier’s Law of Heat Conduction. Boley [2] showed that the addition of the second derivative in time of temperature to the governing equation is the only mathematical way to remove singularities found in the solution to parabolic heat conduction equations.When the relaxation time is zero, the generalized Fourier’s law of heat conduction will revert to the parabolic partial differential equation (PDE) for transient conduction from Fourier’s law. When the rate of temperature change over time is much greater than an exponential rise with time, e , the generalized Fourier’s law of heat conduction equation will revert to the wave equation (Tzou [3] and Sharma [4]). Eq. (1) is a hyperbolic partial differential equation, that is second order with respect to space and second order with respect to time. Reference to the generalized Fourier’s law of heat conduction can be traced back toMaxwell [5],Morse and Feshbach [6]. Cattaneo [7] and Vernotte [8] postulated the generalized Fourier’s law of heat conduction and relaxation equation independently. This equation can be used to account for the finite speed of heat. Reviews in the literature of the use of Eq. (1) have been provided by Joseph and Preziosi [9] andOzisik and Tzou [10]. Sharma [1,4,11–13] discussed the manifestation of the damped-wave transport and relaxation equation in industrial applications and provided bounded solutions within the constraints of the second law of thermodynamics. It was shown that the generalized Fourier’s law of heat conduction can be derived by including the acceleration term in the free-electron theory, the acceleration term in the Stokes–Einstein theory for molecular diffusion, by accounting for the accumulation term in the kinetic theory of gases and combining, in series, Hooke’s elastic element and Newton’s viscous element in the viscoelastic theory. The relaxation time was found to be one-third of the collision time of the electron and the obstacle. The velocity of heat was found to be identical with the velocity of mass derived from kinetic representation of pressure or theMaxwell representation of the speed ofmolecules. They derive a set of equations for length scales comparable with the mean free path of the molecule. Ali [14,15] used statistical mechanics and kinetic theory and derived the generalized Fourier’s law of heat conduction for monatomic and diatomic gases. Glass and McRae [16] looked at the variable specific heat and thermal relaxation parameter. Mitra et al. [17] presented experimental evidence of the wave nature of heat propagation in processedmeat and demonstrated that the hyperbolic heat conduction model is an accurate representation on a macroscopic level of the heat conduction process in such biological material. They report a relaxation time of the order of 16 s. Some investigators have raised some concerns about violations of the second law of thermodynamics by the generalized Fourier’s law of heat conduction (Bai and Lavine [18], Taitel [19], Zanchini [20], and Barletta and Zanchini [21]). Sharma [4] presented an analytical solution for the case of a finite slab subject to constant wall temperature. The final condition in time as the fourth condition for the second-order hyperbolic PDE governing equation was shown to result in well-bounded solutions. This means that caremust go into the choice of the conditions used in the boundaries of space and the initial and final time values. They have to be physically realistic such that equilibrium temperature is attained at steady state. Only for large relaxation times were oscillations found in the solution for temperature. These oscillations were found to be subcritical and damped. The time conditions used by Taitel [19] are unrealistic from the physical realities of heat transfer. That is why his solution exhibited a temperature overshoot. Thus, the equations do not violate the laws of thermodynamics as much as the choice of space and time conditions as constraints. Sharma [13] also showed that a temperature undershoot can occur when Fourier’s law of heat conduction is applied at steady state in the presence of a temperature-dependent heat source. This is in violation of the third law of thermodynamics. Here again, the choice of the space condition at some arbitrary length is not sufficient. The critical length beyond which no heat transfer will occur will have to be identified to keep the solution from violating the third law of thermodynamics. Very little work is reported in the literature on the analytical solutions to the damped-wave conduction and relaxation equation for the cases of a finite sphere and a finite cylinder. Jiang [22] reported a solution to the hyperbolic heat equation in hollow spherical objects by the method of integral transformation. Tsai and Hung [23] studied the dynamic behavior of a bilayered composite sphere subject to sudden change on the outer surface using the Riemann-sum approximation and Laplace transformation. In this study, analytical solutions are developed for the case of a finite sphere and a finite cylinder subject to the constant-wall-temperature boundary condition by themethod of separation of variables. The use of the final condition in time is made in place of the time derivative of temperature at zero time to obtain well-bounded solutions. The steady-state wave temperature has to reach equilibrium. Allowing one component of the solution, the wave temperature, to grow exponentiallywith timemay bewhy the previous investigators found a temperature overshoot under certain conditions.