(225e) Manifestation of Acceleration of Heat Flow in Transient Transport Phenomena

The damped wave conduction and relaxation equation was derived from the free electron theory. The relaxation time is a third of the collision time between the electron and obstacle in a given material. The time to acceleration from the collapsed state to a steady drift velocity is accounted for a ballistic term in the governing equation for temperature. Six different reasons are given to seek a generalized Fourier's law of heat conduction of which the Fourier's law is a particular case. The hyperbolic PDE is solved for by four different methods for 3 different boundary conditions. The reports in the literature of a temperature overshoot is revisited. By the method of separation of variable and the use of a final time condition for the wave temperature bounded solutions given by infinite Fourier series is derived. The solution is bifurcated. For small slab, a t, becomes elliptic from hyperbolic. This equation can be solved for the method of separation of variables. The solution is a infinite Fourier Bessel series in the composite spatio-temporal variable. When separating the two functions after differentiation one is a function in time only and the other is a function of the spatio-temporal variable, eta. Both will be equal when the expression containing the spatio-temporal variable becomes a function of time only and this can be set equal to f. The solution is seen to obey the time and space conditions. The eigen values were solved for the boundary condition at the surface and from the roots of the Bessel function of the zeroth order and first kind. Both solutions are well bounded without any temperature overshoot and disobeyance of Clausius inequality. The periodic boundary condition is evaluated by the method of complex temperature. The system is overdamped. The damped wave conduction and relaxation equation was derived from the free electron theory. The relaxation time is seen to be a 1/3rd of the collision time between the electron and the obstacle in a given material. Free electron theory has been widely accepted in explaining the observations in the theory of electric conduction. By Lorenz analogy the ratio of the electrical conductivity and thermal conductivity and temperature was found to be universal constant. So even for metals, the relaxation time can be a non-negligible parameter. The recent work of Mitra et al, 1995 and Kaminski, 1990 has shown that for materials with a non-homogeneous inner structure the relaxation time can be measured to 10-15 seconds. The relaxation time can be related to the thermal diffusivity by re-arranging the terms obtained from the free electron theory. The electron density can be converted to a density and the Boltzmann constant to heat capacity term for ideal gases to obtain a expression for thermal diffusivity. With the advent of personal computers and continued interest in transient phenomena the damped wave conduction and relaxation need to be studied further. The analytical solution presented by Baumeister and Hamill, 1969 showed a discontinuity at the wave front. This has been improved upon by a substitution, eta = t^2 ? X^2. The solution is in the form of Bessel composite function of the spatio-temporal variable and exhibits three different regimes; a inertial regime of lag and zero transfer, a rising regime described by Bessel composite spatio-temporal function and a third regime of modified Bessel composite function of the spatio-temporal variable and a falling regime in the expression for heat flux. Readily usabe solutions are obtained for the dimensionless heat flux and temperature. Some space time symmetry is seem in the solution. The wavefront solution can be obtained directly from the transformed Bessel differential equation. The manifestation of the relaxation time during a periodic boundary condition is shown in a Figure upon obtaining the solution using the method of complex temperature. It can be seen it is a ovedamped system. A storage coefficient is defined which plays a critical role in the oscillatory regimes and is given by S = rhoCp/tr and has the units of w/m3/K.