On Analytical Solutions During Damped Wave Conduction and Relaxation in a Finite Slab Subject to the Convective Boundary Condition

Abstract

This article describes the use of the final condition in the time domain to obtain bounded and physically reasonable solutions for the convective boundary condition for the case of a finite slab. The temperature overshoot problem is revisited for the convective boundary condition. The use of a physically realistic time condition is shown to remove the overshoot and lead to bounded solutions within Clausius’s inequality. The ramifications of these findings are discussed. The method of separation of variables was used to obtain the analytical solution for the wave temperature. The governing equation for temperature, a hyperbolic partial differential equation (PDE) is multiplied by exp(τ/2) that results in a hyperbolic PDE less the damping component. The wave temperature can be used to better understand the transient phenomena of heat conduction. For materials with large relaxation times, \({\tau_{\rm r} > \frac{\rho C_p}{4h}}\) , the temperature can be expected to undergo subcritical damped oscillations. The analytical solution is presented as an infinite Fourier series solution. The solution was found to be bifurcated. For materials with a small relaxation time, the time domain part of the solution was found to be a decaying exponential and for materials with large relaxation times the time domain part of the solution was found to be cosinuous. Analytical solutions for the average temperature of the finite slab were also obtained. The thermal time constant of the material was found from the solution. The average temperature versus time was found to exhibit convex curvature for systems with large Biot numbers and the average temperature versus time was found to exhibit concave curvature for systems with small Biot numbers. The thermal time constant for the finite slab at different Biot numbers were found and tabulated. The thermal time constant versus Biot number was found to exhibit a maxima. When Fourier parabolic equations are used, the thermal constant decreases monotonically with an increase in Biot number.