The energy-integral method: application to linear hyperbolic heat-conduction problems

This paper utilizes the energy-integral method to obtain approximate analytic solutions to a linear hyperbolic heat-conduction problem for a semi-infinite one-dimensional medium. As for the mathematical formulation of the problem, a time-dependent relaxation model for the energy flux is assumed, leading to a hyperbolic differential equation which is solved under suitable initial and boundary conditions. In fact, analytical expressions are derived for uniform as well as varying initial conditions along with (a) prescribed surface temperature, or (b) prescribed heat flux at the surface boundary. The case when a heat source (or sink) of certain type takes place has also been discussed. Comparison of the approximate analytic solutions obtained by the energy-integral method with the corresponding available or obtainable exact analytic solutions are made; and the accuracy of the approximate solutions is generally acceptable.Nomenclature A,C constants - a₀(t),a₁(t),...,a_n(t) arbitrary time-dependent coefficients, equation (3.2) - b thermal propagation speed - C_p specific heat of solid at constant pressure - g(x) given function, equation (5.1) - h(t) specified function of time - I_n modified Bessel function of the first kind - K thermal conductivity - j,n positive constants - P_n(x,t) polynomial of degreen - q(x,t) heat flux - Q(t),R(t),H(t),E(t) see equations (3.9), (II.d), (4.10), (4.12), respectively - (t) thermal penetration depth - (t,) approximate thermal penetration depth - T(x,t) temperature distribution - t time - y dimensionless time, equation (3.17) - V(y) dimensionless surface heat flux - W(y) dimensionless surface temperature - U-(t) unit-step function - G(x;t,) Green's function - x spatial variable - ()₀ surface value (atx=0)Greek symbols thermal diffusivity - density of solid - parameter, see equations (3.11) and (3.13) - parameter depending onn and - specified parameter, equations (4.5a) and (5.12b) - (t),(t) given functions of time, equations (4.6) and (5.5b) - , dummy variables - relaxation time - energy integral - (y),(y) specified functions ofy; equations (3.22) and (4.19)