Estimation of Finite Heat Distribution Rate in the Process of Intensive Heating of Solids

Fundamental solutions of hyperbolic equation of nonstationary heat conduction have been found and investigated. The mathematical formulation of problems on the influence of a bulk heat source in an unbounded space in the three-dimensional case, as well as the influence of concentrated sources on the surface of a half-space in the three-dimensional and two-dimensional formulation are given. Using integral Laplace and Fourier transformations, spatial and volumetric fundamental solutions (influence functions) for lumped sources and influence function for a surface heat source in the case of plane problem are constructed. The Fourier and Laplace integral transforms have been successively reversed for the case of a bulk source. In the surface heat source problem, it is not possible to reverse integral transforms sequentially. In this case, an approach based on coupling the Fourier transform to a Fourier series on a variable interval is used. In this case the inverse Laplace transform is constructed analytically. The convergence of Fourier series on a variable interval is evaluated. Solution for the problem of concentrated heat source of constant intensity is constructed. The solution is obtained in integral form using the influence function of the volumetric source and the superposition principle. Calculation results are given. Comparison with classical solutions in the case of the heat conduction equation of parabolic type is carried out. It is shown that taking into account finite speed of heat waves essentially influences on heating process only at relatively short initial stage of heat source influence. It is found that the effect of taking into account the finite speed of heat propagation is more essential in problems about surface heating than in problems about volumetric heat sources. For problems involving volumetric sources, we should consider the hyperbolic type of the equation at times comparable to the relaxation time. In the case of surface heat sources the time step of a significant effect of the finite speed of propagation of heat waves is on the order of a few tens of values of the relaxation time of the material.