Heat conduction at a variable heat-transfer coefficient

Some practically important problems of unsteady heat conduction with a time-variable relative heat-transfer coefficient are considered. Various approaches to finding a solution to the analytical problem are systematized: decomposition of the generalized integral Fourier transform, the serial expansion of a sought temperature function, and the reduction of the problem to an integral, second-order Volterra equation. It is demonstrated that the solution is reduced to an infinite series of successive approximations of different functional form in all cases, and the main objective of each of these approaches is to find the most advantageous first approximation. Particular cases of the time dependence of the relative heat-transfer coefficients are considered: linear, exponential, power, and root cases. Analytical solutions and numerical experiments are described, and some specific features of the temperature curves of a number of mentioned dependences are revealed. It is established that the picture of the change in the temperature curve for the linear time law of the heat-transfer coefficient becomes appreciably different from the classic case of a constant coefficient, whereas the exponential dependence does not introduce any essential changes.