A new look at the integral methods for solving heat and mass transfer problems

A new concept of construction of integral relations for approximate solving problems on heat and mass transfer is proposed. This concept is based on the introduction of local functions for the heat flow or the temperature that are determined directly from the differential equation of heat conduction with boundary conditions at the temperature disturbance front. This made it possible to obtain a number of new efficient integral relations, mainly an improved integral relation for the temperature momentum and integrals of the square-law heat flow and the square-law temperature function. New schemes for optimization of the exponent n in the parabolic description of the temperature field with the use of the error norms E₁ and ${E}_{1}^{{\rm{L}}}$ introduced for the first time are proposed. In comparison with the Langford norm E_L (the scheme of T. Myers), the effectiveness of determining optimum parabolic solutions has been substantially increased. On the basis of the integral relations proposed in combination with the new schemes for minimizing an error, optimum solutions of simple parabolic form have been obtained with an error of 1.23% and an integral error E₁ = 0.0301. The solutions obtained are much better in approximation representation and error than the solutions obtained by known integral methods.