Additional boundary conditions in unsteady-state heat conduction problems

Abstract

Using some additional sought function and boundary conditions, a precise analytical solution of the heat conduction problem for an infinite plate was obtained using the integral heat balance method with symmetric first-order boundary conditions. The additional sought function represents the variation of temperature with time at the center of a plate and, due to an infinite heat propagation velocity described with a parabolic heat conduction equation, changes immediately after application of a first-order boundary condition. Hence, the range of its time and temperature variation completely incorporates the ranges of unsteadystate process times and temperature changes. The additional boundary conditions are such that their fulfilment is equivalent the fulfilment of a differential equation at boundary points. It has been shown that the fulfilment of an equation at boundary points leads to its fulfilment inside the region. The consideration of an additional sought function in the integral heat balance method provide a possibility to confine the solution of an equation in partial derivatives to the integration of an ordinary differential equation, so this method can be applied to the solution of equations, which do not admit the separation of variables (nonlinear, with variable physical properties of a medium, etc.).