Unsteady temperature field in slabs with different kinds of time-dependent boundary conditions

A closed solution for one-dimensional heat conduction in a slab with nonhomogenous time-dependent boundary condition at one end and homogenous boundary condition with time-dependent heat transfer coefficient at the other end is proposed. The shifting function method developed by Lee and his colleagues is used to derive the solution of the temperature distribution of the slab. By splitting the Biot function into a constant plus a function and introducing two particularly chosen shifting functions, the system is transformed into a partial differential equation with homogenous boundary conditions only. Consequently, the transformed system can be solved by a series expansion method. Two limiting cases, including time-independent boundary condition and constant heat transfer coefficient, are proved to be identical to those in the literature. Three-term approximation used in numerical examples can always result in an error <1 % in the present study, rendering the proposed methodology efficient and accurate. Finally, the influence of parameters of heat flux function or Biot function on the temperature distribution is presented.