Exact temperature field in a slab with time varying ambient temperature and time-dependent heat transfer coefficient

An analytical solution is presented for one-dimensional heat conduction of a slab with time-varying ambient temperature and time-dependent heat transfer coefficient at the same boundary for the first time. The solution is obtained by using the shifting function method. After a shifting function is specified and series expansion is performed for the boundary value problem, the solution is generated. When limiting studies on either constant ambient temperature or constant heat transfer coefficient are conducted, the present solutions are proven to be identical to those in the literature. Through our investigation on numerical examples, this study shows that the three-term approximation solution can get an error within 1%. Two examples, both exponential and periodical heating or cooling on the slab, are utilized to demonstrate the influence of physical parameters regarding different convective heat transfer on temperature profiles. The difference in temperature field between the slab under time-varying temperature environment and the slab under constant ambient temperature is obvious.