An exact analytical solution is found for unsteady heat transfer, in the first time domain, for a fluid flowing in a laminar, fully developed manner in a duct when the wall temperature is suddenly changed to θw = bxn. The solution to the governing partial differential equation is effected by the use of the Laplace transform and yields the transient surface heat flux as a function of time and position down the duct for n = 1–6. For comparison purposes, solutions are also obtained for a number of approximate models, namely, pure conduction, slug flow, quasi-steady, and a new model proposed by Sucec.