An analysis is made of transient heat transfer in a vapor-heated heat exchanger with an arbitrary timewise-variant flow perturbation. The heat-transfer coefficient between the tube and coolant is assumed to vary like the n power of the coolant velocity. Results are obtained through the use of the perturbation technique. General relations are presented in closed form and their application is illustrated by carrying out some typical examples: step, linear, exponential, and sinusoidal transients in the coolant flow velocity. The influence of the system parameters on the variation of the coolant temperature is investigated. A phenomenon of resonance in the amplitude-ratio and phase-shift is disclosed for the oscillating flow transient. This phenomenon is explained by analyzing the enthalpy change of the coolant particle in the heat exchanger. The results are also compared with the analyses that have assumed a constant heat-transfer coefficient. Heat exchangers to which these results apply include the double-pipe and shell- and-tube type heat exchangers.