In this study, an analytical model for a class of heat storage that utilizes latent heat of a phase-change material (PCM) is developed. Two basic shell-and-tube configurations are considered, one in which the PCM melts inside the tubes while the heat transfer fluid (HTF) flows in the shell along it, and the other in which HTF flows inside the tubes while PCM melts outside. A system of partial differential equations, which describes heat transfer and melting of the PCM and heat transfer in the HTF, is derived with some simplifying assumptions, while still capturing and preserving the essential features of the processes involved. These equations are solved analytically, yielding the overall heat exchange parameters, like instantaneous heat transfer rate, stored energy, and overall operation time of the system. The present work shows that the use of the proposed analytical technique and its modifications for the practical PCM arrangements is beneficial. Proper application of the model makes it possible to obtain the parameters of a real PCM melting process in the form of algebraic formulas, both for the transient values of variables over time, and for the overall process characteristics. A comparison with the results of numerical calculations of transient melting, made using computational fluid dynamics, confirms the validity of analytical findings and allows to assess the degree of accuracy of the results of our analytical method in various practical cases.