We consider the possibilities of numerical simulation for modelling the dynamic processes in a heat transferring device. The principle of operation of considered tube-in-tube heat exchanger is based on the constant contact of the coolant with the treated liquid. The mathematical model of the transient process of transferring heat energy in devices of this type includes a system of three differential equations for the temperatures of cold water (heated), hot water (heating) and dividing walls with corresponding initial and boundary conditions. The numerical algorithm based on the ideas of finite difference method is elaborated. The issue of convergence of the constructed difference scheme based on the study of its approximation and stability is considered. The approximation is easily proved by expanding discrete functions in a Taylor series and, for the problem under consideration, is of the first order with respect to the grid steps. The stability of the constructed difference scheme is proved using the Neumann spectral criterion.