The paper considers a pipe-in-pipe heat exchanger, the principle of operation of which is based on the constant contact of the coolant with the treated liquid. It is used in technological systems for heating or cooling a coolant with a small heat exchange surface in the gas, oil, petrochemical and chemical industries. Heat exchangers with such a design are also used in the food industry, for example, in winemaking and in dairy production. The mathematical model of the dynamic 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. The formulated problem is solved numerically using the ideas of finite difference method. For this purpose a stable and converging difference scheme is constructed, that gives a possibility to find approximate solutions for discrete times for two equations of the system. In this case, the third equation (for temperature of dividing wall) becomes an ordinary differential equation, the solution of which can be obtained in an analytical form. As it follows from the structure of the initial equations, the model contains dissipative terms. This leads to the fact that the solution to the dynamic problem enters a stationary mode determined by the solution of the static problem. The static problem, being a special case of the original dynamic problem, is a system of two ordinary differential equations and one algebraic equation connecting unknown temperatures. The solution of such a system with given boundary conditions is obtained in the analytical form. Using numerical solutions to these two problems we tried to develop an effective approach for establishing the control over maintaining the temperature of cold water at a constant level at the output of the device.