**Background:** The models of multi-flow heat exchangers described in literature sources do not take into account possible phase transition in heat carriers. However, in a number of cases, for example, in multi-flow heat exchangers for utilization of moisture and thermal energy from flue gases at thermal power plants, condensing water vapor changes the phase state. The fact that there are no methods for calculating such devices puts the brakes on the development of effective technological solutions. Thus, the development of modeling of multi-flow heat exchangers for describing the phase transition in the heat carriers is an urgent problem for the energy and related industries.

**Materials and methods:** The model for multi-flow heat exchangers taking into account the phase transition in heat carriers has been constructed as a set of differential equations based on the heat balance for each of the heat carrier flows. The analytical solution to the system of the linear differential equations was obtained by the trial function method, the numerical solution to the same system was found by the Runge-Kutta method.

**Results:** We have developed a mathematical description of a multi-flow heat exchanger taking into account the phase transformation in heat carriers. We have also found and studied analytical and numerical solutions for the contact heat exchanger for moisture and heat energy utilization from flue gases of TPPs, and shown the possibilities of making design calculations in the framework of the proposed model.

**Conclusions: **The developed mathematical model can be used as the basis for more efficient methods of organizing heat transfer processes in technological equipment used for various purposes with an arbitrary number of heat carriers taking into account phase transition in them.