The paper considers the synthesis of heat exchange systems according to the criteria of minimum irreversibility and minimum contact surface. It considers the cases of variable heat capacity of streams, changes in their phase state, and different hydrodynamics. The work takes into account the temperature limits for all or part of the flows at the inlet and outlet of the heat exchange system. To synthesize, it is required to choose the contact structure, free flow parameters, distribution of contact surfaces, and thermal loads between double-flow heat exchangers. KEYWORDSequivalence of two-and multi-threaded systems, finite time thermodynamics, heat exchange systems, heat transfer coefficient, irreversibility, physical implementability MSC CLASSIFICATION 80M50