Along with verification calculations of known designs of heat exchangers, in design engineering and when we develop new technologies, design calculations are necessary to solve the inverse problems of choosing the optimal designs and operating modes of equipment. Previously, the formulation and solution of inverse problems of classification and unsteady heat conduction have been considered, while the inverse problems of heat transfer in the design of heat exchange equipment are poorly presented in the literature. The development of methods to solve inverse problems in the design of heat exchange equipment is an urgent task of power industry. Matrix models of heat transfer based on mass and energy balance equations are used to formulate and solve inverse problems of heat exchange systems. Methods of mathematical programming are applied to solve inverse and optimization problems. For design calculations, a matrix method to solve inverse problems for choosing the design of devices and parameters of heat carriers that ensure the effective operation of the system is proposed. The inverse problem is formulated for the case of the sliding boundary of the beginning of the phase transition with the countercurrent type of movement of heat carriers. The obtained results can be used in power energy, chemical and food industries to improve the efficiency of designing resource-and energy-saving technologies. The solutions obtained can be implemented when developing measures to improve resource and energy saving technologies.