The conditions for minimum dissipation and the dependence of the minimum possible entropy production on the heat load are obtained for heat exchange systems with a given heat load and total heat transfer coefficient. The case of variable heat capacity of streams is considered. Distributions of heat exchange coefficients and heat capacity rates of the streams for which this minimum of irreversibility can be achieved are found. The problem is solved for a wide class of heat transfer kinetic laws in which the heat flow is proportional to the contact surface. The possibility of evaporation and condensation of streams is investigated. The concept of thermodynamic equivalence of dual-stream and multi-stream systems is introduced and the conditions that lead to such equivalence are obtained. Using this equivalence for Newtonian kinetics, optimal synthesis conditions are obtained for dual-stream and multi-stream systems with a given total heat transfer coefficient and heat load. It is shown how the solution can be implemented in a system of dual-stream heat exchangers.