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For many years now, heat exchanger optimization has been a field of research for a lot of scientists. Aims of optimization are different, having in mind heat exchanger networks with different temperatures of certain streams. In this paper mathematical model in dimensionless form is developed, describing operation of one heat exchanger in a heat exchanger network, with given overall area, based on the maximum heat-flow rate criterion. Under the presumption of heat exchanger being a part of the heat exchanger network, solution for the given task is resting in a possibility of connecting an additional fluid stream with certain temperature on a certain point of observed heat exchanger area. The connection point of additional fluid stream determines the exchanging areas of both heat exchangers and it needs to allow the maximum exchanged heat-flow rate. This needed heat-flow rate achieves higher value than the heat-flow rate acquired by either of streams. In other words, a criterion for the existence of the maximum heat-flow rate, as a local extremum, is obtained within this mathematical model. Results of the research are presented by the adequate diagrams and are interpreted, with emphasis on the cases which fulfill and those which do not fulfill the given condition for achieving the maximum heat-flow rate.