As one of the main process equipment, heat exchangers are the subject of intensive research focused on various areas of interest including optimal design, structure improvements, enhanced heat transfer techniques, dynamic response behavior, automatic control, network synthesis, numerical simulation, experimental methods, manufacturing, and development of new types of heat transfer surfaces. For the purpose of this thesis, a counterflow heat exchanger of a given total heat transfer area was selected, in which the required amount of heat load needs to be exchanged, having various mass flow rates and supply temperatures of the streams available from the heat exchanger network. Those inlet parameters are such that by connecting either only stream A or only stream B to the heat exchanger with a given surface it would not be possible to achieve the targeted heat flow rate. Thus, the question arises as to whether the targeted amount of exchanged heat flow rate can be achieved by the simultaneous action of both stream A and stream B on a given overall heat exchange area. This means that such simultaneous operation of both streams does not a priori allow greater exchanged heat flow rate than in the case where these streams act individually. This thesis aims to obtain the general criterion that needs to be met in order not only to achieve an increase in heat flow rate in such connected (networked) heat exchangers, but to maximize the exchanged heat flow rate. Fulfillment of this criterion means finding the optimal position on a heat exchanger for connection of the stream A, so that the maximum heat flow rate can be, along with the stream B, achieved in a single networked heat exchanger. In other words, within this mathematical model, the criterion of the existence of the maximum heat flow rate, as a local extreme, should be found, which is greater than the heat flow rate achieved only with stream A or stream B. Also, an algorithm for entropy generation in a heat exchanger network is developed. Entropy generation due to heat transfer between streams at finite temperature differences is considered in the model. Due to the complexity of the expression for the calculation of entropy generation, it was not possible to extract the analytical criteria for achieving the local maximum entropy, so it is necessary to determine the maximum of the function numerically, for each case separately. The research is based on the following hypotheses: maximum heat flow rate and maximum entropy generation is possible to achieve, as local extrema, if the criteria that can be written in explicit dimensionless form are fulfilled. The goal of this research is to obtain these criteria, which will contain the given overall surface of the heat flow rate, inlet temperatures of the weaker and stronger stream as well as the temperature of the weaker stream which is taken from the heat exchanger network. The first criterion must relate to maximum heat flow rate goal and the second to maximum entropy generation goal. A very important parameter M is formulated, representing the ratio between temperature difference of the weaker inlet connecting stream from heat exchanger network and inlet stronger stream and the temperature difference of the inlet weaker stream and inlet stronger stream of the observed (separated) counterflow heat exchanger. The results of the research are presented in related diagrams and interpreted, with special emphasis on cases fulfilling the maximum heat flow rate criterion for all operating points of the counterflow heat exchanger as well as for certain operating points of the counterflow heat exchanger. Cases where none of the operating points of heat exchanger will meet the hypothesis for achieving the maximum heat flow rate are also presented and interpreted. Finally, the results of dimensionless entropy generation and ratio between dimensionless heat flow rate and dimensionless entropy generation are presented in diagrams and interpreted. An experimental study was also conducted with the aim of confirming the derived mathematical model for developing the criterion for the maximum heat flow rate. The results of the mathematical model are compared with the experimental results with associated calculated composed standard uncertainties. The dissertation confirms the thesis regarding the existence or non-existence of operating parameters of a networked heat exchanger with the aim of achieving the desired maximum heat flow rate, maximum entropy or the desired maximum ratio between dimensionless heat flow rate and dimensionless entropy. For given available (specific) data on flows and stream temperatures as well as the overall exchange surface of the networked heat exchanger, using developed computer program it is possible to give a very quick and easy answer to given demands or criteria.