Inspired by the property diagrams in thermodynamics, which distinctly reflect the performance and characteristics of thermodynamic cycles, we establish a state equation for heat motion and introduce a two-dimension property diagram, T-q diagram, in heat transfer to analyze and optimize the performance of heat exchangers, where heat flow is a state parameter for heat motion. According to the property diagram, it is convenient to obtain the influences of heat exchanger area, heat capacity rate and flow arrangement on the heat transfer performance during the analysis of heat exchangers and their networks. For instance, when analyzing the heat exchanger network in a district heating system, it is obvious to find that: if both the heat demand and the indoor air temperature in each branch of the network are the same, the total area of heat exchangers, the flow rate of water and the return water temperature in each branch are all the same; if the indoor air temperatures in different branches are different, the temperatures of the waters after flowing through different branches are different, which means that the mixing process of return waters with the same temperature is not an essential requirement to realize the best performance of district heating systems. Heat is one of the most common forms during energy utilization [1][2][3][4]. It can be obtained from the chemical energy of fuels through combustion, the atomic energy of radioactive material by either nuclear fission or nuclear fusion, and the solar energy or the geothermal energy by direct collection. Meanwhile, there are two different purposes in heat utilization: one is to generate power, and the other is to directly heat or cool objects. In the former purpose, heat is converted to mechanical energy through the heat absorption, expansion, heat release and compression processes of working substances, where the property diagrams in thermodynamics, e.g. p-v, T-s and h-s diagrams, are wildly used to directly express the variation rules of the state parameters of the working substances during a heat-work conversion process, qualitatively describe the exchanges of work and heat between the working substances and the surrounding environment in a thermodynamic cycle, and optimal design some ideal thermodynamic cycles for promoting the conversion efficiency. A thermodynamically reversible cycle is composed of several quasi-equilibrium processes, i.e. quasi-static processes, which have to progress at an infinitely slow rate in order to be reversible. Therefore, the output power of an ideal reversible thermodynamic cycle is zero. That is, a reversible cycle is a theoretical construct, which cannot actually occur and is seldom even approached in reality. In practical thermodynamic cycles, all real processes take place with finite potential difference and have finite rates, so they are all irreversible, where the property diagram in thermodynamics cannot be directly used to analyze and optimize the performance. For instance, heat transfer processes with finite temperature...