Shell and tube heat exchangers are commonly used in a wide range of practical engineering. The key issue in such a system is the heat exchange between the hot and cold working media. An increased cost of production of these devices has forced all manufacturing companies to reduce the total amount of used materials by better optimizing their construction. Numerous studies on the heat exchanger design codes have been carried out, basically focusing on the use of fully time-dependent partial differential equations for mass, momentum, and energy balance. They are very complex and time-consuming, especially when the designers want to have full information in a full 3D system. The paper presents the 1D mathematical model for analysis of the thermal performance of the counter-current heat exchanger comprised of mixed time-dependent and time-independent equations, solved by the upwind numerical solution method, which allows for a reduction in the CPU time for obtaining the proper solution. The comparison of numerical results obtained from an in-house program called Upwind Heat Exchanger Solver written in a Fortran code, with those derived using commercial software package ASPEN, and those obtained experimentally, shows very good agreement in terms of the temperature and pressure distribution predictions. The proposed method for fast designing calculations appears beneficial for other tube shapes and types of heat exchangers.