## 2 Materials and Methods  
To analytically address the problem, it is essential to examine a closed system of equations that characterizes the movement of coolant flows and heat flows within the cooling system. However, solving this closed system of equations for the desired quantity (in this case, \(M_{\Sigma}\)), even for the simplest cooling system scheme, results in complex and non-differentiable expressions. Therefore, numerical calculations become necessary. The authors conducted calculations for both the simplest cooling systems with various parameters and more complex systems characteristic of modern internal combustion engines. In the direct calculations of the cooling system, the following parameters were established and determined. The following were determined: the efficiency of all heat exchangers of the system; the flow rates of coolants through all heat exchangers; all unknown temperatures of coolants; and the masses of the heat exchange elements (cores) of every heat exchanger and of all exchangers of the system, \(M_{i}\) and \(M_{\Sigma}\). The following parameters were specified: sea water inlet temperature (\(T_{ww1}\)), air temperature behind the compressor (\(T_{1}\)), temperatures of water and oil behind the engine (\(T_{ww2}\), \(T_{on2}\)), air temperature in the engine receiver (\(T_{s}\)), fresh water flow through the engine (\(G_{ww}\), equivalent to the flow through the fresh water pump), sea water flow through the pump (\(G_{ww}\)), oil flow through the pump (\(G_{m}\)), and charge air flow (\(G_{c}\)). The system was designed to remove strictly defined heat flow values into water, oil, and hydraulic fluid (specified as \(Q_{c}\), \(Q_{m}\), and \(Q_{h}\), respectively). The main (stereotypical) parameters of the designs of all heat exchangers and the efficiency level in utilizing the masses of their cores (represented by coefficients \(k_{gj}\)) were also established. To expedite the design process and ensure the requisite accuracy, all elements of the system design need to be integrated into a unified mathematical model, often referred to as a calculation complex. This complex should incorporate both direct and feedback connections between interacting elements. When calculating systems, the following assumptions were made:  
* The heat transfer process in the design mode is steady. * Heat losses in the sections of pipelines connecting the heat exchangers are insignificant. * The ambient temperature is the same for all elements of the cooling system. * The oil used in the engine system has properties that are practically no different from those specified in the standard for grade M-14W2 (interstate standard 12337-84) [38]. System calculations were performed taking into account the conditions of unambiguity. The basis for direct thermal calculation of the circuit is a system of equations consisting of:  
* Heat balance equations for all heat exchangers and the entire cooling system;
* Equations of heat flows in the system;
* Heat balance equations at mixing points of coolant flows;
* Mass balance equations at mixing points of coolant flows;
* Efficiency equations for all heat exchangers;
* Relationship equations between the efficiency of all heat exchangers and their NTU;
* Expressions of the relationship between NTU and the masses of the cores of all heat exchangers. When working with a specific system of equations, a unique solution can be obtained if the number of equations is equal to the number of unknowns. For example, in the scheme presented in Figure 1, it is necessary to determine the values of 21 unknowns based on the original system of 19 equations. Obviously, to achieve an unambiguous solution, it is necessary to additionally specify the values of two unknowns, and the remaining unknowns can then be calculated based on the system of equations. Any two unknowns can be chosen to be set, and it is logical to designate the values of ICC flow rates through the oil cooler and charge air cooler.