## 2 Mathematical Models for Expressing the Fouling Layer Thickness
### Balance Method
After substituting the total heat-transfer surface area and the mean temperature gradient into Equation (3), the overall heat transfer coefficient \(k\) may be calculated using Equation (10):  
\[P=0.97\cdot P_{1} \tag{2}\]0  
Coefficient \(k\) may also be calculated based on a detailed description of the process of heat transport from warm natural gas through the tube wall into the surrounding air. If there is a fouling layer inside the tube, the solution must also take into account the heat transfer through that fouling layer. The authors of this article assumed that the cooler had a fouling layer inside and a clean external heat-transfer surface, and they derived the following equation for calculating the overall heat transfer coefficient:  
\[P=0.97\cdot P_{1} \tag{2}\]1  
wherein \(\alpha_{1}\) is the coefficient of heat transfer inside the tube (W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\)); \(h_{\rm f}\) is the thickness of the fouling layer in the tube (m); \(\lambda_{\rm f}\) is the thermal conductivity coefficient of the fouling layer (W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)); \(\lambda_{\rm p}\) is the thermal conductivity coefficient of the tube (W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\));is the internal surface area of the tube (m\({}^{2}\)); \(d_{1}\) is the internal diameter of the tube (m); \(s_{\rm p}\) is the thickness of the tube wall; \(L\) is the tube length (m); and \(\alpha_{2}\) is the heat transfer coefficient for the heat transfer from the finned tube into the air (W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\)). By comparing Equations (11) and (10), the following equation for calculating the fouling layer thickness was obtained:  
\[P=0.97\cdot P_{1} \tag{2}\]2  
Parameter \(D\) in Equation (12) was as follows:  
\[P=0.97\cdot P_{1} \tag{2}\]3  
The key challenge associated with calculating the fouling layer thickness using Equation (12) was that some of the parameters in Equation (12) are a function of \(h_{\rm f}\). As \(h_{\rm f}\) increases, the thermal resistance on the tube wall also increases, and this leads to a decrease in the cooling capacity \(P\). The fouling layer reduces the internal diameter of the tube and, consequently, the natural gas flow rate increases as well as the value of coefficient \(\alpha_{1}\). The heat transfer coefficient \(\alpha_{1}\) is calculated using a criterial equation that contains, inter alia, the Reynolds number \(Re\). A higher \(\alpha_{1}\) value leads to a lower value of thermal resistance on the interface between the natural gas and the internal wall of the tube. As a result, the cooling capacity increases. Moreover, changes in the fouling layer thickness cause changes in the mean temperature gradient \(\overline{\Delta t}\). Based on the aforementioned facts, it may be stated that the fouling layer thickness cannot be calculated directly and requires numerous iterations. The basic geometric parameters of the analysed cooler are listed in \[P=0.97\cdot P_{1} \tag{2}\]4  
The efficiency of the gas cooling process is significantly affected by the thermal conductivity of the fouling layer. The authors of this article identified the thermal conductivity coefficient of the fouling layer in an experiment. The coefficient value was 0.746 W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\), approximately two orders lower than the value for the tube material [1]. The key settings for the iterations were selected based on the project values of the cooler's parameters. The cooler was designed to cool natural gas at 242.3 kg\(\cdot\)s\({}^{-1}\) and a pressure of 7.45 MPa from a temperature of 75 \({}^{\circ}\)C to 50 \({}^{\circ}\)C. The required quantity of the fan air with a temperature of 28 \({}^{\circ}\)C was 565.23 kg\(\cdot\)s\({}^{-1}\). In the project, it was assumed that the heat-transfer surfaces were clean; therefore, \(h_{\rm j}=0\) was substituted into Equation (11). A similar procedure was applied to the iterations that were used for the temperatures of the cooling air which were different from the projected temperature.