## 2 Mathematical Models for Expressing the Fouling Layer Thickness
### Multiple Regression Analysis  
In a general case, the derived Equation (47) describes the fouling layer thickness with sufficient accuracy. However, the model is not very convenient for common purposes as it includes 12 independent parameters. Therefore, the investigation was carried out with the aim of exploring how to reduce the number of relevant parameters. In any cooler, the individual criteria of Equation (47) contain certain parameters that are of a constant value. In the analysed cooler, those parameters included, for example, the thermal conductivity coefficient for the pipe material and the fouling layer, the internal and external diameters of the pipe, the pipe lengths, etc. In this case, the fouling layer thickness was identified by applying multiple regression analysis, using the MinitabX 18 statistical software and the R package software 4.3.3 [22]. Figure 4: Correlation between the fouling layer thickness obtained from the analytical solution and the thickness obtained from the model created by applying the Buckingham’s theorem. With the use of MinitabX, out of the group of relevant parameters, the following six parameters were selected as those with the most significant impact on the fouling layer thickness: \(Q_{m,a}\); \(T_{2,\text{NG}}\); \(T_{1,a}\); \(\rho_{\text{NG}}\); \(\rho_{a}\); and \(c_{p,\text{NG}}\). Multiple linear regression was applied and the following model equation was created:  
\[h_{\text{f}}=-5784.25-0.00491\cdot Q_{m,a}+1.32825\cdot T_{2,\text{NG}}+0.45745 \cdot T_{1,a}+6.90863\cdot\rho_{\text{NG}}+113.457\cdot\varrho_{\text{a}}++1.70938 \cdot c_{p,\text{NG}} \tag{50}\]  
Gas density \(\rho_{\text{NG}}\) was used in Equation (50) for the mean temperature value (\(T_{1,\text{NG}}-T_{2,\text{NG}}\))/2, and it was identified using Equation (51):  
\[\rho_{\text{NG}}=1.4563\cdot 10^{-3}.t_{\text{mean,NG}}^{2}-3.6377\cdot 10^{-1}.t _{\text{mean,NG}}+63.00 \tag{51}\]  
Air density \(\rho_{\text{a}}\) was calculated using Equation (52):  
\[\rho_{\text{a}}=1.313\cdot 10^{-5}.t_{\text{mean,a}}^{2}+4.628\cdot 10^{-3}.t _{\text{mean,a}}-1.276 \tag{52}\]  
wherein \(t_{\text{mean,a}}=t_{1,a}+\Delta T_{\text{NG}}\). The specific thermal capacity of gas \(c_{p,\text{NG}}\) was calculated using Equation (53):  
\[c_{p,\text{NG}}=1.1696\cdot 10^{-9}.t_{\text{mean,NG}}^{4}-6.3457\cdot 10^{-7}.t_ {\text{mean,NG}}^{3}+1.3139\cdot 10^{-4}.t_{\text{mean,NG}}^{2}--1.0701\cdot 10^{-2} \cdot t_{\text{mean,NG}}+2.9979 \tag{53}\]  
Equation (50) represents the model of a correlation between the fouling layer thickness and the selected input parameters. The regression model parameters were identified by applying the method of least squares, which is sensitive to outliers. Therefore, the input data was also assessed in terms of outliers and influential values, which were excluded from the model. The statistical significance of the regression model, or of the regression model parameters, was verified in the tests of statistical significance at a significance level \(\alpha=0.05\). As a rule, if the \(p\)-value is lower than the significance level \(\alpha\), then the null hypothesis is rejected in favour of the alternative hypothesis. If the \(p\)-value equals or is higher than the selected significance level \(\alpha\), then the null hypothesis is not rejected [23]. The results of the testing showed that the regression model (50) was statistically significant (\(p\)-value = 3 \(\times\) 10\({}^{-122}\) < \(\alpha\)) and that all of the analysed parameters were also statistically significant (\(p\)-value < \(\alpha\)). The coefficient of determination for the multiple linear regression was 0.9985; this means that as much as 99.85% of the fouling layer thickness variability may be explained by the proposed regression model with the given parameters. Heteroscedasticity of the regression model was tested with the use of the Breusch-Pagan test, in which the null hypothesis assumes homoscedasticity.