## 2 Mathematical Models for Expressing the Fouling Layer Thickness
### Buckingham's Method
That rule was applied to obtain the following new criterion from the constant criteria:  
\[\pi_{5}=\frac{S_{\rm e}}{S_{1,\Sigma,c}} \tag{31}\]3The parameter that was being identified was the thickness of the fouling layer in the cooler tubes \(h_{\rm f}\), and since it fell within the \(\pi_{8}\) criterion, it was expressed as a function of other criteria, with the following form:  
\[\pi_{5}=\frac{S_{\rm e}}{S_{1,\Sigma,c}} \tag{31}\]4  
The correlation between the dimensionless arguments in Equation (44) is exponential; therefore, the resulting equation is as follows:  
\[\pi_{5}=\frac{S_{\rm e}}{S_{1,\Sigma,c}} \tag{31}\]5  
In a logarithmic scale, that correlation is linear and in the following form:  
\[\pi_{5}=\frac{S_{\rm e}}{S_{1,\Sigma,c}} \tag{31}\]6  
The individual criteria were calculated from the parameter values that were identified through iterations as described above. For the multiple regression analysis, data from 94 iterations were available. The \(C\) constant and the unknown exponents \(z_{j}\) were identified through multiple linear regression. The coefficient of determination for the multiple linear regression was 0.9999. The regression sum of squares was 13.153, while the residual sum of squares was 0.005. The values of the \(C\) constant and the individual exponents are listed in \[\pi_{5}=\frac{S_{\rm e}}{S_{1,\Sigma,c}} \tag{31}\]7  
For the analysed cooler, the thickness of the fouling layer in the cooler tubes was subjected to the following criteria Equation (47), obtained by breaking down Equation (45):  
\[ \begin{array}{ll}\frac{d_{1}^{2}}{(d_{1}-2\cdot h_{t})^{2}}=&1.3848\cdot 10^{-72 }\cdot\left(\frac{K_{\rm NEG}}{\rm R_{4}}\right)^{0.017396}\cdot\left(\frac{T_ {1,\rm NEG}}{\Delta T_{\rm NEG}}\right)^{1.4880}\cdot\left(\frac{T_{2,\rm NEG} }{T_{1,\rm a}}\right)^{-78.565}\cdot\\ &\cdot\left(\frac{\rho_{\rm NEG}}{\rho_{\rm a}}\right)^{-83.971}\cdot\left( \frac{A_{t}}{\Delta_{\rm NEG}}\right)^{173.23}\end{array}  \tag{47}\]  
Criteria with a zero exponent were not specified in Equation (47). In the linear regression, their impact was automatically included in the value of the \(C\) constant. The parameter that was being identified--the fouling layer thickness \(h_{\rm f}\)--is on the left side of criterial Equation (47) and is affected by the internal diameter of the cooler tube \(d_{1}\). An adjustment of this criterion leads to solving a quadratic equation. Only one of the identified roots of the equation had a physical meaning--the one with a lower value. The values of the fouling layer thickness obtained from the model and from the analytical solution exhibited very good concordance (Figure 4). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Constant** & & & & & & & & \\ \hline \(C\) & \(z_{0}\) & \(z_{1}\) & \(z_{2}\) & \(z_{3}\) & \(z_{9}\) & \(z_{10}\) & \(z_{11}\) \\ \hline \(1.3848\times 10^{-72}\) & 0.0000 & 0.017396 & 1.4880 & \(-78.565\) & \(-83.971\) & 173.23 & 0.0000 \\ \hline \end{tabular}
\end{table} 
Table 3: \(C\) constant and the individual exponents. The correlation may be described by a regression line with a slope approaching 1, more specifically 0.9987, with a reliability value (a squared correlation index) \(R^{2}\) = 0.9999. The standard deviation of the difference was 0.0331, i.e., 3.2%. The results of both solutions were also subjected to a pairwise \(t\)-test at a significance level \(\alpha\) = 0.05. A precondition for using a pairwise \(t\)-test was meeting the assumption of normality of the difference scores, which was verified in a Shapiro-Wilk test of normality. The value of the test criterion \(t\) was identified using the following equation:  
\[t=\frac{\left|\overline{\Delta h_{\mathrm{f}}}\right|\cdot\sqrt{m-1}}{s_{ \Delta}} \tag{48}\]  
wherein \(m\) is the number of values (1) and \(s_{\Delta}\) is the standard deviation (mm).