## 2 Mathematical Models for Expressing the Fouling Layer Thickness
### Buckingham's Method  
In the first approximation, all the parameters that were selected for the purpose of deriving a criterial equation were identical to those used in the balance method. In addition to the parameters listed above, these also included density, the thermal conductivity coefficient and kinematic viscosity of the flowing media, as well as the thermal conductivity coefficients of the tube and fin materials. A total of 18 dimensionless criteria were created and the criterial equation was subjected to an analysis, which revealed the effects of the individual criteria on the resulting \(h_{\mathrm{f}}\) value. The parameters with apparently minor effects were then withdrawn from the model, and a new, simpler model was created. In the new model, some of the geometric dimensions (e.g., the external diameter of the tube, the fin diameter and thickness, and the fin spacing) were replaced with the surface areas calculated on the basis of those parameters. The adjusted list of physical parameters (16 in total) is presented in Table 2.  
In the creation of the model law, all the dimensions of the selected physical parameters were transformed into seven SI base units (kg; m; s; K, A; mol; cd). The relevant parameters in the aforesaid model for the identification of the fouling layer thickness included only four basic dimensions--kg, m, s, and K.  
Figure 3: Correlation between the gas cooling degree and the fouling layer thickness for selected ambient temperatures. A critical equation is normally created by replacing the selected dimensional parameters \(\varphi_{1}\) through \(\varphi_{n}\) with similarity criteria \(\pi_{1}\) through \(\pi_{m}\), while the functional correlations between the individual criteria are identified experimentally or through numerical or analytical calculations. The resulting criterial equation then applies to the entire group of similar processes. When a dimensional analysis is used to describe a process, the number of obtained criteria \(\pi\) is always lower than the number of relevant parameters \(n\) on which the process depends. The basic equation that expresses the corelations between \(n\) relevant parameters \(\varphi_{1}\), \(\varphi_{2}\)... \(\varphi_{i}\)... \(\varphi_{n}\) of various dimensions is as follows [21]:  
\[f(\varphi_{1},\varphi_{2}\ldots\varphi_{i}\ldots\varphi_{n})=0 \tag{15}\]  
Based on the defining equation, each of the \(\varphi\) parameters may be expressed through a specific dimensional equation. It is the product of the base unit symbols with the respective exponents. For the four base units of the selected physical parameters (kg, m, s, K), the defining equation is as follows:  
\[[\varphi]=\mathrm{m}^{x_{1}}\cdot\mathrm{kg}^{x_{2}}\cdot\mathrm{s}^{x_{3}} \cdot\mathrm{K}^{x_{4}} \tag{16}\]  
In Equation (16), dimensional exponents \(x_{1}\) through \(x_{4}\) are rational numbers that were identified as described below. Equation (15) is dimensionally homogenous; therefore, \(\varphi_{i}\) variables cannot be used separately, but only in the form of products:  
\[\pi=\prod_{i=1}^{n}\varphi_{i}^{x_{i}} \tag{17}\]  
wherein  
* \(\pi\)--is the dimensionless variable (similarity criterion) (1);
* \(x_{i}\)--is the exponent (rational number);
* \(\varphi_{i}\)--is the physical parameter with a respective dimension. Pursuant to Equation (15) and considering the physical parameters listed in Table 2, the following correlation must apply:  
\[f(K_{\mathrm{NG}},K_{\mathrm{a}},T_{1,\mathrm{NG}},T_{2,\mathrm{NG}},T_{1, \mathrm{a}},S_{1,\Sigma,\mathrm{c}},d_{1},L,h_{t},S_{\Sigma},S_{\mathrm{e}},S_ {\mathrm{a},\Sigma},\rho_{\mathrm{NG}},\rho_{\mathrm{a}},\lambda_{\mathrm{NG} },\lambda_{\mathrm{f}})=0 \tag{18}\]  
In general, for a certain phenomenon that is described by \(n\) relevant parameters, it is possible to create \(l\) similarity criteria. The number of searched criteria \(\pi\) is identified using the following equation:  
\[l=n-h \tag{19}\]