## 2 Mathematical Models for Expressing the Fouling Layer Thickness
### Buckingham's Method
\begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Physical Parameter** & **Unit** \\ \hline \(K_{\mathrm{NG}}\) & Heat capacity rate of gas & kg m\({}^{2}\cdot\)s\({}^{-3}\cdot\)K\({}^{-1}\) \\ \(K_{\mathrm{a}}\) & Heat capacity rate of air & kg m\({}^{2}\cdot\)s\({}^{-3}\cdot\)K\({}^{-1}\) \\ \(T_{1,\mathrm{NG}}\) & Gas temperature at the inlet into the cooler & K \\ \(T_{2,\mathrm{NG}}\) & Gas temperature at the outlet from the cooler & K \\ \(T_{1,\mathrm{a}}\) & Air temperature at the inlet into the cooler & K \\ \(S_{1,\mathrm{AE}}\) & Net cross-sectional area of all tubes for gas & m\({}^{2}\) \\ \(d_{1}\) & Internal diameter of the cooler tube & m \\ \(L\) & Tube length & m \\ \(h_{t}\) & Fouling layer thickness & m \\ \(S_{\Sigma}\) & Total heat-transfer surface area of the cooler & m\({}^{2}\) \\ \(S_{\mathrm{e}}\) & Surface area of a single filmed tube in the cooler & m\({}^{2}\) \\ \(S_{\mathrm{a},\Sigma}\) & Free surface area of the cooler for air passage & m\({}^{2}\) \\ \(\rho_{\mathrm{NG}}\) & Gas density & kg m\({}^{-3}\) \\ \(\rho_{\mathrm{a}}\) & Air density & kg m\({}^{-3}\) \\ \(\lambda_{\mathrm{NG}}\) & Thermal conductivity of gas & kg m\({}^{-3}\cdot\)K\({}^{-1}\) \\ \(\lambda_{\mathrm{A}}\) & Thermal conductivity of the fouling layer & kg m\({}^{-3}\cdot\)K\({}^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Input parameters for the mathematical model. \[\pi_{5}=\frac{S_{\rm e}}{S_{1,\Sigma,c}} \tag{31}\]  
\[\pi_{6}=\frac{S_{\rm a,\Sigma}}{S_{1,\Sigma,c}} \tag{32}\]  
\[\pi_{7}=\frac{L}{d_{1}} \tag{33}\]  
The fouling layer is affected by the flow surface area; therefore, the following simplex was created:  
\[\pi_{8}=\frac{d_{1}^{2}}{\left(d_{1}-2\cdot h_{f}\right)^{2}} \tag{34}\]  
The simplex created from the gas and air density parameters is as follows:  
\[\pi_{9}=\frac{\rho_{\rm NG}}{\rho_{\rm a}} \tag{35}\]  
The simplex for the thermal conductivity coefficients for the gas and for the fouling layer is as follows:  
\[\pi_{10}=\frac{\lambda_{\rm f}}{\lambda_{\rm NG}} \tag{36}\]  
With regard to the created simplexes, in the set of Equations (22)-(25), \(x_{2}=x_{3}=x_{5}=x_{8}\)\(=x_{9}=x_{10}=x_{11}=x_{12}=x_{14}=x_{16}=0\); therefore, the resulting equations are as follows:  
\[x_{1}+x_{13}+x_{15}=0 \tag{37}\]  
\[2x_{1}+2x_{6}+x_{7}-3x_{13}+x_{15}=0 \tag{38}\]  
\[x_{1}+x_{15}=0 \tag{39}\]  
\[\pi_{5}=\frac{S_{\rm e}}{S_{1,\Sigma,c}} \tag{31}\]0  
The total number of searched criteria is \(l=12\) and the number of created simplexes is 10. The two missing complex criteria had to be identified based on two independent solutions of the set of Equations (37)-(40). The set of Equations (37)-(40) contains six unknowns; therefore, they must be solved by determining two of those unknowns and calculating the remaining four. In the first option, the determined unknowns were \(x_{1}=1\) and \(x_{6}=0\) and the calculated parameters were the following: \(x_{4}=0\); \(x_{7}=-1\); \(x_{13}=0\); and \(x_{15}=-1\). The identified criterion was as follows:  
\[\pi_{5}=\frac{S_{\rm e}}{S_{1,\Sigma,c}} \tag{31}\]1  
In the second option, the determined unknowns were \(x_{6}=1\) and \(x_{1}=0\) and the calculated parameters were the following: \(x_{4}=0\); \(x_{7}=-2\); \(x_{13}=0\); and \(x_{15}=0\). The identified criterion was as follows:  
\[\pi_{5}=\frac{S_{\rm e}}{S_{1,\Sigma,c}} \tag{31}\]2  
Since the criteria equation was to be derived for a particular cooler, the criteria \(\pi_{4}\), \(\pi_{5}\), \(\pi_{6}\), \(\pi_{7}\), and \(\pi_{12}\) were constant. Therefore, they were pooled into a single new criterion \(\pi_{0}\). As a rule, similarity criteria may be transformed into other criteria through multiplication, division, exponentiation with a constant, or multiplication by a constant [21].