## 2 Mathematical Models for Expressing the Fouling Layer Thickness
### Balance Method  
The balance method is based on a comparison of the supplied and removed heat output using Equation (1). The method applicable to a crossflow cooler (Figure 2a) is described in detail in paper [1], while the results related to a double crossflow cooler (Figure 2b) are presented in this article. The equation for calculating the cooling capacity \(P\) is as follows:  
\[P=P_{1}\cdot\eta_{\text{los}}=Q_{m,\text{NG}}\cdot(i_{1,\text{NG}}-i_{2,\text {NG}})\cdot\eta_{\text{los}}=Q_{m,\text{a}}\cdot(i_{2,\text{a}}-i_{1,\text{a}}) \tag{1}\]  
wherein \(P_{1}\) is the heat output withdrawn from natural gas (W); \(\eta_{\text{los}}\) is the coefficient respecting the heat loss into the cooler's structure (1); \(Q_{m,\text{NG}}\) or \(Q_{m,\text{a}}\) is the mass flow rate of gas or air flowing through the cooler (kg\(\cdot\)s\({}^{-1}\)); \(i_{1,\text{NG}}\), \(i_{2,\text{NG}}\) is the specific enthalpy of gas at the entry into and exit from the cooler (J\(\cdot\)kg\({}^{-1}\)); and \(i_{1,\text{a}}\), \(i_{2,\text{a}}\) is the specific enthalpy of air at the entry into and exit from the cooler (J\(\cdot\)kg\({}^{-1}\)). Assuming that, for example, 3% of the output \(P_{1}\) transfers into the cooler's structure, then coefficient \(\eta_{\text{los}}\) equals 0.97 and the following equation is applied:  
\[P=0.97\cdot P_{1} \tag{2}\]  
Concurrently with Equation (1), the following equation is applied:  
\[P=k\cdot S_{\Sigma}\cdot\overline{\Delta t} \tag{3}\]  
wherein \(k\) is the overall heat transfer coefficient (W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\)); \(\overline{\Delta t}\) is the mean temperature gradient (K); and \(S_{\Sigma}\) is the total heat-transfer surface area of the cooler (m\({}^{2}\)). Figure 2: Tube arrangement in the cooler block: (**a**) crossflow; (**b**) double crossflow. The total surface area of the cooler \(S_{\Sigma}\) in Equation (3) is determined by the product of the external surface area of a single tube \(S_{\rm e}\) (m\({}^{2}\)) and the number of all tubes in the cooler \(n_{\rm p}\):  
\[S_{\Sigma}=n_{\rm p}\cdot S_{\rm e} \tag{4}\]  
The surface area of a single tube is calculated using the following equation:  
\[S_{\rm e}=r\cdot\pi\left(\frac{1}{2}\cdot(d_{\rm r}^{2}-d_{2}^{2})+d_{2}\cdot b\right) \tag{5}\]  
wherein \(r\) is the number of fins on a single tube (1); \(d_{\rm r}\) is the fin diameter (m); \(d_{2}\) is the external diameter of the tube (m); and \(b\) is the fin spacing (m). The mean temperature gradient \(\overline{\Delta t}\) (K) in Equation (3) may be identified using the equations presented in detail in paper [1]. The analysed cooler used a double crossflow with two flow runs \(n=2\). The correction coefficient \(\psi\) that is necessary for the identification of the mean temperature gradient \(\overline{\Delta t}\) is therefore calculated using the following equation:  
\[\psi=\frac{\ln A}{n\cdot(1-R)\cdot\ln\left(1+\frac{1}{R}\cdot\ln\frac{R-1}{R \cdot A\frac{R}{R}-1}\right)} \tag{6}\]  
For parameter A, the Equation (6) determines the following:  
\[A=\frac{1-P}{1-R\cdot P} \tag{7}\]  
The \(P\) and \(R\) criteria are identified using the following equations:  
\[P=\frac{K_{\rm NG}}{K_{\rm NG}+K_{\rm a}}\qquad;\qquad R=\frac{K_{\rm NG}\cdot \eta_{\rm los}}{K_{\rm a}} \tag{8}\]  
wherein \(K_{\rm NG}\) is the heat capacity rate of gas; \(K_{\rm a}\) is the heat capacity rate of air (W\(\cdot\)K\({}^{-1}\)). The heat capacity rate expresses the output that is required to heat the mass of the flowing medium by 1 K and is calculated using the following equation:  
\[K=Q_{m}\cdot c_{p} \tag{9}\]  
wherein \(c_{p}\) is the specific heat capacity of the medium (J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\)).