## 2 Materials and Methods
### Jet-Cleaning 3D CFD Modeling
#### 2.1.4 Mathematical Representation  
The Mixture model is widely adopted in multiphase flow simulations due to its capability to resolve interphase coupling with high accuracy while maintaining computational efficiency through reduced resource allocation. This approach requires minimal computational overhead [36]. Consequently, the Mixture model was selected as the multiphase formulation in this study. The jet-cleaning model is typically governed by four governing equations: the mass conservation equation, momentum conservation equation, energy conservation equation, and turbulence model equations [35]. 1. Mass conservation equation In the jet-cleaning model, the mass conservation equation is formulated as follows: \[\frac{\partial\rho_{\text{m}}}{\partial t}+\
abla\cdot\left(\rho_{\text{m}} \overrightarrow{v}_{\text{m}}\right)=0\] (5) where \(\rho_{\text{m}}\) is the density of the mixture, and \(\overrightarrow{v}_{\text{m}}\) is the mass-averaged velocity of the mixture. The expression for the density and the mass-averaged velocity of the mixture are given as follows: \[\rho_{\text{m}}=\sum_{i=1}^{\text{n}}\alpha_{i}\rho_{i}\] (6) \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(D\) & \(d\) & \(L_{1}\) & \(L_{3}\) & \(\alpha\) & \(\beta\) \\ \hline  
\hline \end{tabular}
\end{table} 
Table 2: \(\text{L}_{18}\) orthogonal test table. \begin{table}
\begin{tabular}{c c c} \hline  & **Geometric Parameters** & **Range of Values** \\ \hline Key structures & \(d\) (mm) & 4–6 \\  & \(\beta\) (\({}^{\circ}\)) & 20–50 \\ \hline Position parameters & \(s\) (mm) & 10–20 \\  & \(\theta\) (\({}^{\circ}\)) & 0–20 \\ \hline Operating conditions & \(p\) (Mpa) & 0.1–0.4 \\  & \(r\) (\%) & 0–15 \\ \hline \end{tabular}
\end{table} 
Table 3: Detailed geometric parameters of the jet-cleaning mode. \[\overrightarrow{v}_{\rm m}=\frac{\sum_{i=1}^{n}\alpha_{i}\rho_{i}\overrightarrow{v }_{i}}{\rho_{\rm m}} \tag{7}\]  
where \(\alpha_{i}\) is the volume fraction of the i-th phase. 2. Momentum conservation equation  
The momentum conservation equations govern the equilibrium between momentum variations in fluid parcels and externally applied forces. Within this study, these equations are mathematically expressed as follows:  
\[ \begin{array}{c}\frac{\partial}{\partial t}\left(\rho_{\rm m}\overrightarrow {v}_{\rm m}\right)+\
abla\left(\rho_{\rm m}\overrightarrow{v}_{\rm m} \overrightarrow{v}_{\rm m}\right)=\\ -\
abla P+\
abla\left[\mu_{\rm m}\left(\overrightarrow{v}_{\rm m}+ \overrightarrow{v}_{\rm m}^{\rm T}\right)\right]+\rho_{\rm m}\overrightarrow {g}_{\rm m}+\overrightarrow{F}+\
abla\cdot\left(\sum\limits_{k=1}^{n}\alpha_ {i}\rho_{\rm i}\overrightarrow{v}_{\rm dr,i}\overrightarrow{v}_{\rm dr,i} \right)\end{array}  \tag{8}\]  
where \(\overrightarrow{F}\) is the body force. The mixture viscosity \(\mu_{\rm m}\) and the drift velocity of the i-th phase \(\overrightarrow{v}_{\rm dr,i}\) are mathematically defined as follows:  
\[\mu_{\rm m}=\sum\limits_{i=1}^{n}\alpha_{i}\mu_{i} \tag{9}\]  
\[\overrightarrow{v}_{\rm dr,i}=\overrightarrow{v}_{\rm i}-\overrightarrow{v }_{\rm m} \tag{10}\]
3. Energy conservation equation  
The energy conservation equation governs the balance between the total energy variation in a fluid element and the combined effects of heat transfer, mechanical work, and external work contributions. In this study, this governing equation is expressed as follows:  
\[ \begin{array}{c}\frac{\partial}{\partial t}\sum\limits_{i}(\alpha_{i}\rho_{ \rm E}i_{\rm i})+\
abla\cdot\sum\limits_{i}\left(\alpha_{i}\overrightarrow {v}_{\rm i}(\rho_{\rm i}E_{\rm i}+p)\right)=\\ \
abla\cdot\left(k_{\rm eff}\
abla T-\sum\limits_{i}\sum\limits_{j}h_{\rm j,i} \overrightarrow{\rightarrow}_{j,i}+\left(\overrightarrow{\tau}_{\rm eff} \overrightarrow{v}\right)\right)+S_{\rm h}\end{array}