# Prediction of Boiling Heat Transfer Coefficient for Micro-Fin Using Mini-Channel
## 3 Databases and Methods
### Method  
A summary of each learning method is presented in Table 3. This paper compares three types of learning methods for predicting heat transfer coefficients: (1) deep learning only (Case 1), (2) a combination of deep learning and GPR (Case 2) [1], and (3) the proposed method, which involves building a pre-trained model followed by fine-tuning with deep learning and GPR (Case 3, present). Each proposed method follows a two-phase structure consisting of a training phase and a prediction phase. Figures 6 and 7 illustrate the training and prediction phases of the proposed learning approach. In the training phase, preliminary training is conducted based on standardized mini-channel data. Subsequently, fine-tuning (Case 3) is performed based on micro-fin data. Finally, in the prediction phase, the trained model is used to evaluate uncertainty using GPR, and predictions are adjusted to minimize uncertainty (variance), resulting in the output of predicted values for micro-fin heat transfer coefficients. Furthermore, in the dataset, in addition to refrigerant properties and experimental conditions, common dimensionless numbers have been added to the prediction equations for the heat transfer coefficient to evaluate various phenomena consistently [52]. Dimensionless numbers provide a unified way to evaluate physical phenomena regardless of operating conditions and refrigerant properties, and they are used to describe the governing conditions of each force. Figure 8 illustrates the overview of the Proposed Method Incorporating Dimensionless Numbers. The additional dimensi  
Figure 6: Pre-training and Fine-tuning, with consideration of physical mechanisms (Case 3, Present). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3 (Present)** \\ \hline DNN & ✓ & ✓ & ✓ \\ \hline GPR & - & ✓ & ✓ \\ \hline Pre-training & - & - & ✓ \\ \hline Consideration of physical mechanisms & - & - & ✓ \\ \hline Visualization & - & - & ✓ \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of methods. Bond number \(Bo\), the [PERSON] number \(Fr\), the Weber number \(We\), and the Reynolds numbers \(Re_{V}\) and \(Re_{L}\), which are shown in Equations (4)-(8). \(G\) is the mass flux \(\left[\text{kg}\cdot\text{m}^{-2}\cdot\text{s}^{-1}\right]\), \(\rho_{L}\) is the liquid density \(\left[\text{kg}\cdot\text{m}^{-3}\right]\), \(\rho_{V}\) is the vapor density \(\left[\text{kg}\cdot\text{m}^{-3}\right]\), \(g\) is the gravity \(\left[\text{m}\cdot\text{s}^{-2}\right]\), \(d\) is the inner diameter \(\left[\text{mm}\right]\), \(\sigma\) is surface tension \(\left[\text{mN}\cdot\text{m}^{-1}\right]\), \(\mu_{L}\) is the liquid viscosity \(\left[\text{\mu Pa}\cdot\text{s}\right]\), \(\mu_{V}\) is the vapor viscosity \(\left[\text{\mu Pa}\cdot\text{s}\right]\). \(\rho_{m}\) is average density \(\left[\text{kg}\cdot\text{m}^{-3}\right]\). Here, the \(Bo\), \(Fr\), and \(We\) numbers represent the relationship between the forces acting on the two-phase flow inside the tube, including inertia, volume force, and surface tension. The \(Bo\) number represents the ratio of volume force to surface force, the \(Fr\) number represents the ratio of inertia force to volume force, and the \(We\) number represents the ratio of inertia force to surface tension. When inertia force dominates, the effects of mass velocity and quality are significant, whereas when surface tension dominates, the effects of mass velocity and quality are diminished. Additionally, the \(Re\) number represents the ratio of inertia force to viscous force. By considering the \(Re\) number, it is possible to determine whether viscous or inertia force predominates, thus allowing the flow patterns of two-phase flow to be evaluated according to the operational conditions. \[Bo =\frac{(\rho_{L}-\rho_{V})gd^{2}}{\sigma} \tag{4}\] \[Fr =\frac{G^{2}}{\rho_{L}(\rho_{L}-\rho_{V})gd} \tag{5}\]  
Figure 8: Overview of the Proposed Method Incorporating Dimensionless Numbers.