## 2 Numerical Investigation
Considering the element quality (aspect ratio, angles, etc. ), the element widths were appropriately adjusted (Figure 3). On the basis of the Fourier and Newton law, the local heat transfer coefficient is defined as:  
\[\rho\overrightarrow{v}\cdot\
abla\overrightarrow{v}=-\
abla p+\rho g+\mu \Delta\overrightarrow{v} \tag{2}\]3  
The mean or average heat transfer coefficient over the investigated domain is given as:  
\[\rho\overrightarrow{v}\cdot\
abla\overrightarrow{v}=-\
abla p+\rho g+\mu \Delta\overrightarrow{v} \tag{2}\]4  
The Nusselt number over the investigated domain is defined as [46]:  
\[\rho\overrightarrow{v}\cdot\
abla\overrightarrow{v}=-\
abla p+\rho g+\mu \Delta\overrightarrow{v} \tag{2}\]5  
where \(k_{w}\) is the thermal conductivity at the tube wall and \(A\) is the tube area. Performing our search for the reference temperature location, several variants were tested according to Figure 4, as described in \[\rho\overrightarrow{v}\cdot\
abla\overrightarrow{v}=-\
abla p+\rho g+\mu \Delta\overrightarrow{v} \tag{2}\]6 Except the mentioned variants, some combination trials were built, and the most suitable ones are described later. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Elements in Adjacent Area** & \(\mathbf{y^{+}}\) & **Average \(Nu\)** & **Error (\%)** \\ \hline
960 & 6.153 & 70.012 & - \\
1280 & 3.608 & 57.382 & 18.04 \\
1600 & 2.178 & 56.582 & 1.39 \\
1920 & 2.107 & 56.245 & 0.60 \\
2240 & 1.396 & 56.029 & 0.39 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mesh independence test for the tube-adjacent area. Figure 3: The segment grid. The logarithmic mean temperature difference (LMTD) is computed based on the subdomain inlet temperature \(T_{in}\), outlet temperature \(T_{out}\), and wall temperature \(T_{w}\) as follows:  
\[\Delta T=\frac{(T_{in}-T_{w})-(T_{out}-T_{w})}{\ln\left(\frac{T_{in}-T_{w}}{T_{ out}-T_{w}}\right)} \tag{14}\]  
To provide the reference temperature with a better agreement compared to the other variants, the mean of the selected temperature points on the subdomain bound was considered where the formula can be expressed as follows:  
\[T_{ref}^{(i)}=\frac{T_{em}^{(i)}+T_{em}^{(i+1)}+T_{mf}^{(i)}+T_{w}^{(i)}}{4} \tag{15}\]  
When considering the correlating equations, the Reynolds number is defined with the velocity in the minimum cross-section, while the Prandtl number is defined based on the thermal conductivity (\(k\)) location:  
\[Re_{max}=\frac{\rho U_{max}D}{\mu} \tag{16}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Denotation** & **Description** \\ \hline \(T_{cf}^{(i)}\) & Temperature for \(i\)th row in the free stream of the subdomain inlet. \\ \hline \(T_{cc}^{(i)}\) & Temperature for \(i\)th row in the half of the subdomain inlet. \\ \hline \(T_{em}^{(i)}\) & Temperature for \(i\)th row on the bound of the subdomain inlet. \\ \hline \(T_{mf}^{(i)}\) & Temperature for \(i\)th row in the free stream of the subdomain centre. \\ \hline \(T_{mc}^{(i)}\) & Temperature for \(i\)th row in the half of the subdomain centre. \\ \hline \(T_{w}^{(i)}\) & Temperature for \(i\)th row at the tube wall. \\ \hline \(T_{e,pass}^{(i)}\) & Mass weighted average temperature for \(i\)th row at the subdomain inlet. \\ \hline \(T_{in,mass}^{(i)}\) & Mass weighted average temperature for \(i\)th row at the subdomain centre. \\ \hline \(\Delta T_{cf}^{(i)}\) & LMTD for \(i\)th row based on \(T_{cf}\) \\ \hline \(\Delta T_{cc}^{(i)}\) & LMTD for \(i\)th row based on \(T_{cm}\) \\ \hline \(\Delta T_{e,mass}^{(i)}\) & LMTD for \(i\)th row based on \(T_{e,mass}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The reference temperature characteristics. Figure 4: Temperature location layout.