## 2 Methods
### Preparation of Dataset
#### 2.1.2 Governing Equations and Numerical Simulation  
We suppose an incompressible, viscous, and single-phase nanofluids. The continuity equation and the momentum equation are given as follows:  
\[\mathrm{div}\;\mathbf{u}=0 \tag{1}\]  
\[\frac{\partial\mathbf{u}}{\partial t}+(\mathrm{grad}\mathbf{u})\mathbf{u}=-\frac{1}{\rho} \mathrm{grad}P+\frac{\mu}{\rho}\mathrm{div}(\mathbf{D})+\mathbf{b} \tag{2}\]  
where \(\mathbf{u}\) is the velocity tensor of nanofluid, \(t\) is the time, \(P\) is the fluid pressure, \(\rho\) is the nanofluid density, and \(\mu\) is the nanofluid viscosity. \(\mathbf{b}\) is the body force per unit volume, which is supposed to be zero in this study. \(\mathbf{D}\) is the symmetric part of the velocity gradient:  
\[\mathbf{D}=\frac{1}{2}\Big{[}\mathrm{grad}\;\mathbf{u}+(\mathrm{grad}\;\mathbf{u})^{T} \Big{]} \tag{3}\]  
The energy transport equation for incompressible flow is defined as follows:  
\[\frac{\partial T}{\partial t}+(\mathrm{grad}T).\mathbf{u}=\frac{\lambda}{\rho c} \mathbf{D}:(\mathrm{grad}\mathbf{u})+\frac{S}{\rho c} \tag{4}\]  
The parameter \(S\) represents the volumetric heat source. \(c\), \(\lambda\) represent the specific heat capacity and thermal conductivity of nanofluid, respectively, and both are supposed to be constant at the reference temperature.  
Figure 2: Two geometrical constraints: (**a**) the angle \(\alpha\) is considered superior to \(\theta\); (**b**) the fin height \(h\) is considered to be greater than \(h\_min\).  
We use 4% CuO nanofluids in this study; the values of the material parameters are listed Table 1, according to [49].  
The boundary and initial conditions are listed as below:  
The uniform fixed velocity values are applied at the inner boundary. The zero pressure is applied at the outlet boundary. Nonslip conditions on the walls are considered.  
A temperature of 300 \({}^{\circ}\)C is initially applied to the fluid computational domain, 320 \({}^{\circ}\)C is initially applied to the solid domain, and the temperature of the inlet cross section is fixed at 300 \({}^{\circ}\)C. The thermal boundary condition on the external wall of the absorbent tube is nonuniform heat flux [22].  
Gmsh is used to generate the 3D absorber tube mesh of our problem, as presented in Figure 3. We first mesh the 2D cross section of the structure with unstructured triangular elements, and then convert it into 3D. The element density of near-wall regions in fluid domain is increased where there are high-velocity gradients.  
The mesh independence tests are shown in Table 2, which are carried out for average temperature of the outlet cross section and average pressure of the inlet cross section of three sizes of grids. For all the cases, the temperature and pressure deviations are no more than 0.1%. Mesh 1 meets the precision requirements and takes the least time, so it is adopted to handle the simulation calculations.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\rho\) (kg/m\({}^{3}\)) & \(\mu\) (Pa-s) & \(\lambda\) (W/(m-K)) & \(\epsilon\) (J/(kg-K)) \\ \hline
1212.8 & 8.635 \(\times\) 10\({}^{-5}\) & 0.724 & 3419.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical properties of 4% CuO nanofluid.  
Figure 3: Cross-section of mesh of the rectangular finned absorbent tube.  
The kOmegaSST model [50] in open-source software OpenFOAM is used to solve the turbulence flow problem.  
Furthermore, we validate our CFD method with the experiment [51] and numerical simulation [48] in literature, as shown in Table 3.