# Numerical Investigation of Thermal-Flow Characteristics in Heat Exchanger with Various Tube Shapes
## 2 Problem Description
### Mathematical Details  
#### 2.2.1 Governing Equations  
In the present investigation, the Reynolds number, which is based on the hydraulic diameter of the heat exchanger (\(D_{h}\)) and the maximum velocity in the domain, range from 2600 to 10,200. As a result, incompressible, three-dimensional, steady-state, and turbulent air flow is assumed [23; 58]. For this problem, the 3-D governing equations of continuity and the momentum (RANS equations) and energy equations may be written in Cartesian coordinates as follows:  
\[\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial}{\partial x_{j}}\Bigg{(}\rho u_{i}u_{j}+p\delta_{ij}-\mu\left( \frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}- \frac{2}{3}\delta_{ij}\frac{\partial u_{i}}{\partial x_{i}}\right)+\rho_{a} \overline{u_{i}^{\prime}u_{j}^{\prime}}\Bigg{)}=0 \tag{2}\]  
where:  
\[-\rho_{a}\overline{u_{i}^{\prime}u_{j}^{\prime}}=\mu_{t}\left(\frac{\partial u _{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)-\frac{2} {3}\bigg{(}\rho_{a}k+\mu_{t}\frac{\partial u_{i}}{\partial x_{i}}\bigg{)} \delta_{ij} \tag{3}\]  
According to previous numerical studies, for annular FTHEs [23; 38; 58; 59], the RNG \(k-\varepsilon\) turbulence model of Ansys 18.2 is used in this investigation. The transport equations of this turbulence model can be expressed in the following equations:  
\[\rho_{a}u_{i}\frac{\partial k}{\partial x_{i}}=\frac{\partial}{\partial x_{i} }\bigg{(}\alpha_{k}(\mu+\mu_{t})\frac{\partial k}{\partial x_{i}}\bigg{)}+\mu _{t}S^{2}-\rho_{a}\varepsilon, \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter of Fins** & \(t_{f}\) & \(S_{f}\) & \(D_{f}\) & \(P_{t}\) & \(P_{t}\) \\ \hline Dimensions/mm & 0.5 & 3 & 40 & 46 & 42 \\ \hline Tube Shape & Radius (mm) & Length (mm) & Major axis radius (mm) & Minor axis radius (mm) \\ \hline F1 & & & & \\ F2 & & & & \\ F3 & 3 & 7.335 & & \\ F4 & 3.5 & 5.764 & 7 & 3.345 \\ O2 & 4 & 4.193 & 6.5 & 4.023 \\ O3 & 4.5 & 2.622 & 6 & 4.625 \\ O4 & & & 5.5 & 5.167 \\ O5 (circular) & & & 5.33 & 5.33 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Annular fins and tube shape dimensions.  
Figure 3: 3-D Computational domain.  
\[\rho_{a}u_{i}\frac{\partial\varepsilon}{\partial x_{i}}=\frac{\partial}{\partial x_ {i}}\left(\alpha_{\varepsilon}(\mu+\mu_{t})\frac{\partial\varepsilon}{\partial x _{i}}\right)+C_{1\varepsilon}\frac{\varepsilon}{k}\mu_{t}S^{2}-C_{2\varepsilon }\rho_{a}\frac{\varepsilon^{2}}{k}-R \tag{5}\]  
and are the turbulent kinetic energy and the dissipation rate of \(k\), respectively. The quantities \(\alpha_{k}\) and \(\alpha_{\varepsilon}\) represent, respectively, the inverse of Prandtl numbers for \(k\) and \(\varepsilon\). For more details of RNG theory see Ref. [60].  
Energy equation:  
\[\frac{\partial(u_{i}(\rho_{a}E+p))}{\partial x_{i}}=\frac{\partial}{\partial x _{i}}((k_{air}+k_{t})\frac{\partial T}{\partial x_{i}}) \tag{6}\]  
where, \(E\) is the total energy and \(k_{t}\) is the turbulent thermal conductivity.  
In the solid region, including annular fins, the energy transport equation is represented as:  
\[\frac{\partial}{\partial x_{i}}\left(k_{ss}\frac{\partial T}{\partial x_{i}} \right)=0 \tag{7}\]