# Dynamic and Static Investigation of Ground Heat Exchangers Equipped with Internal and External Fins
## 3 Numerical Setup
### Solver  
The physical properties of circulating fluid, U-tube, backfill material, soil, and aluminum fins are demonstrated in Table 4. For pipes, high-density polyethylene (HDPE) material is considered. The selected grout material is a combination of silica sand and bentonite. Water is used as the fluid inside the pipes.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Density** & **Specific Heat Capacity** & **Thermal Conductivity** \\  & **(kg/m\({}^{3}\))** & **(J/kg-K)** & **(W/m-K)** \\ \hline Fluid & 997 & 4148 & 0.6 \\ U-tube & 950 & 1500 & 0.44 \\ Grout & 1800 & 840 & 0.73 \\ Soil & 2200 & 2000 & 2.5 \\ Aluminum fin & 2700 & 904 & 237 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Physical properties of materials.  
Figure 3: Three-dimensional mesh: (**a**) computational mesh around borehole; (**b**) half of computational domain.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Mesh 1** & **Mesh 2** & **Mesh 3** \\ \hline U-tube with & Element number (\(\times 10^{3}\)) & 300 & 500 & 600 \\ external fins & Fluid outlet temperature (\({}^{\circ}\)C) & 24.64 & 24.71 & 24.72 \\ \hline U-tube with & Element number (\(\times 10^{3}\)) & 300 & 420 & 600 \\ internal fins & Fluid outlet temperature (\({}^{\circ}\)C) & 24.64 & 24.67 & 24.67 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Grid-independency test of finned U-tube GHEs.  
The undisturbed ground temperature is defined as the initial temperature of the entire simulation domain, including fluid, U-tube, grout, and soil. This study is performed for the cooling mode of the system; therefore, the fluid is warmer than the surrounding soil. The constant temperature boundary condition is applied to the outer boundary of the simulation domain where there is no heat flux along this boundary and its temperature should not change until the end of the simulation.  
According to the previous studies [29], since the fluid velocity is an important parameter in the heat transfer and the temperature change of the fluid along the borehole length, the same mass flow rate in pipes as a boundary condition is not a proper condition and the same inlet velocity is applied for the inlet condition of the pipes. The focus is on investigating the effects of the inner and outer fins on ground heat exchangers with the same physical dimensions and same inlet velocity; so, finding the more effective fin cannot be based on the same flow rate or the same hydraulic diameter. Comparing the simple and finned pipes with the same mass flow rate at the inlet of pipes showed that the axial velocity in the finned U-tube (inner surface) will be higher than the simple U-tube because of the hydraulic diameter reduction which changes the heat transfer rate significantly.  
The incompressible Newtonian fluid is used inside the pipes. The soil is considered homogeneous and the physical properties of the fluid, pipe, grout material, and soil are assumed to be constant. The temperature of the ground's surface boundaries is constant and uniform, and any changes in the temperature of the ground over depth and time are ignored. The flow rate is constant along the pipe. A laminar flow is considered for all the simulations. The continuity and momentum equations are solved with the energy equation which is considered as [21; 31]:  
\[\frac{\partial}{\partial t}\big{(}\rho T\big{)}+\frac{\partial}{\partial x_{ j}}\Big{(}\rho Tu_{j}\Big{)}=\frac{\partial}{\partial x_{j}}\bigg{(}\frac{\mu}{ Pr}\frac{\partial u_{i}}{\partial x_{j}}\bigg{)} \tag{1}\]  
where \(u\) is velocity vector, \(\rho\) is fluid density, \(\mu\) is dynamic viscosity, \(T\) is temperature vector and \(Pr\) represents the molecular Prandtl number. The heat conduction in grout and ground is illustrated in Equation (2),  
\[\frac{\partial T}{\partial t}=\alpha\
abla^{2}T \tag{2}\]  
where \(\alpha\) is the thermal diffusivity of grout and ground.