# Dynamic and Static Investigation of Ground Heat Exchangers Equipped with Internal and External Fins
## 5 Results and Discussions
### Axial Velocity and Temperature Contours  
The effects of longitudinal internal fins on the incompressible fluid flow inside U-tube GHE are investigated. The results are presented for different fluid velocities under steady-state conditions. Figure 10 shows axial velocity contours for two cases of a smooth U-tube and internally finned U-tube at different velocities. The two selected cross-sections of the pipe, in which the velocity contours are drawn, are located in the curvature of the inlet and outlet legs of the U-tube at a depth of 24.94 m under the ground surface.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Velocity** & **Amplitude of Wave** & **Mean of Wave** & \\  & **(m/s)** & **(\({}^{\circ}\)C)** & **(\({}^{\circ}\)C)** & **(\({}^{\circ}\)C)** & **\(dT_{bore}\)** \\ \hline  & 0.035 & 0.59 & 19.09 & 27\% \\ Simple U-tube & 0.053 & 0.85 & 19.61 & 30.7\% \\  & 0.070 & 0.99 & 19.87 & 32\% \\ \hline  & 0.035 & 0.62 & 19.21 & 28\% \\ U-tube with outer fins & 0.053 & 0.89 & 19.80 & 32\% \\  & 0.070 & 1.04 & 20.07 & 34\% \\ \hline  & 0.035 & 0.56 & 19.04 & 26\% \\ U-tube with inner fins & 0.053 & 0.82 & 19.64 & 30.9\% \\  & 0.070 & 0.97 & 19.93 & 33\% \\ \hline \end{tabular}
\end{table}
Table 9: Borehole wall temperature with a frequency of 1/4800 Hz at different velocities.  
Figure 9: Borehole wall temperature trends under different velocities: (**a**) simple U-tube; (**b**) U-tube with outer fins; (**c**) U-tube with inner fins.  
Initially, the fluid velocity is uniform at the inlet of the pipe. Then the fluid velocity is increased at the center of the pipe cross-section, while the velocity of the fluid attached to the wall of the pipe approaches zero because of the viscous effects [21]. The flow contours are symmetric until the flow passes the end of the straight pipe. Figure 10 shows that the symmetric patterns of the velocity contours both in simple and finned pipes are changed according to the curvature of the pipes similar to the previous studies of heat exchangers [28; 35]. The velocity contours of the inlet pipe are less affected by the curvature of the pipe than those of the outlet pipe. As the fluid velocity increases, the flow field pattern becomes more complex and the high-velocity flow is pushed to one side of the pipe. The presence of internal fins attenuates the effects of secondary flow and makes the flow patterns close to the straight pipes.  
Figure 11 shows the temperature contours at three different velocities with the same method and selected cross-sections of the pipe mentioned in velocity contours (Figure 10). \(\theta^{*}\) is a dimensionless temperature parameter which is illustrated in Equation (5):  
\[\theta^{*}=\frac{T-T_{min}}{T_{max}-T_{min}} \tag{5}\]  
Similar to the velocity contours, with the increase in fluid velocity, because of strong secondary flow effects in curved pipes [28; 35] the temperature field patterns become more complex.  
Figure 10: Axial velocity contours for smooth (**left**) and finned U-tube (**right**) at different velocities; (**a**) inlet pipe; (**b**) outlet pipe.