# Dynamic and Static Investigation of Ground Heat Exchangers Equipped with Internal and External Fins
## 5 Results and Discussions
### Short Timescale Response  
Transient simulations of BHE show the dynamic behavior of the system in a short timescale [33]. Transient simulation results of simple and finned U-tube GHE are shown in Figure 5. The inlet velocity in three numerical models (simple, inner finned and outer finned U-tube) is 0.053 m/s. The inlet fluid temperature is 30 \({}^{\circ}\)C and the far field temperature (the soil temperature) is considered 15 \({}^{\circ}\)C, which has been used as the initial temperature of whole simulation domains (including the pipe, the fluid, grout materials and the soil). The outlet temperature of all three pipes reached a steady state after 330 min and no change was observed in the graphs after this time. The simulation is for the cooling mode of the system, and the inlet temperature of the fluid is higher than the temperature of the initial temperature and that of the soil around the borehole. The fluid temperature decreases as it passes through the pipe. The outlet fluid temperature has reached 24.86 \({}^{\circ}\)C in a simple pipe, 24.71 \({}^{\circ}\)C in a pipe with external fins and 24.28 \({}^{\circ}\)C in a pipe with internal fins. The difference between the inlet and outlet temperatures of the fluid in the simple U-tube is 5.14 \({}^{\circ}\)C and in the finned U-tube on the outer and inner surfaces is 5.29 \({}^{\circ}\)C and 5.72 \({}^{\circ}\)C, respectively. The temperature reduction in the fluid in the outer and inner finned U-tube has increased, respectively, 2.9% and 11% compared to the simple pipe. Frequency Response Test  
BHEs are affected by temperature fluctuations at different ranges of time scales. Therefore, applying a periodic condition of the fluid inlet temperature instead of constant value is a useful way to \begin{table}
\begin{tabular}{c c c c} \hline
**Velocity (m/s)** & **0.06** & **0.04** & **0.02** \\ \hline \multirow{4}{*}{U-tube with outer fins} & Flow rate (kg/s) & 0.042 & 0.028 & 0.014 \\  & Reynolds number & 2049 & 1366 & 683 \\  & Outlet fluid temperature (\({}^{\circ}\)C) & 25.06 & 23.59 & 20.83 \\  & Borehole wall temperature (\({}^{\circ}\)C) & 20.67 & 20.08 & 18.78 \\ \hline \multirow{4}{*}{U-tube with inner fins} & Flow rate (kg/s) & 0.038 & 0.025 & 0.013 \\  & Reynolds number & 1945 & 1297 & 648 \\ \cline{1-1}  & Outlet fluid temperature (\({}^{\circ}\)C) & 24.66 & 23.13 & 20.42 \\ \cline{1-1}  & Borehole wall temperature (\({}^{\circ}\)C) & 20.50 & 19.89 & 18.55 \\ \hline \end{tabular}
\end{table} 
Table 6: _Cont._  
Figure 5: Outlet temperature of fluid in simple and finned U-tube GHE. describe the thermal response of the borehole heat exchangers in short timescales [33]. To evaluate and compare the thermal responses of three numerical models, the inlet fluid temperature is applied as a sinusoidal function with the same amplitude and different frequencies. The inlet velocity for all three pipes is 0.053 m/s. The mass flow rates for the simple and external finned U-tube are 0.038 kg/s and for the U-tube with internal fins is 0.034 kg/s. The inlet fluid temperature is applied as follows [33]:  
\[T_{in}=T_{1}+a_{in}\ sin(2\pi ft)=T_{1}+a_{in}\ sin\bigg{(}\frac{2\pi}{B}t \bigg{)} \tag{3}\]  
where \(a_{in}\) is the inlet wave amplitude, \(T_{1}\) is the mean of the wave, \(f\) is the frequency and \(B\) is the period. The outlet fluid temperature by variation of amplitude (\(a_{out}\)) and mean number (\(T_{2}\)) is defined as:  
\[T_{out}=T_{2}+a_{out}\ sin(2\pi ft+\varphi)=T_{2}+a_{out}\ sin\bigg{(}\frac{2 \pi}{B}t+\frac{2\pi}{\Delta B}\bigg{)} \tag{4}\]  
where \(\varphi\) is the phase shift and \(\Delta B\) is the time shift of outlet wave that shows delay in the system responses. The outlet temperature of the fluid obtained from the 3D simulation in response to the inlet temperature as a sine wave with 80 and 40 min periods (frequency of 1/4800 Hz and 1/2400 Hz) and the amplitude of 8 \({}^{\circ}\)C for three simple and finned U-tubes are shown by Figures 6 and 7.