# Numerical Analysis on the Performance of a Radiant Cooling Panel with Serpentine-Based Design
## 3 Results and Discussions
### CFD Simulation--ANSYS Fluent
#### 3.3.4 Pressure Drop  
Based on the pressure contour shown in Figure 10, the pressure decreases from the upstream to the downstream region for all of the flow configurations. This is due to pressure losses along the flow in the pipe. Pressure drop is closely related to pumping power, meaning that the higher the pressure drop, the larger the pumping power needed to operate the system. Therefore, it is important for us to determine the total pressure drop in order to design a more efficient cooling system for the RCP.  
The pressure drop is determined by using the Ansys Fluent post-processing method, and is compared to a manual calculation derived from [PERSON]'s equation for pressure drop ([PERSON], 2019), as shown in Equation (16):  
\[\frac{P_{1}}{\rho g}+\frac{v_{1}^{2}}{2g}+Z_{1}\ =\frac{P_{2}}{\rho g}+\frac{v_{2}^ {2}}{2g}+Z_{2}+h_{L} \tag{16}\]  
where \(P_{1}\) is the static pressure at the inlet and \(P_{2}\) is the static pressure at the outlet. \(v\) is the fluid velocity, \(Z\) is the elevation, \(h_{L}\) is the total head loss, \(\rho\) is the fluid density, and \(g\) is the gravitational acceleration. The total head loss, \(h_{L}\) is determined by using Equation (17):  
\[h_{L}=\left[\ f\ \frac{L}{D}+\sum k\ \right]\ \frac{v^{2}}{2g} \tag{17}\]  
where \(f\) is the friction factor, \(L\) is the pipe length, \(D\) is the pipe diameter and \(k\) is the loss coefficient value. Because the flow is laminar and the pipe is circular, the formula used to calculate the friction factor is shown in Equation (18) ([PERSON], 2019):  
\[f=\frac{64}{Re} \tag{18}\]  
\[\frac{P_{1}-P_{2}}{\rho g}=\left[\ f\ \frac{L}{D}+\sum k\ \right]\ \frac{v^{2}}{2g} \tag{19}\]  
Equation (19) is the simplified [PERSON]'s equation, where the inlet velocity is assumed to be the same as the outlet velocity. The pipe configuration is horizontal, thus the change in potential energy is ignored. The loss coefficient value, \(k\), is determined to be 0.2 because the pipe is circular and has a 180\({}^{\circ}\) bend.  
Based on Table 10, the theoretical value of pressure drop is almost identical to the simulated value obtained from the simulation. These data prove that the simulated pressure drop can be validated with the governing [PERSON]'s equation, as the results obtained have a very low percentage of error. Comparing the results among all the three designs, Design 1 has the most pressure drop across the pipe. This is because the pipe configuration in Design 1 has a total of six bends in a single, long pipe. Unlike Design 2, the pipe configuration is divided into two pipes, meaning that it has only three bends for each pipe. For Design 3, the middle pipe is a straight pipe without any bends, while the other two pipes have two bends each. Because of this, the middle pipe of Design 3 has the lowest pressure drop. As can be noticed from the pressure contour figure, the pressure drop of the last serpentine is less than the first serpentine, because of the decreasing pressure  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Design**} & **Room Temperature/** & **Mean Plate** & **q\({}^{\text{m}}\)\({}^{\text{s}}\)\({}^{\text{t}}\)\({}^  
throughout the flow toward the outlet [32; 33]. With the increasing number of bends of the pipe, a larger pressure drop across the pipe is observed, as shown in Figure 14. Therefore, the pressure drop across the pipe is highly related to the number of bends and the path length of the fluid.