# Effect of Different Control Strategies on the Heat Transfer Mechanism of Helical Energy Piles
## 3 Results
### Relationship between Spiral Tube Design and Convection Heat Transfer Coefficient  
Taking the center point \(p_{0}\) of the normal plane at the middle of the spiral as the origin of the coordinates and establishing a polar coordinate system (\(\omega\), \(r\), \(z\)) is shown in Figure 12. The coordinates of \(\omega\), \(r\), and \(z\) in Figure 11 were calculated using Equation (11). \[(\omega,r,z)=(\frac{2\pi rs}{P0}\cos(\frac{2\pi z}{P0}),\frac{2\pi rs}{P0}\sin (\frac{2\pi z}{P0}),z) \tag{11}\]  
where \(\omega\) represents the tangential direction of the spiral line and points downstream, \(r\) is the radial coordinate, and the \(z\) direction is downward along the helix central axis. The velocity fields and temperature fields of the normal plane are shown in Figure 13. The velocity field is shown in the upper part, and the temperature field is presented in the lower part of the subfigure. Figure 13 shows the changes in the secondary circulation in this plane under three conditions: when \(R\) and \(d\) are kept constant and only SP is changed, when \(P\) and \(d\) are kept constant and only SP is changed, and when \(P\) and \(R\) are kept constant and only Sd is changed. It can be observed that the flow rate of the secondary circuit increases with an increase in SP and decreases with an increase in SR and Sd. Additionally, the temperature of the secondary circuit decreases with an increase in SR and \(Sd\) but increases with an increase in Sd. Using the controlled variable method, SP, SR, and Sd were adjusted separately to compare the magnitude of the velocity at point p0 under the three independent influences  
Figure 12: Velocity(\(u_{p0}\)) and velocity components at point \(p_{0}\) in the \(\omega\), \(r\), \(z\) directions. It can be seen in Figure 14 that \(u_{p0}\) increases by 18.1% with an increase in SP; \(u_{p0}\) decreases by 3.24% as SR increases from 4 to 12, then remains stable; an increase in Sd reduces \(u_{p0}\) slightly, but only by 0.11%. Figure 15 shows that when SP increases, \(u_{p0,\omega}\) decreases by 16.87% and \(u_{p0,\omega}\) and \(u_{p0,z}\) increase by 12.7 and 7.2 times; when SR increases, \(u_{p0,\omega}\) increases by 15.3%, and \(u_{p0,r}\) and \(u_{p0,z}\) decrease by 96.7% and 82.5%; and when Sd decreases, there is not much change in \(u_{p0,\omega}\), \(u_{p0,r}\) and \(u_{p0,z}\), less than 5%. Meanwhile, \(SP\) and \(SR\) have significant effects on the flow state of the fluid inside the spiral tube, and the effect of \(SP\) on the flow state of the fluid inside the spiral tube is greater than that of \(SR\), while the effect of \(Sd\) is very small. When \(SP>20\) and when \(SR<5\), the velocity distributions of \(\omega\), \(r\), and \(z\) of \(u_{p0}\) at \(p_{0}\) are similar, and \(u_{p0,\omega}\) varies within \(\pm 20\%\) of the mean flow velocity across the cross-section. \(u_{p0,r}\) and \(u_{p0,z}\) are both greater than 0.75 m/s, resulting in \(u_{p0}\) being much larger than the average flow velocity in the cross-section, which reduces the flow velocity of the fluid near the wall of the spiral tube and increases the thickness of the boundary layer and, therefore, cannot form a stable secondary cycle. According to the Dittus-Boelter formula in the cooling case (see Equation (12)), the convective heat transfer coefficient (\(\alpha_{aff,st}\)) of the inner wall of the smooth circular tube is only related to the tube diameter d, and there is no direct relationship with the length of the circular tube.