# The Design of a High-Intensity Deuteron Radio Frequency Quadrupole Accelerator
## 5 Electromagnetic (EM) Analysis of RFQ
### Multiphysics Analysis  
In our multiphysics analysis, we separately considered numerical simulations involving electromagnetic, thermal, and deformation effects. The steps of our process are outlined in Figure 23. Firstly, the RF electromagnetic field was simulated, yielding the electromagnetic field distribution and heat loss within the cavity. Subsequently, the surface heat loss within the cavity was determined via a simulation of radio frequency effects and used in a thermal analysis module to compute the temperatures within the cavity. From these values, we were able to determine the amount of deformation due to heat loss. For the thermal analysis, the temperature distribution was based on [30]:  
\[\mathrm{T}=T_{0}+\frac{P}{A\cdot h} \tag{5}\]  
where T is the surface temperature, \(T_{0}\) is the initial water temperature, \(A\) is the contact area between the water and cavity, and \(P\) is the RF power loss. The heat transfer coefficient \(h\) describes the convection process on the surface of a conductor and is obtained from the following formulas:  
\[\mathrm{Re}=\frac{\rho vd}{\mu} \tag{6}\]  
Figure 23: Methodology the multiphysics analysis [29]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component Name** & **Section Number** & **Power Loss (kW)** & **Percentage (\%)** \\ \hline \multirow{4}{*}{Cavity wall with electrode} & 1 & 13.2 & 21.69 \\  & 2 & 12.7 & 20.87 \\  & 3 & 12.7 & 20.87 \\  & 4 & 13.5 & 22.18 \\ \hline \multirow{4}{*}{Electrode head} & 1 & 1.12 & 1.84 \\  & 2 & 1.13 & 1.85 \\  & 3 & 1.14 & 1.87 \\  & 4 & 1.16 & 1.91 \\ \hline \multirow{2}{*}{Tuners} & All & 3.39 & 5.57 \\ \hline \multirow{2}{*}{Coupling ring} & 1 & 0.068 & 0.11 \\  & 2 & 0.067 & 0.11 \\ \hline \multirow{2}{*}{End plate} & 1 & 0.29 & 0.48 \\  & 2 & 0.4 & 0.66 \\ \hline \multirow{2}{*}{Sum} & & 60.86 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Power distribution by RFQ part. \[\Pr=\frac{C_{p}\mu}{k} \tag{7}\]  
\[\mathrm{Nu}=0.023\cdot Re^{0.8}.Pr^{0.4} \tag{8}\]  
\[\mathrm{h}=\frac{Nu\cdot k}{d} \tag{9}\]  
where \(k\) is the heat conduction coefficient of water, \(d\) is the diameter of water pipe, \(v\) is the velocity of water, \(\rho\) is the density of water, \(\mu\) is the coefficient of kinetic viscosity, \(C_{p}\) is specific heat capacity, \(Re\) is the Reynolds number, \(Pr\) is the Prandtl number, and \(Nu\) is the Nusselt number. In our design, each wing features three water cooling channels 16 mm in diameter, as illustrated in Figure 24. The water velocity within these channels is 2.5 m/s, which corresponds to 26.5 L/min, a Reynolds number of 39,920, and a heat transfer coefficient of 9010 W/m\({}^{2}\)/K. An additional waterway positioned at each corner of the cavity wall increases the flow rate to 30.2 L/min. In this case, the ambient temperature and water temperature are both 20 \({}^{\circ}\)C. Table 9 lists further parameters of our water-cooling design. Table 10 gives the results of the multiphysics field simulated using CST. The cavity temperature and deformation distributions are shown in Figures 25 and 26. The maximum temperature differential was 18.2 \({}^{\circ}\)C, and the maximum deformation was 86.1 \(\upmu\)m, observed at the end of the electrode head. The corresponding frequency drift was 9.9 kHz, indicating  
Figure 24: Water-cooling channels for the RFQ cavity. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Channel Type** & **Vane** & **Body** \\ \hline Velocity (m/s) & 2.5 & 2.5 \\ Channel number & 48 & 16 \\ Pipe diameter (mm) & 15 & 16 \\ Flow rate (L/min) (each channel) & 26.51 & 30.16 \\ Flow rate (L/min) (all channel) & 1272.48 & 482.55 \\ Fluid density (kg/m\({}^{3}\)) & 998 & 998 \\ Fluid viscosity & 0.001 & 0.001 \\ Fluid thermal conductivity (W/m \(\cdot\) K) & 0.6 & 0.6 \\ Prandtl number & 7 & 7 \\ Reynolds number & 37,425 & 39,920 \\ Nusselt number & 228 & 240 \\ Heat transfer coefficient (W/(m\({}^{2}\) \(\cdot\) K)) & 9128 & 9011 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Parameters of water cooling.