1. TEM3P Simulation of Be Wall Cavity Tianhuan Luo

2. Cavity Model Pillbox cavity with Be wall R=0.36 m, f0~319 MHz, L=0.25m (not exactly 325 MHz, but not important for this study). Be wall 1 mm and 0.1 mm 1mm model: green vacuum part of RF field calculation; yellow and magenta Be wall 0.1 mm model: for thermal only calculate 15 degree of Be wall, which is much easier to mesh; for mechanic, still need to calculate a quarter due to boundary condition limit. With 5mm copper torus: for the mechanical deformation simulation.

3. Check thermal simulation with a simplified model First we check the thermal simulation with analytical calculation, using Be parameters at around 80 K from the previous documents and ignoring their change with the temperature: Thermal conductivity: 300 W/(mK); electrical conductivity: 9.3e7 S/m; Peak E field 21.6 MV/m; duty factor: 0.92e-3. The edge of the wall at r=0.36 m is fixed at 80 K. The RF heating is generated on the wall inner surface. All other surfaces are thermally insulated.

4. 1 mm Be Wall, thermal Radius (m) Temperature (K) Blue: TEM3P Red: Analytical calculation. Analytical calculation is base on Fourier heat transfer equation: \vec_{q}=- \Kappa\cdot\nabla T, cross-checked with the formula in Bob’s slides and Derun’s LINAC98 paper. So far don’t understand the discrepancy between and analytical calculation and TEM3P simulation.

5. 0.1 mm Be Wall, thermal Blue: TEM3P Red: Analytical calculation. Radius (m) Temperature (K) Still shows the same discrepancy with 1 mm wall. From the edge (80 K) to the center (1550 K) of the wall, dt=1475 K, which is 10 times of the dt in the 1mm wall (147 K). For 0.1 mm window, the temperature rise is too large with current parameters setting. Can Bob provides the Be parameters he used in his slides?

6. Mechanical model TEM3P can calculate the deformation from thermal stress and Lawrence Force. We will calculate both and compare their effects. The mechanic parameters we used here for Be are: Lame constant 2e10 Pa; Shear modulus 1.5e11 Pa; thermal expansion coefficient 7.5e-6. The parameters we used here for copper are: Lame constant 4.3e10 Pa; Shear modulus 8.1e10 ; thermal expansion coefficient 1.7e-5; thermal conductivity: 631 W/mK; electric conductivity: 4.35e8 S/m. The outer side of copper torus is defined as fixed position. The reference temperature of thermal stress calculation is 80 k. For the model Be window thinner than 1mm, without any change to the thermal parameters, we reduce the duty factor to keep dT ~ 147 K from wall edge to center.

7. 1 mm model: Thermal Stress dz~0.0004 m at the center of the cavity. Cavity volume expands. d_Z

8. 1mm model: LFdetuning d_Z dz~0.00485 m at the center of the cavity. Cavity volume contracts.

9. Next Need more accurate or realistic thermal and mechanical property curve (as function of T) for beryllium and copper. Then we can get more accurate/realistic temperature results and compare the effect of LFdetuning and thermal stress. So far, the simulation shows a much larger deformation from LFdetuning than thermal stress. Calculate deformation for 0.1 mm window, which is not trivial for meshing and might take long computation time. Add window thickness steps into the model.