1. Longitudinal Expansion of RFQ Vane Ends at Section-to-Section Interface

2. RFQ vanes sit 5mm proud of clamping face and are free to thermally expand longitudinally toward each other Hole(s) for hollow dowel (alignment) + Berillium copper spring (electrical contact). Should holes be vented (virtual leak?). Note, dowel should be weaker than vane tip, hence hollow. This face constrained longitudinally by clamping flange

3. Pete’s Questions How much do the vanes grow longitudinally under normal heat loads? What is the variance in the growth? i.e. what are the tolerances? What stresses do the clamping faces see? What happens if no gap is left to allow growth and the vanes of each section are just butted up to each other?

4. Model Used Section 2 or 3 of RFQ (no cutback etc) –Fully featured major and minor vanes High mesh accuracy at vane ends Cut by three symmetry planes –Expansion prevented across symmetry planes 800kW peak input RF at 10% duty factor –Large over-estimate of heat for safety factor

5. Studies Performed Vary cooling efficiency –Baseline, expected cooling performance –Pessimistic cooling performance –Worst case cooling, far below expectations Vary movement constraints –Fixed clamping face, free vane ends (baseline) –Both clamping face and vane ends fixed –Entire end free to grow longitudinally –Baseline longitudinal constraints, but vanes unable to grow transversely outward

6. Case 1: Cooling System Performs Adequately, as Expected. (Still Allows Plenty of Room for Cooling Pockets to Exceed Expectations, though!)

7. Baseline Performance (Temperature) Cooling pockets do not continue all the way to the end of each section ∴ hotter at ends Might be beneficial to have a loop of cooling in Pete’s section-to-section clamping flange? Asymmetry between major and minor vanes due to vacuum port and its cooling Flow rates adjusted in nearby cooling pockets to make temperatures as uniform and symmetrical as possible

8. Baseline Performance (X-Direction Displacement) Black lines show un-deformed model Overlaid coloured shape shows thermally expanded model, ~100x exaggerated Outward expansion of walls always counteract inward expansion of vane tips Net movement of majority of vanetip is toward beam axis by ≲ 20µm, as before But hotter ends grow away from beam axis To reduce this effect, may be profitable to cool the inter-section clamping flange Y-direction displacements are quite comparable since temperature distribution is reasonably symmetrical

9. Baseline Performance (Y-Direction Displacement) Moves away from beam axis 15µm Moves toward beam axis 10µm

10. Baseline Performance (Longitudinal Displacement) Clamping surfaces held fixed longitudinally End of RFQ vane free to expand longitudinally With anticipated thermal conditions, longitudinal expansion is only 33µm Zero growth on symmetry plane Equal growth either side of symmetry plane

11. Case 2: Cooling System Performs Much Worse Than Expected.

12. Poor Cooling Performance (Temperature)

13. Poor Cooling Performance (X-Direction Displacement) Y-direction displacements are quite comparable since temperature distribution is reasonably symmetrical Vane ends start to move away from beam axis by same amount as rest of vane tips move toward it, and close to our acceptable limits Moves away from beam axis 24µm Moves toward beam axis 21µm

14. Poor Cooling Performance (Longitudinal Displacement) With poor cooling system performance, longitudinal expansion is 54µm

15. Case 3: Cooling System is Terrible. The Cooling Pockets Do Not Provide Anything Like the Water Flow Rates Required.

16. Terrible Cooling Performance (Temperature) Maximum temperature 60°C Even the coolest part is quite warm to the touch!

17. Terrible Cooling Performance (X-Direction Displacement) Y-direction displacements are quite comparable since temperature distribution is reasonably symmetrical Vane tips now moving ±40µm which is far from our requirements, but not bad considering the small amount of cooling applied in this simulation! Moves away from beam axis 47µm Moves toward beam axis 36µm

18. Terrible Cooling Performance (Longitudinal Displacement) Even if the cooling system performs very badly, the longitudinal expansion is still only about 0.13mm

19. Case 4: Baseline Cooling Performance (as in Case 1). No Gap Left to Allow Vane Ends to Expand Longitudinally.

20. Baseline Performance, but No Gap for Longitudinal Growth If the vanes are constrained longitudinally by butting up against each other, they’ll warp transversely Therefore, a gap between sections is necessary even if movements are small Colour bands show longitudinal displacement values Deformed model shows uncontrolled, asymmetric, warped transverse growth

21. Case 5: Baseline Cooling Performance and Free Vanes (as in Case 1). Section-to-Section Clamping Flange is Moderately Cooled.

22. Cooled Clamping Flange (Temperature) Slightly lower temperature of end faces Allows better control of longitudinal temperature distribution

23. Cooled Clamping Flange (X-Direction Displacement) Smaller displacement of end-of-section vane tips away from beam axis Y-direction displacements are quite comparable since temperature distribution is reasonably symmetrical

24. Cooled Clamping Flange (Longitudinal Displacement)

25. Cooled Clamping Flange (von-Mises Stress) Smaller movement of vane ends relieves a bit of stress on clamping face and bolts Peak stress reduced from 66MPa to 55MPa Average stress in clamping area reduced from 25MPa to 15MPa

26. Case 6: Baseline Cooling Performance. Entire End of Section Free to Expand. (This is what would happen to a single 1m RFQ section, with no other sections clamped to it either side.)

27. No Longitudinal Constraint (X-Direction Displacement) Since clamping face is no longer constrained, vane end no longer flexes up and out, away from beam axis Y-direction displacements are quite comparable since temperature distribution is reasonably symmetrical

28. No Longitudinal Constraint (Longitudinal Displacement) RFQ is free to expand longitudinally so it expands a lot more Uniform longitudinal growth of vane end Total longitudinal displacement is 74µm With terrible cooling performance (Case 3), vane expansion is 243µm. Expansion of vanes will never be more than this, so a total gap of 0.5mm is fine.

29. No Longitudinal Constraint (von-Mises Stress) No Longitudinal clamping at all relieves a lot of stress Peak stress reduced from 66MPa to 15MPa and moves from clamping surfaces to cooling pocket

30. Case 7: Baseline Cooling Performance. Vane end unable to grow outward transversely. (This is what would happen if the inner face of the clamping flange is a very snug fit to the protruding vane end)

31. Vane Transversely Constrained (X-Direction Displacement) Vane end cannot grow outward, so it instead is squeezed in closer toward the beam axis than the rest of the vane Y-direction displacements are quite comparable since temperature distribution is reasonably symmetrical

32. Vane Transversely Constrained (Longitudinal Displacement) Transversely constrained vane cannot flex up and out, so only 14µm of longitudinal growth

33. Vane Transversely Constrained (von-Mises Stress) Vane end pressed up against longitudinally fixed clamping face builds up tremendous stresses Likely to result in permanent structure failure Need gaps to allow both transverse and longitudinal growth of vanes

34. Summary Easy to tweak flow rates to get a uniform temperature The cooling system is quite effective at removing heat evenly along the length of the RFQ This results in little longitudinal growth Typical longitudinal growth of a vane end is 40µm Maximum growth of one section will not exceed 250µm ∴ a 0.5mm gap between vane ends of two sections will be more than sufficient even if we need much more RF power than anticipated and the cooling system fails Require longitudinal and transverse gaps for growth Cooled flange brings small benefits but is not essential Transverse growth still <20µm as previously found