1. 1 Calorimeter Thermal Analysis Revision C November 23 2009

2. 2 Conductivity of Copper and Argon at 90°K Rod, Copper Liquid Argon Matrix, Copper Tube, Copper Conductivity at 90°K, W/mm-°K Copper is 0.514 Liquid Argon is 0.00012 Geometry, mm Center to center = 7.5 Matrix hole diameter is 5.851 Tube OD is 5.584 Tube ID is 5.081 Rod OD is 4.881 Gaps are filled with liquid Argon

3. 3 Copper Matrix Finite Element Models The baseline and refined 2D models of the copper matrix are shown.

4. 4 Conduction Results Temperature solution for X, Y and Z gradients Because conduction properties repeat after a 120° rotation about Z, we should find that Kxx = Kyy. The conduction used for these components is therefore the average of the values. The FEA Kyy was 0.26% larger than Kxx. The hand calculation for Kzz is considered to be exact, the FEA is 5% lower. Z is parallel to the rod axes

5. 5 3D Model of Calorimeter Al tube is modeled with shell elements. Its temperature is set to 90°K. It is eccentric. 1.95 mm gap on top 0.10 mm gap here (0.05 shown) Copper Matrix Tungsten Matrix Argon Copper Note: All “gaps” are filled with liquid Argon

6. 6 Al Tube Eccentricity Argon Copper matrix Structural analysis shows that the external aluminum tube wraps around the lower half of the calorimeter. These elements are the same thickness

7. 7 Aluminum Shell and Argon Elements The aluminum shell, above, is wrapped around the OD. It is used as a convenient surface to fix at 90°K. Plate elements are used. They are being plotted here as a surface (no thickness). Liquid argon is modeled with solid conduction elements as shown above.

8. 8 Conductivities Y X Z

9. 9 Deformation of the Aluminum Cylinder The minimum gap between the external aluminum cylinder and the modules is 0.1 mm, the thickness of the Kapton wrap (the Kapton wrap has about the same conductivity as liquid argon so it is not explicitly modeled). The deformation of the aluminum cylinder is predicted using a rigid cylinder to represent the modules and contact elements between the rigid cylinder and the aluminum cylinder (contact elements prevent the aluminum shell from penetrating the rigid cylinder but allow a gap to open between the two).

10. 10 Clearance Between Modules and Cylinder The cylinder deforms to be closer to the modules over the region extending to 52° from the bottom. Above 100° the cylinder deforms to move further from the modules than would be the case with a rigid eccentric cylinder. The net effect on the thermal solution is an increase in conduction between the modules and the aluminum cylinder compared to results for an argon filled gap between two rigid eccentric cylinders. The end of the cylinder is at Z=0 The midpoint is at Z=27.254”

11. 11 Deformation of the Aluminum Cylinder, mm Nodal forces indicating contact Rigid Contact Surface (the modules) The aluminum cylinder with end plates

12. 12 Heat Generation Rate, W/mm 3 Heat flux is higher in the gray zone ranging up to 1.12E-6 W/mm 3

13. 13 Thermal Solution, Peak temperature is 0.1°K above the OD The temperature of the OD (the aluminum tube) is 90°K

14. 14 Convection in the Outer 12 mm Thick Argon Blanket

15. 15 Argon Material Properties, kg-m-sec-°K Numeric values for Ansys density input: dens = Nominal + C 2 *(T- C1) = Nominal(1-CTEvol*(T-Tref)) = 1393-1393*.0044*(T-Tref) = 1393-6.1292*(T-Tref)

16. 16 Simple Conduction (Stagnant Fluid) Radiant heat transfer is low and has not been included in the CFD model.

17. 17 CFD Analysis Most results are from a 2D CFD analysis –3D results for the outer annulus gave only slightly higher net convection. All results of interest are in the transitional regime between laminar and turbulent flow –Laminar flow result gives velocities high enough to justify turning the turbulent flow option* on. –Turbulent results give velocities too low to justify the use of the turbulence option. Laminar flow results were used for thermal analysis (conservative; the convection coefficient, H.. W/m 2 /°K, is 42% greater when turbulence is switched on) * Turning the turbulent flow option on triggers a turbulent energy calculation in the boundary layer that gives increased viscosity and heat transfer.

18. 18 CFD Analysis Results, Outer Annulus Velocity vector plots above and below. Temperatures plotted at left and at bottom These plots show the aspect ratio of the outer annulus. Flow details can only be seen in local blowups.

19. 19 Flow Patterns at the Top and Bottom At the top, convective cells form as would be expected for a fluid layer heated from below. At the bottom, convective cells are not apparent but the fluid is stirred by the cool argon flowing down the OD. The images on the following slides show these patterns

20. 20 Flow Pattern at the Top, 0.5 and 2° K Differential 2°K Differential 0.5°K Differential

21. 21 Flow Pattern at the Bottom, 0.5° K Differential, Turbulent flag on and off 0.5°K Differential Laminar Flow (contour values are not the same as below) 0.5°K Differential Turbulent Flow is turned on, results are similar to those above

22. 22 Flow Pattern at the Bottom, 2 deg K Differential 2°K Differential, Turbulent Flow is turned on

23. 23 Reynold’s Number The effective channel depth for flow is half the thickness of the fluid layer (full thickness flows are possible but have not been observed) R = 2*p*V*h/mu for full thickness flow (the hydraulic diameter is 2*h) Transitional flow for 2000 < R < 4000 Turbulent flow if V>0.062 m/sec Note that in the range of interest Reynolds number with turbulence activated will always be lower than required to justify activation. Laminar values for convection are used. These are expected to be conservative.

24. 24 Three Dimensional Flow Results A 3D model was run to see if convective cells consisting of near through thickness flows would develop. The 3D results hint at slightly increased heat flow compared to 2D results but these coarser mesh models predict about 40% of the heat flux of the fine mesh 2D model. –The 3D results were compared to comparably meshed 2D results to estimate the effect of 3D vs 2D flows. –3D models (and the comparably meshed 2D models) may not have been run long enough to establish a quasi-steady state heat flow (net heat flow tends to increase slowly with time from the intial conditions of zero velocity and linear thermal gradients).

25. 25 Circulation Pattern, Inside and Outside Views at the Same Time Inside View Surface is ¼ thickness Outside View Surface is 3/4 thickness 2°K differential, 4 element layers in model, the inner two layers are shown See the movie files “D3-velocities.avi” and “D3-flux.avi” (use Windows media player)

26. 26 Circulation at an Earlier Time (Eddy’s are not Stable) Inside View Surface is ¼ thickness Outside View Surface is 3/4 thickness

27. 27 Comparison of 2D and 3D Convection Coefficient Results Fine mesh 2D results Fine mesh 2D results earlier runs Coarser mesh 2D and 3D results. Coarse 4 and 8 are 2D results with the coarse 8 mesh being similar to the 3D mesh. 3D results give somewhat higher convection than similarly meshed 2D results. Fine mesh laminar 2D results are used.

28. 28 Outer Annulus, Equation for H The equation fits the laminar results and was used in the runs reported on subsequent slides.

29. 29 Interaction Between Conduction and Convection Solutions Convection Calculations (CFD) –Most convection models were run with uniform temperatures on the inner and outer walls enclosing the 12 mm argon layer. –One convection case used a representative circumferential temperature distribution on the ID and gave 90% of the uniform temperature heat transport. Thermal solutions use an OD convection coefficient that increases with increasing temperature. This is derived from uniform wall temperature results. –The following slides show that, for uniform temperature walls, heat is removed faster from the bottom of the OD of the calorimeter and deposited further up on the heat sink’s ID (for ΔT < 2°K). –The use of the temperature dependent convection coefficient has the effect of increasing heat flow from the bottom of the OD of the calorimeter because that region is warmer. I reduced convection coefficients to 90% of the value obtained for uniform wall temperatures to account for the reduction in film coefficient obtained from the realistic temperature distribution. I think this ensures that the thermal results are conservative. The temperature dependent convection coefficient is needed to accommodate the axial temperature variation seen on the OD of the calorimeter. It also gives a bit more heat removal at the bottom compared to the top but doesn’t bias heat flux as heavily to the lower, warmer, areas at nominal heat loads as the CFD solution.

30. 30 Heat Flux at 0.5°K Differential (Turbulent Option Off) With inner and outer cylinders held at constant temperatures, heat tends to be removed from the lower regions of the ID and deposited at the higher regions of the OD. The variation in heat removal on the inner cylinder is more pronounced for low temperature differences.

31. 31 Heat Flux at 2°K Differential At a 2°K differential the heat flow is more uniformly distributed making the use of a convection coefficient dependent on the local inner wall temperature justifiable.

32. 32 Inner Annulus At Fcal 3 ID and OD are 0.072 and 0.086 m Heat leak is ~ 10 W/m 2 –Mesh used is shown at right –ΔT used was 0.15°K* –Transient analysis is required Flow is laminar and unstable over time –V = 0.0065 m/sec –R = 846 this is < 2000 so flow is laminar * Based on outer annulus thermal analysis Guess net heat transfer is ~77 W/m 2 -°K ΔT ~ 10/77 = 0.13°K (used 0.15°K in run) End result indicates ΔT ~0.25°K so the heat transfer coefficient is underestimated by this analysis.

33. 33 Inner Annulus Results for 0.15 °K Delta Transmitted flux stabilizes at ~1.61 W (average from 400 to 500 sec). This is 5% higher than the average from 225 to 275 sec. Heat transfer is 39.8 W/m 2 -°K Net flux is slowly decaying with some fluctuations. The net flux of 0.21 W at 178 sec can be expected to persist for a long time if the fluid temperature is above the steady state by 10% or more of the net gradient (i.e. 0.015°K)

34. 34 Use of Inner Annulus Results ID temperatures range from about 90.4 to 90.51°K The heat flux (heat leak) at the ID is 10 W/m 2 The film coefficient is 39.8 W/m 2 -°K The surface inside the surface shown must be 0.25°K warmer to drive the heat leak (10 W/m 2 /39.8 W/m 2 -°K) Temperatures at the model ID Convection applies between the ID surface shown and another surface inside this surface (not shown or modeled).

35. 35 Solid Copper Zones Solid copper zones jacket the ID and OD of each module as shown

36. 36 Increased Heat Loads The following models all have: –10 W/m 2 heat leak at the ID (unless otherwise noted) –Ohmic heating of 4.95 W/m 3, scaled with heat load –Temperature dependent Convection at the OD Laminar flow results used Convection coefficient reduced 10% based on evaluating convection with a representative temperature profile on the id. The convection coefficient at each location was assigned based on the temperature at that location. –Several iterations were required to converge on stable temperatures.

37. 37 Summary of Results See above for more information about run parameters The maximum temperature listed is the increase over the outer wall in contact with the 12 mm thick argon layer. This surface was fixed at 90°K. The “OD Annulus Inner T” gives the minimum and maximum temperatures on the aluminum tube that forms the inner wall of the 12 mm outer argon jacket.

38. 38 Nominal Heat Load + Ohmic Heating and 10 W/m 2 ID Heat Leak

39. 39 6xNominal Heat Load + Ohmic Heating and 10 W/m 2 ID Heat Leak

40. 40 10xNominal Heat Load + Ohmic Heating and 10 W/m 2 ID Heat Leak

41. 41 20xNominal Heat Load + Ohmic Heating and 10 W/m 2 ID Heat Leak

42. 42 10xNominal Heat Load + Ohmic Heating and 20 W/m 2 ID Heat Leak