Recent Study Topics

1. Full length model with wafers, hybrids and cable as dead weight 0.173in dia. support pins Clamped pin vertical supports, but with pins at one fixed in Z Core thickness 4.6mm Half length model with wafers, hybrids and cable as dead weight 0.173in support pin By necessity for symmetry the middle is fixed in Z, thus it looks like all pins clamped vertically at ends, but floating in Z Model will be modified to add structural coupling of wafers, and hybrids Core thickness 5.88m Model of pins and end cap alone with stave weight imposed 0.173in diameter 0.25in diameter Significant Changes Calculated apparent density of two phase fluid. For entering and exit quality the mean density is 60kg/m3, whereas liquid density is 1660kg/m3 Previous solutions used an average of 1000kg/m3, so the liquid dead weight is reduced noticeably Round circular tube in half length model Accommodated the change to 5.88mm core Varied core shear modulus, reflected in density change to material 66 to 210kg/m3, CVD carbon foam 56 and 110kg/m3, honeycomb Recent Study Topics Models ATLAS Silicon Tracker Upgrade

2. Sandwich Core Differences in Model • FEA Models • 4.6mm core height model has elongated cooling tubes and the foam does not contact the tubes • Hydraulic diameter 5mm • Less core material than in the half length model, possibly an effect in sag • 5.88mm core height has round tubes and the core comes in contact, except at the very top. • Internal diameter 5.27mm • Intent is to use the core material to improve thermal contact 4.6mm 5.88mm ATLAS Silicon Tracker Upgrade

3. FEA Sandwich Core Summary Based on reduced coolant density ATLAS Silicon Tracker Upgrade

4. Core Thickness • Estimated stave sag for two core thickness based on bending only, no core shear deflection (analytical based on “fixed end supports”) • Foam Shear Modulus (not included) • 4.6mm thick foam (facing separation), δ=35μm • 5.88mm thick foam, δ=29μm • FEA Solution for Shear Modulus=26.9MPa (lowest density foam) with facings 4 to 1 K13D2U fiber orientation (Coolant density 60kg/m3) • 4.6mm, δ=59.4μm (both ends free to move axially) • 5.88mm, δ=62.1μm (1/2 length model, since only one end modeled by necessity it simulates as if both ends free to move axially) • Little difference in solutions ATLAS Silicon Tracker Upgrade

5. Stave Gravity Sag Full Length Model- At One end, pins are Fixed in Z • Conditions • Mass of cable, hybrids, wafers, and chips included in facing density • Mass of two-phase fluid included in tube density • Homogeneous two-phase fluid density average is 60kg/m3 • C3F8 liquid density is 1660kg/m3 • Fluid vapor fraction varies from ~0.3 to 0.8 • Virgin RVC foam • Core foam density is 66kg/m3 K13D2U 4/1 Peak deflection at stave center is 53.7μm ATLAS Silicon Tracker Upgrade

6. Simple BC at Both Ends • Full Length Model- Symmetrical Deflection • Sag increased from 53.7 to 59.4μm (originally one end fixed in Z, now Z fixed in middle) • For same conditions the ½ length model with 5.88mm core thickness was 62.1 μm K13D2U 4/1 Sandwich core thickness 4.6mm ATLAS Silicon Tracker Upgrade

7. Sandwich Core CVD Carbon Full length model- One End, pins fixed Fixed in Z • Core foam density is 210kg/m3 • Other Conditions • Mass of cable, hybrids, wafers, and chips included in facing density • Mass of two-phase fluid included in tube density • Homogeneous two-phase fluid density average is 60kg/m3 • C3F8 liquid density is 1660kg/m3 • Fluid vapor fraction varies from ~0.3 to 0.8 K13D2U 4/1 Peak deflection at stave center is 54.8μm ATLAS Silicon Tracker Upgrade

8. Carbon Foam-No CVD Half Length Model-Pins fixed against vertical motion • Core foam density is 66kg/m3 • Other Conditions • Mass of cable, hybrids, wafers, and chips included in facing density • Mass of two-phase fluid included in tube density • Homogeneous two-phase fluid density average is 60kg/m3 • C3F8 liquid density is 1660kg/m3 • Fluid vapor fraction varies from ~0.3 to 0.8 • Sandwich height • 5.88mm versus 4.6mm K13D2U 4/1 δ=62.1microns ATLAS Silicon Tracker Upgrade

9. Sandwich Core CVD Carbon Half Length Model-Pins free to move axially • Core foam density is 210kg/m3 • Other Conditions • Mass of cable, hybrids, wafers, and chips included in facing density • Mass of two-phase fluid included in tube density • Homogeneous two-phase fluid density average is 60kg/m3 • C3F8 liquid density is 1660kg/m3 • Fluid vapor fraction varies from ~0.3 to 0.8 • Sandwich height • 5.88mm versus 4.6mm K13D2U 4/1 Peak deflection at stave center is 65μm ATLAS Silicon Tracker Upgrade

10. Carbon Foam-No CVD Half Length Model-Includes Silicon Wafers and Hybrids in stiffness simulation • Core foam density is 66kg/m3 • Other Conditions • Mass of cable included in facing density • Mass of two-phase fluid included in tube density • Homogeneous two-phase fluid density average is 60kg/m3 • C3F8 liquid density is 1660kg/m3 • Fluid vapor fraction varies from ~0.3 to 0.8 • Sandwich height • 5.88mm K13D2U 4/1 δ=52microns ATLAS Silicon Tracker Upgrade

11. Detectors and Hybrids Stiffness Contribution • 1/2 Length Model- K13D2U 4/1 fiber orientation, coolant density 60kg/m3 • Silicon modules and hybrids as dead weight-62microns • Silicon modules and hybrids part of stiffness-52microns • Mass of 1st solution 0.1973kg without module stiffness • Mass of second solution 0.1891kg with module and hybrid stiffness • Difference in gravity loading 4.1%; had hoped for same mass • Difference in central deflection 19.2% ATLAS Silicon Tracker Upgrade

12. Fiber Orientation • Comparing 4 to 1 K13d2U versus Quasi-isotropic K13D2U facings • Modulus in direction of stave axis is different by factor of 1.96 • Thermal distortion solutions with the unbalanced lay up was OK • Comparison made for pins free to move in axial direction • Difference between pins fixed on one end and both free is 5.7μm • Sag is reduced by a factor of 1.59 K13D2U Quasi-isotropic δmax=94μm K13D2U 4 to 1 lay up δmax=59.4μm ATLAS Silicon Tracker Upgrade

13. Beryllium End Parts • Conditions • K13D2U quasi-isotropic fiber orientation • 0.173in dia Be pins • Be end cap • Coolant 60kg/m3 • Pins at end free to move in Z, fixed in Y • Z fixed at mid span, X constrained at two ends • Sag decreased from 94.6μm to 80.6 μm through use of Be ATLAS Silicon Tracker Upgrade

14. c h Solve for Effective Core Shear Modulus • 96cm Model of Stave • Use simple edge supports, K13D2U 4/1 • Apply forces at quarter points, ¼ from each end • Extract deflection at Δ4 and Δ2, quarter point and mid-span • Use relationship • Result for 4.6mm core with Al tubes • ~128 MPa versus 26.9 MPa for virgin foam • Tubes contribute most of the shear stiffness, except at very high foam densities P/2 P/2 Division between bending and shear, based 0n estimate of core shear of 128MPa Using sandwich relationships for fixed ends Δbending est=36.7μm Δcore shear est=8.2 μm Combined Δ=45microns (FEA 53.7 μm for one end of the pins fixed) ATLAS Silicon Tracker Upgrade

15. Estimate for 2m Stave • Use analogy of a uniformly loaded beam • G-core shear properties • L- length of beam • c- height of sandwich core • b- width of sandwich • t- facing thickness • h-overall distance across facings • B- expression • w- uniform load • Shear Deflection for 2m stave with 20mm core height quasi K13 facings • G=26.9MPa, δ=56μm • G=212MPa, δ=7μm Based on ~uniform load of 7.9N/m (does include an estimate for mass of 3 internal ribs • Bending Deflection estimate for 2m stave • 81μm for fixed end condition (uniform loading) (fixed) ATLAS Silicon Tracker Upgrade

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17. 2m Stave Core Design ATLAS Silicon Tracker Upgrade

18. End Cap Model Only δ=.20μm Deflection of end cap for ½ stave mass Pin diameter 0.173in δ=.26μm Pin diameter 0.25in ATLAS Silicon Tracker Upgrade