1. CERN Accelerator School Superconductivity for Accelerators Case study 1 Paolo Ferracin ( paolo.ferracin@cern.ch ) European Organization for Nuclear Research (CERN)

2. Case study 1 Low-beta Nb 3 Sn quadrupoles for the HL-LHC Introduction L ARGE H ADRON C OLLIDER (LHC) it will run at 6.5-7 TeV, providing 300 fb - 1 of integrated luminosity within the end of the decade. After 2020, CERN is planning to have an upgrade of the LHC to obtain ten times more integrated luminosity, i.e., 3000 fb -1. Part of the upgrade relies on reducing the beam sizes in the Interaction Points (IPs), by increasing the aperture of the present triplets. Currently, the LHC interaction regions feature NbTi quadrupole magnets with a 70 mm aperture and a gradient of 200 T/m. Goal Design a Nb 3 Sn superconducting quadrupole with an 150 mm aperture for the upgrade of the LHC interaction region operating at 1.9 K Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 2

3. Low-beta Nb 3 Sn quadrupoles for the HL-LHC Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, G ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, G op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 3

4. Additional questions Evaluate, compare, discuss, take a stand (… and justify it …) regarding the following issues High temperature superconductor: YBCO vs. Bi2212 Superconducting coil design: block vs. cos Support structures: collar-based vs. shell-based Assembly procedure: high pre-stress vs. low pre-stress Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 4

5. Low-beta Nb 3 Sn quadrupoles for the HL-LHC Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, G ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, G op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 5

6. Case study 1 solution Maximum gradient and coil size The max. gradient that one could reach is almost 200 T/m …but with a w/r = 2 150 mm thick coil! Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 6

7. Case study 1 solution Maximum gradient and coil size Large aperture need smaller ratio w/r For r=30-100 mm, no need of having w>r Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 7

8. Case study 1 solution Maximum gradient and coil size We assume a value of w/r = 0.5 37 mm thick coil We should get a maximum gradient around 170 T/m Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 8

9. Low-beta Nb 3 Sn quadrupoles for the HL-LHC Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, G ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, G op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 9

10. Case study 1 solution Cable and strand size We assume a strand diameter of 0.85 mm We assume a pitch angle  of 18  Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 10

11. Case study 1 solution Cable and strand size Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 11 We assume Thick. Comp. = -12 % Width. Comp. = -3 % 40 strands Ins. Thick. = 150 μm We obtain Cable width: 18 mm Cable mid-thick.: 1.5 mm

12. Case study 1 solution Cable and strand size Summary Strand diameter = 0.85 mm Cu to SC ratio = 1.2 Pitch angle  = 18  N strands = 40 Cable width: 18 mm Cable mid-thickness: 1.5 mm Insulation thickness = 150 μm Area insulated conductor = 32.7 mm 2 We obtain a filling factor k = area superconductor/area insulated cable = 0.32 Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 12

13. Case study 1 Low-beta Nb 3 Sn quadrupoles for the HL-LHC Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, G ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, G op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 13

14. Case study 1 solution Margins Let’s work now on the load-line The gradient is given by So, for a J sc = 1600 A/mm 2 j o = j sc * k = 512 A/mm 2 G = 142 T/m B peak = G * r * λ = 142 * 75e -3 * 1.15 = 12.2 T Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 14

15. Case study 1 solution Margins Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 15 Nb 3 Sn parameterization Temperature, field, and strain dependence of Jc is given by Summers’ formula where  Nb3Sn is 900 for  = -0.003, T Cmo is 18 K, B Cmo is 24 T, and C Nb3Sn,0 is a fitting parameter equal to 60800 AT 1/2 mm -2 for a Jc =3000 A/mm 2 at 4.2 K and 12 T.

16. Case study 1 solution Margins Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 16 Nb-Ti parameterization Temperature and field dependence of B C2 and T C are provided by Lubell’s formulae: where B C20 is the upper critical flux density at zero temperature (~14.5 T), and T C0 is critical temperature at zero field (~9.2 K) Temperature and field dependence of Jc is given by Bottura’s formula where J C,Ref is critical current density at 4.2 K and 5 T (~3000 A/mm2) and C Nb-Ti (27 T),  Nb-Ti (0.63),  Nb-Ti (1.0), and  Nb-Ti (2.3) are fitting parameters.

17. Case study 1 solution Margins Nb 3 Sn Let’s assume  = 0.000 The load-line intercept the critical (“short-sample” conditions) curve at j sc_ss = 1970 mm 2 j o_ss = j sc_ss * k = 630 mm 2 I ss = j o_ss * A ins_cable = 20600 A G ss = 175 T/m B peak_ss = 15 T Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 17

18. Case study 1 solution Margins Nb 3 Sn The operational conditions (80% of I ss ) j sc_op = 1580 mm 2 j o_op = j sc_op * k = 505 mm 2 I op = 16480 A G op = 140 T/m B peak_op = 12.1 T Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 18

19. Case study 1 solution Margins Nb 3 Sn In the operational conditions (80% of I ss ) 4.6 K of T margin (4000-1580) A/mm 2 of j sc margin (15.8-12.1) T of field margin Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 19

20. Case study 1 solution Margins Nb-Ti “Short-sample” conditions j sc_ss = 1350 mm 2 j o_ss = j sc_ss * k = 430 mm 2 I ss = j o_ss * A ins_cable = 14060 A G ss = 119 T/m B peak_ss = 10.3 T The operational conditions (80% of I ss ) j sc_op = 1080mm 2 j o_op = j sc_op * k = 344 mm 2 I op = 11250 A G op = 95 T/m B peak_op = 8.2 T 2.1 K of T margin (2370-1080) A/mm 2 of j sc margin (11-8.2) T of field margin Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 20

21. Case study 1 Low-beta Nb 3 Sn quadrupoles for the HL-LHC Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, G ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, G op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 21

22. Case study 1 solution Coil layout One wedge coil sets to zero b 6 and b 10 in quadrupoles ~[0°-24°, 30°-36°] ~[0°-18°, 22°-32°] Some examples Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 22

23. Case study 1 Low-beta Nb 3 Sn quadrupoles for the HL-LHC Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, G ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, G op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 23

24. Case study 1 solution E.m. forces and stresses For a quadrupole sector coil, with an inner radius a 1, an outer radius a 2 and an overall current density j o, each block (octant) see Horizontal force outwards Vertical force towards the mid-plan In case of frictionless and “free-motion” conditions, no shear, and infinitely rigid radial support, the forces accumulated on the mid- plane produce a stress of Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 24

25. Case study 1 solution E.m. forces and stresses In the operational conditions (140 T/m) F x (octant) = +1.90 MN/m F y (octant) = -4.02 MN/m The accumulates stress on the coil mid-plane is Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 25

26. Case study 1 Low-beta Nb 3 Sn quadrupoles for the HL-LHC Questions 1. Determine maximum gradient and coil size (using sector coil scaling laws) 2. Define strands and cable parameters 1. Strand diameter and number of strands 2. Cu to SC ratio and pitch angle 3. Cable width, cable mid-thickness and insulation thickness 4. Filling factor κ 3. Determine load-line (no iron) and “short sample” conditions 1. Compute j sc_ss, j o_ss, I ss, G ss, B peak_ss 4. Determine “operational” conditions (80% of I ss ) and margins 1. Compute j sc_op, j o_op, I op, G op, B peak_op 2. Compute T, j sc, B peak margins 5. Compare “short sample”, “operational” conditions and margins if the same design uses Nb-Ti superconducting technology 6. Define a possible coil lay-out to minimize field errors 7. Determine e.m forces F x and F y and the accumulated stress on the coil mid- plane in the operational conditions (80% of I ss ) 8. Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of I ss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 26

27. Case study 1 solution Dimension of the yoke The iron yoke thickness can be estimated with Therefore, being G = 156 T/m (at 90% of I ss ) r = 75 mm and B sat = 2 T we obtain t iron = ~220 mm Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 27

28. Case study 1 solution Dimension of the support structure We assume a 25 mm thick collar Images not in scale Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 28

29. Case study 1 solution Dimension of the support structure Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 29 We assume that the shell will close the yoke halves with the same force as the total horizontal e.m. force at 90% of I ss F x_total = F x_quadrant * 2 * sqrt(2) = +6.8 MN/m Assuming an azimuthal shell stress after cool-down of  shell = 200 MPa The thickness of the shell is t shell = F x_total /2/1000/  shell ~ 17 mm

30. Case study 1 solution Magnet cross-section Coil inner radius: 75 mm Coil outer radius: 112 mm The operational conditions (80% of I ss ) j sc_op = 1580 mm 2 j o_op = j sc_op * k = 505 mm 2 I op = 16480 A G op = 140 T/m B peak_op = 12.1 T Collar thickness: 25 mm Yoke thickness: 220 mm Shell thickness: 17 mm OD: 748 mm Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 30

31. Comparison Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 1 31