CERN Accelerator School Superconductivity for Accelerators Case study 2 Paolo Ferracin (paolo.ferracin@cern.ch) European Organization for Nuclear Research (CERN)

Case study 2 Low-beta Nb-Ti quadrupoles for the HL-LHC Introduction Large Hadron Collider (LHC) it will run at 6.5-7 TeV, providing 300 fb-1 of integrated luminosity within the end of the decade. CERN is planning to have an upgrade of the LHC to obtain significantly higher integrated luminosity. 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-Ti superconducting quadrupole with an 120 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 2

Case study 2 Low-beta Nb-Ti quadrupoles for the HL-LHC Questions Determine maximum gradient and coil size (using sector coil scaling laws) Define strands and cable parameters Strand diameter and number of strands Cu to SC ratio and pitch angle Cable width, cable mid-thickness and insulation thickness Filling factor κ Determine load-line (no iron) and “short sample” conditions Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss Determine “operational” conditions (80% of Iss ) and margins Compute jsc_op, jo_op , Iop , Gop , Bpeak_op Compute T, jsc , Bpeak margins Compare “short sample”, “operational” conditions and margins if the same design uses Nb3Sn superconducting technology Define a possible coil lay-out to minimize field errors Determine e.m forces Fx and Fy and the accumulated stress on the coil mid-plane in the operational conditions (80% of Iss ) Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of Iss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 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 2

Case study 2 Low-beta Nb-Ti quadrupoles for the HL-LHC Questions Determine maximum gradient and coil size (using sector coil scaling laws) Define strands and cable parameters Strand diameter and number of strands Cu to SC ratio and pitch angle Cable width, cable mid-thickness and insulation thickness Filling factor κ Determine load-line (no iron) and “short sample” conditions Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss Determine “operational” conditions (80% of Iss ) and margins Compute jsc_op, jo_op , Iop , Gop , Bpeak_op Compute T, jsc , Bpeak margins Compare “short sample”, “operational” conditions and margins if the same design uses Nb3Sn superconducting technology Define a possible coil lay-out to minimize field errors Determine e.m forces Fx and Fy and the accumulated stress on the coil mid-plane in the operational conditions (80% of Iss ) Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of Iss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Maximum gradient and coil size The max. gradient that one could reach is almost 160 T/m …but with a w/r = 2 120 mm thick coil! Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 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 2

Case study 2 solution Maximum gradient and coil size We assume a value of w/r = 0.5 30 mm thick coil We should get a maximum gradient around 150 T/m Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 Low-beta Nb-Ti quadrupoles for the HL-LHC Questions Determine maximum gradient and coil size (using sector coil scaling laws) Define strands and cable parameters Strand diameter and number of strands Cu to SC ratio and pitch angle Cable width, cable mid-thickness and insulation thickness Filling factor κ Determine load-line (no iron) and “short sample” conditions Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss Determine “operational” conditions (80% of Iss ) and margins Compute jsc_op, jo_op , Iop , Gop , Bpeak_op Compute T, jsc , Bpeak margins Compare “short sample”, “operational” conditions and margins if the same design uses Nb3Sn superconducting technology Define a possible coil lay-out to minimize field errors Determine e.m forces Fx and Fy and the accumulated stress on the coil mid-plane in the operational conditions (80% of Iss ) Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of Iss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Cable and strand size We assume a strand diameter of 0.825 mm We assume a pitch angle  of 17 Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Cable and strand size   We assume Thick. Comp. = -10 % Width. Comp. = -7 % 36 strands Ins. Thick. = 150 μm We obtain Cable width: 15 mm Cable mid-thick.: 1.48 mm Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Cable and strand size Summary Strand diameter = 0.825 mm Cu to SC ratio = 1.5 Pitch angle  = 17 N strands = 36 Cable width: 15 mm Cable mid-thickness: 1.48 mm Insulation thickness = 150 μm Area insulated conductor = 27.2 mm2 We obtain a filling factor k = area superconductor/area insulated cable = 0.28 Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 Low-beta Nb-Ti quadrupoles for the HL-LHC Questions Determine maximum gradient and coil size (using sector coil scaling laws) Define strands and cable parameters Strand diameter and number of strands Cu to SC ratio and pitch angle Cable width, cable mid-thickness and insulation thickness Filling factor κ Determine load-line (no iron) and “short sample” conditions Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss Determine “operational” conditions (80% of Iss ) and margins Compute jsc_op, jo_op , Iop , Gop , Bpeak_op Compute T, jsc , Bpeak margins Compare “short sample”, “operational” conditions and margins if the same design uses Nb3Sn superconducting technology Define a possible coil lay-out to minimize field errors Determine e.m forces Fx and Fy and the accumulated stress on the coil mid-plane in the operational conditions (80% of Iss ) Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of Iss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Margins Let’s work now on the load-line The gradient is given by So, for a Jsc= 1400 A/mm2 jo = jsc * k = 392 A/mm2 G = 110 T/m Bpeak = G * r * λ = 110 * 60e-3 * 1.15 = 7.6 T Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Margins Nb3Sn parameterization Temperature, field, and strain dependence of Jc is given by Summers’ formula where Nb3Sn is 900 for  = -0.003, TCmo is 18 K, BCmo is 24 T, and CNb3Sn,0 is a fitting parameter equal to 60800 AT1/2mm-2 for a Jc=3000 A/mm2 at 4.2 K and 12 T. Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Margins Nb-Ti parameterization Temperature and field dependence of BC2 and TC are provided by Lubell’s formulae: where BC20 is the upper critical flux density at zero temperature (~14.5 T), and TC0 is critical temperature at zero field (~9.2 K) Temperature and field dependence of Jc is given by Bottura’s formula where JC,Ref is critical current density at 4.2 K and 5 T (~3000 A/mm2) and CNb-Ti (27 T), Nb-Ti (0.63), Nb-Ti (1.0), and Nb-Ti (2.3) are fitting parameters. Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Margins Nb-Ti The load-line intercept the critical (“short-sample” conditions) curve at jsc_ss = 1740 mm2 jo_ss = jsc_ss * k = 487 mm2 Iss = jo_ss * Ains_cable= 13250 A Gss = 137 T/m Bpeak_ss = 9.4 T Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Margins Nb-Ti The operational conditions (80% of Iss) jsc_op = 1390 mm2 jo_op = jsc_op * k = 389 mm2 Iop = 10590 A Gop = 110 T/m Bpeak_op = 7.6 T Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Margins Nb-Ti In the operational conditions (80% of Iss) 2.1 K of T margin (2750-1740) A/mm2 of jsc margin (10.1-7.6) T of field margin Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Margins Nb3Sn “Short-sample” conditions jsc_ss = 2570 mm2 jo_ss = jsc_ss * k = 720 mm2 Iss = jo_ss * Ains_cable= 19580A Gss = 202 T/m Bpeak_ss = 14.0 T The operational conditions (80% of Iss) jsc_op = 2060 mm2 jo_op = jsc_op * k = 576 mm2 Iop = 15670 A Gop = 161 T/m Bpeak_op = 11.2 T 4.6 K of T margin (4800-2060) A/mm2 of jsc margin (14.8-11.2) T of field margin Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 Low-beta Nb-Ti quadrupoles for the HL-LHC Questions Determine maximum gradient and coil size (using sector coil scaling laws) Define strands and cable parameters Strand diameter and number of strands Cu to SC ratio and pitch angle Cable width, cable mid-thickness and insulation thickness Filling factor κ Determine load-line (no iron) and “short sample” conditions Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss Determine “operational” conditions (80% of Iss ) and margins Compute jsc_op, jo_op , Iop , Gop , Bpeak_op Compute T, jsc , Bpeak margins Compare “short sample”, “operational” conditions and margins if the same design uses Nb3Sn superconducting technology Define a possible coil lay-out to minimize field errors Determine e.m forces Fx and Fy and the accumulated stress on the coil mid-plane in the operational conditions (80% of Iss ) Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of Iss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Coil layout One wedge coil sets to zero b6 and b10 in quadrupoles ~[0°-24°, 30°-36°] ~[0°-18°, 22°-32°] Some examples Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 Low-beta Nb-Ti quadrupoles for the HL-LHC Questions Determine maximum gradient and coil size (using sector coil scaling laws) Define strands and cable parameters Strand diameter and number of strands Cu to SC ratio and pitch angle Cable width, cable mid-thickness and insulation thickness Filling factor κ Determine load-line (no iron) and “short sample” conditions Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss Determine “operational” conditions (80% of Iss ) and margins Compute jsc_op, jo_op , Iop , Gop , Bpeak_op Compute T, jsc , Bpeak margins Compare “short sample”, “operational” conditions and margins if the same design uses Nb3Sn superconducting technology Define a possible coil lay-out to minimize field errors Determine e.m forces Fx and Fy and the accumulated stress on the coil mid-plane in the operational conditions (80% of Iss ) Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of Iss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution E.m. forces and stresses For a quadrupole sector coil, with an inner radius a1, an outer radius a2 and an overall current density jo , 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 2

Case study 2 solution E.m. forces and stresses In the operational conditions (110 T/m) Fx (octant) = +0.60 MN/m Fy (octant) = -1.26 MN/m The accumulates stress on the coil mid-plane is Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 Low-beta Nb-Ti quadrupoles for the HL-LHC Questions Determine maximum gradient and coil size (using sector coil scaling laws) Define strands and cable parameters Strand diameter and number of strands Cu to SC ratio and pitch angle Cable width, cable mid-thickness and insulation thickness Filling factor κ Determine load-line (no iron) and “short sample” conditions Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss Determine “operational” conditions (80% of Iss ) and margins Compute jsc_op, jo_op , Iop , Gop , Bpeak_op Compute T, jsc , Bpeak margins Compare “short sample”, “operational” conditions and margins if the same design uses Nb3Sn superconducting technology Define a possible coil lay-out to minimize field errors Determine e.m forces Fx and Fy and the accumulated stress on the coil mid-plane in the operational conditions (80% of Iss ) Evaluate dimension iron yoke, collars and shrinking cylinder, assuming that the support structure is designed to reach 90% of Iss Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 solution Dimension of the yoke The iron yoke thickness can be estimated with Therefore, being G = 123 T/m, r = 60 mm and Bsat = 2 T we obtain tiron = ~110 mm Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 Low-beta Nb-Ti quadrupoles for the HL-LHC Questions Determine maximum gradient and coil size (using sector coil scaling laws) Define strands and cable parameters Strand diameter and number of strands Cu to SC ratio and pitch angle Cable width, cable mid-thickness and insulation thickness Filling factor κ Determine load-line (no iron) and “short sample” conditions Compute jsc_ss , jo_ss , Iss , Gss , Bpeak_ss Determine “operational” conditions (80% of Iss ) and margins Compute jsc_op, jo_op , Iop , Gop , Bpeak_op Compute T, jsc , Bpeak margins Compare “short sample”, “operational” conditions and margins if the same design uses Nb3Sn superconducting technology Define a possible coil lay-out to minimize field errors Determine e.m forces Fx and Fy and the accumulated stress on the coil mid-plane in the operational conditions Evaluate dimension iron yoke Evaluate dimension of the support structure (collars and shrinking cylinder) Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 2 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 2

Case study 2 solution Dimension of the support structure We assume that the shell will close the yoke halves with the same force as the total horizontal e.m. force at 90% of Iss Fx_total = Fx_octant * 2 * sqrt(2)= +2.1 MN/m Assuming an azimuthal shell stress after cool-down of shell = 200 MPa The thickness of the shell is tshell = Fx_total /2/1000/ shell ~ 5 mm Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Case study 1 solution Magnet cross-section Coil inner radius: 60 mm Coil outer radius: 90 mm The operational conditions (80% of Iss) jsc_op = 1390 mm2 jo_op = jsc_op * k = 389 mm2 Iop = 10590 A Gop = 110 T/m Bpeak_op = 7.6 T Collar thickness: 25 mm Yoke thickness: 110 mm Shell thickness: 5 mm OD: 460 mm Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2

Comparison Superconductivity for Accelerators, Erice, Italy, 25 April - 4 May, 2013 Case study 2