1. A novel model for Minimum Quench Energy calculation of impregnated Nb 3 Sn cables and verification on real conductors W.M. de Rapper, S. Le Naour and H.H.J. ten Kate CHATS on Appl. Supercond. 12 th October 2011

2. Outline  Introduction  Thermal stability  Conductor design  Model  Geometry  Thermal calculation  Electrical calculation  Solving algorithm  Validation on measurement  Extrapolation  Magnet X-section of measured conductor  Outlook: Full scale conductor and magnet

3. Introduction Thermal stability: A small perturbation (~1 mJ) results in a small normal zone (1-5 mm) in a conductor This normal zone either collapses or results in a thermal run-away (quench) The goal of this model is to accurately predict the energy needed to initiate a thermal run away in high-J c Nb 3 Sn cables and magnets Boundary conditions:  Needs to run on a desktop PC  Being able to evaluate measurements directly  What is the bare minimum of factors that need to be taken into account?

4. Thermal stability Initial perturbation NZ > MPZ No quench Strand NZ < MPZ Cable I re-distribution NZ spreads to neighboring strands Reduced Joule heating NZ: Normal Zone MPZ: Minimum Propagation Zone Magnet No recovery possibility Magnet Quench Cable quench Strand quench Recovery by cable Insufficient Current sharing Recovery by wire High energy Bad cooling Low energy Good cooling Conclusion:  There are only two recovery routes  There is no need to take any magnet effects into account.

5. Conductor: Wire Made up from small Nb 3 Sn filaments imbedded in a pure copper matrix (RRP – PIT) with high RRR Assumptions:  Temperature is homogeneous over wire X-section  Normal current instantly redistributes to Cu (ρ Nb3Sn >>ρ Cu ) This allows to simulate the wire as a 1D object

6. This consists of 14-40 Nb 3 Sn wires, twisted, rolled and impregnated to form a mechanically stable conductor Assumptions:  The cross contacts are negligible  The cable geometry is negligible  Fully adiabatic This allows to simulate the cable as system of equidistantly coupled 1D wires Conductor: Cable

7. The assumption that there are no cross connections is mandatory : Cross contact resistances must be 100 time as small as adjacent contact resistances to keep AC-loss low. Exception:  Coated wires (Poor thermal stability) Any useful conductor will have negligible R c AC-loss in a typical conductor

8. Model: Geometry The model consists of:  1D wires  Parallel wires  Straight  Equidistantly coupled This assumes that the cable geometry is irrelevant to model the thermal stability of a Rutherford cable and therefore a magnet.

9. Model: Thermal calculation 

10. Model: Electrical calculation 

11.  1 2 3 BzBz y

12. Model: Meshing 

13. Model: Solving algorithm The model solves: 1.Current 2.Temperature 3.Material properties Adaptive time stepping to reduce calculation time  Limited ΔI  Limited ΔT  Model runs until all elements are SC or a length longer than preset value is normal  The initial perturbation is varied to find the Minimum Quench Energy (MQE) I T Prop + Δt Δt/2

14. Model: Simulation Transient thermal simulation of a perturbation T (K)I (A)P (W) t = 0.1 ms t = 1.0 ms t = 2.0 ms t = 4.0 ms t = 5.0 ms t = 6.0 ms

15. A measurement over a large field range, 2 currents and 2 temperatures can be fitted with a single parameter set Validation

16. Extrapolation The measured conductor was used in the Small Model Coil 3 (SMC3)  Dipole  Double pancake  14 strands  1.25 mm  13T @ 14.3 kA

17. Extrapolation Measured Extrapolated MQE(B) curve plotted to a field map of the SMC3: Unmeasurable

18. Future work Full-scale magnet with the full-scale conductor: Assuming full-scale conductor has the same MQE(B) curve!

19. Conclusions  To accurately model MQE in High-J c Nb 3 Sn cables the following assumptions are appropriate:  Fully adiabatic  Cross contacts are negligible  Cable geometry is negligible  1D wire approximation is correct  Extrapolations for magnet cross section:  Total number of cables with weak spots (<10µJ) in cos(Θ) design much higher as in block design