Analysis of the Stability Margin of the High Luminosity LHC Superconducting Cables with a Multi-Strand Model

At CERN (European Organization for Nuclear Research), between 1998 and 2008, the world’s largest and most powerful particle collider has been built. The LHC (Large Hadron Collider) is the biggest scientific instrument ever built to explore the new high-energy physic frontiers and it gathers a global user community of 7,000 scientists from all over 60 countries. The accelerated particles are made to collide together approaching the speed of light. This process allows to understand how the particles interact and provides insights into the fundamental laws of nature. After the latest amazing discoveries concerning the Higgs boson and the penta-quarks, another step forward is needed. To extend its discovery potential, the LHC will need a major upgrade around 2020 to increase its luminosity (rate of collisions) by a factor of 10 beyond the original design value (from 300 to 3000 $fb^{−1})$. As a highly complex and optimised machine, such an upgrade of the LHC must be carefully studied and requires about 10 years to implement. The scope of the Large Hadron Collider High Luminosity Project includes a new magnetic design and 16 superconducting inner triplet low-$\beta$ quadrupoles have to be replaced to reach a magnetic peak field of about 12 T. Such a high value of magnetic field requires the use of superconducting magnets wound with $Nb_3Sn$ Rutherford cables, instead of the actual ones made in $NbT i$. The quench level of these magnets (i.e. the maximum energy that a cable can tolerate without quenching) is a key value required to set magnet protection from beam losses, and is expected to be significantly different from the computed and measured levels of the LHC $NbT i$ magnets. In this work, we applied both zero and one-dimensional numerical model of multi-strand Rutherford cables of the low-$\beta$ quadrupole magnets, called MQXF [1], to simulate the electro-thermal instabilities of a beam-induced quench. The heat deposition on the superconducting cable due to the beam losses was obtained with computations performed with the FLUKA code [2]. For the material properties and superconducting model, the ITER $Nb$_{3}$Sn$ critical surface parameterization has been used [3]. iii In the zero-dimensional model, the whole cable is lumped into a single thermal component characterised by uniform temperature and homogenised thermal properties. For thes analyses the CryoSoft ZERODEE [4] code has been used. Increasing the level of complexity of the model, thermal, electric and hydraulic domains are taken into account. Neglecting the cable cross section in comparison with the longitudinal dimension, a one-dimensional model has been considered. The modelling and the simulations are carried out by means of the CryoSoft THEA [5][6] code, that allows to examine not only the thermal phenomena of heat exchange, but also the currents redistribution between different strands and the fluid-dynamic behaviour of the liquid Helium surrounding the cable. For the thermal and electric parameterization the data from [7] have been used, while for the heat exchange between the helium bath and the cable the empirical model presented in [8] is considered. For the THEA code several studies of convergence concerning integration time steps, mesh and tolerance have been carried out, aiming not to lose critical information during the simulations. Two kinds of investigation of the stability margin have been performed, one based on the analysis of the single strand, and the other accounting for all the 40 strands of the multi-strand Rutherford cable. The results of these two models are compared to analyse the effects of heat and current redistribution. The impact on quench energy of a resistive core embedded between the two layers of the Rutherford cables is also studied. The trends of the temperatures and the currents are analysed for each strand both in the space and in the time, in order to better understand the behaviour of the cable during the quench or the recovery phase. A comparison between the quench energy values obtained for the $Nb$_{3}$Sn$ conductor in the working conditions of the Hi-Lumi LHC inner triplet low-$\beta$ quadrupole (MQXF) and those of the $NbT i$ Rutherford cable of the LHC main quadrupole magnet (MQ) [9] is presented. The differences and similarities in quench performance between the impregnated cables for $Nb$_{3}$Sn$ magnets and the non-impregnated ones for NbTi magnets at their respective typical working conditions in superconducting accelerator magnets are highlighted. i