Quench Level of the HL-LHC $Nb_3Sn$ IR Quadrupoles

The scope of the Large Hadron Collider Hi-Lumi Project at CERN includes the installation of several superconducting magnets wound with $Nb_3Sn$ Rutherford cables. 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 NbTi magnets. In this work, we applied a one-dimensional numerical model of multi-strand Rutherford cables to simulate the electro-thermal instabilities caused by the heat released by the particle beam losses. Two models have been applied, one based on the analysis of the single strand, and the other accounting for all the strands in the multi-strand cable. The results of these two models are compared to analyze the effects of heat and current redistribution during quench. A comparison between the quench energy values obtained for the $Nb_3Sn$ conductor in the working conditions of the LHC Hi-Lumi inner triplet low-β quadrupole (MQXF) and those of the NbTi Rutherford cable of the LHC main quadrupole magnet (MQ) is presented