Energy Density Method: An Approach for a Quick Estimation of Quench Temperatures in High-Field Accelerator Magnets

Accelerator magnets for future particle accelerators are designed to work with as high energy densities as possible to achieve high fields and compact magnet designs. A key factor limiting the energy density is given by the protection in case of a quench: If a quench occurs, the stored energy must be first absorbed by the windings, and the magnet temperature shall not exceed a given limit. In this paper, we present a back-of-the-envelope method for estimating the magnet's maximum temperature after a quench based on its stored energy. The method combines the existing concepts of MIIT (Mega*current*current*time), time margin, and protection delay to allow for an easy and direct calculation of the hot-spot temperature. We apply the proposed method to several Nb$_3$Sn dipole and quadrupole magnets developed for the high luminosity Large Hadron Collider and the Future Circular Collider for hadron-hadron collisions and compare the results to a more detailed simulation. The proposed energy density method is a useful tool for fast feedback in the early magnet design phase to ensure that the magnet is not impossible to protect.