Analysis of Uncertainties in Protection Heater Delay Time Measurements and Simulations in Nb$_{3}$Sn High-Field Accelerator Magnets

The quench protection of superconducting high-field accelerator magnets is presently based on protection heaters, which are activated upon quench detection to accelerate the quench propagation within the winding. Estimations of the heater delay to initiate a normal zone in the coil are essential for the protection design. During the development of Nb$_{3}$Sn magnets for the LHC luminosity upgrade, protection heater delays have been measured in several experiments, and a new computational tool CoHDA (Code for Heater Delay Analysis) has been developed for heater design. Several computational quench analyses suggest that the efficiency of the present heater technology is on the borderline of protecting the magnets. Quantifying the inevitable uncertainties related to the measured and simulated delays is therefore of pivotal importance. In this paper, we analyze the uncertainties in the heater delay measurements and simulations using data from five impregnated high-field Nb$_{3}$Sn magnets with different heater geometries. The results suggest that a minimum variation of 3 ms or 20% should be accounted in the heater design for coil outer surfaces and at least 10 ms or 40% in the inner surfaces due to more uncertain heater contact. We also propose a simulation criterion that gives an upper bound enclosing 90% of the measured delays for heaters on the coil outer surface.