Non similarity solution approximation to the thermal hydraulic quenchback in superconductors

When a normal zone appears in a superconducting (s.c.) cable, it propagates axially with a so-called quench velocity, Vq, which depends mainly of the conductor characteristics and very little on the cooling effect of Helium. However, if the ratio of the longitudinal to transversal dimensions of the conductor is quite large, the expansion of the heated Helium in the normal zone may drive fluid elements far from the initial normal zone, which could induce other normal zones in the conductor because of the compression of these fluid elements and their friction with the strands of the cable. The quench velocity would have much higher value than the pure Fourier conduction mechanism; this is called thermohydraulic quench back (THQB) mechanism. After the numerical discovery of the THQB mechanism, an analytical approximation to this phenomenon appeared using the similarity method for the differential equations. Although this solution suggests high quench velocities for the THQB mechanism, it cannot be accepted since the solution predicts much faster quench velocities for short s.c. cables than for long ones. In addition, an experiment seems to contradict its predictions. In this paper, a direct method of integration is used in the differential equation (derived from approximations) in order look for a different solution