Study of the thermal equilibriums of a one-dimensional superconductor wire

The Maddock criterion of equal areas [B. J. Maddock, G. B. James, W. T. Norris, Superconductive composites, heat transfer and steady state stabilization, Cryogenics 9 (1969) 261] gives a necessary and sufficient condition in order that a one-dimensional superconductor of infinite length, submitted to fixed current density and magnetic field, could not evolve towards a resistive state after an accidental local overheating. In this paper, we study the temperature distribution in a portion of a one-dimensional superconductor located between two regions for which the temperature can be perfectly controlled. In consequence, the considered length L of the conductor is supposed finite. In this context we propose, as an extension of Bonzi [B. Bonzi, Etudes des equilibres thermiques d'un supraconducteur existence et stabilite, these de doctorat, INPL, France, 1991; B. Bonzi, H. Lanchon-Ducauquis, Equilibres et stabilite: thermiques d'un supraconducteur, C. R. Acad. Sci. Paris, t317, II, 1993, pp. 899-903.] and El Khomssi [M. Fl Khomssi, Etude des equations paraboliques et elliptiques gerant l'etat thermiques d'un supraconducteur, These de doctorat, INPL, France, 1994; B. Bonzi, M. Fl Khomssi, Practical criteria for the thermal stability of a unidimensional superconductor, J. Phys. III, France 4 (1994) 653.] works: 1. A necessary and sufficient criterion called optimal criterion which allows us to enlarge the superconductor thermal stability conditions; this criterion depends on the heat source term (competition between the heat power caused by the Joule effect and the one absorbed by the cryogenic bath), the length of the superconductor and its thermal conductivity. This criterion makes sure the global thermal stability of the conductor without any condition on the thermal disturbance. 2. A characterisation of the thermal stationary states which can exist when the optimal criterion is not realized. 3. Some numerical realistic applications from which we deduce several important parameters for the superconducting state stability. (C) 1999 Elsevier Science Ltd. All rights reserved.