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. Nor ris, Superconductive composites, heat transfer and steady state stabilizati on, Cryogenics 9 (1969) 261] gives a necessary and sufficient condition in order that a one-dimensional superconductor of infinite length, submitted t o fixed current density and magnetic field, could not evolve towards a resi stive 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 temper ature can be perfectly controlled. In consequence, the considered length L of the conductor is supposed finite. In this context we propose, as an exte nsion of Bonzi [B. Bonzi, Etudes des equilibres thermiques d'un supraconduc teur existence et stabilite, these de doctorat, INPL, France, 1991; B. Bonz i, H. Lanchon-Ducauquis, Equilibres et stabilite: thermiques d'un supracond ucteur, C. R. Acad. Sci. Paris, t317, II, 1993, pp. 899-903.] and El Khomss i [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 allo ws us to enlarge the superconductor thermal stability conditions; this crit erion depends on the heat source term (competition between the heat power c aused by the Joule effect and the one absorbed by the cryogenic bath), the length of the superconductor and its thermal conductivity. This criterion m akes sure the global thermal stability of the conductor without any conditi on 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 impor tant parameters for the superconducting state stability. (C) 1999 Elsevier Science Ltd. All rights reserved.