Analytical formulae for computing the critical current of an Nb₃Sn strand under bending

Abstract

Works on bending strain in Nb 3 Sn wires were initiated in support of the 'react-and-wind' technique used to manufacture superconducting coils. More recently, the bending strains of Nb 3 Sn strands in cable-in-conduit conductors (CICC) under high Lorentz forces have been thought to be partly responsible for the degradation of the conductor performance in terms of critical current and n index, particularly for the international thermonuclear experimental reactor (ITER) conductors. This has led to a new wave of experiments and modelling on this subject. The computation of the current transport capability in an Nb 3 Sn wire under uniform bending used to be carried out through the so-called Ekin's models, and more recently through numerical simulations with electric networks. The flaws of Ekin's models are that they consider only two extreme cases or limits, namely the so-called long twist pitch (LTP) or short twist pitch (STP) cases, and that these models only allow computation of a value for the critical current without reference to the n index of the superconducting filaments (i.e. this index is implicitly assumed to be infinite). Although the numerical models allow a fine description of the wire under operation and can take into account the filament's n index, they need a refined meshing to be accurate enough and their results may be sensitive to boundary conditions (i.e. current injection in the wire), also general intrinsic parameters cannot be easily identified. In this paper, we propose clearly to go further than Ekin's models by developing, from a homogeneous model and Maxwell's equations, an analytical model to establish the general equation governing the evolution of the electric field inside an Nb 3 Sn strand under uniform bending (with possible longitudinal strain). Within the usual strand fabrication limits, this equation allows the definition of one single parameter to discriminate the STP and LTP cases. It is also shown that whereas Ekin's LTP model corresponds well to a limiting solution of the problem when the transverse resistivity tends toward zero (or the twist pitch tends towards infinity), Ekin's STP model must be modified (improved) when the filament's n index is finite. Since the general equation cannot be solved analytically, we start from the LTP model and develop a first order correction to be applied when the transverse resistivity, the twist pitch and the filament's n index are finite. Using a simple but realistic law for depicting the strain dependence of the critical current density in the Nb 3 Sn filaments, we can fully compute the corrected expression and give the result under a general analytical formula for a strand submitted to both bending and compressive/tensile strains. The results are then compared in two different cases with those obtained with the numerical code CARMEN (based on an electrical network) developed at CEA. Last, a semi-empirical formula has been developed to evolve continuously from the LTP limit to the improved STP limit when the transverse resistivity evolves ...

Electromagnetic behaviors of superconducting Nb3Sn wire under a time-dependent current injection

“…2 for an industrial Nb 3 Sn strand as foreseen in the ITER magnets. The fitting formula can be written as: 26 …”

“…Currently the main approaches to study the strain dependence are experimental methods [21][22][23] and theoretical analyses. [24][25][26][27][28][29][30] The analytical models have simple theoretical basis, wide applicability, and accurate prediction of performance degradation for some types of strands under simple loads. However, they cannot include the effect of varying n value on transport current and AC loss.…”

A 3D Model on the Electromechanical Behavior of a Multifilament Twisted Nb3Sn Superconducting Strand

“…Currently, the main approaches to study this problem are experimental [7,8] and theoretical [9][10][11][12][13][14] analyses. The experiments focus on the degradation of strand critical current under various loading conditions: uniaxial tension, periodic bending, and transverse contact.…”

Intra wire resistance and strain affecting the transport properties of Nb3Sn strands in cable-in-conduit conductors

“…The transition between the two limits depends on the ratio of filament resistance and matrix resistance [194], [197].…”

“…In the HRL the strand's internal resistance inhibits current sharing among differently strained filaments, while in LRL all filaments are involved in current sharing. The strand's behavior in-between the two limits may be calculated numerically [191] or empirically [189], [197].…”