Dresner, Stability of superconductors.2002

The curve C3 in Fig. 4.16 is the locus (4.7.6) for αc = 5 and fc = 0.8 (Cu/SC = 4). The arrow on curve C3 points in the direction of decreasing f, i,e., decreasing volume fraction of copper. As we proceed in the direction of the arrow from i = 1, there is at first a rapid increase in e. But eventually the locus (4.7.6) becomes nearly parallel to the contours of e, and further decrease in f brings little gain in stability. In the example under discussion, e = 0.75 at i = 0.6 (Cu/SC = 2) whereas at i = 0.95 (Cu/SC = 3.75), e = 0.10. Thus there is a clear preference for the lower copper-to- superconductor ratio. Trade-off studies similar to this one have also been done by Wipf (1978).

4.8.THE FORMATION ENERGY OF THE MINIMUM PROPAGATING ZONE

Next we estimate the order of magnitude of the MPZ formation energy. To do this we must return from special units to ordinary units. According to Eq. (4.6.7), in special units e = E/2A. Now E/2A has the units PTL-2 and e is dimensionless. Hence in ordinary units we must have

Typical parameter values for a NbTi wire in an 8-T field might be S = 3000 J m-3

K-1, Tc – T b = 1.4 K, D (wire diameter) = 0.8 mm, h = 1000 W m-2 K-1, and k = 200 W m-1 K-1(ρ cu= 5 x 10-10 ohm-m). Then E/ε = 2.67 x 10-5 J. Hence we expect the MPZ energies to lie in the range of a few to a few tens of µ J.

Such energies are very small. The traditional illustration is a 1-gram weight falling through a distance of 1 mm; it gains 9.8 µ J. But such an illustration tells us very little about how to design a superconducting magnet. Instead, we must study the energy released when the conductor slips under the action of the Lorentz force.

Let us consider a magnet with a ventilated winding, i.e., a winding in which spaces are left between conductors to allow the infiltration of liquid helium. Such magnets are typically wound with spiral-wrapped insulation separating adjacent conductors, and the conductors are partly held in place by the frictional force between them and the spacers. If a conductor slips at the point of contact with a spacer, the Lorentz force does work, which eventually is converted into heat. To understand the stability of the magnet, we must compare this work with the MPZ energy calculated above.

A conceptual model that will serve as the basis for our calculations is sketched in Fig. 4.17. The conductor is taken to be square with side a. It is shown supported by several spacers. The arrows show the direction of the Lorentz force per unit length w = JBa2, assumed to be parallel to the surface of contact between the

Figure 4.17. A conceptual modelfor calculating the heat released when the conductor slips at its point of contact with a spacer.

conductor and the spacers (worst case). If the conductor slips suddenly at the center spacer, it suffers an average deflection (pinned boundary condition at adjacent spacers; again a worst case)

where b is the unsupported length of the conductor and YM is its flexural rigidity. Here Y is the Young’s modulus of the conductor (139 GPa for copper at cryogenic temperatures) and M is the geometric moment of inertia of the conductor around the neutral axis, namely, a4/12. The energy E´ released when the conductor slips is wbδ so that

Let us continue our example by taking (Jc)NbTi = 120 kA/cm2 and Cu/SC = 1.5 ( f = 0.6). Then the Stekly number a = 27.4. If we choose i = 0.6, then e = 0.623 and E

= 16.6 µ J. Now J = Jc(1- f )i = 28.8 kA/cm2. Then it follows from Eq. (4.8.3) that b = 5.34 mm. So in this example we must support the conductor at intervals less than about 2.5 mm to insure stability against wire slippage. While these numbers vary somewhat from example to example, their impact is clear: designs must avoid long unsupported spans of conductor.

4.9. THE MAXIMUM ALLOWABLE RESISTIVE FAULT

A design problem that can be studied by a slight extension of the foregoing theory is to determine the maximum allowable resistive fault that can exist in a conductor without quenching it. If we model the fault as a steady power source w (units:W) at z = 0, then we must add a term (w/A )δ(z) to the right-hand side of Eq. (4.4.1); here δ(z) is the Dirac delta function. Now we look for steady, local solutions of Eq. (4.4.1), i.e., time-independent solutions T (z ) forwhich T( + 8)= Tb. We expect two such solutions: one is the analog of the uniform state T = Tb that would exist in the absence of the fault, and the second is the MPZ. The first is stable against small perturbations; the second is not.

Now we continue the numerical example begun at the start of Section 4.8. The second factor in Eq. (4.9.8) is then 44.5 mW. If we take J = 28.8 kA/cm2, as was done following Eq. (4.8.3), then I = 145 A. With a = 27.4 and i = 0.6, the first factor in Eq. (4.9.8) is 0.413 so that the maximum fault resistance is 0.875 mW.

This calculation can thus serve to determine the maximum allowable joint resistance. Since good joints typically have resistances of the order of nW, we do not expect them to have any appreciable effect on stability. This last conclusion can be quantified by comparing the difference De in the formation energies of the upper and lower states with the formation energy of the MPZ in the absence of a resistive fault. This calculation has been carried out by the author (Dresner, 1982). The exact result for De as a function of a, i, and w is rather complicated, but the upshot of the calculations is that De varies almost linearly with w for fixed a and i. Thus we may use the rule of thumb

Clearly, then, in the numerical example given after Eq. (4.9.8), a nW joint will have virtually no effect on the quench energy.

4.10. STABILITY OF PARTLY COVERED CONDUCTORS

Another application of the formation energy of the MPZ as a measure of stability is to the stability of partly covered conductors. Neighboring conductors in a pool-cooled magnet are separated by insulation that partly covers their surfaces and prevents full contact with the liquid helium. In studying the stability of such conductors, the custom was to take the partial occlusion of the surface into account simply by reducing the cooled perimeter. But Meuris and Mailfert (1981) eschewed this simple approximation and studied the effect on stability of point-to-point variation in cooling. Their elegant study gave most surprising results, described below.

Working in the idealized case of constant properties, Meuris and Mailfert considered a long conductor having a single uncooled region of half-length L To

simplify their calculations, they ignored current sharing and took for the function g(t)

according to the values of αi2 and the dimensionless half-length of the uncooled region xs = (Ph/kA)1/2 Ι. Figure 4.21, also redrawn from the paper of Meuris and Mailfert (1981), shows the (αi2, xs)-plane divided into regions according to the steady solutions that occur there.

When the choice (4.10.1) is made for g(t),the equal-area criterion of Maddock, James, and Norris (1969) is simply αi2 = 2. When the conductor is metastable, i.e., when α i2 > 2, the (α i2,xs)-plane is divided into two regions, C and D, distinguished by whether or not xs > 1/αi2. In each of these two regions, one nonzero steady state occurs; these steady states are unstable. In each region, two outcomes are possible following a perturbation: quench or recovery. As usual, Meuris and Mailfert propose the formation energy of the unstable steady state as a measure of conductor stability.

When the conductor is stable (αi 2 < 2), the situation is slightly more complex. In the cross-hatched region A, the conductor always recovers the superconducting state. In the white regions B, E, and F, two steady states occur, one stable, the other unstable. The unstable state is always smaller than the stable state (see Fig. 4.20). Again two outcomes are possible following a perturbation: recovery of the superconducting state or transition to a steady normal zone centered on the uncooled region and described by the stable steady state. Meuris and Mailfert again propose the formation energy of the intermediate unstable steady state as a quantitative criterion of stability.

Figure 4.22 shows the dimensionless formation energy of the unstable state as a function of the dimensionless uncooled length xs for five values of αi2. (N.B.: Meuris and Mailfert’s dimensionless formation energy ec is twice that defined in Eq. (4.8.1).)

The stable normal zones predicted by Meuris and Mailfert have been observed (Claudet et al., 1979). Their existence represents a kind of loss of stability which, though far less severe than a quench, is nonetheless discomfiting.

4.11. TRANSIENT HEAT TRANSFER

I was careful everywhere in the foregoing sections to refer to the formation energy of intermediate unstable steady states only as a figure of merit or a measure or a quantitative criterion of conductor stability. I think it cannot be emphasized too strongly that the MPZ formation energy represents an artificial standard of stability useful only because it allows us to make quantitative decisions aboutconductordesign. Always lurking in the background is the unproven assumption that the conductor with the larger MPZ formation energy has the larger minimum quench energy, too.

By its very definition as a steady state, the MPZ must be calculated using a steady-state heat transfer coefficient. But in reality, the heat transfer coefficient greatly exceeds the steady-state heat transfer coefficient for a brief interval immediately following a pulsed heat addition to the conductor. As we shall see below,

Figure 4.22. The dimensionless formation energy of the unstable state as a function of the dimensionless uncooled length xs for various αi2 (Meuris and Mailfert, 1981). (Redrawn from an original appearing in Meuris and Mailfert (1981) with permission of the IEEE; © IEEE 1981.)

this high but transient heat transfer allows a certain limited deposition of heat in the conductor without creating a normal zone at all.

The phenomenology of transient heat transfer in saturated helium has been elucidated by Tsukamoto and Kobayashi (1975), Schmidt (1978; 1981) and Steward (1978). Although they used different techniques of experimentation, in essence they did the same thing, namely, suddenly energize a heater of low thermal inertia in contact with a saturated helium bath and measure its temperature as a function of time. What they observed is summarized in Fig. 4.23, redrawn from Steward’s paper.

Right atthestart, the heatertemperaturerises suddenly by afew tenths of aKelvin. This modest, early temperature risecorresponds to heat transferlimited by the so-called Kapitza resistance. Kapitza observed that there is a temperature discontinuity at the interface between a solid surface and liquid helium related by the following equation to the heat flux q being transferred from the solid to the helium:

where Ts is the solid temperature, THe is the liquid helium temperature at the interface, and a and n are constants characteristic of the surface. Eq. (4.11.1) implies