Dresner, Stability of superconductors.2002

Figure 9.3. Sketches showing the relation of the Kapitza flux qK and the Joule power qJ in the case of nonrecovery (a) and in two limiting cases (b,c) (Dresner, 1987a). (Redrawn from an original appearing in Dresner (1987) with permission of the IEEE; ©IEEE 1987.)

helium-superconductor interface induced by the Kapitza resistance (cf. Eq. (4.11.1)) is large enough to keep the superconductor temperature above the current sharing threshold. When this happens, the curve of Kapitza flux (Eq.(4.11.1)) and the curve of Joule heat flux qJ produced by the superconductor (cf. Fig. 3.3) must intersect as shown in Fig. 9.3a. Just as in the discussion in Section 4.4 of Fig. 4.8, the intersection 1 represents an unstable steady state and the intersection 2 a stable steady state. If the metal temperature Tm is driven above T1 by the initial heat pulse, it settles down at T2, and recovery of the superconducting state is impossible.

Two limiting cases are possible in each of which the curve of Joule heat flux touches the Kapitza cooling curve in one place only. In the case of Fig. 9.3b,

For Joule heat fluxes < qJmax, the Kapitza resistance will not prevent recovery. The critical temperature Tc of NbTi at 8 T is 5.6 K. If we take THe = 1.8 K, and use the values of a and n suggested in Section 4.4 for rough estimates (a = 40 mW cm–2 K–n, n = 3) we find qJmax = 6.79 W/cm2.

In the case of Fig. 9.3c, the condition of tangency at T = To is

For the case of tangency (Fig. 9.3c) to apply, To must be less than the critical temperature Tc. If it is, then

The critical temperature of NbTi at 3 T is 7.8 K. If we imagine the superconductor to be carrying two-thirds of its critical current, Tcs = 3.8 K (cf. Eq. (4.6.3)). Finally, then, To = 5.7 K. Then qK(To) = 7.41 W/cm2 and from Eq. (9.5.4), qJmax = 15.6

Figure 9.5. A sketch of the stability marginE of the two-dimensional channel of Figure 9.4 as a function of qJ (Dresner, 1987). (Redrawn from an original appearing in Dresner (1987) with permission of the IEEE; ©IEEE 1987.)

When qs is large and d << w, the two-dimensional channel behaves like a one-dimensional channel of length d and Joule heat flux qs = (d/w)qJ. The stability margin of such a channel can be plotted in Fig. 9.5 spanning the asymptotes E = (d/L)Eo and E = Cqs–3 = C(w/d)3qJ–3. If these two curves are faired together, we get the right-hand heavy curve that represents how the stability margin varies with qJ when d << w.

If d >>w, thetwo-dimensional channelwill behavelikeaone-dimensionalchannel only for very large qs when E approaches the asymptote C qs–3 = C(w/d)3q –J3 , which now lies to the left of the asymptote CqJ–3 . We expect E to depart from the asymptote CqJ–3 when it is of the order of (d/L)Eo. The left-hand heavy curve represents how the stability margin varies with qJ when d >> w. Finally, we must add the Kapitza limits (9.5.1) or (9.5.4) to Fig. 9.5 if they apply.

Figure 9.6 shows the experimental points of Pfotenhauer and Sciver. In their paper (1986), they noted that the Joule power did not remain constant during the course of an experiment. They defined qJ in such a way that the experimental points lie slightly to the right of the theoretical curve defined by the limits in Eq. 9.3.8. In Fig. 9.6, I have normalized qJ so that the asymptote passes through the cluster of

Figure 9.6. The experimental values of the stability margin E reported by Pfotenhauer and van Sciver (1986). E0 = 1.43 J/cm2, w = 1.65 mm, L = 50 mm. d1 = 2.5 mm, o) d1 = 0.7 mm. (Redrawn from an original appearing in Dresner (1987) with permission of the IEEE; ©IEEE 1987.)

points near E/Eo = 0.1. Also shown in the figure are the asymptotes C(w/d)3qJ–3 for both sets of points and the corresponding values of (d/L)Eo, shown as horizontal line segments. Both sets of points behave as expected from the preceding argument.

Notes to Chapter 9

1I keep a narrow focus in this chapter by concentrating on superconductor stability. There are other technological problems that have been studied for ordinary (Fourier-law) fluids that must be reconsidered for He-II. One such problem area is the design of heat exchangers, including such details as heat transfer from tube banks to a He-II bath. Another is bubble growth in He-II and the related problem of cavitation in pumps. A third is the motion of He-II through transfer lines in the presence of a heat leak. A fourth is the motion of He-II through porous plugs, including the design of phase separators and thermomechanical pumps. The literature concerned with these technological applications of superfluid He is quite extensive, and a useful starting point for the interested reader is the proceedings of the biennial Space Cryogenics Conferences published as special issues of Cryogenics.

2Eq. (9.3.6) is based on the tacit assumption that the conductor quenches if the He-II in the channel is all converted to He-I. This assumption results in an underestimate of the stability margin since the He-I also has some cooling capacity.

Now, as noted in Section 4.3, Q = rcuJ2A/fP. Since k = fkcu and kcurcu = LoT, Eq. (10.1.1) can be written

where Tmax is the maximum temperature, which occurs at z = 0. From Eq. (10.1.3) and the definition of s it follows that

where r = rcu/f is the effective resistivity of the normal conductor. Integration from z = 0 to z = a yields

(10.1.5)

10.2. THE CRITICAL LENGTH

The functional dependence of Tmax on a is by no means transparent, and to clarify it we consider cases in which r is given by a power law in T:

To simplify the evaluation of the integral in Eq. (10.1.5) we introduce two new variables, y and q, defined by

The integral on the right-hand side can be evaluated easily for n = 0, 1, 2, 3, 4, and 5 (we go no higher, because n = 5 is the maximum exponent expected from the Bloch–Grüneisen law; cf. Fig. 2.6). The results are

other (the lower one), Tmax increases as a increases. It is shown in Section B.6 that the upper state is unstable against small perturbations and so it cannot occur in practice.

By way of example, let us evaluate the critical length for a ceramic-silver superconductor cooled by liquid nitrogen. In the temperature range 77–200 K, the resistivity of silver varies closely as the 3/2 power of the temperature, so we take n = 1.5. The limiting value of Jarb/L1/2o Tb is then 0.845. The resistivity rb of silver at Tb = 77 K is 2.9 x 10–9 ohm-m. If we take the current density in the silver in the normal state to be 5 x 107 A/m2, we find a = 7.0 cm, a comparatively short distance. In a recent EPRI review (1992), Moore has suggested that for high-field applications such as motors, generators, and energy storage, current densities an order of magnitude greater than that given above may be needed. In that case the limiting uncooled half-length would be only 7.0 mm. A cryogen vapor bubble trapped in a tight winding could easily create an uncooled segment this long. The saving grace in this case is the rather large formation energy of the limiting steady state, which has been calculated in Dresner (1994, “Stability”) to be 2.2 J/mm2.

10.3. VAPOR-COOLED LEADS

Soon after the discovery of the high-temperature superconductors, it was suggested that they could be used to make current leads whose heat leak into a helium-filled dewar is much less than that of conventional copper leads. To analyze the thermal behavior of such leads, let us consider the conceptual setup shown in Fig. 10.2. There, a current lead extends from a bath of liquid nitrogen at saturation temperature Tc to a bath of liquid helium at saturation temperature Tb. The current lead is shown penetrated by a central hole through which passes the helium vapor formed inside the dewar. Such a current lead is called a vapor-cooled lead because the upward vapor flow opposes any downward heat flow through the lead.

When the vapor-cooled lead is made of copper, as is the usual practice, the heat that enters the dewar through it and vaporizes liquid helium comes from two sources: (1) Joule power is created in the copper lead by the passage of the current, and (2) heat is conducted down the copper body of the lead from the hot end to the cold end.

If we make the cross-section of the lead smaller to reduce conduction, we increase its resistance and therefore we increase the Joule power. If we make the cross-section larger to decrease the resistance and the Joule power, we increase conduction. Clearly, there is an optimum at which the heat leak into the dewar is minimized. Calculations by Lock (1969) show this minimum to lie close to 1 W/kA, the exact value depending on the temperature Tc (which is usually 300K rather than 77 K) and the residual resistivity of the copper.

Figure 10.2. A schematic drawing of a vapor-cooled lead operating between a helium bath and a nitrogen bath.

If we make the lead out of a high-temperature superconductor and keep Tc = 77 K, the lead is superconducting over its entire length and thus produces no Joule power. If we make it long enough, we can make the conduction heat leak through it as low as we please and thus improve its performance beyond that of a copper lead.

As it happens, if the superconducting lead is too long, it may become unstable and assume a temperature distribution in which part of its length is normal. This partly normal state only occurs when the length L of the lead exceeds some critical length. Leads shorter than the critical length can only exist in the fully superconducting state and are said to be cryostable. The heat leak for the longest possible cryostable lead is smaller than the heat leak of the optimized copper lead by only a modest factor, roughly 4 in the illustrative calculations described below. In these calculations, when the lead is longer than the critical length, two partly normal states, PN1 and PN2, appear in addition to the fully superconducting state, FS (cf. Fig. 10.3). As the length L becomes greater, the peak temperature of PN1 decreases while that of PN2 increases. We surmise, therefore, that PN2 is stable and PN1 unstable. The state PN1 thus appears to act as a bifurcation state, and the difference between its formation energy and that of FS can be used as a measure of the stability of state FS. Noncryostable leads can be operated in practice just as noncryostable magnets can, so the large anticipated reduction in heat leak they promise should be realizable.