Dresner, Stability of superconductors.2002

Figure 4.7. The normal zone propagation velocity v plotted versus current I.

Of the two possible intersections, only intersection 2 is stable against small perturbations. For if the steady state 2 is perturbed by a slight increase in temperature, the cooling curve exceeds the heating curve, and the temperature is restored to T2. Similarly, if steady state 2 is perturbed by a slight decrease in temperature, heating exceeds cooling, and the temperature is again restored to T2. At T1 the situation is just the reverse. A slight positive perturbation will cause the temperature to rise until it reaches T2; a slight negative perturbation will cause the temperature to fall to Tb. So if the initial normal zone does not disappear immediately, its central temperature quickly approaches T2.

The left-hand edge of the normal zone has the form of a traveling wave

Figure 4.8. The case in which the steady-state boiling heat flux intersects the three-part curve of Joule power. The equality of the two stippled areas determines the minimum propagating current.

where x is an abbreviation for z + vt.

The minimum propagating current Im corresponds to v = 0. When v = 0, we

Now we multiply both sides of Eq. (4.4.3) by kdT = k(dT/dx)dx and integrate from x = –∞ to x = ∞:

(4.4.4)

The first term in Eq. (4.4.4) vanishes because far to the left and far to the right of the edge of the normal zone the temperature profile becomes flat, i.e., at x = ± ∞, dT/dx = 0. So the minimum propagating current is determined by the integral condition

(4.4.5)

It has become customary to ignore the temperature variation of the thermal conductivity k and to replace it with a suitable constant value. In that case, the

meaning of Eq. (4.4.5) is that the two stippled areas in Fig. 4.8 must be equal. In other words, the minimum propagating current is chosen to raise the curve Q until the two areas are equal. This is the famous equal-areas theorem of Maddock, James, and Norris.

A rigorous approach requires that the temperature dependence of k be taken into account. As Maddock, James, and Norris point out, at low temperatures the Wiedemann-Franz law tells us that k varies directly as T. In that case, the equal-ar- eas theorem holds when Q and q are plotted against T 2 instead of T.

It is important to note that in applying the equal-areas theorem the transition boiling part of the boiling curve is taken into account. This is because the temperature of the conductor vanes continuously through the normal zone so that the heat transfer is under temperature control.

4.5. lMPROVlNG BOILING HEAT TRANSFER

A typical value for Q for a fully normal cryostable conductor of the Stekly type is 0.15 W/cm2. If cold-end recovery is allowed, this flux (abbreviation: Qn) can perhaps be doubled to 0.30 W/cm2, corresponding to an increase in current density of about 40%. These values refer to conductors with relatively uncomplicated bare copper surfaces. Many workers have tried to improve boiling heat transfer (and thus current density) by roughening the conductor surface, coating it, chemically treating it, or some combination thereof. The literature contains many reports of such attempts. We mention only two in order to give the reader an idea of what can be done.

Butler et al. (1970) coated heat transfer surfaces with thin layers of materials of low thermal conductivity and found that while the burnout point (point P in Fig. 4.1) moved to higher temperatures and lower fluxes, the recovery point R moved upwardsmarkedly. In oneexample workedby Wilson (1983, p. 104), a 7- m coating ofcellulose paint raised the equal-area value of Qn from 0.31 W/cm2 to 0.48 W/cm2.

Nishi et al. (1981) and Ogata and Nakayama (1982) compared heat transfer from a chemically oxidized, roughened surface called Thermo-Excel-C with that from a smooth copper surface. The Thermo-Excel-C surface, produced by making two families of parallel cuts in different directions, has a rasplike appearance. After it has been machined, it is chemically oxidized with an unspecified alkali. Figure 4.9, redrawn from the paper of Nishi et al., shows a marked improvement in heat transfer caused by the mechanical treatment of the surface and a further marked improvement caused by the subsequent chemical treatment. All in all, qr, the

recovery heat flux, is raised roughly fourfold to a value slightly greater than 0.80 W/cm2.

Other factors affecting boiling heat transfer are the shape of the conductor surface, the orientation of the conductor with respect to the direction of gravity, the thickness and orientation of the channels between adjacent conductors, the accu-

Figure 4.9. A diagram showing a marked improvement in heat transfer caused by scoring the surface and and an additional improvement caused by subsequent chemical treatment of the scored surface (Nishi et al., 1981). (Redrawn from an original appearing in Nishi et al. (1981) with permission of the IEEE; ©IEEE 1981.)

mulation of vapor in the channels, and vapor-induced convection of liquid helium. As with attempts to improve heat transfer by surface treatment, the papers reporting on these other factors are legion, and I shall cite only a couple in the discussion below.

Walstrom (1982) studied heat transfer from a conductor with a complex shape, the conductor of the GE/ORNL coil of the IEA Large Coil Task (Beard et al., 1988), shown in Fig. 4.10. The figure also shows some of his results. Two things are noteworthy: first, there is no dip near the recovery point R, and, second, the heat transfer depends strongly on the orientation of the conductor with respect to the direction of gravity. The absence of the dip at R appears to be caused by the surface’s being made up of elements pointing in many directions.

It has long been known that decreasing the thickness of the helium channels degrades boiling heat transfer (Wilson, 1983, pp. 105–6). Nishi et al. (1983) recently found such degradation at channel thicknesses of 1 mm or less. They also studied vapor-induced convection and vapor accumulation near the tops of long, vertical channels, and found them to play countervailing roles, with either able to dominate the other, depending on channel length and thickness and the rate of vapor production. As Christensen and Peck (1982) have pointed out, the deleterious effects of vapor accumulation can be avoided by inclining the channels so as to lead vapor diagonally away from vertical conductors that may have gone normal.

Figure 4.10. Heat transfer from the conductor of the GE/ORNL coil of the IEA Large Coil Task (Walstrom, 1982). The angle is the angle from the horizontal. (Redrawn from an original appearing in Walstrom (1982) with permission of Plenum Publishing Corp., New York.)

Much ingenuity has been exerted in improving boiling heat transfer, and while such improvement has enhanced the performance of cryostable magnets, it has not changed our basic view of cryostability. Accordingly, I close this very brief discussion of boiling heat transfer and continue with the discussion of the problems of stability.

4.6. MINIMUM PROPAGATING ZONES (I)

For large, expensive magnets like those of the international Large Coil Task, the possibility of failure is often deemed unacceptable, and the magnets are designed to be cryostable. Even when magnets are Maddock-stable, their overall current density is still low, but the attendant penalty of large size and high cost is borne in return for the certainty of stable operation.

When the magnets are smaller and the investment lower, designers are often willing to increase the current density beyond the Maddock limit in the hope that if the perturbations are small enough, the magnet will not quench in operation. How this is possible can be seen in principle by recalling our earlier discussion of Fig. 4.8.

Suppose now that the right-hand stippled area is larger than the left-hand stippled area because we have raised Qn. Then we expect from Fig. 4.7 that an initial normal zone whose uniform central temperature approaches T2 will propagate, i.e., grow larger, leading to a quench. If the initial normal zone is very long (so that conditions at the center are not affected by heat conduction at the ends of the zone) and if the initial temperature is > T1, then the central temperature will approach T2, and such a zone will propagate. On the other hand, if the initial temperature is < T1, the central temperature will fall to Tb and the normal zone will disappear (recovery). Thus for long normal zones, the initial temperature T1 marks the bifurcation between quench and recovery.

Long normal zones do not really fit the facts because thermal perturbations are caused most often by local slipping of the conductor under the action of the Lorentz force. What we should like to do is to characterize in some simple way those local perturbations that lead to quenches and those that lead to recovery. This ambition unfortunately exceeds our capabilities, for what it means is that we must determine which initial conditions T(z,0) asymptotically approach propagating solutions of the partial differential Eq. (4.4.1) and which asymptotically approach Tb . This involves solving the partial differential equation is some general way for an arbitrary initial condition T(z,0), and I do not know how to do this.

A less ambitious but more fruitful approach has been suggested by Martinelli and Wipf (1972) and later used by Wilson and Iwasa (1978). These authors surmised the following: In addition to the uniform steady states T = Tb, T = T1, and T = T2, Eq. (4.4.1) has a nonuniform, localized steady state T(z), i.e., a steady state obeying

minimum propagating zone (abbreviation: MPZ), is unstable against small perturbations just like the uniform solution T = T1. Initial conditions T(z,0) that are > TMPZ(z) quench, while initial conditions T(z,0) < TMPZ (z) recover. Of particular interes t t o magne t designer s i s th e formatio n energ y o f th e MPZ ,

EMPZ =A dz S(T´) dT´, which is usually taken as an estimate of the minimum quench energy, i.e., the heat instantaneously deposited at a point that just

causes a quench.

Proving that (1) the MPZ is unstable and (2) that it separates quenching initial conditions from those that recover is not easy. The mathematical details, however, are instructive and have been included as Appendix B.

If we introduce s = k(dT/dz) as a new dependent variable and T as a new independent variable, the time-independent form of Eq. (4.4.1), i.e., with ∂T/∂t = 0, becomes the first-order ordinary differential equation

Figure 4.14. A plot of the dimensionless formationenergy ε of the MPZ plotted against i = I/Ic with the Stekly number α as parameter.

These partial solutions vanish at τ = 0 and τ = τmax and join continuously at τ = 1 – i, as they should.

Astraightforward integration (Elrod et al., 1981) shows that

when αi3 > 1. The quantity ε is a measure of the formation energy of the MPZ. Shown in Fig. 4.14 is e plotted against i with the Stekly a number as a parameter. Also shown is the location of the minimum propagating value of i, iMP, for each a, calculated from the equal-area requirement αi2 = 2 – i (see Fig. 4.15). At i= iMP, ε has a vertical asymptote.

We see from Fig. 4.14 that as i barely exceeds iMP, ε falls precipitously. Thus little is to be gained fromexceeding iMP slightly. If the decision is made to operate the conductor beyond iMP, one should use values well beyond iMP. Otherwise, one might as well use values slightly less than iMP and reap the benefits of cryostability.

A special design problem that sometimes arises in practice is to choose the copper-to-superconductor ratio once the size andshape of the conductorhave been fixed (Elrod et al., 1981). With all else fixed except the volume fraction f of copper, the Stekly number a varies as (1– f )2/f and i varies as 1/(1 –f ).