Dresner, Stability of superconductors.2002

Figure 7.6. The profiles of velocity and pressure in a uniformly heated zone of length b with open ends at time intervals of b/2 over a full cycle of duration 2b. (Redrawn from an original appearing in Dresner (1979, “Heating-induced flow”) with permission of Butterworth-Heinemann, Oxford, England.)

Using the figures given at the beginning of this section for the experiment of

Lue et al., we find vmax = 12.7 m/s and pmax = 0.2 14 MPa. This calculated value of Pmax compares favorably with the maximum pressure rise of 0.15 MPa measured

by Lue et al. However, the measured maximum occurred roughly 20 ms after the heater was energized rather than at the expected b/2c = 6.8 ms. In Dresner, 1979, “Heating-induced flow,” these discrepancies were attributed to the presence of a l-m-long, 1.9-mm-ID tee connecting the cold pressure transducer to the stainless steel tube that encases the conductor. The open area of the tee connector (2.8 mm2) is slightly larger than the open area of the conductor (2.0 mm2), and the volume of the connector is 43% of the void volume of the conductor. When the heater is energized, the helium at the center of the conductor expands into the unheated tee, reducing the maximum pressure rise below what is predicted by Eq. (7.4.7). Furthermore, the diversion of helium into the tee may delay the attainment of the maximum pressure rise. These effects vitiate a detailed comparison of the theory with the experiment of Lue et al., and only the order of magnitude agreement is noteworthy.

7.5. MULTIPLE STABILITY

The discovery of recovery in stagnant helium was a pleasant surprise because it obviated the need for strenuous pumping. The coupling between heat transfer to helium and its state of flow that is the cause of recovery in stagnant helium has another effect that was, at the time of its discovery, even more surprising.

As mentioned in Section 7.1, the item of central interest is how great a perturbation it takes to produce a quench. Using the experimental setup described in the first paragraph of Section 7.4, Lue et al. (1980) observed whether the Cu/NbTi triplet quenched or recovered in response to various heat pulses. They carried out their study by fixing the transport current and the background magnetic field and then, in successive shots, gradually raising the pulse heat. What they expected to observe was recovery at small pulse heats and a transition to quenching at larger pulse heats. The largest pulse heat density that still allowed recovery would then be taken as the stability margin.

In some of their experiments, they did see the expected sequence recovery, quench. But in other experiments, they saw the double sequence recovery, quench, recovery, quench, i.e., they saw recovery following small heat pulses, quench following larger heat pulses, recovery following yet larger heat pulses, and finally quench following the largest heat pulses. Typical experimental results are shown in Figs. 7.7 and 7.8. In both of these figures, the cross-hatched area corresponds to quench and the clear area corresponds to recovery.

In the course of their experiments, Lue et al. explored the dependence of this multivalued stability margin on transport current and externally imposed flow. The qualitative results of their experiments are summarized in Fig. 7.9, which is a schematic representation of the stability margin DH as a function of transport current I and imposed flow velocity v. Points below the surface correspond to recovery; points above the surface correspond to quench. The surface is folded so that certain slices parallel to the (I,∆H)-plane result in a 2-shaped curve like that of Fig. 7.7, while other slices parallel to the (v,∆H)-plane lead to the configuration of Fig. 7.8.

In my 1980 review of this subject (“Stability”), I wrote, “This strange folded surface cries for an explanation . . .” Now to formulate a complete explanation of stability in cable-in-conduit conductors, one must take into account the mutual coupling of the flow-dependent heat transfer and the heating-induced flow. This difficult problem can only be dealt with numerically and has, indeed, attracted the attention of many numerical analysts. In a qualitative analysis designed to avoid this difficulty, Lue et al. (1980) artificially broke the coupling between heat transfer and induced flow by considering two half-problems in which they (1) studied the stability margin imagining the heat transfer coefficient to be externally imposed, and (2) studied the induced flow and associated heat transfer coefficient assuming the heating rate of the helium to be externally imposed. In other words, in problem

Figure 7.7. Typical measurements (Lue et al., 1980) of the stability margin versus transport current. The cross-hatched area corresponds to quench and the clear area to recovery. (Redrawn from an original appearing in Dresner (1984, “Superconductor”) with permission of Butterworth-Heinemann, Oxford, England.)

(1) they studied DH(h) and in problem (2) they studied h(DH), assuming a fixed initial heat pulse duration.

The qualitative nature of the results in problem (1) are sketched in Fig. 7.10. When h is very large, recovery is very rapid, and the total Joule heat produced during recovery is small compared to the initial heat pulse (consider the limiting case when h = 8). Then

(7.5.1)

where AHe is the cross-sectional area of the helium in the cable, Aco is the cross-sectional area of the conductor (metal), S is the volumetric heat capacity of the helium, Tb is the ambient temperature, and Tcs is the current-sharing threshold temperature. The integral in Eq. (7.5.1) is the volumetric enthalpy difference of the

Figure 7.8. Typical measurements (Lue et al., 1980) of the stability margin versus externally imposed mass flow. The cross-hatched area corresponds to quench and the clear area to recovery. (Redrawn from an original appearing in Dresner (1984, “Superconductor”)with permission of Butterworth-Heinemann, Oxford, England.)

helium between Tb and Tcs . Only the enthalpy of the helium is considered in this expression because at low temperatures it greatly exceeds that of the metal.

It only takes a heat density equal to the volumetric enthalpy difference of the metal between Tb and Tcs to warm the metal to the point where the superconductor

Figure 7.9. A schematic representationof the stabilitymargin DH as a function of transport current I and imposed flow velocity v. Points below the surface correspond to recovery, points above the surface to quench. (Redrawn from an original appearing in Dresner (1984. “Superconductor”) with permission of ButterworthHeinemann, Oxford, England.)

Figure 7. 10. A schematic representation of the stability margin as a function of an externally imposed heat transfer coefficient. (Redrawn from an original appearing in Lue et al. (1980) with permission of the Journal of Applied Physics of the American Institute of Physics.)

is fully normal. As just mentioned, this heat density is very small compared to the asymptote (7.5.1). Thus, when a point having this heat density for its ordinate is plotted in Fig. 7.10 it appears to be on the h-axis. If the conductor is to recover, the heat transfer coefficient h must at least be large enough to remove the Joule power in the fully normal state, i.e., it must fulfill the equation

where I is the transport current, rcu is the resistivity of copper, Acu is the cross-sec- tional area of the copper, and P is the wetted perimeterof the cable. Thus the foot of the DH(h)-curve in Fig. 7.10 lies at the point (h,0), where h is given by Eq. (7.5.2).

In problem (2), the pulse duration is held fixed so that q ~ DH. Now from the results of Section 7.3, the induced flow velocity v is proportional to q, and the heat transfer coefficient associated with it is proportional to v 0.8 ~ DH0.8 (curve OAB in Fig. 7.11). Now when DH is very small, both q and the induced flow velocity v are very small. But when q is small, the interval of transient heat transfer is long' and transient heat transfer may endure for the entire heat pulse. Then h will be large (segment CD in Fig. 7.11). As DH (and q) increase, the time to takeoff decreases and so does the effective heat transfer coefficient (arc DA in Fig. 7.11). Eventually, for large DH, the curve of h(DH) joins the curve OAB. The effective heat transfer coefficient h as a function of the pulse heat DH for a fixed pulse duration then looks like curve CDAB in Fig. 7.11.

146 CHAPTER 7

induces lies on the curve h(DH) at point M. The heat transfer coefficient that the pulse requires for recovery lies on the curve DH(h) at point N. Since hM > hN, the conductor recovers. So the segment OP of the DH-axis corresponds to recovery. By identical reasoning, so does the segment RQ. The segment QP, on the other hand, corresponds to quenching, for along fine line 2, the induced hK < the required hL. Finally, points lying above R on the ∆Η-axis correspondto quenching. Here, then,

is the double sequence recovery, quench, recovery, quench. When the minimum of the η(∆Η)-curve lies to the right of the ∆Η(η)-curve and only the intersection R

occurs, we obtain the single sequence recovery, quench.

The qualitative argument just given serves three purposes. First, it strips away the mystery of the folded surface in Fig. 7.9 by showing how the interplay of a few simple phenomena can lead to behavior that at first sight seems incomprehensible. Second, it reveals what ingredients are essential to include in a numerical stability program if one hopes to reproduce multivalued stability. These ingredients are transient heat transfer, takeoff, and augmentation of turbulent heat transfer by heating-induced flow. Third, it provides a basis for estimating the location of the point B in Fig. 7.9.

7.6. THE LIMITING CURRENT

In Fig. 7.9, we call the value of DH on the upper sheet AFCD the upper stability margin and the value on the lower sheet BKEF the lower stability margin. The experimental data in Fig. 7.7 show that the upper stability margin can be many times larger than the lower stability margin. Since we are ignorant of the perturbation spectrum, it is prudent to assume that the effective stability margin in the region of multivalued stability is the lower stability margin. So for practical purposes, there is a sharp drop in stability as the current increases past the current IB at point B. This current has been given the name limiting current and the symbol Ilim.

Operating below the limiting current guarantees the high upper stability margin. When we operate beyond the limiting current in the region of multivalued stability, there are perturbations less than the upper stability margin but larger than the lower stability margin that can quench the magnet. Now magnets have been built that operate successfully beyond the limiting current (Lue and Miller, 1982). In such magnets, the thermal perturbations happily did not exceed the lower stability margin. But there have been other magnets, which, though they worked well below the limiting current, quenched above it (Painter et al., 1992). Whether one should operate below the limiting current or not is a subjective decision up to the individual designer. The size of the limiting current, on the other hand, is a purely technical question.

We cannot determine the limiting current from the elementary considerations discussed so far, but we can use them to obtain a scaling rule that expresses the dependence of the limiting current on various parameters of the conductor (Dresner,

1981, “Parametric study”). The condition determining the limiting current is the confluence of the intersections P and Q in Fig. 7.1 2. If the DH(h)-curve rises sharply near its foot, the heat transfer coefficient at this confluence is close to that at the foot. Hence, Ilim and h satisfy Eq. (7.5.2). To estimate h at the confluence of P and Q, we proceed as follows.

The heat flux qJ into the helium depends on the takeoff time t according to the relation

where C is a constant that depends on the thermodynamic state of the ambient helium. The lower stability margin (intersection P in Fig. 7.12) is characterized by the condition that the duration of the normalizing pulse just equals the time to takeoff. In other words, at point P, the normalizing heat pulse is just finished being drained away by transient heat conduction into the helium at the moment of takeoff. Thus t in Eq. (7.6.1) is the pulse duration.

A heat flux qJ will induce a velocity v of the order of

since qJP/ rHeAHe = q, the power density in the helium (here rHe is the density of the helium). Combining Eqs. (7.6.1) and (7.6.2) and ignoring thermodynamic quantities and numerical constants, we find

where D = 4AHe/P is the hydraulic diameter of the helium space.

The upper branch of the curve h(DH) in Fig. 7.12 represents steady heat transfer in turbulent helium and can be taken to be described by a heat transfer correlation of the form

where Nu is the Nusselt number, Re the Reynolds number, and Pr the Prandtl number (Bird et al., 1960). Again ignoring thermodynamic and numerical constants, we find that Eq. (7.6.4) says

If we combine Eqs. (7.6.5) and (7.6.3) we find

Inserting Eq. (7.6.6) into Eq. (7.5.2), we find, after some manipulation

where Jlim = Ilim/A, A is the cross-sectional area of the cable space, f is the volume fraction of copper in the strands, fco is the volume fraction of the strands in the cable

space, and, as before, rcu is the electrical resistivity of copper.

In the classical Dittus–Boelter correlation m = 4/5, and this value appeared to agree well with the first experimental data on limiting currents reported by Lue and Miller (1981, “Parametric study”; also Dresner, 1981, “Parametric study”). But later experiments (Lue and Miller, 1981, “Heated length”) were better fitted by the value m = 4/15. Arp (1979) has pointed out that Kawamura (1976; 1977) showed that the Nusselt number can vary by almost an order of magnitude from its steady-state value when the Reynolds numbers changes rapidly. Perhaps this may make the different reported values of m seem less troubling; in any case, there is

little more we can say about the value of m.

The proportionality of Jlim to r–cu1/2 (Tc – Tb)1/2 has been tested by Lue and Miller (1981, “Parametric study”; also Dresner, 1981, “Parametric study”) by changing the magnetic field to which the sample was exposed, and quite good agreement was found.

There is a proportionality constant missing from Eq. (7.6.7) that Miller (1985) has determined from a study of the experimental data. In SI units, the constant is close to 1 for 4-K helium. It varies linearly with pressure from 1.2 at 0.3 MPa to 0.95 at 0.6 MPa.

The scaling rule (7.6.7) does not depend on the critical current Ic, and it may happen that the value of Ilim > Ic. What this means is simply that we cannot operate the conductor in the regime of multivalued stability because it becomes resistive before we reach that regime. Such a conductor has single-valued stability for any attainable current, and its stability margin always equals the upper stability margin. An example is the Westinghouse conductor of the Large Coil Task (Beard et al., 1988): the large value of Tc at 8 T forNb3Sn pushes Ilim well beyond Ic. Interestingly, a lower magnetic field may restore multivalued stability because with decreasing field Ilim increases as (T c – Tb)1 / 2 whereas Ic increases faster, as Tc – Tb .

7.7. DISCUSSION OF THE ISOBARIC ASSUMPTION

Eq. 7.5.1. applies to the special limiting case in which all the enthalpy of the helium between Tb and Tcs is devoted to absorbing the initial normalizing pulse. In general, this available enthalpy is split between the heat of the initial normalizing pulse and the Joule heat produced during recovery, as we see now by considering the process of recovery a little more closely. Just after the initial pulse, the conductor is hot, with a temperature above the current sharing threshold, while the helium is still cold (at temperature Tb). Then the conductor and the helium exchange heat, the conductor temperature aways being greater than or equal to that of the helium. As

long as the conductor temperature exceeds the current sharing threshold Tcs, the conductor continues to produce Joule power. If the conductor and the helium equilibrate their temperatures at a temperature < Tcs, the conductor recovers, for as its temperature falls below Tcs, it stops producing Joule power. On the other hand, if the helium temperature reaches Tcs first, the conductor never stops producing Joule power and so quenches. In the limiting case in which the initial normalizing pulse is the largest consistent with recovery, i.e., in which it equals the stability margin, the conductor and helium temperatures equilibrate exactly at Tcs. Thus the helium has been heated from Tb to Tcs by the combined heat of the initial normalizing pulse, DH, and the Joule power produced during the temperature equilibration.

Without solving the time-dependent problem of heat exchange between the conductor and the helium, it is not possible to say how much Joule heat is produced during recovery. But when the heat transfer coefficient is very large, making the process of heat exchange very rapid, we expect little Joule heat to be produced during recovery. This is the situation that corresponds to the upper stability margin, where strong induced flow greatly augments the heat transfer coefficient. Thus we expect the limiting value of DH given in Eq. (7.5.1) to approximate the upper stability margin and to be larger than it.

Eq. (7.5.1) is based on the tacit assumption that recovery is an isobaric process, for only at constant pressure is the heat absorbed by the helium equal to its increase in enthalpy. Now we have already seen in Sections 7.3 and 7.4 that heating-induced flow is accompanied by a pressure rise, so recovery cannot be a strict isobaric process. In the stability experiments of Lue et al. (1980) on single triplets and in those of Miller et al. (1980, “Stability”) on a one-third scale Westinghouse Large Coil Task conductor, the stability margin DH sometimes exceeded the right-hand side of Eq. (7.5.1) by as much as a factor of 2. This seems to me to clear evidence that recovery cannot be treated as isobaric and that the pressure transient associated with heating-induced flow has an important effect on the cooling capacity of the helium. This point was first made by Wilson (1977); Lue et al. (1980) showed that pressure excursions of several atmospheres could account for the discrepancies noted between DH and the right-hand side of Eq. (7.5.1).

We can only argue approximately because we do not know the exact thermodynamic trajectory of the heated zone. Qualitatively, it must look in the pressurespecific volume plane like the curve shown in Fig. 7.13. (N.B.: p represents pressure rise.) Point A is the initial state of the helium; the hatch marks label the part of the curve over which Joule power is produced. According to the earlier discussion, when the normalizing heat pulse equals the stability margin, the helium temperature just reaches Tcs when the Joule heating stops (point B). Thereafter, the helium expands isentropically back to ambient pressure (arc BC).

It can be shown straightforwardly that the heat absorbed by a unit mass of