Dresner, Stability of superconductors.2002

Figure 3.11. A schematic diagram of two twisted filaments in a composite slab.

As we shall see next, if the filaments are twisted within the conductor, the shielding currents decay and the external magnetic flux penetrates uniformly into the composite slab. Then no shielding current sheaths form and no flux jumps occur. This opens the possibility of composite conductors whose thickness is much greater than that just estimated.

Shown in Fig. 3.11 is a schematic drawing of two twisted filaments in the composite slab. The arrows show the direction of the emf induced by the rising external field in the loop shown in the figure. By translational symmetry (which means that any loop is like any other to either side of it) conditions at point B must be the same as those at point A. As one traces the loop, one crosses the matrix at points B and A in the same direction; hence, since the superconductor supports no voltage drop, the voltage difference VB = VA = appears in the matrix. This means that induced currents necessarily flow partly in the matrix and so must decay when the external field stops rising.

Since the transient shielding currents flowing in one loop do not cross into adjacent loops, we can determine their distribution by considering just the piece of composite slab conductor between the point B and the point A as though the rest of the conductor had been sheared off (Fig. 3.12). If the field created by the induced currents is neglected compared to the applied field Bo (we shall see later that this approximation is conservative), then the voltage around path ABCD in Fig. 3.12 is

Since by symmetry, the y-axis is the locus E = 0, and since the

x-direction. This current must not exceed the critical value λJcd or the filaments will become resistive. Thus the pitch width is limited to

Figure 3.12. A schematic diagram ofone pitch width of a composite superconductor showing the many filaments of superconductor embedded in the matrix.

For NbTi at high fields, say 8 T, Jc = 120 kA/cm2. If we take l = 0.4, d = 0.5 mm (1-mm-thick slab), p = 5 x 10¯10 ohm-m (typical for copper at 4.2 K and 8 T), and

= 0.1 T/s (typical for small magnets), we find p < 98 mm.

This estimate is conservative for the following reason. The field created by the induced currents opposes the applied field in direction (Lenz’s law). Thus the actual field change we used to calculate the induced currents is an overestimate and thus overestimates those currents. Hence the pitch width limit given in Eq. (3.8.1) is smaller than it has to be and is thus conservative.

3.9. SELF-FIELD STABILITY

Whereas the external field has the same direction at every point in the slab, the self-field created by the transport current (i.e., the current being driven through the conductor by an external power supply) changes sign in the slab. Indeed, by symmetry, the self-field has equal magnitudes and opposite directions at points that are images under reflection in the midplane of the slab. As a consequence, the self-field creates no net flux linkage with the loops shown in Fig. 3.12 and thus induces no shielding currents. As far as the self-field is concerned, the filaments behave as though they were untwisted. Then, harking back to Fig. 3.10, we see that

slabs for which µoλJcd > B* = [3µ 0S(Tc – T)]1/2 are stable against flux jumps induced by the self-field Bsf only if Bsf < B*.

Now for a slab, the self-field at the edge of the slab is µ oJd, where J is the transport current density in the slab. Hence, self-field stability is assured if J < B*/µod. The same result follows from Eq. (3.5.7), which describes the flux-jump stability of a conductor being charged with current, with the slight change that now Jc is replaced by λJc. Since a =Jd/λJc, we again find for self-field stability that

J < B* /m0d. If we take B* = 0.25 T in accordance with the estimates in Section 3.7, and consider slabs 1 mm thick, we find B* /mod = 40 kA/cm2. This limiting current density for self-field stability is well above the operating current densities in many, though not all, magnets.

3.10. A FINAL WORD ON FLUX JUMPING

As a final word, it is worth remarking that flux-jump and self-field stability are not practical problems any more because modern, commercial conductors are manufactured to be well within the required limits.3 For example, prototype NbTi/Cu strands for the ill-fated Superconducting SuperCollider (SSC) described by Kallsen et al. (1991) have 4.8-µ m or 6.5-µ m filaments of NbTi in a copper matrix. The diameters of the wires themselves are 0.67 mm and 0.91 mm. No twist pitch was reported, but typically pitch widths are between 5 and 10 diameters; thus for these SSC strands, p is in the range 3–9 mm, which is amply small. For these strands, λJc ~ 120 kA/cm2 at 5 T and 4.2 K, so that when they operate at more than about one-half of their critical current, J exceeds the limit B* /µod. But the strands are helium cooled, and since >> for copper, the helium cooling is sufficient to suppress flux jumps.

Notes to Chapter 3

1This statement may be somewhat overdone, especially in the case of the high-temperature superconductors, where the irreversibility line, which may be well separated from the phase boundary, is said by some to separate a vortex glass phase from a vortex liquid phase. This issue is at present hotly debated and the interested reader is referred to Bishop et al., 1992; Bishop et al., 1993; and Huse et al., 1992.

2This solution is one of a type called similarity solutions. In this book such solutions arise in several places: the study underway, the flow of helium induced by heating in cable-in-conduit conductors (Chapter 8), the transfer of heat in turbulent superfluid helium (Chapter 9), and the determination of quench energies of high-temperature superconductors (Chapter 6). Detailed information on the calculation of similarity solutions can be foundin Dresner (1983). For the sake of completeness here, a short appendix on similarity solutions has been included at the end of the main text.

3This is not really the last word by any means for the following reason. It has perhaps occurred to the

reader when he examined Eq. (3.7.1) that as the external field Bo changes with time and the flux moves in and out of the filaments of superconductor,heat is dissipated in the filaments by the induced electric field E acting on the shielding currents. In fact this is so, and in the presence of a changing external field such dissipation, called hysteresis loss, is present in the superconductingfilaments. Furthermore, there is additional dissipation caused by the induced currents that flow partly through the matrix as discussed in Section 3.8. This dissipation, because it arises from currents that cross the matrix from one filament to another, is called coupling loss. Collectively, these losses are known as AC-losses. In applications with alternating magnetic fields, these losses impose a load on the refrigeration system. The hysteresis losses, it turns out, are minimized by making the filaments as fine as possible, and the coupling losses are minimized by making the pitch width as short as possible. The subject of AC-losses is worthy of a book in itself, but it is outside the scope of the present work, which is devoted to stability problems, and will be pursued no further here. The interested reader should consult the relevant chapter of Wilson’s book (1983).

jumps sharply to the higher value at point Q (remember, the flux q is being held fixed). The coalescence of the bubbles into a film and the attendant jump in DT is called burnout or the boiling crisis. The heat flux at the point P is called by several names, viz., the maximum nucleate boiling heat flux, the burnout heat flux, or the first critical heat flux.

If we now reduce the heater power, we do not return at once to the nucleate boiling regime but instead continue in the regime QR of film boiling. In this regime, q is again roughly proportional to DT. Ultimately, as q is reduced, we reach a point at which the rate of vaporization at the liquid-vapor interface is not sufficient to hold the liquid away from the heated surface. Fingers of liquid penetrate and destroy the vapor film, and there is a sudden return to nucleate boiling. This restoration of nucleate boiling, which is accompanied by a sharp drop in the superheat ∆T, is called recovery. The heat flux at point R is called by several names, viz., the minimum film boiling heat flux, the recovery heat flux, or the second critical heat flux.

The description of boiling heat transfer just given corresponds to flux control. Temperature control is also possible although more difficult to arrange. Conceptually, it requires a feedback link between the thermometer and the heater to control the heater power so as to maintain a fixed superheat ∆T. Again as ∆T increases from small values, we begin by traversing the curve OSP. But now there can be no sharp change of ∆T at the burnout point P. Instead the heat flux decreases along the dashed curve PR, approaching the recovery point R. This regime, which can only be realized under temperature control, is called the regime of transition boiling.

When a surface in transition boiling is examined visually, one sees that the surface is covered by patches of bubbles interspersed by patches of vapor film. These patches form and re-form continually, the bubbles here coalescing momentarily while the film there collapses. A fluctuating equilibrium is maintained in which the average fraction of the surface covered by vapor film increases as the superheat increases.

4.2.ADDITIONAL FACTORS AFFECTING BOILING HEAT TRANSFER

The description just given of boiling heat transfer is schematic. The actual position of the q-∆T curve in any specific case depends on a host of factors, among which the most important are:

1. The identity and thermodynamic properties of the coolant (densities, thermal conductivities, specific heats of the liquid and the vapor, the surface tension, the latent heat of vaporization, the saturation temperature).

2.The ambient (saturation) pressure, since all of the above properties vary with position along the saturation line (see Fig. 1.7).

3.The orientation and shape of the surface. For example, in the case of a flat surface, heat transfer is quite different for a surface facing downwards than for a surface facing upwards. In the case of a horizontal wire, heat transfer depends on the wire diameter.

4.The condition of the surface. The shape of the heat transfer curve depends on the surface roughness, the presence of chemical impurities, and the presence of coatings applied to the surface.

5.Gravity. Since buoyancy is a principal driving force, boiling heat transfer is different in strong centrifugal fields from what it is at 1 g. Similarly, there are differences at zero g.

6.Channels. If the boiling coolant resides in a thin channel rather than in an open pool, the heat transfer curve depends on the shape, dimensions, orientation, and length of the channel. Especially important is the possible accumulation of vapor in the upper reaches of the channel, which may seriously diminish heat transfer there.

All these factors are of great importance in determining the stability of superconductors cooled with boiling helium (called pool cooling, even when the helium is in thin channels between adjacent superconductors of a magnet winding). The task in this book, however, is to analyze the stability of the superconductor, assuming its heat transfer characteristics are known. We treat boiling heat transfer much as we treat the solid-state aspect of superconductivity: we assume its results are available to us and try to decide how to use them to design magnets.

The literature of boiling heat transfer is immense. Two excellent references to boiling heat transfer to cryogens that the reader may consult are the article of Brentari and Smith (1965) and Chapter 6 of Sciver’s book (1986).

4.3. CRYOSTABILITY

Stekly, with his coworkers (Kantrowitz, 1965; Stekly and Zar, 1965; Stekly et al., 1966), and Laverick and Lobell (1965) were the first to build superconducting magnets that recovered the superconducting state after a normalizing perturbation. Both groups of workers used boiling helium as the coolant, and both arranged for cooling to exceed Joule heating by adding copper to the conductor until the Joule heating was sufficiently reduced. We can understand how this is done by means of Fig. 4.3, which compares the steady-state boiling heat flux with the three-part curve of Joule power (cf. Fig. 3.3b). When the conductor is in the fully normal state, the Joule power produced per unit surface area of the conductor in contact with the helium Q = ρcu J2A/fP, where J is the overall current density in the conductor (transport current I divided by the total cross-sectional area A), f is the volume fraction of copper, P is the cooled perimeter of the conductor (i.e., the perimeter in contact with the helium coolant), and ρcu is the resistivity of the copper matrix. If

Figure 4.3. A comparison of the steady-state boiling heat flux with the three-part curve of Joule power from Fig. 3.3 when the superheatat recovery is greater than Tc – Tb.

Q is less than qr, the recovery heat flux, then the heating curve (curve b) lies below the cooling curve (curve a) at every temperature. In such a case, the heat balance at every point along the conductor is negative, so that the temperature everywhere ultimately returns to the saturation temperature of the helium pool. (This temperature is most often called the bath temperature and symbolized by Tb.) As we can see from the expression given above for Q, if we increase A by adding copper, we can eventually reduce Q below qr. Stability achieved in this way is called cryogenic stability, cryostability, or unconditional stability.

Figure 4.3 has been drawn assuming the superheat at recovery (point R) is

where, as in the last chapter, λ is the volume fraction of superconductor and Jc is the critical current density of the superconductor alone. The first factor in Eq. (4.3.1) is often eponymously called the Stekly number and symbolized by α. The ratio J/λJc, which gives the ratio of the transport current to the critical current (at Tb), is usually symbolized by i.

As mentioned in Section 1.6, one disadvantage of cryostability is the comparatively low current density J = I/A in the conductor (typically ~3 kA/cm2 at 8 T and ~5 kA/cm2 at 5 T for NbTi/Cu conductors). Low current density means large size, large weight, and high cost for magnets and, by extension, may mean large size and high cost for the rest of the apparatus in which the magnets serve. On the other

Figure 4.6. In a Maddock-stable superconductor, the edges of the normal zone propagate inward and the center is the last point to recover.

ditional stability can be preserved at somewhat higher current densities than the criterion Q = ρcuJ2A/fP < qr allows. In magnets obeying this last criterion, all parts of the conductor driven normal by a thermal perturbation (called a normal zone) recover simultaneously, the temperature falling at every point in the normal zone (see Fig. 4.5). In Maddock-stable conductors, the edges of the normal zone propagate inwards and the center is the last point to disappear (see Fig. 4.6). This mode of recovery, which we analyze in detail below, is called cold-end recovery.

The inward velocity v of the edges of the normal zone depends on the transport current I. As I increases, v gets smaller until at a certain current Im, v becomes zero. When I > Im, the edges of the normal zone propagate outwards, i.e., the normal zone grows instead of shrinking. In this case, of course, the conductor quenches. These relationships are graphically illustrated in Fig. 4.7. The current Im, corresponding to v = 0 that separates the region of cold-end recovery from the region of quenching is called the minimum propagating current.

Maddock, James, and Norris have given a simple, elegant, and practical method of finding the minimum propagating current. They begin with the one-dimensional heat balance equation for a long superconductor in a helium bath:

where S is the heat capacity of the conductor per unit volume, T is the local temperature of the conductor, t is the time, z is the distance along the conductor, k is the thermal conductivity of the conductor, Q is the Joule power per unit cooled surface area, P is the cooled perimeter, A is the cross-sectional area of the conductor, and q is the heat flux being transferred to the helium. The quantities Q and q are the same as those plotted in Fig. 4.3.

We look for a solution of Eq. (4.4.1) that looks like the curves in Fig. 4.6. The uniform central temperature is determined by Q = q, since near the center of the normal zone ∂T/∂t and ∂T/∂z are both zero. This means that the current I must be large enough for the heating and cooling curves of Fig. 4.3 to intersect (see Fig. 4.8).