Dresner, Stability of superconductors.2002

The rather low current density of ~0.5 kA/cm2 of this coil is responsible for its very high stability. Recently, Moore (1992) argued that critical current densities in the range 100–1000 kA/cm2 are necessary for practical application to transmission lines, motors and generators, and magnetic energy storage. Such high current densities would reduce the quench energy by about three orders of magnitude to, say, tens of mJ. Such energies are still roughly three orders of magnitude greater than the energy released by epoxy cracking that we estimated in Sections 6.4 and 6.5. So as mentioned there, potted magnets wound with Ag/BSCCO superconductor should be immune to training.

The preceding work depends on three assumptions, namely, that the temperature variations of S and rAg could be described by power laws, that the thermal conductivity and resistivity of silver obeyed the Wiedemann–Franz law, and that we could take Tb = 0. The first two assumptions are relatively benign, but the last requires some discussion. It had to be made because without it the group-invariance method and the other detailed calculations of Dresner (1994, “Quench energies”) would not have been possible. It can be justified because of the rapid drop of the specific heat at low temperatures. Perhaps the simplest way of seeing this is to employ an argument given in the same reference. Consider the extreme limiting case in which the volumetric heat capacity S falls discontinuously to zero at some finite temperature Tb. Then, as soon as any heat is introduced to the medium, its temperature jumps to Tb. Thus a nascent normal zone behaves as though it were growing in a medium whose initial temperature were Tb, not zero. If we invert this argument, we can say that if the ambient temperature were actually Tb, the normal zone would grow as though the ambient temperature were zero. For these reasons, we expect that if the volumetric heat capacity S falls rapidly with temperature, the quench energy can be adequately approximated by assuming Tb = 0.

6.7.PROPAGATION VELOCITIES OF UNCOOLED SUPERCONDUCTORS

The absence of cooling actually complicated the calculation of the quench energy by forcing us to consider a time-dependent problem where formerly we had only to consider a time-independent one. The calculation of the normal zone propagation velocity is similarly complicated because in the absence of cooling the central temperature no longer approaches a constant value (T 2 of Fig. 4.8; cf. the first paragraph of Section 5.1).

When k, rcu , and S can be taken as independent of temperature, we already know the propagation velocity—it is given by Eqs. (5.1.3) and (5.1.4). As noted above, this assumption of temperature independence is satisfactory for the classical low-temperature superconductors, but not for the newer high-temperature superconductors. To calculate the propagation velocity, we again need to look for traveling wave solutions T(z + vt) of the heat balance equation, now Eq. (6.1.1). As

in Section 4.4, T(–8) = Tb (we no longer need to take Tb = 0 as we did in the determination of the quench energy); but T(8) can no longer be set equal to T2 (indeed, there is no T2). The determination of the temperature profile far behind the propagating front is, as noted above, a complication that does not exist when cooling is present.

To learn how to deal with this complication (Dresner, 1994, “On the connection”), let us first consider the case in which k, rcu , and S can be taken as independent of temperature, even though we already know the answer to this case. In addition, in order not to obscure the principle involved, we simplify the computations by ignoring current sharing; thus we take for QP/A the two-part curve

For a traveling-wave solution that propagates from right to left, T(z + vt), Eq. (6.7.2) becomes

where x = z + vt. We again introduce s = dT/dx as a new dependent variable in order to reduce Eq. (6.7.4) to a first-order differential equation:

Now the advantage of a first-order differential equation is that it may be analyzed graphically by studying its direction field. (The direction field of a first-order differential equation dy/dx = F(x,y) is the diagram one obtains if one draws at points (x,y) short hatch marks having the slope dy/dx calculated from F(x,y). The field of hatch marks then gives at a glance the course of the integral

Figure 6.4. Profiles of s = dT/dx versus x that correspond to the profiles in Figure 6.3. (Redrawn from an original appearing inDresner(1994,“On the Connection”)withpermissionof Butterworth-Heinemann, Oxford, England.)

the initial normal zone approaches a traveling wave solution as time progresses. Figure 6.3 shows profiles of T versus z for various times after the initial normal zone is established. Figure 6.4 shows the corresponding profiles of s versus T corresponding to the left halves of the profiles in Fig. 6.3. These profiles resemble the integral curves in the (s,T)-plane that lie below the horizontal lines = 1/v in Fig. 6.2. As the maximum temperature of the normal zone becomes larger and larger, the value of s at T = Tc approaches 1/v more and more closely. Thus we must match Eq. (6.7.6) to this value at T = Tc . Thus

for in this limit, the three-part curve (4.6.4) for QP/A becomes the two-part curve (6.7.1). (N.B.: As i 0, ci1 / 2 i)

6.8.PROPAGATION WITH TEMPERATURE-DEPENDENT MATERIAL PROPERTIES

Now let us considerwhat happens when kcu, rcu, and S all vary withtemperature and kcu, and rcu ,obey theWiedemann–Franz law.Then atraveling wavesolution T(z + vt) of Eq. (6.1.1) obeys the ordinary differential equation

If we now take s = k(dT/dx) and multiply Eq. (6.8.1) by k, we obtain the first-order differential equation

Since for this solution, s must take the value Tc /v s(Tc) at T = Tc, it follows that

(6.8.6)

or, in ordinary units,

(6.8.7)

From the manner in which this last result has been derived we expect it to overestimate v. This result bears a striking resemblance to an earlier result of Whetstone and Roos (1965). These authors neglected the term k(d2T/dx2) in Eq. (6.8.1) when T > Tc, whereas in the present work it is the entire term d/dx[k(dT/dx)] that is neglected when T > Tc.

6.9.THE EFFECT OF CURRENTSHARING ON THE PROPAGATION VELOClTY

The neglect of current sharing always leads to an underestimate of the propagation velocity (Dresner, 1994, “On the connection”). This is easy enough to understand because current sharing adds additional Joule heating below Tc to the source term given in Eq. (6.7.3). I have calculated the velocity v as a function of (Tc – Tcs)/(Tc – Tb) numerically for several values of Tc/Tb and several values of the exponent m in the assumed expression s = (T/Tb)m. Results are shown in Figs. 6.6, 6.7, and 6.8 for m = 1, 2, and 3, respectively. In each figure, five curves are plotted of v/v0, where v0 is the propagation velocity in the absence of current sharing, i.e., when Tcs = Tc. These curves correspond to values of the ratio Tc/Tb = 3, 2,1.5, 1.2, and 1.1, reading from top to bottom. Note that when Tc/Tb and m are large, and Tcs is near T b (e.g., the right-hand edge of Fig. 6.8), the correction to v due to current sharing can be quite large. The physical reason for this strong increase in v is that under the circumstances just mentioned, the volumetric heat capacity S is quite small at the point on the leading edge of the front at which Joule heating begins.

6.10. AN INTERESTING COUNTEREXAMPLE

It has become deeply ingrained in the minds of many workers in the field of applied superconductivity that traveling-wave solutions always exist, so it may come as a surprise to some to learn that such solutions do not always exist. The following counterexample is given in (Dresner, 1994, “On the connection”).

Figure 6.11. The direction field of Eq. (6.8.3) when S is independent of temperature,i.e., when s(T) = 1, and Tc > 2. (Redrawn from an original appearing in Dresner (1994, “On the Connection”) with permission of Butterworth-Heinemann,Oxford, England.)

6.11. THE APPROACH TO A TRAVELING WAVE

It has also become deeply ingrained in the minds of many workers in the field of applied superconductivity that if a traveling-wave solution exists, it represents the asymptotic limit to which nonrecovering normal zones tend. This turns out not to be generally true, the situation being considerably more complicated. To deal with it, we use the following comparison theorem (which can be proved using the methods described in Sections B.1. and B.2.): If T1(z,t) and T2(z,t) are two solutions of Eq. (6.1.1) that obey the same boundary and initial conditions (T1(a,t) = T2(a,t), T1(b,t) = T2(b,t), and T1(z,0) = T2(z,0), a ≥ z ≥ b) but belong to two different source

terms Q1(T) ≥ Q2(T),then T1(z,t) ≥ T2(z,t) for all t > 0 and a ≥ z ≥ b.

Now let us continue the example of Section 6.10 but add the further assumption that kcu is a constant independent of temperature and that rcu = LoT/kcu. The Wiedemann–Franz law is thus still obeyed, and the traveling-wave solutions are still those of Section 6.10. Now using the special units of Section 6.8, the time-dependent heat balance Eq. (6.1.1) becomes

(6.11.1)

If we add to our list of special units k = 1, then kcu = 1/f, and rcuJ2/f = LoTJ2, which in special units equals T. Thus Eq. (6.11.1) becomes

(6.11.2)

If we take Q1(T) = T and Q2(T) = Tg(T), then Q1(T) Q2(T). Therefore, solutions of the time-dependent equation