Dresner, Stability of superconductors.2002

Figure 7.13. Thermodynamicpath in the pressure-specific volume plane.

(7.7.1)

Now since w on the upper branch of curve ABC, where dp < 0, is > w on the lower branch, where dp > 0, w dp < 0. It follows immediately then that q > –

Now, in strict point of fact, the enthalpy difference – is different from the right-hand side of Eq. (7.5.1). For the temperature at point C is less than Tcs because

stability margins are several hundred mJ/cm3 (cf. Fig. 7.7), so the fractional error incurred by using Tcs as the upper limit in the integral in Eq. (7.5.1) is not large.

The thermodynamic arguments given so far are based on the assumption that a single pair of thermodynamic variables describes the entire normal zone, and from the work of Sections 7.3 and 7.4, we know this is not so. At any single instant, the thermodynamic states of different fluid elements may differ from one another.

In view of all these caveats, it would appear that the estimate of Eq. (7.5.1) has no rigorous interpretation. Nonetheless, it is useful as a figure of merit, for when it is high the upper stability margin is high, and when it is low, the upper stability margin is low. In addition, it is easy to calculate, a property that has great utility in survey calculations.

7.8. THE LOWER STABILITY MARGIN

As mentioned in Section 7.6, the lower stability margin is characterized by the condition that the duration of the normalizing pulse just equals the time t to takeoff. If we ignore the Joule power compared to the heat flux qJ = DHAco /Pt and substitute the last expression into Eq. (7.6.1), we obtain

where Dw is the wire (strand) diameter. The right-hand side of Eq. (7.8.1) should be an upper bound to DH because of the neglect of Joule power generation.

Lue and Miller (1981, “Heated length”) measured the lower stability margin of a triplet of 1-mm strands soldered around a central heater wire. The ambient helium pressure was 0.5 MPa and the ambient temperature was 4.2 K, for which conditions, I have calculated that C = 26 mW cm–2 s1/2 (Dresner, 1984, “Superconductor stability”). For pulse times of 10, 50, and 100 ms, we calculate from Eq. (7.8.1) that DH = 87, 194, and 274 mJ/cm3, respectively. (The results from Eq. (7.8.1) have been reduced by a factor of 5/6 to account for the fact that in a soldered

triplet only 5/6 of the surface is wetted.) The corresponding experimental results are 60, 100, and 140 mJ/cm3 and are slightly lower than the upper bound calculated from Eq. (7.8.1). In Dresner (1984, “Superconductor stability”), an estimate is presented of the Joule power required to reduce the calculated DH to the measured value. It is given as 36%, 25%, and 18%, respectively, of the fully normal heat flux, indicating that the conductor appears only to have been driven into the current sharing range by the initial pulse.

Notes to Chapter 7

1The idea of increasing the surface by subdividing a fixed volume is ancient. To my knowledge, the first suggestion in this direction regarding superconductors came from Chester (1987), who wrote, “Clearly, excellent thermal contact is desirable between the superconductor andthe thermal ballast . . . this is achieved by subdivision of the superconductor to present greater interfacial area.” The present era of development of cable-in-conduit conductors began in 1975 with a reminder from Hoenig, Iwasa, and Montgomery(IEEE Transactions) of the advantages of subdividing the superconductor by cabling. A review of this earlier work can be found in Dresner, 1980, “Stability.”

2If the reader has any doubts about the correctness of leaving the frictional force out of the energy equation, the following observation should settle them. The power dissipated by the frictional force per unit volume is –rFv. If this term were included on the right-hand side of Eq. (7.2.3), the term Fv would not appear on the right-hand side of Eq. (7.2.7). Then the dissipation of kinetic energy into heat would produce no entropy, an obvious contradiction.

3Riemann’s method was invented to deal with hyperbolic partial differential equations, the classical example of which are the equations of compressible flow. The interested reader can find detailed discussions of the method in the books of Courant and Hilbert (1953), or Courant and Friedrichs (1948).

4In the simplest problems of gas dynamics the Riemann invariants do not change along the characteristics and this accounts for their names. We continue to use this name here for convenience even

though the presence of a heat source causes the “invariants” to change along the characteristics.

5Strong heat transfer promoted by transient heat conduction occurs early in supercritical helium as well as in boiling helium. Whereas in boiling heat transfer the transient phase comes to a close (takeoff) when the surface is blanketed by vapor, in supercritical heat transfer the transient phase comes to a close when the temperature of the surface reaches the so-called pseudo-critical line where the density of the helium drops sharply. Takeoff in supercriticalhelium is thus caused by the blanketing of the heat transfer surface by low-density helium, a process sometimes called pseudo-boiling. The time to takeoff in supercritical helium varies as the reciprocal square of the heat flux just as in boiling helium and the two times are not terribly different in magnitude for the same heat flux.

8.2. MAXIMUM QUENCH PRESSURE

To bound the internal pressure that the conductor might suffer, let us assume an entire hydraulic path has gone normal all at once. We presume the ends of the hydraulic path intrude into large plenums held at constant pressure, so that there is no pressure rise at the ends of the hydraulic path. According to Eq. (8.1.1), the pressure rise is largest at the central element, where it equals

where = L/2. We now assume that the velocity has a linear profile, being zero at the center and having a maximum vm at the open ends. Then Eq. (8.2.1) becomes

When po achieves its maximum pm, ∂po/∂t = 0, and then it follows from Eq. (8.2.4) that

where Q = rq is the Joule power density in the helium.

In Miller et al. (1980, “Pressure”), it is shown that the thermodynamic group b2/rc2p depends only weakly on density and can be approximated roughly by 0.45p–1.8 in SI units, where p denotes absolute pressure, rather than pressure rise. Substituting this correlation into Eq. (8.2.5), we obtain the approximate formula

Before we make a direct attack on this problem, it behooves us to inquire if Eqs. (8.1.1) and (8.1.2) and the conditions (8.3.1) are invariant to stretching groups similar to that used in Section 6.2. Such a group is

z´ = lz

v´ = l–1/2v

t´ = l3/2t

q´ = l–3/2q

with the other quantities(p, b, r, cp, c) being unchanged. For a fixed initial thermodynamic state, the expulsion velocity v(0,t) can only depend on q and t, i.e., v(0,t) = (q,t). This relation must be invariant to the transformations (8.3.1), i.e., v´(0,t´) = (q´, t´) or

that if we plot the data in Fig. 8.2 using vt1/3 as ordinate and I2t ~ qt as abscissa, all the curves in Fig. 8.2 should, and do, collapse to a single curve (cf. Fig. 8.3).

The points in Fig. 8.3 correspond to points on the curves in Fig. 8.2 for which t < 1 s. For times t ~ 2 s or longer, the disturbance has reached the center of the conductor and the expanding helium no longer behaves as though the hydraulic half-length were infinite.

The function (qt) has been calculated by the method of similarity solutions (Dresner, 1981, “Thermal expulsion”), discussed in Appendix A. The result of this calculation is

It is important to note here that this explicit form has been attained only at the cost of (1) assuming the thermodynamic properties of the helium remain constant, and

(2) ignoring frictional heating. In fact, r diminishes as helium is expelled from the conductor, so that on both counts Eq. (8.3.5) should underestimate v. As we can see from Fig. 8.3, it does so. The overall slope of the experimental points matches the theoretical slope of 2/3 fairly well, but the similarity solution is low by roughly a factor of 2.

Section 8.4, an infinite pipe was envisioned. We need to know in addition what happens when the initial normal zone is short. Now the problem is a two-region problem and is substantially more complicated than the problems just dealt with. Again we can simplify the problem by artificially breaking the problem into two one-region half-problems. The first half-problem concerns the flow induced in the cold helium by the expanding hot helium in the initial normal zone. To break the coupling we assume that we know how this hot-cold boundary moves. The expanding hot-cold boundary then acts like a piston that accelerates the cold helium, which, for the long times we are interested in (seconds, not milliseconds) is restrained by friction.

8.6. THE PISTON PROBLEM

If the disturbance created by this “piston” has not yet reached the open end of the hydraulic path, we can study the motion induced by the piston as though it were taking place in an infinite tube. Thus we must solve Eqs. (8.1.1) and (8.1.2) with the boundary and initial conditions

where Z(t) is the location of the piston, assumed known. Indeed, in what follows we take Z(t) = Xtn, where X is a known constant of proportionality. Thus we take the displacement of the piston to be proportional to a power n of the elapsed time. Furthermore, in the cold helium q = 0. To make the calculation possible, we restrict ourselves to small velocities v, so that the frictional v3-term on the right-hand side of Eq. (8.1.2) can be neglected. (Later, we shall mention the conditions under which this neglect is permissible, i.e., the conditions under which the solution we shall obtain is self-consistent.) Finally, we treat the thermodynamic properties of the helium as constants independent of temperature and pressure.

If we introduce special units in which r = c = D/4f = 1 (respective dimensions: ML–3, LT–1, L), Eqs. (8.1.1) and (8.1.2) become