Dresner, Stability of superconductors.2002

It will prove convenient in what follows to transform to a frame of reference moving with the piston: z = z – Z(t). Then Eq. (8.6.4) becomes

We shall assume that the second term in the bracket on the left-hand side is small compared with the first—later, we shall discuss this assumption, too. If we drop the second term, we again obtain Eq. (8.6.4) written in terms of z and t, namely,

Equation (8.6.6) is well suited to treatment by the method of similarity solutions, which is described in great generality in Appendix A. Because this method plays such an important role in this book, I shall use it here to solve Eq. (8.6.6) subject to the boundary and initial conditions (8.6.1a) and (8.6.1c). The details of application to a concrete problem may serve as a steppingstone toward full understanding of Appendix A.

We start by exploring the invariance of Eq. (8.6.6) to stretching groups like

where a and d are constants yet to be determined. If we substitute Eqs. (8.6.7) into Eq. (8.6.6), we find that the resulting equations will be precisely Eq. (8.6.6) written in terms of the primed variables if, and only if,

Thus not both of the constants a and d may be chosen at will. The values of a and d are, as we shall see subsequently, determined by the boundary and initial conditions. But for the time being let us consider them arbitrary, save that they obey Eq. (8.6.8).

Now suppose we look for solutions of the form

where y is an as yet undetermined function of the single variable x = z/t 1/d. It is easy to see that Eq. (8.6.9) remains unchanged if we substitute for v, z, and t in terms of v´, z´ , and t´ from Eq. (8.6.7). It is this form that is called a similarity solution.

Since the as yet unknown function y(x) is a function of one variable only, if we substitute Eq. (8.6.9) into the partial differential Eq. (8.6.6), an ordinary differential

When z = 8 or when t = 0, v = 0, according to Eq. (8.6.1a). The first of these

conditions requires that y(∞)= 0; this satisfies the second condition as well.1 Finally, since v(0,t) = nXt n–1, we must have a/d = n – 1 and y(0) = nX. If a/d = n – 1, it

follows from Eq. (8.6.8) that a = 2(n – 1)/(2 – n) and δ = 2/(2 – n). If Eq. (8.6.9) is to represent a disturbance that spreads out with time, d must be > 0, i.e., n must be < 2.

8.7. THE PISTON FORMULA

The stage is now set for the solution of the ordinary differential Eq. (8.6.10). But before we plunge ahead, it pays to imagine what we would do with the solution

y(x) if we already had it. By integrating Eq. (8.6.2) from z = 0 to z = ∞ (N.B.: ∂p/∂ z = ∂p/∂z), we find

x´ = mx

(This is no coincidence and the existence and form of this group could have been predicted from the outset; cf. Appendix A.) It follows then that

(8.7.3)

Since y(0) = nX can be given any value for a fixed y´(0) by varying m, the right-hand side of Eq. (8.7.3) must be a constant independent of y(0) = nX; call it C. Then from Eq. (8.7.1),

Although C is independent of y(0) = nX, it still depends on n because a and d, both of which depend on n, appear in the differential Eq. (8.6.10). The quantity n3/2C, which we henceforth abbreviate as A, varies almost linearly with n and is well approximated by

as numerical solution of Eq. (8.6.10) for several values of n shows (Dresner, 1991, “Theory”). In ordinary units, Eq. (8.7.4) becomes

Except for the numerical value of A, Eq. (8.7.6) has been derived entirely on the basis of group-theoretic arguments.

The two self-consistency conditions, namely that the frictional term in Eq. (8.1.2) be negligible, and that the second term in the bracket in Eq. (8.5.5) be negligible, have been investigated in Dresner, 1991, “Theory,” where it is shown that n must be > 2/3 and p(0,t) < < rc2 for Eq. (8.7.6) to be valid. For supercritical helium at low temperatures rc2 ~ 10 MPa.

8.8. SLUG FLOW

depends on the exponent n, is given by Eq. (8.7.5).

Eq. (8.7.6) applies only as long as the disturbance created by the piston has not yet reached the open end of the hydraulic path. Long after the disturbance has reached the open end of the hydraulic path, we take the helium to be in slug flow (which implies the case n = 1). Slug flow means v = X everywhere. Then, from Eq.

where Dz, is the distance from the instantaneous position of the piston to the open end of the tube. When n ≠ 1, we generalize2 Eq. (8.8.1) to

When faired together, Eqs. (8.7.6) and (8.8.2) represent the pressure rise the piston creates by pushing the cold helium through the conductor and thus constitute the needed solution to the first half-problem.

8.9. THE PROPAGATION VELOCITY

Normal zones expand because heat is transferred from the resistive part of the conductor to the part that is still superconducting. In magnets cooled with boiling helium or potted magnets, the heat is transferred by conduction through the copper matrix, but in cable-in-conduit conductors, it is largely transferred by the expansion of the hot helium. Then, the velocity of normal zone propagation equals the velocity of expansion of the hot–cold front. This last notion implies that the normal zone engulfs no new helium. Therefore, the heated helium consists only of the atoms originally present in the initial normal zone. The picture is thus of a bubble of hot helium expanding against the cold helium that confines it on either side.

We treat the hot helium as a perfect gas having a uniform absolute pressure p and a uniform temperature T. If 2Li is the initial length of the normal zone and 2Lf is its length at a time t, the equation of state of the hot helium is

Eq. (8.9.1) represents the solution to the second half-problem, namely, the state of the hot helium assuming its extent Lffand temperature T are known as functions of the elapsed time t. The two half-problems can now be combined by noting that the pressure p given in Eq. (8.9.1) must equal the pressure p(0,t) + pa, i.e., that the pressure must be continuous across the hot–cold front. If p(0,t) is given by Eq. (8.7.6), if we note that Z = Lf – Li, and if we set h = Lf /Li, we obtain

Here we have used the fact that the density of the cold helium is the same as the initial helium density ρi in the normal zone. If we know T as a function of t (we discuss the function T(t) directly below), we can solve Eq. (8.9.2) to find η(t) and thus the trajectory of the hot–cold front. This must be done numerically, but if the Newton–Raphson method is used, it can easily be done on a programmable pocket calculator,

If p(0,t) is given by Eq. (8.8.2) and we set dZ/dt = (Lf – Li)/t, we find

To utilize Eqs. (8.9.2) and (8.9.3), we need the temperature T as a function of time. A simple, reasonable estimate is the adiabatic hot-spot temperature (cf. Eq. (6.13.2)). The volumetric heat capacity S in Eq. (6.13.2) is that of the copper and

be heated in this way. If the temperature of these fluid elements should rise to the current-sharing threshold Tcs , the strands that they wet become resistive. Then, rather rapidly, quite long segments of the conductor become normal, giving the appearance of a sudden, sharp increase in the propagation velocity. This acceleration of the normal-superconducting front has been dubbed thermal hydraulic quenchback (THQ). Its existence was first discovered by Luongo et al. (1989, “Thermal hydraulic simulation”) in the course of numerical simulations of thermal expulsion from a cable-in-conduit conductor. The partial theory given below for the time of onset of THQ is due to the author (1991, “Theory”). In contrast to the situation before, after the onset of THQ the normal zone engulfs more and more new helium, causing the entire hydraulic path to become normal very rapidly. This adds additional justification for using the quench pressure Eq. (8.2.7) in design.

The first fluid element to reach Tcs is the one next to the hot–cold “piston” because it has the largest velocity and highest pressure of all. In a time dt, the frictional work dW done on a fluid element of length dz next to the piston is

where henceforth we shall use the abbreviation V for dZ/dt, the velocity of the piston. Here the first factor is the shear stress exerted on the fluid by the wetted surface, P is the perimeter of the wetted surface, and Vdt is the displacement of the fluid element. If we divide dW by rAHedz , the mass of the fluid element, and integrate over time, we find

(8.10.2)

where w is the specific frictional work. The temperature rise caused by this work is w/cp. If we add to this temperature rise that caused by compression, we find the overall temperature rise to be

(8.10.3)

where p(0,t) is given by the piston Eq. (8.7.6). When n = 1 we can write Eq. (8.10.3) as

cp DT/c2 = (1/2)(4fV3t/Dc2) + A(∂T/∂p)v rcp (4fV3t/Dc2)1/2 (8.10.4)

which is a quadratic equation for the dimensionless unknown (4fV3t/Dc2)1/2 in which the coefficients are also dimensionless. Let us estimate the size of these

coefficients. Typical values (4.2 K, 1 MPa) of the various parameters are r = 150 kg m-3, cp = 3000 J kg-1 K-1, c = 280 m/s, (∂T/∂p)v = 2 K/MPa, and A = 0.681 ( n

= 1). Then the coefficient A(∂T/∂p)v cp = 0.613; with DT = Tcs – Tb = 1 K, the coefficient cp DT/c 2 = 0.0383. With coefficients of this order, at the positive root of

Eq. (8.10.4), the first term on the right is much smaller than the second. Dropping it, we have as the condition for the onset of THQ

This result means that with the given parameters, the onset of THQ is caused by compression, not frictional dissipation. When n ≠ 1, Eq. (8.10.5) should be replaced by

If we add to the assumed values above, the parameters f= 0.02 and D = 0.4 mm, we then find V3t = 1.53 m3/s2. For velocities much greater than 1 m/s (cf. Fig. 8.4), THQ begins almost at once, supporting in such a case the use of the

conservative quench pressure Eq. (8.2.7).

The left-hand side of Eq. (8.10.6) varies as t(3n/2–1). In the parts of Fig. 8.4 referring to times less than about 2 s, the velocity is decreasing with time, which

means n < 1. For n = 3/4, for example, the exponent 3n/2 – 1 = 1/8. Then, the onset time t varies as (DT)1/(3n/2–1) = (DT)8 .A small change in DT can then be reflected

in quite a large change in the onset time t. In an experiment of finite duration, when t is long, THQ is simply not observed, so that the strong dependence of t on DT can appear as a threshold. Such threshold behavior has been observed in the THQ experiments of Lue et al. (1993; 1994).

Recently, Shaji and Friedberg (1995) have presented a detailed study of thermal hydraulic quenchback.

8.11. HYDRODYNAMIC QUENCH DETECTION

The hydrodynamic disturbances caused by an expanding normal zone can be used to detect an incipient quench by monitoring the efflux of helium from the ends of a hydraulic path (Dresner, 1988). Since we wish to detect a normal zone as early as possible, we are interested in situations in which the velocity of efflux is much

less than the piston velocity. The velocity of efflux veff =t a/dy (Dz/t1/d), where as before Dz is the distance from the instantaneous position of the piston to the open

end of the tube. If the velocity of efflux is to be very much less than the piston velocity, y (Dz/t 1/d) must be < < y(0). Now the asymptotic form of y(x) fo r large x is 6/x2. That y(x) = 6/x2 satisfies Eq. (8.6.10) is easily verified; for a proof that it is the asymptotic form we seek, cf. Appendix A. Substituting this form of y(x) into Eq. (8.6.9) and using Eq. (8.6.8), we find

in special units, or

168 CHAPTER 8

in ordinary units. Using the data given in the previous section for the Westinghouse conductor (c = 280 m/s, D = 0.4 mm, f = 0.02) and taking Dz = 60 m, the half-length of a hydraulic path, we find veff = 33 cm/s at t = 0.5 s. This is roughly twice the ambient flow velocity of 15 cm/s and should be detectable without great trouble. It should be noted that no signal can arrive at the outlet faster than a sound wave, so that no change in the velocity of efflux can be observed until a time t = Dz/c has elapsed. In the foregoing example, this time is 0.21 s.

8.12.RATIONAL DESIGN OF CABLE-IN-CONDUIT CONDUCTORS

The author has evolved a design procedure for cable-in-conduit conductors based on the stability and protection considerations discussed in this and the last chapter (Dresner, 1991, “Rational design”). The goals of this design procedure are the two composition variables f and fco, the volume fraction of copper in the strands and the volume fraction of strands in the cable space, respectively. The procedure is best explained by an example (taken from Dresner, 1991, “Rational design”).

Suppose we wish to design a 4.0-K, 8-T, high-current-density, NbTi/Cu cable-in-conduit conductor for a fusion magnet. We assume the strand diameter to be 0.7 mm, the residual resistance ratio of the copper to be 100, and take the ambient helium pressure to be 5 atm. Since we want high current density, we aim for 20 kA/cm2 over the cable space. We know already that high current density entails high quench pressure, so we choose a hydraulic path length of only 20 m and decide to limit the quench pressure, if possible, to 50 MPa or less.

Figure 8.5 displays a portion of the (fcof)-plane containing a particular triangular region. The region is bounded on the left by a curve that separates compositions (to its left) for which the stability margin is multivalued from compositions (to its right) for which the stability margin is single-valued and equal to the upper stability margin. This curve has been calculated using the scaling law, Eq. (7.6.7).4 The triangular region is bounded on the right by a curve that separates compositions (to its left) for which the maximum quench pressure is < 50 MPa from compositions (to its right) for which the maximum quench pressure is > 50 MPa. This curve has been calculated with Eq. (8.2.7). The triangular region is bounded at the top by a curve above which there is not enough superconductor to carry the current density in the superconducting state. Within the triangle are contours of quench pressure (atm), stability margin (mJ/cm3), and fraction of critical current (dimensionless). We see from the diagram that we can meet the proposed design constraints with f ~0.7 (cu/sc ~ 2.3) and fco ~0.8 (void fraction ~20%), at which point the current density is about two-thirds of critical, the stability margin is ~70 mJ/cm3, and the maximum quench pressure is ~350 atm = 35 MPa.

Figure 8.5. The portion of the (fco f) plane that corresponds to cable-in-conduit conductors that have single-valued stability, a limited quench pressure, and are able to carry a specified transport current. (Redrawn from an original published in Superconducting Materials and Magnets, International Atomic Energy Agency 1991.)

8.13.PERFORATED JACKETS: MODIFIED HYDRODYNAMIC EQUATIONS

The hydraulic path length of 20 m is rather short, and if we hope to maintain a high current density over the cable space, we cannot increase it very much without sending the quench pressure out of bounds. One way to modify the design of the conductor to reduce the quench pressure is to perforate the jacket. Figure 8.6 shows a conceptual arrangement for doing this. The inner cylindrical jacket surrounds the cable and the supercritical helium (region A). Region B, between the two jackets, is space available for pressure relief. The inner jacket is perforated by small holes to bleed helium from region A into region B during a quench.

The following two questions immediately come to mind: (1) How big must the holes be and what must their spacing be to provide substantial pressure relief ? (2) Do the holes seriously affect the heating-induced flow and so reduce the stability margin? To answer these questions we need to modify the hydrodynamic equations (7.2.1–3) to account for the presence of a distributed sink of strength -m [dimensions: ML–3T–1], which is how we shall describe the loss of helium from region A by percolation through the wall (Dresner, 1993, “Reducing”). To make this modification we shall have to derive the hydrodynamic equations from first principles by means of mass, momentum, and energy balances over the fixed control volume lying between z and z + dz in space A.