Dresner, Stability of superconductors.2002

Figure 7.1 . Sketches of the internally cooled conductors of the IEA Large Coil Task (Beard et al., 1988). a) Westinghouse Conductor. Features Nb3Sn transposed compacted cable in a JBK-75 stainless steel sheath, 17.6 kA at 8 T, 20.8 x 20.8 mm. The compaction ratio provides mechanical support for internal radial loads while still permittingaxial slippage during winding, minimizing strain. b) Swiss Conductor. Features NbTi compact high strength solder filled around a central cooling tube, 13 kA at 8 T, 18.5 x 18.5 mm. Low mechanical hysteresis could be demonstrated in low temperature test. Key factors in its choice are low leak risk and quench pressure. c) Euratom Conductor. Features roebel cabled around a kapton insulated stainless steel core, then enclosed in a stainless steel sheath, 11 kA at 8 T, 40 x 10 mm. The strands are fixed mechanically by soft solderingonto the CrNi core with high resistance solder. (Redrawn from an original appearing in Dresner (1984, “Superconductor”) with permission of ButterworthHeinemann, Oxford, England.)

heating of the conductor cannot be extinguished and a quench ensues. Internally cooled superconductors, especially cable-in-conduit conductors, are thus not truly cryostable but rather metastable.

The item of central interest regarding a metastable system is how great a perturbation it takes to produce instability. Hoenig, Iwasa, and Montgomery (Proceedings, 1975) proposed the use of the stability margin (introduced in Section 5.5) as a figure of merit, and it has become the generally accepted standard. As a reminder to the reader, I state the definition again: The stability margin is the uniform heat density instantaneously deposited in a long length of conductor that just causes a quench. Most workers express the stability margin in J/cm3 of conductor, i.e., J/cm3 of metal excluding interstitial helium. This standard has been chosen not so much because uniform perturbations are expected but rather because it provides a simple basis for the comparison of different conductors.

Hoenig et al. (1975, Proceedings) pointed out that the helium in a hollow conductor like the Swiss conductor in Fig. 7.1 would have to flow very fast in order to provide adequate interfacial heat transfer. Only high-speed turbulent flow could provide a large enough heat transfer coefficient to result in a high stability margin. High-speed turbulent flow requires a large pressure drop and can dissipate substantial pumping power at cryogenic temperatures. This dissipation creates very large room-temperature refrigeration loads. To reduce the pressure drop and the pumping power while preserving stability, Hoenig and Montgomery proposed using conductors with very large cooled surfaces, which could be obtained by subdividing the superconductor into many fine strands.

The germinal work of Hoenig and his coworkers spawned a profusion of studies, most of them numerical, of how best to reduce pumping power while preserving stability margin. (Dresner, 1980, “Stability” reviews these early studies.) Most presupposed that the helium flow was imposed by external pumps and that the imposed flow remained unaffected by the heat produced in the normal zone. This early work has become obsolete because, as it turns out, the heat produced in the normal zone does profoundly affect the helium flow. The first hint of this came from an experiment of Iwasa, Hoenig, and Montgomery (1977) in which they measured the stability margin as a function of imposed flow rate. To their (and everyone else’s) surprise, the stability margin was nearly independent of the helium flow rate right down to zero flow! Hoenig and his coworkers (1977; 1979; 1979) as well as others (Miller et al., 1979 and 1980, “Stability”; Lue et al., 1980) confirmed this observation in additional experiments. This meant that vigorous pumping would not be needed to ensure adequate stability.

Iwasa, Hoenig, and Montgomery ascribed recovery in stagnant helium to transient conduction of heat in the supercritical helium. But a drawback of this explanation soon arose out of an experiment of Lue, Miller, and Dresner (1978), who were studying vapor locking in a triplet of superconducting strands sheathed in a stainless steel tube and cooled with stagnant, saturated helium. When the ends

of the stainless steel tube were open, the stability margin was high, but when the ends of the tube were closed, the stability margin fell more than tenfold. Such behavior is incomprehensible if good recovery in stagnant helium is fostered by transient conduction. Instead, it suggests that the good recovery is caused by transient flow induced in the helium by its thermal expansion as it is heated. Plugging the ends of the tube suppresses such flow and strongly diminishes the stability margin.

Strong induced flows had already been suggested by others: Arp (1979) and Krauth (1980) reported calculated induced flows, the Reynolds number of which reached instantaneous values ~ 10 5. In a cable-in-conduit conductor with a hydraulic diameter ~ 1 mm, a Reynolds number ~ 105 corresponds to flow velocities of several meters per second. Such high flow across the heat transfer surface substantially increases the heat transfer through the surface. The calculations of Arp and Krauth were carried out numerically. In keeping with the spirit of this book, which is to obtain simple, widely applicable analytic approximations wherever possible, we follow here an approach of the author’s (1979, “Heating-induced flow”) based on applying Riemann’s method of characteristics to the linearized equations of compressible flow.

7.2.THE ONE-DIMENSIONAL EQUATION OF COMPRESSIBLE FLOW

The full-blown one-dimensional equations of compressible flow in a tube (the continuity, momentum (Euler’s), and energy equations) are three coupled timeand space-dependent partial differential equations. They must be supplemented by the equation of state of the fluid, and together these four equations present a formidable obstacle to analytic solution. However, analytic solutions are attainable in two limiting cases described below.

When the helium in a cable-in-conduit conductor is subjected to a pressure gradient, its motion is restrained by inertia and by friction with the walls and the cable. In the earliest stages of motion, friction with the walls can be neglected. Then the motion can be treated by the same techniques used to analyze motion in a shock tube. In the late stages of the motion, when the disturbance has spread over many tube diameters, the chief restraining force on the fluid is friction, and the inertia of the fluid may be neglected. In both of these limiting cases, analytic solutions are attainable.

The one-dimensional equations of compressible flow in a tube are the following (Taub, 1967):

where r is the fluid density, t is the time, z measures distance along the tube, v is the flow velocity, p is the pressure, F is the frictional force per unit mass of fluid, u is the internal energy per unit mass of fluid, and q is the external (Joule) power input into a unit mass of fluid. The total derivatives dv/dt in Eq. (7.2.2) and d(u + v2/2)/dt in Eq. (7.2.3) are so-called hydrodynamic derivatives, defined by d( )/ = ∂( )/¶t + v∂( )/∂z. The frictional force F = 2fv2/D, where f is the Fanning friction factor and D = 4AHe/P is the hydraulic diameter. Here AHe is the area of helium in the cable cross section and P is the perimeter wetted by helium.

Although the frictional force F appears in Eq. (7.2.2), the momentum balance equation, it does not appear in Eq. (7.2.3), the energy balance equation. The reason for this is that while the frictional force on the fluid decelerates it, i.e., destroys momentum, it does not destroy energy, but merely converts kinetic energy of flow to an equal amount of heat.

If we multiply Eq. (7.2.2) by v and subtract it from Eq. (7.2.3), we obtain

where w = 1/ r is the specific volume of the fluid. (The word specific means per unit mass .) Now, du + p dw = T ds, where T is the temperature and s the specific entropy of the fluid (this is the second law of thermodynamics). Thus Eq. (7.2.6) becomes

which says that the increase in entropy is caused by (1) the conversion of kinetic energy into heat (dissipation term Fv) and (2) heat input into the fluid (term q).2

Now we use Eq. (7.2.7) and the thermodynamic identity

where c is the sonic speed, b is the volume coefficient of thermal expansion, r(∂ w/∂T)p, and cp is the specific heat, to eliminate dp/dt from Eq. (7.2.1). The result is

Equations (7.2.9) and (7.2.2) comprise two partial differential equations in p and v for the determination of the pressure and flow velocity of the fluid. In strict point of fact they are not enough. As time progresses and p and v change, s also changes (according to Eq. (7.2.7)). The change in r must then be calculated using Eq. (7.2.8), and finally the change in T must be calculated from the equation of state. But in the problems we shall treat approximately, Eqs. (7.2.9) and (7.2.2) will be enough. (N.B.: Since p enters Eqs. (7.2.9) and (7.2.2) only in derivatives, we henceforth interpret p as the pressure rise.)

7.3. INDUCED FLOW IN A LONG HYDRAULIC PATH

In the very earliest stages of heating-induced flow, the frictional retarding force can be neglected (shock-tube approximation). Accordingly, we set F = 0 in Eqs. (7.2.2) and (7.2.9). Next we linearize them by dropping second-order terms:

If we now ignore changes in r, c, b, and cp, we can introduce special units in which r = c = brc 2q/cp = 1 (dimensions: ML-3, LT–1, and ML-1T-3, where M is mass, L

is length, and T is time). Then Eqs. (7.3.1) and (7.3.2) become

The quantity v ± p increases by ± dt as we traverse a segment of a (linear) characteristic dz = ± dt.

In the absence of heat source q,

The quantity v ± p is constant along a linear characteristic dz = ± dt.

Figure 7.2. The wave diagram for a uniformly heated zone of length b in the interior of a very long pipe.

These statements can be used to find the velocity v and the pressure rise p as a function of position z and elapsed time t by geometric means using a figure called (in gas dynamics) a wave diagram. The abscissa in a wave diagram is z and the ordinate is t. Fig. 7.2 is the wave diagram for a uniformly heated zone of length b in the interior of a very long pipe (so long, in fact, that conditions at the ends of the pipe (open, closed, restricted) do not matter). The characteristics dz = ± dt are lines of slope ± 1; those with positive slope are called positive characteristics and those with negative slope are called negative characteristics. According to the statements above, we know how the quantity R+ = v + p varies along a positive characteristic and how the quantity R_ = v – p varies along a negative characteristic. Following these quantities, called Riemann invariants in gas dynamics,4 allows us to find v and p at any point.

Consider, for example, the point Q in the interior of the heated zone for any time t > b. The positive and negative characteristics through Q intersect the z-axis (t = 0) at points outside the heated zone. Since v = p = 0 when t = 0, we see by

Figure 7.5. The wave diagram for a uniformly heated zone of length b with open ends showing two points A and E symmetrical with respect to the quarter-point at two times an interval b/2 apart.

Thus the pressure reverses sign every half-cycle. Furthermore, to get the velocity profile at any time (for z > 0), subtract the velocity profile at a time b earlier from the line through the symmetry center having slope 2.

Fig. 7.5 shows two points A and E symmetrical with respect to the quarter-point at times b/2 apart. Using a procedure similar to that above we find that

These equations mean that to get the velocity profile at any time add to the pressure profile at a time b/2 earlier the line of slope –1 through the end of the heated zone and then reflect the resulting curve around the quarter-point. To get the pressure profile at any time subtract the velocity profile at a time b/2 earlier from the line of slope 1 through the origin and then reflect about the quarter point.

In the triangle in Fig. 7.3 corresponding to triangle 1 in Fig. 7.2, v = 0 and p = t, whereas in triangle 2, v = t + z – b/2 and p = b/2 – z. Then using the rules just derived, we can show how the velocity and pressure profiles develop with time. Fig. 7.6 shows these profiles at intervals b/2 over a full cycle of duration 2b. The maximum velocity vmax = b and the maximum pressure pmax = b/2 in special units. Note that these maxima occur at different times. In ordinary units,