Dresner, Stability of superconductors.2002

helium of thickness d in which the temperature rise is uniform at the value given in Eq. (4.11.2), then S ∆ T d = qt, so that

Schmidt postulates that takeoff occurs when the heat transmitted qt equals the latent heat Lδ of the heated layer, for at that time he supposes that there is enough heat present to vaporize the entire heated layer. Thus

(k/S = 2.84 x 10-8 m2/s and L = 2.59 J/cm3 according to Arp and McCarty (1989). The curve in Fig. 4.24 corresponds to a value of qtf1/2 = 51 mW cm-2 s1/2 ; its rather

good agreement with the experimental facts supports Schmidt’s picture of what is going on. (N.B.: Schmidt (1978; 1982) first used a factor of 2, later a factor of π/2 in Eq. (4.11.4) instead of the factor (π/4)1/2.)

Armed with Schmidt’s formula (4.11.4), we can attack the following problem. If an intense disturbance lasts for a short time ∆ t, what is the maximum heat it can produce without causing a transition to film boiling? The maximum allowable heat input per unit area H is clearly that which just makes tf = ∆ t. Now since q = H/∆ t,

Eq. (4.11.5) is based on neglect of any Joule heat produced during transient heat transfer. This is a satisfactory assumption for short enough times ∆ t because H varies as (∆ t)1/2 whereas the Joule heat varies as ∆ t. Table 4.1 shows for several values of ∆ t the heat H calculated from Eq. (4.11.5) and the maximum Joule heat per unit surface area (ρcuJ2A/fP)∆ t for the numerical example of Section 4.8. In this example, for the times given, the neglect of the Joule heat is roughly justified.

The time ∆ t can be estimated for the conceptual model of conductor slippage introduced in Section 4.8. If the conductor slips freely, its motion is restrained only

(b2/a)(δ/Y )1/2, where d is the density of the conductor (8960 kg/m3 for copper,

Southwell, 1969). This time turns out to be about 9 µ s for b = 5 mm and a = (π/4)1/2D = 0.71 mm. The total Joule heat H(A/P)b produced in a 5-mm span

is then about 1.5 µ J. This heat can be added to the MPZ energy 16.6 µ J, since the latter value is predicated on the film-boiling heat transfer coefficient. In this case, transient heat transfer slightly improves the stability of the conductor; its neglect is thus slightly conservative.

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propagation velocity in order to determine its dependence on the various parameters of the conductor.

The basic equation from which we start is Eq. (4.4.2), which describes the left-hand edge of a normal zone, i.e., the edge that propagates from right to left. Again, to obtain a solvable problem, we use Newton’s law of cooling and assume k and S to be independent of temperature. We again introduce the variable s = k(dT/dz) and finally employ the special units of Section 4.7. The result is the first-order ordinary differential equation

the solution of which must obey the boundary conditions s(0) = 0 and s (ai2) = 0 (since τ2 = ai2).

Altov et al. (1973) were the first to solve the problem just formulated using finite differences. Later, the author gave an analytical solution to the problem (1979, “Analytic Solution”). The results of these exact calculations (in the special units of

Section 4.7) is the functional dependence of v on a and i: v = F (a,i). Remembering that (rcuJ 2/f )/(hP tc/A ) = ai 2, we can write

Shown in Fig. 5.1 is the second factor in Eq. (5.1.3) plotted versus i with a as a parameter. It approaches 1 when i approaches 1 for all a. When a →∞,i.e., when h →0 (an uncooled superconductor), F (a,i)/2a1/2→ ci 1/2/2, where c is related to i by the equation (Dresner, 1980, “Propagation”):

The curve ci1/2/2 is the uppermost curve shown in Fig. 5.1. It alone extends down to i = 0. All the other curves cross the axis v = 0 at values of i satisfying the equal-area requirement ai2 = 2 – i. Between this value of i and the Stekly value, which satisfies ai2 = 1, initial normal zones collapse by cold-end recovery, the flanks of the zone moving inwards with the (negative) velocity v. As i approaches the Stekly value from above, v →– ∞.Below the Stekly value of i, all points of the initial normal zone recover simultaneously, and the notion of propagation is inapplicable. (N.B.:

In evaluating the left-hand side of Eq. (5.1.4), use the principal value of the arctan lying in the interval (0,p).)

Figure 5.1. The second factor in Eq. (5.1.3) plotted versus i: it gives the dependence of thepropagation velocity on the current.

The first factor on the right-hand side of Eq. (5.1.3) (callit v*)can be simplified for a matrix material obeying the Wiedemann–Franz law, Eq. (2.9.1). Since the thermal conductivity of the matrix is typically >> than that of the superconductor, k/f= kcu and the first factor in Eq. (5.1.3) becomes

where we have given the quantities ρcu and kcu, which have been assumed independent of temperature, the values they have at the ambient temperature Tb. Using the figures given in the example of Section 4.8, we find v* = 86.8 m/s; since in this example a = 27.4 and i = 0.6, we see from Fig. 5.1. that F(α,i)/2α1/2= 0.32, so that v = 27.8 m/s. This single example gives some idea of the order of magnitude of the propagation velocity. But it can vary widely depending on the circumstances. In the new high-temperature superconductors, it can be as little as a few mm/s, whereas in the cabled conductor of the Superconducting SuperCollider 17-m dipoles, it was as much as several hundred m/s.

5.2.APPROXIMATE CALCULATION OF THE PROPAGATION VELOCITY

The implicit relation between c and i in Eq. (5.1.4) is inconvenient when one wishes to undertake repetitive calculations even on an electronic computer. The analytic solution of reference (Dresner, 1979, “Analytic solution”) for F (a,i) involves an even more complicated relation than that of Eq. (5.1.4) and greater inconvenience. A simple, explicit, and quite accurate approximate result for F (a,i) can be arrived at by replacing in Eq. (5.1.1) the three-part curve (4.7.2) for g(τ) by the two-part curve

This replacement has the desirable property that the differences in area of the right-hand and left-hand stippled regions in Fig. 4.15 are the same for the three-part g(τ) of Eq. (4.7.2) as for the two-part g(τ) of Eq. (5.2.1), namely, αi2(αi2 – 2 + i)/2. Thus the minimum propagating (v = 0) value of i is the same in both cases.

When τ < 1 – i/2 and g = 0,

(Remember, we are calculating the velocity of the left-hand edge of a normal zone for which s = k(dT/dz) > 0.) When t > 1 – i/2 and g = 1,

The partial solutions (5.2.2) and (5.2.4) satisfy the boundary conditions s(0) = 0 and s(αi2) = 0, as they should. They must be equal at t = 1 – i/2, and this condition gives the result

Since this result obtains in special units,

Figure 5.2 shows a plot of the approximate value of F(α,i)/2α1/2 plotted versus i with the same values of as a parameter as used in Fig. 5.1. The agreement is excellent over most of the range of i, with serious deviations only near i = 1. According to the author (Dresner, 1979, “Analytic solution”), considerable im-

provement in agreement can be achieved by applying the empirical factor 1 + 0.561a–1.45 to the result calculated from Eq. (5.2.6).

Figure 5.2. The approximate value of the second factor in Eq. (5.1.3) plotted versus i [cf. Eqs. (5.2.6) and (5.2.7)].

5.3. COMPARISON WITH EXPERIMENTS OF IWASA AND APGAR

The curves in Fig. 5.1 can be compared with experimental propagation velocities in the following way. The intersections of the curves with the i-axis are the dimensionless minimum propagating currents corresponding to the various values of the Stekly number a. Experimental values of the minimum propagating current can thus be used to determine a. Then we can use the value of a so determined to calculate the propagation velocity as a function of i using Eq. (5.1.3) and the data in Fig. 5.1 or Eq. (5.2.7).

Straightforward as this procedure seems, it fails to reproduce the experimentally measured propagation velocities. The experimental and calculated curves have the same intercept on the i-axis, of course, but the theoretical curve is usually several times steeper than the experimental curve. This difficulty is not eliminated by use of a three-part boiling curve and temperature-dependent thermophysical properties, as extensive numerical calculations have shown (Dresner, 1976).

I gave a hint at the resolution of this difficulty, when I became convinced “that steady-state heat transfer coefficients are inadequate to describe the growth of normal zones” (Dresner, 1976). Accordingly, I introduced ad hoc corrections for transient heat transfer that raised the heat transfer coefficient at the propagating wave front but allowed it to approach the steady-state value far behind. Agreement was much improved, and I concluded this work by recommending “a velocity-de- pendent correction that increases with velocity.” Later, I used a velocity-dependent correction that raised heat transfer for expanding zones and lowered it for collapsing zones (Miller et al., 1977). Such a correction slowed both expansion and collapse; thus it decreased the slope of the curve of v versus i in the neighborhood of the minimum propagating current.

This idea proved right: experiments by Iwasa and Apgar (1978) identified the growth and collapse of the vapor film blanketing the conductor surface as the source

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of the velocity-dependent heat transfer. When the normal zone is expanding, the vapor film forms at the advancing wave front. The latent heat of formation is absorbed from the metal, slowing the rate of advance of the front. When the normal zone is contracting, the vapor film condenses at the retreating wave front. The latent heat of the film is then released to the metal, slowing the rate of retreat of the front.

Iwasa and Apgar measured the instantaneous temperature of a helium-cooled copper plate as a function of heat flux. They expressed the heat flux as a sum of two terms, qs (t), the steady-state term, and a(τ)(dT/dt), a term proportional to the time rate of change of the temperature. From their measurements, they obtained the temperature-dependent coefficient a (τ). By fitting curves of propagation velocity versus transport current measured by Miller et al. (1977), I was also able to obtain the coefficient a (τ) (Dresner, 1979, “Transient heat transfer”). The agreement between these latter values and those directly measured by Iwasa and Apgar was excellent. Tsukamoto and Miyagi (1979) and Nick, Krauth, and Ries (1979)

correction, namely, a(τ)(dT/dt), makes correcting the propagation velocity relatively simple. If one adds this term to q in the fundamental Eq. (4.4.1), one finds that the term a(τ)(dT/dt) can be combined with the left-hand side and has the sole effect of increasing the volumetric heat capacity S by Pa/A. One can therefore use the formulas previously derived, e.g., Eq. (5.1.3), merely by increasing S.

5.4. EFFECT OF TRANSIENT HEAT TRANSFER

The formation and decay of the vapor film is a relatively slow process and the work of Iwasa and Apgar applies only for relatively long transits of the front past a fixed point. In the experiments of Miller et al. (1977), which could be fitted well using the Iwasa–Apgar correction, this transit time was of the order of 10 ms. Judging from Steward’s data in Fig. 4.23, we expect the heat transfer coefficient that controls the motion of the front to be that of film boiling. It is worth noting that the Iwasa–Apgar correction affects both the positive (propagation) and negative (cold-end recovery) velocity.

When the time of transit is much shorter than 10 ms, we expect the heat transfer coefficient that controls the motion of the front to be higher than that of film boiling because of the transient heat transfer processes discussed in Section 4.11. In the experiments of Funaki et al. (1985), the transit time was of the order of a few tenths of a ms. If we fit Eq. (5.1.3) to the data of Funaki et al. by suitably choosing the heat transfer coefficient, we find that it lies in the range 0.5–1.5 W cm-2 K-1 when v is in the range 5–20 m/s. This is larger than the film boiling heat transfer coefficient by roughly a factor of 10 and is typical of the lower left-hand corner of Steward’s diagram, Fig. 4.23.