Dresner, Stability of superconductors.2002
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CHAPTER 5 |
with F given by Eq. (5.2.7), choose S and h in accordance with Steward’s diagram, Fig. 4.23, and use Iwasa–Apgar’s correction to S, if applicable.
5.5. TRAVELING NORMAL ZONES (I)
In the simple theory of Sections 5.1 and 5.2, both the Joule heat source Q and the heat flux q being transferred to the helium are functions only of the metal temperature T. In Sections 5.3 and 5.4 we discussed the complexity introduced when the heat flux q at a point depends not only on the temperature there but also on the time elapsed since the normal-superconducting front passed the point. In certain other circumstances described below, it is the Joule heat source Q that depends not only on the temperature but also on the time elapsed since the normal-supercon- ducting front passed.
These circumstances arise in very large composite conductors in which the matrix and the superconductor are sharply segregated. One conductor, designed for use in very large magnets proposed for superconducting magnetic energy storage (SMES) is shown in Fig. 5.4 (Huang and Eyssa, 1991). In this conductor, the superconductor is confined to the circumference of a large cylinder of aluminum. Other similar conductors have been proposed for use in fusion magnets (Mito et al., 1991) and in space applications (Huang, Eyssa, and Hilal, 1989). Typically, such conductors operate with currents in the range 30–100 kA, are of the order of 2.5–5 cm in diameter, and consist of large blocks of high-purity aluminum in which much smaller superconductors are embedded.
The segregation of the aluminum matrix and the superconductor has the following deleterious effect on the stability of the conductor. When the superconductor is first normalized, the current enters the matrix but is confined to the vicinity of the superconductor. Thereafter, the current diffuses throughout the matrix, tending toward a state of uniform current density. In this uniform state, the Joule power is much lower than at the start.
The relaxation time of current redistribution is typically some tens to hundreds of milliseconds. Thus the excess Joule heat (over the uniform state) appears as a short pulse immediately following normalization. If the conductor is not cryostable, this short heat pulse diminishes the external energy it takes to create a propagating normal zone.
Whereas for small (and therefore flexible) conductors, conductor motion is local, for the very large (and therefore stiff) conductors we are considering here, conductor motion may be spread out over many diameters. Then the quench energy (defined in Section 4.6 as the heat instantaneously deposited at a point that just causes a quench) is no longer useful as a measure of stability. Instead, we use the stability margin, defined as the uniform heat density instantaneously deposited in a long length of conductor that just causes a quench.1 The pulse of excess Joule heat produced during current redistribution reduces the stability margin compared to
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Figure 5.4. A large, high-current conductor with segregated matrix and superconductor for superconducting magnetic energy storage (SMES) (Huang and Eyssa, 1991). (Redrawn from an original appearing in Huang and Elyssa (1991) with permission of the IEEE; © IEEE 1991.)
what it would be if the superconductor were homogeneously distributed over the entire matrix.
In addition to affecting the stability margin, the excess Joule heat affects the velocity of propagation of normal zones; since it is released at the head of the propagating wave, it increases the propagation velocity. But an even more remarkable thing happens: a special kind of propagation takes place in conductors that would be unconditionally cryostable if the superconductor were distributed homogeneously throughout the matrix. When a long normal zone is created in such a conductor, the edges propagate outward continually, driven by the release of excess Joule heat from the newly normalized conductor. In the center, the current distribution eventually becomes uniform, after which local cooling exceeds local heating and the superconductor recovers. All that then remains are two vestigial normal zones at the ends that continue to move outwards. The normal zones, called traveling normal zones (TNZ), are finite in extent and move without change of shape at a constant velocity away from the site of the original disturbance.
Boom and his coworkers (Christianson and Boom, 1984) were the first to recognize the adverse influence of current redistribution on the stability of large
92 CHAPTER 5
SMES conductors. Christianson (1986) and Devred and Meuris (1985) studied the increase in the propagation velocity caused by current redistribution, but neither group of authors suggested the existence of traveling normal zones. To my knowledge, the first hint came from Luongo, Loyd, and Chang (1989); but, ironically, after a long and essentially correct discussion, they concluded their conductor would not recover behind the outward moving fronts. But like their predecessors, they did realize that the excess Joule heat produced during current redistribution would increase the propagation velocity and decrease the minimum propagating current.
As far as I know, the first authors to state explicitly that traveling normal zones were possible are Huang et al. (1990; 1991). My own contribution to this subject (1990) was formulation of a simplified model, discussed below, which could be treated analytically. This model predicts a current threshold below which the conductor is cryostable and above which TNZs occur. At this threshold, the propagation velocity jumps to a finite value rather than rising smoothly from zero. When the current becomes high enough, recovery far behind the fronts no longer occurs and instead of TNZs, there is a single expanding zone.
Lately Mints and his coworkers have presented an analytical treatment of TNZs based on equivalent circuits (Kupferman et al., 1991).
TNZs have been seen experimentally (Pfotenhauer et al., 1991) in a SMES proof-of-principle experiment. In addition to verifying the existence of TNZs, the authors saw the expected jump to a finite velocity at the threshold current.
5.6. TRAVELING NORMAL ZONES(II)
To account for the excess Joule heat released during current redistribution, we must add to the left-hand side of Eq. (4.4.2) the term wU(t+z/v), where w is the excess Joule heat density (i.e., the excess Joule heat released when a unit volume of conductor is normalized) and U(t) dt is the fraction of this heat released during the time interval dt beginning a time t after the superconductor becomes normal. The function U(t) is thus normalized so that U(t)dt = 1; for t < 0, of course, U = 0. Finally, note that the function U has the physical dimensions of reciprocal time.
We again (1) assume that k and S are independent of temperature, (2) assume that Newton’s law of cooling applies, (3) employ the special units of Section 4.7, and (4) use Eq. (4.10.1) for g(t). Then Eq. (4.4.2) becomes
d2t/dx2 – vd t/dx + ai 2 – t+ wU(x/v) = 0, |
x > 0 |
(5.6. la) |
d2t/dx2 – vd t/dx – t = 0, |
x < 0 |
(5.6.1b) |
where the origin of x is the point on the traveling-wave profile at which the superconductor first goes normal, namely, the point at which τ = 1.
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The solution we seek must obey the boundary conditions:
τ(– |
) = 0 |
8 |
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τ(+ |
) = 0 (TNZ) |
|
8 |
τ(0) = 1
as well as the condition
dτ/dx continuous at x = 0
(5.6.2a)
(5.6.2b)
(5.6.2c)
(5.6.2d)
When x < 0, τ = exp(λ+x), where λ+ is again the positive root of Eq. (5.2.3). Therefore, τ(0) = 1 and dτ/dx = λ+. These last two conditions plus Eq. (5.6.2b) in general overdetermine the solution to Eq. (5.6.1a), but a solution is possible for certain particular values of v, one of which is the value we seek. If we introduce the Laplace transform c(p) of t(x) defined by
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(5.6.3) |
the transform of Eq. (5.6. la) can then be solved for c to give |
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c = [p +l+ – v – ai2/ p – wvu(vp)]/[p2 – vp – 1] |
(5.6.4) |
where |
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(5.6.5) |
is the Laplace transform of U.
The singularities of the transform (5.6.4) are a pole at p = 0 and poles at p = λ+ and p = λ_, the roots of Eq. (5.2.3). Now the residue at the pole p = λ+ must vanish if τ is not to rise exponentially with distance behind the normal-supercon- ducting front. Thus when p = λ+, the first bracketed term in Eq. (5.6.4) must vanish. Then we find
w = [l (v2 |
+4)1/2 |
– ai2]/[v l |
+ |
u (v l |
+ |
)] |
(5.6.6) |
+ |
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5.7. TRAVELING NORMAL ZONES (III)
According to the initial-value and final-value theorems (Spiegel, 1965)
94 CHAPTER 5
lim pu(p) = U(0+) |
(5.7.1) |
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(5.7.2) |
lim pu(p) = U( ) = 0 |
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the last equality following from the integrability of U. Thus w becomes infinite in |
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the limits v → 0 and v → |
8 |
2 |
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. If ai < 1 (the condition for unconditional cryostability |
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when g(t) is given by Eq. (4.10.1)), so that the numerator of Eq. (5.6.6) is positive for all v, and w > 0 for all v, then w must have a minimum for some value of v.
Shown in Fig. 5.5 is a sketch of w versus v showing the minimum. For any value of w above the minimum, two values of v are possible. The arguments given so far do not determine which of these occur in the laboratory. But because the velocity at point P´´ decreases with increasing w, we are inclined to think the traveling wave corresponding to P´´ does not occur in the laboratory.The reason for this nonoccurrence is that this traveling wave is unstable against small perturbations. Even if once created, the thermal fluctuations that are always present would immediately destroy it. On the other hand, the traveling wave that corresponds to the intersection P´ is stable and does occur in the laboratory. Of course, these explanatory remarks in no way constitute a proof.
We see then that as w increases, a point suddenly occurs at which propagation of TNZs is possible and the velocity jumps suddenly to a finite value. This behavior has been observed in the laboratory (Pfotenhauer et al., 1991). Furthermore, right
at the threshold dv/dw = |
8 |
; a hint of this behavior exists in the experimental data, |
but the spacing of the experimental points is wide and the point is still largely moot.
Figure 5.5. An auxiliary sketch of the dependence of w, the excess Joule heat density, on the propagation velocity v.
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In strict point of fact, Eqs. (5.6. la and 5.6.1b) do not represent the situation we
are trying to model as we can see in the following2way. If we apply the final-value |
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theorem (5.7.2) to Eq. (5.6.4) we see that t( )= ai |
rather than zero as the boundary |
8 |
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condition (5.6.2b) requires. We presume that the temperature distribution far behind the front does not greatly affect the motion of the front. To improve the model for TNZs, we would need a third equation like Eq. (5.6.1 b) that would apply for x > X, where X is the value of x at which τ becomes 1 again.
It has probably not escaped the reader’s notice that the model as it now stands is correct for ai 2 > 2. For, since ai 2 = 2 is the condition for cold-end recovery based on Eq. (4.10.1), there would be no recovery far behind the front. But then the
argument that w has a minimum as a function of v no longer applies since the |
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numerator of Eq. (5.6.6) is not always positive. When ai |
2 |
> 2, w approaches |
8 |
as |
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v → 8 but approaches –8 as v → 0. Of course, negative values of w are unphysical. When w = 0, it follows from Eq. (5.6.5) that
v = (ai2 – 2)/(ai2 – 1)1/2 |
(5.7.3) |
This is the same result as that of Eq. (5.2.6) except that the transition temperature occurs at t = 1 instead of t = 1 – i/2, so that now C = ai2 – 1. When w > 0, the value of v one calculates from Eq. (5.6.6) is always greater than that given by Eq. (5.7.3).
It is not possible to go further with general arguments, so next we turn to the calculation of w and U(t), following the method of Dresner, 1991, “Excess heat”.
5.8.THE EXCESS JOULE HEAT DUE TO CURRENT REDlSTRlBUTION
The equations that govern the diffusion of current in a conducting medium are the two Maxwell equations
D x B = µ0J |
(5.8.1) |
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D x E + |
= 0 |
(5.8.2) |
Ohm’s law |
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E= rJ |
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(5.8.3) |
and the equation of conservation of current |
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D · J = 0 |
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(5.8.4) |
The resistivity r of the matrix is a constant independent of J. From these equations it follows at once that
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CHAPTER 5 |
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(5.8.5) |
Let the axes be so chosen that the z-axis points along the conductor in the direction of flow of the transport current I. Let W denote the cross sectional area of the conductor in the (x,y)-plane and let Wo denote the area of W in which the transport current is initially confined. At t = 0, imagine this confinement to be abrogated. The transport current then diffuses transversely, tending toward a state of uniform current density. During this redistribution, the current density J (now the z-component of J) obeys the diffusion equation
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(5.8.6) |
We wish to solve Eq. (5.8.6) under the boundary and initial conditions |
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J(r,0) = I/Ω o in Wo, 0 elsewhere in W |
(5.8.7) |
J(r ,8) = I /W everywhere in W |
(5.8.8) |
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(5.8.9) |
where dw = dx dy and r is the two-dimensional radius vector (x,y).
It proves useful to expand J in terms of certain eigenfunctions fk of the |
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Helmholtz equation in W, namely, those defined by the equations |
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D2fk +ak2 fk =0 in W |
(5.8.10) |
n · Dfk = 0 on S, the boundary of W |
(5.8.11) |
where n is the outward normal to S. Thus we set |
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(5.8.12) |
which satisfies Eq. (5.8.6). Equation (5.8.12) also satisfies condition (5.8.8). It follows by well-known arguments (Courant and Hilbert, 1953) that
(5.8.13)
(5.8.14)
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With the help of Eq. (5.8.13) we can see that Eq. (5.8.12) satisfies condition (5.8.9). We choose the coefficients to satisfy condition (5.8.7), which can be written
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(l /Wo)g(Wo)= l /W + SAk fk |
(5.8.15) |
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k = 1 |
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whereg(Wo) is the characteristic function of Wo, that is, the function that is 1 inside Wo and 0 outside. Using Eqs. (5.8.13) and (5.8.14) we find
Ak = |
(5.8.16) |
The Joule power expended per unit length of conductor is |
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(5.8.17) |
W |
k = 1 |
The first term on the right in Eq. (5.8.17) is the Joule power produced when the current density is uniform throughout W. The second term is the excess Joule power.
When integrated over t from 0 to |
, the second term gives the excess Joule heat per |
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unit length of conductor. Dividing the result by W gives w, the excess Joule heat density:
k =1
(5.8.18)
k = 1
The summation on the right-hand side of the last equation is a geometric factor, which has been calculated by the author (1991, “Excess heat”) for a variety of situations. We consider here only the case in which W is a circle of radius R and Wo is an infinitely thin annulus at its circumference.
5.9.THE SPECIAL CASE OF A CYLINDRICAL CONDUCTOR
In a cylinder of radius R, for current distributions that depend only on the radius (azimuthal symmetry)
fk =Jo (gkr/R ) |
(5.9.1) |
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CHAPTER5 |
ak = gk/R |
(5.9.2) |
where Jo is the Bessel function of the first kind of order zero and gk is the kth root of J1, the Bessel function of the first kind of order one. Then
[fk ,fk ]= pR2J02 (gk ) |
(5.9.3) |
If Wo is the annulus R1 < r < R2, then |
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[fk ,g]=(2pR2/gk )[(R2/R)J1(gk R2/R) – (R1/R)J 1(gk R1/R)] |
(5.9.4) |
If we set and R2 = R and R1 = R( 1 – e), we find,2 in the limit as e 0, |
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w = (m0J2/2p2R2)S(1/gk2) = mo I 2/16p2R2 |
(5.9.5) |
k =1
The exact value of the sum in the middle term of Eq. (5.9.5) is 1/8, as can be shown by a method of Euler’s (Dresner, 1991, “Excess heat”).
5.10.COMPARISON WITH EXPERIMENT OF PFOTENHAUER ET AL.
According to Eq. (5.8.17), the function U(t) is a sum of decaying exponentials |
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with relaxation times tk= mo/2rak2 = moR |
2/2rgk2 .If we ignore all but the lowest |
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mode, then U(t) = exp(–t/t1)/t1 where t1 |
= moR2/2rg21 and g1 = 3.83171. In the |
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special units of Section 4.7, t |
= m |
o |
R2hP /2rg2SA . When U(t) has this value, u(p) = |
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1 |
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1 |
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(1 + pt1)–1 and Eq. (5.6.6) becomes (still in special units) |
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w = [l+(v 2+4)1/2–ai2][1/v l++t1] |
(5.10.1) |
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Let us use this formula to analyze the experiment reported by Pfotenhauer et al. (1991). Their conductor consisted of a 2.54-cm-diameter rod of high-purity aluminum in the circumference of which eight 2.8-mm-diameter NbTi/Cu conductors were embedded. Their experiment was carried out at Tb = 2.5 K in a self-field of about 1.25 T (1 ~ 50–60 kA). At that field, the critical temperature Tc of NbTi is about 8.5 K, so that Tc – Tb = 6.0 K. Between the temperatures Tc and Tb, the specific heat of aluminum varies between 0.1 and 0.8 J kg-1 K-1, which is a substantial variation. We use the mean value of 0.45 J kg-1 K-1; multiplying by the density of aluminum (2700 kg/m3), we thenfind S = 1200 J m-3 K-1and S(Tc – Tb) = 7200 J m-3.
In their article, Pfotenhauer et al. do not specify the residual resistivity of their aluminum but merely state that it is high-purity aluminum. We guess a residual resistivity ratio rRT/r of 500 and find (since rRT = 2.4 mW-cm)that r = 4.80 x 10-11 W- m. The Wiedemann–Franz law then gives k = 2810 W m-1 K-1 at the average
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temperature (Tc + Tb)/2 = 5.5 K. The normal-state heat flux with a uniformly distributed, 50-kA current, Qn = rI2/AP , is then 2970 W/m2. The relaxation time t1 = 0.144 s. According to Eq. (5.9.5), when I = 50 kA, the excess Joule heat density w = 1.23 x 105 J/m3. Then the average excess Joule heat flux during the time t1, wA/Pt1, = wR/2t1, is 5420 W/m2. Thus the total heat flux during the first 0.144 s is roughly 8390 W/m2. Reference to Steward’s diagram, Fig. 4.23, then shows that
the heat transfer coefficient should be that of film boiling. Accordingly, we assume h = 1000 W m-2 K-1.
The special unit of time, SA/hP, is 7.62 x 10-3 s; the special unit of length, (kA/hP)1/2, is 0.134 m; the special unit of velocity (khP/A)1/2/S is then 17.5 m/s; and the special unit of energy density, S(Tc – Tb), is 7200 J/m3. Then, in special units, w = 17.1 (I=50kA)and t1 = 18.9.Finally,ai2= 1 when I = 71.1 kA(unconditionalstability).
Eq. (5.10.1) can easily be evaluated with a programmable hand-held calculator. Remembering that both w and ai 2 scale as I2, we find after a little trial and error (improved between steps by interpolation) that when I = 67.3 kA, the value of w is equal to the minimum value obtained from Eq. (5.10.1). Thus the threshold for the creation of TNZs is 67.3 kA. At this current, ai 2= 0.896 and the velocity v in special units is 0.217 or in ordinary units 3.8 m/s. At I = 70 kA, ai2 = 0.970 and v = 8.3 m/s, and at I = 75 kA, ai2 = 1.11 and v = 12.9 m/s.
Comparison with the experiment of Pfotenhauer et al. shows only very rough agreement. Their measured threshold was about 55 kA; slightly beyond that their propagation velocity reached values around 20 m/s. The small value of v at the threshold that we have calculated is not so worrisome in view of our expectation
that dv/dw = |
, especially since the experimental points v, plotted versus I, show a |
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downward concavity just beyond the threshold. But well away from the threshold, our calculated velocities are still low by about a factor of 2.
Perhaps the most important conclusion that can be gleaned from the calculations follows from the comparison of the threshold current for the appearance of TNZs (67.3 kA) with the current for unconditional stability (71.1 kA). There is for this conductor a loss of only 5% in threshold current, which seems a rather slight penalty to pay to avoid the cost of distributing the superconductor uniformly throughout the matrix.
Notes to Chapter 5
1It follows from the discussion of Fig. 4.8 in the third paragraph of Section4.6, that the stability margin of a small, pool-cooled conductor that is not cryostable is the volumetric enthalpy of the conductor between Tb and T1.
2The result, Eq. (5.9.5), can teach us a valuable lesson of an unexpected sort: The result w ~ moI2/R2 can easily be obtained from dimensional analysis, the only relevant variables being w itself, mo, I, R, and the matrix resistivity r. It is a common (though in fact indefensible) practice to assume the
dimensionless coefficient in such dimensional formulas to be close to 1. In the case at hand, that constant, 1/16p2, equals 6.333 x 10–3, so that assuming it to be close to 1 makes anerror of more than
two orders of magnitude.