Dresner, Stability of superconductors.2002

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6

Uncooled Conductors

6.1. THE BIFURCATION ENERGY

Strictly speaking, the title of this chapter is a misnomer because all superconductors must be cooled in order to function. But in certain kinds of magnets, most prominently those potted in epoxy, the rate of heat transfer from the conductor to the coolant is slow enough that the issue of quench or recovery following a thermal perturbation is decided as though there were no cooling at all. The mechanism of recovery is longitudinal conduction of heat along the matrix away from the normal zone. In common parlance such magnets are called uncooled.

With the advent of the high-temperature superconductors, particularly the BSCCO conductors, it became possible to operate magnets in the 20–40 K range. Such magnets can be cooled with helium gas. It may happen then that the heat capacity of the gas is small compared with that of the conductor and the magnets may be treated as uncooled in determining their stability. Conductively cooled magnets of the kind recently proposed by Laskaris (1990; Herd et al., 1993) can also be treated as uncooled in determining stability.

As in Chapter 4, we shall focus on the quench energy as the measure of stability. But now, when there is no cooling, there is no minimum propagating zone like that pictured in Fig. 4.11. For when q = 0, it is impossible to fulfill Eq. (4.6.2) since kQ > 0. Thus although some initial conditions (corresponding to large heat depositions) lead to quenches and others (corresponding to small heat depositions) lead to recovery, there is no steady-state solution that separates them. We must therefore turn to the time-dependent heat balance Eq. (4.4.1) without the term qP/A, namely:

+ QP/A

(6.1.1)

to decide whether a given initial condition leads to recovery or to a quench.

The initial condition of greatest interest to us is that corresponding to an instantaneous pulse of energy E per unit area of the plane z = 0 at time t = 0. (The

101

102 CHAPTER 6

pulse energy must be expressed as an energy per unit area because Eq. (6.1.1) is a one-dimensional heat balance equation. In two dimensions, the pulse energy would be expressed as an energy per unit length, and only in three dimensions would it be expressed simply as an energy.) If the pulse is large, we expect a quench; if it is small, we expect recovery. What interests is the value of the pulse energy that divides the two regimes.

Again we take QP/A to be given by Eqs. (4.6.3) and (4.6.4); again we let t = T – Tb be the temperature rise; again we assume k and S are independent of temperature; and again we introduce special units in which k = S = tc = 1. Now since h does not appear in the partial differential Eq. (6.1.1), we can complete our system of special units by taking QnP/A = ρcuJ2/f = 1. The dimensions of these quantities are PL -1Q-1, PTL-3 Q-1, Qand PL-3, where, as before, P is power, L is length, T is time, and is Q temperature. Then Eq. (6.1.1) becomes

	(6.1.2)
where g(t) is given by Eq. (4.7.2).
When t is small, the last term in Eq. (6.1.2) can be neglected and then
t = E(4πt)–1/2exp(–z2/4t)	(6.1.3)

We take this form for the initial condition at very short times.

The evolution of the initial condition is controlled by Eq. (6.1.2) and its ultimate fate (quench or recovery) thus depends on the two parameters E and i (which enters through g(t)). Thus the bifurcation value of E, Ebif = G(i), where G is a dimensionless function yet to be determined. In ordinary units this equation becomes

Ebif = Sk	1/2	3/2		2	/f )	-1/2	(6.1.4)
		tc	(rcu J			G(i )

since E has the dimensions of an energy per unit area (PTL-2).

6.2. GROUP ANALYSIS OF THE BIFURCATION ENERGY

As shown in Dresner (1985), we can go a step further in determining the function G(i) if we assume that when E = Ebif, the temperature rise t of the normal zone drops below t = 1 before much Joule heat is produced. This assumption replaces Eq. (4.7.2) for g(τ) by

= (t + i – 1)/ i 1 – i < t

g(t)	(6.2.1)
= 0	0 < t < 1 – i

Uncooled Conductors

103

This functional dependence of g on t is a member of the family of dependences

= bt – a	a /b < t
g(t)	(6.2.2)

= 0 0 < t < a/b

When g(t) is given by Eq. (6.2.2), Eqs. (6.1.2) and (6.1.3) are invariant to the group of transformations

z´ = lz

t´=l2t

t´= lwt

b´ =l-2b

a´ = lw–2 a

E´= lw+1E

0 < l <

8

, –

8

< w <

8

(6.2.3)

What this means is that if we use Eqs. (6.2.3) to replace z, t, t, b, a, and E in Eqs. (6.1.2) and (6.1.3) by their primed counterparts, we obtain Eqs. (6.1.2) and (6.1.3) again. Thus if we have a solution τ(z, t, b, a, E) of Eq. (6.1.2) with the initial condition (6.1.3), then τ´(z´, r´, b´, a´, E´) is also a solution corresponding to the value E´ of the pulse energy. Solutions that quench clearly transform into solutions that quench and solutions that recover transform into solutions that recover.

Now the bifurcation value of E can only depend on the values of a and b: Ebif = G(a,b). Since this relation must be valid for any consistent values of E, a, and b, it must be invariant to the transformations (6.2.3). Thus

lw+1Ebif =G(lw–2 a,l-2b)	(6.2.4)
The only possible form for G that satisfies this functional equation is
G = Cab–3/2	(6.2.5)

where C is an as yet undetermined constant (Cohen, 1931; Dresner, 1983). When a = (1 – i)/i and b = l/i, so that Eqs. (6.2.2) and (6.2.1) are the same,

104 CHAPTER 6

		G = C(1 – i)i1/2				(6.2.6)
Then Eq. (6.1.4) becomes
E	=CSk	1/2t3/2	(r	cu	J 2/f )-1/2(1 – i)i 1/ 2	(6.2.7)
	bif	c

in ordinary units. As already remarked in Note 2 of Chapter 5, it is dangerous to assume that a constant like C is close to 1. A more detailed but still approximate derivation of Eq. (6.2.7), given below in Section 6.3, yields the value C = p.

6.3. ESTIMATION OF THE UNDETERMINED CONSTANT

Suppose we try an approximate solution to Eqs. (6.1.2), (6.1.3), and (6.2.1) of the form

t(z,t )=H(t) F(z,t ); F(z,t )=(4pt )-1/2exp(–z2/4t)

(6.3.1)

where H(t) is an as yet to be determined function for which H(0) = E. If we substitute this trial solution into Eq. (6.1.2), we find

H´(t) F(z,t )= g[H(t) F(z,t )]	_	(6.3.2)
	(H´ = dH/dt)

There is no solution to Eq. (6.3.2), which is the same thing as saying that Eq. (6.3.1) is not an exact solution of Eq. (6.1.2). But we expect Eq. (6.3.1) to be a good approximation, so we determine H(t) by requiring Eq. (6.3.2) to be satisfied at z = 0 (method of collocation):

H´(t)(4pt )-1/2=g[H(t)(4pt )-1/2]

(6.3.3)

Since we are only interested in times for which t > 1 – i, we take g(t) = (τ + i – 1)/i in Eq. (6.3.3). Then we can solve for H(t) and find

t(0,t )=(4pt -1/2 H(t )

=(4pt )-1/2et /i[E– ((1 –i)/i)(4p)1/2	t
		(6.3.4)
	0
If
8
E = ((1	– i)i1/2	(6.3.5)
0

then

Uncooled Conductors

105

(6.3.6)

If E is larger than the value given in Eq. (6.3.5), t(0,t ) diverges exponentially toward

8	with time. If E is smaller than the value given in Eq. (6.3.5), t(0,t ) diverges
exponentially toward –		8
		) with time. But because we have taken g(t) = (τ + i – 1)/i,

the expression (6.3.4) fails to apply once t falls below 1 – i. The value (6.3.5) of E is the bifurcation value; comparing Eq. (6.3.5) with Eq. (6.2.6), we see that C = p. This is, of course, an approximate value, but we expect it to be of the right order of magnitude.

6.4. SIZEOFTHE BIFURCATION ENERGY

How big is the bifurcation energy? I estimate it here for the conductor of a small laboratory magnet I recently built using potted NbTi/Cu strands (diameter, 0.81 mm; f = 0.6). The magnet was designed to produce a field of 7 T at 4.2 K, which it did when the strand current density J was 16.5 kA/cm2 (corresponding to i = 0.23). The critical temperature of NbTi at 7 T is 6.0 K so that tc = 1.8 K. The resistivity of the copper, including magnetoresistivity, is about 5 x 10-10 Wm. Using an average temperature of 5.1 K, we find from the Wiedemann–Franz law that kcu =250Wm-1K-1andk=f kcu = 150Wm-1 K-1. The average volumetric heat capacity S of the conductor in the range 4.2–6.0 K is 3400 J m-3 K-1. Then we find from Eq. (6.2.7) that the bifurcation energy is 24.5 J/m2. Since the cross-sectional area of the strands is 5.15 x 10-7 m2, the total bifurcation energy EA is 12.6 mJ .

The magnet in question trained twice (at 6.3 and 6.4 T) before reaching design field. These training quenches were presumably caused by local cracking of the epoxy potting at points of high stress (e.g., at unpurged gas bubbles in the epoxy). When the epoxy cracks and initiates a quench, it releases an amount of strain energy exceeding the bifurcation energy. The fact that there were only two training quenches suggests that when the epoxy cracks it does not always release an amount of energy exceeding the bifurcation energy. There is some evidence of another sort, coming from acoustic emission studies, that this is so. When epoxy cracks, it makes noise that can be detected. In these studies the noise was sometimes accompanied by a voltage across the magnet terminals, signifying the creation of a normal zone, and sometimes unaccompanied by voltage. The tentative conclusion is that epoxy cracking liberates energies of the order of a few to a few tens of mJ. Since this energy release is independent of temperature, training should not occur in potted magnets wound with high-temperature superconductors because for the latter the bifurcation energy is much larger than for the low-temperature superconductors (as we shall see later).

106

CHAPTER 6

6.5.THE EFFECT OF TRANSVERSE HEAT CONDUCTION ON THE BIFURCATION ENERGY

So far we have treated the epoxy as though it were a perfect insulator of heat. In strict point of fact, this is not so, and transverse heat conduction through the epoxy to adjacent turns may need to be considered. To judge whether or not transverse conduction is important, we need to estimate how far heat can diffuse through the epoxy during the time in which the issue of quench or recovery is decided. The unit of time in our special system of units is Stc /(rcu J 2/f ), which equals 0.27 ms in the example of the last section. According to Reed and Clark (1983), for epoxies k ~0.1 W m-1 K-1 and Cp ~1 J kg-1 K-1. If we take the density of epoxy to be around 1000 kg/m3, we find the thermal diffusivity k/S of epoxy to be ~10-4 m2/s. In a unit of time, then, heat can penetrate about 0.16 mm into the epoxy. If the separation of adjacent turns is greater than 0.16 mm, transverse conduction of heat from one strand to its neighbor may be neglected. In the example of Section 6.4 the strands are close packed. The epoxy-filled interstices have a volume fraction of 1 – 0.0931 and the average thickness of epoxy between turns is only about 0.04 mm. In this case, it seems likely that transverse conduction cannot be neglected.

When similar estimates show that transverse heat conduction cannot be neglected, Eq. (6.1.1) may be replaced by the following three-dimensional heat balance equation for a homogeneous, anisotropic medium:

(6.5.1)

where kt is the thermal conductivity in the transverse direction. If we introduce the new transverse coordinates x' = xk1/2/k1t /2 and y´ = yk1/2/k 1/2t , Eq. (6.5.1)

becomes

(6.5.2)

where r= (x´ 2 + y´2 + z2)1/2 is theradial coordinate in(x´,y´,z) space. In special units, this becomes

(6.5.3)

The initial condition we need to fulfill is

(6.5.4)

which is the instantaneous point-source solution to Eq. (6.5.3) when the last term is neglected. Note now that E1 has the units of energy. When g(t) is given by Eq. (6.2.2), Eqs. (6.5.3) and (6.5.4) are invariant to the transformation

Uncooled Conductors

107

r´= λr

t´ = λ2t

t = lwt

b´ = l-2b

a´= lw–2 a

E1´ = lw+3E1

0 < l <

8

, –

8

< w <

8

(6.5.5)

Then G = Cab-5/2 so that when a = (1 – i)/i and b = l/i, G = C( 1 – i)i3/2 .

We have to be careful at this juncture about the meaning of E1. According to

Eq. (6.5.4)
			tdx dy dz = (k/kt )E	(6.5.6)
where E is the pulse energy we are seeking. In ordinary units, then,
E = CSk1/2k	τ	c	5/2(r J 2/f )-3/2(1 – i)i 3/ 2	(6.5.7)
t			cu

A calculation exactly like that described in Section 6.3 yields the result C = 6p2 (Dresner, 1985).

6.6.BIFURCATION ENERGIES OF HIGH-TEMPERATURE SUPERCONDUCTORS

The work of the foregoing sections has been based on the assumption that the quantities k, rcu , and S could be taken independent of temperature. This assumption is satisfactory for the old, low-temperature superconductors for which the bath and critical temperatures are close together. But it is not satisfactory for the new, high-temperature superconductors. A Ag/BSCCO composite operated at 30 K in fields of a few tesla, for example, may have a critical temperature of 70 or 80 K. Over the range 30–80 K, the physical properties k, rcu , and S vary strongly with temperature, and we cannot assume their constancy. But we can salvage the group-invariance method of Section 6.2 if we fit the temperature variations of the physical properties with power laws. Accordingly, we take

S=Sc (T/Tc )m

(6.6.1)

108	CHAPTER 6
rAg = rAge (T/Tc )n	(6.6.2)

Furthermore, we assume that the thermal conductivity of the silver and its resistivity obey the Wiedemann–Franz law. (It is worth remembering here the results of Chapter 2, according to which 0 < m < 3 and 1 < n < 5.)

These assumed temperature variations make Eq. (6.1.1) nonlinear, and an immediate consequence of this is that we cannot usefully introduce the temperature rise t = T – Tb as a new dependent variable. To attain a solvable problem, we therefore assume that the bath temperature Tb is zero. This is a drastic assumption, but as we shall see later, it probably does not vitiate the utility of the results.

If we now introduce special units in which Sc = kc = T c = rAgeJ2/f = 1 (dimensions PTL-3Θ-1, PL-1Θ-1, Q, and PL-3, respectively), then Eq. (6.1.1) becomes

(6.6.3)

where g(T) is given (literally) by Eq. (4.7.2).

The nonlinear Eq. (6.6.3) can be dealt with in the same group-theoretic way as we dealt with the linear Eq. (6.1.2) in Section (6.2), but the calculations are more complicated (Dresner, 1994, “Quench energies”). We quote only the results here. The bifurcation energy per unit cross-sectional area Ebif, defined by

						(6.6.4)
is given by
E	= CS	T 2 L1/2f (ρ		J	)-1i -1/2 (1– i)m-n+3/2	(6.6.5)
bif	c	c o	Age	c

where Lo is the Lorenz constant and C is a dimensionless constant that depends on m and n. Detailed calculations (Dresner, 1994, “Quench energies”) show that Eq. (6.6.5) is only valid if m + 3/2 > n so that the exponent of 1 – i in Eq. (6.6.5) is always positive.

An approximate calculation similar to that in Section 6.3 shows that when m = n, C is close to 4.5.

Let us use this formula to estimate the quench energy of a small experimental coil described by Schwenterly et al. (1993). The conductor was a 2.5 mm x 0.2 mm multifilamentary Ag/BSCCO tape with a silver-to-BSCCO volume ratio of 6. At 5.5 T, the extrapolated critical temperature was ~65 K; at Tb = 20 K, the critical current was ~2.5 A.

Figure 6.1 shows a plot of the parameter cluster ScT 2c / rc , as a function of temperature for silver for residual resistance ratios of 100, 500, and 2500. There is

Uncooled Conductors

109

Figure 6.1. A plot of the parameter cluster ST2/ρ as a function of temperature for silver for residual resistance ratios of 100, 500, and 2500. (Redrawn from an original appearing in Dresner (1993, “Stability”) with permission of Butterworth-Heinemann, Oxford, England.)

a knee at about 20 K, beyond which the parameter cluster depends only slightly on temperature and almost not at all on the residual resistance ratio. At 65 K, we use the value 3 x 1018 A2 m-4 s K. Since the critical current density in the tape is 5 x 106 Am-2, the first factor in Eq. (6.6.5) is 362 J/mm2. When multiplied by the area of the tape, 0.5 mm2, this gives 181 J, which is an enormous perturbation energy.

The value just calculated needs to be multiplied by the second factor in Eq. (6.6.5). Reasonable values for m and n are m = 2 and n = 3 (see Table 6.1). When i = 0.9, the second factor in Eq. (6.6.5) is 1/3, and the quench energy is roughly 60 J, which is still very large.

Table 6.1. Approximate Values of m and n over

Various Temperature Ranges for Ag /BSCCO

Conductors

Temperature(K)	m	n
10–20	3	< 5, depends on RRR
20–65	2	3
65–130	0.67	1.5
130–200	0.33	1.2
200	0	1