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A critical current-margin design criterion for high performance magnet stability
A "critical-current-margin. (CCM) design criterion for superconducting magnet stability is presented. It is applicable to high-performance magnets that operate at high current densities; these magnets have cooled conductor surfaces, but their calculated heat fluxes are weft beyond recovery values. For stability o f a magnet based on this CCM design criterion, disturbances must be limited in size but need not be eliminated nor localized. The CCM criterion is compared with the MPZ/Cold-End Theory. It is shown that this criterion offers a much higher confidence level of stability than the MPZ/Cold-End Theory. Furthermore, and unique to this theory, stability is independent of current density in the stabilizer. The CCM criterion is based on two magnet winding elements: heat transfer and composite conductor. In this criterion heat transfer must always remain in the nucleate boiling region throughout the operation of a magnet. The design emphasis is on more superconductor and less stabilizer. It is also demonstrated that the CCM theory works particularly well with conductors such as cabled conductors that have high ratios o f cooled perimeter to cross-section. Experimental results are presented supporting this CCM design criterion.
A critical current-margin design criterion for high performance magnet stability Y. Iwasa The effect of current density on the economics of a superconducting magnet diminishes with magnet size, economically permitting the operation of large superconducting magnets at lower current densities. Nevertheless, there always exists an economic incentive to operate the magnet at higher current densities. Specifically, we shall call magnets which operate at high-current densities so tlmt their normal-state heat generations exceed 1 W cm, -2 high-performance magnets. In this paper we introduce a new design criterion, which we call the critical-current-margin, CCM, criterion, for stability of high-performance magnets. First, two stability criteria currently in use are briefly reviewed.
Cold-end stability The stability criterion based on Maddock's equal area theory I is sometimes called the cold-end stability criterion. The criterion permits higher current densities than those permissible with Stekly's classic cryostatic stability criterion ~ owing to the addition of conduction cooling along the composite conductor. The criterion limits the maximum normal-state heat generation per unit cooled surface in the conductor to approximately 0.3 W cm -2 Based on this design criterion, the magnet is immune to any disturbances, localized or global, that might occur in structurally sound magnets. The only design requirement with regard to stability is to ensure heat transfer conditions implicit in the theory within the magnet winding. The author is at the Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology Cambridge, Massachusetts 02139, USA. The Francis Bitter National Magnet Laboratory is supported by the National Science Foundation, The work was supported by the Fossil Energy/MHD Program of the Department of Energy. Paper received April 30 1979.
MPZ/Cold-end theory Recently, the 0.3 W cm -2 barrier was broken by Wilson's MPZ/Cold-End theory,3 which combined Wipf's MPZ theory4 and the cold-end theory of Maddock. 1 According to the MPZ/Cold-End theory, a magnet can recover from a limited disturbance even if it operates at a normal-state heat generation above 0.3 W cm-;. The theory appears to account for the successful performance of a large, high-performance MHD magnet recently put into operation, s The limitation of the theory, however, are: permissible disturbances must be localized and limited in amplitude. This is a severe limitation in large magnets where global disturbances such as widespread conductor movements are not improbable. Also the maximum value of disturbances in the magnet winding must be known accurately for the theory to be applicable with any degree of confidence.
Critical-current-margin (CCM) stability criterion Application The critical-current-margin (CCM) stability criterion is intended ctfiefly for high-performance magnets. As stated above we def'me a high-performance magnet as one in which the normal-state heat generation rate exceeds 1.0 W cm -2 As we shall see later, disturbances in the magnet need not be localized. A high-performance magnet based on this design criterion either stays completely and continuously superconducting during its operation despite disturbances or is driven normal; there is no recovery from a localized quench. We shall demonstrate that, when designed according to this stability criterion, a high-performance magnet can be
0011-2275/79/012705-10 $02.00 © 1979 IPC Business Press CRYOGENICS . DECEMBER 1979
705
expected to perform under energy inputs far greater than those permitted by the MPZ/Cold-End criterion. Unquestionably the chances of success improve as we learn more about disturbances in the winding and can design to limit or eliminate them. Theory - qualitative description
There are two important temperature rises that must be considered when a conductor receives a disturbance, either locally or along its entire tength. One is a temperature rise necessary in the conductor to transfer to the coolant the excess heat received, and the other is a temperature margin accessible in the conductor between operating and transition temperatures. If this temperature margin is greater than the temperature rise required for heat transfer, the conductor may remain completely superconducting even in the presence of a disturbance. This is the condition sought in the new stability criterion. This criterion is thus akin to the adiabatic stability criterion, with one important difference: unlike the adiabatic stability criterion, which provides only limited cooling due to the heat capacity of the winding and thus permits only minute disturbances, 6 the CCM permits sizable disturbances in the winding because of the presence of liquid helium. Theory - quantitative derivation
The two important factors in the theory, heat transfer and transition temperature, are reviewed briefly first.
Heat transfer: nucleate boiling region Fig. 1 illustrates the main features of a relationship between conductor surface temperature, measured from bath temperature, vs heating flux under steady-state condition. For heating up to qe, a critical heat flux, heat transfer is in the nucleate boiling region and wall temperature remains relatively small. Beyond qe, heat transfer enters the film boiling region and wall temperature climbs rapidly. The dotted lines represent a simplified model useful in the discussion below. In saturated liquid helium at 4.2 K, qc typically ranges from 0.5 to 0.8 W cm, -2 although it could be smaller in narrow channels. 7
Essentially the same characteristics have been observed und under transient conditions when heating lasts only a short while. 8'9 In transient heating, where heating is usually less than 50 ms, however, a more meaningful parameter for the abscissa is energy flux, e, rather than heat flux, as indicated by the parentheses in Fig. 1. A vital point is that in transient heating the nucleate boiling heat transfer can be sustained for heating rates well above the steady-state critical value qc, provided energy deposited does not exceed ec. The value of e¢ depends principally on heating flux, and is smaller for high flux (~10 W cm -2) and greater for low flux (~1 W cm-2). [Obviously it is infinite for heat fluxes less than qc.] Specifically, based on recent experimental observations,a, lo, it the product of heat flux (assuming it is constant during heating) and ec appears to be approximately constant for a given cooling surface, although it varies over a wide range from experiment to experiment. Tsukamoto and Kobayashi8 obtained values for this product ranging from 20 to 80 x 10 -3 WJ cm -4, while values obtained by Nick, Krauth, and Ries 1° and Arp ~ were about 3 x 10-3 WJ cm. -+ The values of ec also depends on the amount of liquid helium available in the immediate vicinity of a heated surface? A typical e e range is 10 "~ 50 x 10 -3 j cm-2 8, 12,13 The crucial quantity for the present CCM theory of stability however, is the maximum temperature rise, represented by ON in Fig. 1, that occurs in the nucleate boiling region, Under steady-state condition, ON, measured on a copper surface in the present experiment, is 0.2 ~ 0.4 K; it appears to be the same under transient condition, based on both present and previous experiments.~2 According to the recent transient heat transfer measurements of Schrnidt 14 ON (K) actually varies with heat flux, Q(W cm -2) as ON = 0.2 Q; namely, 0.2 K at 1 W cm -2 and 1 K at 5 W cm -2. As will become evident in later discussion, if this relationship is indeed valid, it becomes more difficult to satisfy the CCM criterion with disturbances transmitting large heat fluxes at the conductor surface. A practical remedy for this will be discussed later.
Transition Temperature. The transition temperature of a conductor is a decreasing function of current and field. Based on NbTi data of Hampshire, Sutton, and Taylor ~s typical curves showing dependences of transition temperature on currcmt at four levels of magnetic field are shown in Fig. 2. For simplicity on computation, they are all approximated by straight lines. In the figure each 0e, measured from the bath temperature, is a transition temperature in a given field with zero transport current; I c is the critical current in a given field at bath temperature. Note the location of 0N (Fig. 1) in the temperature abscissa. Region of stable operation. Suppose a conductor,
8h
f
I qc (ec) q (e)
Fig. 1 Temperature 8, measured f r o m the bath temperature, vs heat f l u x (and energy f l u x ) . The dotted curve shows a simplified function
706
characterized by Fig. 2, is subject, on a unit surface basis, to a transient disturbance that deposits energy flux e. Let us further suppose that it is carrying a transport current equal to It(B4): lop = It(B4). For the purpose of discussion, the simplified heat transfer curve (the dotted curve in Fig. 1) is used. When the conductor is in field Bz, its transition temperature is clearly greater than ON. Thus the conductor is completely undisturbed and remains completely superconducting for disturbances up to ec; above ec it is driven normal. The same is true in field B2.
CRYOGENICS. DECEMBER 1979
]'c(B 0
of attraction (BOA)' discussed recently by Wipf. 16 Thus a magnet designed to generate a field of B1, fbr example, can be operated with the same degree of stability either with an operating current of It(B2) or Ic(Bs). If a maximum disturbance in the winding exceeds ec, both designs will fail; if not, both will succeed. Thus it obviously pays to operate the magnet with Ic (B2) irrespective of a resultant current density in the stabilizer. This conclusion will be further discussed below.
~ 1
/~(82) /¢(83 )
The same argument applies to steady-state disturbances. Fig. 3, therefore, is also applicable when energy flux is replaced with heat flux.
I 8 I
2"o(B4}
I
B2
Critical current margin. In addition to requiting a nucleate boiling heat transfer in the magnet winding, this design criterion requires what we now det~me as the critical current margin. This is best explained graphically, as illustrated in Figs. 4 and 5.
83 I 8.
°o
ON
0c(84)
0~(83)
0~(82)
0°(80
eo Fig. 2 Critical current vs transition temperature lines at d i f f e r e n t magnetic fields
In field B3, the transition temperature just equals ON and above B3 it is less than ON. Thus in fields above Ba the conductor cannot tolerate any disturbances. Accordingly, the region of stable operation may be represented in an e vs B plot. For this particular example the region is a rectangle, bounded by lines of ec and B3, as illustrated by the dotted lines in Fig. 3. When a more realistic heat transfer curve is used, the solid curve in Fig. 1, the boundary is modified, as indicated by the solid curve in Fig. 3. At a higher operating current, for example Iop= Ic (B3), the boundary of stability is reduced in the B axis to//2, while in the e axis it is virtually unchanged. The crossed line in Fig. 3 represents the new boundary line.
Fig. 4 shows the magnet load line and two possible critical current curves of the conductor. This load line actually relates current and the maximum field at the conductor at some point in the magnet. As for the critical current curves, in curve 1 the critical current equals operating current at Bin; in curve 2 it exceeds the operating current by a certain amount. Given a disturbance, according to the above discussion, with curve 1 the magnet will fail to generate a designed field of Bm; with curve 2 it will succeed, provided there is a sufficient temperature margin. This point is more clearly illustrated in Fig. 5, which shows I vs 0 plots in Bm for curves 1 and 2. Namely, our new design criterion for stability is able to provide a sufficient temperature margin at the operating point (Bm, lop) so that the transition temperature of the conductor at the operating point is at least 0N above bath temperature. To have this temperature margin, the conductor critical current at the operating point must at least be I:.
A unique conclusion about stability can be drawn based on this criterion. Namely, stability, if it can be quantified in terms of maximum permissible disturbance flux, ec, is independent of operating current up to a certain level. This maximum permissible disturbance is analogous to the 'basin
fop =/e (B4) . . . .
(Simplified)
zop=z~(B 3) ++++ "° (q¢)-
Steble
°i
+ "+ + + + + + -4+ 4+
8', ÷
I
I
I I
8
B
B
Fig. 3 Disturbance energy vs field plots showing the region o f stable operation
CRYOGENICS.DECEMBER1979
Bm
B Fig. 4
Load line and t w o short-sample current plots (1) and (2)
707
I~ ~
At Bm iM =
o©
#=(am) -as 8©{Bm)
/
S'lable region in field
Bm
/
\N 0
0
/M
I
e Fig. 5 Current vs transition temperature curves for two shortsample current plots of Fig, 4 at field Bm. The dotted line is for short-sample current plot (1) at field B~n< Brn
Fig. 6
An analytical expression for ~ is derived below using a linear relationship between transition temperature and transport current in a given B. Namely in B = Bin;
Oe(I,B i n )
=
Oe(Bm)Ie(Bm)
(Bm)
This is because ON, which depends chiefly upon bubble dynamics and growth, decreases with bath temperature. Still it is recommended that NbTi magnets be operated at 4.22 K rather than at 4.45 K. (1)
Our design criterion requires:
Oc(I=Iop, Bm) ;* ON
(2)
Combining (1) and (2) and solving forfc, we obtain /c (Bin) ~ Iop
0c(Bm) - - ON
(3)
0c(Bm)
The critical current margin, (i~ -Iop), is thus given: Ic (Bin) - I o p ;~ Iop
ON 0e (Bm) -- ON
(4)
The maximum permissible operating current may be expressed in terms of critical current: /°--Z -- /m = 0 ° ( S m ) - 0S ,r • 0c(Bm)
(5)
The region of stable operation can also be represented in an e vs (Iop[I*c)plot, as shown in Fig. 6. Again this is a simplified picture based on the dotted heat transfer curve of Fig. 1. An operating current that corresponds to point A in Fig. 6 should certainly be avoided; a slight reduction in operating current will result in point B, with a large gain in the degree of stability. Note that when 0c(Bm) = 0N, which is reached at some high field, this criterion cannot be used. For 0N ~ 0.3 K and the conductor is NbTi, this field is about 10 T when the bath temperature is 4.22 K. In a bath temperature of 4.45 K, which is the case when the system is connected directly to a liquefier, this limiting field becomes 9.5 T, although there is a likelihood that ON may be slightly less,17 and therefore the limiting field could be higher than 9.5 T.
708
The region of stable operation in an e vs (Iop/l¢) plot
From the above, the necessity of using Nb3 Sn in high fields becomes quite obvious; it may even be advantageous in fields as low as 8 T. Returning to Fig. 4 if the magnet under discussion is wound with a curve 1 type conductor, it satisfied the criticalcurrent-margin and thus is stable for fields up to Bin< Bin; at this point 0c(~op ' Bm) = ON. This is illustrated by the dotted lines in Figs 4 and 5. In essence, this criterion traces its origin to the stability work of Stekly,Is in which this mode of fully-super. conducting operation under steady-state heating was first examined in detail. However, Stekly's work still placed an emphasis on stable recovery from the normal state and matching an operating current with a critical current.2 To achieve stable recovery, a generous amount of normal metal stabilizer was added to the superconductor. In contrast, in this design criterion the emphasis is placed totally on the fully superconducting region; consequently, more superconductor is used and less stabilizer. This criterion is quite different from the nucleate cooling stability criterion of Whetstone and Boom, proposed in 1967.19 Their criterion, like Stekly's and many other criteria that followed, emphasizes recovery, and therefore it its maximum permissible normal-state heat generation is limited and not high. In their case it is about 0.8 W cm-,2 equal to a maximum possible cooling rate in the nucleate boiling region.
Comparison with the MPZ/Cold-EndStability Criterion. For stability of high-performance magnets, in addition to the adiabatic theory either the MPZ/Cold-End theory or this CCM design criterion may be used. We shall demonstrate here that, given an uncertainty in disturbance size in the magnet, the chances of success for a magnet are far greater
C R Y O G E N I C S . DECEMBER 1979
based on this criterion than on the MPZ/Cold-End theory. This is done by comparing the size of a permissible energy input for the two cases at a field of 8 T for a given conductor/cooling configuration. Implicit in the argument is the idea that the chances of success are better the greater the size of a permissible disturbance energy. For the sake of comparison, we shall assume a disturbance to be limited to an MPZ length, and compare permissible energies derived from the two criteria on a unit volume basis. According to the MPZ/Cold-End energy computation of Wilson, a'which neglects any transient cooling, the maximum permissible energy density, at a given ambient field, is a sharply decreasing function of normal-state heat generation. Table 1 gives maximum permissible energy densities for a one-dimensional conductor at 8 T at selected values of normal-state heat generation. Note that the total permissible energy over an MPZ length, which is given by the product of this energy density and MPZ length and conductor cross-section, decreases even more sharply because MPZ length itself shrinks with normal-state heat generation. In contrast, the maximum permissible energy density according to the CCM criterion is given by:
Ec - eeP A
(6)
where P is the conductor cooled perimenter, A is the conductor cross-section, and ec is the critical energy flux previously discussed. The above equation shows that: the permissible energy density for the CCM criterion is independent of normalstate heat generation, and it increases as P/A, favouring small conductors or cabled conductors, such as those of the Rutherford type or bundle conductors. 2° Although not explicit in the equation, in the CCM criterion disturbances need not be restricted to an MPZ length. If a nucleate boiling heat transfer can be maintained and a maximum energy density is limited to Ec, disturbances can occur globally in a magnet. To complete the numerical comparison with the MPZ/ColdEnd criterion, we shall choose a relatively large conductor with P = 2 cm and A -- 0.25 cm 2, which results in Ec of 80 ~ 400 x 10-3 J cm. Based on the above idea that the amount of permissible energy input is a measure of stability, we conclude that the present criterion offers a much higher degree of stability than the MPZ/Cold-End theory.
Table 1. Maximun permissible energy densities, at 8 T and 4.2 K, for selected normal-state heat generations, based on the MPZ/Cold-End Criterion Normal-State Heat Generation, W cm -2
Energy Density, 10-3 J cm--3
0.3 0.5
40 6
1
2
3
1
CRYOGENICS. DECEMBER 1979
Remarks on conductor, cooling, and disturbance decoupling
Conductor configuration. As remarked above with regard to (6), the degree of stability can be improved by increasing the ratio of cooled conductor perimeter to cross.section. Thus the CCM criterion would work particularly well with cabled conductors such as those of the Rutherford type or bundle configurations. 2° Matrix material. Since the success of this criterion depends critically on keeping a temperature rise in the superconductor low, it might be beneficial to put NbTi f'daments into a matrix material of low thermal conductivity such as stainless steel or cupronickel to reduce the effect of heating at surfaces. A new composite conductor, of course, must still have a layer of copper at the surface to disperse heat. The proposed conductor works only when there is no appreciable heat dissipation within filaments. Surface condition. Surface condition affects ON.. As pointed out earlier, a thin layer of insulation seems to increase ON drastically, making it more expensive, if not impossible, to meet the CCM criterion, because an increased ON means a higher critical current margin. For this design criterion to be most effective it is suggested that the conductor surface should be bare and be kept clean. The thickness of an oxidation layer, for example, should be kept to a minimum. In this regard, it may be prudent to put a layer of copper on aluminum-stabilized conductors. Also, the effect of a long exposure to the atmosphere on conductor surface conditions should be examined, especially during a long production period. For large magnets such as those for MHD and fusion that are expected to remain in the 4.2 K environment for a long time, we can expect surface condition to remain unchanged once the magnets are energized. Other cooling schemes. Operation in the supercritical state becomes appealing in the CCM design criterion, particularly with Nb3 Sn. 21 In the supercritical state there is no sudden change in the wall temperature owing to a single-phase operation and heat transfer rate is roughly proportional to the temperature difference between the conductor wall and coolant. By driving the wall temperature higher, substantially more energy may be removed; such a scheme is more practical with Nb3 Sn conductors. Another attractive scheme in this regard is operation in pool-boiled helium II. 22-24 According to Van Sciver's17 heat transfer measurements, a large heat flux (up to 2 W cm -2) is possible for ON ~ 1 K in subcooled He II at 1.9 K. Even with such a high ON, the CCM criterion is applicable for NbTi at fields as high as 10 T, because at a bath temperature of 1.9 K, NbTi at 10 T has a temperature margin of ~3 K.
Disturbance decoupling. As mentioned earlier in connection with the Schmidt's results on 0 vs Q, it becomes increasingly difficult to satisfy the CCM criterion with disturbances transmitting a large heat flux on conductor surface because of a large.resultant ON. A large heat flux may be transformed to a small heat flux when it is first transmitted through a medium having-a low thermal diffusion constant. That is, disturbances can be decoupled from the conductor surface. One practical suggestion might be to attach rigidly to the conductor an insulating material such that all the disturbances may be deposited to this insulating material rather than directly to the conductor surface.
709
Experiments and discussion
Rotatalde shutter plate
Experiments were carried out to confirm some key assumptions made and to verify the conclusions reached concerning use of the new design criterion.
Test conductor and sample holder Starting with a composite-copper.NbTi conductor of square cross-section (4.6 mm by 4.6 mm), we simultaneously wrapped sixteen 0.53 mm diameter bare copper wires tightly, leaving no gap between wires, over the superconductor. Pitch length was 9.8 mm (Fig. 7a). After soft-soldering the wire to the conductor, we removed every other wire, creating helical channels, each 0.53 mm wide and 0.53 mm deep, around the conductor (Fig. 7b). The original square conductor had a copper to superconductor ratio of 1.2. As indicated in Fig. 7c, two insulated heater wires, each of 0.45 nun diameter were wrapped, and epoxied, around the conductor by placing each wire in a channel. The two heater wires were connected at one end, thereby creating a noninductively wound heater. The total conductor length between terminals was 600 mm and the heated region which covered the middle section was 110 mm long. Over the heated section, computed cooled surface per unit conductor length, including contributions by the wrapped wires, is 2.3 cm 2/era. Measured normal-state resistances per unit length near 4.2 K were, in/aL2 cm -1, 0.17, 0.24, 0.33, and 0.38 respectively, in fields of 2 T, 4 T, 6 T, and 8 T. The test conductor was placed in a 188 mm diameter circular channel machined in a phenolic plate (Fig. 8). The channel was of a square cross-section (5.7 mm x 5.7 mm) and a series of 5 mm diameter vent holes was machined at 20 mm interval in the bottom of the channel along its entire length to provide liquid helium access.
Clarr
tog plat(
S0rnl
Fig. 8 Sketch o f the test conductor sample holder showing t w o cover plates
Similarly, a phenolic cover plate had matching vent holes machined for liquid helium. In addition, to investigate the effect of the helium flow on cooling, a rotatable second cover plate with an identical set of vent holes was placed on top of the first cover plate (Fig. 8). The relative position of the holes in the two cover plates controlled the helium flow. A mechanical linkage outside of the dewar permitted this control during the experiment. During the experiment two positions were used for the vent holes: open, when the two plate holes were lined up; and the other, closed, when they were completely blocked. The bottom vent holes were always open. A set of voltage taps was distributed along the test conductor length. Four Au-Fe/chromel thermocouples were attached directly to the conductor at a helical channel midway between the heated channels as indicated in Fig. 7c. Each thermocouple was held to the conductor surface by a drop of Stycast 2850 epoxy.
Procedures, results, and discussion Two sets of data were collected on the test conductor: heat transfer and quench.
Heat transfer (transient). The test conductor was heated by passing a pulsed current through the heater and its temperature response was measured with the thermocouples. To eliminate current and field noise, measurements were done in the absence of field and transport current.
Thermocouples
Heater - - - - -
J
Fig. 7 Sketch o f the test conductor, a -- Sixteen copper wires wrapped; b -- every other copper wire removed; c -- heat wires placed in helical channels with one set of thermocouples attached at the middle channel
710
Fig. 9 shows a typical oscillogram of three heater current traces, labelled A, B, C, and corresponding temperature traces, labelled a, b, c, all as a function of time. Because of a non-zero diffusion time between heater and test conductor, a heating rate applied in the heater was always greater than a heating rate in the conductor while the heater current was on. Heat thus accumulated in the heater and was released to the conductor even after heating was stopped in the heater. This accounts for the delayed
CRYOGENICS. DECEMBER 1979
0.3K. The diffusion time and other factors such as the exposure of a portion of the heater wire to helium may very well affect ec, and thus when heating is applied differently, for example by frictional means, a different e c may be obtained. The difference however, is likely to be smaller than an uncertaintly in an estimated size of distrubance: a precise value of ec is not too crucial.
Heat transfer (steady-state). Fig. 11 shows actual steadystate temperature vs heater current traces obtained under both open-hole and closed-hole conditions. Temperature traces are plotted against a linear heating scale for the two cases i~ the figure inset. Here the importance of helium access is clearly demonstrated.
Quench (critical current measurements}. Critical currents of the test conductor were measured in three fields using a standard four-probe technique, and are shown in Fig. 12. Fig. 9 Oscillograms showing heater current traces, labelled A,B, C, and corresponding temperature response traces, labelled a, b, c. Vertical scale temperature is 0.75 K/div; time scale is 10.7 ms/div
temperature responses observed in Fig. 9. The time scale was 10.7 ms/div. Temperature vs input energy flux traces, like the one shown in Fig. 1, can be plotted from data such as that presented in Fig. 9. Energy is expressed per unit conductor cooled surface. Fig. 10 shows results obtained with heating pulses each lasting 20 ms. The open circles correspond to results with the helium vent holes open, while the solid circles refer to results with the holes closed. Helium flow appears to have little effect on the results. Apparently only the liquid helium within the channel contributed to cooling. As we shall see below, this is not the case in steady-state heating. The dotted line shows an apparent curve inferred from quench results shown in Fig. 15. The discrepancy will be discussed later.
1.0 - ~ 0.6
II ~ 0.6 -
°21-
0.4-
o0
c~ 0.1
02
0.5
0.4
0.5
~[0~
0.6[,/
~ '
0.2 O(
2
4
6 8 Heater current, A I 0.1
0
The critical energy flux is 25 ~ 30 x 10 -3 J cm -2 with ON
I0
l J 02 0.3 Heat flux, WCm2
I 0.4
I~
I I [ 0.5 0.6 0.7
Fiq. 11 Conductor temperature vs steady-state heater current traces for vent holes open and vent holes closed. Inset shows 0 vs heat flux plots
6--
5--
5--
o Open
4-
• Closed
/4b q¢/ /
^
4 ~t
2
I-
,l? 1 ~ "
%
l
K3
I
20
I
30
I
4O
I
50
I
60
I
70
I
80
I I
Energy flux, mJ/cm 2 Fig. 10 Conductor temperature vs energy deposited into conductor, expressed per unit conductor cooled surface. Open circles are with vent holes open; closed circles are with vent holes closed
CRYOGENICS.
DECEMBER
1979
I 2
I 3
I 4
t 5
1 6
I 7
Field, T Fig. 12 4.22 K
Critical current vs field data for the test conductor at
711
Three current levels indicated by the dotted lines in the figure are those used in quench measurements.
Quench (transient heating): In these measurements, a maximum permissible input energy flux was measured for a given set of field and transport current. Fig. 13 shows typical oscillogram traces corresponding to a field of 6 T and a transport current of 4 kA. The helium vent holes were open. There are four traces: the top two nearly identical traces, labelled A and B, are heater currents, while the bottom two traces, labelled a and b, are conductor voltages. Two time scales were used: 10.7 ms/div until the end of heating and 53.5 ms/div after the end of heating. With the heating current corresponding to trace A, the test conductor remained completely superconducting (trace a); at a slightly higher heating rate (trace B), the conductor was driven normal at the heated section and the normal zone propagated, followed by self heating. The normal-state heat generation corresponding to this field and current was 2.3 W cm -2 before the self heating had taken place. The computed critical energy flux for this set is 27.5 mJ cm -2, in terms of energy density, it is 300 mJ cm. -° This is very close to the critical value found in the transient heat transfer measurements. In a field of 8 T with 4 kA, according to the earlier discussion, we expect critical energy flux to drop drastically, because 4 kA is too close to the critical current of 4.3 kA at 8 T. This was indeed proven in the experiment as demonstrated in Fig. 14. Here the critical energy density is 32w mJ cm -3" Fig. 15 summarizes the critical energy fluxes obtained in different fields with transport currents of 3 kA, and 5 kA. Open data points are for the experiment with open vent holes, and the solid data points are for that with closed vent holes. Note that Fig. 15 closely resembles Fig. 3.
Fig. 14 Oscillogram similar to ~:ig. 13 except here data was taken at 8 T with 4 kA. The conductor quenched at a much smaller heat input compared with the case shown in Fig. 13
about 65 x 10 -s J cm-2; instead it was 33.5 x 10 -9 J cm-2; with a transport current of 3 kA, which was, although not measured, much less than a corresponding critical current. A heat transfer curve more consistent with quench results is shown by the dotted line in Fig. 10. Apparently, the conductor surface temperature in the film boiling region measured at a location not exactly at the heated surface is less, especially at high transient heating rates. As an illustration of the application of this design concept, let us consider this conductor for an 8-T magnet. Based on results of Fig. 15 and our discussion, maximum permissible disturbance energy vs operating current, similar to Fig. 6, may be plotted for the conductor at 8-T; this is shown in Fig. 16. The abscissa also indicates normal-state heat generation g.
Consistent with the heat transfer data of Fig. 10, the vent holes have no effect on these results. According to the data of Hampshire, et al.,is the transition temperature at 2 T, 0c (2 T), is about 4K. Thus, based on transient heat transfer data of Fig. 10, ec should have been
O
30
O
0 0
•
A
¢M E
0 O
O 0
x 2
~zo-
O
(Open)
(C~ed)
O
•
5kA
O
•
4kA
10--
o
0 Fig. 13 Oscillogram showing heater current traces, labelled A and B, and corresponding test conductor voltage traces labelled a and b. Traces A and a show when the conductor remained fully superconducting, while traces B and b show when it was driven normal. D Data were taken at 6 T with 4 k A
712
0
L
I
I
I
[
I
I
I
2
3
4
5
6
7
o
I 8
B
Field, T Critical energy, expressed per unit conductor cooled Fig. 15 surface, vs field data f o r transport currents 3 k A , 4 k A , and 5 k A , with vent holes open (open data points) and vent holes closed (solid data points)
C R Y O G E N I C S . DECEMBER 1979
- ----
.re (813 = 4.SkA z= (eT)=5.6~
~=--
MPZ
-3OO at)
%
will have computed normal-state resistance of 0.5/z~2 cm -1. Because of this increased resistance, normal-state heat generation will also be increased for a given operating current, as indicated by the abscissa g' in Fig. 16. Note that the amount of copper may be reduced as long as heat transfer is unaffected and the conductor thermal diffusivity remains close to that of copper. The expanded stability regions is shown by a dashed line in Fig. 16.
Quench (steady-state heating). The experiment was also Absolute stability
._= E - 1 0 0 ~'
.w_ F ® 0-
I0--
o
0
I
2 3 4 Operating current, (kA) ~).3
0
~2
ji
i
013
7
0
j4
~5 ~, ~7 ~ (N~ core)
5
4
~3
2
6
5
5
6
g
Normal - state generation, (W/cm2) Fig. 16 Permissible energy disturbance vs operating current. Energy is given in terms of per unit c o n d u c t o r cooled surface in the left ordinate and in terms o f per unit c o n d u c t o r volume in the right ordinate. Corresponding normal-state heat generations are also given in the abscissa: g for this test conductor and g" for modified conductor
Based on (5) and 0c(8 T) = 1.4 K, 0N - 0.3 K, and Ic(8 T) = 4.3 kA, the range operating current must be limited to 3.35 kA, as indicated by the solid vertical line in Fig. 16. Up to this operating current, a maximum permissible disturbance pulse, according to the data point of 3 kA, is 24 mJ cm -2 for each conductor cooled surface (left ordinate in Fig. 16) and 270 mJ cm -3 per unit conductor volume (right ordinate in Fig. 16). Beyond 3.35 kA, disturbances must be limited drastically, to zero according to the simple model of Fig. 6. For comparison, maximum permissible disturbances according to the MPZ/Cold-End theory are also plotted in the figure. Note that for operating currents below 1.5 kA, the magnet is also in the Cold-End stability region.
performed with steady-state heating. At a given field and transport current, a slowly increasing current was passed through the heater until a sudden transition from a completely superconducting state to a xompletely normalstate was induced in the test conductor. Results, similar to those of Fig. 15 except that here the ordinate is critical heat flux, are summarized in Fig. 17. The two sets of results resemble each other remarkably, as predicted earlier in the text. Here the effect of helium circulation is clearly evident. The implication of the CCM design criterion to steadystate disturbances, however, is less crucial, because, unlike transient disturbances, steady-state disturbances are more predictable and tractable, and thus amenable to the designer's control. These sets of results together with those of heat transfer measurements (Figs 9 and I 0) and critical current measurements (Fig. 11) conclusively support the basic concept of the CCM criterion. Other comments Although much less so than with the MPZ/Cold-End theory, the amplitude of disturbances is critical in this critical-current-margin design theory. Therefore, advances in understanding the nature of disturbances that enable one to control or limit disturbances in the magnet winding should greatly benefit this criterion. Recently one such step in this direction has been achieved concerning frictional heating, considered to be a principal source of disturbances in the winding. A series of measurements of frictional qc 0.6--
Two points from Fig. 15 are shown in Fig. 16. Certainly a complete version of this e vs lop plot at a maximum design field should be a useful set of stability data for the magnet designer. Suppose we wish to operate this 8-T magnet a 4 kA and thus wish to extend the range of stable operation to 4.4 kA. According to this design criterion, the conductor critical current at 8 T must be increased from 4.3 kA. Since 0c(8 T) ~ 1.4 K, and ON ~ 0.3 K, we find, from (3): 1~ (8 T) ~ 4400 1.4
1.4 - 0.3
0
0.5--
o
o
[3 o
D
o
0.4--
o o E L)
O~
•
0.3--
•
O
0 A
0
0.2-
Open o
Closed •
5kA
o
•
4kA
•
3kA
0.1-
I2 (8 T)/> 5600 A Oo
This means the amount of superconductor must be increased by 30%. If the conductor cross-section is also to remain fixed, the copper to superconductor ratio must be reduced: in this case from 1.2 to 0.7. The new conductor
CRYOGENICS.
D E C E M B E R 1979
I
I
I
2
1
3
I
I
4 5 Field. T
I
6
I
7
I
8
6'
Fig. 17 Critical heat f l u x vs field data for transport currents 3 k A , 4 k A , and 5 k A , w i t h vent holes open (open data points) and vent holes closed (solid data points)
713
properties of winding materials in the 4.2 K environment 2s-27 has revealed certain material combinations to be unquestionably superior from the consideration of frictional heating. Another relevant topic concerning this criterion is magnet protection. Since recovery does not take place in a magnet of this design, a normal zone, however small, will propagate rapidly throughout the entire magnet. The protection problem, therefore, might be as straight forward as it is for adiabatically stable magnets. One point that should be stressed is that a normal voltage within the winding could become excessively high if too much copper is removed. For those die-hard conservatives, a belt-and-suspender approach is recommended: a design criterion that combines the present criterion and the absolute stability criterion. In this approach enough superconductor is used to satisfy one criterion and at the same time enough copper is used to satisfy the other criterion. Fig. 18 summarizes the five stabilization criteria discussed above - cryostatic, cold-end, MPZ[cold-end, CCM, and adiabatic. The abscissa represents normal-state heat generation per unit cooled conductor surface, which is used to def'me the region of operation for each stabilization technique. The ordinate represents, qualitatively, the size of disturbance permissible under each stability criterion. For heat generations up to 0.3W cm, -2 the size of disturbance can be large. Beyond 0.3 W crn -2 and perhaps up to 1 W cm -2, the MPZ/cold-end criterion may be used for localized disturbances. As shown by the horizontal line that extends ~ 10 W cm -2,the CCM criterion can be used alone or in combination with the cryostatic, cold-end, or MPZ/coldend criteria. As remarked above, up to 0.3 W era,-2 the CCM criterion joins the cryostatic stability criterion to become a beltand.suspender approach, and beyond 0.3 W cm -2 and up to 1 W cm -2, the contribution of the MPZ/cold-end may be included, provided disturbances are localized. The dotted curve that joins the horizontal line at around 1 W cm-2 indicates this combination. Beyond ~ 10 W cm -2 , the adiabatic criterion seems the only practical approach.
Although it does not guarantee absolute recovery, the degree of stability provided with this criterion is much higher than that with either the MPZ/cold-end criterion or the adiabatic criterion. Its most unique and interesting feature is that stability is independent of operating current density in the stabilizer. It was shown experimentally that cooling channels have little effect on transient disturbances. Perhaps for the immediate future, this CCM criterion may most profitably be used for large test magnets required for MHD and fusion programmes. The cost of such magnets, intermediate in size, could be substantially reduced by basing designs on this criterion. Finally this work has demonstrated the power of a 'stability' experiment' that integrates both heat transfer and quench measurements. I wish to express my thanks to J.F. Maguire, J.F. Johnson, and L. Lyon for assistance in the experiments; D.B. Montgomery, M.J. Leupold, R.J. Thome, J.E.C. Williams, and V. Arp for helpful conversations; and A.M. Dawson for improvement of the manuscript.
References
1 2 3 4 5 6 7 8 9 10
Conclusions
The critical-current-margin design criterion presented here is recommended for magnets of high current densities.
11 12 13 14 15 16 17 18 19
•
:
(
MPZK.old-l, nd/¢CM
i :=i:i:;:.zo
20
X
"-
' ....
~CM
21 22 l~diabotic
0, I
l
0,2 0,:~
Cryoltotl¢-~-~l r
I
CMl~d.b~dd
g~ Wcm-~
'I
.L
I0
CCM-----~1
I(30
Adlol~,ic
Cold-end Fig. 18 Permissibledisturbance vs normal-state heat generation per unit cooled conductor surface under various stability criteria
714
23
24 25 26 27
Maddock,B.J., James, G.B., Norris, W.T., Cryogenics 9 261 (1969) Kantrowitz A.R., Stekly, Z.J.J.,Appl. Phys. Lett. 6 (1965) 56 Wilson,M.N., lwasa, Y., Cryogenics 18 (1978) 17 Martinelli,A.P., Wipf, S.L., Proceedings of the 1972 Applied Superconductivity Conference, IEEE 72 CHO682-5-TABSC, 331 (1973) Wang,S.T., et al., Proceedings 6th International Conference on Magnet Technology, Bratislava, Czechoslovakia, Bratislava, (1978) 157 Superczynski,MJ., IEEE Trans. Magn. MAG-15 (1979) 325 Wilson,M.N., Liquid helium technology Pergamon Press, Oxford (1966) 109 Tsukamoto, O, Kobayashi, S.,J. Appl. Phys. 46 (1975) 1359 Iwasa,Y., Leupold, MJ., Williams,J.E.C., IEEE Trans Magn MAG-13 (1977) 20 Nick, W., Krauth, H., Ries, G., IEEE Tram Magn MAG-15 (1979) 359 Arp, V., (private communications). Iwasa,Y., Apgar, B.A., Cryogenics 18 (1978) 267 Iwasa,Y., Apgar, B.A., unpublished data (1977) Schmidt,C., Appl Phys Lett 32 (1978) 827 Hampshire,R.G., Sutton, J., Taylor, M.T., Supplement au Bulletin de L'Institut International du Froid Commission I. Londres, Annexe, (1969) 1 Wipf,S.L., IEEE Tram Magn MAG-15 (1979) 379 Van Seiver, S.W., Cryogenics 18 (1978) 521 Steldy, ZJ.J.,JAppl Phys 37 (1966) 324 Whetstone,C.N., Boom, R.W.,Advances in Cryogenic Engineering, 13 K.D. Timmerhaus, Ed, Plenum Press, New York, (1968) Hoenig,M.O., Montgomery, D.B., IEEE Tram Magn MAG-I1 (1974) 569 Hoenig,M.O., Montgomery, A.G., Waldman, SJ.,IEEE Trans Magn MAG-15 (1979) 792 WisconsinSuperconductive Energy Project, 1 University of Wisconsin, Madison (1974) Kobayashi, H., Yasukoehi, K., Tokuyama, K., Proc ICEC 6, Grenoble (1976) 307 Claudet,G., Meuris, C., Parain, H., Turck, B., IEEE Trans Magn MAG-15 (1979) 340 lwasa,Y., Kensley, R., Williams, J.E.C., IEEE Trans Magn MAG-15 (1979) 36 Kensley, R.S., S.M. Thesis MIT (1979) unpublished Kensley, R.S., Iwasa, Y., (to be published)
C R Y O G E N I C S . DECEMBER 1979
A critical current-margin design criterion for high performance magnet stability Y. Iwasa The effect of current density on the economics of a superconducting magnet diminishes with magnet size, economically permitting the operation of large superconducting magnets at lower current densities. Nevertheless, there always exists an economic incentive to operate the magnet at higher current densities. Specifically, we shall call magnets which operate at high-current densities so tlmt their normal-state heat generations exceed 1 W cm, -2 high-performance magnets. In this paper we introduce a new design criterion, which we call the critical-current-margin, CCM, criterion, for stability of high-performance magnets. First, two stability criteria currently in use are briefly reviewed.
Cold-end stability The stability criterion based on Maddock's equal area theory I is sometimes called the cold-end stability criterion. The criterion permits higher current densities than those permissible with Stekly's classic cryostatic stability criterion ~ owing to the addition of conduction cooling along the composite conductor. The criterion limits the maximum normal-state heat generation per unit cooled surface in the conductor to approximately 0.3 W cm -2 Based on this design criterion, the magnet is immune to any disturbances, localized or global, that might occur in structurally sound magnets. The only design requirement with regard to stability is to ensure heat transfer conditions implicit in the theory within the magnet winding. The author is at the Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology Cambridge, Massachusetts 02139, USA. The Francis Bitter National Magnet Laboratory is supported by the National Science Foundation, The work was supported by the Fossil Energy/MHD Program of the Department of Energy. Paper received April 30 1979.
MPZ/Cold-end theory Recently, the 0.3 W cm -2 barrier was broken by Wilson's MPZ/Cold-End theory,3 which combined Wipf's MPZ theory4 and the cold-end theory of Maddock. 1 According to the MPZ/Cold-End theory, a magnet can recover from a limited disturbance even if it operates at a normal-state heat generation above 0.3 W cm-;. The theory appears to account for the successful performance of a large, high-performance MHD magnet recently put into operation, s The limitation of the theory, however, are: permissible disturbances must be localized and limited in amplitude. This is a severe limitation in large magnets where global disturbances such as widespread conductor movements are not improbable. Also the maximum value of disturbances in the magnet winding must be known accurately for the theory to be applicable with any degree of confidence.
Critical-current-margin (CCM) stability criterion Application The critical-current-margin (CCM) stability criterion is intended ctfiefly for high-performance magnets. As stated above we def'me a high-performance magnet as one in which the normal-state heat generation rate exceeds 1.0 W cm -2 As we shall see later, disturbances in the magnet need not be localized. A high-performance magnet based on this design criterion either stays completely and continuously superconducting during its operation despite disturbances or is driven normal; there is no recovery from a localized quench. We shall demonstrate that, when designed according to this stability criterion, a high-performance magnet can be
0011-2275/79/012705-10 $02.00 © 1979 IPC Business Press CRYOGENICS . DECEMBER 1979
705
expected to perform under energy inputs far greater than those permitted by the MPZ/Cold-End criterion. Unquestionably the chances of success improve as we learn more about disturbances in the winding and can design to limit or eliminate them. Theory - qualitative description
There are two important temperature rises that must be considered when a conductor receives a disturbance, either locally or along its entire tength. One is a temperature rise necessary in the conductor to transfer to the coolant the excess heat received, and the other is a temperature margin accessible in the conductor between operating and transition temperatures. If this temperature margin is greater than the temperature rise required for heat transfer, the conductor may remain completely superconducting even in the presence of a disturbance. This is the condition sought in the new stability criterion. This criterion is thus akin to the adiabatic stability criterion, with one important difference: unlike the adiabatic stability criterion, which provides only limited cooling due to the heat capacity of the winding and thus permits only minute disturbances, 6 the CCM permits sizable disturbances in the winding because of the presence of liquid helium. Theory - quantitative derivation
The two important factors in the theory, heat transfer and transition temperature, are reviewed briefly first.
Heat transfer: nucleate boiling region Fig. 1 illustrates the main features of a relationship between conductor surface temperature, measured from bath temperature, vs heating flux under steady-state condition. For heating up to qe, a critical heat flux, heat transfer is in the nucleate boiling region and wall temperature remains relatively small. Beyond qe, heat transfer enters the film boiling region and wall temperature climbs rapidly. The dotted lines represent a simplified model useful in the discussion below. In saturated liquid helium at 4.2 K, qc typically ranges from 0.5 to 0.8 W cm, -2 although it could be smaller in narrow channels. 7
Essentially the same characteristics have been observed und under transient conditions when heating lasts only a short while. 8'9 In transient heating, where heating is usually less than 50 ms, however, a more meaningful parameter for the abscissa is energy flux, e, rather than heat flux, as indicated by the parentheses in Fig. 1. A vital point is that in transient heating the nucleate boiling heat transfer can be sustained for heating rates well above the steady-state critical value qc, provided energy deposited does not exceed ec. The value of e¢ depends principally on heating flux, and is smaller for high flux (~10 W cm -2) and greater for low flux (~1 W cm-2). [Obviously it is infinite for heat fluxes less than qc.] Specifically, based on recent experimental observations,a, lo, it the product of heat flux (assuming it is constant during heating) and ec appears to be approximately constant for a given cooling surface, although it varies over a wide range from experiment to experiment. Tsukamoto and Kobayashi8 obtained values for this product ranging from 20 to 80 x 10 -3 WJ cm -4, while values obtained by Nick, Krauth, and Ries 1° and Arp ~ were about 3 x 10-3 WJ cm. -+ The values of ec also depends on the amount of liquid helium available in the immediate vicinity of a heated surface? A typical e e range is 10 "~ 50 x 10 -3 j cm-2 8, 12,13 The crucial quantity for the present CCM theory of stability however, is the maximum temperature rise, represented by ON in Fig. 1, that occurs in the nucleate boiling region, Under steady-state condition, ON, measured on a copper surface in the present experiment, is 0.2 ~ 0.4 K; it appears to be the same under transient condition, based on both present and previous experiments.~2 According to the recent transient heat transfer measurements of Schrnidt 14 ON (K) actually varies with heat flux, Q(W cm -2) as ON = 0.2 Q; namely, 0.2 K at 1 W cm -2 and 1 K at 5 W cm -2. As will become evident in later discussion, if this relationship is indeed valid, it becomes more difficult to satisfy the CCM criterion with disturbances transmitting large heat fluxes at the conductor surface. A practical remedy for this will be discussed later.
Transition Temperature. The transition temperature of a conductor is a decreasing function of current and field. Based on NbTi data of Hampshire, Sutton, and Taylor ~s typical curves showing dependences of transition temperature on currcmt at four levels of magnetic field are shown in Fig. 2. For simplicity on computation, they are all approximated by straight lines. In the figure each 0e, measured from the bath temperature, is a transition temperature in a given field with zero transport current; I c is the critical current in a given field at bath temperature. Note the location of 0N (Fig. 1) in the temperature abscissa. Region of stable operation. Suppose a conductor,
8h
f
I qc (ec) q (e)
Fig. 1 Temperature 8, measured f r o m the bath temperature, vs heat f l u x (and energy f l u x ) . The dotted curve shows a simplified function
706
characterized by Fig. 2, is subject, on a unit surface basis, to a transient disturbance that deposits energy flux e. Let us further suppose that it is carrying a transport current equal to It(B4): lop = It(B4). For the purpose of discussion, the simplified heat transfer curve (the dotted curve in Fig. 1) is used. When the conductor is in field Bz, its transition temperature is clearly greater than ON. Thus the conductor is completely undisturbed and remains completely superconducting for disturbances up to ec; above ec it is driven normal. The same is true in field B2.
CRYOGENICS. DECEMBER 1979
]'c(B 0
of attraction (BOA)' discussed recently by Wipf. 16 Thus a magnet designed to generate a field of B1, fbr example, can be operated with the same degree of stability either with an operating current of It(B2) or Ic(Bs). If a maximum disturbance in the winding exceeds ec, both designs will fail; if not, both will succeed. Thus it obviously pays to operate the magnet with Ic (B2) irrespective of a resultant current density in the stabilizer. This conclusion will be further discussed below.
~ 1
/~(82) /¢(83 )
The same argument applies to steady-state disturbances. Fig. 3, therefore, is also applicable when energy flux is replaced with heat flux.
I 8 I
2"o(B4}
I
B2
Critical current margin. In addition to requiting a nucleate boiling heat transfer in the magnet winding, this design criterion requires what we now det~me as the critical current margin. This is best explained graphically, as illustrated in Figs. 4 and 5.
83 I 8.
°o
ON
0c(84)
0~(83)
0~(82)
0°(80
eo Fig. 2 Critical current vs transition temperature lines at d i f f e r e n t magnetic fields
In field B3, the transition temperature just equals ON and above B3 it is less than ON. Thus in fields above Ba the conductor cannot tolerate any disturbances. Accordingly, the region of stable operation may be represented in an e vs B plot. For this particular example the region is a rectangle, bounded by lines of ec and B3, as illustrated by the dotted lines in Fig. 3. When a more realistic heat transfer curve is used, the solid curve in Fig. 1, the boundary is modified, as indicated by the solid curve in Fig. 3. At a higher operating current, for example Iop= Ic (B3), the boundary of stability is reduced in the B axis to//2, while in the e axis it is virtually unchanged. The crossed line in Fig. 3 represents the new boundary line.
Fig. 4 shows the magnet load line and two possible critical current curves of the conductor. This load line actually relates current and the maximum field at the conductor at some point in the magnet. As for the critical current curves, in curve 1 the critical current equals operating current at Bin; in curve 2 it exceeds the operating current by a certain amount. Given a disturbance, according to the above discussion, with curve 1 the magnet will fail to generate a designed field of Bm; with curve 2 it will succeed, provided there is a sufficient temperature margin. This point is more clearly illustrated in Fig. 5, which shows I vs 0 plots in Bm for curves 1 and 2. Namely, our new design criterion for stability is able to provide a sufficient temperature margin at the operating point (Bm, lop) so that the transition temperature of the conductor at the operating point is at least 0N above bath temperature. To have this temperature margin, the conductor critical current at the operating point must at least be I:.
A unique conclusion about stability can be drawn based on this criterion. Namely, stability, if it can be quantified in terms of maximum permissible disturbance flux, ec, is independent of operating current up to a certain level. This maximum permissible disturbance is analogous to the 'basin
fop =/e (B4) . . . .
(Simplified)
zop=z~(B 3) ++++ "° (q¢)-
Steble
°i
+ "+ + + + + + -4+ 4+
8', ÷
I
I
I I
8
B
B
Fig. 3 Disturbance energy vs field plots showing the region o f stable operation
CRYOGENICS.DECEMBER1979
Bm
B Fig. 4
Load line and t w o short-sample current plots (1) and (2)
707
I~ ~
At Bm iM =
o©
#=(am) -as 8©{Bm)
/
S'lable region in field
Bm
/
\N 0
0
/M
I
e Fig. 5 Current vs transition temperature curves for two shortsample current plots of Fig, 4 at field Bm. The dotted line is for short-sample current plot (1) at field B~n< Brn
Fig. 6
An analytical expression for ~ is derived below using a linear relationship between transition temperature and transport current in a given B. Namely in B = Bin;
Oe(I,B i n )
=
Oe(Bm)Ie(Bm)
(Bm)
This is because ON, which depends chiefly upon bubble dynamics and growth, decreases with bath temperature. Still it is recommended that NbTi magnets be operated at 4.22 K rather than at 4.45 K. (1)
Our design criterion requires:
Oc(I=Iop, Bm) ;* ON
(2)
Combining (1) and (2) and solving forfc, we obtain /c (Bin) ~ Iop
0c(Bm) - - ON
(3)
0c(Bm)
The critical current margin, (i~ -Iop), is thus given: Ic (Bin) - I o p ;~ Iop
ON 0e (Bm) -- ON
(4)
The maximum permissible operating current may be expressed in terms of critical current: /°--Z -- /m = 0 ° ( S m ) - 0S ,r • 0c(Bm)
(5)
The region of stable operation can also be represented in an e vs (Iop[I*c)plot, as shown in Fig. 6. Again this is a simplified picture based on the dotted heat transfer curve of Fig. 1. An operating current that corresponds to point A in Fig. 6 should certainly be avoided; a slight reduction in operating current will result in point B, with a large gain in the degree of stability. Note that when 0c(Bm) = 0N, which is reached at some high field, this criterion cannot be used. For 0N ~ 0.3 K and the conductor is NbTi, this field is about 10 T when the bath temperature is 4.22 K. In a bath temperature of 4.45 K, which is the case when the system is connected directly to a liquefier, this limiting field becomes 9.5 T, although there is a likelihood that ON may be slightly less,17 and therefore the limiting field could be higher than 9.5 T.
708
The region of stable operation in an e vs (Iop/l¢) plot
From the above, the necessity of using Nb3 Sn in high fields becomes quite obvious; it may even be advantageous in fields as low as 8 T. Returning to Fig. 4 if the magnet under discussion is wound with a curve 1 type conductor, it satisfied the criticalcurrent-margin and thus is stable for fields up to Bin< Bin; at this point 0c(~op ' Bm) = ON. This is illustrated by the dotted lines in Figs 4 and 5. In essence, this criterion traces its origin to the stability work of Stekly,Is in which this mode of fully-super. conducting operation under steady-state heating was first examined in detail. However, Stekly's work still placed an emphasis on stable recovery from the normal state and matching an operating current with a critical current.2 To achieve stable recovery, a generous amount of normal metal stabilizer was added to the superconductor. In contrast, in this design criterion the emphasis is placed totally on the fully superconducting region; consequently, more superconductor is used and less stabilizer. This criterion is quite different from the nucleate cooling stability criterion of Whetstone and Boom, proposed in 1967.19 Their criterion, like Stekly's and many other criteria that followed, emphasizes recovery, and therefore it its maximum permissible normal-state heat generation is limited and not high. In their case it is about 0.8 W cm-,2 equal to a maximum possible cooling rate in the nucleate boiling region.
Comparison with the MPZ/Cold-EndStability Criterion. For stability of high-performance magnets, in addition to the adiabatic theory either the MPZ/Cold-End theory or this CCM design criterion may be used. We shall demonstrate here that, given an uncertainty in disturbance size in the magnet, the chances of success for a magnet are far greater
C R Y O G E N I C S . DECEMBER 1979
based on this criterion than on the MPZ/Cold-End theory. This is done by comparing the size of a permissible energy input for the two cases at a field of 8 T for a given conductor/cooling configuration. Implicit in the argument is the idea that the chances of success are better the greater the size of a permissible disturbance energy. For the sake of comparison, we shall assume a disturbance to be limited to an MPZ length, and compare permissible energies derived from the two criteria on a unit volume basis. According to the MPZ/Cold-End energy computation of Wilson, a'which neglects any transient cooling, the maximum permissible energy density, at a given ambient field, is a sharply decreasing function of normal-state heat generation. Table 1 gives maximum permissible energy densities for a one-dimensional conductor at 8 T at selected values of normal-state heat generation. Note that the total permissible energy over an MPZ length, which is given by the product of this energy density and MPZ length and conductor cross-section, decreases even more sharply because MPZ length itself shrinks with normal-state heat generation. In contrast, the maximum permissible energy density according to the CCM criterion is given by:
Ec - eeP A
(6)
where P is the conductor cooled perimenter, A is the conductor cross-section, and ec is the critical energy flux previously discussed. The above equation shows that: the permissible energy density for the CCM criterion is independent of normalstate heat generation, and it increases as P/A, favouring small conductors or cabled conductors, such as those of the Rutherford type or bundle conductors. 2° Although not explicit in the equation, in the CCM criterion disturbances need not be restricted to an MPZ length. If a nucleate boiling heat transfer can be maintained and a maximum energy density is limited to Ec, disturbances can occur globally in a magnet. To complete the numerical comparison with the MPZ/ColdEnd criterion, we shall choose a relatively large conductor with P = 2 cm and A -- 0.25 cm 2, which results in Ec of 80 ~ 400 x 10-3 J cm. Based on the above idea that the amount of permissible energy input is a measure of stability, we conclude that the present criterion offers a much higher degree of stability than the MPZ/Cold-End theory.
Table 1. Maximun permissible energy densities, at 8 T and 4.2 K, for selected normal-state heat generations, based on the MPZ/Cold-End Criterion Normal-State Heat Generation, W cm -2
Energy Density, 10-3 J cm--3
0.3 0.5
40 6
1
2
3
1
CRYOGENICS. DECEMBER 1979
Remarks on conductor, cooling, and disturbance decoupling
Conductor configuration. As remarked above with regard to (6), the degree of stability can be improved by increasing the ratio of cooled conductor perimeter to cross.section. Thus the CCM criterion would work particularly well with cabled conductors such as those of the Rutherford type or bundle configurations. 2° Matrix material. Since the success of this criterion depends critically on keeping a temperature rise in the superconductor low, it might be beneficial to put NbTi f'daments into a matrix material of low thermal conductivity such as stainless steel or cupronickel to reduce the effect of heating at surfaces. A new composite conductor, of course, must still have a layer of copper at the surface to disperse heat. The proposed conductor works only when there is no appreciable heat dissipation within filaments. Surface condition. Surface condition affects ON.. As pointed out earlier, a thin layer of insulation seems to increase ON drastically, making it more expensive, if not impossible, to meet the CCM criterion, because an increased ON means a higher critical current margin. For this design criterion to be most effective it is suggested that the conductor surface should be bare and be kept clean. The thickness of an oxidation layer, for example, should be kept to a minimum. In this regard, it may be prudent to put a layer of copper on aluminum-stabilized conductors. Also, the effect of a long exposure to the atmosphere on conductor surface conditions should be examined, especially during a long production period. For large magnets such as those for MHD and fusion that are expected to remain in the 4.2 K environment for a long time, we can expect surface condition to remain unchanged once the magnets are energized. Other cooling schemes. Operation in the supercritical state becomes appealing in the CCM design criterion, particularly with Nb3 Sn. 21 In the supercritical state there is no sudden change in the wall temperature owing to a single-phase operation and heat transfer rate is roughly proportional to the temperature difference between the conductor wall and coolant. By driving the wall temperature higher, substantially more energy may be removed; such a scheme is more practical with Nb3 Sn conductors. Another attractive scheme in this regard is operation in pool-boiled helium II. 22-24 According to Van Sciver's17 heat transfer measurements, a large heat flux (up to 2 W cm -2) is possible for ON ~ 1 K in subcooled He II at 1.9 K. Even with such a high ON, the CCM criterion is applicable for NbTi at fields as high as 10 T, because at a bath temperature of 1.9 K, NbTi at 10 T has a temperature margin of ~3 K.
Disturbance decoupling. As mentioned earlier in connection with the Schmidt's results on 0 vs Q, it becomes increasingly difficult to satisfy the CCM criterion with disturbances transmitting a large heat flux on conductor surface because of a large.resultant ON. A large heat flux may be transformed to a small heat flux when it is first transmitted through a medium having-a low thermal diffusion constant. That is, disturbances can be decoupled from the conductor surface. One practical suggestion might be to attach rigidly to the conductor an insulating material such that all the disturbances may be deposited to this insulating material rather than directly to the conductor surface.
709
Experiments and discussion
Rotatalde shutter plate
Experiments were carried out to confirm some key assumptions made and to verify the conclusions reached concerning use of the new design criterion.
Test conductor and sample holder Starting with a composite-copper.NbTi conductor of square cross-section (4.6 mm by 4.6 mm), we simultaneously wrapped sixteen 0.53 mm diameter bare copper wires tightly, leaving no gap between wires, over the superconductor. Pitch length was 9.8 mm (Fig. 7a). After soft-soldering the wire to the conductor, we removed every other wire, creating helical channels, each 0.53 mm wide and 0.53 mm deep, around the conductor (Fig. 7b). The original square conductor had a copper to superconductor ratio of 1.2. As indicated in Fig. 7c, two insulated heater wires, each of 0.45 nun diameter were wrapped, and epoxied, around the conductor by placing each wire in a channel. The two heater wires were connected at one end, thereby creating a noninductively wound heater. The total conductor length between terminals was 600 mm and the heated region which covered the middle section was 110 mm long. Over the heated section, computed cooled surface per unit conductor length, including contributions by the wrapped wires, is 2.3 cm 2/era. Measured normal-state resistances per unit length near 4.2 K were, in/aL2 cm -1, 0.17, 0.24, 0.33, and 0.38 respectively, in fields of 2 T, 4 T, 6 T, and 8 T. The test conductor was placed in a 188 mm diameter circular channel machined in a phenolic plate (Fig. 8). The channel was of a square cross-section (5.7 mm x 5.7 mm) and a series of 5 mm diameter vent holes was machined at 20 mm interval in the bottom of the channel along its entire length to provide liquid helium access.
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Fig. 8 Sketch o f the test conductor sample holder showing t w o cover plates
Similarly, a phenolic cover plate had matching vent holes machined for liquid helium. In addition, to investigate the effect of the helium flow on cooling, a rotatable second cover plate with an identical set of vent holes was placed on top of the first cover plate (Fig. 8). The relative position of the holes in the two cover plates controlled the helium flow. A mechanical linkage outside of the dewar permitted this control during the experiment. During the experiment two positions were used for the vent holes: open, when the two plate holes were lined up; and the other, closed, when they were completely blocked. The bottom vent holes were always open. A set of voltage taps was distributed along the test conductor length. Four Au-Fe/chromel thermocouples were attached directly to the conductor at a helical channel midway between the heated channels as indicated in Fig. 7c. Each thermocouple was held to the conductor surface by a drop of Stycast 2850 epoxy.
Procedures, results, and discussion Two sets of data were collected on the test conductor: heat transfer and quench.
Heat transfer (transient). The test conductor was heated by passing a pulsed current through the heater and its temperature response was measured with the thermocouples. To eliminate current and field noise, measurements were done in the absence of field and transport current.
Thermocouples
Heater - - - - -
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Fig. 7 Sketch o f the test conductor, a -- Sixteen copper wires wrapped; b -- every other copper wire removed; c -- heat wires placed in helical channels with one set of thermocouples attached at the middle channel
710
Fig. 9 shows a typical oscillogram of three heater current traces, labelled A, B, C, and corresponding temperature traces, labelled a, b, c, all as a function of time. Because of a non-zero diffusion time between heater and test conductor, a heating rate applied in the heater was always greater than a heating rate in the conductor while the heater current was on. Heat thus accumulated in the heater and was released to the conductor even after heating was stopped in the heater. This accounts for the delayed
CRYOGENICS. DECEMBER 1979
0.3K. The diffusion time and other factors such as the exposure of a portion of the heater wire to helium may very well affect ec, and thus when heating is applied differently, for example by frictional means, a different e c may be obtained. The difference however, is likely to be smaller than an uncertaintly in an estimated size of distrubance: a precise value of ec is not too crucial.
Heat transfer (steady-state). Fig. 11 shows actual steadystate temperature vs heater current traces obtained under both open-hole and closed-hole conditions. Temperature traces are plotted against a linear heating scale for the two cases i~ the figure inset. Here the importance of helium access is clearly demonstrated.
Quench (critical current measurements}. Critical currents of the test conductor were measured in three fields using a standard four-probe technique, and are shown in Fig. 12. Fig. 9 Oscillograms showing heater current traces, labelled A,B, C, and corresponding temperature response traces, labelled a, b, c. Vertical scale temperature is 0.75 K/div; time scale is 10.7 ms/div
temperature responses observed in Fig. 9. The time scale was 10.7 ms/div. Temperature vs input energy flux traces, like the one shown in Fig. 1, can be plotted from data such as that presented in Fig. 9. Energy is expressed per unit conductor cooled surface. Fig. 10 shows results obtained with heating pulses each lasting 20 ms. The open circles correspond to results with the helium vent holes open, while the solid circles refer to results with the holes closed. Helium flow appears to have little effect on the results. Apparently only the liquid helium within the channel contributed to cooling. As we shall see below, this is not the case in steady-state heating. The dotted line shows an apparent curve inferred from quench results shown in Fig. 15. The discrepancy will be discussed later.
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CRYOGENICS.
DECEMBER
1979
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Critical current vs field data for the test conductor at
711
Three current levels indicated by the dotted lines in the figure are those used in quench measurements.
Quench (transient heating): In these measurements, a maximum permissible input energy flux was measured for a given set of field and transport current. Fig. 13 shows typical oscillogram traces corresponding to a field of 6 T and a transport current of 4 kA. The helium vent holes were open. There are four traces: the top two nearly identical traces, labelled A and B, are heater currents, while the bottom two traces, labelled a and b, are conductor voltages. Two time scales were used: 10.7 ms/div until the end of heating and 53.5 ms/div after the end of heating. With the heating current corresponding to trace A, the test conductor remained completely superconducting (trace a); at a slightly higher heating rate (trace B), the conductor was driven normal at the heated section and the normal zone propagated, followed by self heating. The normal-state heat generation corresponding to this field and current was 2.3 W cm -2 before the self heating had taken place. The computed critical energy flux for this set is 27.5 mJ cm -2, in terms of energy density, it is 300 mJ cm. -° This is very close to the critical value found in the transient heat transfer measurements. In a field of 8 T with 4 kA, according to the earlier discussion, we expect critical energy flux to drop drastically, because 4 kA is too close to the critical current of 4.3 kA at 8 T. This was indeed proven in the experiment as demonstrated in Fig. 14. Here the critical energy density is 32w mJ cm -3" Fig. 15 summarizes the critical energy fluxes obtained in different fields with transport currents of 3 kA, and 5 kA. Open data points are for the experiment with open vent holes, and the solid data points are for that with closed vent holes. Note that Fig. 15 closely resembles Fig. 3.
Fig. 14 Oscillogram similar to ~:ig. 13 except here data was taken at 8 T with 4 kA. The conductor quenched at a much smaller heat input compared with the case shown in Fig. 13
about 65 x 10 -s J cm-2; instead it was 33.5 x 10 -9 J cm-2; with a transport current of 3 kA, which was, although not measured, much less than a corresponding critical current. A heat transfer curve more consistent with quench results is shown by the dotted line in Fig. 10. Apparently, the conductor surface temperature in the film boiling region measured at a location not exactly at the heated surface is less, especially at high transient heating rates. As an illustration of the application of this design concept, let us consider this conductor for an 8-T magnet. Based on results of Fig. 15 and our discussion, maximum permissible disturbance energy vs operating current, similar to Fig. 6, may be plotted for the conductor at 8-T; this is shown in Fig. 16. The abscissa also indicates normal-state heat generation g.
Consistent with the heat transfer data of Fig. 10, the vent holes have no effect on these results. According to the data of Hampshire, et al.,is the transition temperature at 2 T, 0c (2 T), is about 4K. Thus, based on transient heat transfer data of Fig. 10, ec should have been
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712
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C R Y O G E N I C S . DECEMBER 1979
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will have computed normal-state resistance of 0.5/z~2 cm -1. Because of this increased resistance, normal-state heat generation will also be increased for a given operating current, as indicated by the abscissa g' in Fig. 16. Note that the amount of copper may be reduced as long as heat transfer is unaffected and the conductor thermal diffusivity remains close to that of copper. The expanded stability regions is shown by a dashed line in Fig. 16.
Quench (steady-state heating). The experiment was also Absolute stability
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Based on (5) and 0c(8 T) = 1.4 K, 0N - 0.3 K, and Ic(8 T) = 4.3 kA, the range operating current must be limited to 3.35 kA, as indicated by the solid vertical line in Fig. 16. Up to this operating current, a maximum permissible disturbance pulse, according to the data point of 3 kA, is 24 mJ cm -2 for each conductor cooled surface (left ordinate in Fig. 16) and 270 mJ cm -3 per unit conductor volume (right ordinate in Fig. 16). Beyond 3.35 kA, disturbances must be limited drastically, to zero according to the simple model of Fig. 6. For comparison, maximum permissible disturbances according to the MPZ/Cold-End theory are also plotted in the figure. Note that for operating currents below 1.5 kA, the magnet is also in the Cold-End stability region.
performed with steady-state heating. At a given field and transport current, a slowly increasing current was passed through the heater until a sudden transition from a completely superconducting state to a xompletely normalstate was induced in the test conductor. Results, similar to those of Fig. 15 except that here the ordinate is critical heat flux, are summarized in Fig. 17. The two sets of results resemble each other remarkably, as predicted earlier in the text. Here the effect of helium circulation is clearly evident. The implication of the CCM design criterion to steadystate disturbances, however, is less crucial, because, unlike transient disturbances, steady-state disturbances are more predictable and tractable, and thus amenable to the designer's control. These sets of results together with those of heat transfer measurements (Figs 9 and I 0) and critical current measurements (Fig. 11) conclusively support the basic concept of the CCM criterion. Other comments Although much less so than with the MPZ/Cold-End theory, the amplitude of disturbances is critical in this critical-current-margin design theory. Therefore, advances in understanding the nature of disturbances that enable one to control or limit disturbances in the magnet winding should greatly benefit this criterion. Recently one such step in this direction has been achieved concerning frictional heating, considered to be a principal source of disturbances in the winding. A series of measurements of frictional qc 0.6--
Two points from Fig. 15 are shown in Fig. 16. Certainly a complete version of this e vs lop plot at a maximum design field should be a useful set of stability data for the magnet designer. Suppose we wish to operate this 8-T magnet a 4 kA and thus wish to extend the range of stable operation to 4.4 kA. According to this design criterion, the conductor critical current at 8 T must be increased from 4.3 kA. Since 0c(8 T) ~ 1.4 K, and ON ~ 0.3 K, we find, from (3): 1~ (8 T) ~ 4400 1.4
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CRYOGENICS.
D E C E M B E R 1979
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713
properties of winding materials in the 4.2 K environment 2s-27 has revealed certain material combinations to be unquestionably superior from the consideration of frictional heating. Another relevant topic concerning this criterion is magnet protection. Since recovery does not take place in a magnet of this design, a normal zone, however small, will propagate rapidly throughout the entire magnet. The protection problem, therefore, might be as straight forward as it is for adiabatically stable magnets. One point that should be stressed is that a normal voltage within the winding could become excessively high if too much copper is removed. For those die-hard conservatives, a belt-and-suspender approach is recommended: a design criterion that combines the present criterion and the absolute stability criterion. In this approach enough superconductor is used to satisfy one criterion and at the same time enough copper is used to satisfy the other criterion. Fig. 18 summarizes the five stabilization criteria discussed above - cryostatic, cold-end, MPZ[cold-end, CCM, and adiabatic. The abscissa represents normal-state heat generation per unit cooled conductor surface, which is used to def'me the region of operation for each stabilization technique. The ordinate represents, qualitatively, the size of disturbance permissible under each stability criterion. For heat generations up to 0.3W cm, -2 the size of disturbance can be large. Beyond 0.3 W crn -2 and perhaps up to 1 W cm -2, the MPZ/cold-end criterion may be used for localized disturbances. As shown by the horizontal line that extends ~ 10 W cm -2,the CCM criterion can be used alone or in combination with the cryostatic, cold-end, or MPZ/coldend criteria. As remarked above, up to 0.3 W era,-2 the CCM criterion joins the cryostatic stability criterion to become a beltand.suspender approach, and beyond 0.3 W cm -2 and up to 1 W cm -2, the contribution of the MPZ/cold-end may be included, provided disturbances are localized. The dotted curve that joins the horizontal line at around 1 W cm-2 indicates this combination. Beyond ~ 10 W cm -2 , the adiabatic criterion seems the only practical approach.
Although it does not guarantee absolute recovery, the degree of stability provided with this criterion is much higher than that with either the MPZ/cold-end criterion or the adiabatic criterion. Its most unique and interesting feature is that stability is independent of operating current density in the stabilizer. It was shown experimentally that cooling channels have little effect on transient disturbances. Perhaps for the immediate future, this CCM criterion may most profitably be used for large test magnets required for MHD and fusion programmes. The cost of such magnets, intermediate in size, could be substantially reduced by basing designs on this criterion. Finally this work has demonstrated the power of a 'stability' experiment' that integrates both heat transfer and quench measurements. I wish to express my thanks to J.F. Maguire, J.F. Johnson, and L. Lyon for assistance in the experiments; D.B. Montgomery, M.J. Leupold, R.J. Thome, J.E.C. Williams, and V. Arp for helpful conversations; and A.M. Dawson for improvement of the manuscript.
References
1 2 3 4 5 6 7 8 9 10
Conclusions
The critical-current-margin design criterion presented here is recommended for magnets of high current densities.
11 12 13 14 15 16 17 18 19
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714
23
24 25 26 27
Maddock,B.J., James, G.B., Norris, W.T., Cryogenics 9 261 (1969) Kantrowitz A.R., Stekly, Z.J.J.,Appl. Phys. Lett. 6 (1965) 56 Wilson,M.N., lwasa, Y., Cryogenics 18 (1978) 17 Martinelli,A.P., Wipf, S.L., Proceedings of the 1972 Applied Superconductivity Conference, IEEE 72 CHO682-5-TABSC, 331 (1973) Wang,S.T., et al., Proceedings 6th International Conference on Magnet Technology, Bratislava, Czechoslovakia, Bratislava, (1978) 157 Superczynski,MJ., IEEE Trans. Magn. MAG-15 (1979) 325 Wilson,M.N., Liquid helium technology Pergamon Press, Oxford (1966) 109 Tsukamoto, O, Kobayashi, S.,J. Appl. Phys. 46 (1975) 1359 Iwasa,Y., Leupold, MJ., Williams,J.E.C., IEEE Trans Magn MAG-13 (1977) 20 Nick, W., Krauth, H., Ries, G., IEEE Tram Magn MAG-15 (1979) 359 Arp, V., (private communications). Iwasa,Y., Apgar, B.A., Cryogenics 18 (1978) 267 Iwasa,Y., Apgar, B.A., unpublished data (1977) Schmidt,C., Appl Phys Lett 32 (1978) 827 Hampshire,R.G., Sutton, J., Taylor, M.T., Supplement au Bulletin de L'Institut International du Froid Commission I. Londres, Annexe, (1969) 1 Wipf,S.L., IEEE Tram Magn MAG-15 (1979) 379 Van Seiver, S.W., Cryogenics 18 (1978) 521 Steldy, ZJ.J.,JAppl Phys 37 (1966) 324 Whetstone,C.N., Boom, R.W.,Advances in Cryogenic Engineering, 13 K.D. Timmerhaus, Ed, Plenum Press, New York, (1968) Hoenig,M.O., Montgomery, D.B., IEEE Tram Magn MAG-I1 (1974) 569 Hoenig,M.O., Montgomery, A.G., Waldman, SJ.,IEEE Trans Magn MAG-15 (1979) 792 WisconsinSuperconductive Energy Project, 1 University of Wisconsin, Madison (1974) Kobayashi, H., Yasukoehi, K., Tokuyama, K., Proc ICEC 6, Grenoble (1976) 307 Claudet,G., Meuris, C., Parain, H., Turck, B., IEEE Trans Magn MAG-15 (1979) 340 lwasa,Y., Kensley, R., Williams, J.E.C., IEEE Trans Magn MAG-15 (1979) 36 Kensley, R.S., S.M. Thesis MIT (1979) unpublished Kensley, R.S., Iwasa, Y., (to be published)
C R Y O G E N I C S . DECEMBER 1979