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Normal zone propagation in adiabatic superconducting magnets Part 1: Normal zone propagation velocity in superconducting composites
Normal zone propagation in adiabatic superconducting magnets Part 1: normal zone propagation velocity in superconducting composites* Z.P. Zhao* and Y. Iwasa Francis Bitter National Magnet Laboratory and Plasma Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 31 January 1991 A normal zone propagation model has been developed for superconducting composites under adiabatic conditions. It is based on the Whetstone-Roos model, originally developed for normal zone propagation in adiabatic wires of unclad superconductor. The model takes into account the temperature and magnetic field dependent material properties, for both superconductor and matrix metal. Analytical results agree well with experimental data.
Keywords: superconducting magnets; superconducting composites; normal zone propagation Normal zone propagation in superconductor wires depends on the balance between Joule heating within the normal zone and cooling at the normal zone surface. Consequently, a normal zone may: 1, grow because the Joule heating exceeds the cooling; 2, shrink because the cooling exceeds the Joule heating; or 3, remain constant because the Joule heating is equal to the cooling. Case 1 is generally true for adiabatic magnets. Case 2, on the other hand, is generally applicable to cryostable magnets. Several general reviews of the theoretical analyses of normal zone propagation are available ~-5. There are many solutions for normal zone propagation velocity under a variety of conditions. However, most models neglect the temperature and magnetic field dependences of material properties. An exact model, proposed by Whetstone and Roos 6'7 in 1965 for unclad superconductors (conductors without a normal metal stabilizer) under adiabatic conditions, takes into account the temperature dependence of material properties. The Whetstone-Roos model treats single wires of unclad superconductor. Consequently, it is not of direct use for most applications because superconductors of technological interest are composites consisting of superconducting filaments in a matrix of normal stabilizing metal such as copper. Recently Joshi and Iwasa ~'2 applied the model to a magnet quench simulation code.
Extending further the work of Whetstone and Roos and of Joshi and Iwasa, we propose a model applicable to superconductor composites under adiabatic conditions. This model takes into account the temperature and magnetic field dependent material properties. The analytical results of normal zone propagation velocity are compared with experimental data on N b - T i and Nb3Sn composites.
Normal zone propagation velocity The Whetstone-Roos model, among others ~-5, assumes that the normal-superconducting boundary moves at a steady velocity along the wire, with the boundary moving continuously into the superconducting region where the transport current and magnetic field are below the critical values. The model assumes the process to be adiabatic. The Joule heat is conducted forward along the wire as a travelling temperature wave and the overall rise in the wire temperature causes the normal-superconducting boundary to advance into the superconducting region. By neglecting the second-order derivative of temperature in the normal region near the normalsuperconducting boundary, Whetstone and Roos derived an explicit expression for normal zone propagation velocity UI6
U~=j p(T)-K.(T) C.(T) *Based in part on the Doctoral Thesis of Z.P. Zhao submitted to the Graduate School of Engineering, Tokyo Denki University, Tokyo, Japan, December 1990 *Visiting scientist from Tokyo Denki University, Tokyo, Japan
×
f ,,
7"=
C~(T)d
,]S ,,
7"=
K.(T)
dT
C~(T)dT~
(1)
) T = T¢
0011 - 2 2 7 5 / 9 1 / 0 9 0 8 1 7 - 09 © 1991 B u t t e r w o r t h - Heinemann Ltd
Cryogenics 1991 Vol 31 September
817
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa
where K, C and P are, respectively, the thermal conductivity, volumetric specific heat and normal state electrical resistivity of the superconductor. Subscripts n and s stand for the normal state and superconducting state, respectively. Tc is the superconductor's critical temperature and T® is the ambient temperature. If K and C are taken as constant, Equation (1) represents the special case of the constant-property problem which has been investigated extensively4. In the original analysis of Whetstone and Roos 7, they derived the minus sign for Equation (1) in their appendix, but a plus sign was used in their text 7 where velocity calculations and comparison with experiments were presented. The velocity formula given by Joshi and Iwasa, who based their work on the formula given in the main text of Whetstone and Roos' paper, uses a plus sign instead of the minus sign in Equation (1). The Joshi and Iwasa formula has appeared in later publications g-H. The WhetstoneRoos model is analysed herein and a detailed derivation of Equation 1 is presented in Appendix A. In the temperature ranges T,~ OD and T > OD (Reference 12), where Oa is the Debye temperature, the relationship between thermal conductivity, Kn, and electrical resistivity, p, for ordinary metals is A T = K~p, where A is the Lorenz number. This relation is known as the Wiedemann-Franz law. In the Sommerfeld models for metals in zero magnetic field, the value is A = x2k2/3e2 = 2.45 x 10 -s V 2 K -2, where ks is the Boltzmann constant. In order to obtain a more explicit expression for velocity, U~, than that given by Equation (1), the temperature dependence of electrical resistivity, p, can be neglected for a small temperature range. Thus, the following substitution can be obtained if p is taken as constant 1 d K . = _1
g. d r
(2)
r
Applying this substitution to Equation (1), we obtain
Vl=j
[A[
c.(r)
r.
jr=to
(3)
From the analysis in Appendix A, it is clear that this solution is obtained under the implied assumption that (CnUt) 2 -- 4pj2 dK. ~ AWR > 0
dr.
(4)
where AwR is a parameter used in the following discussion. The condition given in Equation (4), however, has been ignored in earlier analyses. If AwR = 0, the propagation velocity may be expressed as
where - -
I f
c"(rc)= Kn(Tc) dT r=r, r.
Cs dT
(6)
If AWR < 0, then Equation (A6) of Appendix A will yield two imaginary solutions. This means that Equation (A6) cannot be used to describe the thermal conduction process in the normal region. The inequality AWR < 0 is equivalent to a negative value under the square root in Equation (1). Therefore the condition required in this model for describing normal zone propagation in an adiabatic superconducting wire may be expressed as
r f
-Cs OT C,(T¢) > Kn(T) dT r= ro r=
(7)
or, by substituting Equation (2) co(to)
>
Cs dT
(8)
From these analyses of the Whetstone-Roos model, it is clear that the solution of normal zone propagation velocity under adiabatic conditions is obtained through the following assumptions 1 The condition required in Equations (7) or (8) is implied. This inequality, however, is not a unique result of the model. 2 The normal state resistivity, p, of the superconductor wire is taken as constant for Equation (2). The thermal and electrical transport properties of superconductors at low temperatures, along with their effects on normal zone propagation, have to be further investigated. 3 The model is limited to unclad superconductors. Superconductors used in magnets are composites. These composites consist of multiffiaments of superconductor clad in a copper or aluminium matrix. Joshi and Iwasa extended this model to composites but, as mentioned above, their velocity formula was identical with Equation (1) except that a plus sign instead of a minus sign was used. In the absence of a magnetic field or transport current, the superconductor transition occurs at the critical temperature, Tc. In the presence of a magnetic field or transport current, the transition occurs at a temperature below Tc and the latent heat density is TdzoHc(dItc/ dT), where p-0 is the free space permittivity. By neglecting the latent heat, we might underestimate the propagation velocity for small transport currents where the Joule heating, pj', is comparable to the latent heat. This is, however, generally not a problem for normal zone growth in superconducting magnets. Validity of Equation (8)
Ui
=
818
Cn(T)
(,
dT ]
I
T=r,
Cryogenics 1991 Vol 31 September
(5)
Equation (8) gives the condition that must be satisfied for the Whetstone-Roos model. To verify the validity of this condition for N b - T i and NbaSn superconduc-
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa tors, the specific heats of the superconductors in both the normal and superconducting states are considered. Specific heats of superconductors. In the absence of an ambient magnetic field, the observed volumetric specific heats of a superconductor in the normal state, Cpn, and superconducting state, Cp, are, respectively Con = 3"T +/3T 3
(9)
Cp~ = aexp ( _ . b£BI,/~ + ~ T 3
(10)
where 3'T and BT3 represent, respectively, the electronic contribution and the lattice contribution to volumetric specific heat. As given in Equations (9) and (10), the lattice specific heat is the same in both normal and superconducting states, but the contribution from the conduction electrons is quite different. In the presence of a magnetic field, the superconducting specific heat will be dependent on both temperature and magnetic field. By using the G6rter-Casimir theory and by approximating the magnetization curves of type H superconductors with straight lines, Elrod et al. ~3 have derived an expression that relates the specific heat in the superconducting state to 3', To/3T 3 and the upper critical magnetic field, Hc2(0)
3"T +
3 BT2]BT
(11)
Specific h e a t s of Nb - Ti a n d Nb3Sn. Most specific heat data of Nb3Sn z4-~9 are those measured under zero magnetic field. There appear to be significant uncertainties associated with specific data measured in the presence of a magnetic field. The only available experimental data at high magnetic fields for N b - T i seem to be those of Elrod et al. Applying their data to Equation (11), we may express the volumetric specific heats of N b - T i in the superconducting and normal states in magnetic fields as
experiments 26-2s. However, a more recent study by Khlopkin 3° over the temperature range 4 . 5 - 7 0 K in magnetic flux densities up to 19 T gives the coefficient 3' = 0.988 mJ cm -3 K -2 for the Debye temperature, On = 232 K. One of the consequences of the smaller value of 3' proposed by Stewart et al. and by Khlopkin is that the parameter AC/3"Tc is usually large, 3.5 4- 0.3. Khlopkin's latest data are employed in the present analysis. The value of 3, given here is similar to that of Junod et al. 17.18 (1.28 mJ cm -3 K -2) and of Vieland et al. 26 (1.18 mJ cm -3 K-z), both measured in the absence of ambient magnetic fields. Applying Khlopkin's data to Equation (11), the temperature and magnetic field dependence of the volumetric specific heat of Nb3Sn may be described by the following polynomial Csb~Sn[J cm -3 K - t ] 0.988 x 10-3T + 1.388 x 10-ST 3, lO
T<-L
+
(13) The normalized value of the specific heat jump at the superconducting transition, A C / T T ~, is 2.5 in Khlopkin's calculation, which is considerably higher than the value of 1.43 given by BCS theory. Verification o f Equation (8). Based on the above discussion of the specific heats of N b - T i and Nb3Sn, two methods are used to analyse the validity of Equation (8). A b s e n c e o f m a g n e t i c field. Applying Equations (9) and (10) to Equation (8) yields ,aw. = C.(Tc) - ~1 l ro ~= Cs(T) dt
=
+
oo
-- 1
rc a • e-b/ks r dt
CNb-Ti [J cm -3 K -1]
~
0.81 x 10-3T + 1.29 x 10-ST 3, /4 = ]0.81 x I0-3T ,~- + 4.27 x 10-ST 3, r1,2
T>L
(14) T > T~ T_< Tc (12)
There are many theoretical 17.ta,2°,2t and experimental 22 studies on the thermodynamic properties of Nb3Sn. However, the results on the electronic specific heat, 3"T, vary considerably. The value of 7 deduced by Ghosh and Strongin 23 from the upper critical field measurements is 4.49 4- 0.18 mJ cm -3 K -2. It was suggested by Stewart et al. 24,25, on the basis of specific heat measurements in magnetic fields, that 3, = 3.14 4- 0.27 mJ cm -3 K -2. Both of these values are much lower than the value of 4.71 mJ cln -3 K -2 previously determined 26-2s. The size of the jump in the specific heat, AC, at Tc was found by Mitrovic et al. 29 to be the same (ACIr= rc = 0 . 1 9 7 J c m - 3 K -2) as the values of both Stewart et al. and those found in earlier
Because the electronic specific heat in the superconducting state, Ces, decreases exponentially, the maximum value of specific heat occurs at the transition. Consequently, by using the BCS value of ICeslr=rc = 2.433,Tc to replace (a • e-b/kBr) in the last term of Equation (14), the following inequality can be obtained to estimate the minimum magnitude of AwR Aw"-->Vrc+41 B ( 3 T 3 + T4" rc,/
- 2.433'(Tc- Too) (15)
Table 1 summarizes the values of AWRcalculated for N b - T i and Nb3Sn. Although the above mentioned data on the specific heats of N b - T i and Nb3Sn range over almost an order of magnitude, the condition AwR > 0 holds. Even if the specific heat of the BCS theory is replaced by the much greater experimental values mentioned above, the inequality condition is still satisfied.
Cryogenics 1991 Vol 31 September
819
Normal zone propagation. Part I: Z.P. Zhao and Y. Iwasa 1
Table
Values of AWR for N b - T i and Nb3Sn
Materials
AWR (mJ c m - 3 K-1)
Nb-Ti
Remarks
5.1
3" = 0.81 mJ cm -3 K -2 (from Elrod et al. 13) B = 1 2 . 8 5 x 10 -3 mJ cm -3 K -4 Tc = 9 . 1 K
5.5
3' = 0.98 mJ c m - 3 K - 2 (from Zbasnik et al. 31) x 10 -3 mJ c m - 3 K - 4 To= 9.0 K
B = 14.52
Nb3Sn
45.9
= 0.99 mJ c m - 3 K - 2 (from Khlopkin 3°)
Oo = 232 K T¢ = 18.0 K 3' = 1.28 mJ c m - 3 K - 2 (from Junod et al. is) OD = 232 K Tc = 17.97 K
41.3
46.4
= 1.18 mJ cm -3 K -2 (from Vieland e t a / . 26)
Oo = 228 K Tc= 18.0K
Presence of magnetic field. Applying Equations (9) and (11) to Equation (8) gives
H
of 4.2 K. The upper critical magnetic fields are Vj-/a(0) of 14.5 T for N b - T i and #oHc2(0) of 22.0 T for Nb3Sn. Equation (8) is therefore generally satisfied. Modification of the model for composites
+
3' - ~ +
B+
4T~
(16)
Figures 1 and 2 represent the values of AwR for N b - T i and Nb3Sn at different ambient magnetic fields for To= 60
'
I
'
I
'
I
'
I
'
I
In a superconductor composite, the normal state thermal conductivity, Kn, electrical conductivity, p, and specific heat, {7,, depend not only on the respective properties of the normal state superconductor but also on those of the matrix metal. Thus the normal zone propagation velocity [Equation (3)] is affected by the properties of the matrix metal.
'
300
2T
'
I
'
I
'
I
'
I
'
I
I
I
'
I
L
/
50
250 4O 200 30 150 IT
d
I
d
20
100 OT
10
I
50
0
,
4
I 5
,
I 6
Critical
,
I 7
I 8
,
Temperature
,
I 9
J ,
I0
(K)
0 I---F---T-
4
6
I
I 8
Critical Calculated results of the unity for N b - T i at magnetic flux densities operating temperature, T=, is 4.2 K. The and upper magnetic flux density, Bc=, respectively Figure 1
820
criterion [Equation (8)] of O, 1 and 2 T. The critical temperature, T¢, are 9.2 K and 1 4 . 5 T ,
Cryogenics 1991 Vol 31 September
I
I
I0
I
t2
Temperature
t4
I
16
I
18
(K)
Calculated results of the unity criterion [Equation ( 8 ) ] for Nb3Sn at magnetic flux densities of 0 and 1 T. The operating temperature, 7"=, is 4.2 K. The critical temperature, To, and upper magnetic flux density, Bc2, are 18.0 K and 22.0 T, respectively Figure 2
Normal zone propagation. Part I: Z.P. Zhao and Y. Iwasa
In previous investigations L2's-tl the normal state volumetric specific heat, C., of a superconductor/ copper composite was represented entirely by that of the copper. From the velocity formula [ Equation (3)], the volumetric specific heat of the copper matrix has to be included when calculating the normal state specific heat, C., of a superconductor composite. The volumetric specific heatof a composite may be calculated by superposition of the two components
number must be taken into account in calculating normal zone propagation velocity.
(17)
B ce a n d Tc. Many data are available on the relationship between Be2 and Tc for N b - T i and Nb3Sn 34-39. The following relation has been derived 38
Cp = (1 - g')CF + ~'CM
where CF is the volumetric specific heat of the superconductor filaments and CM is the volumetric specific heat of the copper. ~" is the copper-to-superconductor volume ratio. Copper at 4.2 K has the volumetric heat capacity of Cp ~ 10 -3 J cm -3 K -1, which is equally shared between electronic specific heat, Ce ~- T, and lattice specific heat, Cg ---T3. Above 4.2 K, the lattice specific heat rapidly dominates. The volumetric specific heat of copper used in this study is based on the data published in NBS Monograph 21 (1960) 32. The normal zone propagation velocity [ Equation (3)] is also strongly affected by the normal state thermal conductivity and electrical resistivity. Since the normal state thermal conductivity of the superconductor is much lower than that of copper, the normal state thermal conductivity of the composite, K,, may be approximated by the copper thermal conductivity. Similarly, since the normal state electrical resistivity of the superconductor is about 1000 times greater than that of copper, the normal state electrical resistivity of the composite may be given by copper resistivity. Arenz et al. 33 demonstrated that in magnetic flux densities up to 12.5 T between 3 and 15 K, the transverse magnetoresistivity data of polycrystalline copper (residual resistivity ratio, RRR = 108) yield a good Kohler plot, whereas there is considerable deviation in the longitudinal field Kohler plot. Both the electrical resistivity and thermal conductivity of electrons in copper are affected more by a transverse field than by a longitudinal field. Thus the Lorenz number will also be dependent on magnetic field. It has been shown 3a that the longitudinal Lorenz number is magnetic field independent, but the transverse Lorenz number varies linearly with field, dA/d(#oH) = 0.112 x 10 -s V 2 K -2 T-~, which is, on a relative basis, 5 % per tesla. This is a significant field dependence as it implies that at #oH = 16 T the transverse Lorenz number is doubled. From the experiments of Arenz et al. the formula relating Lorenz number with temperature, T, and magnetic flux density, #oH, was derived.
The normal zone propagation velocity [Equation (3)] depends also on the superconductor's critical properties, Tc, H~2 and J~. The relationships between these properties are summarized below.
Bc2(T) 1_ [Tc(B)117 Bc2(0~ -
t T~O)J
(19)
with Tc(0)= 9.2 K and Bc2(0)= 14.5 T for N b - T i conductors of nominal composition 4 4 - 4 8 wt% Ti and Tc(0) = 18.0 K and Bc2(0) = 22.0 T for Nb3Sn. J ~ a n d B. There is a vast amount of data on the relationship between Jc and B. However, Jc values are often found to range from one to several orders of magnitude. For accurate analysis of normal zone propagation velocity, the data must be measured for each set of conductors. The experimental data on Jc versus B for N b - T i and NbaSn composites used in this work are shown in Figure 3. Jc a n d To. For most applications, the relationship between J¢ and T at a given flux density, B, may be represented by a straight line
L(T) Jc(B, T=)
L(B) - T
(20)
Tc(B) - T=
This linearity has been observed in many experiments 4° and is consistent with the Anderson-Kim t h e o r y 41 . Jc(B, To.) is the critical current density, which is a function of field, B, and operating temperature, T®.
~ 0
,
,
,
,
,
,
,
,
,
,
,
15 '
,
,
,
,
,
,
NbTi:
Jo
=
10 5 A/cm
NbsSn:
Jo
=
10 4 A/era2
,
,
2
0
A(T, /z0H) = (2.184 - 0.0353T + 0.112/z0H ) x 10 -s [V 2 K -2]
Quench criteria
(18)
Although Equation (18) is valid only for the temperature range 5 - 12 K 33, we used it in applying Equation (3) to calculate normal zone propagation in N b - T i and Nb3Sn composites over the temperature range 1 . 8 - 1 8 K. In solenoidal magnets, the magnetic field is principally perpendicular to the conductor axis. Therefore the effects of magnetic field on resistivity and Lorenz
o
10
0
i
0
i
i
,
[
5
,
h
i
,
I
,
I0 B (T)
i
i
i
I
i
i
15
i
i
20
Figure 3 Experimental data for Jc versus B for N b - T i and Nb3Sn composites at 4.2 K
Cryogenics 1991 Vol 31 September
821
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa Results
6
Over the last five years, experiments have been performed at the Francis Bitter National Magnet Laboratory to examine the applicability of the modified Whetstone-Roos model to normal zone propagation velocity in compositest'2"8-]t. However, all these experiments have been compared with an expression for normal zone propagation velocity which is identical to Equation (3) except that a plus sign instead of the minus sign is used. This section re-examines the experimental results using the expression of Equation (3) that incorporates material properties given in the previous sections.
Velocity
versus
15
/,'I'I/
/,'///
./
////,,,/,./
"-- 10
/ V / / +,'V,.S /
///://Y
_0
• 1
0
0
I 50
-
/ I i+,Y
I I00
I 150
I 200
I 250
Operating Current (At)
Figure 4 Comparison of the analytical results (
) and the experimental data ~'2 for Ui at 7"= = 4.2 K for an N b - T i composite under several ambient magnetic flux densities (in T): x, 0; + , 1.2; El, 2.4; A , 3.5; O, 4.8; I I , 6.0. - - - - - - , Analytical results presented in earlier work I'2. The wire diameter is 0 . 9 6 5 mm and the copper-to-superconductor ratio is 2.0
822
//
i/I
//
iI
/
sS
_ /'
I ~
3
2 /V
1
V//
~
operating currents
Velocity, UI, is calculated as a function of operating current at an operating temperature, T=, of 4,2 K and various ambient magnetic fields. Figure 4 shows the experimental points and computed plots at six ambient flux densities for an N b - T i wire with a diameter of 0.965 mm and copper-to-superconductor ratio of 2. Figure 5 shows similar results for an Nb3Sn wire with a diameter of 0.9 mm and copper-to-superconductor ratio of 1. In both figures, the data points are from previous experiments i.~.$ , , . .9 Agreement between theory and experiment is excellent. For these computed curves, no adjustable coefficient was used. The agreement also indicates that an adiabatic condition was maintained in these experiments, as theory assumes the adiabatic condition. The dashed curves in Figures 4 and 5 represent the analytical results obtained previously 1,2,8,9. Agreement between analytical results and experimental points in the
8
// ,1 / /,," ,, / / / ,/ /," / ,'/
5
Cryogenics 1991 Vol 31 September
o
0
50
100
150
200
250
Operntin K Current (A)
Figure 5 Comparison of the analytical results (
) and the experimentalldataSlfor Ui at To. = 4.2 K for an Nb3Sn composite under several ambient magnetic flux densities (in T): O, 0; zx, 2; El, 4; x, 6; e , 8; A , 10; I I , 12. - - - , Analytical results presented in earlier work s. The wire diameter is 0 . 9 0 mm and the copper-to-superconductor ratio is 1.0
present work appears to be better than in the previous work. The present work indicates that the modification of the model and the material properties given above are quite reasonable.
Non-linearity of Uz
Figures 4 and 5 indicate the approximate linearity between velocity, Ul, and operating current, lop, at low ambient magnetic fields. The velocity formula [Equation (3)] also appears to suggest that U] and lop have a linear relationship. However, as shown in the quench criteria, an increase in one of the critical properties invariably produces a decrease in the other two. This in turn changes the term under the square root in Equation (3) and results in non-linearity. This effect is shown in Figures 6 and 7. The calculations were performed on an N b - T i wire with a diameter of 0.76 mm and copper-to-superconductor ratio of 3.0. Figure 6 shows the relationship between UI and lop at different ambient magnetic flux densities, and the relationship between Ul and magnetic flux density at different operating currents, both at To* = 4.2 K. Also included in Figure 6 are the curves for ambient temperature T= = 1.8 K. The velocities at T® = 1.8 K are generally greater than those at To, = 4.2 K. The increase in velocity for each pair of solid and dashed lines varies with operating current and ambient temperature. Figure 7 compares U2 values at operating current lop = 100 A (dashed lines) with values at lop = 300 A (solid lines), when varying the ambient temperature, To,, in various magnetic flux densities. From Figure 7 it can be seen that a three-fold increase in lop results in a three- to ten-fold increase in velocity, U~. The nonlinearity of U~ is due both to the fact that the critical surface B - J - Tof a superconductor is non-linear and to the fact that the specific heat of a solid is not a linear function of temperature.
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa 750
I
I
I
T~ = 1.8 K
0T
T~ = 4.2 K
500
Conclusions
I
#//#
,,
/
/
,'/
lJ
A normal zone propagation model has been developed for superconductor composites under adiabatic conditions. The model incorporates the temperature and magnetic field dependent conductor properties of both superconducting materials and matrix metals. The material property dependent condition [Equation (8)] required in the propagation velocity formula has been demonstrated to be satisfied for N b - T i and Nb3Sn composites. The model gives normal zone propagation velocities that are in good agreement with experimental data for N b - T i and Nb3Sn composites. The analytical results indicate that the propagation velocity is not a linear function of operating current, magnetic field and temperature.
/
o
o 250
'/'/,'/
s 0
~
t
I
300
0
(8)
- " /
600
i
Operating C u r r e n t
500
, "
-
II
-
•~ 200
--
/
200 /
/
I
I
/
/
.,'/J~f/
/I
/j//J
. "
i
I
I
I
t
/
100A
I
I
i
I
i
2 4 6 8 M a g n e t i c F l u x D e n s i t y (T)
0
(b)
Figure 6 Non-linearity of velocity, U~, at operating temperatures, T= = 4.2 K ( ) and 1 . 8 K ( - - - ) . (a) U= versus operating current at five different magnetic flux densities; (b) UI versus magnetic flux density at five different operating currents
Application of the model The normal zone propagation model developed in this work has been incorporated into a computer code to analyse the quench behaviour of epoxy resin impregnated multisection solenoids. The analytical quench results for an eight-coil magnet and a 12-coil magnet will be described in Parts 2 and 3 of this work.
200
'
\'
,
'
,
\\\ ~
' ~c..'i , ' / , ~-.w..~ 4 T /I
150
. . . .
1
S,T
E I00 0
>'~
50
0 1
I
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I
I
2
3
4
5
6
7
8
9
Operating
Temperature
Acknowledgements Z.P. Zhao would like to acknowledge the financial support of the Niwa Memorial Foundation at Tokyo Denki University during his stay at MIT. This work was supported in part by the National Laboratory for High Energy Physics, Tsukuba, Japan. The authors are grateful to A. M. Dawson for useful comments on the manuscript.
/ I
/
100 0
-
T= = 4.2 K
II
I /
I
300
I #
(A)
I
:1400
--
1200
T= = 1.8 K
'/'1
-
~300
I
500
..~ 400 _
I
900
10
(K)
Figure 7 Velocity, Ui, v e r s u s operating t e m p e r a t u r e for operating currents, lop = 1 0 0 A ( - - - ) and 3 0 0 A ( ), each at five different magnetic flux densities
References 1 Joshi, C.H. and Iwasa, Y. Prediction of current decay and terminal voltages in adiabatic superconducting magnets Cryogenics (1989) 29 157 2 Joshi, C.H. Thermal and electrical characteristics of adiabatic superconducting solenoids during a spontaneous transition to the resistive state ScD Thesis MIT, USA (1987) 3 Dresner, L. Propagation of normal zones in composite superconductors Cryogenics (1976) 16 675 4 Dresner, L. Propagation of normal zones in thermally insulated superconductors Adv Cryog Eng Mat (1980) 26 647 5 Turck, B. About propagation velocity in superconducting composites Cryogenics (1980) 20 146 6 Whetstone, C.N. and Roo~, C.E. Thermal phase transition in superconducting N b - Z r alloys J Appl Phys (1965) 36 783 7 Whetstone, C.N. Thermal phase transitions in superconducting N b - Z r alloys PhD Thesis Vanderbilt University, USA (1964) 8 Ishiyama, A. and Iwasa, Y. Quench propagation velocities in an epoxy-impregnated Nb3Sn superconducting winding model 1EEE Trans Magn (1988) MAG-24 1194 9 Ishiyama, A., Matsumura, H., Takita, T. and lwasa, Y. Quench propagation analysis in adiabatic superconducting windings IEEE Trans Magn (1991) MAG-27(3) 2092 10 Brown, J.N. and lwasa, Y. Temperature-dependent stability and protection parameters in an adiabatic superconducting magnet Cryogenics (1991) 31 341 11 Brown, J.N., Tahara, Y., Williams, J.E.C. and Iwasa, Y. Temperature dependent parameters of stability and protection in an adiabatic niobium titanium coil IEEE Trans Magn (1991) MAG-27(3) 2144 12 Hust, J.G. and Sparks, L.L. Lorenz ratios of technically important metals and alloys, NBS Technical Note NBS-TN-634, USA (1973) 13 EIrod, S.A., Miller, J.R. and Dresner, L. The specific heat of NbTi from 0 to 7 T between 4.2 and 20 K Adv Cryog Eng Mat (1981) 28 601 14 Morin, F.J. and Malta, J.P. Specific heats of transition metal superconductors Phys Rev (1963) 129 1115 15 Knapp, G.S., Bader, S.D. and Fisk, Z. Phonon properties of A-15 superconductor obtained from heat-capacity measurements Phys Rev B (1976) 13 3783 16 Karkin, A.E., Mirmeishtein, A.V., Arldhipov, V.E. and Goshchitsldi, B.N. Specific heats of Nb3Sn irradiated by fast neutrons Phys Stat Sol (1980) 61 Kl17
Cryogenics 1991 Vol 31 September
823
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa 17 Juned, A., Muller, J., Rletschel, H. and Schneider, E. Chaleur sp6cifique kt transformation martensitique dans le syst~me Nb I -x Snx J Phys Chem Sol (1978) 39 317 (in French) 18 Junod, A., Jarlborg, T. and Muller, J. Heat-capacity analysis of a large number of A15-type compounds Phys Rev B (1983) 27 1568 19 Shamrai, V., Bohnhammel, K. and Wolf, G. The heat capacities of NbaSnHx in dependence on the hydrogen content Phys Stat Sol B (1982) 109 511 20 D u m s , J.M. and Cm'lmtte, J.P. 1laermodynamics of superconducting Nb3Sn Sol State Commun (1979) 29 501 21 Wolf, E.L., Zmlulzinskl, J., Arnold, G.B., Rowell, J.M. and Beledey, M.R. Tunneling and the eleetron-phonon-coupled superconductivity of NbaSn Phys Rev B (1980) 22 1214 22 Shen, L.Y.L. Tunneling into a high Tc superconductor-Nb3Sn Phys Rev Lett (1972) 29 1082 23 Ghmh, A.K. and Strongin, M. Density of states and Tc of disordered A15 compounds, in: Superconductivity in d- and f-Bank Metals (Eds SUbl, H. and Maple, M.B.) Academic Press, New York, USA (1980) 305 24 Steward, G.R., Cort, B. mid Webb, G.W. Specific heat of AI5 Nb3Sn in fields to 18 tesla Phys Rev B (1981) 24 3841 25 Stewm'd, G.R. and llrandt, B.L. High-field specific heats of AI5 V3Si and Nb3Sn Phys Rev B (1984) 29 3908 26 Vlehmd, L.J. lind Wkklund, A.W. Specific heat of niobium-tin Phys Rev (1968) 166 424 27 Chmg, G.K., L/m, T.L. and Khan, W.Y. Specific heat anomaly of N'bsSn at the normal-superconducting transition Acta Phys Sin (1965) 21 817 28 Viswanatl~m, R., Lno, H.L. and Vieland, L.J. Low-temperature heat Calmcity of Nb3Sn .Proc Low Temp Phys Conf Plenum Press, New York, USA (1974) 472 29 Mitrovi¢, B., Sehaehinger, E. and Carbotte, J.P. Thermodynamics of superconducting Nb3AI, Nb3C-e, Nb3Sn and V3Ga Phys Rev B (1984) 29 6187 30 Khiopkln, M.N. The specific heat of NbaSn in magnetic fields up to 19 T SOy Phys JETP (1986) 63 164 31 Z b e m t , J. cited as private communication in work by Iwasa, Y., Weggel, D., Montgomery, D.IL, Weggel, R. and Hale, J.R. Prediction of transient stability limits for composite superconductors subject to flux jumping J Appl Phys (1969) 40 2006 32 Cormedni, R.J. and Gniewek, J.J. Specific heats and enthalpies of technical solids at low tempertures bIBS Monograph 2l (1960) 33 A r e ~ , R.W., Clark, C.F. and Lawless, W.N. Thermal conductivity and electrical resistivity of copper in intense magnetic fields at low temperatures Phys Rev B (1982) 26 2727 34 Ed~m-ri, A. and Slmdoni, M. Superconducting Nb3Sn: a review Cryogenics (1971) 11 274 35 Larlmlestler, D.C. N b - T i alloy superconductors - present status and potential for improvements Adv Cryog Eng Mat (1979) 26 10 36 Larbalestier, D.C. Niobium-titanium superconducting materials, in: Superconductor Materials Science: Metallurgy, Fabrication and Applications (Eds Foner, S. and Schwartz, B.B.) Plenum Press, New York, USA (1981) 133 37 Iwasa, Y. and Leulmld, M.J. Critical current data of NbTi conductors at sub-4.2 K temperatures and high felds Cryogenics (1982) 22 477 38 Lubell, M.S. Empirical scaling formulas for critical current and critical field for commercial NbTi IEEE Trans Magn (1983) MAG-19 754 39 Hawksworth, D.G. and Larlmle~er, D.C. Enhanced values of H¢2 in NbTi ternary and quarternary alloys Adv Cryog Eng Mat (1979) 26 479 40 Suenga, M. Metallurgy ofcontinnous filamentary A15 superconductors, in: Superconductor Materials Science: Metallurgy, Fabrication and Applications (Eds Foner, S. and Schwartz, B.B.) Plenum Press, New York, USA (1981) 201 41 Tinkham, M. Introduction to Superconductivity R.E. Krieger Publishing Co., Inc., Malabar, Florida, USA (1975) 175
conducting regions, respectively, by
~xO(K n --~x OTn~] - Ca cgTnot hnPA_( r n __ ro,) +
pj2
The general power density equations, which include cooling terms, are given for the normal and super-
824
Cryogenics 1991 Vol 31 September
O (A1)
OxC9(Ks OTs'~/Ox_ Cs otOTs ~-hsP(7',
-
To,) = 0
(A2)
where: C is the specific heat; k the thermal conductivity; j the current density; p the resistivity; h the convective heat transfer coefficient on the perimeter, P; A the wire cross-section; and T** the temperature of the bath. The subscripts n and s refer to the normal and superconducting regions, respectively, and C, K and P are all functions of temperature. A one-dimensional travelling wave originating at x = 0 with constant propagation velocity, U~, can then be described by using the transformation
q = x - Ult
(A3)
with Equations (A1) and (A2). The normal-superconducting boundary is assumed to be located at q = 0, the point where the temperature is equal to the superconducting transition temperature, To. Assuming that the heat conducted across the boundary out of the normal region is equal to that conducted into the superconducting region and the latent heat of transition is negligible, we may write the boundary conditions as K~ dT. dTs dq q=O- = Ks dq-lq=o÷
(A4)
T. = T~ Iq=O- ,
(A5)
\
T s = T¢ Iq=O+
By neglecting the cooling term in Equations (A1) and (A2), a linear temperature profile can be obtained in the normal region for the case of constant K, C and p 6'7. This suggests that it might be possible to neglect the second-order derivative of T, in Equation (A1) near q = 0 in the normal region. Equations (A1) and (A2) are then reduced to 6'7
dT~
--
+ UlC.
K~ d2T~ + dK~ dq 2 ~
dq
+ pj2=o
(dT~'~2 dT~ = 0 \-d-qq,] + UIC~ d---q
Equation (A6) can be solved algebraically for
Appendix A: Solution of normal zone propagation velocity
=
dT~_
dq
[
(A6)
(A7)
dTn/dq
-C~Vi 4- (C.Ul)2-4pj 2 dTJ dK. 2--
dTo
(A8)
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa
To solve Equation (A7) a substitution, Q, has
to
be used
-UI
Cs dT 7"=
Q-
dT~ dq
(A9) - C . UI
±
[( )2 C.UI
-
1/2
4pj2- ~ (AI4)
2dKn Kn dT
Another substitution can be obtained from Equation (A9) Q dQ _ d2Ts dTs dq 2
(A10) which can be solved for the velocity, U1, at T = T¢ as follows
Equation (A7) can then be converted into the standard Bernoulli's equation
dQ + ( ~ dKs~ Q = dTs dTs/
CsUI Ks
E 2 .1
Ul Ca Kn dT
(All) =
Combined with the boundary condition, Equation (A5) gives the solution for the superconducting region from temperature T= of the superconductor to T¢ of the boundary
,,jr= Ks dTs
dTs
)]
dTs
= --
Ks r=
Cs dT~
]
r=ro (A15)
Ul C~ dT
Kn dT
r®
dT
T=
Cs dr
.J )r=r,
(A12) =
Eliminating dTs in the exponentials gives the final solution of Equation (A7)
Q
Cs dT
Squaring both sides and eliminating the common terms on both sides give the final results for velocity,
+
(f
r~
"I-[(CnUI)2-4pj2~-~]1/2r=ro
r° 1 d K s d T s ) I l l [(_CsUl~
x exp
T=T~
(A13)
Applying boundary condition Equation (A4) to Equations (A8) and (A13) yields
-4pj2 dKn r= n
(A16)
that is
Ul=j
~n
r= C~dT -1/2
x
Ca K. dT
r® Cs d
T/]
(A171
T=T,
Cryogenics 1991 Vol 31 September
825
Received 31 January 1991 A normal zone propagation model has been developed for superconducting composites under adiabatic conditions. It is based on the Whetstone-Roos model, originally developed for normal zone propagation in adiabatic wires of unclad superconductor. The model takes into account the temperature and magnetic field dependent material properties, for both superconductor and matrix metal. Analytical results agree well with experimental data.
Keywords: superconducting magnets; superconducting composites; normal zone propagation Normal zone propagation in superconductor wires depends on the balance between Joule heating within the normal zone and cooling at the normal zone surface. Consequently, a normal zone may: 1, grow because the Joule heating exceeds the cooling; 2, shrink because the cooling exceeds the Joule heating; or 3, remain constant because the Joule heating is equal to the cooling. Case 1 is generally true for adiabatic magnets. Case 2, on the other hand, is generally applicable to cryostable magnets. Several general reviews of the theoretical analyses of normal zone propagation are available ~-5. There are many solutions for normal zone propagation velocity under a variety of conditions. However, most models neglect the temperature and magnetic field dependences of material properties. An exact model, proposed by Whetstone and Roos 6'7 in 1965 for unclad superconductors (conductors without a normal metal stabilizer) under adiabatic conditions, takes into account the temperature dependence of material properties. The Whetstone-Roos model treats single wires of unclad superconductor. Consequently, it is not of direct use for most applications because superconductors of technological interest are composites consisting of superconducting filaments in a matrix of normal stabilizing metal such as copper. Recently Joshi and Iwasa ~'2 applied the model to a magnet quench simulation code.
Extending further the work of Whetstone and Roos and of Joshi and Iwasa, we propose a model applicable to superconductor composites under adiabatic conditions. This model takes into account the temperature and magnetic field dependent material properties. The analytical results of normal zone propagation velocity are compared with experimental data on N b - T i and Nb3Sn composites.
Normal zone propagation velocity The Whetstone-Roos model, among others ~-5, assumes that the normal-superconducting boundary moves at a steady velocity along the wire, with the boundary moving continuously into the superconducting region where the transport current and magnetic field are below the critical values. The model assumes the process to be adiabatic. The Joule heat is conducted forward along the wire as a travelling temperature wave and the overall rise in the wire temperature causes the normal-superconducting boundary to advance into the superconducting region. By neglecting the second-order derivative of temperature in the normal region near the normalsuperconducting boundary, Whetstone and Roos derived an explicit expression for normal zone propagation velocity UI6
U~=j p(T)-K.(T) C.(T) *Based in part on the Doctoral Thesis of Z.P. Zhao submitted to the Graduate School of Engineering, Tokyo Denki University, Tokyo, Japan, December 1990 *Visiting scientist from Tokyo Denki University, Tokyo, Japan
×
f ,,
7"=
C~(T)d
,]S ,,
7"=
K.(T)
dT
C~(T)dT~
(1)
) T = T¢
0011 - 2 2 7 5 / 9 1 / 0 9 0 8 1 7 - 09 © 1991 B u t t e r w o r t h - Heinemann Ltd
Cryogenics 1991 Vol 31 September
817
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa
where K, C and P are, respectively, the thermal conductivity, volumetric specific heat and normal state electrical resistivity of the superconductor. Subscripts n and s stand for the normal state and superconducting state, respectively. Tc is the superconductor's critical temperature and T® is the ambient temperature. If K and C are taken as constant, Equation (1) represents the special case of the constant-property problem which has been investigated extensively4. In the original analysis of Whetstone and Roos 7, they derived the minus sign for Equation (1) in their appendix, but a plus sign was used in their text 7 where velocity calculations and comparison with experiments were presented. The velocity formula given by Joshi and Iwasa, who based their work on the formula given in the main text of Whetstone and Roos' paper, uses a plus sign instead of the minus sign in Equation (1). The Joshi and Iwasa formula has appeared in later publications g-H. The WhetstoneRoos model is analysed herein and a detailed derivation of Equation 1 is presented in Appendix A. In the temperature ranges T,~ OD and T > OD (Reference 12), where Oa is the Debye temperature, the relationship between thermal conductivity, Kn, and electrical resistivity, p, for ordinary metals is A T = K~p, where A is the Lorenz number. This relation is known as the Wiedemann-Franz law. In the Sommerfeld models for metals in zero magnetic field, the value is A = x2k2/3e2 = 2.45 x 10 -s V 2 K -2, where ks is the Boltzmann constant. In order to obtain a more explicit expression for velocity, U~, than that given by Equation (1), the temperature dependence of electrical resistivity, p, can be neglected for a small temperature range. Thus, the following substitution can be obtained if p is taken as constant 1 d K . = _1
g. d r
(2)
r
Applying this substitution to Equation (1), we obtain
Vl=j
[A[
c.(r)
r.
jr=to
(3)
From the analysis in Appendix A, it is clear that this solution is obtained under the implied assumption that (CnUt) 2 -- 4pj2 dK. ~ AWR > 0
dr.
(4)
where AwR is a parameter used in the following discussion. The condition given in Equation (4), however, has been ignored in earlier analyses. If AwR = 0, the propagation velocity may be expressed as
where - -
I f
c"(rc)= Kn(Tc) dT r=r, r.
Cs dT
(6)
If AWR < 0, then Equation (A6) of Appendix A will yield two imaginary solutions. This means that Equation (A6) cannot be used to describe the thermal conduction process in the normal region. The inequality AWR < 0 is equivalent to a negative value under the square root in Equation (1). Therefore the condition required in this model for describing normal zone propagation in an adiabatic superconducting wire may be expressed as
r f
-Cs OT C,(T¢) > Kn(T) dT r= ro r=
(7)
or, by substituting Equation (2) co(to)
>
Cs dT
(8)
From these analyses of the Whetstone-Roos model, it is clear that the solution of normal zone propagation velocity under adiabatic conditions is obtained through the following assumptions 1 The condition required in Equations (7) or (8) is implied. This inequality, however, is not a unique result of the model. 2 The normal state resistivity, p, of the superconductor wire is taken as constant for Equation (2). The thermal and electrical transport properties of superconductors at low temperatures, along with their effects on normal zone propagation, have to be further investigated. 3 The model is limited to unclad superconductors. Superconductors used in magnets are composites. These composites consist of multiffiaments of superconductor clad in a copper or aluminium matrix. Joshi and Iwasa extended this model to composites but, as mentioned above, their velocity formula was identical with Equation (1) except that a plus sign instead of a minus sign was used. In the absence of a magnetic field or transport current, the superconductor transition occurs at the critical temperature, Tc. In the presence of a magnetic field or transport current, the transition occurs at a temperature below Tc and the latent heat density is TdzoHc(dItc/ dT), where p-0 is the free space permittivity. By neglecting the latent heat, we might underestimate the propagation velocity for small transport currents where the Joule heating, pj', is comparable to the latent heat. This is, however, generally not a problem for normal zone growth in superconducting magnets. Validity of Equation (8)
Ui
=
818
Cn(T)
(,
dT ]
I
T=r,
Cryogenics 1991 Vol 31 September
(5)
Equation (8) gives the condition that must be satisfied for the Whetstone-Roos model. To verify the validity of this condition for N b - T i and NbaSn superconduc-
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa tors, the specific heats of the superconductors in both the normal and superconducting states are considered. Specific heats of superconductors. In the absence of an ambient magnetic field, the observed volumetric specific heats of a superconductor in the normal state, Cpn, and superconducting state, Cp, are, respectively Con = 3"T +/3T 3
(9)
Cp~ = aexp ( _ . b£BI,/~ + ~ T 3
(10)
where 3'T and BT3 represent, respectively, the electronic contribution and the lattice contribution to volumetric specific heat. As given in Equations (9) and (10), the lattice specific heat is the same in both normal and superconducting states, but the contribution from the conduction electrons is quite different. In the presence of a magnetic field, the superconducting specific heat will be dependent on both temperature and magnetic field. By using the G6rter-Casimir theory and by approximating the magnetization curves of type H superconductors with straight lines, Elrod et al. ~3 have derived an expression that relates the specific heat in the superconducting state to 3', To/3T 3 and the upper critical magnetic field, Hc2(0)
3"T +
3 BT2]BT
(11)
Specific h e a t s of Nb - Ti a n d Nb3Sn. Most specific heat data of Nb3Sn z4-~9 are those measured under zero magnetic field. There appear to be significant uncertainties associated with specific data measured in the presence of a magnetic field. The only available experimental data at high magnetic fields for N b - T i seem to be those of Elrod et al. Applying their data to Equation (11), we may express the volumetric specific heats of N b - T i in the superconducting and normal states in magnetic fields as
experiments 26-2s. However, a more recent study by Khlopkin 3° over the temperature range 4 . 5 - 7 0 K in magnetic flux densities up to 19 T gives the coefficient 3' = 0.988 mJ cm -3 K -2 for the Debye temperature, On = 232 K. One of the consequences of the smaller value of 3' proposed by Stewart et al. and by Khlopkin is that the parameter AC/3"Tc is usually large, 3.5 4- 0.3. Khlopkin's latest data are employed in the present analysis. The value of 3, given here is similar to that of Junod et al. 17.18 (1.28 mJ cm -3 K -2) and of Vieland et al. 26 (1.18 mJ cm -3 K-z), both measured in the absence of ambient magnetic fields. Applying Khlopkin's data to Equation (11), the temperature and magnetic field dependence of the volumetric specific heat of Nb3Sn may be described by the following polynomial Csb~Sn[J cm -3 K - t ] 0.988 x 10-3T + 1.388 x 10-ST 3, lO
T<-L
+
(13) The normalized value of the specific heat jump at the superconducting transition, A C / T T ~, is 2.5 in Khlopkin's calculation, which is considerably higher than the value of 1.43 given by BCS theory. Verification o f Equation (8). Based on the above discussion of the specific heats of N b - T i and Nb3Sn, two methods are used to analyse the validity of Equation (8). A b s e n c e o f m a g n e t i c field. Applying Equations (9) and (10) to Equation (8) yields ,aw. = C.(Tc) - ~1 l ro ~= Cs(T) dt
=
+
oo
-- 1
rc a • e-b/ks r dt
CNb-Ti [J cm -3 K -1]
~
0.81 x 10-3T + 1.29 x 10-ST 3, /4 = ]0.81 x I0-3T ,~- + 4.27 x 10-ST 3, r1,2
T>L
(14) T > T~ T_< Tc (12)
There are many theoretical 17.ta,2°,2t and experimental 22 studies on the thermodynamic properties of Nb3Sn. However, the results on the electronic specific heat, 3"T, vary considerably. The value of 7 deduced by Ghosh and Strongin 23 from the upper critical field measurements is 4.49 4- 0.18 mJ cm -3 K -2. It was suggested by Stewart et al. 24,25, on the basis of specific heat measurements in magnetic fields, that 3, = 3.14 4- 0.27 mJ cm -3 K -2. Both of these values are much lower than the value of 4.71 mJ cln -3 K -2 previously determined 26-2s. The size of the jump in the specific heat, AC, at Tc was found by Mitrovic et al. 29 to be the same (ACIr= rc = 0 . 1 9 7 J c m - 3 K -2) as the values of both Stewart et al. and those found in earlier
Because the electronic specific heat in the superconducting state, Ces, decreases exponentially, the maximum value of specific heat occurs at the transition. Consequently, by using the BCS value of ICeslr=rc = 2.433,Tc to replace (a • e-b/kBr) in the last term of Equation (14), the following inequality can be obtained to estimate the minimum magnitude of AwR Aw"-->Vrc+41 B ( 3 T 3 + T4" rc,/
- 2.433'(Tc- Too) (15)
Table 1 summarizes the values of AWRcalculated for N b - T i and Nb3Sn. Although the above mentioned data on the specific heats of N b - T i and Nb3Sn range over almost an order of magnitude, the condition AwR > 0 holds. Even if the specific heat of the BCS theory is replaced by the much greater experimental values mentioned above, the inequality condition is still satisfied.
Cryogenics 1991 Vol 31 September
819
Normal zone propagation. Part I: Z.P. Zhao and Y. Iwasa 1
Table
Values of AWR for N b - T i and Nb3Sn
Materials
AWR (mJ c m - 3 K-1)
Nb-Ti
Remarks
5.1
3" = 0.81 mJ cm -3 K -2 (from Elrod et al. 13) B = 1 2 . 8 5 x 10 -3 mJ cm -3 K -4 Tc = 9 . 1 K
5.5
3' = 0.98 mJ c m - 3 K - 2 (from Zbasnik et al. 31) x 10 -3 mJ c m - 3 K - 4 To= 9.0 K
B = 14.52
Nb3Sn
45.9
= 0.99 mJ c m - 3 K - 2 (from Khlopkin 3°)
Oo = 232 K T¢ = 18.0 K 3' = 1.28 mJ c m - 3 K - 2 (from Junod et al. is) OD = 232 K Tc = 17.97 K
41.3
46.4
= 1.18 mJ cm -3 K -2 (from Vieland e t a / . 26)
Oo = 228 K Tc= 18.0K
Presence of magnetic field. Applying Equations (9) and (11) to Equation (8) gives
H
of 4.2 K. The upper critical magnetic fields are Vj-/a(0) of 14.5 T for N b - T i and #oHc2(0) of 22.0 T for Nb3Sn. Equation (8) is therefore generally satisfied. Modification of the model for composites
+
3' - ~ +
B+
4T~
(16)
Figures 1 and 2 represent the values of AwR for N b - T i and Nb3Sn at different ambient magnetic fields for To= 60
'
I
'
I
'
I
'
I
'
I
In a superconductor composite, the normal state thermal conductivity, Kn, electrical conductivity, p, and specific heat, {7,, depend not only on the respective properties of the normal state superconductor but also on those of the matrix metal. Thus the normal zone propagation velocity [Equation (3)] is affected by the properties of the matrix metal.
'
300
2T
'
I
'
I
'
I
'
I
'
I
I
I
'
I
L
/
50
250 4O 200 30 150 IT
d
I
d
20
100 OT
10
I
50
0
,
4
I 5
,
I 6
Critical
,
I 7
I 8
,
Temperature
,
I 9
J ,
I0
(K)
0 I---F---T-
4
6
I
I 8
Critical Calculated results of the unity for N b - T i at magnetic flux densities operating temperature, T=, is 4.2 K. The and upper magnetic flux density, Bc=, respectively Figure 1
820
criterion [Equation (8)] of O, 1 and 2 T. The critical temperature, T¢, are 9.2 K and 1 4 . 5 T ,
Cryogenics 1991 Vol 31 September
I
I
I0
I
t2
Temperature
t4
I
16
I
18
(K)
Calculated results of the unity criterion [Equation ( 8 ) ] for Nb3Sn at magnetic flux densities of 0 and 1 T. The operating temperature, 7"=, is 4.2 K. The critical temperature, To, and upper magnetic flux density, Bc2, are 18.0 K and 22.0 T, respectively Figure 2
Normal zone propagation. Part I: Z.P. Zhao and Y. Iwasa
In previous investigations L2's-tl the normal state volumetric specific heat, C., of a superconductor/ copper composite was represented entirely by that of the copper. From the velocity formula [ Equation (3)], the volumetric specific heat of the copper matrix has to be included when calculating the normal state specific heat, C., of a superconductor composite. The volumetric specific heatof a composite may be calculated by superposition of the two components
number must be taken into account in calculating normal zone propagation velocity.
(17)
B ce a n d Tc. Many data are available on the relationship between Be2 and Tc for N b - T i and Nb3Sn 34-39. The following relation has been derived 38
Cp = (1 - g')CF + ~'CM
where CF is the volumetric specific heat of the superconductor filaments and CM is the volumetric specific heat of the copper. ~" is the copper-to-superconductor volume ratio. Copper at 4.2 K has the volumetric heat capacity of Cp ~ 10 -3 J cm -3 K -1, which is equally shared between electronic specific heat, Ce ~- T, and lattice specific heat, Cg ---T3. Above 4.2 K, the lattice specific heat rapidly dominates. The volumetric specific heat of copper used in this study is based on the data published in NBS Monograph 21 (1960) 32. The normal zone propagation velocity [ Equation (3)] is also strongly affected by the normal state thermal conductivity and electrical resistivity. Since the normal state thermal conductivity of the superconductor is much lower than that of copper, the normal state thermal conductivity of the composite, K,, may be approximated by the copper thermal conductivity. Similarly, since the normal state electrical resistivity of the superconductor is about 1000 times greater than that of copper, the normal state electrical resistivity of the composite may be given by copper resistivity. Arenz et al. 33 demonstrated that in magnetic flux densities up to 12.5 T between 3 and 15 K, the transverse magnetoresistivity data of polycrystalline copper (residual resistivity ratio, RRR = 108) yield a good Kohler plot, whereas there is considerable deviation in the longitudinal field Kohler plot. Both the electrical resistivity and thermal conductivity of electrons in copper are affected more by a transverse field than by a longitudinal field. Thus the Lorenz number will also be dependent on magnetic field. It has been shown 3a that the longitudinal Lorenz number is magnetic field independent, but the transverse Lorenz number varies linearly with field, dA/d(#oH) = 0.112 x 10 -s V 2 K -2 T-~, which is, on a relative basis, 5 % per tesla. This is a significant field dependence as it implies that at #oH = 16 T the transverse Lorenz number is doubled. From the experiments of Arenz et al. the formula relating Lorenz number with temperature, T, and magnetic flux density, #oH, was derived.
The normal zone propagation velocity [Equation (3)] depends also on the superconductor's critical properties, Tc, H~2 and J~. The relationships between these properties are summarized below.
Bc2(T) 1_ [Tc(B)117 Bc2(0~ -
t T~O)J
(19)
with Tc(0)= 9.2 K and Bc2(0)= 14.5 T for N b - T i conductors of nominal composition 4 4 - 4 8 wt% Ti and Tc(0) = 18.0 K and Bc2(0) = 22.0 T for Nb3Sn. J ~ a n d B. There is a vast amount of data on the relationship between Jc and B. However, Jc values are often found to range from one to several orders of magnitude. For accurate analysis of normal zone propagation velocity, the data must be measured for each set of conductors. The experimental data on Jc versus B for N b - T i and NbaSn composites used in this work are shown in Figure 3. Jc a n d To. For most applications, the relationship between J¢ and T at a given flux density, B, may be represented by a straight line
L(T) Jc(B, T=)
L(B) - T
(20)
Tc(B) - T=
This linearity has been observed in many experiments 4° and is consistent with the Anderson-Kim t h e o r y 41 . Jc(B, To.) is the critical current density, which is a function of field, B, and operating temperature, T®.
~ 0
,
,
,
,
,
,
,
,
,
,
,
15 '
,
,
,
,
,
,
NbTi:
Jo
=
10 5 A/cm
NbsSn:
Jo
=
10 4 A/era2
,
,
2
0
A(T, /z0H) = (2.184 - 0.0353T + 0.112/z0H ) x 10 -s [V 2 K -2]
Quench criteria
(18)
Although Equation (18) is valid only for the temperature range 5 - 12 K 33, we used it in applying Equation (3) to calculate normal zone propagation in N b - T i and Nb3Sn composites over the temperature range 1 . 8 - 1 8 K. In solenoidal magnets, the magnetic field is principally perpendicular to the conductor axis. Therefore the effects of magnetic field on resistivity and Lorenz
o
10
0
i
0
i
i
,
[
5
,
h
i
,
I
,
I0 B (T)
i
i
i
I
i
i
15
i
i
20
Figure 3 Experimental data for Jc versus B for N b - T i and Nb3Sn composites at 4.2 K
Cryogenics 1991 Vol 31 September
821
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa Results
6
Over the last five years, experiments have been performed at the Francis Bitter National Magnet Laboratory to examine the applicability of the modified Whetstone-Roos model to normal zone propagation velocity in compositest'2"8-]t. However, all these experiments have been compared with an expression for normal zone propagation velocity which is identical to Equation (3) except that a plus sign instead of the minus sign is used. This section re-examines the experimental results using the expression of Equation (3) that incorporates material properties given in the previous sections.
Velocity
versus
15
/,'I'I/
/,'///
./
////,,,/,./
"-- 10
/ V / / +,'V,.S /
///://Y
_0
• 1
0
0
I 50
-
/ I i+,Y
I I00
I 150
I 200
I 250
Operating Current (At)
Figure 4 Comparison of the analytical results (
) and the experimental data ~'2 for Ui at 7"= = 4.2 K for an N b - T i composite under several ambient magnetic flux densities (in T): x, 0; + , 1.2; El, 2.4; A , 3.5; O, 4.8; I I , 6.0. - - - - - - , Analytical results presented in earlier work I'2. The wire diameter is 0 . 9 6 5 mm and the copper-to-superconductor ratio is 2.0
822
//
i/I
//
iI
/
sS
_ /'
I ~
3
2 /V
1
V//
~
operating currents
Velocity, UI, is calculated as a function of operating current at an operating temperature, T=, of 4,2 K and various ambient magnetic fields. Figure 4 shows the experimental points and computed plots at six ambient flux densities for an N b - T i wire with a diameter of 0.965 mm and copper-to-superconductor ratio of 2. Figure 5 shows similar results for an Nb3Sn wire with a diameter of 0.9 mm and copper-to-superconductor ratio of 1. In both figures, the data points are from previous experiments i.~.$ , , . .9 Agreement between theory and experiment is excellent. For these computed curves, no adjustable coefficient was used. The agreement also indicates that an adiabatic condition was maintained in these experiments, as theory assumes the adiabatic condition. The dashed curves in Figures 4 and 5 represent the analytical results obtained previously 1,2,8,9. Agreement between analytical results and experimental points in the
8
// ,1 / /,," ,, / / / ,/ /," / ,'/
5
Cryogenics 1991 Vol 31 September
o
0
50
100
150
200
250
Operntin K Current (A)
Figure 5 Comparison of the analytical results (
) and the experimentalldataSlfor Ui at To. = 4.2 K for an Nb3Sn composite under several ambient magnetic flux densities (in T): O, 0; zx, 2; El, 4; x, 6; e , 8; A , 10; I I , 12. - - - , Analytical results presented in earlier work s. The wire diameter is 0 . 9 0 mm and the copper-to-superconductor ratio is 1.0
present work appears to be better than in the previous work. The present work indicates that the modification of the model and the material properties given above are quite reasonable.
Non-linearity of Uz
Figures 4 and 5 indicate the approximate linearity between velocity, Ul, and operating current, lop, at low ambient magnetic fields. The velocity formula [Equation (3)] also appears to suggest that U] and lop have a linear relationship. However, as shown in the quench criteria, an increase in one of the critical properties invariably produces a decrease in the other two. This in turn changes the term under the square root in Equation (3) and results in non-linearity. This effect is shown in Figures 6 and 7. The calculations were performed on an N b - T i wire with a diameter of 0.76 mm and copper-to-superconductor ratio of 3.0. Figure 6 shows the relationship between UI and lop at different ambient magnetic flux densities, and the relationship between Ul and magnetic flux density at different operating currents, both at To* = 4.2 K. Also included in Figure 6 are the curves for ambient temperature T= = 1.8 K. The velocities at T® = 1.8 K are generally greater than those at To, = 4.2 K. The increase in velocity for each pair of solid and dashed lines varies with operating current and ambient temperature. Figure 7 compares U2 values at operating current lop = 100 A (dashed lines) with values at lop = 300 A (solid lines), when varying the ambient temperature, To,, in various magnetic flux densities. From Figure 7 it can be seen that a three-fold increase in lop results in a three- to ten-fold increase in velocity, U~. The nonlinearity of U~ is due both to the fact that the critical surface B - J - Tof a superconductor is non-linear and to the fact that the specific heat of a solid is not a linear function of temperature.
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa 750
I
I
I
T~ = 1.8 K
0T
T~ = 4.2 K
500
Conclusions
I
#//#
,,
/
/
,'/
lJ
A normal zone propagation model has been developed for superconductor composites under adiabatic conditions. The model incorporates the temperature and magnetic field dependent conductor properties of both superconducting materials and matrix metals. The material property dependent condition [Equation (8)] required in the propagation velocity formula has been demonstrated to be satisfied for N b - T i and Nb3Sn composites. The model gives normal zone propagation velocities that are in good agreement with experimental data for N b - T i and Nb3Sn composites. The analytical results indicate that the propagation velocity is not a linear function of operating current, magnetic field and temperature.
/
o
o 250
'/'/,'/
s 0
~
t
I
300
0
(8)
- " /
600
i
Operating C u r r e n t
500
, "
-
II
-
•~ 200
--
/
200 /
/
I
I
/
/
.,'/J~f/
/I
/j//J
. "
i
I
I
I
t
/
100A
I
I
i
I
i
2 4 6 8 M a g n e t i c F l u x D e n s i t y (T)
0
(b)
Figure 6 Non-linearity of velocity, U~, at operating temperatures, T= = 4.2 K ( ) and 1 . 8 K ( - - - ) . (a) U= versus operating current at five different magnetic flux densities; (b) UI versus magnetic flux density at five different operating currents
Application of the model The normal zone propagation model developed in this work has been incorporated into a computer code to analyse the quench behaviour of epoxy resin impregnated multisection solenoids. The analytical quench results for an eight-coil magnet and a 12-coil magnet will be described in Parts 2 and 3 of this work.
200
'
\'
,
'
,
\\\ ~
' ~c..'i , ' / , ~-.w..~ 4 T /I
150
. . . .
1
S,T
E I00 0
>'~
50
0 1
I
I
I
I
I
I
I
I
2
3
4
5
6
7
8
9
Operating
Temperature
Acknowledgements Z.P. Zhao would like to acknowledge the financial support of the Niwa Memorial Foundation at Tokyo Denki University during his stay at MIT. This work was supported in part by the National Laboratory for High Energy Physics, Tsukuba, Japan. The authors are grateful to A. M. Dawson for useful comments on the manuscript.
/ I
/
100 0
-
T= = 4.2 K
II
I /
I
300
I #
(A)
I
:1400
--
1200
T= = 1.8 K
'/'1
-
~300
I
500
..~ 400 _
I
900
10
(K)
Figure 7 Velocity, Ui, v e r s u s operating t e m p e r a t u r e for operating currents, lop = 1 0 0 A ( - - - ) and 3 0 0 A ( ), each at five different magnetic flux densities
References 1 Joshi, C.H. and Iwasa, Y. Prediction of current decay and terminal voltages in adiabatic superconducting magnets Cryogenics (1989) 29 157 2 Joshi, C.H. Thermal and electrical characteristics of adiabatic superconducting solenoids during a spontaneous transition to the resistive state ScD Thesis MIT, USA (1987) 3 Dresner, L. Propagation of normal zones in composite superconductors Cryogenics (1976) 16 675 4 Dresner, L. Propagation of normal zones in thermally insulated superconductors Adv Cryog Eng Mat (1980) 26 647 5 Turck, B. About propagation velocity in superconducting composites Cryogenics (1980) 20 146 6 Whetstone, C.N. and Roo~, C.E. Thermal phase transition in superconducting N b - Z r alloys J Appl Phys (1965) 36 783 7 Whetstone, C.N. Thermal phase transitions in superconducting N b - Z r alloys PhD Thesis Vanderbilt University, USA (1964) 8 Ishiyama, A. and Iwasa, Y. Quench propagation velocities in an epoxy-impregnated Nb3Sn superconducting winding model 1EEE Trans Magn (1988) MAG-24 1194 9 Ishiyama, A., Matsumura, H., Takita, T. and lwasa, Y. Quench propagation analysis in adiabatic superconducting windings IEEE Trans Magn (1991) MAG-27(3) 2092 10 Brown, J.N. and lwasa, Y. Temperature-dependent stability and protection parameters in an adiabatic superconducting magnet Cryogenics (1991) 31 341 11 Brown, J.N., Tahara, Y., Williams, J.E.C. and Iwasa, Y. Temperature dependent parameters of stability and protection in an adiabatic niobium titanium coil IEEE Trans Magn (1991) MAG-27(3) 2144 12 Hust, J.G. and Sparks, L.L. Lorenz ratios of technically important metals and alloys, NBS Technical Note NBS-TN-634, USA (1973) 13 EIrod, S.A., Miller, J.R. and Dresner, L. The specific heat of NbTi from 0 to 7 T between 4.2 and 20 K Adv Cryog Eng Mat (1981) 28 601 14 Morin, F.J. and Malta, J.P. Specific heats of transition metal superconductors Phys Rev (1963) 129 1115 15 Knapp, G.S., Bader, S.D. and Fisk, Z. Phonon properties of A-15 superconductor obtained from heat-capacity measurements Phys Rev B (1976) 13 3783 16 Karkin, A.E., Mirmeishtein, A.V., Arldhipov, V.E. and Goshchitsldi, B.N. Specific heats of Nb3Sn irradiated by fast neutrons Phys Stat Sol (1980) 61 Kl17
Cryogenics 1991 Vol 31 September
823
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa 17 Juned, A., Muller, J., Rletschel, H. and Schneider, E. Chaleur sp6cifique kt transformation martensitique dans le syst~me Nb I -x Snx J Phys Chem Sol (1978) 39 317 (in French) 18 Junod, A., Jarlborg, T. and Muller, J. Heat-capacity analysis of a large number of A15-type compounds Phys Rev B (1983) 27 1568 19 Shamrai, V., Bohnhammel, K. and Wolf, G. The heat capacities of NbaSnHx in dependence on the hydrogen content Phys Stat Sol B (1982) 109 511 20 D u m s , J.M. and Cm'lmtte, J.P. 1laermodynamics of superconducting Nb3Sn Sol State Commun (1979) 29 501 21 Wolf, E.L., Zmlulzinskl, J., Arnold, G.B., Rowell, J.M. and Beledey, M.R. Tunneling and the eleetron-phonon-coupled superconductivity of NbaSn Phys Rev B (1980) 22 1214 22 Shen, L.Y.L. Tunneling into a high Tc superconductor-Nb3Sn Phys Rev Lett (1972) 29 1082 23 Ghmh, A.K. and Strongin, M. Density of states and Tc of disordered A15 compounds, in: Superconductivity in d- and f-Bank Metals (Eds SUbl, H. and Maple, M.B.) Academic Press, New York, USA (1980) 305 24 Steward, G.R., Cort, B. mid Webb, G.W. Specific heat of AI5 Nb3Sn in fields to 18 tesla Phys Rev B (1981) 24 3841 25 Stewm'd, G.R. and llrandt, B.L. High-field specific heats of AI5 V3Si and Nb3Sn Phys Rev B (1984) 29 3908 26 Vlehmd, L.J. lind Wkklund, A.W. Specific heat of niobium-tin Phys Rev (1968) 166 424 27 Chmg, G.K., L/m, T.L. and Khan, W.Y. Specific heat anomaly of N'bsSn at the normal-superconducting transition Acta Phys Sin (1965) 21 817 28 Viswanatl~m, R., Lno, H.L. and Vieland, L.J. Low-temperature heat Calmcity of Nb3Sn .Proc Low Temp Phys Conf Plenum Press, New York, USA (1974) 472 29 Mitrovi¢, B., Sehaehinger, E. and Carbotte, J.P. Thermodynamics of superconducting Nb3AI, Nb3C-e, Nb3Sn and V3Ga Phys Rev B (1984) 29 6187 30 Khiopkln, M.N. The specific heat of NbaSn in magnetic fields up to 19 T SOy Phys JETP (1986) 63 164 31 Z b e m t , J. cited as private communication in work by Iwasa, Y., Weggel, D., Montgomery, D.IL, Weggel, R. and Hale, J.R. Prediction of transient stability limits for composite superconductors subject to flux jumping J Appl Phys (1969) 40 2006 32 Cormedni, R.J. and Gniewek, J.J. Specific heats and enthalpies of technical solids at low tempertures bIBS Monograph 2l (1960) 33 A r e ~ , R.W., Clark, C.F. and Lawless, W.N. Thermal conductivity and electrical resistivity of copper in intense magnetic fields at low temperatures Phys Rev B (1982) 26 2727 34 Ed~m-ri, A. and Slmdoni, M. Superconducting Nb3Sn: a review Cryogenics (1971) 11 274 35 Larlmlestler, D.C. N b - T i alloy superconductors - present status and potential for improvements Adv Cryog Eng Mat (1979) 26 10 36 Larbalestier, D.C. Niobium-titanium superconducting materials, in: Superconductor Materials Science: Metallurgy, Fabrication and Applications (Eds Foner, S. and Schwartz, B.B.) Plenum Press, New York, USA (1981) 133 37 Iwasa, Y. and Leulmld, M.J. Critical current data of NbTi conductors at sub-4.2 K temperatures and high felds Cryogenics (1982) 22 477 38 Lubell, M.S. Empirical scaling formulas for critical current and critical field for commercial NbTi IEEE Trans Magn (1983) MAG-19 754 39 Hawksworth, D.G. and Larlmle~er, D.C. Enhanced values of H¢2 in NbTi ternary and quarternary alloys Adv Cryog Eng Mat (1979) 26 479 40 Suenga, M. Metallurgy ofcontinnous filamentary A15 superconductors, in: Superconductor Materials Science: Metallurgy, Fabrication and Applications (Eds Foner, S. and Schwartz, B.B.) Plenum Press, New York, USA (1981) 201 41 Tinkham, M. Introduction to Superconductivity R.E. Krieger Publishing Co., Inc., Malabar, Florida, USA (1975) 175
conducting regions, respectively, by
~xO(K n --~x OTn~] - Ca cgTnot hnPA_( r n __ ro,) +
pj2
The general power density equations, which include cooling terms, are given for the normal and super-
824
Cryogenics 1991 Vol 31 September
O (A1)
OxC9(Ks OTs'~/Ox_ Cs otOTs ~-hsP(7',
-
To,) = 0
(A2)
where: C is the specific heat; k the thermal conductivity; j the current density; p the resistivity; h the convective heat transfer coefficient on the perimeter, P; A the wire cross-section; and T** the temperature of the bath. The subscripts n and s refer to the normal and superconducting regions, respectively, and C, K and P are all functions of temperature. A one-dimensional travelling wave originating at x = 0 with constant propagation velocity, U~, can then be described by using the transformation
q = x - Ult
(A3)
with Equations (A1) and (A2). The normal-superconducting boundary is assumed to be located at q = 0, the point where the temperature is equal to the superconducting transition temperature, To. Assuming that the heat conducted across the boundary out of the normal region is equal to that conducted into the superconducting region and the latent heat of transition is negligible, we may write the boundary conditions as K~ dT. dTs dq q=O- = Ks dq-lq=o÷
(A4)
T. = T~ Iq=O- ,
(A5)
\
T s = T¢ Iq=O+
By neglecting the cooling term in Equations (A1) and (A2), a linear temperature profile can be obtained in the normal region for the case of constant K, C and p 6'7. This suggests that it might be possible to neglect the second-order derivative of T, in Equation (A1) near q = 0 in the normal region. Equations (A1) and (A2) are then reduced to 6'7
dT~
--
+ UlC.
K~ d2T~ + dK~ dq 2 ~
dq
+ pj2=o
(dT~'~2 dT~ = 0 \-d-qq,] + UIC~ d---q
Equation (A6) can be solved algebraically for
Appendix A: Solution of normal zone propagation velocity
=
dT~_
dq
[
(A6)
(A7)
dTn/dq
-C~Vi 4- (C.Ul)2-4pj 2 dTJ dK. 2--
dTo
(A8)
Normal zone propagation. Part 1: Z.P. Zhao and Y. Iwasa
To solve Equation (A7) a substitution, Q, has
to
be used
-UI
Cs dT 7"=
Q-
dT~ dq
(A9) - C . UI
±
[( )2 C.UI
-
1/2
4pj2- ~ (AI4)
2dKn Kn dT
Another substitution can be obtained from Equation (A9) Q dQ _ d2Ts dTs dq 2
(A10) which can be solved for the velocity, U1, at T = T¢ as follows
Equation (A7) can then be converted into the standard Bernoulli's equation
dQ + ( ~ dKs~ Q = dTs dTs/
CsUI Ks
E 2 .1
Ul Ca Kn dT
(All) =
Combined with the boundary condition, Equation (A5) gives the solution for the superconducting region from temperature T= of the superconductor to T¢ of the boundary
,,jr= Ks dTs
dTs
)]
dTs
= --
Ks r=
Cs dT~
]
r=ro (A15)
Ul C~ dT
Kn dT
r®
dT
T=
Cs dr
.J )r=r,
(A12) =
Eliminating dTs in the exponentials gives the final solution of Equation (A7)
Q
Cs dT
Squaring both sides and eliminating the common terms on both sides give the final results for velocity,
+
(f
r~
"I-[(CnUI)2-4pj2~-~]1/2r=ro
r° 1 d K s d T s ) I l l [(_CsUl~
x exp
T=T~
(A13)
Applying boundary condition Equation (A4) to Equations (A8) and (A13) yields
-4pj2 dKn r= n
(A16)
that is
Ul=j
~n
r= C~dT -1/2
x
Ca K. dT
r® Cs d
T/]
(A171
T=T,
Cryogenics 1991 Vol 31 September
825