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Transverse temperature distribution and heat generation rate in composite superconductors subjected to constant thermal disturbance
Pergamon
Int. Comm. HeatMass Transfer, Vol. 22, No. 4, pp. 461--474, 1995 Copyright © 1995 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/95 $ 9 . 5 0 + . 0 0
0735-1933(95)00031-3
TRANSVERSE TEMPERATURE DISTRIBUTION AND HEAT GENERATION RATE IN COMPOSITE SUPERCONDUCTORS SUBJECTED TO CONSTANT THERMAL DISTURBANCE
Y.S. Cha Energy Technology Division Argonne National Laboratory Argonne, IL 60439 W.J. Minkowycz Department of Mechanical Engineering The University of Illinois at Chicago Chicago, IL 60607 S.Y. Seol Department of Mechanical Engineering Chonnam National University Kwangju 500-757, Korea
(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT Analytical solution of one-dimensional, transient heat conduction with a distributed heat source is obtained to predict the transverse temperature distribution and heat generation rate per unit volume of the composite superconductor. The solution indicates that temperature distribution and heat generation rate depend on three dimensionless parameters: the dimensionless external disturbance w0, the dimensionless interface temperature 0 " , and the dimensionless parameter ~ that is dependent on the thickness and the thermal conductivity of the superconductor. Results of transient and steady-state solutions are presented. It is shown that the heat generation rate per unit volume of the composite, Q/Qc, is directly proportional to the current in the stabilizer.
Introduction Cryogenic stabilization is important for the design and superconducting magnets and is a subject of continuing research. 461
operation of Essential to
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Y.S. Cha, W.J. Minkowycz and S.Y. Seol
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stabilization of superconducting m a g n e t s is the prediction of h e a t generation rate per unit volume of the composite conductor. An example is the well-known e q u a l - a r e a theory of M a d d o c k et al. [1]. H e a t g e n e r a t i o n r a t e per u n i t volume of the composite conductor is u s u a l l y c a l c u l a t e d by a s s u m i n g t h a t (a) c u r r e n t in the superconductor is equal to the critical c u r r e n t with all the excess t r a n s f e r r e d to the stabilizer, and (b) critical c u r r e n t decreases linearly with t e m p e r a t u r e [2]. A more realistic model was proposed by Wilson [2] to calculate the h e a t generation rate by t a k i n g into account the effects of finite t h e r m a l conductivity a n d thickness of the superconductor. The solution r e p o r t e d by Wilson is for s t e a d y - s t a t e condition w i t h no e x t e r n a l d i s t u r b a n c e . The effect of finite t h e r m a l c o n d u c t i v i t y on h e a t g e n e r a t i o n r a t e is even more p r o n o u n c e d for h i g h - t e m p e r a t u r e superconductors in view of t h e i r relatively low t h e r m a l conductivities. Recently, Askew et al. [3] reported the results of m e a s u r e m e n t of flux flow r e s i s t i v i t y in YBa2Cu307 wires. The c e n t r a l e x p e r i m e n t a l p r o b l e m is m a i n t e n a n c e of a r e a s o n a b l e c o n s t a n t s a m p l e t e m p e r a t u r e over the d u r a t i o n of a given m e a s u r e m e n t cycle. Since the s a m p l e has a finite specific h e a t a n d t h e r m a l conductivity, any d i s s i p a t i o n in the b u l k will cause a t e m p e r a t u r e g r a d i e n t to appear. The t e m p e r a t u r e response of the s a m p l e is clearly not s t e a d y - s t a t e a n d t r a n s i e n t t h e r m a l response is n e e d e d to i n t e r p r e t the e x p e r i m e n t a l results. In this paper, we revisit the problem of h e a t g e n e r a t i o n r a t e in a composite conductor by c o n s i d e r i n g t h e m o r e g e n e r a l case of t r a n s i e n t condition w i t h c o n s t a n t t h e r m a l disturbance. We t r e a t the problem as one-dimensional and obtain a closed-form solution that predicts the t r a n s i e n t t r a n s v e r s e t e m p e r a t u r e d i s t r i b u t i o n in the superconductor and the h e a t generation rate per unit volume of the composite conductor. Most of the papers on s t a b i l i t y deal with p r o p a g a t i o n of the n o r m a l zone in the l o n g i t u d i n a l direction a n d a s s u m e t h a t the t e m p e r a t u r e of the composite conductor is uniform in the t r a n s v e r s e direction [4-6]. The only exception is the two-dimensional model r e p o r t e d by Chyu and O b e r l y [7]; t h e i r solution, however, is n u m e r i c a l and t h e r e f o r e does not identify the r e l e v a n t d i m e n s i o n l e s s (scaling) p a r a m e t e r s . In this paper, we begin by deriving the governing equations.
Then the r e s u l t s of
s t e a d y - s t a t e and t r a n s i e n t solutions are presented. Finally, we discuss the limitations of t h e p r e s e n t m o d e l w i t h p a r t i c u l a r e m p h a s i s on its a p p l i c a t i o n to h i g h - t e m p e r a t u r e superconductors. Analvsis The objective is to d e t e r m i n e the t r a n s i e n t t r a n s v e r s e t e m p e r a t u r e distribution and h e a t g e n e r a t i o n r a t e caused by c u r r e n t r e d i s t r i b u t i o n as a r e s u l t of a c o n s t a n t t h e r m a l d i s t u r b a n c e in composite conductors. The geometry is shown in Fig. 1, together with the a s s u m e d l i n e a r r e l a t i o n s h i p b e t w e e n critical c u r r e n t d e n s i t y a n d t e m p e r a t u r e .
We
a s s u m e t h a t t h e c h a r a c t e r i s t i c t i m e of c u r r e n t diffusion is s h o r t c o m p a r e d to t h a t of
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T H E R M A L ANALYSIS O F COMPOSITE SUPERCONDUCTORS
463
Stabilizer
Superconductor
Superconductor Ti To ~
Stabilizer
i i
dc
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,'
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Jc(Tg )= Jo(To)
I
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-
I I t
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T o Ti Tg
Jt;(T) = TG-T Jc(Ti) Tc-Ti'
T
Tc
Temp
J~(Ti) = J(;(Ti) = Tc -T i Jc(Tg) Jo(To) Tc-Tg
FIG. 1 Schematic d i a g r a m of model geometry, and a s s u m e d r e l a t i o n s h i p s between critical c u r r e n t d e n s i t y and t e m p e r a t u r e .
t h e r m a l diffusion so t h a t t h e c u r r e n t d i s t r i b u t i o n is c o m p l e t e l y d e t e r m i n e d by t h e t e m p e r a t u r e distribution [2]. We also a s s u m e t h a t the length of the t h e r m a l disturbance in the axial direction is m u c h g r e a t e r t h a n the t h i c k n e s s of the s u p e r c o n d u c t o r in the t r a n s v e r s e direction so t h a t the problem can be t r e a t e d as o n e - d i m e n s i o n a l in space by neglecting the edge effect. The governing equation in the superconductor then becomes 1 0T
W -
a Ot
02T t
k
OX 2
(1)
The h e a t source p e r unit volume of the superconductor W is composed of two parts, or W = W 0 + Wi(T)
(2)
where W 0 is the e x t e r n a l l y imposed h e a t i n g r a t e p e r u n i t volume of the s u p e r c o n d u c t o r and r e p r e s e n t s a c o n s t a n t t h e r m a l d i s t u r b a n c e , a n d Wi(T) is the i n t e r n a l l y g e n e r a t e d heating rate per unit volume of the superconductor as a r e s u l t of c u r r e n t r e d i s t r i b u t i o n in the superconductor and the stabilizer. I f we f u r t h e r a s s u m e t h a t c u r r e n t d e n s i t y in the s u p e r c o n d u c t o r is e q u a l to t h e c r i t i c a l c u r r e n t d e n s i t y c o r r e s p o n d i n g to t h e local
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Vol. 22, No. 4
t e m p e r a t u r e with all the excess c u r r e n t t r a n s f e r r e d to the stabilizer, the internal generation rate is [2]: Wi(T) = EJ(T) = EJc(T) = E J c ( T i ) ( T c - T)/(T c - T i)
(3)
The electrical field strength developed in the composite conductor as a result of current flow in the stabilizer, E, is assumed to be uniform across the entire conductor. The initial and boundary conditions are t = 0, T = T i
(4)
OT X=0, - - ~ = 0; X = a, T = T i
(5)
If we assume that the thermal conductivity of the metal stabilizer is much higher than that of the superconductor (which is applicable to high-temperature ceramic superconductors) and t h a t the thickness of the superconductor is not small compared to that of the stabilizer, we need not solve the heat conduction equation in the stabilizer. By substituting Eqs. (2) and (3) into Eq. (1), the resulting equation can be solved by Laplace transform [8]. After some manipulation, the solution becomes: 0=
cosh(~)
Q/Qc)
+ 2(w0(~0g + ~ 2) ~
Q/Qc
(-1)n e x p [ - ( N 2 + ~2)~] cOs(Nx)
n=0
(6)
N(~ 2 + N 2 )
where N = (2n+l)x/2 and the dimensionless variables are defined as 0 = ( T c - T ) / ( T c - T i)
(7)
x = X/a
(8)
= at/a 2
(9)
~2 = EJc(Ti ) a2/[k(Tc _ Ti)]
(10)
Og = (T c - Tg)/(T c - T i)
(11)
w 0 = W0L/Qc
(12)
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465
The p a r a m e t e r ~. = a/(a + b) is t h e s u p e r c o n d u c t o r fraction in t h e composite, a n d the critical g e n e r a t i o n r a t e Qc ( h e a t g e n e r a t i o n r a t e per u n i t volume of the composite when most of the c u r r e n t is in the stabilizer) is defined as Qc = P J02(T0 ) ),2/( ! _ ~)
(13)
The p a r a m e t e r Q in Eq. (6) is the i n t e r n a l h e a t g e n e r a t i o n r a t e p e r u n i t volume of the composite conductor (superconductor plus s t a b i l i z e r ) , which is u n k n o w n a n d m u s t be determined. The p a r a m e t e r Q can be expressed in t e r m s of t h e electric field s t r e n g t h E and the c u r r e n t d e n s i t y J0(T0), or Q= ),EJ0(T 0)
(14)
Using Eqs. (10), (13), and (14), we can show t h a t ~2 = (Q/Qc)0
(15)
where = Qca(a + b) / [k(T c - Tg)]
(16)
An additional condition t h a t m u s t be satisfied as a r e s u l t of c u r r e n t r e d i s t r i b u t i o n is t h a t the sum of the c u r r e n t in the superconductor a n d the s t a b i l i z e r m u s t be equal to the total current, or s a Jc(T ) dX + bE / P = a J0(T0) 0
(17)
Based on the dimensionless p a r a m e t e r s defined previously, it can be shown t h a t Eq. (17) is equivalent to
~o1 0 d x
+ (Q/Qc)Og = Og
(18)
S u b s t i t u t i n g Eq. (6) into Eq. (18) r e s u l t s in the following r e l a t i o n s h i p between 0g a n d Q/Qc:
tanh(~) + S 0g =
(I_Q/Qc)+I 1 tanh( )- w 0 /_2w0O S (19)
where
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Y.S. Cha, W.J. Minkowycz and S.Y. Seol
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(~2 +N2)N2
n=0
(20)
We have now completed the formulation of the problem. F o r given values of w 0, 0g, and 0, we w a n t to calculate 0 as functions of z a n d x, a n d Q/Q c a n d I] as function x, by solving Eqs. (6), (15), and (19) simultaneously. Numerically, it is e a s i e r to c a r r y out the calculations
in a slightly different m a n n e r .
I n s t e a d of w0, 0g, a n d ~, one can use w0,
Q/Qc , a n d ¢ as inputs, t h e n ~ can be calculated explicitly from Eq. (15).
0g can then be
c a l c u l a t e d by u s i n g Eq. (19). Finally, the dimensionless t e m p e r a t u r e 0 is calculated with Eq. (6). The calculations are s t r a i g h t f o r w a r d because no i t e r a t i o n is involved. T h e r e is a simple r e l a t i o n s h i p between h e a t g e n e r a t i o n r a t e p e r u n i t volume of the composite a n d c u r r e n t in the s t a b i l i z e r (Ist). S u b s t i t u t i n g the electric field s t r e n g t h E = PJst into Eq. (14), it can be shown that Jst/J0(T0 ) = (Q/Qc) a/b
or Ist / I0 = Q / Qc
(21)
Thus, the fraction of c u r r e n t in the stabilizer is equal to the fraction of the h e a t generated in the composite conductor. A n o t h e r i n t e r e s t i n g p a r a m e t e r is the total i n t e r n a l h e a t g e n e r a t e d per u n i t volume in the superconductor, W t = ~:Wi (T)dX / a
(22)
Because total h e a t g e n e r a t e d in the composite is equal to the sum of the h e a t generated in the superconductor and the stabilizer, it follows t h a t Q(a + b) = W t a + p(Jst)2b
(23)
S u b s t i t u t i n g Eq. (21) into Eq. (23) and solving for the d i m e n s i o n l e s s p a r a m e t e r wt, we obtain wt = Wt UQc = (Q/Qc)(1 - Q/Qc).
(24)
Both Eqs. (21) and (24) a p p e a r to be quite simple a n d depend only on h e a t generation rate per unit volume of the composite Q/Qc. However, because Q/Qc depends on w 0, 0, and 0g, so do Ist/I 0 and w t. E q u a t i o n s (21) and (24) are applicable to both t r a n s i e n t and steadys t a t e situations.
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467
Before presenting the results, we will point out a special case of the general solution given by Eq. (6). Because the a r g u m e n t in the exponential term is negative, a steadystate solution always exists as • becomes large. Under steady-state condition, Eqs. (6) and (19) are reduced to 0=
[ c, °W s h (°~ xO) l (g l ~- r -. - - cosh(~) Q/Qc)
W00g Q/Qc
tanh(~) /
0g --
( 1 - Q / Qc) + [ 1 - t a n h ( ~ ) / ~ ]
(25)
(w0) (26)
If there is no external thermal disturbance (w0=0), the steady-state solution becomes 0 = cosh(l~x)/cosh(~)
(27)
0g = tanh(~)/~/(1 - Q/Qc)
(28)
Equations (27) and (28) are identical to the solution reported by Wilson [2], who calculated the steady-state, transverse temperature distributions and h e a t generation rate, and applied the results to determine the cryogenic stability of a composite conductor. Results and Discussion~ The results of the previous analysis will be presented in two parts. The first part describes the numerical results of the steady-state solution, and the second describes the results of the transient solution. Instead of Og, we will define a dimensionless parameter 0* as 0* = (T i - Tg)/(T c - Tg)
(29)
The relationship between 0g and 0* is 0* = 1 - 1/0g
(30)
The results will be expressed in terms of 0", the parameter employed by Wilson [2], so that a direct comparison can be made. Steady-State Solution In this case, the exponential terms in Eqs. (6) and (19) approach zero as z becomes very large. Figure 2 shows the calculated 0* as a function of Q/Qc (or Ist/I 0) for various values of ¢. For the case of no external disturbance (w 0 = 0), the result is shown in Fig. 2(a) and is identical to that reported by Wilson [2]. The case of ¢ = 0 corresponds to the
468
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
1.o ~ , , , , , , ,
~''
06
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./~t
~o
1 0.0 -0.2
I
0.0
,
0.2
,
I
,
,
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q
~
0.4 0.6 Q/O orl /I c
st
~
I
,
J
0.8
1.0
0
(a) 1.0
r
I
0.8 0.6 0.4 °CD
0.2 0.0 -0.2 " 0 ' 4 0 O' •
=
,
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0.2
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,
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0.4
= , ,
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0.6
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0.8
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CYO C or I t / I 0
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0.8 0.6 0.4 *¢:D
0.2 0.0 -0.2 0.0
0.2
0.4 0.6 Q/Q orl /I C
st
0.8
1.0
0
(c) FIG. 2 V a r i a t i o n of 0* w i t h Q/Qc or I s t / I o for v a r i o u s v a l u e s of ¢ u n d e r s t e a d y - s t a t e c o n d i t i o n . (a) w 0 = 0; (b) w 0 = 0.1; (c) w 0 = 0.2.
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THERMAL ANALYSIS OF COMPOSITE SUPERCONDUCTORS
469
special s i t u a t i o n where the superconductor is at a uniform t e m p e r a t u r e because t h e r m a l conductivity k approaches infinity. It can be seen from Fig. 2(a) that e* = 0 (T i = T g ) a t Ist/I0=0 and e* = 1 (T i = T c) at Ist/I 0 = 1. This m e a n s t h a t in the absence of a n external disturbance, all of the c u r r e n t is in the superconductor when T i = Tg, a n d all of the c u r r e n t is in the stabilizer when T i = Tc.
The s i t u a t i o n is different when there is an
external d i s t u r b a n c e (w 0 > 0), as shown in Figs. 2(b) and 2(c). Except for ~ = 0, it can be observed that e* < 0 (T i < Tg) when Ist/I0 = 0 and {}* < 1 (T i < T c) when Ist/I 0 = 1. To m a i n t a i n all the c u r r e n t in the superconductor in the presence of a n external disturbance, the interface t e m p e r a t u r e T i m u s t be lower t h a n the generation t e m p e r a t u r e Tg. This is because the external disturbance causes c u r r e n t redistribution within the superconductor even though all of the current is still in the superconductor. On the other hand, when all of the c u r r e n t is transferred to the stabilizer, Figs. 2(b) and 2(c) show t h a t the interface temperature T i is lower t h a n the critical t e m p e r a t u r e T c. This result is incorrect physically because if the interface t e m p e r a t u r e is lower t h a n the critical temperature, a portion of the superconductor (near the interface) is still capable of c a r r y i n g a c e r t a i n a m o u n t of c u r r e n t and consequently not all of the c u r r e n t is in the stabilizer. This unrealistic result arises because the analytical model could not account for the s i t u a t i o n when the t e m p e r a t u r e of the s u p e r c o n d u c t o r exceeds the critical temperature. This can be explained by e x a m i n i n g Eq. (3), which is the a s s u m e d i n t e r n a l g e n e r a t i o n r a t e i n the superconductor.
W h e n T is g r e a t e r t h a n T c, the i n t e r n a l
generation rate becomes negative, but this is obviously incorrect.
W h e n T i is slightly
lower t h a n or equal to T c at Ist/I0 = 1, as shown in Fig. 2, the bulk of the superconductor is at t e m p e r a t u r e s g r e a t e r t h a n Tc, a n d Eq. (3) fails to describe the physical s i t u a t i o n correctly. We discuss this f u r t h e r in a later section. Another important feature that can be observed in Fig. 2 is that when ~ > 3, a relative m i n i m u m appears in each of the curves in these figures. For example, when ¢ = 5, Q/Q c increases with decreasing ~* for Q/Qc < 1.5 and increases with i n c r e a s i n g ~* for Q/Qc > 1.5. The portion of the curve with negative slope is most likely u n s t a b l e because as the interface t e m p e r a t u r e T i is reduced, the superconductor should be able to carry more current, and therefore less c u r r e n t is transferred to the stabilizer. Typical t e m p e r a t u r e distribution in the superconductor is shown in Fig. 3, where the d i m e n s i o n l e s s temperature e is plotted a g a i n s t x for various values of Q/Qc (or Ist/I0) with w 0 = 0.2 a n d ¢ = 5.
As Q/Qc is i n c r e a s e d , t e m p e r a t u r e i n the core of the
superconductor increases and approaches T c (e = 0). However, when Q/Qc is greater t h a n 0.8, {} becomes negative and T becomes greater t h a n T c.
This is a g a i n the r e s u l t of
employing Eq. (3), which is not valid when the temperature of the superconductor exceeds the critical temperature.
470
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
1,5
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Vol. 22, No. 4
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~ '
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1.0
a~ 0 . 5
0.0
-0.5
;
,
0.2
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L
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0.6
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FIG. 3 V a r i a t i o n of 0 w i t h x for v a r i o u s v a l u e s of Q / Q c or Ist/I 0 with w 0 = 0 a n d ¢ = 5 u n d e r s t e a d y - s t a t e condition.
We d e t e r m i n e d how t h e d i m e n s i o n l e s s total i n t e r n a l g e n e r a t i o n r a t e w t v a r i e s w i t h the dimensionless temperature
0* a n d t h e h e a t g e n e r a t i o n r a t e p e r u n i t v o l u m e of t h e
c o m p o s i t e Q/Qc" F i g u r e 4 is a plot o f w t v e r s u s Q / Q c (or Ist/I0) c a l c u l a t e d by Eq. (24). T h e total i n t e r n a l
generation
reaches a maximum
a t Ist/I 0 = 0.5.
T h e v a l u e of w t is at
m a x i m u m b e c a u s e t h e i n t e r n a l g e n e r a t i o n r a t e d e p e n d s on t h e p r o d u c t of two c o m p e t i n g factors, t o t a l c u r r e n t in t h e s u p e r c o n d u c t o r a n d e l e c t r i c field s t r e n g t h .
When the total
c u r r e n t in t h e s u p e r c o n d u c t o r is high, electric field s t r e n g t h is low b e c a u s e the c u r r e n t in t h e s t a b i l i z e r is small.
W h e n t o t a l c u r r e n t in t h e s u p e r c o n d u c t o r is low, e l e c t r i c field
s t r e n g t h is h i g h b e c a u s e t h e c u r r e n t in t h e s t a b i l i z e r is high.
T h i s c h a r a c t e r i s t i c is
c l e a r l y d e m o n s t r a t e d in Fig. 4 and by Eq. (24). F i g u r e 5 shows t h e v a r i a t i o n of O* w i t h w t for v a r i o u s v a l u e s of ¢. A g a i n , w t r e a c h e s a m a x i m u m at some v a l u e of O* t h a t d e p e n d s on t h e v a l u e s of ¢ a n d w 0. Transient Solution F i g u r e 6 s h o w s t h e r e s u l t s of the t r a n s i e n t s o l u t i o n c a l c u l a t e d by Eqs. (19) a n d (30). It a p p e a r s t h a t t h e s t e a d y - s t a t e is a p p r o a c h e d w h e n • _= 1, w h i c h m e a n s t h a t t h e t i m e r e q u i r e d to r e a c h s t e a d y s t a t e is t -_- a2/a.
T h i s c h a r a c t e r i s t i c t i m e for t r a n s v e r s e h e a t
c o n d u c t i o n in t h e s u p e r c o n d u c t o r c a n be c o m p a r e d to t h a t of c u r r e n t diffusion in t h e stabilizer.
T h e c h a r a c t e r i s t i c t i m e of c u r r e n t diffusion in t h e s t a b i l i z e r is tst_= P0L2/p,
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THERMAL ANALYSIS OF COMPOSITE SUPERCONDUCTORS
471
0.30 0.25 0.20 5"0,15 0,10 0.05 0.000,0
I
I
I
I
~
0.2
L
I
I
I
J
J
I
I
I
I
J
0.4 0.6 ~Q orl/I ¢
st
I
I
~
1.0
0.8 0
FIG. 4 Variation of w t with Q/Q c or I st/I 0.
1.0
1.0
0.8
0.8
E
,
,
i
,
r
,
,
i
,
,
,
,
i
,
,
,
,
i
,
,
,
,
i
,
,
,
,
0.6
0.6
0.4 0
*¢= 0.4
0.2
0.2
0.0
:
0.0 5 -02
~ , ~ ,
• 0.00
J J , , , I L J , , J
0.05
0.10
....
0.15
I ....
0.20
Ii
,~
0,25i , ' 0".30
-0.2 ! 0 -0.~.
o'15
030
W
W
t
t
(a)
(b)
FIG. 5 Variation of 0* with w t for various values of~. (a) w 0 = 0; (b) w 0 = 0.1. where P0 (= 4 ~ x 10 -7 H/m) is the p e r m e a b i l i t y of free space, a n d L is the characteristic length in the longitudinal direction. The ratio of the two characteristic times is t / tst = pa2/~u0 L2
(31)
T h e r m a l diffusivity for h i g h - t e m p e r a t u r e superconductors at 77 K is ¢x= 10-6-10 -7 m2/s, and the resistivity of silver at 77 K is p --- 3.65 x 10-9 ~2-m. In the a n a l y t i c a l model, we a s s u m e d t h a t L is much g r e a t e r t h a n a so t h a t edge effect can be neglected, a n d the problem can be treated as one-dimensional. If L/a = 10, t / t s t =0.58 x 102 >> 1
472
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
Vol. 22, No. 4
1.0
1.0
0.8
0.00 - -0.25
0.8
//
- 0.50
0.6
0.75 1 O0
",= 0.4
-
0.6 -
/.~
0.4 L
/ /
-
---
>/
o,~o 0.25 o.5o
.... / , ~ ..... ~ f f
1.00
..
/ - i f "
0.2 0.2
oo I
-0,0 -0.2
-0.2 0.0
0.2
0.4 0.6 Q/Qc or I t / I °
0.8
10
.
-0,4
(a)
0.0
0,2
.
0.4 0.6 Q/Qc or I t / I °
.
.
0.8
.0
(b)
FIG. 6 Variation of O* with Q/Qc or Ist/I0 with ¢ = 5 u n d e r t r a n s i e n t condition. (a) w 0 = 0; (b) w 0 = 0.1. a n d the time required for current redistribution in the stabilizer is much shorter t h a n that required for the t r a n s v e r s e t e m p e r a t u r e in the superconductor to reach steady state.
A
steady-state c u r r e n t in the stabilizer and the electric field s t r e n g t h can be assumed to establish i n s t a n t l y . But as one can see, the m a r g i n is relatively small.
If L/a > 10 is
required, t/tst will no longer be much greater t h a n 1, and the error introduced by the oned i m e n s i o n a l model will increase. As m e n t i o n e d previously, the t r a n s i e n t t h e r m a l response of a composite conductor may be p a r t i c u l a r l y r e l e v a n t to the i n t e r p r e t a t i o n of e x p e r i m e n t a l results on flux flow resistivity for high t e m p e r a t u r e superconductors because of their relatively low t h e r m a l conductivities. Summ~rv A closed-form solution of transient heat conduction with a distributed heat source is employed to predict transient transverse temperature distribution and heat generation rate per unit volume of a composite conductor subjected to a constant thermal disturbance. The heat source includes a constant external disturbance and an internal generation term that is a function of temperature distribution in the superconductor (Eq. (3)). W e show that the dimensionless temperature distribution 0 (Eq. (6)) and the heat generation rate per unit volume of the composite (~Qc depend on the three dimensionless parameters 0g, ¢, and w 0. Results of the transient and steady-state solutions are presented in Figs. 26. Heat generation rate per unit volume of the composite, Q/Qe, is directly proportional to current in the stabilizer (Eq. (21)). The dimensionless total internal generation rate w t in the superconductor is shown to reach a m a x i m u m at Q/Qc = 0.5 (Fig. 4) because w t depends on two competing factors: the current in the superconductor and the electric field strength which is proportional to the current in the stabilizer. It is demonstrated that steady state is approached w h e n the dimensionless time • becomes greater than i. In the absence of an external disturbance, the steady-state solution reduces to that reported by
Vol. 22, No. 4
T H E R M A L ANALYSIS OF COMPOSITE SUPERCONDUCTORS
473
Wilson [2]. The t r a n s i e n t t h e r m a l response of composite conductor m a y be relevant to the m e a s u r e m e n t on flux flow resistivity s i m i l a r to t h a t reported by Askew et al. [3]. A basic a s s u m p t i o n m a d e in this analysis is t h a t the c h a r a c t e r i s t i c time of c u r r e n t r e d i s t r i b u t i o n in the s u p e r c o n d u c t o r and t h e s t a b i l i z e r is much s h o r t e r t h a n t h a t of t e m p e r a t u r e redistribution, so t h a t h e a t conduction in the superconductor is i n d e p e n d e n t of the c u r r e n t diffusion process in the s t a b i l i z e r .
This a s s u m p t i o n is shown to be
m a r g i n a l for h i g h - t e m p e r a t u r e superconductors. A n o t h e r basic a s s u m p t i o n m a d e in t h e p r e s e n t a n a l y s i s is t h e d i s t r i b u t i o n of i n t e r n a l h e a t g e n e r a t i o n r a t e described by Eq. (3).
I t is e v i d e n t t h a t Eq. (3) cannot
accommodate a t e m p e r a t u r e g r e a t e r t h a n T c because dissipation becomes negative when T > T c and t h u s Eq. (3) gives unrealistic results. In reality, T can be g r e a t e r t h a n T c a n d the local c u r r e n t d e n s i t y should be reduced to zero if T > T c. Consequently, local h e a t generation rate should be zero. In o t h e r words, W i = 0 and J = 0 w h e n T > T c. This additional c o n s t r a i n t will complicate the analytical solution b u t can be h a n d l e d easily by n u m e r i c a l solution. We are c u r r e n t l y c a r r y i n g out the n u m e r i c a l solution, in which we can include the more general condition of a t i m e - d e p e n d e n t e x t e r n a l d i s t u r b a n c e i n s t e a d of the constant external disturbance used in the present analysis.
The p r e s e n t a n a l y t i c a l
solution can be employed to v a l i d a t e the r e s u l t s of the n u m e r i c a l solution u n d e r special circumstances. Acknowledgments This work has been supported by the U.S. D e p a r t m e n t of Energy, E n e r g y Efficiency a n d Renewable Energy, as p a r t of a p r o g r a m to develop electric power technology, u n d e r C o n t r a c t W-31-109-Eng-38.
S.Y. Seol was s u p p o r t e d by the r e s e a r c h f o u n d a t i o n of
Chonnam National University.
a
one-half the thickness of the superconductor, m
b
thickness of the stabilizer, m
E
electric field strength, V/m
I
current, A
I0 J Jc J0 k
o p e r a t i n g current, A c u r r e n t density, A/m 2 critical c u r r e n t density, A/m 2 o p e r a t i n g c u r r e n t density at T = T 0, A/m 2 t h e r m a l conductivity of the superconductor, W/m-K
L
c h a r a c t e r i s t i c l e n g t h in t h e l o n g i t u d i n a l direction for c u r r e n t diffusion in the
Q Qc
stabilizer, m h e a t g e n e r a t i o n r a t e per unit volume of the composite conductor, W/m 3 critical h e a t generation r a t e per u n i t volume of the composite conductor (Eq. (13)), W/m 3
474
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
T
temperature, K
Tc
critical t e m p e r a t u r e , K
Vol. 22, No. 4
Ti
interface t e m p e r a t u r e between the superconductor and the stabilizer, K
T0
o p e r a t i n g t e m p e r a t u r e before a t h e r m a l disturbance, K
t
time or c h a r a c t e r i s t i c time for t h e r m a l diffusion, s
tst
c h a r a c t e r i s t i c time for c u r r e n t diffusion in the stabilizer, s
W
h e a t i n g r a t e per unit volume of the superconductor (Eq. (2)), W/m 3
Wi
i n t e r n a l g e n e r a t i o n r a t e per unit volume of the superconductor (Eq. (3)), W/m 3
W0
e x t e r n a l h e a t i n g r a t e per unit volume of the superconductor, W/m 3
Wt
total i n t e r n a l generation rate per unit volume of the superconductor defined in Eq. (22), W/m 3
w0
dimensionless e x t e r n a l h e a t i n g rate p e r unit volume of the superconductor defined in Eq. (12), W/m 3
wt
dimensionless total i n t e r n a l generation r a t e p e r unit volume of the superconductor defined in Eq. (24), W/m 3
X
t r a n s v e r s e coordinate, m
x
d i m e n s i o n l e s s t r a n s v e r s e coordinate, X/a t h e r m a l diffusivity of the superconductor, m2/s d i m e n s i o n l e s s p a r a m e t e r defined in Eq. (10)
p
electrical resistivity of the stabilizer, ~ - m fraction of superconductor in the composite, a/(a+b)
¢
a d i m e n s i o n l e s s p a r a m e t e r defined in Eq. (16) d i m e n s i o n l e s s time, ¢zt/a 2
0 0g
d i m e n s i o n l e s s t e m p e r a t u r e defined in Eq. (7) d i m e n s i o n l e s s t e m p e r a t u r e defined in Eq. (11)
0*
d i m e n s i o n l e s s t e m p e r a t u r e defined in Eq. (30) References
1.
B.J. Maddock, G.B. J a m e s and W.T. Norris, Cryogenics, 9,261(1969).
2.
M.N. Wilson, Su~erconductin~ M a g n e t s , Oxford U n i v e r s i t y P r e s s , Chap. 6, New York (1983).
3.
T.R. A s k e w , J.G. N e s t e l l , R.B. F l i p p e n , D.M. Groski a n d N.McN. Alford, I E E E Trans. on Appl. Superconductivity, 3, 1398 (1993).
4.
V.R. Romanovskii, Cryogenics, 28, 756 (1988).
5.
A. Abeln, E. K l e m t and H. Reiss, Cryogenics, 32, 269 (1992).
6.
R. K u p f e r m a n , R.G. Mints and E. Ben-Jacob, J. Appl. Phys., 70, 7484 (1991).
7.
M.K. Chyu a n d C.E. Oberly, Cryogenics, 3 1 , 6 8 0 (1991).
8.
H.S. C a r s l a w and J.C. J a e g e r , QQnduction of H e a t in Solids, 2nd Ed., p. 404, Oxford U n i v e r s i t y P r e s s (1980).
Received February 22, 1995
Int. Comm. HeatMass Transfer, Vol. 22, No. 4, pp. 461--474, 1995 Copyright © 1995 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/95 $ 9 . 5 0 + . 0 0
0735-1933(95)00031-3
TRANSVERSE TEMPERATURE DISTRIBUTION AND HEAT GENERATION RATE IN COMPOSITE SUPERCONDUCTORS SUBJECTED TO CONSTANT THERMAL DISTURBANCE
Y.S. Cha Energy Technology Division Argonne National Laboratory Argonne, IL 60439 W.J. Minkowycz Department of Mechanical Engineering The University of Illinois at Chicago Chicago, IL 60607 S.Y. Seol Department of Mechanical Engineering Chonnam National University Kwangju 500-757, Korea
(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT Analytical solution of one-dimensional, transient heat conduction with a distributed heat source is obtained to predict the transverse temperature distribution and heat generation rate per unit volume of the composite superconductor. The solution indicates that temperature distribution and heat generation rate depend on three dimensionless parameters: the dimensionless external disturbance w0, the dimensionless interface temperature 0 " , and the dimensionless parameter ~ that is dependent on the thickness and the thermal conductivity of the superconductor. Results of transient and steady-state solutions are presented. It is shown that the heat generation rate per unit volume of the composite, Q/Qc, is directly proportional to the current in the stabilizer.
Introduction Cryogenic stabilization is important for the design and superconducting magnets and is a subject of continuing research. 461
operation of Essential to
462
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
Vol. 22, No. 4
stabilization of superconducting m a g n e t s is the prediction of h e a t generation rate per unit volume of the composite conductor. An example is the well-known e q u a l - a r e a theory of M a d d o c k et al. [1]. H e a t g e n e r a t i o n r a t e per u n i t volume of the composite conductor is u s u a l l y c a l c u l a t e d by a s s u m i n g t h a t (a) c u r r e n t in the superconductor is equal to the critical c u r r e n t with all the excess t r a n s f e r r e d to the stabilizer, and (b) critical c u r r e n t decreases linearly with t e m p e r a t u r e [2]. A more realistic model was proposed by Wilson [2] to calculate the h e a t generation rate by t a k i n g into account the effects of finite t h e r m a l conductivity a n d thickness of the superconductor. The solution r e p o r t e d by Wilson is for s t e a d y - s t a t e condition w i t h no e x t e r n a l d i s t u r b a n c e . The effect of finite t h e r m a l c o n d u c t i v i t y on h e a t g e n e r a t i o n r a t e is even more p r o n o u n c e d for h i g h - t e m p e r a t u r e superconductors in view of t h e i r relatively low t h e r m a l conductivities. Recently, Askew et al. [3] reported the results of m e a s u r e m e n t of flux flow r e s i s t i v i t y in YBa2Cu307 wires. The c e n t r a l e x p e r i m e n t a l p r o b l e m is m a i n t e n a n c e of a r e a s o n a b l e c o n s t a n t s a m p l e t e m p e r a t u r e over the d u r a t i o n of a given m e a s u r e m e n t cycle. Since the s a m p l e has a finite specific h e a t a n d t h e r m a l conductivity, any d i s s i p a t i o n in the b u l k will cause a t e m p e r a t u r e g r a d i e n t to appear. The t e m p e r a t u r e response of the s a m p l e is clearly not s t e a d y - s t a t e a n d t r a n s i e n t t h e r m a l response is n e e d e d to i n t e r p r e t the e x p e r i m e n t a l results. In this paper, we revisit the problem of h e a t g e n e r a t i o n r a t e in a composite conductor by c o n s i d e r i n g t h e m o r e g e n e r a l case of t r a n s i e n t condition w i t h c o n s t a n t t h e r m a l disturbance. We t r e a t the problem as one-dimensional and obtain a closed-form solution that predicts the t r a n s i e n t t r a n s v e r s e t e m p e r a t u r e d i s t r i b u t i o n in the superconductor and the h e a t generation rate per unit volume of the composite conductor. Most of the papers on s t a b i l i t y deal with p r o p a g a t i o n of the n o r m a l zone in the l o n g i t u d i n a l direction a n d a s s u m e t h a t the t e m p e r a t u r e of the composite conductor is uniform in the t r a n s v e r s e direction [4-6]. The only exception is the two-dimensional model r e p o r t e d by Chyu and O b e r l y [7]; t h e i r solution, however, is n u m e r i c a l and t h e r e f o r e does not identify the r e l e v a n t d i m e n s i o n l e s s (scaling) p a r a m e t e r s . In this paper, we begin by deriving the governing equations.
Then the r e s u l t s of
s t e a d y - s t a t e and t r a n s i e n t solutions are presented. Finally, we discuss the limitations of t h e p r e s e n t m o d e l w i t h p a r t i c u l a r e m p h a s i s on its a p p l i c a t i o n to h i g h - t e m p e r a t u r e superconductors. Analvsis The objective is to d e t e r m i n e the t r a n s i e n t t r a n s v e r s e t e m p e r a t u r e distribution and h e a t g e n e r a t i o n r a t e caused by c u r r e n t r e d i s t r i b u t i o n as a r e s u l t of a c o n s t a n t t h e r m a l d i s t u r b a n c e in composite conductors. The geometry is shown in Fig. 1, together with the a s s u m e d l i n e a r r e l a t i o n s h i p b e t w e e n critical c u r r e n t d e n s i t y a n d t e m p e r a t u r e .
We
a s s u m e t h a t t h e c h a r a c t e r i s t i c t i m e of c u r r e n t diffusion is s h o r t c o m p a r e d to t h a t of
Vol. 22, No. 4
T H E R M A L ANALYSIS O F COMPOSITE SUPERCONDUCTORS
463
Stabilizer
Superconductor
Superconductor Ti To ~
Stabilizer
i i
dc
Ji f i i
\
|~,,_. 11";
,'
_.
Jc(/i}
i i
, i
J
I
-
i i
I
I t
-J.-
Jc(Tg )= Jo(To)
I
I I I
J c (T)
-
I I t
-
I I I
- - -
T o Ti Tg
Jt;(T) = TG-T Jc(Ti) Tc-Ti'
T
Tc
Temp
J~(Ti) = J(;(Ti) = Tc -T i Jc(Tg) Jo(To) Tc-Tg
FIG. 1 Schematic d i a g r a m of model geometry, and a s s u m e d r e l a t i o n s h i p s between critical c u r r e n t d e n s i t y and t e m p e r a t u r e .
t h e r m a l diffusion so t h a t t h e c u r r e n t d i s t r i b u t i o n is c o m p l e t e l y d e t e r m i n e d by t h e t e m p e r a t u r e distribution [2]. We also a s s u m e t h a t the length of the t h e r m a l disturbance in the axial direction is m u c h g r e a t e r t h a n the t h i c k n e s s of the s u p e r c o n d u c t o r in the t r a n s v e r s e direction so t h a t the problem can be t r e a t e d as o n e - d i m e n s i o n a l in space by neglecting the edge effect. The governing equation in the superconductor then becomes 1 0T
W -
a Ot
02T t
k
OX 2
(1)
The h e a t source p e r unit volume of the superconductor W is composed of two parts, or W = W 0 + Wi(T)
(2)
where W 0 is the e x t e r n a l l y imposed h e a t i n g r a t e p e r u n i t volume of the s u p e r c o n d u c t o r and r e p r e s e n t s a c o n s t a n t t h e r m a l d i s t u r b a n c e , a n d Wi(T) is the i n t e r n a l l y g e n e r a t e d heating rate per unit volume of the superconductor as a r e s u l t of c u r r e n t r e d i s t r i b u t i o n in the superconductor and the stabilizer. I f we f u r t h e r a s s u m e t h a t c u r r e n t d e n s i t y in the s u p e r c o n d u c t o r is e q u a l to t h e c r i t i c a l c u r r e n t d e n s i t y c o r r e s p o n d i n g to t h e local
464
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
Vol. 22, No. 4
t e m p e r a t u r e with all the excess c u r r e n t t r a n s f e r r e d to the stabilizer, the internal generation rate is [2]: Wi(T) = EJ(T) = EJc(T) = E J c ( T i ) ( T c - T)/(T c - T i)
(3)
The electrical field strength developed in the composite conductor as a result of current flow in the stabilizer, E, is assumed to be uniform across the entire conductor. The initial and boundary conditions are t = 0, T = T i
(4)
OT X=0, - - ~ = 0; X = a, T = T i
(5)
If we assume that the thermal conductivity of the metal stabilizer is much higher than that of the superconductor (which is applicable to high-temperature ceramic superconductors) and t h a t the thickness of the superconductor is not small compared to that of the stabilizer, we need not solve the heat conduction equation in the stabilizer. By substituting Eqs. (2) and (3) into Eq. (1), the resulting equation can be solved by Laplace transform [8]. After some manipulation, the solution becomes: 0=
cosh(~)
Q/Qc)
+ 2(w0(~0g + ~ 2) ~
Q/Qc
(-1)n e x p [ - ( N 2 + ~2)~] cOs(Nx)
n=0
(6)
N(~ 2 + N 2 )
where N = (2n+l)x/2 and the dimensionless variables are defined as 0 = ( T c - T ) / ( T c - T i)
(7)
x = X/a
(8)
= at/a 2
(9)
~2 = EJc(Ti ) a2/[k(Tc _ Ti)]
(10)
Og = (T c - Tg)/(T c - T i)
(11)
w 0 = W0L/Qc
(12)
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T H E R M A L ANALYSIS O F COMPOSITE SUPERCONDUCTORS
465
The p a r a m e t e r ~. = a/(a + b) is t h e s u p e r c o n d u c t o r fraction in t h e composite, a n d the critical g e n e r a t i o n r a t e Qc ( h e a t g e n e r a t i o n r a t e per u n i t volume of the composite when most of the c u r r e n t is in the stabilizer) is defined as Qc = P J02(T0 ) ),2/( ! _ ~)
(13)
The p a r a m e t e r Q in Eq. (6) is the i n t e r n a l h e a t g e n e r a t i o n r a t e p e r u n i t volume of the composite conductor (superconductor plus s t a b i l i z e r ) , which is u n k n o w n a n d m u s t be determined. The p a r a m e t e r Q can be expressed in t e r m s of t h e electric field s t r e n g t h E and the c u r r e n t d e n s i t y J0(T0), or Q= ),EJ0(T 0)
(14)
Using Eqs. (10), (13), and (14), we can show t h a t ~2 = (Q/Qc)0
(15)
where = Qca(a + b) / [k(T c - Tg)]
(16)
An additional condition t h a t m u s t be satisfied as a r e s u l t of c u r r e n t r e d i s t r i b u t i o n is t h a t the sum of the c u r r e n t in the superconductor a n d the s t a b i l i z e r m u s t be equal to the total current, or s a Jc(T ) dX + bE / P = a J0(T0) 0
(17)
Based on the dimensionless p a r a m e t e r s defined previously, it can be shown t h a t Eq. (17) is equivalent to
~o1 0 d x
+ (Q/Qc)Og = Og
(18)
S u b s t i t u t i n g Eq. (6) into Eq. (18) r e s u l t s in the following r e l a t i o n s h i p between 0g a n d Q/Qc:
tanh(~) + S 0g =
(I_Q/Qc)+I 1 tanh( )- w 0 /_2w0O S (19)
where
466
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
Vol. 22, No. 4
(~2 +N2)N2
n=0
(20)
We have now completed the formulation of the problem. F o r given values of w 0, 0g, and 0, we w a n t to calculate 0 as functions of z a n d x, a n d Q/Q c a n d I] as function x, by solving Eqs. (6), (15), and (19) simultaneously. Numerically, it is e a s i e r to c a r r y out the calculations
in a slightly different m a n n e r .
I n s t e a d of w0, 0g, a n d ~, one can use w0,
Q/Qc , a n d ¢ as inputs, t h e n ~ can be calculated explicitly from Eq. (15).
0g can then be
c a l c u l a t e d by u s i n g Eq. (19). Finally, the dimensionless t e m p e r a t u r e 0 is calculated with Eq. (6). The calculations are s t r a i g h t f o r w a r d because no i t e r a t i o n is involved. T h e r e is a simple r e l a t i o n s h i p between h e a t g e n e r a t i o n r a t e p e r u n i t volume of the composite a n d c u r r e n t in the s t a b i l i z e r (Ist). S u b s t i t u t i n g the electric field s t r e n g t h E = PJst into Eq. (14), it can be shown that Jst/J0(T0 ) = (Q/Qc) a/b
or Ist / I0 = Q / Qc
(21)
Thus, the fraction of c u r r e n t in the stabilizer is equal to the fraction of the h e a t generated in the composite conductor. A n o t h e r i n t e r e s t i n g p a r a m e t e r is the total i n t e r n a l h e a t g e n e r a t e d per u n i t volume in the superconductor, W t = ~:Wi (T)dX / a
(22)
Because total h e a t g e n e r a t e d in the composite is equal to the sum of the h e a t generated in the superconductor and the stabilizer, it follows t h a t Q(a + b) = W t a + p(Jst)2b
(23)
S u b s t i t u t i n g Eq. (21) into Eq. (23) and solving for the d i m e n s i o n l e s s p a r a m e t e r wt, we obtain wt = Wt UQc = (Q/Qc)(1 - Q/Qc).
(24)
Both Eqs. (21) and (24) a p p e a r to be quite simple a n d depend only on h e a t generation rate per unit volume of the composite Q/Qc. However, because Q/Qc depends on w 0, 0, and 0g, so do Ist/I 0 and w t. E q u a t i o n s (21) and (24) are applicable to both t r a n s i e n t and steadys t a t e situations.
Vol. 22, No. 4
THERMAL ANALYSIS OF COMPOSITE SUPERCONDUCTORS
467
Before presenting the results, we will point out a special case of the general solution given by Eq. (6). Because the a r g u m e n t in the exponential term is negative, a steadystate solution always exists as • becomes large. Under steady-state condition, Eqs. (6) and (19) are reduced to 0=
[ c, °W s h (°~ xO) l (g l ~- r -. - - cosh(~) Q/Qc)
W00g Q/Qc
tanh(~) /
0g --
( 1 - Q / Qc) + [ 1 - t a n h ( ~ ) / ~ ]
(25)
(w0) (26)
If there is no external thermal disturbance (w0=0), the steady-state solution becomes 0 = cosh(l~x)/cosh(~)
(27)
0g = tanh(~)/~/(1 - Q/Qc)
(28)
Equations (27) and (28) are identical to the solution reported by Wilson [2], who calculated the steady-state, transverse temperature distributions and h e a t generation rate, and applied the results to determine the cryogenic stability of a composite conductor. Results and Discussion~ The results of the previous analysis will be presented in two parts. The first part describes the numerical results of the steady-state solution, and the second describes the results of the transient solution. Instead of Og, we will define a dimensionless parameter 0* as 0* = (T i - Tg)/(T c - Tg)
(29)
The relationship between 0g and 0* is 0* = 1 - 1/0g
(30)
The results will be expressed in terms of 0", the parameter employed by Wilson [2], so that a direct comparison can be made. Steady-State Solution In this case, the exponential terms in Eqs. (6) and (19) approach zero as z becomes very large. Figure 2 shows the calculated 0* as a function of Q/Qc (or Ist/I 0) for various values of ¢. For the case of no external disturbance (w 0 = 0), the result is shown in Fig. 2(a) and is identical to that reported by Wilson [2]. The case of ¢ = 0 corresponds to the
468
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
1.o ~ , , , , , , ,
~''
06
Vo|. 22, No. 4
./~t
~o
1 0.0 -0.2
I
0.0
,
0.2
,
I
,
,
,
q
~
0.4 0.6 Q/O orl /I c
st
~
I
,
J
0.8
1.0
0
(a) 1.0
r
I
0.8 0.6 0.4 °CD
0.2 0.0 -0.2 " 0 ' 4 0 O' •
=
,
I
0.2
,
,
I
0.4
= , ,
L
0.6
,
I
0.8
,
,
,
1.0
CYO C or I t / I 0
(b)
0.8 0.6 0.4 *¢:D
0.2 0.0 -0.2 0.0
0.2
0.4 0.6 Q/Q orl /I C
st
0.8
1.0
0
(c) FIG. 2 V a r i a t i o n of 0* w i t h Q/Qc or I s t / I o for v a r i o u s v a l u e s of ¢ u n d e r s t e a d y - s t a t e c o n d i t i o n . (a) w 0 = 0; (b) w 0 = 0.1; (c) w 0 = 0.2.
Vo|. 22, No. 4
THERMAL ANALYSIS OF COMPOSITE SUPERCONDUCTORS
469
special s i t u a t i o n where the superconductor is at a uniform t e m p e r a t u r e because t h e r m a l conductivity k approaches infinity. It can be seen from Fig. 2(a) that e* = 0 (T i = T g ) a t Ist/I0=0 and e* = 1 (T i = T c) at Ist/I 0 = 1. This m e a n s t h a t in the absence of a n external disturbance, all of the c u r r e n t is in the superconductor when T i = Tg, a n d all of the c u r r e n t is in the stabilizer when T i = Tc.
The s i t u a t i o n is different when there is an
external d i s t u r b a n c e (w 0 > 0), as shown in Figs. 2(b) and 2(c). Except for ~ = 0, it can be observed that e* < 0 (T i < Tg) when Ist/I0 = 0 and {}* < 1 (T i < T c) when Ist/I 0 = 1. To m a i n t a i n all the c u r r e n t in the superconductor in the presence of a n external disturbance, the interface t e m p e r a t u r e T i m u s t be lower t h a n the generation t e m p e r a t u r e Tg. This is because the external disturbance causes c u r r e n t redistribution within the superconductor even though all of the current is still in the superconductor. On the other hand, when all of the c u r r e n t is transferred to the stabilizer, Figs. 2(b) and 2(c) show t h a t the interface temperature T i is lower t h a n the critical t e m p e r a t u r e T c. This result is incorrect physically because if the interface t e m p e r a t u r e is lower t h a n the critical temperature, a portion of the superconductor (near the interface) is still capable of c a r r y i n g a c e r t a i n a m o u n t of c u r r e n t and consequently not all of the c u r r e n t is in the stabilizer. This unrealistic result arises because the analytical model could not account for the s i t u a t i o n when the t e m p e r a t u r e of the s u p e r c o n d u c t o r exceeds the critical temperature. This can be explained by e x a m i n i n g Eq. (3), which is the a s s u m e d i n t e r n a l g e n e r a t i o n r a t e i n the superconductor.
W h e n T is g r e a t e r t h a n T c, the i n t e r n a l
generation rate becomes negative, but this is obviously incorrect.
W h e n T i is slightly
lower t h a n or equal to T c at Ist/I0 = 1, as shown in Fig. 2, the bulk of the superconductor is at t e m p e r a t u r e s g r e a t e r t h a n Tc, a n d Eq. (3) fails to describe the physical s i t u a t i o n correctly. We discuss this f u r t h e r in a later section. Another important feature that can be observed in Fig. 2 is that when ~ > 3, a relative m i n i m u m appears in each of the curves in these figures. For example, when ¢ = 5, Q/Q c increases with decreasing ~* for Q/Qc < 1.5 and increases with i n c r e a s i n g ~* for Q/Qc > 1.5. The portion of the curve with negative slope is most likely u n s t a b l e because as the interface t e m p e r a t u r e T i is reduced, the superconductor should be able to carry more current, and therefore less c u r r e n t is transferred to the stabilizer. Typical t e m p e r a t u r e distribution in the superconductor is shown in Fig. 3, where the d i m e n s i o n l e s s temperature e is plotted a g a i n s t x for various values of Q/Qc (or Ist/I0) with w 0 = 0.2 a n d ¢ = 5.
As Q/Qc is i n c r e a s e d , t e m p e r a t u r e i n the core of the
superconductor increases and approaches T c (e = 0). However, when Q/Qc is greater t h a n 0.8, {} becomes negative and T becomes greater t h a n T c.
This is a g a i n the r e s u l t of
employing Eq. (3), which is not valid when the temperature of the superconductor exceeds the critical temperature.
470
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
1,5
'
'
'
i
'
I
I
'
'
i
;
'
I
,
,
'
t
'
;
'
Vol. 22, No. 4
i
~ '
I
,
;
1.0
a~ 0 . 5
0.0
-0.5
;
,
0.2
0.0
L
L
0.4
,
[
0.6
,
,
I
i
0.8
,
1.0
X
FIG. 3 V a r i a t i o n of 0 w i t h x for v a r i o u s v a l u e s of Q / Q c or Ist/I 0 with w 0 = 0 a n d ¢ = 5 u n d e r s t e a d y - s t a t e condition.
We d e t e r m i n e d how t h e d i m e n s i o n l e s s total i n t e r n a l g e n e r a t i o n r a t e w t v a r i e s w i t h the dimensionless temperature
0* a n d t h e h e a t g e n e r a t i o n r a t e p e r u n i t v o l u m e of t h e
c o m p o s i t e Q/Qc" F i g u r e 4 is a plot o f w t v e r s u s Q / Q c (or Ist/I0) c a l c u l a t e d by Eq. (24). T h e total i n t e r n a l
generation
reaches a maximum
a t Ist/I 0 = 0.5.
T h e v a l u e of w t is at
m a x i m u m b e c a u s e t h e i n t e r n a l g e n e r a t i o n r a t e d e p e n d s on t h e p r o d u c t of two c o m p e t i n g factors, t o t a l c u r r e n t in t h e s u p e r c o n d u c t o r a n d e l e c t r i c field s t r e n g t h .
When the total
c u r r e n t in t h e s u p e r c o n d u c t o r is high, electric field s t r e n g t h is low b e c a u s e the c u r r e n t in t h e s t a b i l i z e r is small.
W h e n t o t a l c u r r e n t in t h e s u p e r c o n d u c t o r is low, e l e c t r i c field
s t r e n g t h is h i g h b e c a u s e t h e c u r r e n t in t h e s t a b i l i z e r is high.
T h i s c h a r a c t e r i s t i c is
c l e a r l y d e m o n s t r a t e d in Fig. 4 and by Eq. (24). F i g u r e 5 shows t h e v a r i a t i o n of O* w i t h w t for v a r i o u s v a l u e s of ¢. A g a i n , w t r e a c h e s a m a x i m u m at some v a l u e of O* t h a t d e p e n d s on t h e v a l u e s of ¢ a n d w 0. Transient Solution F i g u r e 6 s h o w s t h e r e s u l t s of the t r a n s i e n t s o l u t i o n c a l c u l a t e d by Eqs. (19) a n d (30). It a p p e a r s t h a t t h e s t e a d y - s t a t e is a p p r o a c h e d w h e n • _= 1, w h i c h m e a n s t h a t t h e t i m e r e q u i r e d to r e a c h s t e a d y s t a t e is t -_- a2/a.
T h i s c h a r a c t e r i s t i c t i m e for t r a n s v e r s e h e a t
c o n d u c t i o n in t h e s u p e r c o n d u c t o r c a n be c o m p a r e d to t h a t of c u r r e n t diffusion in t h e stabilizer.
T h e c h a r a c t e r i s t i c t i m e of c u r r e n t diffusion in t h e s t a b i l i z e r is tst_= P0L2/p,
Vol. 22, No. 4
THERMAL ANALYSIS OF COMPOSITE SUPERCONDUCTORS
471
0.30 0.25 0.20 5"0,15 0,10 0.05 0.000,0
I
I
I
I
~
0.2
L
I
I
I
J
J
I
I
I
I
J
0.4 0.6 ~Q orl/I ¢
st
I
I
~
1.0
0.8 0
FIG. 4 Variation of w t with Q/Q c or I st/I 0.
1.0
1.0
0.8
0.8
E
,
,
i
,
r
,
,
i
,
,
,
,
i
,
,
,
,
i
,
,
,
,
i
,
,
,
,
0.6
0.6
0.4 0
*¢= 0.4
0.2
0.2
0.0
:
0.0 5 -02
~ , ~ ,
• 0.00
J J , , , I L J , , J
0.05
0.10
....
0.15
I ....
0.20
Ii
,~
0,25i , ' 0".30
-0.2 ! 0 -0.~.
o'15
030
W
W
t
t
(a)
(b)
FIG. 5 Variation of 0* with w t for various values of~. (a) w 0 = 0; (b) w 0 = 0.1. where P0 (= 4 ~ x 10 -7 H/m) is the p e r m e a b i l i t y of free space, a n d L is the characteristic length in the longitudinal direction. The ratio of the two characteristic times is t / tst = pa2/~u0 L2
(31)
T h e r m a l diffusivity for h i g h - t e m p e r a t u r e superconductors at 77 K is ¢x= 10-6-10 -7 m2/s, and the resistivity of silver at 77 K is p --- 3.65 x 10-9 ~2-m. In the a n a l y t i c a l model, we a s s u m e d t h a t L is much g r e a t e r t h a n a so t h a t edge effect can be neglected, a n d the problem can be treated as one-dimensional. If L/a = 10, t / t s t =0.58 x 102 >> 1
472
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
Vol. 22, No. 4
1.0
1.0
0.8
0.00 - -0.25
0.8
//
- 0.50
0.6
0.75 1 O0
",= 0.4
-
0.6 -
/.~
0.4 L
/ /
-
---
>/
o,~o 0.25 o.5o
.... / , ~ ..... ~ f f
1.00
..
/ - i f "
0.2 0.2
oo I
-0,0 -0.2
-0.2 0.0
0.2
0.4 0.6 Q/Qc or I t / I °
0.8
10
.
-0,4
(a)
0.0
0,2
.
0.4 0.6 Q/Qc or I t / I °
.
.
0.8
.0
(b)
FIG. 6 Variation of O* with Q/Qc or Ist/I0 with ¢ = 5 u n d e r t r a n s i e n t condition. (a) w 0 = 0; (b) w 0 = 0.1. a n d the time required for current redistribution in the stabilizer is much shorter t h a n that required for the t r a n s v e r s e t e m p e r a t u r e in the superconductor to reach steady state.
A
steady-state c u r r e n t in the stabilizer and the electric field s t r e n g t h can be assumed to establish i n s t a n t l y . But as one can see, the m a r g i n is relatively small.
If L/a > 10 is
required, t/tst will no longer be much greater t h a n 1, and the error introduced by the oned i m e n s i o n a l model will increase. As m e n t i o n e d previously, the t r a n s i e n t t h e r m a l response of a composite conductor may be p a r t i c u l a r l y r e l e v a n t to the i n t e r p r e t a t i o n of e x p e r i m e n t a l results on flux flow resistivity for high t e m p e r a t u r e superconductors because of their relatively low t h e r m a l conductivities. Summ~rv A closed-form solution of transient heat conduction with a distributed heat source is employed to predict transient transverse temperature distribution and heat generation rate per unit volume of a composite conductor subjected to a constant thermal disturbance. The heat source includes a constant external disturbance and an internal generation term that is a function of temperature distribution in the superconductor (Eq. (3)). W e show that the dimensionless temperature distribution 0 (Eq. (6)) and the heat generation rate per unit volume of the composite (~Qc depend on the three dimensionless parameters 0g, ¢, and w 0. Results of the transient and steady-state solutions are presented in Figs. 26. Heat generation rate per unit volume of the composite, Q/Qe, is directly proportional to current in the stabilizer (Eq. (21)). The dimensionless total internal generation rate w t in the superconductor is shown to reach a m a x i m u m at Q/Qc = 0.5 (Fig. 4) because w t depends on two competing factors: the current in the superconductor and the electric field strength which is proportional to the current in the stabilizer. It is demonstrated that steady state is approached w h e n the dimensionless time • becomes greater than i. In the absence of an external disturbance, the steady-state solution reduces to that reported by
Vol. 22, No. 4
T H E R M A L ANALYSIS OF COMPOSITE SUPERCONDUCTORS
473
Wilson [2]. The t r a n s i e n t t h e r m a l response of composite conductor m a y be relevant to the m e a s u r e m e n t on flux flow resistivity s i m i l a r to t h a t reported by Askew et al. [3]. A basic a s s u m p t i o n m a d e in this analysis is t h a t the c h a r a c t e r i s t i c time of c u r r e n t r e d i s t r i b u t i o n in the s u p e r c o n d u c t o r and t h e s t a b i l i z e r is much s h o r t e r t h a n t h a t of t e m p e r a t u r e redistribution, so t h a t h e a t conduction in the superconductor is i n d e p e n d e n t of the c u r r e n t diffusion process in the s t a b i l i z e r .
This a s s u m p t i o n is shown to be
m a r g i n a l for h i g h - t e m p e r a t u r e superconductors. A n o t h e r basic a s s u m p t i o n m a d e in t h e p r e s e n t a n a l y s i s is t h e d i s t r i b u t i o n of i n t e r n a l h e a t g e n e r a t i o n r a t e described by Eq. (3).
I t is e v i d e n t t h a t Eq. (3) cannot
accommodate a t e m p e r a t u r e g r e a t e r t h a n T c because dissipation becomes negative when T > T c and t h u s Eq. (3) gives unrealistic results. In reality, T can be g r e a t e r t h a n T c a n d the local c u r r e n t d e n s i t y should be reduced to zero if T > T c. Consequently, local h e a t generation rate should be zero. In o t h e r words, W i = 0 and J = 0 w h e n T > T c. This additional c o n s t r a i n t will complicate the analytical solution b u t can be h a n d l e d easily by n u m e r i c a l solution. We are c u r r e n t l y c a r r y i n g out the n u m e r i c a l solution, in which we can include the more general condition of a t i m e - d e p e n d e n t e x t e r n a l d i s t u r b a n c e i n s t e a d of the constant external disturbance used in the present analysis.
The p r e s e n t a n a l y t i c a l
solution can be employed to v a l i d a t e the r e s u l t s of the n u m e r i c a l solution u n d e r special circumstances. Acknowledgments This work has been supported by the U.S. D e p a r t m e n t of Energy, E n e r g y Efficiency a n d Renewable Energy, as p a r t of a p r o g r a m to develop electric power technology, u n d e r C o n t r a c t W-31-109-Eng-38.
S.Y. Seol was s u p p o r t e d by the r e s e a r c h f o u n d a t i o n of
Chonnam National University.
a
one-half the thickness of the superconductor, m
b
thickness of the stabilizer, m
E
electric field strength, V/m
I
current, A
I0 J Jc J0 k
o p e r a t i n g current, A c u r r e n t density, A/m 2 critical c u r r e n t density, A/m 2 o p e r a t i n g c u r r e n t density at T = T 0, A/m 2 t h e r m a l conductivity of the superconductor, W/m-K
L
c h a r a c t e r i s t i c l e n g t h in t h e l o n g i t u d i n a l direction for c u r r e n t diffusion in the
Q Qc
stabilizer, m h e a t g e n e r a t i o n r a t e per unit volume of the composite conductor, W/m 3 critical h e a t generation r a t e per u n i t volume of the composite conductor (Eq. (13)), W/m 3
474
Y.S. Cha, W.J. Minkowycz and S.Y. Seol
T
temperature, K
Tc
critical t e m p e r a t u r e , K
Vol. 22, No. 4
Ti
interface t e m p e r a t u r e between the superconductor and the stabilizer, K
T0
o p e r a t i n g t e m p e r a t u r e before a t h e r m a l disturbance, K
t
time or c h a r a c t e r i s t i c time for t h e r m a l diffusion, s
tst
c h a r a c t e r i s t i c time for c u r r e n t diffusion in the stabilizer, s
W
h e a t i n g r a t e per unit volume of the superconductor (Eq. (2)), W/m 3
Wi
i n t e r n a l g e n e r a t i o n r a t e per unit volume of the superconductor (Eq. (3)), W/m 3
W0
e x t e r n a l h e a t i n g r a t e per unit volume of the superconductor, W/m 3
Wt
total i n t e r n a l generation rate per unit volume of the superconductor defined in Eq. (22), W/m 3
w0
dimensionless e x t e r n a l h e a t i n g rate p e r unit volume of the superconductor defined in Eq. (12), W/m 3
wt
dimensionless total i n t e r n a l generation r a t e p e r unit volume of the superconductor defined in Eq. (24), W/m 3
X
t r a n s v e r s e coordinate, m
x
d i m e n s i o n l e s s t r a n s v e r s e coordinate, X/a t h e r m a l diffusivity of the superconductor, m2/s d i m e n s i o n l e s s p a r a m e t e r defined in Eq. (10)
p
electrical resistivity of the stabilizer, ~ - m fraction of superconductor in the composite, a/(a+b)
¢
a d i m e n s i o n l e s s p a r a m e t e r defined in Eq. (16) d i m e n s i o n l e s s time, ¢zt/a 2
0 0g
d i m e n s i o n l e s s t e m p e r a t u r e defined in Eq. (7) d i m e n s i o n l e s s t e m p e r a t u r e defined in Eq. (11)
0*
d i m e n s i o n l e s s t e m p e r a t u r e defined in Eq. (30) References
1.
B.J. Maddock, G.B. J a m e s and W.T. Norris, Cryogenics, 9,261(1969).
2.
M.N. Wilson, Su~erconductin~ M a g n e t s , Oxford U n i v e r s i t y P r e s s , Chap. 6, New York (1983).
3.
T.R. A s k e w , J.G. N e s t e l l , R.B. F l i p p e n , D.M. Groski a n d N.McN. Alford, I E E E Trans. on Appl. Superconductivity, 3, 1398 (1993).
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Received February 22, 1995