The longitudinal propagation of the normal zone in superconducting wires was experimentally investigated in order to evaluate existing analytical expressions which attempt to describe the propagation velocity in a more or less simple manner. The availability of a reliable expression is important for application in computer codes that calculate quench evolutions in superconducting magnets. We measured the propagation velocity as function of transport current and magnetic field in five different insulated NbTi superconductors having a copper, a copper-nickel or a mixed matrix. The comparison to calculated velocities using five existing models showed that large quantitative and qualitative differences exist

Boschman, H.

Kate, H.H.J. ten

Klundert, L.J.M. van de

English

ten Kate, H.H.J.

van de Klundert, L.J.M.

University of Twente Research Information

IEEE TRANSACTIONS ON MAGNETICS, VOL.  MAG-^^, NO. 2, MARCH i987 LONGITUDINAL PROPAGATION VELOCITY OF THE NORMAL ZONE I N  SUPERCONDUCTING  WIRES H.H. J. ten Kate, H. Boschm~in, L. J.M. van de Klundert, University of Twente, Faculty of Applied Physics, Low Temperature Laboratory, P.O.  Box 217, 7500 AE Enschede. The Netherlands 1551  Abstract rhe longitudinal propagation of the normal zone i n  superconducting wires was experimentally  inves,tigated in   order   to   evaluate   exis t ing  ar ia lyt ical .   expressions which attempt to describe the propagation velocity in a more or l e s s  simple manner. The ava i lab i l i ty   o f  a reliable  xpression is important for   appl ica t ion   in  computer  codes tha t   ca lcu la te  quench evolut ions  in  superconducting  magnets. We measured the  propagation ve loc i ty  as function of transport current ,and magnetic f i e l d  i n  f i v e  d i f f e r e n t  i n s u l a t e d  NbTi superconductors having a copper, d copper-nickel o r  a mixed matrix. ?e comparison to   ca lcu la ted   ve loc i t ies   us ing   f ive   ex is t ing  models showed tha t   l a rgequan i ta t ive  and qual i ta t ive differences exis t .  Introduction The r e s i s t a n c e   r a t e   i n  a superconducting mabet after the  occurrence  of a quench is of a grea t  importance for  the  design  of an  appropriate quench protection system. The usual method for determining, the growth of the coil  resistance in impregnated c o i l s  is to   calculate   the  longi tudinal   veloci ty   with which the  normal  zone in the superconductor.  expands a f t e r  which the  different   t ransverse components of the  quench propagation  velocity  can  be calculated  assuming  an estimate f o r   t h e   r a t i o   o f   t r m s v e r s e   t o   l o n g i t u d i n a l  thermal  conductivities. The ve loc i t i e s   i n  either d i rec t ion  so obtained are used i n  computer coded tha t  simulate, among other  things,  the growth of the normal conducting  part of  the magnet and the  fur ther  expiration  of a quench [lj. It is the  aim of  the  experiments  o compare and contrast  experimentally found ve loc i t ies   wi th   f ive  theore t ica l  models. Present analytic formulae The normal  zone propagation is determined by the  amount of  heat  production i n   t h e  normal zone, t he  enthalpy and heat  flow from the  zone t o   t h e  environment. The heat-balance equation i n  one dimension ( the  longi tudinal  z direction)  that   describes  uch a process and the  temperature  profile  along  the wire is given by: where c is the  mean heat capacity, k the mean thermal conductivity, 0 the   a t  flow in   the  t ransverse  d i rec t ion  and q the heat production. While t r ea t ing  the  case as one dimensional, the basic assumptions are t h a t  the  temperature  profile and the  heat  production are uniform  over  the  cross-section. A s  a consequence the  heat capacity and thermal conductivity are averaged f o r  each  ross-section. It is obvious tha t   t he  one- dimensional  treatment  neglects  the  effects  due  to  the uniform dis t r ibut ion of  the f i laments ,  the non-uniform current  dis t r ibut ion as  well a s  inhomogeneous matrices such as the presence of copper-nickel barriers. After adoption  of  equation (1). the  r maining assumptions concern the mechanisms of heat transfer and heat production. The term @ i n  (1) can include steady- state heat  conduction  expressed by a t ransfer  Manuscript received September 30, 1986 coef f ic ien t  h [W/m2 lk]. but   a l so  a ,  t rans ien t  component. The latter term caxi usually be neglected i f  the conductor has no, direct contact with liquid helium. The reader is refer red  to  Lvovsky [6 ]  f o r  more d e t a i l s  about  trangient terms. Here the  models. assuming a constant  heat  transfer are t rea ted  which attempt t.0 describe propagations in enclosed superconductors. This means 0 = h*p/A*(T-Ti), where p is the perimeter and A the  cross-section  f   the wire and T, the   bath temp9ratut-e. The heat production term q at  cur ren t  I is d i f fe ren t .   in   , the .   th ree   reg ions   in   the   t empera ture  p r o f i l e   t h a t  can be distinguished: q = 0 ,  , superconducting s t a t e ;   (2 )  T-Ti I q = (m) p ( ~ )  , curren t   shar ing   s ta te ;  (3)  4 = P ( I / A ) ~  , normal state; (4) i n  which Tc is the  f ie ld-dependent   cr i t ical  temperature, and p the me? r e s i s t i v i ty  o f  t he -wi re .  Equation ( l ) . c a n  be solved analytically provided k and c are constant and $&en at a cer ta in  intermediate  temperature. Three,authors solved, the model neglecting the  current-sharing state. They considered  the  joule heating  to  begin  d scontinuously at  a ,  ce r t a in  transition  temperature Ti. Then, the  general  solution of (1) using  the,appropriate boundary conditions  iS found i n  t h e  form o f  a t rave l l ing  wave with a reduced velocity v i n  terms of  the  Stekly  parameter a,. the  reduced transition  temperature ti = (Ti-T, )/(Tc-T, ) , and the  reauced  transport current i = I / I c ( B ) .  The physical  propagation ve loc i ty  V is now: c o  and the  Stekly  paraineter a = oIi/(Aph(TC-T,,)). (7) The choice of the  trans  ion  temperature  has  led  to th ree  models. The corisequ ces. of these ,models f o r  t h e  reduced  velocity v, the inimum propagation  current (also called.cold-end  recovery  current) i, (v-0) as well as t h e  f u l l  recovery  current if (v=--) are given in  the  next  tab le .  I )  provided physical parameters c .  k taken at  Tc. The moaels are discussed below with the help of Fig. 1 t h a t  shows the temperature profile and the temperature 0018-9464/87/0300-1557$01.0001987 IEEE 1558 // I Dr I L u l  j Fig. 1. Temperature p ro f i l e  and the heating and cooling curves vs. temperature, assuming a constant h. dependence  of the  heating and cooling  curves.  For convenience the calculated velocities according to five models a re  shown in Fig. 2. Cherry and Gittleman [2] describe  the  solution where the  joule   heat ing  s tar ts   tep-wise at  t h e   c r i t i c a l  temperature. This leads to propagation velocities which are too small and a minimum propagation  current which is too  large.  Keilin e t  a l .  [3] predict   he   veloci ty  when the   jou le   hea t ing   s ta r t s  a t  the  current-sharing temperature Tcs. The found velocit ies  are  obviously too large and the minimum propagation  velocity is too small. Fig. 2. Comparison of  calculated  velocit ies  according t o  f i v e  models (-) and measured da ta  (0) f o r  a cer ta in  wire and a f i e l d  of 6 t e s l a .  A more r e a l i s t i c  behaviour is given by Dresner [4]. He takes  the  transit ion  temperature  as  the mean value between the  current  sharing and t h e   c r i t i c a l  temperature and finds a cor rec t  minimum propagation current  but a too  large  full-recovery  current.  For a less than 20,  Dresner states a correct ion factor  on v of (1+0.561a-1*45). The f 4 r t h  model as  described by Turck et  al. [5] includes  equation 3 in   the  current-  sharing  region. The third  i f ferent ia l   equat ion  thus obtained  introduces, compared to  Dresner's model, qn increase  of   the   veloci ty   in   the  vicini ty   of   the  c r i t i c a l  c u r r e n t .  The solution is: It can  be  s en from Fig. 2 tha t   t he  formulae of Dresner and Turck coincide  in  the main current  range between the minimum propagation current and 80% of the c r i t i c a l   c u r r e n t .  Lvovsky proposed an ana ly t ica l  expression  that  pproximates  his  numerical  solution [TI. He  s ta ted :  i-i v = v [l-{l-T 1n(1--)} -VI (1 -0 .4~1)*  v1 max m -2  m i-i 1-i (-)I t m m where and This solution gives a minimum propagation current which is identical   to  those  of  Dresner and Turck. However the maximum reduced velocity v following this formulae is larger  than  thativen by Turck. Since Lvovsky takes  the  physical  parameters a t  Tc and Turck a t  Tav, the  deviation between the maximum physical ve loc i t i e s  is a fac tor  ( T o / T , ) ~ * ~ .  This  difference can be c lear ly  seen  in  F ig .  2 .  The physical model parameters A serious  handicap  with  calculating  propagation ve loc i t i e s  is the  lack  of  precise knowledge of  the various physical properties of the wire materials. Many authors fit  ca lcu la ted  ve loc i t ies  to  measured ones and ascr ibe   the   f i t t ing   fac tor   to   the   l a rge   uncer ta in t ies  i n  t h e  model constants such as the thermal conductivity and heat  capacity. Here we attempt  to compute them using  h ,   the   heat   rar isfer   coeff ic ient ,   as   the  only f i t t i n g  parameter. The measured quant i t ies  are the velocity V,  the wire diameter D ,  the wire r e s i s t i v i t y  as function  of f i e l d  p ( B ) ,  t he   c r i t i ca l   cu r ren t  I c ( B ) ,  the  current I and the appl ied f ie ld  Bext. The magnetic f i e l d  on the  wire is given by Bext + fb*I,  where t h e   l a t t e r  term represents  the self-f ie ld  and the  f i e ld  due to  adjacent  wires.   For  the  cri t ical   current  the  formulation  of Lube11 was adopted: Tc(B) = Tc(B=O)* (1-B/l4.5)Oa59. The current  sharing  temperature Tcs(B, I )  = To+(Tc(B)-To)*( l - i )  and the  average transition  temperature Tav = (Tc+Tcs)/2. The heat capacity is c(T,B) = [(6.75*f+50.55)*T3+ (97.43*f+69.81*B)*T]/(l+f), which includes a f ie ld-  dependent term [81 . The mean thermal  conductivity is eliminated from the calculat ion by using the Wiedemann- Franz Law s t a t i n g  k = Lo*T/p(B), where Lo = 2.44*10-a Wfi/K' . The temperature  dependent  parameters are taken at  the proposed transition temperatures which are given as Tc(B) ,  T c ( B , I )  and Tav(B.I)  by Cherry,  Keilin and Dresner  espectively.  In  Turck's model they are a lso  taken a t  Tav(B, I), while Lvovsky uses them a t  A more serious .problem ar ises .   wi th   the  cr i t ical  current which has a large  influence on the  velocity especially at  low magnetic f i e lds .  The effect   of   the  currently  applied  definit ions  uch  as 1 uV/cm, lo-" fim and 10-l' nm may resu l t   in   devia t ions   as   l a rge  as 1 t o  5%. . Therefore  the  ultimate  accuracy  of  the calculated  v loci t ies  w i l l  be  about  the same percentage.   In our  calculations  the  10-l '   cri terion has been applied. Tc(B) - Measurements on five superconductors The propagation  velocities  are  d termined by measuring the time-lag between the s tar t  of  the vol tage r i s e  a t  two adjacent  voltage  taps on the wire. During the measurement the current is kept almost constant by using a choking coi l   wi th  an  appropriate  inductance, which value is chosen i n  such a way that  the wire can not burn out due to excessive heating. This protection is very important in wires with a copper-nickel matrix, s ince ,  for  example, a t  a current density of 1000 A/mm2 the  maximum time constant is 2 m s .  Approximately  one meter of the sample is wound on a former (dia.  70 m m ) ,  covered  with  about 1 mm STYCAST 2 8 5 0 ~ ~   r e s i n  and 1559 i n s e r t e d  i n  a 7 tesla magnet. In   order  to prevent  an undesired  heat  exchange  the  s nsing wires are of nichrome. So we obtain an enclosed superconductor with no d i r e c t  helium  contact. The m a x i m u m  current  during the  tests was 400 A and t h e  f i e l d  was varied between 0 and 6 tesla. The specif icat ions  of   the   invest igated wires are given i n  t a b l e  1. The calculated  v loci t ies   using these assumptions were already shown i n  Fig. 2 .  The differences between Kei l in ' s  and Cherry's formulae with those  of Dresner and Turck are- reduced by approximately 50% i f  t he  cons t an t s  c and k are taken a t  Tav. The e f fec t   o f   us ing   the  mean heat  capacity between Tcs and Tc instead  of  the  capacity a t  Tav is 1 o r  2 percent'and thus negligibly small. Now the   resu l t s  w i l l  be  discussed  consecutively. F i r s t   bo th  wires with a copper  matrix (1 and 2 )  are considered. In Figures 3 and 4, the  comparison between measured da ta  and the  models of Dressner and Turck is shown. A s  already shown i n  Fig.  2,  these models f i t  qual i ta t ively and.quant i ta t ively very well. A t  4 and 6 tesla with the thick wire, and at 2, 4 and 6 tesla with the  th in  wire the   ve loc i t ies  were measured up t o  t h e  c r i t i c a l   c u r r e n t ,  as ind ica ted   in   the  figures. Note that   both models almost  coincide between the  minimum propagation  current and  80% of   the   c r i t i ca l   cur ren t .  Fig. 3. Wire 1. t ab l e  1, Cu matrix. Comparison between measured ve loc i t i e s  and the formulae  of  Turck and Dresner. Parameters: Bext 0-6 T, h = 650 W/m2 /K. t 80 - I $ > 60 I 40 20 0 100 200 i Fig. 4. Wire 2 ,  t ab l e  1, Cu matrix. A s  i n   f i g u r e  3. h = 850 W/mz /K. The ex t ra   increase   o f   the   ve loc i ty   near   the   c r i t i ca l  current as given by Turck  cannot  be  proven by t h i s  experiment i f  we remind the  limited  accuracy  of  the c r i t i c a l   c u r r e n t .  Moreover, this   di f ference  has  no practical sense because superconductors usually do not operate   near   the  cr i t ical   current .   Further  i t  appears t h a t  a t  zero   ex terna l   f ie ld  and large  currents   the models underestimates  in  both wires the veloci ty  with 10%. Note t h a t  t h e  minimum propagation current and the  slope a t  th i s  po in t  is correctly given a t  a l l  f i e lds .  Obviously a constant   heat   t ransfer   coeff ic ient .  which is 650 and 850 W/m'/K fo r  these  wires respectively,  cah describe  the  f eld-dependent minimum propagation current as given i n   t a b l e  1 very well. Wire 3 and 4.  have a high-resis t ivi ty   matr ix   of  CuNi. This implies a bad thermal conductivity as well, which may have its inf luence  on  the qual i ta t ive behaviour  of  the  velocity. As a consequence the  heat t ransfer   coef f ic ien t  h has  almost no e f f e c t  on the ve loc i t i e s  a t  least in  the  practical   range  of 0 t o  10 W,m2K, In Fig. 5 the   r e su l t s   fo r  wire 3 are shown. It I - I .. I LA1 -c Fig. 5. Wire 3 .  t ab l e  1. CuNi matri. 3 Measured - -  ve loc i t i e s  a t  0. 1 &d 2.5 tesla, compared t o  Dresner's and Turck's  formulae. h = 650 W/m' /k. 1560 L 4 1 LAJ ~ Fig. 6. Wire 4, t ab l e  1, ve loc i t i e s  between 0 and 6 T compared to  Dresner's and Turck's  formulae. h = 650 W/mZ /k. is surpr i s ing   to  see tha t   there  is only a weak f i e l d  dependency,  and tha t   t he  measured ve loc i t ies   increase  almost  linear  with  current up t o  80% of IC although in  both models they are non-linear. The r e s u l t s  of wire 4 indicate  a simular bahaviour, see f igure 6. Nor does any of the  o ther  th ree  models provide a  good solut ion as i l l u s t r a t ed  in  F ig .  7. Therefore, we must conclude tha t  t he  f ive  models under consideration cannot explain the propagat ion veloci ty  in  this  type of conductors i n  a sa t i s f ac to ry  manner though the formulae  of  Dresner and  Turck  can  be  used t o  g e t  an indicative value with an accuracy of about 30%. Fig. 8. Wire 5. t ab l e  1, mixed matrix. Measured ve loc i t i e s  between 0 and 6 tesla. Comparison t Fig. 7. Wire 4. Comparison of a l l  measured data   with a l l   f i v e  models computed a t  4 t e s l a .  The measured d a t a   a t  4 T are indicated by (0). Fina l ly  the  resu l t s  of wire 5 are considered. They are p ic tured  in  F ig .  8 .  It is a superconductor  for ac applications and therefore i t  has a  mixed matrix whereby the f i laments  are  surrounded by CuNi while the core is made of Cu with CuNi ba r r i e r s .  The influence of the copper content is c l e a r  i n  s p i t e  of the low copper content  of 28%. A s ign i f i can t  minimum propagation current i s  present and i t  is well explained by Dresner's  formula. The f i t t i n g   h e a t   t r a n s f e r  coeff ic ient  is 500 W/m2/K.  Conclusions Propagation  velocit ies were measured as  function of  applied fiel and current  in  enclosed superconductors. It appears that the models of  Dresner and  Turck provide   cor rec t   ve loc i t ies   in   the   case  of wires having a copper  matrix. The ex t r a  ve loc i ty  r a t e  i n   t he   v i c in i ty   o f   t he   c r i t i ca l   cu r ren t  as s t a t e d  i n  Turck's model could not be proven though i n  a few cases such an increase was measured. The absence  of a well defined cr i t ical  current  causes  considerable  errors which prohibi ts  a definit  conclusion on this  point .  The assumptions i n   t h e  models of Cherry and Keilin a re   un rea l i s t i c  and cause  large  deviations. It is surpr i s ing   to   no te   tha t   the  formula  of Lvovsky which ac tua l ly  is an analyt ical  approximation of a numerical solut ion  under .es t imates   the  veloci ty   in   the main current range whi le   near   the   c r i t i ca l   cur ren t   the  veloci ty  is much too  large.  The minimum propagation current  and the   ve loc i ty   ra te   near   th i s   cur ren t ,  as function  of  the  f ield  but  sing a constant  heat t r ans fe r   coe f f i c i en t ,  is well described by Dresner's and Turck's model.  The veloci t ies   near   the cr i t ical  cu r ren t  a r e  f in i t e .  The formulae  of Dresner and Turck  have a very limited  accuracy  in  the  case  of wires with a copper- nickel  or a mixed matrix. They can be used to  es t imate(  the velocity within approximately 30%. For th i s  t ype  of wires with a bad thermal conductivity it is recommended to   so lve   the  set of heat-balance  equations  in two dimensions. References 1. H.H.J .  ten  Kate, H. Boschman, H.G. Knoopers, L.J.M. van de Klundert, Proc. MT-9,  1985, Ed. Marinucci. Weymuth,  SIN Switzerland, 588-591. 1986. 2. W.H. Cherry,  J.T.  Gittleman,  Solid  State Electr. 1, 287,  1960. 3. V.E. Keilin,  E. Yu. Klimenko, M.G. Kremlev, N.B. Samoilov, Les Champs Magnetiques Intenses ,  Par is ,  231,  1960. 4. L. Dresner,  Proc. A S C  1978, IEEE MAG-15,  328, 5. B. Turck,  Cryogenics 20, 145.  1980. 6. Yu. M. Lvovsky, M.O. Lutset,  Cryogenics 22, 581-587 7. Yu. M. Lvovsky, Cryogenics 24,  691-696,  1984. 8. M.S. Lubell,  Proc. ASC 1982, IEEE MAG-19,  754-757. 1979 + and 639-647,  1982. 1983. 

Longitudinal propagation velocity of the normal zone in superconducting wires

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Les Champs Magnetiques Intenses,

https://doi.org/10.1109/TMAG.1987.1065020

Longitudinal propagation velocity of the normal zone in superconducting wires

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