Peculiarities of 2-D normal zone propagation in one-layer helical superconducting solenoid

ICEC 14 Proceedings

PECULIARITIES SOLENOID

OF

2-D NORMAL

A.V. G a v r l l l n ,

A.I.Ruslnov

P.N. L e b e d e v P h y s i c a l

Institute,

ZONE PROPAGATION

53 L e n l n s k l l

IN ONE-LAYER

prospect,

HELICAL

SUPERCONDUCTING

117924 Moscow, R u s s i a ,

CIS

The analytical and computer solutions of problem of 2-D propagation of normal zone in one-layer adiabatic solenoid, with taking into account discrete helical structure of winding, is given. Normal zone evolution is analyzed, beginning from its origination due to initial local impulse heat disturbance to final steady regime of propagation along solenoid. It is shown that in intermediate time interval another quasi-steady regime could exist.

INTRODUCTION Mathematical modelling of heat process of normal zone propagation during quenching in adiabatic solenoidal composite windings of direct current superconducting magnets meets serious difficulties even by the use of modern computers. The main one consists in that the winding usually has complicated structure and represents the anisotropic heterogeneous medium, containing helix-shaped metallic composite wires divided by "thin" streaks of insulating material. The simplified method of theoretical investigation of normal zone propagation in so complicated structure has been suggested in works of Z.J.J. Stekly [I] and M.N. Wilson [2], who have treated the winding as an anisotropic continuum (a homogeneous anisotropic solid). Naturally, such approximative approach has not permitted to model fine physical effects connected with discrete nature of winding and helicity of its layers. As the results quoted below indicate, these effects may be of a great importance in practice and are conditioned by dynamic interdependence of processes of longitudinal (along helical wire) and transverse (across inter-turn insulation) heat propagation. Empirical models [3,4], in which discrete structure of winding has been taken into account, have been based on approximate assumptions which have not enabled to observe abovesaid effects in detail. Suggested in the present paper mathematical formulation of 2-D heat problem for helical layer of single discrete winding is strict enough to describe fine peculiarities of 2-D normal zone propagation minutely. Mathematical formulation of the problem To investigate from the first principles the influence of helical discrete nature of solenoid layers on heat process of normal zone propagation (nzp), model of one layer in the form of infinite adiabatic helicoid wound on cylinder of radius r with pitch D=dt+~ and turn length I is considered; dt is the axial cross size of wire , ~ is the inter-turn insulation thickness. To have an opportunity to compare our results with analytical ones in continuum model [1,2], the thermal conductivity kw, the volumetric specific heat ~cw and the electric resistivity p of wire are considered as constant. Such assumption is justified in strong magnetic field when nzp velocity is great. Analysis of heat transfer conditions in helicoid enables to reduce the governing 2-D heat balance equation to practically equivalent I-D differential equation [5]: ~ c . -aT ~ ( x , z ) = - - ~8 (k. aT 8x )(x,T) + pjaE + q + q tI

2

~'(2Rc+~/kls)-1 {T(x+(-I) " "I,z)-T(x,T)+ m=l

m

w

8T

tiT

+2~exp(-Anw) /exp(Ant) [B.--@~ (x+(-l) m'l, t ) - C a - ~ (x, t)]dt} , n=l

T(x,O)

(1)

0

=

Tb

,

TC-~,x)

= TCw, x ) = Tb ,

(21

where x is the space coordinate along helical wire, the point x=O corresponds to the initial normal zone origination point; T is the process tlme~ T(x,T) is the wire

402

Cryogenics 1992 Vol 32 ICEC Supplement

ICEC 14Proceedings cross section temperature, Tb is the helium bath temperature; g=l for TzTc and g=0 for T>dt). The last integral-difference term in equation (I) represents the difference (divided by dt) of heat fluxes Qf-(x,T) and Qf+(x,T) on left and right boundaries of wire matrix with inter-turn insulation, respectively; Rc is the contact thermal resistance :of the boundary; An(kls,~cis,~), Bn(kts,Rc,~), Cn(kls,Rc,~) are some coefficients. Expression for Q£ is obtained in the result of analytical solution of I-D non-stationary thermal conductivity equation with constant coefficients kls, ~ci,, Rc for insulation layer, separating adjacent turns [5]. Insulation layer is assumed the plane (6<
T (~) -Tb

Tc-Tb

vw ( -v-, 2

-

~ i1 - - I exp[2iuy(~/l)] -® [1+~f 2 ] y - i

dy 2' c = +0, (y+ic)

(3)

where ~=x-v,T, f = s i n ( u y ) / u y , u=(v,/vw)(~/2). The unknown v e l o c i t y v, i s the root of transcendental equation:

( v,

)2

-

vw

1 ~ ( l + n f 2) 2 ~ o (l+wf2)2y ~ 1

I n t e g r a l (3) parameters: zone

front

is

calculated

dy

+

1+~

explicitly

(4)

2

'

by means o f r e s i d u e s . 6, ~ a r e t h e problem where lp=[kw(Tc-Tb)/pj 51/~ i s t h e normal

~=[pj212/(kw(Tc-Tb))]ll2=l/lp, length

in

wire (usually, ~>>1); ~=(kt/kw)'{l~/(D'dt)}, is the effective transverse (turn-to-turn) conductivity. General (for any ~-values) equation (I), for arbitrary times Ts>>TI), beginnin E from the moment of normal zone origination, is solved by the computer code [5] (j=const, xa0). The numerical solution coincides analytical one for the second steady regime of nzp (c<
kt=D/(dt/kw+~/kis)=D.kls/~

where thermal (0STSTs, means of with the

Normal zone propaRation in solenoid layer Initially, local normal zone, having originated in finite region of one turn due to impulse heat disturbance (xgTI-I>..>TI>To>0; Fig. l, si=xl/l). Local normal zones longitudinal propagation is characterized by multiciphered function xc=xc(T), giving the positions of local normal zones fronts in dependence on time, xc(T)=0 if T0, xtSxc
Cryogenics 1992 Vol 32 ICEC Supplement

403

ICEC 14 Proceedings on elliptic line). The regime of nzp with non-steady velocity vt
-

Vz -

¢"

vw

= ~*{(~cw/~ct)'(kt/kw~/2"(jt/j)},

(5)

¢$I as (Vls(1)/vw)m(vts(i)/vz), ~ml in work [2]. Just after the amalgamation of all local normal zones and the formation of continuous (along wire) normal region the first steady regime of nzp is changed for the second transient regime in which longitudinal velocity vl begins to exhibit damping oscillations, and transverse velocity increases slowly (Fig.6,7; with the purpose to decrease computations time, computer simulation has been carried out for 8=60). Temperature profile along wire is complicated enough (Fig.8). As the oscillations disappear, the slow transfer to the second steady regime of nzp is realized. For small ~-values (~<vw). F o r large ~-values (~>I) the first steady regime of nzp and the both transient ones may be practically absent; longitudinal steady velocity is relatively low (FIE.6, ~=5) due to the transfer of considerable part of Joule heat into inter-turn insulation with high specific heat. REFERENCES Stekly, Z.J.J., Behaviour of superconducting coil subjected to steady local heating within the windings Jornal of Applied Physics (1966) 37 324-332 2

Wilson, M.N., Superconducting magnets Clarendon Press Oxford (1983)

3

Joshi, C.H. and Iwasa, Y., Prediction of current decay and terminal voltages in adiabatic superconducting magnets CryoKenics (1989) 29 157-167

4

Oshima, M., Thome, R.J., Mann, W.R. and Pillsbury, R.D., PQUENCH - a 3-D quench propagation code using a logical coordinate system IEEE Transactions on Magnetics (1982) 27 2096-2099

5

Gavrilin, A.V., Computer code for simulation of thermal processes durlnE quench in superconducting magnets windings This Conference Rusinov, A.I. and Krivolutskaya, N.V., Calculation of normal zone propagation velocity and kinetics of quench in solenoid with thin winding ProceedinKs of LPI Moscow NAUKA (1984) 150 70-91 (in Russian) 110{ 10011

~ II

/t II

~

~-, g_oll I I

II II

II I1

,, [SU-II I I I

~o

Fo=0.31 Fo=0.53

A []

A

-

dashed

local

normal

zones|

8 =300

'A

~ 40

~

w=15o0

s1~

~

151 1,-J 13J

12J

6'"i'"f"Y"4"'~'"~'"Y"~'"{"i'6'l'1"1'2'i',3'

s Figure I Longitudinal (along wire) temperature profile in solenoid layer Cryogenics

1992

\

11-I Origination of

V o l 3 2 ICEC S u p p l e m e n t

~:::z=ZZ[2[1

k

9~ }:,'--,

~oB~ ?,,"

~

~-----:~-.. "" ....

~1 --

7

":'-,

"-..

54 I ; k

4~ ~ "

Cth

3-~ , ',

",3rd~

2~

404

~8=300W=1500°':0.01"J Amalgamation of ~:q adjacent local normal zones

sl~" ',,

x

30

] J

Temperatureprofiles I of '"

7° eo

curve

solid curve

t u ~

~ "~.d',~- -

~ "-.------~

",A_ N"

~,'~

.

"-

c--

. . . . . .

I I [

]

_~ze~ tur--~" " , --I ~.b ' ' 'O.'l ' ' '0.'2 ' ' '0.'3 ' ' '0.'¢ ' ' '0.'5 ' ' '0.~

Fo Figure 2 Longitudinal coordinate of ns-boundary (in wire) vs time

ICEC 14 Proceedings

~=300

4

n=I5°°

3

~=0.01

0.9

>~

V[,(,~=0.896

Co- -~u-rA" -

Vts(1 ) = 0,782

I

oc~ 08

0.3 0.2

1st

I I

>~.,

2

1

Approximot[on

0.8 0.7 0.6 0.5

3 4t

2

~=300 n = 1 5 0 0 ~=0.01

1.0

5

oo

~.___ ~)

0.1 |

(~0

,

,

, ) ,

C1

,

,

,

,

,

,

02

,

,

,

,

,

03

,

,

,

04

FO

,

,

D5

,

,

,

o.oo

0

,6. h e l i c o a x i s _Zc:_ _-:=-1. . . . . . . . . .

/ ' " " t .... -. . . . . • ;., ~

/ellipse

/

-o.~ '-o'., -0.3 '-oi= '-o.,

12 i

~.?--..~Fo=O.5g

o.o

>_10! 8i

-"~__.',

FO=0.31 .-.. "

~. -' v~ o.~ 'a.'=

\

Figure 5 Evolution of normal region f o r m on c y l i n d r i c a l surface of solenoid layer in time

°~ ~N

~ ....

'

3 ....

ON

4 ....

'o.'¢'

'

'o,'5

'

'

'o.6

lllllllrlI j Ill

II/llill/I

~ ~ °

The 2nd trans;ent ragime of nzp

IIIllllll IIIA ,,

~ re-~ ~-'-

ooo,,oo,,o,u,,oo

~8C

i n

....

'o,3'

,=6o ~=,0 ~=0.0,

IflWJHilf

~=60 n=60

'.40 '.2O :00 8O

'i

'o.'2'

Figure 6 Normal zone velocity along helical wire v s time

~=60 W=60 c=O. 01

''

'

~1 ~i~ ...~=~o~=~o~=~ °~-:=-i:: ~;:: ~-::" ~:-:-::~:-:-:i-::-:: Fo

o.~ ' G.~' -o.s

Local dimensionless coordinate along turn, s - E - l / 2

1.0 0.9 0.8 0,7 0.8 0.5 0,4 0,3 O,2 0.1 O,O

'oh'

Figure 4 Transverse (turn-to-turn) velocity o f n o r m a l zone vs t i m e

~]=300 ~=1500 ~ = 0 . 0 1

12'

'

Fo

Figure 3 Longitudinal velocities of local normal zones at turns v s time

15: 14:

'

g ....

Fo

Figure 7 Transverse velocity of normal zone vs time

~

~

'

~

zo

2~ oo ~,% 80 60 40. 20. 0

•,._

.... Fo=2

"

.~=0.01-

~

Fo=5

analytical solution

S

Figure 8 Longitudinal (along wire) temperature profile in solenoid layer

Cryogenics 1992 Vol 32 ICEC Supplement

405