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Transient cooling of an internally cooled superconducting magnet
Transient cooling of an internally cooled superconducting magnet R.J. van der Linden and C.J. Hoogendoorn Department of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
Received 6 June 1988 Large forced convection-cooled superconducting magnets can locally become normal conducting due to a temperature rise initiated by an energy disturbance. If this disturbance is too large the conductor will quench. For a small disturbance the coolant, a supercritical helium flow, will be able to remove the dissipated energy and the conductor will return to its superconducting state. In this paper the critical disturbance from which recovery is still possible is calculated with the simulation program STABIL, for the Sultan conductor as developed by the Netherlands Energy Research Foundation ECN.
Keywords: superconducting magnets; cooling systems; thermal properties
tq
Nomenclature A a B Cp D
cross section thermal diffusivity magnetic field specific heat tube diameter Ot timestep Dz mesh size G heat generation per unit length h heat transfer coefficient H enthalpy flow solid I transport current J current density Lm mixing length Lq extension of disturbance P pressure qhf momentary heat flux qdiss momentary dissipation Q disturbance Qcr critical disturbance (stability margin) R tube radius r radial coordinate S source term in heat equation(s) T temperature Tcrml. lower current sharing temperature (Equation 4) Tcrm3x upper current sharing temperature t time t, take-over time
0011-2275/89/030179-09 $03.00 @ 1989 Butterworth& Co (Publishers)Ltd
[m 2] [m 2 s-1] IT] [J kg- l K - 1] Ira] [s] [m] [W m- 1] [W m-1 K - 1 ] [J kg- 1] [J m- 3] [A] [A m-2] [m] [m] [N m- 2] [W] [W] [J] [J] [m] [m] [W m - 3] [K]
U V X, Z
Y y+ c
duration of disturbance axial velocity radial velocity axial coordinate distance from wall ( R - r) y(pz.)~/tl penetration parameter
[(J m-ZK -2) s-1]
(~.p%) Cq
2 P P©l q qT Tw Z
Is] [m s-1] [m s - i ] [m] [m]
error in critical disturbance due to the trial and error procedure thermal conductivity density specific electrical resistivity viscosity turbulent viscosity wall shear stress heat transfer parameter
((,/cp)°'*). °'6)
[ J m - 1 K - 1] [kg m- 3]
[f2m] [Ns m - 2] [Ns m-z] [N m-z]
[W m - 1 K - 1]
pO
Re
Reynolds number - r/
Nu
Nusselt n u m b e r 2
Pr
Prandtl number a
hD v
Indices [K] [K] I-s] Is]
r
cr cu sc
He
reference critical copper superconductor helium
Cryogenics 1989 Vol 29 March
179
Coofing of an internally cooled magnet." R.J. van der Linden and C.J. Hoogendoorn
In stable operation of a superconducting magnet no heat dissipation occurs. Under these circumstances no cooling other than to compensate for heat leaks from the surroundings is required. However, due to magnetic or mechanical instabilities locally small heat releases can occur. At these places the conductor temperature can rise above the critical temperature. This means that locally the conductor loses its superconducting property and becomes normal conducting. The heat, dissipated in this area, causes the temperature to rise even more and the normal conducting zone expands. To stop this quenching process cooling of the magnet is required. For large magnets internal cooling of the windings with a forced flow of supercritical or superfluid helium is often applied. Extensive stability studies based on static criteria are reported x-5. The stability margins found in these studies are too conservative. The thermal stability of large internally cooled superconductors depends strongly on dynamic effects of the heat transfer to the coolant. The critical disturbance from which a conductor, internally cooled with a force flow of supercritical helium, can still recover, can be calculated with a simulation program called STABIL. This program, originally developed by Cornelissen 6- 8 gives a fully transient description of the processes in the conductor and the coolant flow. In a preceding study 9 the heat transfer results calculated with this code were validated by comparing them with experimental results found at the Netherlands Energy Research Foundation (ECN). The results were found to be in good agreement. Cornelissen 6 used this program to study the stability of the Sultan conductor, developed at ECN. In this paper this previous work will be extended. The influence of numerical and physical parameters on the critical disturbance will be studied.
The simulation model The processes in a conductor cable, internally cooled with a turbulent flow of supercritical helium, are simulated with the aid of a numerical program called STABIL 7. In this program the transient, variable property phenomena in both the solid (the copper/superconductor matrix) and the fluid (the helium flow) have been described. For the solid a one-dimensional model has been used. For the temperature dependent properties 2, p and cp of the conductor the mass weighted averages over all the materials in the matrix have been taken. The helium channel inside the copper tube has been taken circular. The fluid equations can then be described by a twodimensional, axisymmetric model. These two models are coupled through the boundary conditions at the wall. At this place there must be temperature continuity and the heat flux which leaves the solid must be equal to the heat flux which enters the fluid.
where a = 2/pcp is the thermal diffusivity. These properties depend on temperature: (2)
pcr, = c 1T + c 2 T 3
2 = c3 T
(c~, c 2, c3 constants)
The source term S contains the local heat dissipation, the applied disturbance and the heat flux to the helium flow. The local heat dissipation is calculated from a so-called current sharing model 7 (Figure 1). For temperatures above the upper current sharing temperature Tcr . . . . which depends on the magnetic field B, the conductor loses its superconducting property completely. The whole operation current I flows through the copper backup which has a lower resistance than the now normal conducting 'superconductor'. Below Tcrm~, the superconductor is able to carry a current density J~r(B, T), For a fixed magnetic field J~, is linear related to the temperature difference (Tcrma,T): Jer(B, T ) = J~,(B, TR)
Tcrma x -
T
Tcrma x -
TR
(3)
where TR is a reference temperature. The lower current sharing temperature is defined as the temperature at which the operating current density J = I/Asc is equal to the maximum current density J¢r" The value of Tcrmi, follows from (3):
Tcrmi.(B, J)= Tcr,...(B)- (Tcrmax(B)- T.)*J/Jc,(B, TR) (4) Below this temperature Tcrmi. the superconductor is able to carry the whole current. The conductor is then in its fully superconducting state. In the intermediate region between Tcrmi. and Tcrma, the superconductor can only carry a part of the total operating current, the surplus will flow through the copper backup. The heat generation G in the copper/superconductor matrix is: (W m - ')
G = VI = p,,.culcul/Acu
T
(5)
-FF[
1-r J
<3
<:3 I I I
J I
d (IB,T)
I-
cr
I
The governing equations
The one-dimensional energy equation in terms of the enthalpy for the copper/superconductor matrix is: -
180
a
+s
Cryogenics 1989 Vol 29 March
(1)
T
. (B,J)
crmln
T
T
(B)
crmax
Figure 1 Current sharing model (fixed magnetic ~ield). I, Superconducting region: /cu = 0 ; II, current sharing region: /cu =1-1~; III, normal conducting region: /cu = /
Coofing of an internally cooled magnet."R.J. van der Linden and C.J. Hoogendoom where V is the voltage drop per unit length. This voltage is assumed to prevail in both copper and superconductor. I , , the current through the copper, follows from the above described current sharing model:
~ ~.]111 r i l l I I I l l l l l l l
-
E ~lllltl
T < Tcrmin
Tcrm~, < T
I, = I
(6)
The fluid flow can be described by the Navier-Stokes equations and the continuity equation 1°. Because the fluid properties depend on temperature these equations are coupled to the energy equation, which is used in terms of the enthalpy H because of some computational advantages. In cylindrical coordinates these equations read:
Navier-Stokes equations at-
\
az
-t-r
Or
a(aU)
+~ apV radial
+ -r a t
rti -~r
apU Oz
~OpUV
(7)
1 aprVV'~q_ d_a_(~arv'~ + r
Or ]
a(av)
aP
+at ti aTz
az
Or
aE-z]
1a r O~ (prV)
(8)
Illlllllll]
4,.._
v
/
y
-/
o.)+o,+
r T;
at
-X-
I
Figure 2 Staggered grid. O, Grid point; I~, u grid point; A , v grid point; . . . . . , control volume; , u control volume; ~ , t , , , v control volume; area indicated with *, grid point ij
accuracy was found to be possible by assuming a hyperbolic instead of a linear temperature profile between the wall and the first gridpoint 7. At any time at least one point of the radial grid was located in the laminar sublayer (y + = 5).
The effects of turbulence on the flow are accounted for by adding a turbulent viscosity tit to the laminar viscosity ti in the Reynolds averaged Navier-Stokes equations. Comparison of three eddy viscosity models showed that in our case of a pipe flow the simple mixing length model was to be preferred to the more complex k and k - ~ models 12'13. In the mixing length model as developed by Prandtl, the turbulent viscosity is defined as the product of a length scale Lm, the mixing length, and a velocity scale. The velocity scale is taken to be the local velocity gradient multiplied by the mixing length, so
tit = pL* Lm ~y
aPHat=-\-~--z{OpUHO rlaprVH~q_l H-~r J) -r-fir HO (2-~pr--~r
o
q
OU
Energy equation
+~
~
Turbulence model
aP Oz
ti ~
at = - \--~--z
]
Continuity equation Op at
. . . . . .
-
J
I~ = I - J,(B, T)*A~
--
~
,
i~-
I~, = 0
Tcrmi. < T < Tcrm~
axial
v
I
--
o,
(9)
o,
v ~+ v ~+ s
Numerical method To solve Equations (1) and (7) to (9) a fully implicit finite difference scheme has been employed. For the spatial derivatives a central difference scheme has been used in the solid. In the fluid a hybrid scheme on a staggered grid (Figure 2), based on a control volume approach, has been employed for the Navier-Stokes and energy equations. The continuity equation has been used to calculate the pressure field by employing the Simpler algorithm of Patankarlk In order to deal with a compressible flow some improvements of this algorithm suggested by Cornelissen 7 have been used. The heat flux from the cop~er matrix to the helium flow has been calculated from the helium temperature gradient near the wall. To be able to calculate the flow and the temperature gradient in the wall region the radial grid was chosen to be dense near the wall and rather coarse in the bulk of the fluid. A reduction of the number of radial gridpoints from 20 to 13 with conservation of
(10)
For a pipe flow the expression for the mixing length is given by Nikuradse 14
L,,,/R = 0 . 1 4 - 0 . 0 8 (1 - y/R) z - 0 . 0 6 (1 - y / R ) 4
(11)
Damping effects on turbulence in the vicinity of a wall have been incorporated by applying the van Driest correction on the mixing length 15 L~ = L * (1 - e x p (-y+/A+)), A + = 26
y=R-r (Zwp) 1/2
(12)
is the distance from the wall and y + =
y/q. Turbulent heat transfer has been accounted
for by taking the turbulent Prandtl number in the Reynolds averaged energy equation equal to unity.
Fluid properties The fluid properties 2, p, cp and ti depend on temperature and pressure. Unlike the properties of the solid, no simple expressions exist for the fluid properties. Especially near the pseudo-critical point ( P = 2.2 atm, T = 5 K), the properties vary strongly. The values of the properties have been calculated from correlations found in litera-
Cryogenics 1989 Vol 29 March
181
Cooling of an internally cooled magent: R.J. van der Linden and C.J. Hoogendoorn ture 16"~s, and have been stored in a look-up table as a function of pressure and enthalpy. Momentary values of the properties at a gridpoint can be found by a simple linear interpolation in this table.
Conductor specification The conductor on which the stability analysis has been performed is the Sultan innercoil, developed at ECN 19'2°. This coil is the Dutch contribution to the S U L T A N project, in which SIN (Switzerland) and C N E N (Italy) also participate. C N E N developed a 6 T outer coil and SIN delivered the instrumentation. The ECN innercoil was developed to enhance the magnetic field up to 8 T. At this moment a 12 T insert coil for further upgrading of the system is under construction. The main objective of this project is to realise a high field test facility in which samples of prototype conductors can be tested. In particular, the forced flow Toroidal Field conductors for the Next European Torus (NET) are planned to be developed and tested with the aid of the Sultan test facility2 i. In the present paper we consider the 8 T innercoil. This coil is a sixteen-strand, NbTi, Rutherford cable, soldered to a rectangular copper tube (Figure 3). The relevant specifications are listed in Table 1. In the simulations the rectangular tube has been replaced by a circular tube with the same wetted perimeter.
Stability calculation results First a reference situation will be defined. For this particular situation the critical disturbance is determined by trial and error. If for some disturbance Q the conductor returns after some time to its superconducting state, the conductor is called stable for this disturbance. The next
-~
CRUTHAERFOBRD L ~ (16strands muftlfllament NbTI ) "
{ eupetctltl¢llhelium) Figure 3 mm 2) Table 1
Hollow conductor for the ECN 8 T innercoil (8.4 x 8.4
Conductor specifications - Sultan 8T innercoil
Wetted perimeter Conductor cross section Helium cross section Fraction superconductor Transport current Critical current density (Jc at 8T) Critical temperature (Tcrma x at 8T) Metal properties Superconductor
0.0191 m 0.463 x 10 - 4 m 2 0.290 x 10 -4 m 2 0.13 1.86 x 103 A 0,678 x 109 A m -2 5.5 K Copper
Composite
p (kg m -3) p% ( j m - 3 K ) 2 (W(mK)-I)
6700 8900 8607 1507 T + 14.3 T 3 96 T + 6.6 T3 284 T + 7.7 T 3 0.02T 45T 39T
Pel (~m)
0.3 x 10 -6
182
0.5 x 10 - 9
Cryogenics 1989 Vol 29 March
-
calculation is done when the disturbance has been raised. If the calculated normal conducting zone (T > Tcrmi,) then expands with time the conductor quenches. The conductor is then unstable for this disturbance and a lower disturbance must be assumed to find the critical value. The procedure is stopped when AQ = Qunstable - Qstable is less than a predefined value. The critical energy is then defined as Qor = ½ (Qstable + Qunstable) with an error eQ = (½AQ/Qc,) x 100%. A variation of some numerical parameters starting from the reference situation has been performed to find the inaccuracy in the value of these parameters. This inaccuracy should at least be less than the above defined e r r o r ~Q.
To find the influence of system pressure and Reynolds n u m b e r on the critical disturbance, the trial and error procedure described above has been repeated for different values of these flow parameters. The influence of the extension and the duration of the disturbance are also studied in this way. Reference situation As a reference situation a 200 mm section of the Sultan innercoil has been chosen, operating at a pressure of 12.5 atm and a temperature of 4.2 K. The mass flow has been set at 2.5 g s-1 which equals a mean velocity of 0.578 m s-1 (Re = 10s). The tube was divided into 40 cross sections resulting in an axial mesh size of 5 mm between two adjacent cross sections. Every cross section is divided into 13 radial gridpoints. The timestep between two time levels is taken as 1 ms. The applied disturbance, situated in the middle of the tube, is extended over two gridpoints and lasts one timestep. The relevant parameters in this reference situation are given in Table 2. In Figure 4 the length of the normal conducting zone, T > Tcrmi,, is shown as it develops in time after a disturbance Q has been applied between 0 and I ms. In case the applied disturbance is 17 mJ or below, this normal length becomes zero after some time. The total conductor is then back in its superconducting state and is thus called stable for this disturbance. Unstable situations, in which the normal zone expands with time, are found when the applied disturbance was taken 18 mJ or above. The critical disturbance for this reference situation, therefore, is Qc, = 17.5 m J, with an inaccuracy of 0.5 eO=l~.5 x 1 0 0 % , ~ 3 % . In this case not only were the 17 and 18 mJ cases studied, but a calculation was also performed at the critical disturbance of 17.5 mJ derived from these two cases. The result as plotted in Figure 4 (dotted line) shows that the normal length is nearly constant for a long time. This means that this situation is indeed quite near the critical situation in which the normal zone does not change because dissipation and cooling are in equilibrium.
Numerical parameter variation The sensitivity of the solution of the reference situation to numerical parameters, as there are the tube length,
Cooling of an internally cooled magent." R.J. van der Linden and C.J. Hoogendoorn Table 2
Reference situation parameters
Parameter
Symbol
Unit
Value
Pressure Temperature Mass flow
P T m D Re
M Pa K g s 1 m s 1 -
1.25 4.2 2.5 0.587 105
Tube l e n g t h Axial mesh size N u m b e r of axial gridpoints N u m b e r of radial gridpoints
L Dz Nz Nr
mm mm -
200 5 40 13
L e n g t h of disturbance
Duration of disturbance
Lq tq
mm ms
10 1
Time step
Dt
ms
1
- mean velocity - Reynolds number
[0.
C-
9.
Mesh size. In Figure 5 the reference situation with mesh size Dz = 5 ram, is compared with the results for mesh sizes of 2 and 10 mm. Decreasing the mesh size from 5 to 2 mm gave only minor changes in the calculated results. The difference was well within the 3 % limits found in the stability analysis for the reference situation indicated by the results for the reference situation with disturbances of 17 and 18 mJ. Increasing the mesh size to 10 mm gave larger differences with the reference situation, even for short times. Thus, the 10 mm mesh size was judged to be too coarse, and the 5 mm mesh size was found to be accurate enough.
7. G.
vo
5
E L.
5. 4. 3. 2. 1. 0.
~.
~o.
i0. Time
Figure 4
gridpoints 2 0 + 21 time step 1
tube ends be seen. Because of this, the tube length was doubled when the influence of the mesh size or the extension of the disturbance was studied.
~o
.E E
Comment
~o.
40.
Timestep. Another important numerical parameter is the timestep Dt. Three different timesteps are compared in Figure 6, viz. Dr= 0.5 ms, D t = 1.0 ms (the reference
(ms) IO.
Reference situation - normalzone developmentin time.
x,Q=20W;F'I,Q=18W;A,Q=17W;O,Q=15W
the mesh size and the timestep, has been studied. The applied disturbance has been taken equal to 17.5 mJ, the critical disturbance. In this case small differences in the calculated heat flux lead to large changes in the solution for long times because of the critically stable character of the solution. The influence of the numerical parameters is therefore more pronounced than it would be in a more stable situation.
Tube length. First of all the influence of the limited tube length has been examined. Keeping the mesh size fixed the tube length is doubled by doubling the number of axial mesh nodes. This leads to identical results. In the case of mesh size of 2 mm instead of 5 mm, doubling of the tube length has been found not to change the results. From this it is concluded that the tube length has been taken large enough for this problem and increasing it will not change the calculated critical disturbance significantly. Only if the mesh size or the extension of the disturbance is much increased may an influence of the
w
I
.I 0.
]e.
20.
Time Figure5
I
38.
40.
(ms)
Meshsizevariation. A , A : D z = 2 m m , x , B : D z = 5 m m ,
O , C: D z = 10 m m
Cryogenics 1 9 8 9 Vol 29 March
183
Cooling of an internally cooled magent." R.J. van der Linden and C.J. Hoogendoorn 35..
10. 9. C •~
B. ~ _
~"
•-
15.
5.
~?.
0.
1.
I
3.
1
~.
I
9.
b
12.
15.
Pressure {arm)
0.
0.
10.
20. Time
30.
40.
(ms)
Figure 6 Time step variation. I-I, A : O t = 2 . 0 ms; + , C : D t = 0 . 5 ms
ms; O , B: D t =
or
Flow parameter variation As described for the reference situation, the critical disturbance has been calculated in case of different flow Reynolds numbers and system pressures. To explain the results gathered in Figure 7, the two heat transfer mechanisms involved in the cooling process, penetration and turbulence, will be described first. For short times heat has to penetrate through the laminar boundary layer by conduction. Turbulence influence will start after some time once a considerable amount of heat has penetrated the transition region. The penetration heat transfer coefficient for a fixed temperature step at the wall and constant fluid properties is given byT: (13)
where the parameter e = 2pcp. The stationary turbulent heat transfer coefficient can be described by the McAdams correlation 22 N u = c R e °'8 P r ° 4 ; c ,~ 0.023
184
Cryogenics 1989 Vol 29 March
7 Critical disturbance dependence on pressure and Reynolds number. I:lq, R e = 1.5 x 1 05; C), Re = 1.3 x 1 -05; x , R e = 1.0 x 105; [Z, R e = 0.5 x 105; /~, R e = 0.0 x 105
1.0
situation) and D t = 2 . 0 ms. In all three cases the disturbance itself has been applied in the same timespan of 2 ms. For times below 15 ms curves B and C (Figure 6) are almost identical. Curve A, for the largest timestep, is initially somewhat lower than the other two. It is important that the transient effects in the first milliseconds are described accurately, because small errors in these moments will accumulate in time. A 2 ms timestep was therefore found too large. For large times curves B and C deviate. With a 0.5 ms timestep a stable solution was found when the 1.0 ms timestep gave an unstable solution. However, as stated before these differences occur due to the critically stable character of the solution at 17.5 mJ. The effect on the critical disturbance is again well within the 3% limits given by the 17 and 18 mJ disturbance cases calculated with a timestep of 1 ms.
h(t) = (2pcp/nt) '/2 = n - 1/2(e/t)l/2
Figure
h = (c/D) Re°'a Z
(14)
where the parameter ;~ = 2°6(r/cp)°4 The transition from penetration to turbulent heat transfer is said to take place after a take-over time t t. Cornelissen 7 estimated this time by calculating the time at which the heat transfer coefficients from Equations (13) and (14) are equal. He found: D2 t t = DZ/(anc2Re l"6Pr°'a~ = - - R - 1.6 e ' nc" -Z~
(15)
R e y n o l d s n u m b e r e f f e c t s . The fact that, for a constant
system pressure, the critical energy increases with the Reynolds number is easily explained. At high Reynolds numbers the heat accumulated in the conductor at short times will later be cooled away more efficiently because the stationary heat transfer coefficient is larger than for small Reynolds numbers (Equation 14). Further, because the take-over time t t decreases with Reynolds number (Equation 15), the influence of turbulence starts earlier. The still rather high value of Qcr at zero Reynolds number, about 85% of the value at R e = 0.5 x 105, is not unrealistic. It must be noted that already an amount of about 5 mJ is needed to raise the conductor temperature to the lower current sharing temperature Tcrml.. Disturbances below 5 mJ will cause no dissipation and therefore always lead to a stable solution. The surplus of the critical disturbance is due to the influence of heat transfer to the coolant and conduction in the solid. The heat transfer is enhanced by an induced flow due to the thermal expansion of the helium. This enhanced stability by heat-induced flow is already observed by other authors23'24et al. The possible multivalued behaviour of a conductor due to this phenomenon as described by Dressner ~a has not been found in our calculations. Increasing the Reynolds number to 1.5 x 105 increases the critical disturbance considerably. In the reference situation P = 12.5 atm, the critical disturbance of 17.5 mJ in
Cooling of an internally cooled magent. R.J. van der Linden ant C.J. Hoogendoorn £psilon
the Re = 105 case is almost doubled in the Re = 1.5 x l0 s case (Q, = 33 m J). By taking large enough mass flows it is thus possible to increase the conductor stability considerably.
System pressure. If at constant Reynolds number the system pressure is varied between 3 and 12.5 atm, a minimum for the critical disturbance is found at about 6 atm (see Figure 7). To explain this behaviour the heat balance is considered. If the critical disturbance Q,, is applied at time t = 0, the total momentary heat flux qH=(t) to the coolant is in balance after some time to with the total momentary dissipation qdi,, (t) in the conductor. The energy stored in the conductor is then constant for t > to. This amount of energy AEo depends on the turbulent heat transfer. Since the Reynolds number is constant the main parameter determining AEo is X. For times smaller than to the heat flux will be larger than the heat dissipated in the conductor. The conductor energy will decrease in this period. The total energy Qo cooled away for times up to to is equal to Qo =
(16)
(qhf(t) -- qdi,,(t)) dt
This energy depends strongly on the penetration heat transfer described by the parameter e. The critical disturbance can be found by adding this energy decrease for t < t o, Qo, to the conductor energy AEo in the equilibrium state: (17)
Q=,=AEo + go
In Table 3 the values of Q., AEo and Qo are gathered as computed for the reference case of Re = 105, Q = Q~,, at different pressures. From Table 3 it can be seen that both the conductor energy AEo, depending on Z, and the energy decrease Qo at small times, depending on e, have a minimum at a pressure of about 6 atm. In Figure 8 the parameters e and Z, depending on the fluid properties only, have been plotted as a function of temperature for different values of the system pressure. Both e and X are low over a large temperature range at a pressure of 6 atm. This accounts for the low critical disturbance at this pressure. The large peak values of e and Z at a pressure of 3 atm have a large positive influence on the cooling. This explains the enhanced heat transfer at low pressures. Disturbance parameter variation
In practice little is known about the disturbances which can occur in a superconductor. In our analysis the disturbance has been applied in 1 ms, extended over 10 mm. This resulted in a critical disturbance of 17.5 mJ (reference situation). However, varying the extension or Table
3
Heat balance for critical disturbance
P(atm)
Oct (m J)
AE o (m J)
Oo(mJ )
3 4 5 6 9 12.5
20.5 19 17 15 15.5 17.5
9 9 8 7.5 7.5 8
11.5 10 9 7.5 8 9.5
Ghi
[S2/m4K2s)
(N/mK)
~E-~ F
/ •
tIt
/..-.
i
,/
/ ÷..
£
25
+\ ~
I
I
5~
7~
.~?"-~;"--
r'~
....
•
L~-~
• "'.. ÷\ %
~£m i
~
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~
75
T~rlture (K)
Telperature (K]
F i g u r e 8 Heat transfer parameters. , P = 3 . 0 atm; - - ~ P = 6 . 0 atm; + - +, P = 9 . 0 atm; . . . . P = 12.5 atm
duration of the disturbance will change this value. The influence of these parameters is discussed here.
Disturbance extension. If the extension of the disturbance is varied the initial conductor temperature changes. This will influence both the total heat flux to the coolant, qhf (At), and the total dissipation in the conductor, qd~,,(At), just after the disturbance has been applied. This will thus influence the stability of the conductor for this disturbance. To examine the influence of the disturbance extension Lq qualitatively, a parameter Rq is defined: qdiss(At) with At ~ 0 Rq = qhf(At)
(18)
If this parameter increases, the dissipation grows compared to the cooling just after the disturbance has been applied, and the conductor will be more unstable. This means that the critical disturbance decreases for increasing Rq and vice versa. Consider the initial copper temperature to be raised uniformly to a temperature T over the total disturbance length Lq. Since both qai,s and qhf are then proportional to the disturbed length Lq the only variable in Equation (16) is this initial temperature T. The heat flux to the coolant is proportional to the temperature difference (T-THe). The expression for the dissipation in the conductor depends on the value of the conductor temperature T. From Equations (5) and (6), it can be found that in the current sharing region, Tcrmi, < T < Tcrmax the dissipation is proportional to the temperature difference (T-Tcrmi,). For higher temperatures the dissipation is constant, for lower temperatures the dissipation is zero. Substituting this information in Equation (16) gives: c1
T - TH------~
Rq =
c2
T-
Tcrmin
T - THe
Tcrmax < T T.o < Tcr,,~. < T < Tcr,,ax (19)
0
THe < T < Tcrmi n
Cryogenics 1989 Vol 29 March
185
Cooling of an internally cooled magent: R.J. van der Linden and C.J. Hoogendoorn Table 4
Disturbance extension variation results
Extension Lq (ram)
Critical disturbance Qcr (m J)
5 10 20 50 75 85 90 95
17.5 17.5 17 16.5 16.5 17 1 7.5 18
100
18.5
For decreasing conductor temperature the parameter Rq will increase as long as T > Tcrmax. If T drops below Tcr . . . . Rq starts to decrease with T until it becomes zero at T = Tcrmin. Since the initial conductor temperature decreases for increasing extension of the disturbance this means that the critical disturbance will decrease for increasing values of Lq until the extension becomes too large and T drops below Tcrmax. For larger extensions Q , will increase with Lq; the value of Lq for which T = Tcrma~ is calculated to be about 100 mm. In T a b l e 4 the critical disturbances calculated for different values of the disturbance extension are gathered. The critical disturbance decreases for extensions up to 75 m m and grows for larger extensions. This result is in good agreement with the simplified description using the Rq parameter.
Due to the variation of the fluid properties the critical disturbance depends on the system pressure. A minimal stability is found at a pressure of about 6 atm. The largest disturbances are allowed at a pressure of 3 atm. By increasing the flow Reynolds number, turbulence influence becomes more important and the critical disturbance is increased. Still, a relative large critical disturbance has been found in the case of a stagnant flow. One reason for this is the enhanced heat transfer due to a heat-induced flow. In this study no sign of a possible multivalued behaviour of the conductor was found. This subject will be examined in further work. The influence of the duration and the extension of the disturbance was found to be small for small values of these parameters. However, if the disturbance is spread over a large extension or a large time interval the stability of the conductor increases.
Acknowledgement These investigations in the program of the Foundation for Fundamental Research on Matter (FOM) have been supported by the Netherlands Technology Foundation (STW).
References 1 Stekly, L.J.J. and Zar. J . L Stable superconducting coils IEEE Trans Nucl Sci (1965) 12 367-372 2 Maddock, B.J., James, G.B. and Norris, W.T. Superconductive composites, heat transfer and steady state stabilization, Cryogenics
(1969) 9 261-273
Disturbance duration. When the duration tq of the disturbance was increased, keeping the total energy input constant, no significant change in the critical disturbance was found in the interval 0.5 ms < tq < 4 ms. For larger disturbance durations an increase of the critical disturbance was observed. The critical disturbance distributed over 10 ms was found to be about 19 mJ, which is 1.5 mJ or about 13 % higher than the critical disturbance in the reference situation (tq = 1 ms). The reason for this enhanced stability when a disturbance is spread out over a larger period in time is that the coolant has more time to remove the applied disturbance before the dissipation in the conductor starts. When the duration of the disturbance is long enough the coolant will remove so much of the heat before the end of the disturbance that the conductor never reaches the lowercurrent sharing temperature. Since in this case no dissipation occurs the conductor is stable and the critical disturbance will be much higher than in the case of a short disturbance.
Conclusions The critical disturbance for the Sultan innercoil in the reference situation was found to be 17.5 mJ (Qcr/As¢+ ~u = 38 mJ cm-3). From the parameter study it is found that the used numerical timestep and mesh size are well chosen. The inaccuracy in the determination of the critical disturbance due to numerical parameters is less than the 3 % inaccuracy due to the used trial and error procedure.
186
Cryogenics 1989 Vol 29 March
3 Keilin,V.L, Klimenko, E.J., Kremly, M.G. and Samoilov, B.N.
4 5 6 7
Les champs magnetiques intenses, leurs production, leurs applications Coil Intern Grenoble CNRS (1967) 231-236 lwasa, Y. A critical current-margin design criterion for high performance magnet stability Cryogenics (1979) 19 705-714 Cornelissen,M.C.M. and Hoogendoorn, C.J. Thermal stability of superconducting magnets: static criteria Cryogenics (1984) 24 669-675 Cornelissen,M.C.M. and Hoogendoorn, C.J. Thermal stability of superconducting magnets: dynamic criteria Cryogenics (1985) 25 3-9 Cornelissen, M.C.M. The Thermal Stability of Superconductors Cooled by a Turbulent Flow of Supercritical Helium Thesis Delft University of Technology, Delft, Holland (1984)
8 Cornelissen, M.C.M., Hoogendoorn, C.J. and Franken, W.M.P.
Thermal stability of a superconductor cooled by a turbulent flow of supercritical helium Proc 18~hInt Conf Heat and Mass Transfer in Refrigeration and Cryogenics Dubrovnic (1986) 641--643 9 Bloem,W.B., Linden, R.J. van der, Hoogendoorn, C.J. Postma, H.
Experimental and numerical results on transient heat transfer to supercritical helium W,~rme- und StofffibertragunO (1988) 22, 315-323 10 Bird, B.R., Stewart, W.E. and Lightfoot, E.N. Transport Phenomena John Wiley & Sons, New York (1960) 11 Patanker, S.V. Numerical Heat Transfer and Fluid Flow Hemisphere Publishing Corporation, Washington (1980) i2 Cornelissen,M.C.M. and Hoogendoorn,C.J. A numerical study of heat transfer for a turbulent flow of supercritical helium Proc 3rd lnt Conf on Numerical Methods in Laminar and Turbulent Flow Seattle (1984) 831-841
13 Cornelissen, M.C.M. and Hoogendoorn, C.J. Forced convection heat transferto supercritical helium Appl Sci Res (1985)42 161-183 14 Schlichfing,H. Boundary Layer Theory McGraw Hill, New York (1960) 15 Driest,E.R. van On turbulent flownear a wall(1956)J Aeronautical Sci 1007-1011 16 Arp, V. New forms of state equations for helium Cryogenics (1974) 14 593-598 17 McCarty, R.D. Thermophysical properties of helium-4 from 4 to
Cooling of an internally cooled magent." R.J. van der Linden and C.J. Hoogendoom 3000 R with pressures to 15000 PSIA, NBS technical note 62 (1972) 18 MeCarty, R.D. Thermodynamic properties of helium-4 from 2-1500 K at pressures to 10s Pa J Phys Chem RefData (1973) 2 924-957 19 Franken, W.M.P. and Spoorenberg, C.d.G. The design of the SULTAN inner coil External Report ECN-I09 (1981) 20 Horvath,I., Vecsey, G., Weymuth, P., ZeUweger,J., Balsamo, E.P.,
Pasotti, G., Ricci, M.V., Sacchetti, N., Spadoni, M., Elen, J.D. and Franken, W.M.P. Status report on the forced flow high field test facility SULTAN IEEE Trans Ma#n (1983) 19 668
21
Plaum, J.M., Roeterdink, J.A., Eikelboom, J.A., Lijbrink, E. and
KIok, J. Development toroidal field conductor for NET, lOth Int Conf on Magnet Technology Boston (1987) 22 Schnurr, N.M. Numerical predictions of heat transfer to supercritical helium in turbulent flow through circular tubes J Heat Transfer Trans ASME (1977) 99 580-585 23 Dressner, L. Heating induced flow in cable in conduit conductors Cryogenics (1979) 19 653-658 24 Krafft, G. Recent progress on heat transfer to liquid helium, 8th Int Cryogenics Engineering Conf Geneva (1980) 754-768
Cryogenics 1989 Vol 29 March
187
Received 6 June 1988 Large forced convection-cooled superconducting magnets can locally become normal conducting due to a temperature rise initiated by an energy disturbance. If this disturbance is too large the conductor will quench. For a small disturbance the coolant, a supercritical helium flow, will be able to remove the dissipated energy and the conductor will return to its superconducting state. In this paper the critical disturbance from which recovery is still possible is calculated with the simulation program STABIL, for the Sultan conductor as developed by the Netherlands Energy Research Foundation ECN.
Keywords: superconducting magnets; cooling systems; thermal properties
tq
Nomenclature A a B Cp D
cross section thermal diffusivity magnetic field specific heat tube diameter Ot timestep Dz mesh size G heat generation per unit length h heat transfer coefficient H enthalpy flow solid I transport current J current density Lm mixing length Lq extension of disturbance P pressure qhf momentary heat flux qdiss momentary dissipation Q disturbance Qcr critical disturbance (stability margin) R tube radius r radial coordinate S source term in heat equation(s) T temperature Tcrml. lower current sharing temperature (Equation 4) Tcrm3x upper current sharing temperature t time t, take-over time
0011-2275/89/030179-09 $03.00 @ 1989 Butterworth& Co (Publishers)Ltd
[m 2] [m 2 s-1] IT] [J kg- l K - 1] Ira] [s] [m] [W m- 1] [W m-1 K - 1 ] [J kg- 1] [J m- 3] [A] [A m-2] [m] [m] [N m- 2] [W] [W] [J] [J] [m] [m] [W m - 3] [K]
U V X, Z
Y y+ c
duration of disturbance axial velocity radial velocity axial coordinate distance from wall ( R - r) y(pz.)~/tl penetration parameter
[(J m-ZK -2) s-1]
(~.p%) Cq
2 P P©l q qT Tw Z
Is] [m s-1] [m s - i ] [m] [m]
error in critical disturbance due to the trial and error procedure thermal conductivity density specific electrical resistivity viscosity turbulent viscosity wall shear stress heat transfer parameter
((,/cp)°'*). °'6)
[ J m - 1 K - 1] [kg m- 3]
[f2m] [Ns m - 2] [Ns m-z] [N m-z]
[W m - 1 K - 1]
p
Re
Reynolds number - r/
Nu
Nusselt n u m b e r 2
Pr
Prandtl number a
hD v
Indices [K] [K] I-s] Is]
r
cr cu sc
He
reference critical copper superconductor helium
Cryogenics 1989 Vol 29 March
179
Coofing of an internally cooled magnet." R.J. van der Linden and C.J. Hoogendoorn
In stable operation of a superconducting magnet no heat dissipation occurs. Under these circumstances no cooling other than to compensate for heat leaks from the surroundings is required. However, due to magnetic or mechanical instabilities locally small heat releases can occur. At these places the conductor temperature can rise above the critical temperature. This means that locally the conductor loses its superconducting property and becomes normal conducting. The heat, dissipated in this area, causes the temperature to rise even more and the normal conducting zone expands. To stop this quenching process cooling of the magnet is required. For large magnets internal cooling of the windings with a forced flow of supercritical or superfluid helium is often applied. Extensive stability studies based on static criteria are reported x-5. The stability margins found in these studies are too conservative. The thermal stability of large internally cooled superconductors depends strongly on dynamic effects of the heat transfer to the coolant. The critical disturbance from which a conductor, internally cooled with a force flow of supercritical helium, can still recover, can be calculated with a simulation program called STABIL. This program, originally developed by Cornelissen 6- 8 gives a fully transient description of the processes in the conductor and the coolant flow. In a preceding study 9 the heat transfer results calculated with this code were validated by comparing them with experimental results found at the Netherlands Energy Research Foundation (ECN). The results were found to be in good agreement. Cornelissen 6 used this program to study the stability of the Sultan conductor, developed at ECN. In this paper this previous work will be extended. The influence of numerical and physical parameters on the critical disturbance will be studied.
The simulation model The processes in a conductor cable, internally cooled with a turbulent flow of supercritical helium, are simulated with the aid of a numerical program called STABIL 7. In this program the transient, variable property phenomena in both the solid (the copper/superconductor matrix) and the fluid (the helium flow) have been described. For the solid a one-dimensional model has been used. For the temperature dependent properties 2, p and cp of the conductor the mass weighted averages over all the materials in the matrix have been taken. The helium channel inside the copper tube has been taken circular. The fluid equations can then be described by a twodimensional, axisymmetric model. These two models are coupled through the boundary conditions at the wall. At this place there must be temperature continuity and the heat flux which leaves the solid must be equal to the heat flux which enters the fluid.
where a = 2/pcp is the thermal diffusivity. These properties depend on temperature: (2)
pcr, = c 1T + c 2 T 3
2 = c3 T
(c~, c 2, c3 constants)
The source term S contains the local heat dissipation, the applied disturbance and the heat flux to the helium flow. The local heat dissipation is calculated from a so-called current sharing model 7 (Figure 1). For temperatures above the upper current sharing temperature Tcr . . . . which depends on the magnetic field B, the conductor loses its superconducting property completely. The whole operation current I flows through the copper backup which has a lower resistance than the now normal conducting 'superconductor'. Below Tcrm~, the superconductor is able to carry a current density J~r(B, T), For a fixed magnetic field J~, is linear related to the temperature difference (Tcrma,T): Jer(B, T ) = J~,(B, TR)
Tcrma x -
T
Tcrma x -
TR
(3)
where TR is a reference temperature. The lower current sharing temperature is defined as the temperature at which the operating current density J = I/Asc is equal to the maximum current density J¢r" The value of Tcrmi, follows from (3):
Tcrmi.(B, J)= Tcr,...(B)- (Tcrmax(B)- T.)*J/Jc,(B, TR) (4) Below this temperature Tcrmi. the superconductor is able to carry the whole current. The conductor is then in its fully superconducting state. In the intermediate region between Tcrmi. and Tcrma, the superconductor can only carry a part of the total operating current, the surplus will flow through the copper backup. The heat generation G in the copper/superconductor matrix is: (W m - ')
G = VI = p,,.culcul/Acu
T
(5)
-FF[
1-r J
<3
<:3 I I I
J I
d (IB,T)
I-
cr
I
The governing equations
The one-dimensional energy equation in terms of the enthalpy for the copper/superconductor matrix is: -
180
a
+s
Cryogenics 1989 Vol 29 March
(1)
T
. (B,J)
crmln
T
T
(B)
crmax
Figure 1 Current sharing model (fixed magnetic ~ield). I, Superconducting region: /cu = 0 ; II, current sharing region: /cu =1-1~; III, normal conducting region: /cu = /
Coofing of an internally cooled magnet."R.J. van der Linden and C.J. Hoogendoom where V is the voltage drop per unit length. This voltage is assumed to prevail in both copper and superconductor. I , , the current through the copper, follows from the above described current sharing model:
~ ~.]111 r i l l I I I l l l l l l l
-
E ~lllltl
T < Tcrmin
Tcrm~, < T
I, = I
(6)
The fluid flow can be described by the Navier-Stokes equations and the continuity equation 1°. Because the fluid properties depend on temperature these equations are coupled to the energy equation, which is used in terms of the enthalpy H because of some computational advantages. In cylindrical coordinates these equations read:
Navier-Stokes equations at-
\
az
-t-r
Or
a(aU)
+~ apV radial
+ -r a t
rti -~r
apU Oz
~OpUV
(7)
1 aprVV'~q_ d_a_(~arv'~ + r
Or ]
a(av)
aP
+at ti aTz
az
Or
aE-z]
1a r O~ (prV)
(8)
Illlllllll]
4,.._
v
/
y
-/
o.)+o,+
r T;
at
-X-
I
Figure 2 Staggered grid. O, Grid point; I~, u grid point; A , v grid point; . . . . . , control volume; , u control volume; ~ , t , , , v control volume; area indicated with *, grid point ij
accuracy was found to be possible by assuming a hyperbolic instead of a linear temperature profile between the wall and the first gridpoint 7. At any time at least one point of the radial grid was located in the laminar sublayer (y + = 5).
The effects of turbulence on the flow are accounted for by adding a turbulent viscosity tit to the laminar viscosity ti in the Reynolds averaged Navier-Stokes equations. Comparison of three eddy viscosity models showed that in our case of a pipe flow the simple mixing length model was to be preferred to the more complex k and k - ~ models 12'13. In the mixing length model as developed by Prandtl, the turbulent viscosity is defined as the product of a length scale Lm, the mixing length, and a velocity scale. The velocity scale is taken to be the local velocity gradient multiplied by the mixing length, so
tit = pL* Lm ~y
aPHat=-\-~--z{OpUHO rlaprVH~q_l H-~r J) -r-fir HO (2-~pr--~r
o
q
OU
Energy equation
+~
~
Turbulence model
aP Oz
ti ~
at = - \--~--z
]
Continuity equation Op at
. . . . . .
-
J
I~ = I - J,(B, T)*A~
--
~
,
i~-
I~, = 0
Tcrmi. < T < Tcrm~
axial
v
I
--
o,
(9)
o,
v ~+ v ~+ s
Numerical method To solve Equations (1) and (7) to (9) a fully implicit finite difference scheme has been employed. For the spatial derivatives a central difference scheme has been used in the solid. In the fluid a hybrid scheme on a staggered grid (Figure 2), based on a control volume approach, has been employed for the Navier-Stokes and energy equations. The continuity equation has been used to calculate the pressure field by employing the Simpler algorithm of Patankarlk In order to deal with a compressible flow some improvements of this algorithm suggested by Cornelissen 7 have been used. The heat flux from the cop~er matrix to the helium flow has been calculated from the helium temperature gradient near the wall. To be able to calculate the flow and the temperature gradient in the wall region the radial grid was chosen to be dense near the wall and rather coarse in the bulk of the fluid. A reduction of the number of radial gridpoints from 20 to 13 with conservation of
(10)
For a pipe flow the expression for the mixing length is given by Nikuradse 14
L,,,/R = 0 . 1 4 - 0 . 0 8 (1 - y/R) z - 0 . 0 6 (1 - y / R ) 4
(11)
Damping effects on turbulence in the vicinity of a wall have been incorporated by applying the van Driest correction on the mixing length 15 L~ = L * (1 - e x p (-y+/A+)), A + = 26
y=R-r (Zwp) 1/2
(12)
is the distance from the wall and y + =
y/q. Turbulent heat transfer has been accounted
for by taking the turbulent Prandtl number in the Reynolds averaged energy equation equal to unity.
Fluid properties The fluid properties 2, p, cp and ti depend on temperature and pressure. Unlike the properties of the solid, no simple expressions exist for the fluid properties. Especially near the pseudo-critical point ( P = 2.2 atm, T = 5 K), the properties vary strongly. The values of the properties have been calculated from correlations found in litera-
Cryogenics 1989 Vol 29 March
181
Cooling of an internally cooled magent: R.J. van der Linden and C.J. Hoogendoorn ture 16"~s, and have been stored in a look-up table as a function of pressure and enthalpy. Momentary values of the properties at a gridpoint can be found by a simple linear interpolation in this table.
Conductor specification The conductor on which the stability analysis has been performed is the Sultan innercoil, developed at ECN 19'2°. This coil is the Dutch contribution to the S U L T A N project, in which SIN (Switzerland) and C N E N (Italy) also participate. C N E N developed a 6 T outer coil and SIN delivered the instrumentation. The ECN innercoil was developed to enhance the magnetic field up to 8 T. At this moment a 12 T insert coil for further upgrading of the system is under construction. The main objective of this project is to realise a high field test facility in which samples of prototype conductors can be tested. In particular, the forced flow Toroidal Field conductors for the Next European Torus (NET) are planned to be developed and tested with the aid of the Sultan test facility2 i. In the present paper we consider the 8 T innercoil. This coil is a sixteen-strand, NbTi, Rutherford cable, soldered to a rectangular copper tube (Figure 3). The relevant specifications are listed in Table 1. In the simulations the rectangular tube has been replaced by a circular tube with the same wetted perimeter.
Stability calculation results First a reference situation will be defined. For this particular situation the critical disturbance is determined by trial and error. If for some disturbance Q the conductor returns after some time to its superconducting state, the conductor is called stable for this disturbance. The next
-~
CRUTHAERFOBRD L ~ (16strands muftlfllament NbTI ) "
{ eupetctltl¢llhelium) Figure 3 mm 2) Table 1
Hollow conductor for the ECN 8 T innercoil (8.4 x 8.4
Conductor specifications - Sultan 8T innercoil
Wetted perimeter Conductor cross section Helium cross section Fraction superconductor Transport current Critical current density (Jc at 8T) Critical temperature (Tcrma x at 8T) Metal properties Superconductor
0.0191 m 0.463 x 10 - 4 m 2 0.290 x 10 -4 m 2 0.13 1.86 x 103 A 0,678 x 109 A m -2 5.5 K Copper
Composite
p (kg m -3) p% ( j m - 3 K ) 2 (W(mK)-I)
6700 8900 8607 1507 T + 14.3 T 3 96 T + 6.6 T3 284 T + 7.7 T 3 0.02T 45T 39T
Pel (~m)
0.3 x 10 -6
182
0.5 x 10 - 9
Cryogenics 1989 Vol 29 March
-
calculation is done when the disturbance has been raised. If the calculated normal conducting zone (T > Tcrmi,) then expands with time the conductor quenches. The conductor is then unstable for this disturbance and a lower disturbance must be assumed to find the critical value. The procedure is stopped when AQ = Qunstable - Qstable is less than a predefined value. The critical energy is then defined as Qor = ½ (Qstable + Qunstable) with an error eQ = (½AQ/Qc,) x 100%. A variation of some numerical parameters starting from the reference situation has been performed to find the inaccuracy in the value of these parameters. This inaccuracy should at least be less than the above defined e r r o r ~Q.
To find the influence of system pressure and Reynolds n u m b e r on the critical disturbance, the trial and error procedure described above has been repeated for different values of these flow parameters. The influence of the extension and the duration of the disturbance are also studied in this way. Reference situation As a reference situation a 200 mm section of the Sultan innercoil has been chosen, operating at a pressure of 12.5 atm and a temperature of 4.2 K. The mass flow has been set at 2.5 g s-1 which equals a mean velocity of 0.578 m s-1 (Re = 10s). The tube was divided into 40 cross sections resulting in an axial mesh size of 5 mm between two adjacent cross sections. Every cross section is divided into 13 radial gridpoints. The timestep between two time levels is taken as 1 ms. The applied disturbance, situated in the middle of the tube, is extended over two gridpoints and lasts one timestep. The relevant parameters in this reference situation are given in Table 2. In Figure 4 the length of the normal conducting zone, T > Tcrmi,, is shown as it develops in time after a disturbance Q has been applied between 0 and I ms. In case the applied disturbance is 17 mJ or below, this normal length becomes zero after some time. The total conductor is then back in its superconducting state and is thus called stable for this disturbance. Unstable situations, in which the normal zone expands with time, are found when the applied disturbance was taken 18 mJ or above. The critical disturbance for this reference situation, therefore, is Qc, = 17.5 m J, with an inaccuracy of 0.5 eO=l~.5 x 1 0 0 % , ~ 3 % . In this case not only were the 17 and 18 mJ cases studied, but a calculation was also performed at the critical disturbance of 17.5 mJ derived from these two cases. The result as plotted in Figure 4 (dotted line) shows that the normal length is nearly constant for a long time. This means that this situation is indeed quite near the critical situation in which the normal zone does not change because dissipation and cooling are in equilibrium.
Numerical parameter variation The sensitivity of the solution of the reference situation to numerical parameters, as there are the tube length,
Cooling of an internally cooled magent." R.J. van der Linden and C.J. Hoogendoorn Table 2
Reference situation parameters
Parameter
Symbol
Unit
Value
Pressure Temperature Mass flow
P T m D Re
M Pa K g s 1 m s 1 -
1.25 4.2 2.5 0.587 105
Tube l e n g t h Axial mesh size N u m b e r of axial gridpoints N u m b e r of radial gridpoints
L Dz Nz Nr
mm mm -
200 5 40 13
L e n g t h of disturbance
Duration of disturbance
Lq tq
mm ms
10 1
Time step
Dt
ms
1
- mean velocity - Reynolds number
[0.
C-
9.
Mesh size. In Figure 5 the reference situation with mesh size Dz = 5 ram, is compared with the results for mesh sizes of 2 and 10 mm. Decreasing the mesh size from 5 to 2 mm gave only minor changes in the calculated results. The difference was well within the 3 % limits found in the stability analysis for the reference situation indicated by the results for the reference situation with disturbances of 17 and 18 mJ. Increasing the mesh size to 10 mm gave larger differences with the reference situation, even for short times. Thus, the 10 mm mesh size was judged to be too coarse, and the 5 mm mesh size was found to be accurate enough.
7. G.
vo
5
E L.
5. 4. 3. 2. 1. 0.
~.
~o.
i0. Time
Figure 4
gridpoints 2 0 + 21 time step 1
tube ends be seen. Because of this, the tube length was doubled when the influence of the mesh size or the extension of the disturbance was studied.
~o
.E E
Comment
~o.
40.
Timestep. Another important numerical parameter is the timestep Dt. Three different timesteps are compared in Figure 6, viz. Dr= 0.5 ms, D t = 1.0 ms (the reference
(ms) IO.
Reference situation - normalzone developmentin time.
x,Q=20W;F'I,Q=18W;A,Q=17W;O,Q=15W
the mesh size and the timestep, has been studied. The applied disturbance has been taken equal to 17.5 mJ, the critical disturbance. In this case small differences in the calculated heat flux lead to large changes in the solution for long times because of the critically stable character of the solution. The influence of the numerical parameters is therefore more pronounced than it would be in a more stable situation.
Tube length. First of all the influence of the limited tube length has been examined. Keeping the mesh size fixed the tube length is doubled by doubling the number of axial mesh nodes. This leads to identical results. In the case of mesh size of 2 mm instead of 5 mm, doubling of the tube length has been found not to change the results. From this it is concluded that the tube length has been taken large enough for this problem and increasing it will not change the calculated critical disturbance significantly. Only if the mesh size or the extension of the disturbance is much increased may an influence of the
w
I
.I 0.
]e.
20.
Time Figure5
I
38.
40.
(ms)
Meshsizevariation. A , A : D z = 2 m m , x , B : D z = 5 m m ,
O , C: D z = 10 m m
Cryogenics 1 9 8 9 Vol 29 March
183
Cooling of an internally cooled magent." R.J. van der Linden and C.J. Hoogendoorn 35..
10. 9. C •~
B. ~ _
~"
•-
15.
5.
~?.
0.
1.
I
3.
1
~.
I
9.
b
12.
15.
Pressure {arm)
0.
0.
10.
20. Time
30.
40.
(ms)
Figure 6 Time step variation. I-I, A : O t = 2 . 0 ms; + , C : D t = 0 . 5 ms
ms; O , B: D t =
or
Flow parameter variation As described for the reference situation, the critical disturbance has been calculated in case of different flow Reynolds numbers and system pressures. To explain the results gathered in Figure 7, the two heat transfer mechanisms involved in the cooling process, penetration and turbulence, will be described first. For short times heat has to penetrate through the laminar boundary layer by conduction. Turbulence influence will start after some time once a considerable amount of heat has penetrated the transition region. The penetration heat transfer coefficient for a fixed temperature step at the wall and constant fluid properties is given byT: (13)
where the parameter e = 2pcp. The stationary turbulent heat transfer coefficient can be described by the McAdams correlation 22 N u = c R e °'8 P r ° 4 ; c ,~ 0.023
184
Cryogenics 1989 Vol 29 March
7 Critical disturbance dependence on pressure and Reynolds number. I:lq, R e = 1.5 x 1 05; C), Re = 1.3 x 1 -05; x , R e = 1.0 x 105; [Z, R e = 0.5 x 105; /~, R e = 0.0 x 105
1.0
situation) and D t = 2 . 0 ms. In all three cases the disturbance itself has been applied in the same timespan of 2 ms. For times below 15 ms curves B and C (Figure 6) are almost identical. Curve A, for the largest timestep, is initially somewhat lower than the other two. It is important that the transient effects in the first milliseconds are described accurately, because small errors in these moments will accumulate in time. A 2 ms timestep was therefore found too large. For large times curves B and C deviate. With a 0.5 ms timestep a stable solution was found when the 1.0 ms timestep gave an unstable solution. However, as stated before these differences occur due to the critically stable character of the solution at 17.5 mJ. The effect on the critical disturbance is again well within the 3% limits given by the 17 and 18 mJ disturbance cases calculated with a timestep of 1 ms.
h(t) = (2pcp/nt) '/2 = n - 1/2(e/t)l/2
Figure
h = (c/D) Re°'a Z
(14)
where the parameter ;~ = 2°6(r/cp)°4 The transition from penetration to turbulent heat transfer is said to take place after a take-over time t t. Cornelissen 7 estimated this time by calculating the time at which the heat transfer coefficients from Equations (13) and (14) are equal. He found: D2 t t = DZ/(anc2Re l"6Pr°'a~ = - - R - 1.6 e ' nc" -Z~
(15)
R e y n o l d s n u m b e r e f f e c t s . The fact that, for a constant
system pressure, the critical energy increases with the Reynolds number is easily explained. At high Reynolds numbers the heat accumulated in the conductor at short times will later be cooled away more efficiently because the stationary heat transfer coefficient is larger than for small Reynolds numbers (Equation 14). Further, because the take-over time t t decreases with Reynolds number (Equation 15), the influence of turbulence starts earlier. The still rather high value of Qcr at zero Reynolds number, about 85% of the value at R e = 0.5 x 105, is not unrealistic. It must be noted that already an amount of about 5 mJ is needed to raise the conductor temperature to the lower current sharing temperature Tcrml.. Disturbances below 5 mJ will cause no dissipation and therefore always lead to a stable solution. The surplus of the critical disturbance is due to the influence of heat transfer to the coolant and conduction in the solid. The heat transfer is enhanced by an induced flow due to the thermal expansion of the helium. This enhanced stability by heat-induced flow is already observed by other authors23'24et al. The possible multivalued behaviour of a conductor due to this phenomenon as described by Dressner ~a has not been found in our calculations. Increasing the Reynolds number to 1.5 x 105 increases the critical disturbance considerably. In the reference situation P = 12.5 atm, the critical disturbance of 17.5 mJ in
Cooling of an internally cooled magent. R.J. van der Linden ant C.J. Hoogendoorn £psilon
the Re = 105 case is almost doubled in the Re = 1.5 x l0 s case (Q, = 33 m J). By taking large enough mass flows it is thus possible to increase the conductor stability considerably.
System pressure. If at constant Reynolds number the system pressure is varied between 3 and 12.5 atm, a minimum for the critical disturbance is found at about 6 atm (see Figure 7). To explain this behaviour the heat balance is considered. If the critical disturbance Q,, is applied at time t = 0, the total momentary heat flux qH=(t) to the coolant is in balance after some time to with the total momentary dissipation qdi,, (t) in the conductor. The energy stored in the conductor is then constant for t > to. This amount of energy AEo depends on the turbulent heat transfer. Since the Reynolds number is constant the main parameter determining AEo is X. For times smaller than to the heat flux will be larger than the heat dissipated in the conductor. The conductor energy will decrease in this period. The total energy Qo cooled away for times up to to is equal to Qo =
(16)
(qhf(t) -- qdi,,(t)) dt
This energy depends strongly on the penetration heat transfer described by the parameter e. The critical disturbance can be found by adding this energy decrease for t < t o, Qo, to the conductor energy AEo in the equilibrium state: (17)
Q=,=AEo + go
In Table 3 the values of Q., AEo and Qo are gathered as computed for the reference case of Re = 105, Q = Q~,, at different pressures. From Table 3 it can be seen that both the conductor energy AEo, depending on Z, and the energy decrease Qo at small times, depending on e, have a minimum at a pressure of about 6 atm. In Figure 8 the parameters e and Z, depending on the fluid properties only, have been plotted as a function of temperature for different values of the system pressure. Both e and X are low over a large temperature range at a pressure of 6 atm. This accounts for the low critical disturbance at this pressure. The large peak values of e and Z at a pressure of 3 atm have a large positive influence on the cooling. This explains the enhanced heat transfer at low pressures. Disturbance parameter variation
In practice little is known about the disturbances which can occur in a superconductor. In our analysis the disturbance has been applied in 1 ms, extended over 10 mm. This resulted in a critical disturbance of 17.5 mJ (reference situation). However, varying the extension or Table
3
Heat balance for critical disturbance
P(atm)
Oct (m J)
AE o (m J)
Oo(mJ )
3 4 5 6 9 12.5
20.5 19 17 15 15.5 17.5
9 9 8 7.5 7.5 8
11.5 10 9 7.5 8 9.5
Ghi
[S2/m4K2s)
(N/mK)
~E-~ F
/ •
tIt
/..-.
i
,/
/ ÷..
£
25
+\ ~
I
I
5~
7~
.~?"-~;"--
r'~
....
•
L~-~
• "'.. ÷\ %
~£m i
~
L5
I
I
~
75
T~rlture (K)
Telperature (K]
F i g u r e 8 Heat transfer parameters. , P = 3 . 0 atm; - - ~ P = 6 . 0 atm; + - +, P = 9 . 0 atm; . . . . P = 12.5 atm
duration of the disturbance will change this value. The influence of these parameters is discussed here.
Disturbance extension. If the extension of the disturbance is varied the initial conductor temperature changes. This will influence both the total heat flux to the coolant, qhf (At), and the total dissipation in the conductor, qd~,,(At), just after the disturbance has been applied. This will thus influence the stability of the conductor for this disturbance. To examine the influence of the disturbance extension Lq qualitatively, a parameter Rq is defined: qdiss(At) with At ~ 0 Rq = qhf(At)
(18)
If this parameter increases, the dissipation grows compared to the cooling just after the disturbance has been applied, and the conductor will be more unstable. This means that the critical disturbance decreases for increasing Rq and vice versa. Consider the initial copper temperature to be raised uniformly to a temperature T over the total disturbance length Lq. Since both qai,s and qhf are then proportional to the disturbed length Lq the only variable in Equation (16) is this initial temperature T. The heat flux to the coolant is proportional to the temperature difference (T-THe). The expression for the dissipation in the conductor depends on the value of the conductor temperature T. From Equations (5) and (6), it can be found that in the current sharing region, Tcrmi, < T < Tcrmax the dissipation is proportional to the temperature difference (T-Tcrmi,). For higher temperatures the dissipation is constant, for lower temperatures the dissipation is zero. Substituting this information in Equation (16) gives: c1
T - TH------~
Rq =
c2
T-
Tcrmin
T - THe
Tcrmax < T T.o < Tcr,,~. < T < Tcr,,ax (19)
0
THe < T < Tcrmi n
Cryogenics 1989 Vol 29 March
185
Cooling of an internally cooled magent: R.J. van der Linden and C.J. Hoogendoorn Table 4
Disturbance extension variation results
Extension Lq (ram)
Critical disturbance Qcr (m J)
5 10 20 50 75 85 90 95
17.5 17.5 17 16.5 16.5 17 1 7.5 18
100
18.5
For decreasing conductor temperature the parameter Rq will increase as long as T > Tcrmax. If T drops below Tcr . . . . Rq starts to decrease with T until it becomes zero at T = Tcrmin. Since the initial conductor temperature decreases for increasing extension of the disturbance this means that the critical disturbance will decrease for increasing values of Lq until the extension becomes too large and T drops below Tcrmax. For larger extensions Q , will increase with Lq; the value of Lq for which T = Tcrma~ is calculated to be about 100 mm. In T a b l e 4 the critical disturbances calculated for different values of the disturbance extension are gathered. The critical disturbance decreases for extensions up to 75 m m and grows for larger extensions. This result is in good agreement with the simplified description using the Rq parameter.
Due to the variation of the fluid properties the critical disturbance depends on the system pressure. A minimal stability is found at a pressure of about 6 atm. The largest disturbances are allowed at a pressure of 3 atm. By increasing the flow Reynolds number, turbulence influence becomes more important and the critical disturbance is increased. Still, a relative large critical disturbance has been found in the case of a stagnant flow. One reason for this is the enhanced heat transfer due to a heat-induced flow. In this study no sign of a possible multivalued behaviour of the conductor was found. This subject will be examined in further work. The influence of the duration and the extension of the disturbance was found to be small for small values of these parameters. However, if the disturbance is spread over a large extension or a large time interval the stability of the conductor increases.
Acknowledgement These investigations in the program of the Foundation for Fundamental Research on Matter (FOM) have been supported by the Netherlands Technology Foundation (STW).
References 1 Stekly, L.J.J. and Zar. J . L Stable superconducting coils IEEE Trans Nucl Sci (1965) 12 367-372 2 Maddock, B.J., James, G.B. and Norris, W.T. Superconductive composites, heat transfer and steady state stabilization, Cryogenics
(1969) 9 261-273
Disturbance duration. When the duration tq of the disturbance was increased, keeping the total energy input constant, no significant change in the critical disturbance was found in the interval 0.5 ms < tq < 4 ms. For larger disturbance durations an increase of the critical disturbance was observed. The critical disturbance distributed over 10 ms was found to be about 19 mJ, which is 1.5 mJ or about 13 % higher than the critical disturbance in the reference situation (tq = 1 ms). The reason for this enhanced stability when a disturbance is spread out over a larger period in time is that the coolant has more time to remove the applied disturbance before the dissipation in the conductor starts. When the duration of the disturbance is long enough the coolant will remove so much of the heat before the end of the disturbance that the conductor never reaches the lowercurrent sharing temperature. Since in this case no dissipation occurs the conductor is stable and the critical disturbance will be much higher than in the case of a short disturbance.
Conclusions The critical disturbance for the Sultan innercoil in the reference situation was found to be 17.5 mJ (Qcr/As¢+ ~u = 38 mJ cm-3). From the parameter study it is found that the used numerical timestep and mesh size are well chosen. The inaccuracy in the determination of the critical disturbance due to numerical parameters is less than the 3 % inaccuracy due to the used trial and error procedure.
186
Cryogenics 1989 Vol 29 March
3 Keilin,V.L, Klimenko, E.J., Kremly, M.G. and Samoilov, B.N.
4 5 6 7
Les champs magnetiques intenses, leurs production, leurs applications Coil Intern Grenoble CNRS (1967) 231-236 lwasa, Y. A critical current-margin design criterion for high performance magnet stability Cryogenics (1979) 19 705-714 Cornelissen,M.C.M. and Hoogendoorn, C.J. Thermal stability of superconducting magnets: static criteria Cryogenics (1984) 24 669-675 Cornelissen,M.C.M. and Hoogendoorn, C.J. Thermal stability of superconducting magnets: dynamic criteria Cryogenics (1985) 25 3-9 Cornelissen, M.C.M. The Thermal Stability of Superconductors Cooled by a Turbulent Flow of Supercritical Helium Thesis Delft University of Technology, Delft, Holland (1984)
8 Cornelissen, M.C.M., Hoogendoorn, C.J. and Franken, W.M.P.
Thermal stability of a superconductor cooled by a turbulent flow of supercritical helium Proc 18~hInt Conf Heat and Mass Transfer in Refrigeration and Cryogenics Dubrovnic (1986) 641--643 9 Bloem,W.B., Linden, R.J. van der, Hoogendoorn, C.J. Postma, H.
Experimental and numerical results on transient heat transfer to supercritical helium W,~rme- und StofffibertragunO (1988) 22, 315-323 10 Bird, B.R., Stewart, W.E. and Lightfoot, E.N. Transport Phenomena John Wiley & Sons, New York (1960) 11 Patanker, S.V. Numerical Heat Transfer and Fluid Flow Hemisphere Publishing Corporation, Washington (1980) i2 Cornelissen,M.C.M. and Hoogendoorn,C.J. A numerical study of heat transfer for a turbulent flow of supercritical helium Proc 3rd lnt Conf on Numerical Methods in Laminar and Turbulent Flow Seattle (1984) 831-841
13 Cornelissen, M.C.M. and Hoogendoorn, C.J. Forced convection heat transferto supercritical helium Appl Sci Res (1985)42 161-183 14 Schlichfing,H. Boundary Layer Theory McGraw Hill, New York (1960) 15 Driest,E.R. van On turbulent flownear a wall(1956)J Aeronautical Sci 1007-1011 16 Arp, V. New forms of state equations for helium Cryogenics (1974) 14 593-598 17 McCarty, R.D. Thermophysical properties of helium-4 from 4 to
Cooling of an internally cooled magent." R.J. van der Linden and C.J. Hoogendoom 3000 R with pressures to 15000 PSIA, NBS technical note 62 (1972) 18 MeCarty, R.D. Thermodynamic properties of helium-4 from 2-1500 K at pressures to 10s Pa J Phys Chem RefData (1973) 2 924-957 19 Franken, W.M.P. and Spoorenberg, C.d.G. The design of the SULTAN inner coil External Report ECN-I09 (1981) 20 Horvath,I., Vecsey, G., Weymuth, P., ZeUweger,J., Balsamo, E.P.,
Pasotti, G., Ricci, M.V., Sacchetti, N., Spadoni, M., Elen, J.D. and Franken, W.M.P. Status report on the forced flow high field test facility SULTAN IEEE Trans Ma#n (1983) 19 668
21
Plaum, J.M., Roeterdink, J.A., Eikelboom, J.A., Lijbrink, E. and
KIok, J. Development toroidal field conductor for NET, lOth Int Conf on Magnet Technology Boston (1987) 22 Schnurr, N.M. Numerical predictions of heat transfer to supercritical helium in turbulent flow through circular tubes J Heat Transfer Trans ASME (1977) 99 580-585 23 Dressner, L. Heating induced flow in cable in conduit conductors Cryogenics (1979) 19 653-658 24 Krafft, G. Recent progress on heat transfer to liquid helium, 8th Int Cryogenics Engineering Conf Geneva (1980) 754-768
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