Stability and Quench. Contents. Stabilisation criteria - preventing quench to happen

Transcription

1 Stability and Quench Antti Stenvall Tampere University of Technology Electromagnetics FINLAND Antti Stenvall Stability and Quench p.1/67 Contents Motivation and introduction Introduction and motivation Overview what does stability mean Perspective What is a quench? Stabilisation criteria - preventing quench to happen Flux jump Minimum propagation zone concept Quench analysis - preventing damages Acknowledgements and references Antti Stenvall antti@stenvall.fi Stability and Quench p.2/67

2 Introduction The origins of stability considerations are in Training and Degradation Training of a NbZr solenoid [3] NbZr solenoid I c vs. short-sample I c [3] Antti Stenvall antti@stenvall.fi Stability and Quench p.3/67 Overview of stability S t a b l e o p e r a t i o n D I S T U R B A N C E E x t e r n a l e n e r g y i n p u t : - c o n d u c t o r m o t i o n - i n s u l a t i o n c r a c k - f l u x j u m p - A C l o s s - h e a t l e a k - f a l s e b e a m -... T e m p e r a t u r e i n c r e a s e Partial transition to n o r m a l s t a t e a n d J o u l e h e a t g e n e r a t i o n C o n s i d e r in stability analysis to a n s w e r s a n d m a x i m i s e a n s w e r N O Q U E N C H Y E S H e a t g e n e r a t i o n N O > H e a t r e m o v a l S i m u l a t i o n D e t e c t i o n P r o t e c t i o n P e r f o r m i n g q u e n c h a n a l y s i s Antti Stenvall antti@stenvall.fi Stability and Quench p.4/67

3 Overview of stability cont. In stability considerations stability of superconducting system is scrutinised The main purpose of stability considerations is to quarantee safe operation and take care of risks What stable operation means? Operation of a superconducting device is stable when typical disturbances do not cause failures to normal operation Antti Stenvall antti@stenvall.fi Stability and Quench p.5/67 Perspective Domestic fuse burns around 10 A/mm 2 Superconductor can carry losslessly 1000 A/mm 2 at 4.2 K What would happen if superconducting state is lost? How easily is it lost? 1 cal equals to the energy required to raise temperature of 1 g of water by 1 K around room temperature (roughly 4.2 J). Is this nothing? At 4.2 K same energy rises temperature of 4 kg of NbTi superconductor by 2 K. In a typical application that is considerably over the maximum tolerated disturbance. When considering disturbances of J we talk about mj or even µj Antti Stenvall antti@stenvall.fi Stability and Quench p.6/67

4 Progress Motivation and introduction What is a quench? Informal definition for quench When quench origins, margins to quench Quench classifications Devred, Wilson Disturbance spectrum Stabilisation criteria - preventing quench to happen Flux jump Minimum propagation zone concept Quench analysis - preventing damages Acknowledgements and references Antti Stenvall antti@stenvall.fi Stability and Quench p.7/67 What is quench? Quench is an event in a superconducting coil/conductor where the orignated, typically propagating, normal zone causes a current decay in the coil/conductor. Without the current decay, the device will be damaged. Current decay may be automated, intrinsic or then an operator just turns of the power Automated automated quench detection system causes current decay Intrinsic in a persistent coil the growing normal zone resistance causes current decay, also variations may exist Operator NOT RECOMMENDED, an operator follows meters and keeps hand on a power source switch Antti Stenvall antti@stenvall.fi Stability and Quench p.8/67

5 Critical surface Quench origins when critical surface is punctured C u r r e n t d e n s i t y [ A / m m 2] Critical t e m p e r a t u r e s M g B N b T i N b S n 3 Critical J - B - T s u r f a c e s M a g n e t i c f l u x d e n s i t y [T] 4 0 T e m p e r a t u r e [K] Data gathered from [5, 32] Antti Stenvall antti@stenvall.fi Stability and Quench p.9/67 Margins to quench C u r r e n t Critical current characteristic C u r r e n t I c I o p C u r r e n t m a r g i n B m a x I o p Critical current characteristic T e m p e r a t u r e m a r g i n L o a d l i n e R e a l c u r r e n t m a r g i n B o p B c T o p T c s T c I c Critical current I op Operation current B op Max B directed to coil at I op B max Max B directed to coil at coil s I c B c Upper critical B at given temperature T op Operation temperature T cs Current sharing temperature T c Critical temperature at given B Antti Stenvall antti@stenvall.fi Stability and Quench p.10/67

6 Quench classifications: Devred Presented in [4] (excellent publication, get it for you) The B seen by the conductor is B max. Thus at operation temperature T op conductor can reach at maximum I max = I c (B max, T op ). Then magnet quenches either at I quench = I max or I quench < I max. If I quench = I max we have a conductor-limited quench. Then if I max really is I c (B max, T op ) we have short-sample quench. This is a success. If I max < I c (B max, T op ), degradation is said to have taken place. If a quench occur at I quench < I max, it is due to an energy release which increases the conductor temperature enough in an enough big volume. This kind of a quench is called as energy-deposited quench or premature quench. Antti Stenvall antti@stenvall.fi Stability and Quench p.11/67 Devred s quenches summary Q u e n c h c u r r e n t X X C o i l 1 C o i l 2 P l a t e a u s, C o n d u c t o r - l i m i t e d q u e n c h e s S h o r t - s a m p l e T r a i n i n g X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X D e g r a d a t i o n P r e m a t u r e q u e n c h e s X M a g n e t r e a c h e d s h o r t - s a m p l e c u r r e n t S U C C E S S! Q u e n c h n u m b e r Antti Stenvall antti@stenvall.fi Stability and Quench p.12/67

7 Quench classifications: Wilson Presented in [35] Based on disturbance spectrum Concentrated on how quenches originate Space Point Distributed Time Transient Joules Joules/m 3 Continuous Watts Watts/m 3 Antti Stenvall antti@stenvall.fi Stability and Quench p.13/67 Scales of some disturbances Picture from [2] Antti Stenvall antti@stenvall.fi Stability and Quench p.14/67

8 Progress Motivation and introduction What is a quench? Stabilisation criteria - preventing quench to happen Heat balance Heat generation: current sharind model, power-law model Cooling Cryogenic stabilisation Equal-area theorem Flux jump Minimum propagation zone concept Quench analysis - preventing damages Acknowledgements and references Antti Stenvall antti@stenvall.fi Stability and Quench p.15/67 Stabilisation - preventing quench to happen Starting point: Heat diffusion equation λ(t) T(x) + Q(t, x) Q cooling (T) + Q ext = C p (T) T(x), t λ Heat conductivity Q Heat generation Q cooling Cooling term Q ext External heat input C p Volumetric specific heat Antti Stenvall antti@stenvall.fi Stability and Quench p.16/67

9 Heat generation - Current sharing model C u r r e n t Critical current characteristic T < T c s S u p e r c o n d u c t o r M a t r i x M a t r i x T < T < T c s c C u r r e n t C u r r e n t i n s u p e r c o n d u c t o r I o p C u r r e n t i n m a t r i x T > T c T e m p e r a t u r e T o p T c s T c Antti Stenvall antti@stenvall.fi Stability and Quench p.17/67 Current sharing model cont. Heat generation in a superconductor is computed as Q }{{} = fj sc E } {{ } + (1 f)j m E } {{ } Heat generation in superconducting region in matrix f Area of superconductor / area of conductor J sc Current density in the superconducting region J m Matrix current density E Electric field: E = ρ m J m ρ m matrix resistivity Heat generation in the matrix occurs due to its resistance, whereas in superconductor ρ = 0, but E there must agree with E in electric field. This arises from changes in current distribution, flux flow resistance... From the critical current characteristic I c (T) heat generation can be determined as (see next slide) Antti Stenvall antti@stenvall.fi Stability and Quench p.18/67

10 Current sharing model - equations The operation current is J sc A cond f = I c (T) J sc = I c(t) A cond f. (1) I op = J sc A cond f + J m A cond (1 f). (2) Thus current density in matrix is determined from I op using (1) as { } Iop I c (T) J m = max A cond (1 f), 0 (3) by substituting (1) and (3) to (18) heat generation can be computed as Q = ρ m(t)i op A 2 cond (1 f) max {I op I c (T), 0}. (4) Antti Stenvall antti@stenvall.fi Stability and Quench p.19/67 Current sharing model - equations cont. If we expect a linear dependence between I c and T as { } Tcs T I c (T) = I c0 max, 0 T cs T 0 C u r r e n t I o p Critical current c h a r a c t e r i s t i c L i n e a r e s t i m a t i o n I c0 is critical current at T 0 T c s T e m p e r a t u r e T c we get heat generation as Q = 0 ( ) T < T cs Q = ρ m(t)i op T A 2 cond (1 f) I op I cs T c0 T cs T 0 T cs T < T c Q = ρ mi 2 op T T A cond (1 f) c Antti Stenvall antti@stenvall.fi Stability and Quench p.20/67

11 Current sharing model summary What heat generation looks like? I c (4.2 K,B 0 )=200 A Heat generation [ /m 3 ] I op =100 A I op =150 A I =200 A op T cs (0,B 0 ) Temperature [K] Antti Stenvall antti@stenvall.fi Stability and Quench p.21/67 Another approach: Power law Power-law fit can be used to characterise E I curve of a conductor and to compute heat generation. According to power-law ( ) n I E(I) = E c, I c (T, B) where n is called superconductor index number or just n-value. Then heat generation is Average electric field [µv/cm] Ec 0 Sample #4 B=12.5 T I c =667 A n value=41 Measured V I curve V I curve for determination of n value Power law fitting Electric field criterion Q = min I op E } A {{ cond } E I c op n+1 A cond I c (T,B) n ( Iop ρ norm (T) A cond ) Current [A] A Nb 3 Sn conductor ρ norm Conductor normal state resistivity Antti Stenvall antti@stenvall.fi Stability and Quench p.22/67

12 Power-law heat generation I c (4.2 K,B 0 )=200 A Heat generation [ /m 3 ] I op =100 A I op =150 A I op =200 A Temperature [K] In logarithmic scale Temperature [K] Heat generation as a function of temperature computed for few operation currents according to the power-law model. Dashed line presents n = 15 and solid n = 50. Both figures present same data but right is in logarithmic scale to illustrate subcritical heat generation. Constant normal state resistivity was expected. Antti Stenvall antti@stenvall.fi Stability and Quench p.23/67 Cooling modes of superconductor Indirect Contact to a heat sink through conduction (most likely cryocooler) From stability perspective there are no cooling on the time scale of interest Bath cooling Pool of liquid (e.g. helium) at atmospheric pressure and saturation temperature (4.2 K), could also be hydrogen, neon, nitrogen, oxygen? Boiling heat transfer Force-flow cooling Supercritical (pressurised) or two-phase flow, warm or cold circulation Superfluid cooling (cooling with He II T 1.9 K) He II is the superfluid phase of He (no other liquids have this phase). Then very high heat transfer can be achieved due to nonviscous flow of He II. Stagnant bath, heat removal through counterflow heat exchange Achieved in small cryostats by underpressurising, or by using e.g. lambda refrigerator Reference [2] Antti Stenvall antti@stenvall.fi Stability and Quench p.24/67

13 Convective cooling - mathematical model Simple model for heat flux q [W/m 2 ] q = h T h Convective heat transfer coefficient T Temperature difference between coolant and cooled surface In heat diffusion equation Q cooling is in [W/m 3 ], thus cooling is added as (for single conductor) Q cooling = hp A cond T p wetted perimeter A cond Conductor cross-section area Antti Stenvall antti@stenvall.fi Stability and Quench p.25/67 But... h is not simple function General shape of pool boiling curve [38] (both h and q) Antti Stenvall antti@stenvall.fi Stability and Quench p.26/67

14 In practice things are even more complicated Liquid helium [2] Liquid nitrogen [23] Orders of difference in h or q They depend at least on surface orientation, roughness (porosity, effective area), material and history Antti Stenvall antti@stenvall.fi Stability and Quench p.27/67 Cryogenic stabilisation Presented in 1965 [15, 27] The ultimum stabilisation criterion even though all the current flows in the matrix temperature is constant, neglects heat conduction At steady state T t C p T t = Q Q cooling. = 0. To fulfil stability it must be Q Q cooling ρ m ( I op A cond (1 f) ) 2 hp A cond (T c T op ) This condition of cryostability can be formulated as Stekly s α Stekly parameter as operation is stable when α Stekly < 1 and cryostable at α Stekly = 1 α stekly = ρ m I 2 op hpa cond (1 f) (T c T op ). Antti Stenvall antti@stenvall.fi Stability and Quench p.28/67

15 Cryostability Finally we get equation for cryostable operation current I Stekly as H e a t g e n e r a t i o n I stekly = hpa cond (T c T op ) ρ m. ( A ) C U R R E N T S H A R I N G ( B ) P O W E R L A W P o o l b o i l i n g H e a t g e n e r a t i o n P o o l b o i l i n g C o o l i n g N o t c r y o s t a b l e C o o l i n g N o t c r y o s t a b l e N o t c r y o s t a b l e N o t c r y o s t a b l e C r y o s t a b l e C r y o s t a b l e S t a b l e S t a b l e T e m p e r a t u r e T e m p e r a t u r e T o p T c s T c T o p T c s T c Schematic view of determination of cryostability when heat generation is computed according to (A) current sharing model and (B) power law. Antti Stenvall antti@stenvall.fi Stability and Quench p.29/67 Equal-area theorem Introduced in 1969 by Maddock et al [18] Takes into account also heat conduction along the conductor T e m p e r a t u r e [ K ] C o o l i n g [ W / u n i t a r e a ] H e a t g e n e r a t i o n [ W / u n i t a r e a ] A 2 A 1 P o s i t i o n a l o n g c o n d u c t o r E q u a l - a r e a t h e o r e m : O p e r a t i o n i s s t a b l e w h e n A 1 = A 2 Antti Stenvall antti@stenvall.fi Stability and Quench p.30/67

16 Equal-area theorem cont. Dependence on conductor length is nullified by mathematical tricks. See equations yourself. In one dimensions steady state heat diffusion equation states ( d λ(t) dt ) = Q cooling (T) Q(T), dx dx where A is the cross-sectional area of the conductor. Then we multiply this equation with λ(t) and integrate from T 0 to T and get T T 0 d dx Next step is to do substitution S = λ dt dx ( λ(t) dt ) T ( λ(t)dt = Qcooling (T) Q(T) ) λ(t)dt. (5) dx T 0 which leads to dt = 1 λ(t) Sdx. Antti Stenvall antti@stenvall.fi Stability and Quench p.31/67 Equal-area theorem cont. While the limits of the integration are T 0 S 0 = k dt dx and T S = k dt T0 dx By substituting this to (5) it is equivalent to S S 0 SdS = T T T 0 ( Qcooling (T) Q(T) ) λ(t)dt, T if λ(t) 0, which is of course natural because heat conductivity does not go to zero. Then if we assume that the conductor is long and end temperatures are T 1 and T 2, for which Q cooling = Q, S 0 and S are 0. If it is further expected that λ(t) is independent of temperature we get the equal-area condition T2 T 1 Q cooling (T) Q(T)dT = 0. (6) Antti Stenvall antti@stenvall.fi Stability and Quench p.32/67

17 Equal-area theorem, what to conclude Thus, finally we have eliminated the conductor length at which temperature is not that of the coolant. It would be useful to compare this to the criterion given by Stekly. For that we need to expect constant h and need to solve (6). C o o l i n g H e a t g e n e r a t i o n A 2 A 1 T e m p e r a t u r e T o p = T 1 T c s T c T 2 = T e q Schematic view of finding operation current which stabilisates in framework of equal-area theorem. Antti Stenvall antti@stenvall.fi Stability and Quench p.33/67 Equal-area theorem, what to conclude cont. As it was mentioned T 1 and T 2 correspond to temperatures where cooling equals to heat generation. Then T 1 must be T op and T 2 is called T eq and is given as ( ph (T eq T op ) = ρ m A cond T eq = T op + ρ m A cond ph I op A cond (1 f) ( ) 2 Iop. 1 f As we see this goes to quite heavy algebra. But the critical current to quarantee stability in the framework of equal-area thereom I Maddock is I Maddock = hpa cond [(T c T op ) + (T cs T op )] ρ m. (7) This is higher than I Stekly. See details of the derivation of (7) from [18]. ) 2 Antti Stenvall antti@stenvall.fi Stability and Quench p.34/67

18 Progress Motivation and introduction What is a quench? Stabilisation criteria - preventing quench to happen Flux jump Minimum propagation zone concept Quench analysis - preventing damages Acknowledgements and references Antti Stenvall antti@stenvall.fi Stability and Quench p.35/67 Flux jump - current penetration models Bean s critical state model as a starting point for current penetration [1] Ampére s law B = µ 0 J A p p l i e d m a g n e t i c f l u x d e n s i t y (in +y direction) J c Slab is infinite in y a n d z d i r e c t i o n s a n d h a s t h i c k n e s s 2 a y x 0 - J c C u r r e n t d e n s i t y (in +z direction) z x - a 0 a Antti Stenvall antti@stenvall.fi Stability and Quench p.36/67

19 Flux jump - feedback loop A case of transport current, similarly with screening currents and applied field Consider also Faraday s law E = B t Q = J E 0 M a g n e t i c f l u x d e n s i t y in y- direction I n t e g r a t e h e a t g e n e r a t i o n Q 0 B E P O S I T I V E F E E D B A C K O F F L U X J U M P P H E N O M E N O N - J T C o m p u t e t e m p e r a t u r e r i s e a n d a s s u m e i s o t h e r m a l s l a b Electric field in z- direction J (T ) c 1 0 J (T ) c 2 C u r r e n t d e n s i t y in z- direction ( n o w t r a n s p o r t c u r r e n t ) Antti Stenvall antti@stenvall.fi Stability and Quench p.37/67 Progress Motivation and introduction What is a quench? Stabilisation criteria - preventing quench to happen Flux jump Minimum propagation zone concept Minimum quench energy MQE MQE models: analytical, numerical MQE measurements Magnitudes of MQE Training - origins of minimum quench energy Wire motion Epoxy failures Filament microyielding Quench analysis - preventing damages Acknowledgements and references Antti Stenvall antti@stenvall.fi Stability and Quench p.38/67

20 Minimum propagation zone concept Wilson and Iwasa presented concept of minimum propagation zone (MPZ) [36] MPZ is normal zone in superconductor. Any normal zone larger than MPZ will grow and a smaller one collapses and full superconductivity recovers [35, p.76] This concept is important because it allows to study superconductor s tolerance against disturbances in certain operation conditions Conductors/coils which stability has been considered within MPZ framework are Meta-stable conductors Steady state temperature distribution in Wilson s minimum propagation zone Disturbance 1 Disturbance 2 T c 0 l 0W /2 3 l 0W /2 x T op l 0W Antti Stenvall antti@stenvall.fi Stability and Quench p.39/67 Minimum quench energy (MQE) Minimum quench energy is the energy required to create MPZ When minimum quench energy is known it can be computed if certain disturbances exceed it or not After having researched many stability aspects, Wilson pointed out that an essential design parameter is the size of the disturbance against which a coil is stabilised [37] epoxy tape x 2 x 1 2R mz2 2R mz1 x 3 2R mz3 Antti Stenvall antti@stenvall.fi Stability and Quench p.40/67

21 MQE analytic model Iwasa s model [14] to give size of 3D MPZ (V MPZ ) R mz3 = 3λ l (T c T op ) ρ m J 2 m R mzi = λ l longitudinal heat conductivity λ xi effective heat conductivity in transverse direction λ xi λ l R mz3 MQE = V MPZ Tc T op C p (T)dT Antti Stenvall antti@stenvall.fi Stability and Quench p.41/67 Numerical modelling of MQE In 1D solve heat diffusion equation in transient conditions Can be extended to 2D and 3D to include also effects of transverse heat conduction Heat generation according to the power-law or current sharing model D i s t u r b a n c e e n e r g y Q T x T R e c o v e r y Q u e n c h T x x M o r e p l a u s i b l e Q for next iteration Antti Stenvall antti@stenvall.fi Stability and Quench p.42/67

22 Numerical modelling of MQE cont. MQE exists between (a) and (b) [13] Find computationally MQE with relevant tolerance Antti Stenvall Stability and Quench p.43/67 Numerical modelling of MQE cont. MQE is a function of disturbance duration [13] length [13] References [13, 19, 20, 30] Antti Stenvall antti@stenvall.fi Stability and Quench p.44/67

23 Minimum quench energy measurement Try to minimize heat capacity of auxiliary stuff near the disturbance volume Sample holder example for MQE measurement [26] Measurement results for NbTi wire with sample holder presented [40] Antti Stenvall Stability and Quench p.45/67 Magnitudes of MQE MQE for NbTi wire [41] MQE for several Cu-Nb reinforced Nb 3 Sn wire [40] Antti Stenvall antti@stenvall.fi Stability and Quench p.46/67

24 Magnitudes of MQE cont. MQE [mj] Measured Computed T op =25.5 K, I c =286 A T op =28.0 K I c =221 A Current [A] Computed and measured MQE for a MgB 2 wire [28] MQE for YBCO conductor [33] Antti Stenvall antti@stenvall.fi Stability and Quench p.47/67 MQE summary MQE is very important stability parameter at coil level and also at conductor level It allows building meta-stable devices Comparison of stability of conductors is simple with MQE computations Next I consider the origins of MQE also as causes of training Small heater for MQE and thermally and electrically insulated conductor at interesting area Antti Stenvall antti@stenvall.fi Stability and Quench p.48/67

25 Wire motion C a b l e I F B. C o n d u c t o r View of Rutherford cable used in LHC at CERN [17]. Lorentz force directed to individual conductor in a Rutherford cable. F = IBl, I conductor current, l length under consideration. If a conductor moves a distance δ the energy deposition W W = IBlδ. Does this exceed MQE or not? Source of training. Antti Stenvall antti@stenvall.fi Stability and Quench p.49/67 Use pre-stressing for making tight winding Low pre-stressing wire movements High pre-stressing microcracking in filaments find optimum Solid and dashed bars show coils manufactured with low prestress (10-20 MPa) and high prestress (70-80 MPa) [4] Antti Stenvall antti@stenvall.fi Stability and Quench p.50/67

26 Epoxy failures The elastic energy that an epoxy crack can release can be computed as W = σ2 2Y V, where σ is the average stress in the epoxy, Y its Young modulus and V the volume of the crack. When this equals to MQE quench is ignited. Thus, minimum epoxy volume V min which ignits a quench is V min = 2 Y W MQE σ 2. When l 0 is the length of MPZ, it can be roughly said that the critical epoxy coating thickness e c is e c = 2 Y W MQE σ 2 πdl 0, Antti Stenvall antti@stenvall.fi Stability and Quench p.51/67 Filament microyielding Even though the previous causes could be eluded, there can still be microyielding in filaments which can not be entirely eliminated. Also, conductor short samples can train. One origin of training are local plastic deformations of NbTi at the surface of a filament. [4] These plastic deformations can be observed from an acoustic emission measurement. In a single conductor measurements training effect due to filament microyielding can be easily seen. Next slide shows the rate of acoustic emissions during three consecutive strain tests of a NbTi conductor. When the limit of the preceding cycle is exceeded the acoustic emissions are generated again. This effect is known as Kaiser effect [9] and is closely related to training because acoustic emissions are of mechanical origin. Antti Stenvall antti@stenvall.fi Stability and Quench p.52/67

27 Filament microyielding - Kaiser effect Rate of acoustic emissions in a strain test of NbTi wire with diameter of 0.26 mm. [24] Antti Stenvall antti@stenvall.fi Stability and Quench p.53/67 Progress Motivation and introduction What is a quench? Stabilisation criteria - preventing quench to happen Flux jump Minimum propagation zone concept Quench analysis - preventing damages Normal zone propagation velocity General algorithm for quench simulations Approaches: Wilson s, Finite element discretisation Quench protection External dump resistor Inductively coupled secondary Quench back Subdivision Acknowledgements and references Antti Stenvall antti@stenvall.fi Stability and Quench p.54/67

28 Quench analysis - preventing damages Threats of quench Insulation breakdown [8] Melting [25, p ] Mechanical strains mechanical faults [12] In a quench analysis, the magnet designer confirms that a coil can quench safely Analysis includes quench simulations, design of protection system and possibly also considerations of quench detection Numerical approach necesasry One of the first programs was Wilson s QUENCH [35, p.218] Some references [6, 7, 10, 11, 16, 21, 22, 29, 31, 39] Antti Stenvall antti@stenvall.fi Stability and Quench p.55/67 Normal zone propagation velocities Wilson used normal zone propagation velocities v NZP in his approach to control spreading of the normal zone and thus computed current decay Quench propagation is essentially important P r o p a g a t i n g n o r m a l z o n e f r o n t T c Whetstone-Roos formula [34] [ ( 1 v NZP =J op ρ(t)λ(t) Tc (B,J op ) T op C p (T)dT T e m p e r a t u r e a l o n g c o n d u c t o r C p (T) 1 λ(t) ] 1/2 T=T c (B,J op ) λ(t) T T o p Tc (B,J op ) T op C p (T)dT ) Antti Stenvall antti@stenvall.fi Stability and Quench p.56/67

29 Normal zone propagation velocity 2D Instead of closed formula for v NZP heat diffusion equation can be solved in transient conditions Here 2D MQE model was used to compute v NZP Click figure to see animation To be presented in Applied Superconductivity Conference 2008 Chicago Antti Stenvall antti@stenvall.fi Stability and Quench p.57/67 Algorithm for solving quench See details of implementation from [29] Antti Stenvall antti@stenvall.fi Stability and Quench p.58/67

30 Wilson s approach Modelling 3D quench propagation with propagation velocities [35, p.209]. Growth of normal zone in two different solenoids [35, p.211]. Antti Stenvall antti@stenvall.fi Stability and Quench p.59/67 Approach with finite element discretisations Solve heat diffusion equation in transient conditions No need to know propagation velocities Click figure to see animation Quench in an MgB 2 racetrack [29] Antti Stenvall antti@stenvall.fi Stability and Quench p.60/67

31 Quench protection (QP) Active protection Needs efficient quench detection Can be made very effective Dump resistor, heaters,... Passive protection Good to have together with active protection Necessary in coils operating at persistent mode Inductively coupled secondary, subdivision, quench back,... Antti Stenvall Stability and Quench p.61/67 QP: External dump resistor S R s L I s D R norm With ideal diode compute current decay as: I k+1 = I k if S is closed I k+1 = I k I k ( R k norm + R s ) t L if S is open Antti Stenvall antti@stenvall.fi Stability and Quench p.62/67

32 QP: Inductively coupled secondary Not suitable for AC magnets Losses in secondary during loading if not opened Effective only when current decay is fast R norm L p M L s R s Antti Stenvall antti@stenvall.fi Stability and Quench p.63/67 QP: Quench back High thermal conductivity of coil body / thermal interface High electrical conductivity to allow eddy currents (only applicable with DC magnets) Coil Quench origin Large normal zone Coil body or thermal interface Antti Stenvall antti@stenvall.fi Stability and Quench p.64/67

33 QP: Subdivision R 1 L 1 R e s i t o r s D i o d e s M M a g n e t I s R 2 L 2 R norm Protection of OXFORD characterisation magnet 12 T bore field Antti Stenvall antti@stenvall.fi Stability and Quench p.65/67 End of show Thank you for your attention Antti Stenvall antti@stenvall.fi Stability and Quench p.66/67

34 Acknowledgements Following book, publication, presentation and course material were of great help while making this presentation Wilson M N 1983 Superconducting Magnets (Oxford: Oxford University Press) Devred A 1992 AIP Conf. Proc "Quench origins" DOI: / Bottura L "Stability (and cooling)" Presented at Summer School on Materials and Applications of Superconductivity, July, 2007, Forschungszentrum Karlsruhe, Germany Prestemon S, Ferracin P and Todesco E Course material: Superconducting accelerator magnets, 4-8 June, 2007, Lawrence Berkeley National Laboratory, Berkeley, USA See next pages for other references. Antti Stenvall antti@stenvall.fi Stability and Quench p.67/67 References [1] Bean C P 1962 Phys. Rev. Lett Magnetization of hard superconductors doi: /physrevlett [2] Bottura L Stability (and cooling) Presented at Summer School on Materials and Applications of Superconductivity, July, 2007, Forschungszentrum Karlsruhe, Germany [3] Chester P F 1967 Rep. Prog. Phys Superconducting magnets doi: / /30/2/305 [4] Devred A 1992 AIP Conf. Proc Quench origins doi: / [5] Ekin J W 2004 in Gaseous Dielectrics X (edited by Christophorou L C, Olthoff J K and Vassiliou P, Springer) Superconductors: An emerging power technology [6] Eyssa Y M and Markiewicz W D 1995 IEEE Trans. Appl. Supercond Quench simulation and thermal diffusion in epoxy-impregnated magnet system doi: / [7] Eyssa Y M, Markiewicz W D and Miller J 1997 IEEE Trans. Appl. Supercond Quench, thermal, and magnetic analysis computer code for superconducting solenoids doi: / [8] Fujino H 1990 IEEE Elec. Ins. Mag. 6 7 Electrical insulation technology for superconducting devices in Japan doi: / [9] Fujita H, Takaghi T, Bobrov E S, Tsukamoto O and Iwasa Y 1985 IEEE Trans. Magn The training in epoxy-impregnated superconducting coils [10] Gavrilin A V 1993 IEEE Trans. Appl. Supercond Computed simulation of thermal process during quench in superconducting winding solenoid doi: / [11] Gavrilin A V, Dudarev A V and ten Kate H H J 2001 IEEE Trans. Appl. Supercond Quench modeling of the ATLAS superconducting toroids doi: / [12] Imbasciati L, Bauer P, Ambrosio G, Lamm M J, Miller J R, Miller G E, Zlobin A V 2003 IEEE Trans. Appl. Supercond Effect of thermo-mechanical stress during quench on Nb 3 Sn cable performance doi: /tasc [13] Ishibashi K, Wake M, Kobayashi M and Katase A 1979 Cryogenics Thermal stability of high current density magnets doi: / (79)90064-x [14] Iwasa Y 1994 Case Studies in Superconducting Magnets (New York: Plenum Press) [15] Kantrowitz A R and Stekly Z J J 1965 Appl. Phys. Lett A