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Electromagnetic effects on first wall and structural components
Fusion Engineering and Design 16 (1991) 217-227 North-Holland
217
Electromagnetic effects on first wall and structural components K. Miya a, Y.R. C r u t z e n b, T. Takagi c, L.V. Boccacini d a n d L. B o t t u r a
e
a The University of Tokyo, Japan b ISEI Institute, SER Division JRC, ISPRA, CEC, Italy c Institute of Fluid Science, Tohoku University, Sendai, Japan d Kernforschungszentrum Karlsruhe, Karlsruhe, Germany e NET Team, Max Plank Institute, Garching, Germany
In a tokamak-type nuclear fusion reactor, the plasma disruption instabilities and the large magnetic fields generate electromagnetic-type phenomena in transient conditions which lead to strong induced eddy currents and large body loads that must be correctly accommodated in the engineering design of the first wall and structural components. During the last few years, many efforts at the TESLA Laboratory of the JRC-Ispra and at other Institutions have been actively made in terms of the comprehension of the real impact of the accidental events of plasma disruption on the mechanical design of the internal device components such as the multi-segmented first wall and their supports. Fully three-dimensional transient dynamic analyses have been performed due to the geometry complexity of the "conducting" and "plasma" regions both from the electrical and mechanical standpoints. Actually, there is a great interest to extend the computational strategies in magneto-structural analysis in order to investigate the impact of the huge electromagnetic loads on the integrity of the fusion structural system from the view point of magnetic damping.
1. Introduction
Components and devices in a magnetic confinement fusion reactor are operated under very strong magnetic field environment. This requires fusion reactor designers not only to understand qualitatively but also to evaluate quantitatively electromagnetic effects on fusion reactor components. For the purpose one has to deal with two main subjects, eddy current analysis based on electromagnetism and stress analysis based on elasticity and plasticity. Furthermore electromagnetic coupling effects between not only plasmas and conducting components near the plasmas but also electromagnetic field and mechanical deformation should be preferably taken into account. As to the level of research in both fields it is no doubt that the theory of elastoplasticity has already been established and large computer codes are now commercially available for
stress analysis of fusion reactor components, but the theory of eddy current has been progressed only in these years being assisted with numerical verification by computer codes. This is evident from proceedings of international conferences on computational electromagnetics of which typical one is a series of COMPUMAG [1] where eddy current analysis is a major subject and various ideas and innovative computational techniques were presented so far. As far as studies on the electromagnetic effects on fusion engineering design are concerned, several modelling efforts have been performed at several institutions in the world in order to account for and to correctly simulate the following situations [2,3]: - the plasma disruptive instability scenarios characterized by fast quench of a large current and coupled with a rapid plasma movement away from its equilibrium position;
0 9 2 0 - 3 7 9 6 / 9 1 / $ 0 3 . 5 0 © 1991 - Elsevier Science Publishers B.V. All rights reserved
218
K. Miya et al. / Electromagnetic effects on first wall
- the complex magnetic field distribution in devices in order to drive and to confine the plasma column inside a vacuum vessel; - the 3D in-vessel multi-segmented conducting bodies with narrow gaps between them allowing remote maintenance; - the inductive coupling between first wall segments, the vacuum vessel and coils including the complex electromagnetic interactions between the "electrically conducting" regions and the "plasma source" part; - the effects of the electrical breaks/insulations which modify the circulation of eddy currents and of the additional saddle loops incorporated for improving the plasma position control. Furthermore, from the viewpoint of magnetomechanical design of fusion reactor components it is desired to establish design methodology based on numerical evaluation of magneto-mechanical coupling and thus the following items should be added for future studies: - code development for analysis of coupling effect between electromagnetic field and elastoplastic deformation of components. - code development for control technique of eddy current and thereby electromagnetic force by optimizing design configuration of components and proper selection of material constants. With regard to the magnetic stiffness observed in various electromagnetic structures, researches have already been made by many investigators in the light of magnetoelastic buckling of a ferromagnetic beam plate first in the decade of 1970 [4,5,6] and then magnetoelastic buckling and vibration of superconducting toroidal coils nested in a full or partial torus for a fusion nuclear reactor in the early 1980's [7,8,9,10]. However, these studies dealt with magnetomechanics of bodies placed in a static magnetic field, thus have nothing to do with eddy current. Fundamental studies on magnetic damping have been made recently [11,12,13] and the approach should be applied to electromagnetomechanical design of fusion reactor components for accurate design in the theoretical and numerical ways despite of very few examples of its application to a fusion reactor conceptual design. Here it is instructive to note that magnets-mechanical coupling can be divided into such three aspects as static magnetic stiffness, dynamic magnetic stiffness and magnetic damping. The first two of them are caused by magnetic energy change due to static and dynamic deformation of current carrying elastic bodies, respectively and the last by velocity induced voltage. The static magnetic
stiffness is observed in deformation of superconducting toroidal coils, the dynamic magnetic stiffness is due to interaction between transient applied field and deformation of any structural components through eddy current, and the magnetic damping is ascribed to eddy current caused by vibration of the body in static magnetic field. On the other hand, designers of fusion reactor components seem to have not shown keen interest in the active control of eddy current and electromagnetic force. It could be predicted that the control technique will be developed significantly and effectively through progress of design technology in the near future. One can point out one example that an insert of insulation between conducting plates causes generally concentration of eddy current near it leading to localized large electromagnetic force and increasing possibility of arcing between components, and contrarily an insert of conducting connections between them could reduce peak values of eddy current, which would be more favorable from the viewpoint of structural design. In this sense electromagnetic force is dependent on structural design as well as material constants suggesting potential capability of eddy current control. The governing equations of eddy current receive a small modification when an effect of moving body in a static and dynamic magnetic fields should be included in the system. The coupling effect has been proved, in particular, important from the viewpoint of magnetomechanical design of fusion reactor components because it creates, in many cases, positive magnetic stiffness in the dynamical system, namely the magnetic damping constrains motion of a conducting body as a result of coupling between eddy current, magnetic field and elastoplastic deformation [10].
2. Governing equations for eddy current
Eddy current phenomenon is governed by two of Maxwell's equations that are Ampere's law and Faraday's induction law supplemented with constitutive equations between B and H, and E and J taking different constants depending on physical properties of materials concerned, i.e. non-magnetic, magnetic and superconducting material. Recently eddy current induced in low and high T. superconductors was successfully computed by constructing a new constitutive equation between electrical field E and eddy current J and validity of the new method was verified experimentally in terms of A.C. loss evaluation in a low Tc superconductor [14] and
K~Miya et al. / Electromagneticeffects on first wall evaluation of electromagnetic force generated by a high Tc superconductor [15]. This numerical approach could be applied to more accurate evaluation of A.C. loss and electromagnetic force in poloidal and toroidal coils of a fusion reactor in future. For solutions of eddy current based on Maxwell's equations there are two methods i.e. indirect and direct methods where field variables such as magnetic vector and scalar potentials (A, ~b), current vector potential (T) in the former case and magnetic field ( H ) and eddy current ( J ) in the latter case are adopted. In what to follow essential parts of the theory will be introduced in the light of A-~b method, T-to method, T-method and J - m e t h o d focussing on derivation of the governing equation and starting from the following Maxwell's equations and constitutive ones, I7 X H = J ,
17 x E = - OB/Ot,
(1,2)
J = o-E,
B =/~H,
(3,4)
where notations are conventional. The magnetic vector potential A defined as B = V X A is a fundamental field variable in the A - ¢ method. From eq. (2) and B = V X A we find a relation E = - a A / ~ t - V¢ and then the following governing equation from eq. (1),
1
V X - - VxA+o/z
(o
t)
+V~b =Js,
(5)
we find the governing equation for the T-to method as follows, 1 0 VX-o. V X T + / z - ~ ( T - V t o ) = -
V.tr(OA/at + V&) = O.
(6)
Equations (5) and (6) are the governing equation in the A - & method while one should notice that eq. (6) is derived directly from eq. (5) by taking rotation of eq. (5) and in this sense eq. (6) is not independent. An inclusion of eq. (6) is related to an introduction of a gage as a constraint to the vector potential A. Two kinds of gage, Coulomb gauge and Lorentz gauge, are common. The readers refer to ref. [16] as to discussion about the meaning and usage of the gage. In the similar way to the above one can use the current vector potential defined as J = V x T as the field variable. Combining eqs. (1), (2) and H = T - Vto
aB 0 0t '
(7)
where Bo is the applied magnetic field. From the condition V. B = 0 or by taking divergence of eq. (7), we find,
v.~ ( r - v,o) =0.
(8)
A suitable gauge in the method is T. u = 0 where u is an arbitrary vector field which does not have closed field lines. Eqs. (7) and (8) define governing equations for the T-~o method. Here it is possible to express Vto in terms of T under the Coulomb gauge V. T = 0 and the boundary condition n × T = 0. For this purpose the following Helmholtz theorem is convenient [17,18] T =
x V ' ± d V ' + f ( v ' . T) r ' 1 dV' R R
1 [f(v, xr)
- f(n x T) x V'IR dS'- f(n. T) V 'IR
as']. (9)
Introducing conditions, V ' . T = 0, n x T = 0, J ' = V' x T into eq. (9) and applying Biot-Savart's law, we obtain the following important result, 1 ~
where Js is a source current. Since displacement current can be negligible in the present problems, continuity condition of the eddy current, V. J = 0, is required to satisfy at any moment but this is the natural consequence of eq. (1). It is rewritten as follows,
219
1
n = T+ --~--~J T.17 -~ as',
(10)
where a prime in the above equations denotes differential operator with regard to a source and T. = n • T is a normal component of T at a boundary. Comparing eqs. (9) and (10), we find,
1 r
,1
17oJ= - - ~ J T n V ~ dS'.
(11)
Introducing eq. (11) into eq. (7), we find
a 17X--1 ~7XT+p,~-~T+ ~/ z f -rnV ,1 dS' = -aBo/at.
(12) Equation (12) is the governing equation in the Tmethod [18] whose figure of merit is to limit analysis region to a conductor similarly to the following direct method. It is easily to derive a governing equation in a 2-D form (shell model) from eq. (12) that is equivalent to the stream function method.
220
K. Miya et al. / Electromagnetic effects on first wall
Eddy current J is directly connected to the electrical field E and the magnetic vector potential A as follows,
3. Electromagnetomechanical behavior of fusion reactor components
~fJ'
Many publications are found in proceedings of COMPUMAG conferences [1] dealing with analysis of eddy current induced in electrical machines. Researchers on eddy current analysis of the fusion machines were performed mostly with use of computer codes developed recently and have been increasing in the number in these years in particular in terms of first wall technology in the fusion conceptual designs such as FER, NET and ITER as introduced in the present review paper. In order to introduce the state of the art of electromagnetic effects on fusion reactor components, the review will be made in the light of the following subjects, (1) measurement of eddy current contributing to
1 E = --J-E o-
A =
o,
4~JR
dV',
(13,14)
w h e r e E 0 is an external electrical field [19]. Combining eqs. (13), (14) and E = - 8 A / S t we find,
~ . f aj' 1
J + - ~ J -~t --R d V ' + o-Vcb = E 0,
Vc~,
(15)
where the continuity condition V. J = 0 and the boundary condition Jn = 0 should be supplemented on solving eq. (15). This method could be named the ./-method [191.
oo[ Outer saddle loop
4oo~ (c) Gap Plasma
Supporting plate
/ 0
Copper conducto[
inner saddleIOOp
I
2 5 4 Time (ms)
xlO4 75
(a)
50 25
LB ~ ,
~ o Analysis -Experiment
O0
U
25
~> -5.0 i -75 -I0.o -125 0.o
0.5
I 0
15
20
2.5
5.0
55
Distance from "A"
(d)
(b) Fig. 1. Eddy current measurement of model of ASDEX; (a) model of ASDEX, (b) flow pattern of current, (c) decay of plasma current, (d) comparison of experiment and analysis.
K. Miya et al. / Electromagnetic effects on first wall verification of the theory and numerical codes, (2) eddy current analysis of fusion reactor components, (3) coupling effect of electromagnetic field and elastoplastic one appended with magneto-elastic buckling of a conducting cylindrical shell by pulsed magnetic field. 3.1. Measurement o f eddy current There are few publications dealing with measurement of eddy current [20,21,22]. However, it is no doubt that numerical results should sometimes be verified favorably with experimental ones because the latter might not be accurate due to coarse geometrical and long time meshes. At present three methods are established, which are a tap probe, a solenoid type of probe and the split ring type of probe. Hashimoto et al. [21] have verified experimentally that the split ring type of probe is superior to the conventional solenoid type of probe especially when magnetic flux due to eddy current does not distribute uniformly as in the case of eddy current at a vacuum vessel corner. Fig. 1 shows a schematic picture of the ASDEX vacuum vessel [23]. From an electrical point of view the vessel is divided into two halves separated by gaps. The copper conductors of the inner and outer saddle loops as well as the plasma position are shown in fig. l(a). Evaluation of the magnetic forces caused by a plasma disruption was performed using a model (scale 1 : 10) of the vacuum vessel. Eddy currents were measured on this model, scaled to the ASDEX facility and used to calculate a load distribution suitable for the stress analysis. The measurement on the ASDEX model was performed in 1980 by Preis [24]. In the model the plasma ring is replaced by coils. The induced currents were measured by a split Rogowski coil. A calculation of the eddy current distribution on the ASDEX vacuum vessel has been performed with the computer program C A R I D D I [19]. As such half of the vessel is symmetric in respect to the equatorial plane and to the vertical middle plane, only a quarter of these structures needs to be modelled. Symmetry and periodicity conditions have been used to take into account the whole of the vessel. The time behavior of the plasma current used as input is shown in fig. l(c). Fig. l(b) illustrates schematically the distribution of the eddy current on the vessel. Particularly the "gap current" - the current that flows in poloidal direction near the gaps - is shown. In fig. l(d) the calculated and measured "gap current" at the time of maximum current are compared. The "gap current" is shown as function of the gap coordinate "s". The current flows on the vessel wall (Line A - B - C - D - E - F - G ) , on the inner saddle
221
loop (Line B - H ) and on the outer saddle loop (Line E-I). The continuous lines show the measured values, the dotted lines limit a field of + 10 kA in respect to the measured values. Finally the squares show the values calculated by CARIDDI. The calculated eddy current distribution agrees satisfactorily with the measured ones. From the above experiments we can have stout confidence as to validity of numerical results of eddy current followed by electromagnetic force evaluation. 3.2. Eddy current analysis o f fusion reactor components In the context of protection of magnetic fusion reactor systems against plasma disruption damage, numerical simulations of the electro-mechanical interactions on a first wall and in-vessel structural components have been performed. The effects of reference plasma disruption scenarios on the first wall structural integrity for next generation tokamaks such as NET and ITER have been investigated in detail. As to studies relating to the NET design of the double-null plasma configurations foreseen during the initial physics phase for the inboard first wall and during the final technology phase when a breeding blanket would be installed in the outboard region, it is demonstrated how the high segmentation of the internal structural components reduces the amplitude of the induced loads during electromagnetic transients [25,26]. The inboard/outboard regions were consequently split into several internal removable segments (from 32 to 48) allowing remote maintenance from the access ports of the 16 vessel sectors. At JRC Ispra, different attempts have been made to perform crossexaminations between computer tools based on different formulations and distinct modelling techniques (shell and solid approaches) and have permitted to compare their strengths and limits of applicability [2,25]. The major conclusion was that the shell models were very attractive due to their "flexibility" and "simplicity" of use but the real 3D effects encountered could only be reproduced accurately by a solid approach. The comparison was extended to the complex evaluation of the magnetic transient forces and torques acting on the first wall components. Good agreement was obtained for the force resultant and for the electrical time constant (ratio between the stored magnetic energy and the dissipated ohmic power). The large induced radial forces were mostly responsible for the generation of twisting deformation and overturning situation that were balanced only by appropriate me-
222
K Miya et al. / Electromagnetic effects on first wall
chanical boundary conditions. Uncoupled mechanical computations were performed in that context [26,27]. Complete 3D electromagnetic analysis of the inboard blanket segments of the ITER device has then been performed. The use of both the "three-current components" tools C A R I D D I and T R I F O U has demonstrated that the estimation of the eddy current distribution and the magnetic body force intensity in complex 3D sitiuations could be predicted accurately [28,29]. As far as the recent ITER outboard blanket design is concerned, the effects of the twin saddle loops (copper closed rings) have been investigated in detail. They produce an increase of the pressure intensity acting on the plasma-facing wall and of the resultant magnetic force (from 10 to 20 MN) which have to be transmitted to the blanket support system, mainly for the dynamic plasma disruption scenario with plasma vertical movement [29]. From KfK, complementary investigations of the electromechanical effects in the ceramic breeder outboard blanket designed for N E T / I T E R have been carried out, analyzing the additional saddle loop (lateral copper plates) effects [30]. More recently, improved models have been defined considering the inboard a n d / o r a first wall segments together with the vacuum vessel and the poloidal field coils. Detailed electromagnetic calculations including the mutual interactions between all the passive conductors have been performed [29,30]. This approach with refined meshes of the internal first wall components is complementary to previous approaches from the NET Team. In order to deal with the uncertainties due to the limited database presently available, several models of plasma disruption scenarios, parametrizing some characteristics have been applied [30-32]. Structural design proposals of locking systems needed for the next tokamak first wall components have been discussed. Both the understanding of the impact of the huge transient electromagnetic forces transmitted to the vessel from the fastening/guiding support systems [26] and the knowledge of the remote handing operations with related connection/disconnection of the internal component supports are involved. Among the various systems proposed as attaching locks, the "twin-belt" concept for the inboard blanket region, retained by the N E T / I T E R Teams as one of the options to be studied and developed in detail in the TESLA Laboratory, and is presented at this conference. In fig. 2 is shown the comparison of measurement and numerical prediction performed at the University
10
'
v
i'
gap
"
'
i
2"
0
.....
-5
I,.
1( 0A
10[)
B
200
C
Location X [ram] Fig. 2. Measurement of eddy current in a first wall module.
of Tokyo [22] for eddy current of the sectored first wall module with use of a split ring type of probe. We can seen a fairly good agreement. Bottura et al. [32] performed the eddy current analysis of the vacuum vessel of NET during a disruption in conjunction with calculation of Ohmic power and energy deposited in structures, induced forces and magnetic field changes in superconducting coils, using the 3D code " C A R I D D I " . One of the results that they obtained is shown in fig. 3 showing the effects of the timescale of the plasma disruption. The external voltage produced by the plasma scales with the inverse of the disruption time, as the initial plasma current is in all cases 22 MA. The effect of the inductance of the components becomes stronger as the current decay is
o
~ ~ ~ ~ '
I I ~ '
'
I
~ I [
~ i ~ ~ ~ ~ 1 ~ I '
It'.
.......
-CC =
H \ , A,
.......
-~c:
tt\
~ I '
~ [ ~-
22 ms 5ms
sel
0
V
0
~
I
I'~-~-~
1 oo Time
I
v t
~ I
T I
200
t
t
i
t
I
300
(ms)
Fig. 3. Eddy currents induced in the ITER twin loops and vacuum vessel.
K~ Miya et al. / Electromagnetic effects on first wall faster. The induced current tends to an asymptotic value when the decay time is much smaller than the time constant of the induced mode, the limit being the initial plasma current. Miya and coworkers investigated the change of eddy current distribution depending on different state of connections between three pieces constituting an upper half of a sectored first wall as shown in fig. 4. Seven cases were investigated but only the eddy current distributions at the time when it reaches a peak are shown in the figure for models 1-4. Distributions at other times are similar to the above results and thus omitted. As it is evident from the figure, the eddy current strongly depend on the connection indicating the possibility to control the eddy current. As far as numerical results computed here are concerned, lowest eddy currents are obtained for the models 4 and 6, in terms of peak value. Also, it can be said that the overturning torque is the largest for model 1 with two insulations and lowest for the models 4 and 6. Therefore, it seems that the models 4 and 6 are the best among various options of connection, at leasts from the viewpoint of structural design.
3.3. Coupling effect-magnetic damping Takagi et al. [33] performed an experiment with a shallow arch to investigate the dynamic behavior under
223
t
1
Iq
.Iw
I
Fig. 5. Experimental set-up of shallow arch; coil current (A), maximum displacement (mm). external transient coil current. An arch test piece is set with both ends clamped as shown in fig. 5. Table 1 shows the size of the aluminium test pieces. The deflection of the arch was measured by an optical noncontact displacement sensor. A pulse coil was made from 100-turn copper wires (1.1 mmO). A typical result showing the magnetic damping is given in fig. 6. The figure shows the relation between the coil current and the maximum deflection. Both numerical and experimental results might show a kind of snap-through buckling behavior. The critical current
[MA/m z] ,
lol
--y
1 mo~el 3
3y 50
100
B 250 [mm]
-5 --~-..... .... ×....
model model model model
I 2 3 4
mo~ed 4
-10
Fig. 4. Eddy current control by modification of connection between components.
224
K. Miya et al. / Electromagnetic effects on first wall
Table 1 Geometric description of tested arch
L (mm) W (mm) h (ram) R (mm) H (mm) /3 y
18 ¢
A2
100.0 20.0 0.2 500.0 2.51 5.74 25.0
141.4 20.0 0.2 5(10.0 5.03 8.13 50.0
i --.~-- experiment ! i • coupled onalysis -.-~---- uncoupled enolysis
~.-.~'.
15
A1
E
v12 g ~- 9
TBy: 1000 55"~-exp.(_ L ~
0
600
400
.
.
o= 0
ical
200
With ..... I
0
:
2 Ht2.
resuIls coupling
Wilhoul
t)~J
x
is defined so as to correspond to the maximum deflection of the arch. In the figure a scatter of experimental measurement by the optical non-contact displacement sensor is observed and large discrepancy is also noticed between the numerical results with the coupling effect and without the effect. The discrepancy becomes larger as the maximum displacement as easily expected from increase of penetrating magnetic flux. The validity of the analysis is assured by the observation that the numerical results with the coupling effect are within the scatter band of the experimental results. Takagi et al. [34,35] also performed a numerical analysis to investigate the coupling effect between eddy current and vibration of a conducting plate. In this study the thin plate is set under a moving magnetic field generated by four moving circular coils. Results showed that the magnetic damping coefficient is a function of the coil velocity and a negative damping effect yielding an attractive force appears at a certain velocity. This interesting fact suggests the possibility of electromagnetic force control in either positive and
/f
66
E 6 ~
Experiment al results
r - -
r
coupling
I
I
I
4
6
8
0
Maximum displacement (mm)
Fig. 6. Comparison between experiment and analysis with regard to magnetic damping.
k
o
[
I
Bx
,
i
I
2 3 magnetic field, Bx (T)
4
Fig. 7. Relation between Bx and maximum deflection.
negative direction and the necessity for accurate analysis of the plasma motion during plasma disruption. Miya et al. [36] performed an experiment similar to that of Hua et al. [11] to compare the numerical results with the experimental ones. The magnetomechanical behavior of a plate was analyzed under strong and static magnetic field along the plate [36]. In fig. 7 is shown a very interesting result: the maximum deflection reaches some peak at B x = 0.8 T and then decreases significantly with an increase of the applied magnetic field BX. This fact is very favoarable from the viewpoint of structural design because the magnetic field where thin structures are placed is as high as 5 T and elastic deformation can be expectedly constrained very much. The coupled behavior predicted by the numerical analysis should be desired to be verified by experiment for advanced technology design of fusion reactor components. The decrease can be explained by considering that the repulsive force between the exciting coil and the plate is proportional to the applied static field B 0 while the positive damping resisting deflection is proportional to B~ since the eddy current is almost constant regardless of deflection. Such a drastic constraint of deflection is very favorable from the viewpoint of structural design and thus this kind of analysis should be made systematically for more complicated structures in the near future. Nishio of JAERI performed a similar computation for a square plate (50 m m × 1 0 0 c m × l cm) under similar magnetic field situation. Results are shown in fig. 8 where the decrease of the maximum displacement is not observed. Therefore it can be said from both results that the decrease of the deflection dc-
225
K. Miya et al. / Electromagnetic effects on first wall lO
I00 ----D--
A
(Fixed-Fixed)
(Rounded-Rounded)
•
E E
Experimental buckling data - O - 10 kHz --q]-- 1.25kHz
',~
Iz i11
'",,~,.~
uJ
~'r
V
o<
~, 1o
~',,$
--.I I1.
Q
rg,,: a"
x
<
', _
lok.z
xp. 1.25kHz 2
4
6
8
Elast!c-~-,, plastic / buckling (static)
STATIC MAGNETIC FIELD ( Tesla )
Fig. 8. Static magnetic field (Tesla).
pends primarily on the amount of the eddy current distribution due to the size effect.
3. 4. Magneto-elasto-plastic buckling of cylindrical shell At present it is not clear whether or not structural instability of fusion reactor components is observed under impulsive electromagnetic force while it seems to depend on the shape and dimension of the components. However, it will be of use to investigate the buckling behavior of various types of components. Nemoto et al. [37] performed an experiment of buckling of a cylindrical shell applying impulsive electromagnetic force. The shell was surrounded by a solenoidal coil and an eddy current was induced in the shell resulting in the impulsive force due to repulsive force between the eddy current and the coil current. Figures 9 and 10 show the buckling loads experimentally obtained in the tests of aluminium alloy shells (A1070 and A5052). The buckling loads are shown as a function of the radius to thickness ratio ( a / h ) of the cylindrical shell, with the parameter of impulsive current frequency. They are compared to calculated resuits of elastic, plastic and elastic-plastic buckling for static loads. Elastic-plastic buckling analysis has been carried out. Buckling of A1070 aluminium shells occurs in the plastic range as shown in fig. 9. On the other hand, buckling of A5052 aluminium shells occurs in the elastic range. Experimental results are higher than estimated static buckling loads at high frequency. Exact analysis is required to be made in the future taking
0.1
' ~/~".k--"-
'/~). "",. '~,~,~
....,
' '' ~oo
a/h Fig. 9. Impulsive electromagnetic buckling characteristics (A1070-0). 1o
t00 ",,,,,,
[ A5052-H14 Oy=220MPa
"''
Experimental buckling data --0-- 1.25kHz
10
4
/
E.lastic- / " ~"/"',~x~ .],, C~.,.
/ |
buckling (static)
~i;'f/~ 2 ~ ~ ~d":(&
a; 10 loo a/h Fig. 10. Impulsive electromagnetic buckling characteristics (A5052-H14).
226
K. Miya et aL / Electromagnetic effects on first wall
into account rigorous formulation of electromagnctomechanical coupling.
4. Concluding remarks The state of the art concerning electromagnetic effects on fusion reactor components is summarized as follows: (1) C o m p u t e r tools based on the shell approach and complete 3D formulation are available for eddy current analysis of various types of components. The validity of the tools is verified fully by experiment. (2) Significant coupling effects between electromagnetic forces and mechanical displacements are observed and should be taken into consideration in the structural design of components in anticipation of positive magnetic damping especially because fusion devices are placed under very strong magnetic fields. In order to utilize the favorable effects more advanced numerical tools should be developed in the near future. (3) The final goal is to develop control techniques for electromagnetomechanical p h e n o m e n a and thereby to establish an optimal design methodology.
Nomenclature B B0 H E E0 J Tc A T 4' p~ ~o n R
magnetic flux density, applied magnetic flux density, Magnetic field, electrical field, applied electrical field, current density, critical temperature of superconductor, magnetic vector potential, current vector potential, electrical scalar potential, electrical conductivity, magnetic permeability, magnetic scalar potential, normal unit vector on a boundary, distance between source and field points.
References [1] IEEE Transactions on Magnetics, Vol. 26, No. 2 (COMPUMAG-Tokyo), (1990).
[2] L. Turner, Electromagnetic computations for fusion devices, ibid. [3] L. Bottura, A. Astapkovich, S. Chiocchio, E. Coccorese, Y. Crutzen, Y. Gribov, J. Leuer, S. Nishio, A. Portone, S. Sadaoov, R. Sayer, and J. Wesley, Transient electromagnetics in ITER, IAEA-CN-53/F-3-10 Washington (USA), (1990) [4] F.C. Moon and Y.W. Pao, Magnetoelastic buckling of a thin plate, ASME J. Appl, Mech. 35 (1968) 53-58. [5] K. Miya, K. Hara and K. Someya, Experimental and theoretical study on magnetoelastic buckling of a ferromagnetic cantilevered beam-plate, ASME J. Appl. Mech. 45 (1978) 355-360. [6] K. Miya, T. Takagi and Y. Ando, Finite element analysis of magnetoelastic buckling of ferromagnetic beam plate, ASME J. Appl. Mech. 47 (1980) 377-382. [7] F.C. Moon, Experiments on magnetic buckling in superconducting toms, ASME J. Appl. Mech. 46 (1979) 145150. [8] K. Miya, T. Takagi and Y. Ando, Finite element analysis of magnetoelastic buckling of eight coil superconducting full torus, ASME J. App. Mech. 49 (1982) 180-185. [9] K. Miya, M. Uesaka and F.C. Moon, Finite element analysis of vibration of toroidal field coils coupled with laplace transform, ASME J. Appl. Mech. 49 (1982) 594600. [10] K. Miya and M. Uesaka, An application of finite element method to magnetomechanics of superconducting magnets for magnetic fusion reactors, Nucl. Engrg. Des. 72 (1982) 540-543. [ll] T. Hua, M.J. Knott, L.R. Turner, and R.B. Wehrle, Experimental modelling of eddy currents and deflection for tokamak limiters, Fusion Technology, 10 (1986) 10471052. [12] J.S. Lee, J.H. Prevost, and P.C.Y, Lee, Finite element analysis of magnetically induced vibrations of conductive plates, Fusion Eng. Design, 13 (1990) 125-141. [13] T. Morisue, Analysis of a coupled problem; the FELIX Cantilevered beam, IEEE Trans. Magnetics, 26 (1990) 540-543. [14] H. Hashizume, T. Sugiura, K. Miya, Y. Ando and S. Akita, Numerical analysis of A.C. loss in superconductor, Cryogenics 31 (1991) 601-606. [15] T. Sugiura, H. Hashizume and K. Miya, Numerical electromagnetic field analysis of Type-II superconductors, Proc. of Int. Symp. Applied Electromagnetics, Sendai. (Elsevier, 1991), p. 76. [16] C.W. Trowbridge, Overview of the 'State of the Art" in electromagnetic analysis and design, in: Industrial Application of Electromagnetic Computer Codes, edited by Y.R. Crutzen, G. Molinari and G. Rubinacci (Kluwer Academic Publishers, 1990), pp. 1-27. [17] P.M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw-Hill, N.Y., 1953). [18] K. Miya, T. Sugiura and H. Hashizume, Three dimensional analysis of an eddy current by the T-method, in: Electromagnetomechanical Interactions in Deformable
K. Miya et al. / Electromagnetic effects on first wall
[19]
[20]
[21]
[22]
[23] [24]
[25]
[26]
[27]
[28]
Solids and Structures, edited by Y. Yamamoto and K. Miya (North-Holland, 1987), pp. 183-189. R. Albanese and G. Rubinacci, Integral formulation for 3D eddy current computation using edge elements, IEE Proc. A, 135 (1988) 457-462. T. Takahashi, Numerical and experimental analysis of eddy currents induced in tokamak machine, Proc. of 7th Symp. Eng. Problems of Fusion Research, (1977), pp. 1393-1397. M. Hashimoto, H. Hashizume, T. Sugiura, T. Takagi and K. Miya, Experiments and numerical analysis of eddy currents in the first wall of a fusion reactor, IEEE Trans. Magnetics 26 (1990) 896-872. M. Tsuchimoto, M. Hashimoto, M. Fukaya, K. Miya and M. Seki, Numerical and experimental eddy current analysis of complex first wall components of a fusion reactor, ISFN-2, Karlsruhe, June 1991, Fusion Engrg. Des. 16 (1991) 369-375, in these Proceedings, Part A. The ASDEX Group, Divertor Tokamak ASDEX, IPP Report III/27 (1976) H. Preis and F. Schneider, Bestimmung yon Wirbelstr6men in Vakuumgefziss einer Apparatur zur Erforschung der kontrollierten Kernfusion, Archiv fiir Electrotechnik 62 (1980) 181-186. Y. Crutzen, S. Rigo and E. Castilla, Evaluation of the electromagnetic consequences of a plasma disruption on the magnetic fusion device in-vessel system, IEEE Trans. Magnetics 26 (1990) 857-869. Y. Crutzen and F. Farfaletti-Cassli, Structural design problems related to plasma disruption events of the removable NET first wall segments, Fusion Eng. Design 9 (1989) 193-199. Y. Crutzen and G. Rubinacci, Evaluation of the electromagnetic effects on a tokamak first wall caused by a plasma disruption using a thin shell formulation, Fusion Eng. Design 11 (1989) 293-303. P. Chaussecourte, A. Bossavit, J.-C. Verit6 and Y. Crutzen, Electromagnetic transients induced in the tokamak first wall by a plasma disruption using TRIFOU, IEEE Trans. Magnetics 26 (1990) 861-864.
227
[29] S. Chiocchio, Y. Crutzen, E. Castillo and J. Wu, Eddy currents, electromagnetic forces and related mechanical effects in the NET plasma-facing components, Proc. SOFT-16 1990 (Elsevier, Amsterdam, 1991). [30] L.V. Boccanici and M. Dalle Donne, Electromagnetic forces and stresses in ceramic breeder blankets caused by disruptions, Proc. SOFT-16 1990 (Elsevier, Amsterdam, 1991). [31] Y. Crutzen, S. Papadopoulos, N. Richard, N. Siakavellas and J. Wu, Recent developments in 3D transient magneto-structural integrated analysis, COMPUMAG-Sorrento (1991) pp. 235-238. [32] L. Bottura and A. Fleischer, Vacuum vessel eddy current analysis during a disruption, Proc. of SOFT-16, 1990 (Elsevier, Amsterdam, 1991). [33] T. Takagi, Y. Mohri and J. Tani, Dynamic behaviour of shallow arch under electromagnetic force considering coupling effect, Proc. of the 2nd Joint Seminar between Japan and Poland, Oita, Japan, Jan., 1991, pp. 85-88, to be published in Int. Journal of Applied Electromagnetic in Materials (Elsevier, 1991). [34] T. Takagi, J. Tani and S. Matsuda, Dynamic behavior of a thin plate under a moving magnetic field, J. Applied Electromagnetics in Materials, Vol. 1 (Nos. 2-4) (1990) 147-154. [35] T. Takagi, S. Matsuda, J. Tani and M. Saigo, Dynamic behavior analysis of a thin plate under moving magnetic field considering a coupling between magnetic field and vibration, Proc. of Int. Symp. Applied Electromagnetics, Sendai (Elsevier, 1991), p. 159. [36] K. Miya, H. Hashizume and M. Arita, Experimental and theoretical study on magnetomechanical coupled behavior of first wall for a fusion reactor, to appear in Int. J. Applied Electromagnetics in Materials (1991). [37] M. Nemoto, Y. Kawamoto, K. Inoue, K. Ioki, M. Hashimoto, M. Tsuchimoto and K. Miya, Study on impulsive electromagnetic buckling of cylindrical shell, Proc. of Int. Symp. Applied Electromagnetics, Sendai (Elsevier, 1991).
217
Electromagnetic effects on first wall and structural components K. Miya a, Y.R. C r u t z e n b, T. Takagi c, L.V. Boccacini d a n d L. B o t t u r a
e
a The University of Tokyo, Japan b ISEI Institute, SER Division JRC, ISPRA, CEC, Italy c Institute of Fluid Science, Tohoku University, Sendai, Japan d Kernforschungszentrum Karlsruhe, Karlsruhe, Germany e NET Team, Max Plank Institute, Garching, Germany
In a tokamak-type nuclear fusion reactor, the plasma disruption instabilities and the large magnetic fields generate electromagnetic-type phenomena in transient conditions which lead to strong induced eddy currents and large body loads that must be correctly accommodated in the engineering design of the first wall and structural components. During the last few years, many efforts at the TESLA Laboratory of the JRC-Ispra and at other Institutions have been actively made in terms of the comprehension of the real impact of the accidental events of plasma disruption on the mechanical design of the internal device components such as the multi-segmented first wall and their supports. Fully three-dimensional transient dynamic analyses have been performed due to the geometry complexity of the "conducting" and "plasma" regions both from the electrical and mechanical standpoints. Actually, there is a great interest to extend the computational strategies in magneto-structural analysis in order to investigate the impact of the huge electromagnetic loads on the integrity of the fusion structural system from the view point of magnetic damping.
1. Introduction
Components and devices in a magnetic confinement fusion reactor are operated under very strong magnetic field environment. This requires fusion reactor designers not only to understand qualitatively but also to evaluate quantitatively electromagnetic effects on fusion reactor components. For the purpose one has to deal with two main subjects, eddy current analysis based on electromagnetism and stress analysis based on elasticity and plasticity. Furthermore electromagnetic coupling effects between not only plasmas and conducting components near the plasmas but also electromagnetic field and mechanical deformation should be preferably taken into account. As to the level of research in both fields it is no doubt that the theory of elastoplasticity has already been established and large computer codes are now commercially available for
stress analysis of fusion reactor components, but the theory of eddy current has been progressed only in these years being assisted with numerical verification by computer codes. This is evident from proceedings of international conferences on computational electromagnetics of which typical one is a series of COMPUMAG [1] where eddy current analysis is a major subject and various ideas and innovative computational techniques were presented so far. As far as studies on the electromagnetic effects on fusion engineering design are concerned, several modelling efforts have been performed at several institutions in the world in order to account for and to correctly simulate the following situations [2,3]: - the plasma disruptive instability scenarios characterized by fast quench of a large current and coupled with a rapid plasma movement away from its equilibrium position;
0 9 2 0 - 3 7 9 6 / 9 1 / $ 0 3 . 5 0 © 1991 - Elsevier Science Publishers B.V. All rights reserved
218
K. Miya et al. / Electromagnetic effects on first wall
- the complex magnetic field distribution in devices in order to drive and to confine the plasma column inside a vacuum vessel; - the 3D in-vessel multi-segmented conducting bodies with narrow gaps between them allowing remote maintenance; - the inductive coupling between first wall segments, the vacuum vessel and coils including the complex electromagnetic interactions between the "electrically conducting" regions and the "plasma source" part; - the effects of the electrical breaks/insulations which modify the circulation of eddy currents and of the additional saddle loops incorporated for improving the plasma position control. Furthermore, from the viewpoint of magnetomechanical design of fusion reactor components it is desired to establish design methodology based on numerical evaluation of magneto-mechanical coupling and thus the following items should be added for future studies: - code development for analysis of coupling effect between electromagnetic field and elastoplastic deformation of components. - code development for control technique of eddy current and thereby electromagnetic force by optimizing design configuration of components and proper selection of material constants. With regard to the magnetic stiffness observed in various electromagnetic structures, researches have already been made by many investigators in the light of magnetoelastic buckling of a ferromagnetic beam plate first in the decade of 1970 [4,5,6] and then magnetoelastic buckling and vibration of superconducting toroidal coils nested in a full or partial torus for a fusion nuclear reactor in the early 1980's [7,8,9,10]. However, these studies dealt with magnetomechanics of bodies placed in a static magnetic field, thus have nothing to do with eddy current. Fundamental studies on magnetic damping have been made recently [11,12,13] and the approach should be applied to electromagnetomechanical design of fusion reactor components for accurate design in the theoretical and numerical ways despite of very few examples of its application to a fusion reactor conceptual design. Here it is instructive to note that magnets-mechanical coupling can be divided into such three aspects as static magnetic stiffness, dynamic magnetic stiffness and magnetic damping. The first two of them are caused by magnetic energy change due to static and dynamic deformation of current carrying elastic bodies, respectively and the last by velocity induced voltage. The static magnetic
stiffness is observed in deformation of superconducting toroidal coils, the dynamic magnetic stiffness is due to interaction between transient applied field and deformation of any structural components through eddy current, and the magnetic damping is ascribed to eddy current caused by vibration of the body in static magnetic field. On the other hand, designers of fusion reactor components seem to have not shown keen interest in the active control of eddy current and electromagnetic force. It could be predicted that the control technique will be developed significantly and effectively through progress of design technology in the near future. One can point out one example that an insert of insulation between conducting plates causes generally concentration of eddy current near it leading to localized large electromagnetic force and increasing possibility of arcing between components, and contrarily an insert of conducting connections between them could reduce peak values of eddy current, which would be more favorable from the viewpoint of structural design. In this sense electromagnetic force is dependent on structural design as well as material constants suggesting potential capability of eddy current control. The governing equations of eddy current receive a small modification when an effect of moving body in a static and dynamic magnetic fields should be included in the system. The coupling effect has been proved, in particular, important from the viewpoint of magnetomechanical design of fusion reactor components because it creates, in many cases, positive magnetic stiffness in the dynamical system, namely the magnetic damping constrains motion of a conducting body as a result of coupling between eddy current, magnetic field and elastoplastic deformation [10].
2. Governing equations for eddy current
Eddy current phenomenon is governed by two of Maxwell's equations that are Ampere's law and Faraday's induction law supplemented with constitutive equations between B and H, and E and J taking different constants depending on physical properties of materials concerned, i.e. non-magnetic, magnetic and superconducting material. Recently eddy current induced in low and high T. superconductors was successfully computed by constructing a new constitutive equation between electrical field E and eddy current J and validity of the new method was verified experimentally in terms of A.C. loss evaluation in a low Tc superconductor [14] and
K~Miya et al. / Electromagneticeffects on first wall evaluation of electromagnetic force generated by a high Tc superconductor [15]. This numerical approach could be applied to more accurate evaluation of A.C. loss and electromagnetic force in poloidal and toroidal coils of a fusion reactor in future. For solutions of eddy current based on Maxwell's equations there are two methods i.e. indirect and direct methods where field variables such as magnetic vector and scalar potentials (A, ~b), current vector potential (T) in the former case and magnetic field ( H ) and eddy current ( J ) in the latter case are adopted. In what to follow essential parts of the theory will be introduced in the light of A-~b method, T-to method, T-method and J - m e t h o d focussing on derivation of the governing equation and starting from the following Maxwell's equations and constitutive ones, I7 X H = J ,
17 x E = - OB/Ot,
(1,2)
J = o-E,
B =/~H,
(3,4)
where notations are conventional. The magnetic vector potential A defined as B = V X A is a fundamental field variable in the A - ¢ method. From eq. (2) and B = V X A we find a relation E = - a A / ~ t - V¢ and then the following governing equation from eq. (1),
1
V X - - VxA+o/z
(o
t)
+V~b =Js,
(5)
we find the governing equation for the T-to method as follows, 1 0 VX-o. V X T + / z - ~ ( T - V t o ) = -
V.tr(OA/at + V&) = O.
(6)
Equations (5) and (6) are the governing equation in the A - & method while one should notice that eq. (6) is derived directly from eq. (5) by taking rotation of eq. (5) and in this sense eq. (6) is not independent. An inclusion of eq. (6) is related to an introduction of a gage as a constraint to the vector potential A. Two kinds of gage, Coulomb gauge and Lorentz gauge, are common. The readers refer to ref. [16] as to discussion about the meaning and usage of the gage. In the similar way to the above one can use the current vector potential defined as J = V x T as the field variable. Combining eqs. (1), (2) and H = T - Vto
aB 0 0t '
(7)
where Bo is the applied magnetic field. From the condition V. B = 0 or by taking divergence of eq. (7), we find,
v.~ ( r - v,o) =0.
(8)
A suitable gauge in the method is T. u = 0 where u is an arbitrary vector field which does not have closed field lines. Eqs. (7) and (8) define governing equations for the T-~o method. Here it is possible to express Vto in terms of T under the Coulomb gauge V. T = 0 and the boundary condition n × T = 0. For this purpose the following Helmholtz theorem is convenient [17,18] T =
x V ' ± d V ' + f ( v ' . T) r ' 1 dV' R R
1 [f(v, xr)
- f(n x T) x V'IR dS'- f(n. T) V 'IR
as']. (9)
Introducing conditions, V ' . T = 0, n x T = 0, J ' = V' x T into eq. (9) and applying Biot-Savart's law, we obtain the following important result, 1 ~
where Js is a source current. Since displacement current can be negligible in the present problems, continuity condition of the eddy current, V. J = 0, is required to satisfy at any moment but this is the natural consequence of eq. (1). It is rewritten as follows,
219
1
n = T+ --~--~J T.17 -~ as',
(10)
where a prime in the above equations denotes differential operator with regard to a source and T. = n • T is a normal component of T at a boundary. Comparing eqs. (9) and (10), we find,
1 r
,1
17oJ= - - ~ J T n V ~ dS'.
(11)
Introducing eq. (11) into eq. (7), we find
a 17X--1 ~7XT+p,~-~T+ ~/ z f -rnV ,1 dS' = -aBo/at.
(12) Equation (12) is the governing equation in the Tmethod [18] whose figure of merit is to limit analysis region to a conductor similarly to the following direct method. It is easily to derive a governing equation in a 2-D form (shell model) from eq. (12) that is equivalent to the stream function method.
220
K. Miya et al. / Electromagnetic effects on first wall
Eddy current J is directly connected to the electrical field E and the magnetic vector potential A as follows,
3. Electromagnetomechanical behavior of fusion reactor components
~fJ'
Many publications are found in proceedings of COMPUMAG conferences [1] dealing with analysis of eddy current induced in electrical machines. Researchers on eddy current analysis of the fusion machines were performed mostly with use of computer codes developed recently and have been increasing in the number in these years in particular in terms of first wall technology in the fusion conceptual designs such as FER, NET and ITER as introduced in the present review paper. In order to introduce the state of the art of electromagnetic effects on fusion reactor components, the review will be made in the light of the following subjects, (1) measurement of eddy current contributing to
1 E = --J-E o-
A =
o,
4~JR
dV',
(13,14)
w h e r e E 0 is an external electrical field [19]. Combining eqs. (13), (14) and E = - 8 A / S t we find,
~ . f aj' 1
J + - ~ J -~t --R d V ' + o-Vcb = E 0,
Vc~,
(15)
where the continuity condition V. J = 0 and the boundary condition Jn = 0 should be supplemented on solving eq. (15). This method could be named the ./-method [191.
oo[ Outer saddle loop
4oo~ (c) Gap Plasma
Supporting plate
/ 0
Copper conducto[
inner saddleIOOp
I
2 5 4 Time (ms)
xlO4 75
(a)
50 25
LB ~ ,
~ o Analysis -Experiment
O0
U
25
~> -5.0 i -75 -I0.o -125 0.o
0.5
I 0
15
20
2.5
5.0
55
Distance from "A"
(d)
(b) Fig. 1. Eddy current measurement of model of ASDEX; (a) model of ASDEX, (b) flow pattern of current, (c) decay of plasma current, (d) comparison of experiment and analysis.
K. Miya et al. / Electromagnetic effects on first wall verification of the theory and numerical codes, (2) eddy current analysis of fusion reactor components, (3) coupling effect of electromagnetic field and elastoplastic one appended with magneto-elastic buckling of a conducting cylindrical shell by pulsed magnetic field. 3.1. Measurement o f eddy current There are few publications dealing with measurement of eddy current [20,21,22]. However, it is no doubt that numerical results should sometimes be verified favorably with experimental ones because the latter might not be accurate due to coarse geometrical and long time meshes. At present three methods are established, which are a tap probe, a solenoid type of probe and the split ring type of probe. Hashimoto et al. [21] have verified experimentally that the split ring type of probe is superior to the conventional solenoid type of probe especially when magnetic flux due to eddy current does not distribute uniformly as in the case of eddy current at a vacuum vessel corner. Fig. 1 shows a schematic picture of the ASDEX vacuum vessel [23]. From an electrical point of view the vessel is divided into two halves separated by gaps. The copper conductors of the inner and outer saddle loops as well as the plasma position are shown in fig. l(a). Evaluation of the magnetic forces caused by a plasma disruption was performed using a model (scale 1 : 10) of the vacuum vessel. Eddy currents were measured on this model, scaled to the ASDEX facility and used to calculate a load distribution suitable for the stress analysis. The measurement on the ASDEX model was performed in 1980 by Preis [24]. In the model the plasma ring is replaced by coils. The induced currents were measured by a split Rogowski coil. A calculation of the eddy current distribution on the ASDEX vacuum vessel has been performed with the computer program C A R I D D I [19]. As such half of the vessel is symmetric in respect to the equatorial plane and to the vertical middle plane, only a quarter of these structures needs to be modelled. Symmetry and periodicity conditions have been used to take into account the whole of the vessel. The time behavior of the plasma current used as input is shown in fig. l(c). Fig. l(b) illustrates schematically the distribution of the eddy current on the vessel. Particularly the "gap current" - the current that flows in poloidal direction near the gaps - is shown. In fig. l(d) the calculated and measured "gap current" at the time of maximum current are compared. The "gap current" is shown as function of the gap coordinate "s". The current flows on the vessel wall (Line A - B - C - D - E - F - G ) , on the inner saddle
221
loop (Line B - H ) and on the outer saddle loop (Line E-I). The continuous lines show the measured values, the dotted lines limit a field of + 10 kA in respect to the measured values. Finally the squares show the values calculated by CARIDDI. The calculated eddy current distribution agrees satisfactorily with the measured ones. From the above experiments we can have stout confidence as to validity of numerical results of eddy current followed by electromagnetic force evaluation. 3.2. Eddy current analysis o f fusion reactor components In the context of protection of magnetic fusion reactor systems against plasma disruption damage, numerical simulations of the electro-mechanical interactions on a first wall and in-vessel structural components have been performed. The effects of reference plasma disruption scenarios on the first wall structural integrity for next generation tokamaks such as NET and ITER have been investigated in detail. As to studies relating to the NET design of the double-null plasma configurations foreseen during the initial physics phase for the inboard first wall and during the final technology phase when a breeding blanket would be installed in the outboard region, it is demonstrated how the high segmentation of the internal structural components reduces the amplitude of the induced loads during electromagnetic transients [25,26]. The inboard/outboard regions were consequently split into several internal removable segments (from 32 to 48) allowing remote maintenance from the access ports of the 16 vessel sectors. At JRC Ispra, different attempts have been made to perform crossexaminations between computer tools based on different formulations and distinct modelling techniques (shell and solid approaches) and have permitted to compare their strengths and limits of applicability [2,25]. The major conclusion was that the shell models were very attractive due to their "flexibility" and "simplicity" of use but the real 3D effects encountered could only be reproduced accurately by a solid approach. The comparison was extended to the complex evaluation of the magnetic transient forces and torques acting on the first wall components. Good agreement was obtained for the force resultant and for the electrical time constant (ratio between the stored magnetic energy and the dissipated ohmic power). The large induced radial forces were mostly responsible for the generation of twisting deformation and overturning situation that were balanced only by appropriate me-
222
K Miya et al. / Electromagnetic effects on first wall
chanical boundary conditions. Uncoupled mechanical computations were performed in that context [26,27]. Complete 3D electromagnetic analysis of the inboard blanket segments of the ITER device has then been performed. The use of both the "three-current components" tools C A R I D D I and T R I F O U has demonstrated that the estimation of the eddy current distribution and the magnetic body force intensity in complex 3D sitiuations could be predicted accurately [28,29]. As far as the recent ITER outboard blanket design is concerned, the effects of the twin saddle loops (copper closed rings) have been investigated in detail. They produce an increase of the pressure intensity acting on the plasma-facing wall and of the resultant magnetic force (from 10 to 20 MN) which have to be transmitted to the blanket support system, mainly for the dynamic plasma disruption scenario with plasma vertical movement [29]. From KfK, complementary investigations of the electromechanical effects in the ceramic breeder outboard blanket designed for N E T / I T E R have been carried out, analyzing the additional saddle loop (lateral copper plates) effects [30]. More recently, improved models have been defined considering the inboard a n d / o r a first wall segments together with the vacuum vessel and the poloidal field coils. Detailed electromagnetic calculations including the mutual interactions between all the passive conductors have been performed [29,30]. This approach with refined meshes of the internal first wall components is complementary to previous approaches from the NET Team. In order to deal with the uncertainties due to the limited database presently available, several models of plasma disruption scenarios, parametrizing some characteristics have been applied [30-32]. Structural design proposals of locking systems needed for the next tokamak first wall components have been discussed. Both the understanding of the impact of the huge transient electromagnetic forces transmitted to the vessel from the fastening/guiding support systems [26] and the knowledge of the remote handing operations with related connection/disconnection of the internal component supports are involved. Among the various systems proposed as attaching locks, the "twin-belt" concept for the inboard blanket region, retained by the N E T / I T E R Teams as one of the options to be studied and developed in detail in the TESLA Laboratory, and is presented at this conference. In fig. 2 is shown the comparison of measurement and numerical prediction performed at the University
10
'
v
i'
gap
"
'
i
2"
0
.....
-5
I,.
1( 0A
10[)
B
200
C
Location X [ram] Fig. 2. Measurement of eddy current in a first wall module.
of Tokyo [22] for eddy current of the sectored first wall module with use of a split ring type of probe. We can seen a fairly good agreement. Bottura et al. [32] performed the eddy current analysis of the vacuum vessel of NET during a disruption in conjunction with calculation of Ohmic power and energy deposited in structures, induced forces and magnetic field changes in superconducting coils, using the 3D code " C A R I D D I " . One of the results that they obtained is shown in fig. 3 showing the effects of the timescale of the plasma disruption. The external voltage produced by the plasma scales with the inverse of the disruption time, as the initial plasma current is in all cases 22 MA. The effect of the inductance of the components becomes stronger as the current decay is
o
~ ~ ~ ~ '
I I ~ '
'
I
~ I [
~ i ~ ~ ~ ~ 1 ~ I '
It'.
.......
-CC =
H \ , A,
.......
-~c:
tt\
~ I '
~ [ ~-
22 ms 5ms
sel
0
V
0
~
I
I'~-~-~
1 oo Time
I
v t
~ I
T I
200
t
t
i
t
I
300
(ms)
Fig. 3. Eddy currents induced in the ITER twin loops and vacuum vessel.
K~ Miya et al. / Electromagnetic effects on first wall faster. The induced current tends to an asymptotic value when the decay time is much smaller than the time constant of the induced mode, the limit being the initial plasma current. Miya and coworkers investigated the change of eddy current distribution depending on different state of connections between three pieces constituting an upper half of a sectored first wall as shown in fig. 4. Seven cases were investigated but only the eddy current distributions at the time when it reaches a peak are shown in the figure for models 1-4. Distributions at other times are similar to the above results and thus omitted. As it is evident from the figure, the eddy current strongly depend on the connection indicating the possibility to control the eddy current. As far as numerical results computed here are concerned, lowest eddy currents are obtained for the models 4 and 6, in terms of peak value. Also, it can be said that the overturning torque is the largest for model 1 with two insulations and lowest for the models 4 and 6. Therefore, it seems that the models 4 and 6 are the best among various options of connection, at leasts from the viewpoint of structural design.
3.3. Coupling effect-magnetic damping Takagi et al. [33] performed an experiment with a shallow arch to investigate the dynamic behavior under
223
t
1
Iq
.Iw
I
Fig. 5. Experimental set-up of shallow arch; coil current (A), maximum displacement (mm). external transient coil current. An arch test piece is set with both ends clamped as shown in fig. 5. Table 1 shows the size of the aluminium test pieces. The deflection of the arch was measured by an optical noncontact displacement sensor. A pulse coil was made from 100-turn copper wires (1.1 mmO). A typical result showing the magnetic damping is given in fig. 6. The figure shows the relation between the coil current and the maximum deflection. Both numerical and experimental results might show a kind of snap-through buckling behavior. The critical current
[MA/m z] ,
lol
--y
1 mo~el 3
3y 50
100
B 250 [mm]
-5 --~-..... .... ×....
model model model model
I 2 3 4
mo~ed 4
-10
Fig. 4. Eddy current control by modification of connection between components.
224
K. Miya et al. / Electromagnetic effects on first wall
Table 1 Geometric description of tested arch
L (mm) W (mm) h (ram) R (mm) H (mm) /3 y
18 ¢
A2
100.0 20.0 0.2 500.0 2.51 5.74 25.0
141.4 20.0 0.2 5(10.0 5.03 8.13 50.0
i --.~-- experiment ! i • coupled onalysis -.-~---- uncoupled enolysis
~.-.~'.
15
A1
E
v12 g ~- 9
TBy: 1000 55"~-exp.(_ L ~
0
600
400
.
.
o= 0
ical
200
With ..... I
0
:
2 Ht2.
resuIls coupling
Wilhoul
t)~J
x
is defined so as to correspond to the maximum deflection of the arch. In the figure a scatter of experimental measurement by the optical non-contact displacement sensor is observed and large discrepancy is also noticed between the numerical results with the coupling effect and without the effect. The discrepancy becomes larger as the maximum displacement as easily expected from increase of penetrating magnetic flux. The validity of the analysis is assured by the observation that the numerical results with the coupling effect are within the scatter band of the experimental results. Takagi et al. [34,35] also performed a numerical analysis to investigate the coupling effect between eddy current and vibration of a conducting plate. In this study the thin plate is set under a moving magnetic field generated by four moving circular coils. Results showed that the magnetic damping coefficient is a function of the coil velocity and a negative damping effect yielding an attractive force appears at a certain velocity. This interesting fact suggests the possibility of electromagnetic force control in either positive and
/f
66
E 6 ~
Experiment al results
r - -
r
coupling
I
I
I
4
6
8
0
Maximum displacement (mm)
Fig. 6. Comparison between experiment and analysis with regard to magnetic damping.
k
o
[
I
Bx
,
i
I
2 3 magnetic field, Bx (T)
4
Fig. 7. Relation between Bx and maximum deflection.
negative direction and the necessity for accurate analysis of the plasma motion during plasma disruption. Miya et al. [36] performed an experiment similar to that of Hua et al. [11] to compare the numerical results with the experimental ones. The magnetomechanical behavior of a plate was analyzed under strong and static magnetic field along the plate [36]. In fig. 7 is shown a very interesting result: the maximum deflection reaches some peak at B x = 0.8 T and then decreases significantly with an increase of the applied magnetic field BX. This fact is very favoarable from the viewpoint of structural design because the magnetic field where thin structures are placed is as high as 5 T and elastic deformation can be expectedly constrained very much. The coupled behavior predicted by the numerical analysis should be desired to be verified by experiment for advanced technology design of fusion reactor components. The decrease can be explained by considering that the repulsive force between the exciting coil and the plate is proportional to the applied static field B 0 while the positive damping resisting deflection is proportional to B~ since the eddy current is almost constant regardless of deflection. Such a drastic constraint of deflection is very favorable from the viewpoint of structural design and thus this kind of analysis should be made systematically for more complicated structures in the near future. Nishio of JAERI performed a similar computation for a square plate (50 m m × 1 0 0 c m × l cm) under similar magnetic field situation. Results are shown in fig. 8 where the decrease of the maximum displacement is not observed. Therefore it can be said from both results that the decrease of the deflection dc-
225
K. Miya et al. / Electromagnetic effects on first wall lO
I00 ----D--
A
(Fixed-Fixed)
(Rounded-Rounded)
•
E E
Experimental buckling data - O - 10 kHz --q]-- 1.25kHz
',~
Iz i11
'",,~,.~
uJ
~'r
V
o<
~, 1o
~',,$
--.I I1.
Q
rg,,: a"
x
<
', _
lok.z
xp. 1.25kHz 2
4
6
8
Elast!c-~-,, plastic / buckling (static)
STATIC MAGNETIC FIELD ( Tesla )
Fig. 8. Static magnetic field (Tesla).
pends primarily on the amount of the eddy current distribution due to the size effect.
3. 4. Magneto-elasto-plastic buckling of cylindrical shell At present it is not clear whether or not structural instability of fusion reactor components is observed under impulsive electromagnetic force while it seems to depend on the shape and dimension of the components. However, it will be of use to investigate the buckling behavior of various types of components. Nemoto et al. [37] performed an experiment of buckling of a cylindrical shell applying impulsive electromagnetic force. The shell was surrounded by a solenoidal coil and an eddy current was induced in the shell resulting in the impulsive force due to repulsive force between the eddy current and the coil current. Figures 9 and 10 show the buckling loads experimentally obtained in the tests of aluminium alloy shells (A1070 and A5052). The buckling loads are shown as a function of the radius to thickness ratio ( a / h ) of the cylindrical shell, with the parameter of impulsive current frequency. They are compared to calculated resuits of elastic, plastic and elastic-plastic buckling for static loads. Elastic-plastic buckling analysis has been carried out. Buckling of A1070 aluminium shells occurs in the plastic range as shown in fig. 9. On the other hand, buckling of A5052 aluminium shells occurs in the elastic range. Experimental results are higher than estimated static buckling loads at high frequency. Exact analysis is required to be made in the future taking
0.1
' ~/~".k--"-
'/~). "",. '~,~,~
....,
' '' ~oo
a/h Fig. 9. Impulsive electromagnetic buckling characteristics (A1070-0). 1o
t00 ",,,,,,
[ A5052-H14 Oy=220MPa
"''
Experimental buckling data --0-- 1.25kHz
10
4
/
E.lastic- / " ~"/"',~x~ .],, C~.,.
/ |
buckling (static)
~i;'f/~ 2 ~ ~ ~d":(&
a; 10 loo a/h Fig. 10. Impulsive electromagnetic buckling characteristics (A5052-H14).
226
K. Miya et aL / Electromagnetic effects on first wall
into account rigorous formulation of electromagnctomechanical coupling.
4. Concluding remarks The state of the art concerning electromagnetic effects on fusion reactor components is summarized as follows: (1) C o m p u t e r tools based on the shell approach and complete 3D formulation are available for eddy current analysis of various types of components. The validity of the tools is verified fully by experiment. (2) Significant coupling effects between electromagnetic forces and mechanical displacements are observed and should be taken into consideration in the structural design of components in anticipation of positive magnetic damping especially because fusion devices are placed under very strong magnetic fields. In order to utilize the favorable effects more advanced numerical tools should be developed in the near future. (3) The final goal is to develop control techniques for electromagnetomechanical p h e n o m e n a and thereby to establish an optimal design methodology.
Nomenclature B B0 H E E0 J Tc A T 4' p~ ~o n R
magnetic flux density, applied magnetic flux density, Magnetic field, electrical field, applied electrical field, current density, critical temperature of superconductor, magnetic vector potential, current vector potential, electrical scalar potential, electrical conductivity, magnetic permeability, magnetic scalar potential, normal unit vector on a boundary, distance between source and field points.
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