TRANSIENT ELECTROMAGNETIC STUDIES
FOR THE TFCX PRE-CONCEPTUAL DESIGN
R.J. Thome, R.D. Pillsbury, Jr., W.R.
M.I.T. Plasma Fusion Center
May 1984
PFC/RR-84-7
Mann
TABLE OF CONTENTS
PAGE
1.0
INTRODUCTION AND SUMMARY
1
2.0
START-UP FLUX PENETRATION
3
2.1
2.2
2.3
3.0
DISRUPTION EDDY CURRENTS
3.1
3.2
3.3
4.0
Model Description
Flux Penetration Lag
Component Eddy Currents
Model Description
Limiter Carrying Net Current
Limiter Carrying No Net Current
PLASMA VERTICAL STABILITY
3
5
13
21
21
27
37
58
APPENDIX A - PF Coil Fields at the First Wall and Vacuum Vessel
75
APPENDIX B - Normal and Tangential Pressure Distributions for
First Wall and Vacuum Vessel - Net Limiter
80
APPENDIX C - Normal and Tangential Pressure Distributions for
First Wall and Vacuum Vessel - No Net Limiter
106
1.0
INTRODUCTION
During the pre-conceptual design for TFCX, transient electromagnetic
effects were investigated in three main areas; delay in start-up field and
flux penetration due to eddy currents induced in the machine
structure;
disruption induced eddy currents and electromagnetic loads; and requirements for
passive
and
active
vertical
stabilization
elements
for
the
plasma.
Start-up and disruption effects were estimated with a finite element
transient em model based on a nominal performance,
with superconducting
PF
coils and an internal
copper TF coil machine,
vacuum
vessel.
Several
variations on wall thickness and materials were run to develop brackets
on the electromagnetic effects due to design options under consideration.
PF coil current profiles for start-up were based on a 60 s ramp time.
Model variations
were run covering different construction options which
resulted in time delays to reach an equivalent flux (plasma loop voltage)
level which ranged from 60 ma to 3.2 s.
From this set,
design configurations have time lags between
the most likely
245 ms and 275 ms.
considered acceptable in view of the long flux swing interval.
This is
A more de-
tailed discussion is given in Section 2.0.
The components which are influenced most by the rapid magnetic field
change associated with a disruption are the first wall, limiter and vacuum vessel.
The plasma was assumed to ramp linearly from
-
10 MA to zero
in 10 ms and the finite element model was used for several construction
options in terms
uity.
of material
thickness
and toroidal
electrical
contin-
The disruption induces toroidally directed or saddle type currents
-2-
current magnetic
toroidal
fields,
loads which are in general,
sheets.
These currents
geometry.
depending on component
field
interact with the eddy
and poloidal
field to develop
normal and tangential to the induced current
In many cases, the loads are non-trivial (eg., - up to
and,
conditions)
under certain
structural analysis.
require
therefore,
-
consideration
0.3 MPa
in the
A summary of maximum currents and loads is given in
and more detailed load distributions are in Appendices B and
Section 3.0
C.
The negative field index associated with production of an elongated
plasma leads to a condition which is inherently unstable to perturbations
in plasma vertical position.
a very rapid time scale
and passive
elements.
The unrestrained growth rate would occur on
( ~ 1 us), but is stabilized through the active
carry
The latter
eddy
currents
induced
by
the
plasma motion and create a stabilizing field until the control coils can
be activated.
This reduces the power requirements for the control coils.
The analysis during this phase utilized a simplified model of the plasma
and structures
which
led
to
the
following
preliminary
conclusions.
in
1.
Active control coils are necessary and will require power
the 0.1 to 10 NVA range depending on location;
2.
The proximity and electrical continuity of the machine structures is sufficient for passive stabilization without addition of high conductivity elements to augment this effect.
The em transient effects described in this report were performed for
the pre-conceptual design of TFCX.
of machine dimensions,
but are
They were based on a particular set
appropriate
as indicators
minor dimensional variations about the base point.
of trends
for
-3-
2.0
START-UP FLUX PENETRATION
codel
A finite element transient electromagnetics
was used to gain
an insight into the effects of prospective TFCX shell geometries and wall
thicknesses on start-up field and flux penetration
to the plasma.
Sev-
eral cases were run with selected changes in features which could effect
the eddy current patterns and magnitudes.
2.1
Model Description
The PF coil locations used for this analysis (and for the disruption
analysis in Section
3.0)
are defined in Table
2.1-1.
Field components
along the first wall and vacuum vessel due to the PF coils at end. of burn
are given in figures in Appendix A and used in Section 3.0 to determine
part of the wall em loading following disruption.
The amp-turn changes
used for the start-up cases were based on a linear ramp from the values
in the
column labelled
"start-up" to the values
in the
column labelled
"start of burn" in 60 seconds.
Figure 2.1-1 shows an outline of the regions used in the finite element model
together
with
representative
thicknesses,
materials,
and
whether the region is capable of carrying a net current through the rzplane or whether it
is a no-net region.
capable of carrying current,
The latter refers to a section
but which is restricted to return
the same
1 R.D. Pillsbury, "A Two-Dimensional or Axisymmetric Finite Element Program for the
Solution of Transient or Steady-State, Linear or Non-Linear, Magnetic Field Problems", COMPUMAG, Chicago, September 1981.
NOWNWAOMM6000
-4-
amount of current toroidally (eg., as in sectors where currents can flow
toroidally,
but which must return in saddle current fashion at the ends of
the sector).
it
The model is axially symmetric in all cases.
Geometrically,
represents a case with an internal vacuum vessel, a dewar surrounding
the central solenoid and inner PF coils, and copper TF coils in stainless
steel cases.
The plasma was modelled as four nested regions with a speci-
fied uniform current density in each region.
Figure 2.1-2 shows the field
lines produced by the PF coils at start of burn and the boundaries of the
finite element plasma region.
The close fit
of the flux surfaces to the
plasma region boundary is a good cross-check on model characteristics and
adequate for the purposes of a study of this type.
Table 2.1-2 defines the model materials and thicknesses for the Case
21 * baseline.
In some instances shell thicknesses have been combined for
ease of modelling in this preliminary
cylinder" is shown as 38 cm thick,
study.
For example,
the "bucking
but is intended to model the combined
effect of a 26 cm bucking cylinder plus a 2 cm dewar wall plus 10 cm of
outer TF coil case which is toroidally continuous in this region.
Simi-
larly, the "15 cm vacuum vessel" on the inboard side models 5 cm of vacuum vessel plus 10 cm of inner TF coil case ring which is toroidally continuous in this region.
The port and outboard vacuum vessel are not actually
toroidally continuous in an axially
Case number for authors reference only
symmetric sense,
but the latter
was
-5-
treated as a net current region since this was expected to be conservative in estimating the time lag effects under consideration.
case
for another
defines the materials and thicknesses
Table 2.1-3
used in parametric studies of start-up flux penetration.
Results will be
discussed in Sections 2.2 and 2.3.
2.2
Flux Penetration Lag
The models described in the previous section were used to calculate
the flux produced in the z plasma current
for
density,
toroidal continuity.
o plane at r - 3.89 m,
variations
several
the point of maximum
on
wall
and
thickness
No plasma was included in these cases.
Figure 2.2-1 shows the flux as a function of time at the point indicated, for the case
well as
the
of
that
"no eddy currents",
The
results with vessels present.
the delay
this figure
as
vessels are
present
and
in achieving
the
system
is, PF coils only,
as
time lag is defined
in
the desired
flux level when the
has progressed
beyond
initial
the
transients and established a "stationary" eddy current pattern.
Table 2.2-1
summarizes
the
time
lags
for
the
cases
Cases 14 and 20 are by far the longest at approximately
considered.
3 seconds,
are unrealistic extremes since they are dominated by 2 cm thick first
which are
copper.
All
other cases
range
from 60 to
275 ms.
but
walls
The most
realistic configuration corresponding to designs as of this writing would
lie between Cases 21 and 25 which imply a flux lag in the 245 to 275 ms
range.
Since the start-up flux swing is expected to occur on the 40 to
60 second time scale, this type of lag is probably not significant.
How-
ever, a plasma MHD analysis which accounts for the current distribution in
-6-
1
1 1 I1 1
I
i
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It
ti
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z
z
co
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CD
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Do
C)
z
C.'
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w
C.
(W)7
Fig. 2.1-1
Outline of regions used in finite element model.
C)
-
I
-7-
@1/1
HLC
0
a
L
9
9
t
C
1
0
CW)Z
Fig. 2.1-2
Finite element model plasma boundary superimposed
on field lines produced by the PF coils at start
of burn.
-8TABLE 2.1-1
TFCX - PF COIL POSITIONS AND CURRENTS
COIL
#
Rc(m)
Zc(m)
Ar(m)
AZ(m)
1
0.810
± 0.625
0.390
1.250
2
0.810
1.875
0.390
3.400
NI(MA-T)
START OF BURN
NI(MA-T)
END OF BURN
6.210
4.076
- 7.411
1.250
8.776
7.469
- 8.766
0.530
0.530
1.488
5.154
2.389
4.200
0.530
0.530
1.488
5.154
2.389
3.700
0.600
0.530
0.294
- 6.102
- 6.647
NI(MA-T)
START-UP
±
3
1.480
±
4
2.400
±
5
6.950
I
-9TABLE 2.1-2
TFCX -
COMPONENT
START-UP LAG MODEL -
MATERIAL
NAME
CASE 21 * BASELINE
THICKNESS
NET
NET
(cm)
CURRENT
CURRENT
BUCKING CYLINDER
as
38.0
DEWAR 1
su
18.0
DEWAR 2
sa
2.0
CAP
as
2.0
PORT
as
4.0
VACUUM VESSEL INBOARD
so
VACUUM VESSEL OUTBOARD
so
SHIELD INBOARD
80% as
10.0
SHIELD OUTBOARD
80% ss
20.0
FIRST WALL INBOARD
as
2.0
FIRST WALL OUTBOARD
as
2.0
BOX BEAM
as
200 in
For authors reference only
15.0
5.0
2
-10-
TFCX -
COMPONENT
NAME
BUCKING CYLINDER
TABLE 2.1-3
START-UP LAG MODEL - CASE 12
MATERIAL
THICKNESS
(cm)
as
15.3
DEWAR 1
s
2.0
DEWAR 2
so
2.0
CAP
us
2.0
PORT
88
4.0
VACUUM VESSEL INBOARD
so
2.0
VACUUM VESSEL OUTBOARD
88
2.0
SHIELD INBOARD
80%
SHIELD OUTBOARD
80% sa
FIRST WALL INBOARD
as
1.0
FIRST WALL OUTBOARD
as
1.0
For authors reference only
s
10.0
20.0
BASELINE
NET
CURRENT
NO NET
CURRENT
-11-
13.9
Case 20
Case 14
13.8
Case 25
Case 21
I
No -Eddy
Currents
Case 16 & 19
Cage
12 &
15
13.7
S
time lag
4
;2
(Typical)-
13.6
--
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time (sec)
Fig. 2.2-1
Flux at plasma ( r = 3.89 m) vs time for selected
TFCX shell characteristics.
0.9
-12-
TABLE 2.2-1
VOLT-SEC
- 0.5
START-UP FLUX LAG AT (3.89,0);
SEC
No EC
CASE *
NO.
a
VACUUM VESSEL
(cm)
TIME
(cm)
FIRST WALL
(matl.) -(Inboard) (Outboard)
OTES
LAG
(s)
21
15/5
2
s
Net
Net
0.245
Case 21 (See Table
2.1-1)
25
15/5
2
so
Net
Net
0.275
Case 21 Plus 2 cm
Cu Net Limiter
(See Fig. 2.0-1)
12
1
as
No Net
0.065
Case 12 (See Table
2.1-2)
14
2
cu
No Net
3.2
2
cu
No Net
0.065
"t
"t
"
15
2
s5
No Net
0.180
"
"
"I
16
p + Net
0.060
"t
I
0.180
"
I
"II
I
",
"
S
I
ALL
18
OTHER
No Net
Net
No Net
Net
REGIONS
19
2
sa
Net
Net
20
2
cu
Net
Net
For authors reference only
~ 3.2
" with FW
variations
"
S
-13-
the plasma and variations in the fields over the plasma cross-section remains to be done to verify this conclusion.
2.3
Component Eddy Currents
In a system with a complex geometry and many components and materials,
the eigenvalues which govern the transients are combinations of time constants for
individual
single "time
elements
constant"
in the
for the
conducting
Although
system.
There is no
regions.
the
current
patterns
will shift within and between components during the transient, it is useful to plot the total current in each component as a function of time to
gain insight
into the
relative
importance
of
the different
components.
Figure 2.3-1 shows the total current in the net current carrying components in Case 21
initial
of the
dominance
coil (dewar 1).
(see Table 2.2-1
box beam,
and dewar
2.0-1)
and illustrates the
section near the inner PF
These are followed by the bucking cylinder which becomes
the port,
shield out-
and shield inboard are no net current carrying regions.
They carry
the dominant component eventually.
board,
and Fig.
currents toroidally
which
The net current
one
in
For this case,
must return
direction
toroidally within
must turn
in the
their section.
saddle direction
order to return toroidally in the opposite direction.
in
The total saddle
current magnitudes for this case are shown in Fig. 2.3-2.
In Case
25,
the effects of a limiter were evaluated
section of the first
wall as being copper (see Fig. 2.0-1).
by modeling a
To maximize
the influence of the limiter, it was assumed to be a net current carrying
region and,
therefore,
toroidally continuous.
Component currents versus
time for this case are shown in Figs. 2.3-3 and 2.3-4 in a format similar
-14-
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Fig.
2.3-1
AOO
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Lii
I AAAAIIAIIIIIIIIIIIII
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Component eddy currents vs time for the Case 21 Baseline.
Currents are for the components above the midplane. Component
names correspond to those in Fig. 2.1-1.
-L....
-15-
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A
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ft
ft
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ft
ft
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A
ft
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ft
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ft
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A
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ft
A
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A
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ft
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ft
A
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A
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A
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A
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ft
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I
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C'J
A
A
ft
ft
A
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ft
ft
ft
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ft
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ft
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ft
A
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ft
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ft
I
A
A
ft
A-------4.---4---ft---ft~---+--4......
ft
ft
A
ft
ft
ft
ft
ft
A
ft
ft
ft
A
ft
ft
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ft
A
ft
A
A
A
A
ft
ft
A
A
ft
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ft
ft
ft
ft
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A
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A
ft
A
ft
A
v-~
C\I
-
-
(0
CJ
c~J
-
W) ~IN~HdflO ~]GGU3
Fig. 2.3-2
Component saddle currents vs time for the Case 21 baseline.
Saddle currents occur only in no net regions. Region names
correspond to those in Fig. 2.1-1.
Wi
-16-
I
I
9
9
9
9
o
9
1-4
9
9
'2
-~
:
-
9
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i
9
9
9
9
79
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9
9
9
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79
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C)9
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9
9
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9
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9
9
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9
9
9
9
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9
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9
9
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9
----------------------------------------
:
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9
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9
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9
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9
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9
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p
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9
9
9
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9
9
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9
9
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9
9
9
-
-
C
9
9
9
9
A-1
I-
I
(.0
cm
INOJVIOO
Fig. 2.3-3
Component qddy currents vs time for Case 25.
are for the components above the midplane.
correspond to those in Fig. 2.1-1.
Currents
Component names
-17-
IF-I
a
a
a
~--I I
I
I
a
a
a
a
a
a
a
a
a
a
a
-
a
a
a
a
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a
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a
a
I
a
a
I.----------
I
a
a
a
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a
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a
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a
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a
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a----------a--------------------4-----a
a
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------
a
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2.3-4
I
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a
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a a a at a a a a
I
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a
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a
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-
Fig.
I
a
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.
I
a
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S4----a--------------a
a
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I
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--------------------------a
a
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a
a a a
a
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a a
a
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a a
a
a
a
a
a
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a
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(_)
CD
I
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
Ia
~
I
a I
I
c0
Component saddle currents vs time for Case 25.
Saddle currents
occur only in no net regions. Region names correspond to those
in Fig. 2.1-1.
LIi
-18-
to that used previously.
Note that even though the limiter is toroidally
continuous in the model and copper, it is not initially dominant because,
during start-up,
the
field penetration
occurs from the
outside
of the
vessel inward.
The previous cases were run without any plasma current present.
ures 2.3-5 and 2.3-6 are a repeat of Case 25 but with a plasma
ramping up to 10.3 MA during the start
up PF swing.
Fig-
current
The current density
was uniform and restricted to the innermost region in Fig. 2.0-1.
A com-
parison with the previous two figures indicates that the primary impact is
to reduce the current induced in the limiter.
The start-up cases which have been run in this preliminary analysis
imply that flux penetration to the plasma is acceptable and are useful as
an indicator of the dominant components during this part of the operating
scenario.
-19-
L
I
A
A
A
A
A
A
A
A
A
A
A
A
AA
A
A
A
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II
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A
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A
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-±-----------------------------------------A
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I i...ii..iI ~
C,,
CM :11
T
(H) iIH n3df )WG3 iINNOJVIOO
Fig.
2. 3-5
Component eddy currents vs time for Case 26. Currents are for
the components above the 'midplane. Component names correspond
to those in Fig. 2.1-1.
r
-20-
Iii
I
5 I'll'
a a at 5
ii:
-
-
a
a
a
a
a
a
a
a
I
I
I
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e~.
------..
a
a
a
a
a
a
a
a
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a
a
a
a
a
a
a
a
Sa
a
a
a
a
a
a
a
a
a
a
SI
a
a
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Component saddle currents vs time for Case 26. Saddle currents
occur only in no net regions. Region names correspond to those
in Fig. 2.1-1.
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-21-
3.0
DISRUPTION EDDY CURRENTS
The axially symmetric transient electromagnetics
code was used with
several variations on the models used in the start-up analysis to produce
preliminary estimates
of the loads produced on selected components as a
result of a disruption.
In particular,
tributions were estimated for the first
The latter
induced current and em load diswall,
vacuum vessel and limiter.
was considered for cases which were either toroidally contin-
uous or segmented to prevent net current flow.
maximum load results are
Current distributions and
given in this section to summarize the
Appendices B and C contain load distributions along the
first
study.
wall and
vacuum vessel throughout the transients.
3.1
Model Description
An outline of the finite element regions used in this study is shown
in Fig.
3.1-1.
Attention was focused on currents and loads on the first
wall, the section
vacuum vessel.
of the first
wall which modeled the limiter,
and the
The numbered points in Fig. 3.1-1 will be useful in iden-
tifying peak load locations when used with some of the tables which follow.
Note that the outboard vacuum vessel was modeled as a net current carrying
region and the port -as a no net region.
metric as assumed in the model.
Neither are actually axially sym-
This will be rectified in future models
using the SPARK1 code, but is not expected to be a major influence on the
trends indicated in this study.
"SPARK Version One: Reference Manual",
1 D.W. Weisenburger,
Plasma Physics Laboratory, Princeton, NJ, October 1983.
PPPL-2040,
Princeton
-22-
-
II
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II
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I~
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I
,
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t
i
.
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(w F1
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W
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to4
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to
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V)
0
a
2
Z
Ix
Li-
~LJ
;1F1
~2
~
7
(~.
U
~i
'-'I'
-
w
(W)
Fig. 3.1-1
*31-
'2
Finite element model for disruption used a stationary plasma
ramped to zero current in 10 ms and a limiter which had either
"net" or "no net" current carrying ability.
-23-
The plasma carried 10.3 MA before disruption.
The plasma current was
distributed over each of four regions and was motionless as the current was
ramped to zero in 10 ms.
were held
constant
at
column in Table 2.1-1).
are defined in Table
Throughout the transient the PF coil currents
levels
corresponding
The characteristics
3.1-1.
Figure
3.1-2
to
end-of-burn
(see
last
of each region in the model
illustrates the
steady-state
(t<0) field line distribution produced by the plasma alone.
Figure 3.1-3 is a sketch showing typical current and load directions
on both net and no-net current carrying passive components.
The top sketch
shows a section of a toroidally continuous shell segment capable of carrying a net current through an rz-plane.
The induced currents in this type
of section as a result of a disruption will be in the toroidal direction
and will interact with fields from the plasma,
eddy current fields.
the PF coil system and other
The result will be em loads per unit shell surface
area which have components normal and tangential to the shell surface.
In
the figures which follow these will be referred to as normal and tangential
pressures and have the dimensions of force per unit shell surface area.
Normal pressures are defined positive outward and tangential pressures are
defined positive clockwise in the rz - plane.
The lower sketch in Fig. 3.1-3 shows a no-net current carrying shell
In
segment.
This is also subject to the type of loads described above.
addition,
however, the toroidally directed currents must return within the
same shell, hence, there must be saddle components of current.
The model
-24-
TABLE 3.1-1
TFCX - PLASMA DISRUPTION MODEL
COMPONENT
NAME
MATERIAL
THICKNESS
(cm)
BUCKING CYLINDER
so
38
DEWAR 1
s
18
DEWAR 2
s8
2
CAP
s
2
PORT
88
4
VACUUM VESSEL INBOARD
88
15
VACUUM VESSEL OUTBOARD
88
5
SHIELD INBOARD
80% sa
10
SHIELD OUTBOARD
80% so
20
FIRST WALL INBOARD
S
2
FIRST WALL OUTBOARD
us
2
LIMITER
Cu
2
BOX BEAM
S8
1290 cm 2
NET
NO NET
CURRENT
CURRENT
or
-25-
CE
z
CI:
CO
by
thCpas a
A~itBMSFF/li:D
ANI~ifuM~f~ff/CD
e
lo
-f
I00
Co
Fig. 3.1-2
:zc-
5-
=V
Nj
Steady state (t < 0) field line distribution produced
by the plasma alone.
rojr~roij
C)o
CF
1-1SZ TMv9
AL
t
OAV4Ad.4
-26-
e
O
Qt
---
>i.
~Et~
-
~
CU~eAi1
~J~(g~OA.)
4
________
Y--x
I
x
00
4 Mi-
cuperpr
2A,
1Z.&C
OAJ
A
N)
iEATT
= (AY- ADICLE
5A.
~R
AY
~><-
Fig. 3.1-3
Definition of positive current and load directions for
disruption eddy current analysis.
-27-
is axially symmetric so the saddle current distribution cannot be determined.
However,
the magnitude of the total saddle current for a no net
region must be equal to the total toroidal current flow in one direction
and are important since they interact with the toroidal field to produce
equal and opposite loads on each end of the shell with the loads lying in
rz-planes.
The same saddle currents interact with the poloidal field and
the eddy current fields to produce equal and opposite
loads in the tor-
oidal direction on each end of the segment.
3.2
Limiter Carrying Net Current
This case is based on the configuration in Fig.
3.1-1
and the as-
sumption that the limiter can carry net current in the toroidal direction.
The latter
may be expected to produce the maximum possible current in the
limiter.
The plasma current decays linearly to zero in 10 ms and the steadystate field pattern from the plasma which is shown in Fig.
3.1-2 changes
as shown in Figs. 3.2-1 to 3.2-4 which correspond to times of 10, 30, 50,
and 100 me, respectively.
the figures
shells are
by
"kinks" in the field
essentially
local discontinuities
field.
The presence of eddy currents is indicated in
lines
toroidally directed
in the tangential
because the
current
components
currents
sheets
which
in the
cause
of poloidal magnetic
The strong influence of the net current carrying limiter is evi-
dent.
The flux through the z-o plane is shown in Fig.
instants of time.
3.2-5 for selected
The t<0 curve is due to the steady-state plasma alone.
-28-
CO
C.0
CC
z
CE
z
C-)
CD
cc'
co
(W
Fig. 3.2-1
Lf7
-
)z
Fieldline distribution at 10 ms for a 10 ms plasma disruption
with a net current carrying limiter.
-29-
(M
CO)
z
C)
LL
CO
z
C)
C"Jq~
06
Cfl
ri
(C
(W)Z
Fig. 3.2-2
Fieldline distribution at 30 ms for a 10 ms plasma disruption
with a net current carrying limiter.
-30-
CE
L.
z
CC)
CO)
z
C)
C\i
CD+
(WL )
Fig. 3.2-3
z
Fieldline distribution at 50 ms for a 10 ms plasma disruption
with a net current carrying limiter.
-31-
C6cc
z
CI:
z
C-)
ULC)
D
z
CD
Fig. 3.2-4
Fieldline distribution at 100 ms for a 10 ms plasma
disruption with a net current carrying limiter.
- atmemomm
-32-
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Fig. 3.2-5
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xflhi
Flux vs radial position along the midplane (Z = 0) for various
times during and after a 10 ms plasma disruption for a net
current carrying limiter.
A
I
I IAI
-33-
The plasma decays in 10 ms which leads to a collapse of the center of the
pattern on the same time scale since the major portion of this part of the
flux corresponds to those field lines which were totally within the near(e.g.,
est conducting wall and did not link any conducting material
For t>10 ms, the decay of the flux pattern is much slower
3.1-2).
Fig.
see
because of the eddy currents induced in the vessel walls.
The total currents in the individual regions in the model are given
Comparison of the latter with the former shows
in Figs. 3.2-6 and 3.2-7.
that the net current regions carry substantially higher current levels.
Fig.
Furthermore,
the
(i.e.,
shows
that
the
components
relatively rapid
decay
inboard
rate
which
eventually becomes dominant.
because
is
they
further
the
nearest
are
from
thin
a
with
the
thicker
and
compared
the plasma,
plasma
but have
respond first,
wall inboard and outboard)
first
vacuum vessel
3.2-6
Note that the vacuum vessel outboard is not
actually toroidally continuous as assumed in the model, hence, it will be
less influential and somewhat higher currents will probably result in the
inboard vessel and limiter.
Furthermore,
if
the first
wall inboard and
outboard are not net current carriers as assumed then the other components can be expected to respond faster than shown.
The current sheet amplitude is shown in Fig. 3.2-8 as a function of
distance along the first
plane.
wall measured from the inboard side in the z=o
The plot is for the instant at 10 ms when the plasma current has
just reached zero and shows relatively high currents along the first wall
and a concentration
of current in the limiter.
A similar plot at 100 ms
-34-
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C\J
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Fig. 3.2-6
S1NiH~no
CO
:rD
Ce
AGG1 1NINO5NO3
Component eddy currents vs time for a 10 ms plasma disruption
and a net current carrying limiter. Currents are for the
components above the midplane. Component names correspond to
those in Fig. 3.1.1.
-.1
-35-
ft f
ft
ft
f-ft
t
t
ft f
ft
ft
*
t ft
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ftft
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Fig
3.-
omoet
an
a
ft
ne reios
curet
t
ftft
I
f
ft
f
ft
t
x
U
f
tim
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ft
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fo
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-36-
33
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1
M
-
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- - r -- - -
at
limiter.
10m:o
0m
lsadsuto
n
e
urn
-37-
is shown in Fig.
wall
3.2-9 and indicates that the currents in the first
sections have decayed,
but the current in the limiter has increased
sub-
Plots of this type for the vacuum vessel are shown in Figs.
stantially.
3.2-10 and 3.2-11
which are for 10 and 25 me,
these
are
close
The time of
wall example and 25 ma for the vacuum
100 me was selected for the first
vessel since
respectively.
to the time
for maximum
currents
in the
limiter and vessel, respectively.
Tables 3.2-1 and 3.2-2 summarizes the maximum currents and loads for
this case,
together with the times at which they occur,
tions keyed to the points identified in Fig. 3.1-1.
and their loca-
In general,
the loads
are extreme in the components away from the plasma at later times ('
me).
10
The extreme loads on the first wall/limiter occur on the limiter at
100 me.
The peak shield loading occurs at 10 ms (consistent with the much
shorter time
constant of the
positive and
negative
no-net region).
pressures
indicative
All
of
a
components have
non-uniform
both
loading.
More detailed distributions may be found in Appendix B which gives the
loads plotted along the first
wall and vacuum vessel peripheries at se-
lected instants of time.
3.3
Limiter Carrying No Net Current
This case is based on the configuration in Fig. 3.1-1 and the assump-
tion that the limiter can not carry net current in the toroidal direction.
The latter may be expected to produce much lower toroidal currents in the
limiter than the previous case,
but will result in saddle currents which
are necessary to satisfy the no net restriction.
-38-
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a
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Toroidal current sheet amplitude vs position along the first wall
and limiter at 100 ms for a 10 ms plasma disruption and a net
current carrying limiter.
S
-39-
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urn
-41TFCX -
10 me PLASMA DISRUPTION - NET LIMITER - CASE 23
Extrema In The Current Sheet Magnitude
time
I
Maximum
K-theto
(MA/m)
0
29
0.946
lee
0
16
3.830
70
10
30
0.063
60
e
1
1.711
60
0.020
60
point
Minimum
K-theto
time
point
#
(MA/m)
(me)
DEWAR
0
FIRST WALL - LIMITER
0
Reg i on
16
SHIELD
VACUUM VESSEL
1
PORT
1
-0.103
0
-0.009
45
22
(me)
Extremo For The Normal Pressure Magnitude
Maximum
Minimum
Region
point
#
Normal
Pressure
(MPa)
time
(ms)
point
#
Normal
Pressure
(MPG)
time
(ms)
DEWAR
15
-0.185
100
29
0.987
100
FIRST WALL - LIMITER
16
-0.625
60
13
0.468
100
SHIELD
30
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10
16
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100
10
VACUUM VESSEL
1
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60
13
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lee
PORT
8
-0.017
60
12
0.005
60
Extrema For The Tangential Pressure Magnitude
Region
point
Minimum
Tangential
Pressure time
(MPG)
(ms)
#
Maximum
Tangential
Pressure
(MPG)
point
time
(me)
22
-0.414
100
15
0.071
100
16
-3.386
100
13
0.247
100
SHIELD
20
-0.012
60
16
0.046
10
VACUUM VESSEL
16
-0.318
100
9
0.275
90
7
-0.014
80
13
0.009
60
DEWAR
FIRST WALL -
PORT
Table 3.2-1
LIMITER
Summary of maximum
Net Current Carrying Limiter induced currents, normal pressures and tangential
pressures.
-42TFCX -
10 me PLASMA DISRUPTION - NET LIMITER - CASE 23
Extrema In The Saddle Current Magnitude
point
Region
Minimum
Current
t ime
(me)
point
Maximum
Current
#
(kA)
t ime
(me)
11.49
60
#
(kA)
SHIELD INBOARD
15
0.00
0
12
SHIELD OUTBOARD
5
-66.33
10
15
13
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40
22
PORT
128.0
76.64
60
60
Extrema In The Toroidal Force Magnitude
point
Region
SHIELD INBOARD
SHIELD OUTBOARD
PORT
Mi nimum
F-theta
(kN/m)
Maximum
t ime
(ms)
point
#
(kN/m)
t ime
(ms)
F-theto
14
-6.34
80
15
6.34
60
3
-24.95
15
7
5.53
100
13
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60
7
35.99
80
Extrema In The Normal Force Magnitude
point
Reg i on
SHIELD INBOARD
SHIELD OUTBOARD
Table 3.2-2
#
Minimum
F-normal
(MN/m)
t ime
point
(me)
#
9
-0.072
10
15
15
-0.347
60
4
Max imum
F-Normal
t ime
(MN/m)
(ma)
0.0
0.218
0
10
Summary of maximum
Net Current Carrying Limiter saddle currents and saddle current loads on the no
net current carriers.
-43-
The plasma current decays linearly to zero in 10 ms and the steadystate field pattern from the plasma which is shown in Fig.
3.1-2
changes
as shown in Figs. 3.3-1 to 3.3-3 which correspond to times of 10, 30, and
50 ms, respectively.
The presence of eddy currents is indicated in the
figures by "kinks" in the field lines because the currents in the shells
are essentially
current
sheets which cause local discontinuities
tangential components of magnetic field.
The influence of the limiter is
induced current cannot
not as strong as in the previous case because its
link the main flux through the
in the
center of the machifte due to the no net
restriction.
3.3-4 for selected
The -flux through the z=o plane is shown in Fig.
instants of time.
The t<0 curve is due to the steady-state plasma alone.
The plasma decays in 10 ms which leads to a collapse of the center of the
pattern on the same time scale since the major portion of this part of the
flux corresponds to those field lines which were totally within the nearest
conducting wall and did not link any conducting material.
For t>10 me, the
decay of the flux pattern is. much slower because of the eddy currents inThe two abrupt changes in slope on the out-
duced in the vessel walls.
board side are primarily due to the first wall and outboard vacuum vessel
which are net current carrying regions in the model.
actually designed
as
a no net region,
then the
If the first
inner
wall is
"kink" would es-
sentially disappear.
The total currents in the individual regions in the model are given
in Figs. 3.3-5 and 3.3-6.
that the net
Comparison of the latter with the former shows
current regions
carry substantially higher current levels.
-44-
C-
V-
CE
*
Fig. 3.3-1
Fieldline distribution at 10 ms for a 10 ms plasma disruption
with a no net current carrying limiter.
C)
z
-45-
C!)
z
CC
L-
m CD)
CD
z
CD)
Fig.
3.3.2
Fieldline distribution at 30 ms for a 10 ms plasma disruption
with a no net current carrying limiter.
-46-
c-o:
H-
CE
0-
z
C\JD
LI)
mLL.
(W
3
msfora 1 ms
limter
nonet urret
withacaryin
Fig.3.33
istibuionat
Feldine 5
Z
lasa dsrutio
-47-
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Fig. 3.3-4
B
B
I
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I
I
B
B
B
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Flux vs radial position along the midplane (Z = 0) for
various times during and after a 10 ms plasma disruption
for a no net current carrying limiter.
-48-
U.,
I
-
-- -
- -
- -
- -
--
Ii
I
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Fig. 3.3-5
CCoC
C.
suejjno ,Kpp] 4uquodwoo
Component eddy currents vs time for a 10 ms plasma disruption
and a no net current carrying limiter. Currents are for the
components above the midplane. Component names correspond to
those in Fig. 3.1-1.
-49-
9
-- -
96
9
I
9
99
- - -- -
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9
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9
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9
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Fig 3.-
r
9
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- - - -
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corsodt
hsei
nyi
i.321
-50-
Furthermore,
Fig.
3.3-5
shows that components nearest the plasma
wall inboard and outboard) respond first,
the first
(i.e.,
but have a relative-
ly rapid decay rate because they are thin compared with the vacuum vessel
inboard which eventually
Note that the vacuum vessel
becomes dominant.
outboard is not actually toroidally
continuous as assumed in the model,
hence, it will be less influential and higher currents will result in the
inboard vessel.
If the first
wall inboard and outboard are not net cur-
rent carriers as assumed then the other components can be expected to respond faster than shown.
Note that the limiter carries substantially less
current than in the previous
case
(e.g.,
compare
Fig.
3.3-6
with Fig.
3.2-6).
The current sheet amplitude is shown in Fig. 3.3-7 as a function of
distance along the first
plane.
wall measured from the inboard side in the z-o
The plot is for the instant at 10 ms when the plasma current has
just reached zero and shows relatively high currents along the first
wall.
The shaded region illustrates the current carried by the no net limiter.
A similar plot at 30 ms is shown in Fig. 3.3-8 and indicates that the current in the limiter has increased somewhat.
Plots of this type for the
vacuum vessel are shown in Figs. 3.3-9 and 3.3-10 which are for 10 and 30
ms, respectively.
Tables 3.3-1
this case
and 3.3-2 summarize the mazimum currents and loads for
together with
the
times at which
they occur and their loca-
tions keyed to the points identified in Fig. 3.1-1.
In general, the peak
-51-
loads are uniformly lower for the no net limiter than for the net limiter
(See Table 3.2-1).
Similarly times at which the extrema occur are smaller.
The point at which the peak loadings occur does not change significantly
for the dewar, shield, and vacuum vessel.
More detailed distributions may be found in Appendix C which gives
the loads plotted along the first
selected instants of time.
wall and vacuum vessel peripheries at
-52-
1111
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z 4e (W/HJ) eL44q-
Toroidal current sheet amplitude vs position along the first wall
and limiter at 30 ms for a 10 ms plasma disruption and a no net
current carrying limiter.
-54-
--------------------- -- - -I
- -
-
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-55-
1'1'1 1 3F~"F~rjum
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Fig 3.1
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-56-
TFCX -
10 me PLASMA DISRUPTION -
NO NET LIMITER - CASE 24
Extrema In The Current Sheet Magnitude
point
1
Region
1
DEWAR
Minimum
K-theto
(MA/M)
time
(me)
0
point
#
Maximum
K-theta
(MA/m)
time
(ms)
0
29
0.287
50
13
-0.586
30
16
0.727
35
SHIELD
16
-0.072
15
30
0.034
12.5
VACUUM VESSEL
17
0.041
5
1
1.057
22.5
1
-0.006
50
22
0.006
40
FIRST WALL -
LIMITER
PORT
Extrema For The Normal Pressure Magnitude
point
A
Region
Minimum
Normal
Pressure
(MPG)
0.356
50
-0.356
5
12
0.067
10
30
-0.026
10
17
0.024
12.5
1
-0.853
15
13
0.142
50
22
-0.002
40
8
0.004
30
-0.102
FIRST WALL - LIMITER
30
SHIELD
PORT
time
(ms)
29
15
VESSEL
point
#
50
DEWAR
VACUUM
time
(ms)
Maximum
Normal
Pressure
(MPG)
.
Extremo For The Tangential Pressure Magnitude
point
Region
#
Minimum
Tangential
Pressure time
(ms)
(MPa)
point
#
Maximum
Tangential
Pressure
(MPa)
time
(Ms)
22
-0.133
50
15
0.037
50
16
-0.310
50
13
0.184
50
SHIELD
15
-0.002
10
16
0.032
15
VACUUM VESSEL
16
-0.258
50
10
0.143
50
PORT
12
-0.002
30
1
0.005
50
DEWAR
FIRST WALL -
LIMITER
Table 3.3-1
Summary of
No Net Current Carrying Limiter maximum induced currents, normal pressures and
tangential pressures.
-57TFCX -
10 ms PLASMA DISRUPTION - NO NET LIMITER - CASE 24
Extrema In The Saddle Current Magnitude
Minimum
point
Current
#
(kA)
2
Region
LIMITER
SHIELD INBOARD
12
5
SHIELD OUTBOARD
13
PORT
time
point
Maximum
Current
(kA)
time
(me)
(me)
#
-180.0
40
4
0.07
22.5
-4.06
5
7
2.09
15
-51.09
12.5
15
6.06
15
3
0.12
5
time
-9.89
40
Extremo In The Toroidal Force Magnitude
point
Reg i on
#
1
LIMITER
SHIELD
INBOARD
SHIELD
OUTBOARD
14
PORT
time
point
(ms)
#
Maximum
F-theto
(kN/m)
-48.48
50
4
0.02
22.5
-0.44
5
1.36
12.5
Minimum
F-theta
(kN/m)
6
-18.57
10
7
-13.54
45
14
6
12
326.0
(ms)
12.5
10.67
40
Extrema In The Normal Force Magnitude
Max
Mi n imum
F-no rmal
t ime
point
#
(MN/m)
(me)
LIMITER
4
-0.000
SHIELD INBOARD
7
-0.014
point
Reg i on
SHIELD OUTBOARD
Table 3.3-2
15
0.000
imum
#
F-Normal
(MN/m)
ti me
(ms)
22.5
2
0.834
40
15
12
0.023
5
15
5
0.165
12.5
Summary of maximum
No Net Current Carrying Limiter saddle currents and saddle current loads on the no net
current carriers.
-58-
4.0
The negative
PLASMA VERTICAL STABILITY
field index associated with the poloidal field neces-
sary to create an elongated plasma produces a configuration which is inherently unstable
vertically
leads to a net
since a small displacement
force on the plasma in a direction tending to increase the displacement.
strained.
the
However,
rapid
displacement
eddy currents induced in passive
growth rate until an active
is
10-
(~
The growth rate would occur on a fast time scale
coil system can
s.)
if
uncon-
stabilized
initially
conducting elements
6
by
which restrain the
be actuated to produce the
radial field required to stabilize the plasma position.
The inherent passive stabilization characteristics
of the TFCX con-
figuration and preliminary estimates of the active coil power requirements
as a function of location were carried out with a simplified model.
The
plasma consists of a rigid circular filament carrying a constant current.
coupled to any number
The plasma can move vertically and is inductively
of passive
or active
circuits
lumped parameter
of
specified
geometry.
The most effective locations for passive stabilization elements can
be determined by first
assuming that they are a pair of conducting loops
symmetrically located relative to the z
-
o plane.
Contours of constant
effectiveness* for loop locations can then be superimposed on the normalized geometry of TFCX as illustrated in Fig. 4.0-1.
This shows the most
useful locations and the rapid degradation in passive stabilization ability
with distance
from the plasma.
For this type
of
system it
may be
R.J. Thome, R.D. Pillsbury, Jr., D.B. Montgomery, et al., "Passive and Active
Circuits for Vertical Plasma Stabilization", PFC/RR-83-32, December 1983.
-59-
C14
0
C-,
F-4
c~cc
-
I-
cc
X:
LU
MO%
NOI.LBOO1 IHIXHi 0QZI1UNWON
Fig. 4.0-1
-
Contours of Constant Passive Element Effectiveness
Superimposed on TFCX (ro = 3.9 in).
-60-
shown that a necessary but not sufficient condition for stabilization is:
z/ro) > -flf
G(r/ro,
-
(4.1)
(-Bz)ro
4w x 107 [H/nm]
110
where:
In TFCX,
BZ
M
vertical field
ro
-
plasma radius
n
-
field index
G
-
a function of passive element normalized
location
the vertical field is -0.64
10.3 MA,
the plasma current is
T, the field index is -0.5, and the radius to the current center is about
3.9 m.
With these parameters Eq.
lie on G contours with values of 0 0.1
shaded region
in Fig.
which simultaneously
4.0-1
includes
lie outside the
stantial sections of the first
implies that passive loops must
(4.1)
or greater to be effective.
contours
first
with
wall.
the desired
They
The
values
imply that sub-
wall and shield may be expected to be ef-
fective.
The contours were based on the use of a single pair of passive loops.
They need not be toroidally continuous because of the up-down asymmetry of
the required
stabilization
currents.
Hence,
saddle
type
currents
(as
shown in Fig. 4.0-2 which are induced in sectors which are not toroidally
continuous may be expected to perform the same function to first
order.
The previous plot implies that we need only model conducting sections
relatively close to the plasma to gain insight into the passive stabilization characteristics of the system although other conducting sections may
-61-
L
a..<
0LiJ
uQQ
0
2
LLI
2
C-
00
Fig. 4.0-2
Vertical stabilizing coils may have toroidal
continuity (ie - complete loops) or may be
segmented (ie - saddle coils).
-62-
influence the
performance
(i.e.,
field
prevent
from
penetration)
the
active coil system.
A filament type model consisting of the plasma, 24 passive elements,
and a pair of control coils located at one of three locations was construcThe top half of the model is illu-
ted to investigate vertical stability.
4.0-3 with an outline of
strated in Fig.
Passive coils are labelled P1,
clarity.
either A4,
A5,
of TFCX for
selected features
P2,
and the active coils are
..
Note that
or A6 depending on the case under consideration.
A4, A5 and some of the passive elements are saddle shaped.
The procedure used by the computer program is outlined in Table 4.01.
After
input of the
characteristics,
if
eigenvalues
specified,
such as
and
feedback
The system is stable
If
is negative.
output variables
field,
stationary
plasma,
the system eigenvalues are found.
the real part of all
are then
circuit,
initial
coil currents,
conditions
voltages,
plasma displacement, and active coil power can be determined.
Cases were
studied with element resistances
as
indicated in Table
These correspond to the equivalent resistances of the first wall,
4.0-2.
vacuum vessel and box beam cross-sections as indicated in Fig. 4.0-3.
order to bracket the power requirements,
extremes: toroidally
shield.
Figs.
cases).
the shield was considered for two
continuous and 80% stainless steel/20% water or no
The active coil power requirements for these cases are shown in
4.0-4 and 4.0-5 as a function
plasma to
In
10% of its
initial
The figures indicate:
of the time
displacement
required to return the
(assumed
to be 0.01 m in all
-63-
I
II
I
I
I
i
I
I
I
IV)
..
-J
-j
V4V?
1
I,,
-J
%J
r
'
'1
0
x
-C.
r
.1
'p
-4
I
I
'
I
I
I
I
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I
I
CD
(W
Fig. 4.0-3
) Z
Passive and active elements for preliminary vertical
stability study of TFCX.
4-
-64-
TABLE 4.0-1
IN COMPUTER PROGRAM
COMPUTATIONAL PROCEDURE USED
INPUT:
[Mijj,
[Rij,
Brk;
CIRCUIT CHARACTERISTICS
(Inductance, Resistance, Radial
Field Per Unit Current)
PLASMA CHARACTERISTICS
ro, Ip
(Plasma Radius, Plasma Current)
11f, Bz
STATIONARY FIELD CHARACTERISTICS
(Field Index, Vertical Field)
taitbi, tcij
ci;
FEEDBACK CHARACTERISTICS
(Lead, Lag Delay, Gain)
V + (tb + tc) V + tbtc V
a (Z + ta
-
DETERMINE EIGENVALUES,
Xi;
NOTE: STABILITYY REQUIRES
Re[Xi] < 0 FOR ALL i
DETERMINE:
Ii(t)
CURRENT
V i ( t)
NOTE: V = 0 IF COIL IS PASSIVE
Z(t)
DISPLACEMENT
P(t)
POWER
-65-
TABLE 4.0-2
ELEMENT RESISTANCES FOR MODEL IN FIGURE 4.0-3
ELEMENT
REGION MODELED
R(Q)
P 1
Box Beam
5.04 E - 04
P 2
Port
5.12 E - 03
P 3
Port
4.98 E - 03
P 4
Vacuum Vessel
6.72 E - 04
P 5
Vacuum Vessel
1.44 E - 04
P 6
Vacuum Vessel,
Shield & F.W.
6.65 E - 05
P 7
Shield
2.32 E - 04
P 8
Shield
2.16 E - 04
P 9
Shield
1.63 E - 04
P10
F.W.
2.03 E - 03
P11
F.W.
1.79 E - 03
P12
F.W.
1.41 E - 03
-66-
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Peak control coil power as a function of time to return the plasma
to 10% of its initial displacement for the extreme case of a
toroidally continuous shield.
-67-
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>Ulc1
Peak control coil power vs time to return the plasma to 10%
of its initial displacement for the extreme case of no shield.
-68-
(1)
An active coil location on the face of a removable sector,
wall has a power requirement in the
but behind the first
.01 to 1.0 MVA range unless return times faster than 0.1
seconds are desired in which case the power requirement
would be higher.
(2)
Active coil locations further out from the plasma have
power requirements which increase rapidly for equivalent
plasma position return times. Coils at locations like
A6 may have prohibitive power requirements depending on
structural characteristics (e.g., presence of the box
beam) or shield electrical conductivity.
The figures show that AS requires more power than A6 which seems inconsistent with its
relative position.
subsequent
However,
cases shown
in Fig. 4.0-6, which assumes no shield and no passive elements at P2 or P3,
1
implies that it is effectively shielded by these elements.
The controller
ta
-
time
constants
for
cases
all
tc M 3 ma and tb - 0. When the value of ta
-
chosen
were
to
be
tc was varied between 0
and 10 ms the required gain and power to return the plasma to 10% of its
initial displacement in 1 second did not change significantly.
for ta M tc
-
Similarly
3 ms and varying tb from 0 to 6 ms also had little
on power for a return of 1
second.
return was lowered to 10 ms,
significant effect.
However,
impact
when the required time to
the choice of controller time constants had
For example,
ta - tc
-
3 ms and tb
-
0 required 74
kVa, when tb was 3 ms the power was 190 kVa.
A typical set of output curves for plasma displacement,
current,
active coil voltage,
active coil
and active coil power as functions
are shown in Figs. 4.0-7 to 4.0-10, respectively.
of time
-69-
III
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NORMAL AND TANGENTIAL PRESSURE DISTRIBUTIONS FOR THE
FIRST WALL AND LIMITER; NET CURRENT CARRYING LIMITER
APPENDIX B -
Section 3.2 contained the induced current distributions in the first
wall and vacuum vessel at selected instants of time for the case of the
net current carrying limiter.
peak presure
the
These points
were deter-
and pressure distributions
at selected
locations and times.
amplitudes,
mined by evaluating
Tables were also included to summarize the
current
times throughout the transients.
This Appendix
contains a selection
of
those distributions to allow an insight to be gained into the development
of the pressure profiles along the first
wall and vacuum vessel and the
manner in which the profiles change with time.
Figures are numbered in this Appendix according to the following outline.
Each figure corresponds to a specific instant of time and they are
grouped in sets giving profiles at t-10, 20,
40, 60,
80, and 100 ms.
Cap-
tions are not included on the figures since the required information is
contained in the axis labels on each plot.
Figures B-1 to B-6:
Normal Pressure Along First Wall
Figures B-7 to B-12:
Tangential Pressure Along First Wall
Figures B-13 to B-18:
Normal Pressure Along Vacuum Vessel
Figures B-19
to
B-24:
Tangential
Pressure
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APPENDIX C - NORMAL AND TANGENGIAL PRESSURE DISTRIBUTIONS FOR THE
FIRST WALL AND LIMITER; NO NET CURRENT CARRYING LIMITER
Section 3.3 contained the induced current distributions in the first
wall and vacuum vessel at selected instants of time for the case of the net
Tables were also included to summarize the peak
current carrying limiter.
pressure amplitudes, locations and times.
evaluating the
current
and
throughout the transients.
pressure
These points were determined by
distributions
at
selected
times
This Appendix contains a selection of those
distributions to allow an insight to be gained into the development
the pressure
profiles
along
the
first
wall
and
vacuum
of
vessel and the
manner in which the profiles change with time.
Figures are numbered in this Appendix according to the following outline.
Each figure corresponds to a specific instant of time and they are
grouped in sets giving profiles at t-10, 20, 30, 40, and 50 ms.
Captions
are not included on the figures since the required information is contained in the axis labels on each plot.
Figures C-1 to C-5:
Normal Pressure Along First Wall
Figures C-6 to C-10:
Tangential Pressure Along First Wall
Figures C-11 to C-15:
Normal Pressure Along Vacuum Vessel
Figures C-16 to C-20:
Tangential Pressure Along Vacuum Vessel
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