CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
OD MODELISATION OF THE MAGNETIC FLUX COMPRESSION
SCHEME FOR ISENTROPIC COMPRESSION EXPERIMENTS
P. L'Eplattenier !, G. Avrillaud 2, J. Vanpoperynghe3
1
Centre d'Etudes de Gramat, 46500 Gramat, France
2
ITHPP, 46500 Thegra, France
3
CEA/DAM Bruyeres le Chatel, France
Abstract This paper deals with the advantages of using a flux compression scheme in High Pulsed
Power (HPP) generators in order to produce isentropic magnetic high pressure ramps. Our field of
interest are pressures in the range of several Mbars.For isentropic compression above 1 Mbar, the main
advantage of the flux compression is to give considerable freedom to shape the current waveform and
thus the magnetic pressure waveform. The optimizations of the shape of this magnetic pressure
waveform are performed using improved OD codes. Some physical models have been added to the initial
OD circuit code to reproduce the experimental results obtained on Z at Sandia National Laboratories, and
on ECF1 and ECF2 at Centre d'Etudes de Gramat. Moreover, a simple hydrodynamic model is used to
determine the isentropicity of the compression in a given material. We present some comparisons
between experimental results obtained on the Z generator and our models. We will also give potential
improvement of those results based on the use of a coil as a stator in order to have a better shape of the
current waveform.
use in our OD models to judge the isentropicity of a
given pressure waveform. Section II presents the
magnetic flux compression scheme as an
intermediate stage to improve ICE experiments. It
also introduces the numerical tools developed at
CEG to optimize this stage. Finally, section III
shows an example of such an optimization on the Z
generator at Sandia National Laboratories (SNL).
INTRODUCTION
Isentropic compression loading in a sample allows
to determine the isentrope of the material Pso(p)
issued from the initial condition P0, T0, poSince a few years, there has been a new interest in
High Pulsed Power (HPP) generators for the
generation of quasi isentropic compression
experiments (ICE/11 In those experiments, the
pressure is generated by a magnetic field created by
a current flowing on one side of a sample. In order
to get quasi-isentropic compression above IMbar,
the shape of the pressure waveform and thus of the
current waveform is critical.
The magnetic flux compression scheme has first
been introduced on pulsed power generators as a
power amplification scheme[2]. As shown in the
following, it can also be used as a current waveform
shaper with no changes on the generator itself.
Section I exposes the hydro techniques that we
L HYDRO TECHNICS FOR PRESSURE
OPTMISATION
A current I(t) generates a pressure P(t) on one side
of a sample (referenced as the loaded side). At the
back side of the sample, the free surface velocity
can then be measured by the VIS AR technique. The
simultaneous measurement of the free surface
velocity on 2 samples with 2 different thicknesses
allows to compute the pressure versus the density
along the isentrope, Pso(p), for 0 < Pso ^ PexPi_isen,
1188
formation of a shock in the thicker sample limiting
PexpLisen to Pshock=P(tShock) at time Uock corresponding
to the first characteristics where the shock occurs.
The C0" could also reach the loaded side of the
thinner sample at a time tretum limiting Pexpi_isen to
Pretum=P(tretum). Finally, the free surface velocity
measured with the VISAR does not allow too fast a
variation of this velocity (and thus of the pressure
on the loaded side) limiting Pexpi_isento Pmes(tmes).
The hydro model allows, for a given shape P(t) of
the pressure, to calculate, as functions of the
thickness of a sample, PShock> Pretum and Pmes, as well
as Pisen^niinOPshoc^Pretum^Pmes). The model then finds
two thicknesses TI and T2 with AT=T2-Ti large
enough, at least 50jam so that the VISAR method
can be practically exploitable, and such that Pexpi_isen
= minffisen (TI), Pisen (T2)) is as high as possible.
This is shown on figure 2, where Pmes has been
omitted for clarity reasons.
where PeXpi_isen is the maximal exploitable pressure
in isentropic compression. This pressure is extracted
from the so called characteristics method well
known in the hydrodynamic field (fig. 1).
When a pressure is applied on a sample, the
information propagates from the loaded side to the
back side on characteristics. The first characteristic
generated from the loaded side is called C0+. It
reflects on the back side and creates the Co". In the
area below C0", the characteristics are straight lines
with a slope given by the Lagrangian sound
velocity, C i = C s ( p ) _^.
Po
Using Us=Co+S.Up
and by assuming f _ ]
=f
_] ,
Uujs UuJH
one can derive the following equation for Q :
c, = c
JL with
P*
P* =
PQC o
4S
;
with Cs the sound velocity, p0 the density at P=Po,
Us the shock velocity, UP the particle velocity, C0
the sound velocity at P=P0 and S a material constant.
In the area above C0~, the characteristics are not
straight lines any more. However, they can be well
approximated by straight lines or parabolas. The
flow can thus be analytically resolved for all the
characteristics coming from the loaded side before
the C0" reaches it. A shock forms in the sample when
2 characteristics intersect before the back side.
A T •.
Sample thickness
FIGURE 2. Schematic of the determination of Pexpijsen
In the frame of this model, for a given thickness,
one can find the ideal pressure waveform to get
isentropic compression as high as possible. An ideal
pressure waveform is computed so that all the
characteristics converge on the same point at a
distance from the loaded side XQ larger than T2. One
is then ensured there is no shock formation in the
sample. The highest value of the pressure is then
limited by the time tretum when the Co" in a sample
with a thickness Ti=T2-AT reaches the loaded side.
One thus have with such a pressure waveform
Pexpi_isen = PT2 (tretum) in the thinner sample.
FIGURE 1. Illustration of the characteristics method, in
lagrangian coordinates
Two VISAR diagnostics can be used to
simultaneously measure the free surface velocity on
2 samples with different thicknesses TI < T2 for
which the same pressure waveform P(t) has been
applied on the loaded side. The maximal exploitable
pressure in isentropic compression Pexpi_isen is not
necessarily the maximal value Pmax of P(t).
Depending on the shape of P(t), there could be a
EL THE FLUX COMPRESSION SCHEME
The magnetic flux compression scheme (fig.3) is
usually used as a power amplification scheme on
HPP generators as an alternative to pulse forming
water lines or plasma opening switches. It can also
be used as an intermediate stage in between the
1189
main generator and the load that allows to shape the
current waveform.
Two different currents are needed. The first one is
generated using typically 3/4 of the whole stored
energy and is called the primary current. It is used to
implode a primary liner called armature, usually
made with an aluminum wire array for conductivity
reasons.
The
armatures
already
tested
experimentally were between 6 and 15 cm long with
an initial radius between 4 and 5 cm and made with
108 to 428 wires. The other current, generated with
the rest of the stored energy is then injected inside
the armature through a secondary injection gap. It
thus flows in the central rod, usually called stator,
on the load and on the inner part of the armature. It
is called the secondary circuit. When the liner
passes through the injection gap, the flux injected in
the secondary circuit is trapped and since the
inductance between the armature and the stator
decreases with the radius of the armature, the
current in the secondary circuit will rises at the same
time.
compression. They are fast running codes that give
promising results in good agreement with the
experimental ones, as shown on figure 4. This figure
shows the experimental secondary current compared
to the one given by the OD code for different flux
compression shots on generator Z at SNL.______
FIGURE 4. Comparison of experimental and numerical
secondary currents for different shots on generator Z
We then have developed optimization procedures
in multi dimensional spaces based on the gradient
method driving the OD codes, allowing to complete
our basic analytical optimization of the scheme.
m. APPLICATION: OPTIMIZATION OF AN
ISENTROPIC COMPRESSION EXPERIMENT
Those procedures can be used to optimize the flux
compression parameters in order to get Pexpi_isen as
high as possible for a given generator. Once such an
optimization is done, one can compare the obtained
pressure waveform with the ideal pressure
waveforms presented in section I. We now present
an example of optimization of the magnetic pressure
on a sample with isentropic compression on the Z
generator at SNL. We started with a set of
parameters corresponding to the next flux
compression shot on Z (scheduled for July 2001)
and which will be called case 1 in the following. For
that shot, for experimental reasons, we decided to
keep the central rod as a full cone. We thus have
optimized the liner mass, the small and big radius of
the cone, and the gap of the secondary area. We
numerically
obtained
for
that
shot
Pexpi_isen=2.7Mbars with 2 sample thicknesses
Ti=555^im and T2=605fim. figure 5 shows how the
pressure waveform compares to an ideal pressure
waveform for a 500 fj,m thick sample. The general
Injection gap
FIGURE 3. Schematic of the flux compression scheme
The advantage of that scheme is that it introduces
many parameters that allow to shape the current
waveform in the load: the armature initial radius,
length, mass, the radius of the injection gap, the
shape of the stator that can be a cylinder, a cone or a
coil, and, depending on the generator, the levels of
primary and secondary currents. 2D MHD codes can
be used to simulate the flux compression scheme.
However, their long CPU times make them
practically not usable for the optimization all the
parameters of the flux compression. For the purpose
of optimizing those parameters, we have thus
developed at CEG improved OD codes. Those are
circuit codes coupled with some physical models
reproducing the main feature of the flux
1190
shape is already pretty good, but the first part
corresponding to the injection of the secondary
current and the time before the amplification is far
from being optimal yet.______________
This figure shows that we get pressure waveforms
much closer to the optimal ones than with a solid
stator. The concept of a coil as a stator is still to be
tested experimentally, in particular for magnetic
insulation problems at the secondary injection gap,
but this example shows how efficient it could be to
use such a stator.
CONCLUSION
We have presented in this paper the use of HPP
generator coupled with the flux compression scheme
for isentropic loading on material samples. The flux
compression is an intermediate stage with many
degrees of freedom allowing to shape the pressure
waveform in order to get the highest possible
pressure in isentropic compression. We also
presented the numerical tools developed at CEG in
order to optimize those degrees of freedom.
The first numerical studies gave very promising
results. However, they have to be confronted with
more experimental results and 2D MHD
simulations. In particular, the magnetic field
diffusion in the sample have to be checked
carefully. The use of a coil as a stator should be very
efficient but could also present possible
experimental issues that have to be checked.
The ECF2 generator will be on line at CEG by the
fall of 200 P3]. This 3.6 MJ energy stored generator
using a flux compression stage will allow to get
many experimental results on isentropic
compression loading.
FIGURE 5. comparison of actual and optimal pressure
waveforms for case 1
We then introduced a coil as the stator to show
how case 1 could be improved in further
experiments. The dependence of the inductance in
the secondary circuit with the radius of the armature
is then completely different than with a solid conical
stator. There is now a large inductance at crowbar
time, and a very fast varying one when the armature
gets closer to the stator. We first have chosen a coil
with a constant pitch and made an optimization on
the flux compression parameters, including the
pitch. This case will be referenced as case 2 in the
following. We obtained after optimization
PexPi_isen:=3.21Mbars,
with
T1=420jam
and
T2=470|a,m. After that we made a 3rd case where the
coil has a linearly varying pitch and made again the
optimization. We obtained then Pexpi isen=3.55Mbars
with Ti=430^tm and T2=480|o,m. Figure 6 shows the
actual and ideal pressure waveforms for both cases 2
and 3.
ACKNOWLEDGMENTS
The authors are very indebted to J.F. Leon and the
ECF1/ECF2 team at CEG; and also to R.B.
Spielman, M. MazaraMs, and the Z Team at SNL
for their support.
REFERENCES
^ S.L Krivosheev, « Pulsed Current Generatorfor
Microsecond Duration Pressure Pulse Generation », 12*
Symposium on High Current Electronics, Tomsk, 24-29
Sept. 2000.
[21
L J
J.F. Leon, « Flux Compression Experiments on Z
accelerator », in Pulsed Power Conference- 1999
^ P. Monjaux, in Pulsed Power Conference- 2001.
FIGURE 6. Comparison of actual and optimal pressure
waveforms for cases 2 and 3
1191