IPFS
knowledge should be freely
accessible to all
Institute for Plasma Focus Studies
Internet Workshop on Plasma Focus Numerical Experiments
Module 4; (Follow the instructions in the following notes.
You may also wish to refer to the supplementary notes part4supplementary.htm.
Summary:
This module looks at variation of neutron yield with
pressure; running PF1000-from short circuit (very high pressure), through
optimum pressure to low pressure. The very high pressure of the short circuit
shot stops all current sheath motion thus simulating a short circuit. The aim
is just to obtain the short-circuit current waveform for comparison with the
focusing waveforms. In a second example we also look at variation of SXR with
pressure; operating NX2 from short circuit (very high pressure), through
optimum pressure to low pressure. In the course of these numerical experiments
we take a small detour (during the NX2 experiments) to determine circuit
parameters from a short circuit discharge; something very basic, but often
overlooked. At the end of the course two additional exercises are given, one
comparing computed and measured Yn vs
For the PF1000 neutron experiments:
Steps: (a) Configure the code for the PF1000 at 27kV 3.5 Torr Deuterium using model parameters which we had fitted earlier.
(b) Fire the PF1000 at very high pressure, effectively a short circuit.
(c) Fire the PF1000 at lower pressures from 19 Torr down to 1 Torr; looking for optimum neutron yield.
(d) Place current waveforms at different pressures on the same chart; for comparative study. Completing this chart forms part of Exercise 5.
(e) Tabulate results at different pressures; for comparative study; including speeds, dimensions, duration, temperature and neutron yield. This completes Exercise 5. Discussion.
For the NX2 SXR experiments:
Steps: (a) Configure the NX2 at 11kV 2.6 Torr Neon using fitted model parameters
(b) Fire the NX2 at very high pressure, effectively a short circuit; first introduction to macro code modification.
(c) Detour:
Use this short circuit waveform as though it were a measured current waveform, to analyse the lightly damped L-C-R discharge; to fix bank parameters.
(d) Fire NX2 at 5 Torr; as an example of insufficient
current drive; over-riding the model’s time-match guard.
(e) Fire NX2 at lower pressures down to 0.5 Torr, looking for optimum SXR yield.
(f) Place current waveforms at different pressures on the same chart, for comparison.
(g) Tabulate results at different pressures; for comparative study; including speeds, dimensions, duration, temperature and neutron yield. Discussion.
The sessions ends with a general consideration of plasma
focus yield scaling.
The material:
You need RADPF5.15dd.xls(called RADPF5.15dd.xls) for the following work. Copy and Paste on your Desktop. You also need the files PF1000pressureblank.xls and NX2pressureblank.xls.These files contain also tabulation blanks for your convenience.
Also provided is file HiRepHiPerformPF.doc from which NX2 Ysxr
vs
Three additional files are provided for two additional exercise which you may complete at your leisure later. These are: PF400Yncomparison.xls and an accompanying paper pf400_soto.pdf for the first additional exercise for you to duplicate. The other paper (Saturation….pdf) goes with the final open exercise suggested as an epilogue to this course.
Part 1: Neutron Yield
of PF1000 vs Pressure
(a) Preparing Sheet 2 and Configure the code for PF1000
Open RADPF5.15dd.xls. Copy PF1000pressureblank.xls to Sheet 2 using procedure which we have done before; repeated briefly as follows.
With RADPF5.15dd.xls open; open PF1000pressureblank.xls;
click the Edit Tab; scroll down and click 'Move or Copy file'. A window pops
out. In the 'To book’: choose ‘RADPF5.15dd.xls;
then choose ‘move to end’; click ‘OK’. Rename ‘Sheet1(2)’ as Sheet2.
The Current waveforms are now displayed in the chart in Sheet2 of RADPF5.15dd.xls.
[PF1000pressureblank.xls has time-current data for several traces, and scrolling to the right, a table of plasma focus properties at various pressures to be filled in, and below that a normalized table; and there are also two charts; one for the current traces at various pressures and one for Yn, Ipeak, Ipinch vs pressures. Have a close look at the opened
sheet to see the locations of the supplied time-current data, the blank spaces for you to fill in the other computed time-current data, the tables with the blank spaces to be filled in, and the partially filled in charts.]
Use the data in PF1000pressureblank.xls to configure.
Bank: Lo=33.5
nH, Co=1332 mF, ro=6.1mOhm
Tube: b=16
cm, a=11.55 cm, zo=60 cm
Operation: Vo=27kV,
Model: fm=0.13, fc=0.7, fmr=0.35,
fcr=0.65
(b) Firing a very high pressure shot.
Key in 100,000 Torr at B9. [In the laboratory it is of course impossible
to fire such a shot and a physical short-circuit may need to be used at the
insulator end of the plasma focus; or fire at the highest safe pressure in argon. In the lab we have used 50
Torr argon, to obtain very approximate results.]
{In the numerical experiment at this high pressure the
current sheath only moves a little down the tube, adding hardly any inductance
or dynamic loading to the circuit. So it is equivalent to short-circuiting the
plasma focus at its input end. In the code there is a loop during the axial
phase, computing step- by- step the variables as time is incremented. The loop
is broken only when the end of the anode (non-dimensionalised z=1) is reached.
In this case we do not reach the end of the anode. However there is an alternative
stop placed in the loop that stops the run when (non-dimensionalised time=6 ie
nearly 1 full cycle time, 2p, of
the short- circuited discharge) is reached. Moreover at the start of the run,
the code computes a quantity ALT= ratio of
characteristic capacitor time to sum of characteristic axial & radial
times. Numerical tests have shown that when this quantity is less than 0.65,
the total transit time is so large (compared to the available current drive
time) that the radial phase will not be efficiently completed. Moreover because
of the large deviation from normal focus behaviour, the numerical scheme and
‘house keeping’ details incorporated into the code may become subjected to
numerical instabilities leading to error messages. To avoid these problems a
time-match guard feature has been incorporated to stop the code from being run
when ALT<0.65. When this happens one
can over-ride the stop; and continue running unless the run
is then terminated by Excel for e.g. ‘over-flow’ problems. In that case one has
to abandon the run and reset the code.}
Fire the high pressure shot. The
Visual Basics Code appears at Statement 430 Stop; with a warning message that
pressure is too high. In this case we know what we are doing, and over-ride as
follows: Click on ‘Run’ (above the code sheet), and ‘continue’. Another ‘Stop’
appears just below line 485; with a warning about transit time. Click on ‘Run’
and ‘continue’; another ‘Stop’ appears below Line 488.. Click on ‘Run’ and
‘Continue’.
In a little while the run has proceeded and
finally the statement “If T > 6 Then Stop”
appears; indicating we have completed nearly one cycle of the capacitor
discharge.
Now, locate the ‘x’ at the extreme
right hand corner of the screen. Click on this ‘x’; pop-up appears with the
message ‘This command will stop the debugger’. Click on OK, which brings you
back to the worksheet 1.
Copy the data in columns A & B
from A20 and B20 to the end of the computed current data (several thousand
cells down); ‘paste’ the copied
time-current data onto Columns E & F (in the labeled space provided in
Sheet 2. Locate the data table by scrolling to the right. Fill in the value of
Ipeak [read from Fig 1 or from the relevant cell] onto the table
against 100,000 Torr; all the other quantities (Ipinch, peak va,
S, peak vs …. T and Yn….being zero. )
(c) Fire at different pressures Place
Fire the next shot at 19 Torr+. As the ALT value is over 0.65, the
run proceeds as normal. Copy the time-current data from Columns A & B (from
rows 20 down) to Sheet 2 columns G & H. Fill in the table [Ipeak,
Ipinch, peak va, S, peak vs … T… Yn…ni**
& EINP )
for the data from shot 19 Torr. [+Note: The waveform and data for this
point, is already copied for you, to save you some time.]
** The data for ni is
output in column AK in Sheet 1 (you need to scroll way down as these are
outputted only for the pinch phase).
Repeat for pressures 14,10,
9, 8, 7.5 , 7, 6, 3.5 , 2 and 1; tabulating the data for all
these shots; but copy and paste the time-current data for only selected shots
e.g 14, 10, 6 and 2 [some of the shots are pre-filled for you to save you some
time]. The list of pressures had been
chosen as above, as I carried out the numerical experiments. It was clear that
Yn was increasing rapidly from 14 Torr to 10 Torr. More points were
chosen between 10 Torr & 6 Torr until it was obvious that the optimum
pressure (for Yn) was between 8 and 7 Torr.
(d) Place the current waveforms at
different pressures on the same chart.
Suggested
procedure: To save you time, the
comparison chart has already been created for you, and pre-filled with several
waveforms namely 19, 7.5, 3.5 and 1 Torr. You only have to fill in the ones for
1,000,000 and 14, 10,6 and 2 Torr in the correct columns indicated by the
column headings already placed on Sheet 2. You will note that the computed
current waveform for 3.5 Torr falls neatly on the measured current waveform (as
you have seen during an earlier exercise precisely with this PF1000 27kV 3.5
Torr current waveform.
(e) Tabulate results at different pressures ; for comparative study; including speeds, dimensions, duration, temperature and neutron yield. Discussion.
This tabulation has already been
done as step (c) proceeded above.
In order to chart some of the computed
data on one comparative chart, below the table you have filled in, there is
another table with each data column normalized to the data at 7.5 Torr, which
was found to be the pressure with the highest Yn. Thus the values of
all the data in the normalized table is in the region of 1.
Plot normalized Yn, Ipeak,
Ipinch, and radial EINP against
[As you fill in the table, the normalized quantities are
automatically computed, and the chart begins to take the correct shape. At the start the chart is in a jumble because
many points have not been filled in, and thus there are erratic zero points all
over the place.]
Discussion
Note 1: Look at the change of
current waveforms from very high pressures to low pressures. At very high
pressures the waveform is a damped sinusoid. At 19 torr the characteristic
flattening of the current waveform due to dynamics is already clearly evident.
The current peak comes earlier than the unloaded (high pressure) case, the
current then droops until the rollover into the dip (due to the increased
radial phase loading) at around 15 us. At lower pressures these characteristics
remain the same except that the current trace is depressed more and more as
speed increases. The peaking (reaching maximum current) also comes earlier and
earlier, as does the radial phase rollover of the current trace. At 2.6 Torr,
there is hardly any droop, the current waveform showing a distinct flat top
leading to the rollover. At 1 Torr the axial speed is now so high that the
axial phase is completed in less than 5 us and the current is still rising when
it is forced down by the radial phase dynamics.
Note2 : A very important point to note in neutron scaling
is that there exist a great deal of confusion and even misleading information in
published literature because of sloppy practice with regards to Ipeak
and Ipinch. These quantities are sometimes treated as one and the
same or when a distinction is attempted there is then a confusion between the
total current at the time of pinch and Ipinch. For example in the
case of PF1000, there appears to be some disappointment(in their publications)
that (at 35kV) with the current at more than 2 MA, Yn is still at
best in the mid 10^11; and not at least
an order of magnitude higher that one might expect for currents around
2MA. However if you numerically run
PF1000 at 35kV you will find that Ipinch is only 1MA; so we are not
surprised that the measured yield is at best an order of magnitude down from
what you would expect thinking that your current is around 2MA. (scaling at Yn~I4
, a factor of 2 in current gives a factor of 16 in the yield). So it is
important that the thinking of yield should be in terms of Ipinch as
the relevant scaling parameter. When using the model, the distinction of Ipinch
and Ipeak is clear.
Coming to the detailed
tabulations: As Po decreases, Ipeak
decreases, and continues to decrease, because the increasing axial speeds
increases the circuit loading, throughout the whole range of pressures. However
it is noticed that Ipinch
increases from high pressures, peaking in a flat manner at 6 Torr and then
drops more sharply towards 1 torr. One factor contributing to the increase is
the shift of the pinch time from very late in the discharge
(when discharge current has
dropped greatly) to earlier in the
discharge (when current has dropped less). That is the main factor for Ipinch
increasing despite a decreasing Ipeak. At low pressures (e.g.1
Torr), the radial phase now occurs so early that it is forcing the current down
early in the discharge. That lowers both the Ipeak as well as the Ipinch.
These points are clear when you look at the comparative chart of current traces
at various pressures.
The radial EINP follows the same
pattern as Ipinch, and for the same reasons. The radial EINP
computes the cumulative work done by the current sheath in the radial phases.
Looking at the other quantities,
we note that the speeds (axial, radial shock and radial piston) and temperature
all continue to rise as pressure lowers; similarly S and maximum induced
voltage V also increase as pressure is decreased. Pinch length zmax
is almost a constant. Minimum pinch radius and pinch duration continue to
decrease; the former due to better compression at higher speeds and the latter
due to the increased T. The number density progressively drops, due to the
decreasing starting numbers, despite the increasing compression.
From the tabulations of the above
numerical experiments, it might be useful to consider the beam-target mechanism
which we are using to compute the neutron yield. This is summarized in the
following note.
[ note: From part4supplementary.htm
Yb-t= Cn ni Ipinch2zp2((lnb/rp))s/Vmax1/2
where s is the D-D fusion cross section. In the range we are
considering we may take s~Vmaxn where n~2-3; say we take n=2.5; then we have
Yb-t ~ ni Ipinch2zp2((lnb/rp))
Vmax2
The factor zp2((lnb/rp))
is practically constant.
Thus we note that it is the behaviour of ni ,Ipinch and Vmax as pressure changes
that determines the way Yn increases to a maximum and then drops as
pressure is changed.}
[An additional experiment is
suggested at the end of these notes, in which you can see how numerical
experiments on Yn vs operating pressure compare with measured
results in the case of PF400]
Part 2: Soft X-ray
Yield of NX2 with Operating Pressure
Prepare the worksheets for the experiment. Open RADPF5.15dd.xls. Insert Sheet 2 & Sheet 3.. Copy NX2pressureblank.xls to Sheet 2.
[NX2pressureblank.xls has 2 worksheets, Sheet 2 and Sheet 3. Sheet 2 has time-current data for several traces, and scrolling to the right, a table of plasma focus properties at various pressures to be filled in, and below that a normalized table; and there are also two charts; one for the current traces at various pressures and one for YSXR, Ipeak, Ipinch vs pressures. Have a close look at the opened sheet to see the locations of the supplied time-current data, the blank spaces for you to fill in the other computed time-current data, the tables with the blank spaces to be filled in, and the partially filled in charts. Sheet 3 has labeled spaces for the computed high pressure current data, a chart and spaces to be filled in for data to be measured from the current waveform.]
Using the same procedures as suggested for the previous PF1000 experiments, copy and paste NX2pressureblank.xls Sheet 2 into Sheet 2 of RADPF5.15dd.xls Then copy the data and chart of NX2pressureblank.xls Sheet 3 into Sheet 3 of RADPF5.15dd.xls
Sheet 2 and Sheet 3 are now ready to receive the data of the numerical experiments.
(a) Configure the NX2 at 11kV 2.6 Torr Neon using fitted model parameters
We use an earlier version of the NX2 with a lower inductance of 15nH.
The parameters for that version of NX2 were successfully fitted as:
Bank: Lo=15
nH, Co=28 mF, ro=2.2 mOhm
Tube: b=4.1cm,
a=1.9 cm, zo=5 cm
Operation: Vo=11kV,
Model: fm=0.1, fc=0.7, fmr=0.12,
fcr=0.68
(b) Fire the NX2 at very high pressure, effectively a short circuit; first introduction to macro code modification.
Key in 1,000,000 Torr at B9. [In the laboratory it is of course
impossible to fire a shot at such high pressure] {In the numerical experiment at this high pressure the current sheath
only moves a little down the tube, adding hardly any inductance or dynamic
loading to the circuit. So it is equivalent to short circuiting the plasma
focus at its input end. In the code there is a loop during the axial phase,
computing step by step the variables as time is incremented. The loop is broken
only when the end of the anode (non-dimensionalised z=1) is reached. In this
case we do not reach the end of the anode. However
there is an alternative stop placed in the loop that stops the run when
(non-dimensionalised time=6 ie nearly 1 full cycle time, 2p, of
the short circuited discharge) is reached. Moreover at the start of the run, the code
computes a quantity ALT= ratio of
characteristic capacitor time to sum of characteristic axial & radial
times. Numerical tests have shown that when this quantity is less than 0.65,
the total transit time is so large (compared to the available current drive
time) that the radial phase will not be efficiently completed. Moreover because
of the large deviation from normal focus behaviour, the numerical scheme and
‘house keeping’ details incorporated into the code may become subjected to
numerical instabilities leading to error messages. To avoid these problems a
time-match guard feature has been incorporated to stop the code from being run
when ALT<0.65. When this happens one can over-ride the stop; and continue
running unless the run is then terminated by Excel for e.g. ‘over-flow’
problems. In that case one has to abandon the run and reset the code.}
We want to use the NX2 in
short-circuit mode to illustrate the basic but often overlooked treatment of a
lightly damped L-C-R circuit for determining circuit parameters. The method we
use requires determining the reversal ratio of the lightly damped discharge.
For this purpose we would like to have say 3 cycles of the lightly damped
discharge ie we should continue computing until normalized time reaches 6p~20.
Since the code has a stop placed at t=6, we need to make a change in this
statement in the code.
We have RADPF5.15dd.xls opened. We will now
‘step into’ the code to edit it.
Above the worksheet, locate the
control button ‘Tools’. Click on ‘Tools’ then ‘Macro’, then ‘Macros’. Then highlight ‘radpf005’ and click
on the button ‘Step Into’. The program code in Visual Basic appears. We have
entered the code.
Scroll down to line 580. Just
below this line is the Statement “If T
> 6 Then Stop”. Change the number ‘6’
to the number ‘20’. Then Exit the code by clicking the ‘x’ at the extreme
top right hand corner above the spreadsheet. When drop-down appears with
message “This command will stop the debugger” click on the button ‘OK’;
bringing us back to Sheet 1.
The code is now configured to run
the discharge short-circuited for 3 cycles before stopping.
Fire the high pressure shot. The
Visual Basics Code appears at Statement 430 Stop; with a warning message that
pressure is too high. In this case we know what we are doing, and over-ride as
follows: Click on ‘Run’ (above the code sheet), and ‘continue’. Another ‘Stop’
appears just below line 485; with a warning about transit time. Click on ‘Run’
and ‘continue’; another ‘Stop’ appears below Line 488.. Click on ‘Run’ and
‘Continue’.
In a little while the run has proceeded and
finally the statement “If T > 20 Then Stop”
appears; indicating we have completed more than 3 cycles of the capacitor
discharge.
Now, locate the ‘x’ at the extreme
right hand corner of the screen. Click on this ‘x’; pop-up appears with the
message ‘This command will stop the debugger’. Click on OK, which brings you
back to the worksheet 1.
(c) Use this short circuit waveform as
though it were a measured current waveform, to
analyse the lightly damped L-C-R discharge; to measure discharge period T and reversal
ratio f; hence determine Lo and ro. Only Co
& Vo are assumed to be known.
Copy the current waveform data
from Columns A & B and paste to Sheet 3 into the columns A & B starting
from A5 & B5; so that we may carry out our little ‘detour’ experiment. To
save you some time the chart has been prepared in advance and the current
waveform should appear; once the data is pasted correctly starting at A5 and
B5.
From the
current waveform: measure 3 T (to 3 decimal places); hence obtain T.
Measure the successive peak
currents, recording all as positive values. Thus measure:
f1=I2/I1,
f2=I3/I2, f3=I4/I3,
f4=I5/I4 and f5=I6/I5;
and f=(1/5)(f1+f2+f3+f4+f5).
We are given Co and Vo.
With the measured T and f (measured from the current waveform) we calculate Lo
and ro and Io using the following approximations
applicable to slightly damped L-C-R discharges:
Lo=T2/(4p2Co)
ro=-(2/p)Ln(f)(Lo/Co)0.5
Io=pCoVo(1+f)/T
We note from this little ‘detour’ that
this method gives highly accurate results for lightly damped discharges. In
practice the accuracy is limited by experimental features such as electrical
noise and electrostatic shielding of the coil which may result in a tilted zero
baseline. We also note that it is important for every plasma focus to establish
reliable baseline data. First, the capacitance Co should be reliably
known or determined. Then from the value of Co, Lo and ro
may be fixed; and further Io deduced to calibrate the monitoring coil.
Also copy the 1,000,000 Torr
time-current data to Sheet2 to into the columns provided for this purpose (E
& F)
(d) Fire NX2 at 5 Torr; as an
example of insufficient current drive; over-riding the model’s time-match
guard.
We now proceed to the NX2 SXR vs
pressure experiment.
Key in 5 Torr in B9. Fire a shot.
The Visual Basics Code appears at
Statement 430 Stop; with a warning message that pressure is too high. In this
case we know what we are doing and over-ride as follows: Click on ‘Run’ (above
the code sheet), and ‘continue’. Another ‘Stop’ appears just below
line 485; with a warning about
transit time. Click on ‘Run’ and ‘continue’; another ‘Stop’ appears below Line
488. Click on ‘Run’ and ‘Continue’.
In a little while the run has
completed successfully. In this manner we force the code to run even though the
code warns us that the pressure is too high for a good shot.
Copy the time-current data
(A20-B20 to several thousand rows down) for this shot and paste into the
reserved and labeled space (already done for you in columns Q &R) in Sheet
2. Add the data ( Ipeak, Ipinch, Peak va, S,
Peak vs, vp, amin, zmax, pinch
duration etc ) for this shot to the table prepared for this purpose (scroll a
little to the right for this table.).
(e) Fire NX2 at lower pressures down to 0.5 Torr, looking for optimum YSXR
In a similar way, force the code
to run for 4.5 Torr (with an ALT=0.64; so need to force).
Add data to table.
Continue with the following shots: 4 Torr (ALT=0.68, so code runs without ‘Stop’ breaks) 3.5, 3.2, 3, 2.9, 2.8, 2.7, 2.6, 2.4, 2, 1.5, 1, 0.5; adding the data for each shot to the table; but transferring the time-current data to sheet 2 of only those shots in bold [we want to plot a few current traces to see the way the traces evolve with pressure]
(f) Place current waveforms at different pressures on the same chart, for comparison.
The selected current traces are plotted onto the same chart in Sheet 2. When we plot the curve for 2.6 Torr, note that the computed current trace falls neatly over the measured; as these have already been pre-fitted.
(g) Tabulate results at different pressures; for comparative study; including currents, speeds, dimensions, duration, temperature and neutron yield.
Discussion.
We note the way we are computing
the neon SXR line radiation; with power of:
Hence the SXR energy generated
within the plasma pinch depends on the properties:
Number density ni
Effective charge number Z
Pinch radius rp
Pinch length zf and
Temperature T
and Pinch duration ; since the
power is integrated over the pinch duration.
This generated energy is then
reduced by the plasma self-absorption which depends primarily on density and
temperature; the reduced quantity of energy is then emitted as the SXR yield.
It was first pointed by Liu Mahe
in his PhD thesis “Soft X-rays from Compact Plasma Focus” NTU/NIE 1996, that a
temperature around 300 eV is optimum for SXR production. Our subsequent
experience through numerical experiments suggest that around 2x106 K
(below 200eV) seems to be better.
Important note: Unlike the case of neutron scaling, for SXR scaling
there is an optimum small range of temperatures (T window) to operate. This could be the most important
point to observe for SXR scaling.
With these complicated coupled
effects and the small T window I have doubts about such simplistic scaling laws
as put forward from time to time: Ysxr~Ipinch4/rmin2
???-doubtful
In this present series of
experiments on the NX2 we note that a peak yield of 21J is obtained at 2.9 Torr
Neon at a temperature of 1.5x106 K (computed at the middle of the
pinch duration). This compares well with experimental data in Zhang Guixin’s
1999 PhD thesis, in his series of yield vs pressure experiments at 11.5 kV
using the NX2 (in the configuration of our numerical experiments; our measured
current waveform was taken from his series of experiments). In that series He
obtained a peak yield of 20J at 3.3 Torr with yield fall-off similar to our
numerical experiments, although his curve peaks less sharply as our results.
Zhang’s experimental results are
plotted as black points on the chart for comparison with the computed Ysxr
vs pressure. Note that the computed yield at optimum pressure is comparable with
the measured optimum yield; that the optimum pressure also compare well as is
the falloff of yield to either side of the optimum pressure.
General notes on fitting, Yield Scaling and applications of
the code
On fitting: We
soon learn that one is not able to get a perfect fit; in the sense that you can
defend it as absolutely the perfect fit. The way to treat it is that one has
got a working fit; something to work with; which gives comparable results with
experiments; rather than perfect agreement. There is no such thing anyway;
experiments in Plasma Focus (i e on one
PF under consistent conditions) give a range of results; especially in yields
(factor of 2-5 range is common). So a working fit should still give results
within the range of results of the hardware experiment.
Even though a fit may only be a 'working' fit (as opposed to the hypothetical perfect fit) when one runs a
series of well planned numerical experiments one can then see a trend e.g. how
properties, including yields, change with pressure or how yields scale with Ipinch,
or with Lo etc. And if carefully carried out, the numerical
experiments can provide, much more easily, results just like hardware
experiments; with the advantage that after proper reference to existing
experiments, then very quickly one can extend to future experiments and predict
probable results.
On scaling: Data used
for scaling should be taken from yield-optimized (or at least from near
optimized) situations. If one takes from the worst case situations e.g. way out
in the high pressure or low pressure regions, the yield would be zero for a
non-zero Ipinch. Such data would completely distort the scaling
picture.
Not only should the pressure be
changed, but there should be consideration for e.g. suitable (or even optimized)
Ipinch/a; as the value of Ipinch/a
would affect the pressure at which optimized S is achieved.
On directions of work
and applications: Efforts on the
model code may be applied in at least two directions. The first direction is in
the further development of the code; e.g. trying to improve the way the code
models the reflected shock region or the pinch region.
The second direction is to apply
the model to provide a solution to a particular problem. An example was when it
was applied to look at expected improvements to the neutron yield of the PF1000
when Lo is reduced.
Using the model code it was a
relatively easy procedure, firing shots as Lo was reduced in steps;
optimizing the various parameters and then looking for the optimized neutron yield
at the new value of Lo. When this exercise was carried out in late
2007, for PF1000 at 35 kV, unexpectedly it was found that as Lo was
reduced from 100nH in steps, in the region around 35nH Ipinch
achieved a limiting value; in the sense that as Lo was reduced
further towards 5 nH, whilst Ipeak continued to increase to above
4MA, Ipinch dropped slightly from its maximum value of 1.05 MA to
just below 1 MA. This Pinch Current Limitation Effect could have considerable
impact on the future development of the plasma focus. No longer should one
think in terms of Ipeak, instead it is always Ipinch that
matters. Increasing Ipeak does not necessarily mean increasing Ipinch.
On numerical experiments
to enhance experience & intuition: Moreover the relationship between Ipeak
and Ipinch is implicit in the coupling of the equations of circuit
and motion within the code which is then able to handle all the subtle
interplay of static and dynamic inductances and dynamic resistances and the rapid
changes in distributions of various forms of energies within the system. Whilst
the intuitive feel of the experienced focus exponents are stretched to the
limit trying to figure out isolated or integrated features of these interplays,
the simplicity of the underlying physics is captured by the code which then
produces in each shot what the results should be; and over a series of shots
then reveal the correct trends; provided of course the series is well planned.
So the code may also be useful to
provide the numerical experimenter time-compressed experience in plasma focus
behaviour; enhanced experience at much reduced time. At the same time the
numerical experimenter can in a day fire a number of different machines,
without restrictions by time, geography or expense. The problem then becomes
one of too much data; sometimes overwhelming the experience and intuition of
the numerical experimenter.
On versatility: of the work you have carried out in these 8
sessions plus the additional experiments given below, one of which is
practically completed for you to duplicate; the other is completely open for
you to explore. Your numerical experiments have included
examining plasma focus behaviour
comparing BIG, medium size and small plasma focus, looking for common and
scalable parameters. You studied neutron and SXR yields as functions of
pressure, comparing computed with experimental data. In 8 sessions involving
some 20-30 hours of hands-on work you have ranged over a good sampling of
plasma focus machines and plasma focus behaviour.
This was all done with one code
the RADPF05.13.9b, the universal plasma focus laboratory facility. We should
have the confidence that if we explore the open experiments suggested in 2.
below, that could lead us to new areas and new ideas.
Additional Exercises:
1. As an
additional exercise which you can look at later, you are provided with an additional file PF4000YnComparison.xls.
This file records data of measured Yn (from Leopoldo Soto’s paper, also attached)
and comparison with computed data using
RADPF5.15dd.xls You will see
that the agreement between our computed data and Soto’s published data of neutron yield vs
pressure may be considered to be good;
features of comparison include the magnitude of the optimum yield, the optimum
pressure and the fall-off on each side of the optimum pressure. You may wish to
verify the comparison by running the numerical experiments yourself.
2. An open exercise: V Yu Nukulin & S
N Polukhin recently (2007) published a paper (attached)
discussing the saturation of neutron yield from megajoule PF facilities. Using an analytical method they surmised
that in big plasma focus devices if storage energy is increased by essentially
increasing storage capacitance Co
then Ipeak reaches a limiting value of around 2MA. This is because
as Co increases, the current risetime increases and of necessity the
anode length has to be increased. Thus the increased effective inductance on
the circuit balances out the increase in
Reference to this course and the Lee model code should be
given as follows:
Lee S. Radiative Dense Plasma Focus Computation
Package (2008) : RADPF www.plasmafocus.net http://www.intimal.edu.my/school/fas/UFLF/
End Part 4-End of course- Internet Workshop on
Plasma Focus Numerical Experiments
14th
April to 19th May 2008
[Comments and interaction on the
course work and other matters related to plasma focus are welcome at anytime]
Postscript: Numerical experiments at IPFS in connection with the open exercise has resulted in a paper on extending Plasma Focus scaling to multi-megajoule level. Also see list of papers
S Lee
Institute for Plasma Focus Studies,
This activity is
carried out in association with AAAPT and
the Plasmas Groups of