International Workshop On Plasma Diagnostics & Applications
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INTERNATIONAL WORKSHOP ON PLASMA DIAGNOSTICS & APPLICATIONS Scaling Laws for Plasma Focus Machines from Numerical Experiments Date: 2nd – 3rd July 2009 Venue: National Institute of Education, NTU, Singapore By: S H Saw & S Lee IPFS
Contents
Introduction The Lee Model Code Computation of Neutron yield Computation of Neon SXR yield Numerical Experiments Scaling laws for neutrons from 10kJ - 25 MJ Scaling laws for neon SXR from 0.2 kJ - 1 MJ Conclusion Acknowledgement
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
2
Introduction (1/2)
Plasma focus machines: and soft x-rays.
sources of neutrons
A simple machine - UNU/ICTP PFF 3 kJ machine - 108 neutrons (in D2)
The biggest machine - PF1000 1MJ machine – 1013 neutrons (in D2)
Repetitive machine –NX2 – intense soft x-ray source (in neon) with potential applications
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
3
Introduction (2/2)
x-ray pulses from larger plasma focus devices extending to the MJ regime – Filippov et al.
Numerical experiments simulating neutron yield and soft x-ray pulses from plasma focus devices.
Institute of Plasma Focus Studies o International Internet Workshop on Plasma Focus Numerical Experiments - Lee model code computes realistic focus pinch parameters, neutrons, absolute values of soft x-ray yield Ysxr consistent with those measured experimentally.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
4
The Lee Model Code (1/8)
Realistic simulation of all gross focus properties
Couples the electrical circuit with plasma focus dynamics, thermodynamics and radiation (1984,1990)
Incorporates the vital role of a finite small disturbance speed - „communication delay effect‟ (1996, 2000)
Includes plasma self-absorption for SXR yield (2000)
Includes neutron yield, Yn, using a beam–target mechanism(2007) International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
5
The Lee Model Code - Five Phases (2/8)
Axial Phase
Radial Inward Shock Phase
Radial Reflected Shock (RS) Phase.
Slow Compression (Quiescent) or Pinch Phase
Expanded Column Phase
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
6
The Lee Model Code - Five Phases (3/8) Axial Phase
◦ Snowplow model: an equation of motion coupled to a circuit equation with axial phase model parameters: mass factor fm and current factor fc.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
7
The Lee Model Code - Five Phases (4/8) Radial Inward Shock Phase ◦ Elongating slug model:
4 coupled equations with circuit equation, thermodynamics effects(ionization and excitation), communication delay effect (finite small disturbance speed);
with radial model parameters: radial mass swept-up factor fmr and radial current factor fcr ; computes the radial inward shock speed, axial elongation speed of the column, speed of the current sheath (magnetic piston), temperature and number densities.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
8
The Lee Model Code - Five Phases (5/8) Radial Reflected Shock (RS) Phase ◦ Plasma is collisional:
Four coupled equations thermodynamics effects(ionization and excitation); with radial model parameters: radial mass swept-up factor fmr and radial current factor fcr; computes the radial outward reflected shock speed, radial inward piston speed, the axial elongation speed of the column and the circuit. The plasma temperature behind the reflected shock undergoes a jump by a factor about 2.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
9
The Lee Model Code - Five Phases (6/8) Slow Compression (Quiescent) or Pinch Phase Joule Heating and Radiation Emission: ◦ Three coupled equations:
The piston radial motion equation, the pinch column elongation equation and the circuit equation with thermodynamics effects(ionization and excitation) ; With the radial model parameters: radial mass swept-up factor fmr and radial current factor fcr; thermodynamic effects incorporated, Duration sets as the transit time of small disturbances across the pinched plasma column
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
10
The Lee Model Code - Five Phases (7/8) Expanded Column Phase ◦ Snowplow model:
two coupled equations as in the axial phase; the column attains the radius of the anode, use the expanded column inductance to compute the current. This phase is not considered important as it occurs after the focus pinch.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
11
The Lee Model Code (8/8) Institute for Plasma Focus Studies ◦ http://www.plasmafocus.net/
Internet Workshop on Plasma Focus Numerical Experiments (IPFS-IBC1) 14 April-19 May 2008
◦ http://www.plasmafocus.net/IPFS/Papers/IWPCAkeynot e2ResultsofInternet-basedWorkshop.doc
• Lee S
Radiative Dense Plasma Focus Computation Package: RADPF o
o
http://www.intimal.edu.my/school/fas/UFLF/File1RADPF. htm http://www.plasmafocus.net/IPFS/modelpackage/File1R ADPF.htm International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
12
Computation of Neutron yield (1/2) Adapted from Beam-target neutron generating mechanism (ref Gribkov et al) • A beam of fast deuteron ions close to the anode • Interacts with the hot dense plasma of the focus pinch column • Produces the fusion neutrons Given by: Yb-t= Cn niIpinch2zp2(ln(b/rp))σ /U0.5 where ni = ion density b = cathode radius, rp = radius of the plasma pinch column with length zp, σ = cross-section of the D-D fusion reaction, n- branch, U= beam energy, and Cn = calibration constant NOTE International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
13
Computation of Neutron yield (2/2) Note: •
The D-D cross-section is sensitive to the beam energy in the range 15-150 kV; so it is necessary to use the appropriate range of beam energy to compute σ.
•
The code computes induced voltages (due to current motion inductive effects) Vmax of the order of only 15-50 kV. However it is known, from experiments that the ion energy responsible for the beam-target neutrons is in the range 50-150keV, and for smaller lower-voltage machines the relevant energy could be lower at 3060keV.
•
In line with experimental observations the D-D cross section σ is reasonably obtained by using U= 3Vmax.
•
The model uses a value of Cn =2.7x107 obtained by calibrating the yield at an experimental point of 0.5 MA.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
14
Computation of Neon SXR yield (1/2) Neon SXR energy generated YSXR = Neon line radiation QL dQL 2 QL calculated from: 4.6 x10 31 ni ZZ n4 rp2 z f / T dt where : Zn = atomic number, ni = number density , Z = effective charge number, rp = pinch radius, zf = pinch length and T = temperature QL is obtained by integrating over the pinch duration.
NOTE International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
15
Computation of Neon SXR yield (2/2) Note:
•
The SXR yield is the reduced quantity of generated energy after plasma self-absorption which depends primarily on density and temperature
•
The model computes the volumetric plasma self-absorption factor A derived from the photonic excitation number M which is a function of the Zn,ni, Z and T.
•
In our range of operation the numerical experiments show that the self absorption is not significant.
•
Liu Mahe (1999) first pointed out that a temperature around 300 eV is optimum for SXR production. Shan Bing‟s (2000) subsequent work and our experience through numerical experiments suggest that around 2x106 K (below 200 eV) or even a little lower could be better.
•
Hence for SXR scaling there is an optimum small range of temperatures (T window) to operate.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
16
Numerical Experiments (1/2) The Lee code is configured to work as any plasma focus: Input:
o bank parameters: L0, C0 and stray circuit resistance r0; o tube parameters: b, a and z0 o operational parameters: V0 and P0 and the fill gas.
Standard practice: fit the computed total current waveform to an experimentally measured total current waveform using four model parameters ◦ For the axial phase: mass swept-up factor fm; plasma current factor fr;; ◦ For the radial phases radial mass swept-up factor, fmr; and radial plasma current factor fcr
Important information apparent from the current trace: ◦ Axial and radial phase dynamics ◦ Crucial energy transfer into the focus pinch
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
17
Numerical Experiments (2/2) The exact time profile of the total current trace: ◦ depends on the bank parameters, tube parameters, operational parameters, fraction of mass swept-up, fraction of sheath current in the axial and radial phases. ◦ determines the axial and radial speeds which in turn affect the profile and magnitudes of the discharge current. ◦ reflects the Joule heating and radiative yields. ◦ reflects the sudden transition of the current flow from a constricted pinch to a large column flow (at the end of pinch phase). ◦ powers all dynamic, electrodynamic, thermodynamic and radiation processes in the various phases of the plasma focus. ◦ contains information on all the dynamic, electrodynamic, thermodynamic and radiation processes that occur in the various phases of the plasma focus.
This explains the importance attached to matching the computed current trace to the measured current trace in the procedure adopted by the Lee model code. International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
18
Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (1/4)
To study the neutrons emitted by PF1000-like bank energies from 10kJ to 25 MJ. 1) Apply the Lee model code to fit a measured current trace of the PF1000: C0 = 1332 μF,V0 = 27 kV, P0 = 3.5 torr D2; b = 16 cm, a = 11.55 cm or c=1.39; z0 = 60 cm; external (or static) inductance L0= 33.5 nH and; damping factor RESF= 1.22 (or stray resistance r0=6.1 mΩ). 2) Apply the Lee model code to the PF1000 like bank energies for a range of C0 ranging from 14 µF (10 kJ) to 39960 µF (25 MJ): Voltage, V0 = 35 kV; P0 = 10 torr deuterium; RESF = 1.22; ratio c=b/a is 1.39. For each C0, anode length z0 is varied to find the optimum z0. For each z0, anode radius a0 is varied to get end axial speed of 10 cm/µs. International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
19
Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (3/5)
Fitted model parameters : fm = 0.13, fc = 0.7, fmr = 0.35 and fcr=0.65.
Computed current trace agrees very well with measured trace through all the phases: axial and radial, right down to the bottom of the current dip indicating the end of the pinch phase as shown below. PF1000: C0 = 1332 μF; V0 = 27 kV; P0 = 3.5 Torr D2; b = 16 cm; a = 11.55 cm; z0 = 60 cm; L0= 33.5 nH; r0 = 6.1 mΩ or RESF=1.22.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
20
Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (4/5)
Voltage,V0 = 35 kV; P0 = 10 torr deuterium; RESF = 1.22; ratio c=b/a is 1.39. Numerical experiments: C0 ranging from 14 µF(8.5 kJ) to 39960 µF (24 MJ) For each C0, anode length z0 is varied to find the optimum z0. For each z0, anode radius a0 is varied to get end axial speed of 10 cm/µs. Yn scaling changes: • Yn~E02.0 at tens of kJ • Yn~E00.84 at the highest energies (up to 25MJ)
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
21
Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (5/5) Scaling of Yn with Ipeak and Ipinch:
Yn vs Ipinch (higher line), Yn vs Ipeak (lower line)
Log Yn, Yn in 10 10
10000.0
Yn=3.2x1011 Ipinch4.5
y = 10-12x4.5
and
100.0
y = 7x10-12x3.8
Yn=1.8x1010 Ipeak3.8 where Ipeak = (0.3-5.7)MA
1.0 100
1000
10000
and
Ipinch = (0.2 -2.4)MA.
0.0
Log I, I in kA
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
22
Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (1/4) To study the neon SXR emitted by a modern fast bank energies from 0.2 kJ to 1 MJ. Apply the Lee model code to a proposed modern fast plasma focus machine: 1) With optimised values: c=b/a =1.5 V0 = 20 kV L0= 30 nH RESF = 0.1 Model parameters : fm=0.06, fc=0.7, fmr=0.16, fcr=0.7. 2)
For C0 varying from 1 μF (0.2 kJ) to 5000 μF (1MJ): For each C0, vary P0, z0, and a0 to find the optimum Ysxr International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
23
Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (2/4)
Computed Total Current versus Time For L0 = 30nH; V0 = 20 kV; C0 = 30 uF; RESF = 0.1; c=1.5 Model parameters : fm = 0.06, fc = 0.7, fmr =0.16, fcr = 0.7 Optimised a=2.285cm; b=3.43 cm and z0=5.2 cm.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
24
Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (3/4)
Ysxr scales as:
•E01.6 at low energies in the 0.2 to several kJ region. •E00.76 at high energies towards 1MJ.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
25
Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (4/4)
• Ysxr~Ipeak3.2 (0.1–2.4 MA) and • Ysxr~Ipinch3.6 (0.07-1.3 MA) • Black data points with fixed parameters RESF=0.1; c=1.5; L0=30nH; V0=20 kV and model parameters fm=0.06, fc=0.7, fmr=0.16, fcr=0.7. • White data points are for specific machines with different values for the parameters :c, L0, V0 etc.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
26
Conclusion (1/2)
The scaling laws obtained (at optimized condition) for Neutrons:
Yn~E02.0 at tens of kJ to Yn~E00.84 at the highest energies (up to 25MJ)
Yn =3.2x1011Ipinch4.5 (0.2-2.4 MA) Yn=1.8x1010Ipeak3.8 (0.3-5.7MA)
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
27
Conclusion (2/2)
The scaling laws obtained (at optimized condition) for neon SXR:
Ysxr~E01.6 at low energies Ysxr~E00.8 towards 1 MJ
Ysxr~Ipeak3.2 (0.1–2.4 MA) and Ysxr~Ipinch3.6 (0.07-1.3 MA)
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
28
Acknowledgement The authors acknowledge the contributions of: • Paul Lee and • Rajdeep Singh Rawat
to various parts of this paper.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
29
Papers from Lee model code
S Lee and S H Saw, “Pinch current limitation effect in plasma focus,” Appl. Phys. Lett. 92, 2008, 021503. S Lee and S H Saw, “Neutron scaling laws from numerical experiments,” J Fusion Energy 27, 2008, pp. 292-295. S Lee, P Lee, S H Saw and R S Rawat, “Numerical experiments on plasma focus pinch current limitation,” Plasma Phys. Control. Fusion 50, 2008, 065012 (8pp). S Lee, S H Saw, P C K Lee, R S Rawat and H Schmidt, “Computing plasma focus pinch current from total current measurement,” Appl. Phys. Lett. 92 , 2008, 111501. S Lee, “Current and neutron scaling for megajoule plasma focus machine,” Plasma Phys. Control. Fusion 50, 2008, 105005, (14pp). S Lee and S H Saw, “Response to “Comments on “Pinch current limitation effect in plasma focus,” Appl. Phys. Lett.94, 2009, 076102. S Lee, S H Saw, L Soto, S V Springham and S P Moo, “Numerical experiments on plasma focus neutron yield versus pressure compared with laboratory experiments,” Plasma Phys. Control. Fusion 51, 2009, 075006 (11 pp). S H Saw, P C K Lee, R S Rawat and S Lee, “Optimizing UNU/ICTP PFF Plasma Focus for Neon Soft X-ray Operation,” accepted for publication in IEEE Trans. on Plasma Science. Lee S, Rawat R S, Lee P and Saw S H. “Soft x-ray yield from NX2 plasma focuscorrelation with plasma pinch parameters” accepted for publication in JAP. S Lee, S H Saw, P Lee and R S Rawat, “Numerical experiments on plasma focus neon soft x-ray scaling”, (to be published).
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
30
THANK YOU Email: [email protected] Website: 1) http://www.intimal.edu.my/school/fas/uflf/ 2) http://www.plasmafocus.net/ 3) http://iwpca2008.intimal.edu.my
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
The Plasma Dynamics in Focus
Axial Accelaration Phase Inverse Pinch Phase HV
30 mF, 15 kV
Computation & experiment
a
experiment Pinch dissembles
Radial inward shock phase
RS phase
Expanded column phase
rp rs tcomp
tf
radius rmin t Fig 3. Schematic of radial phases
Contents
Introduction The Lee Model Code Computation of Neutron yield Computation of Neon SXR yield Numerical Experiments Scaling laws for neutrons from 10kJ - 25 MJ Scaling laws for neon SXR from 0.2 kJ - 1 MJ Conclusion Acknowledgement
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
2
Introduction (1/2)
Plasma focus machines: and soft x-rays.
sources of neutrons
A simple machine - UNU/ICTP PFF 3 kJ machine - 108 neutrons (in D2)
The biggest machine - PF1000 1MJ machine – 1013 neutrons (in D2)
Repetitive machine –NX2 – intense soft x-ray source (in neon) with potential applications
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
3
Introduction (2/2)
x-ray pulses from larger plasma focus devices extending to the MJ regime – Filippov et al.
Numerical experiments simulating neutron yield and soft x-ray pulses from plasma focus devices.
Institute of Plasma Focus Studies o International Internet Workshop on Plasma Focus Numerical Experiments - Lee model code computes realistic focus pinch parameters, neutrons, absolute values of soft x-ray yield Ysxr consistent with those measured experimentally.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
4
The Lee Model Code (1/8)
Realistic simulation of all gross focus properties
Couples the electrical circuit with plasma focus dynamics, thermodynamics and radiation (1984,1990)
Incorporates the vital role of a finite small disturbance speed - „communication delay effect‟ (1996, 2000)
Includes plasma self-absorption for SXR yield (2000)
Includes neutron yield, Yn, using a beam–target mechanism(2007) International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
5
The Lee Model Code - Five Phases (2/8)
Axial Phase
Radial Inward Shock Phase
Radial Reflected Shock (RS) Phase.
Slow Compression (Quiescent) or Pinch Phase
Expanded Column Phase
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
6
The Lee Model Code - Five Phases (3/8) Axial Phase
◦ Snowplow model: an equation of motion coupled to a circuit equation with axial phase model parameters: mass factor fm and current factor fc.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
7
The Lee Model Code - Five Phases (4/8) Radial Inward Shock Phase ◦ Elongating slug model:
4 coupled equations with circuit equation, thermodynamics effects(ionization and excitation), communication delay effect (finite small disturbance speed);
with radial model parameters: radial mass swept-up factor fmr and radial current factor fcr ; computes the radial inward shock speed, axial elongation speed of the column, speed of the current sheath (magnetic piston), temperature and number densities.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
8
The Lee Model Code - Five Phases (5/8) Radial Reflected Shock (RS) Phase ◦ Plasma is collisional:
Four coupled equations thermodynamics effects(ionization and excitation); with radial model parameters: radial mass swept-up factor fmr and radial current factor fcr; computes the radial outward reflected shock speed, radial inward piston speed, the axial elongation speed of the column and the circuit. The plasma temperature behind the reflected shock undergoes a jump by a factor about 2.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
9
The Lee Model Code - Five Phases (6/8) Slow Compression (Quiescent) or Pinch Phase Joule Heating and Radiation Emission: ◦ Three coupled equations:
The piston radial motion equation, the pinch column elongation equation and the circuit equation with thermodynamics effects(ionization and excitation) ; With the radial model parameters: radial mass swept-up factor fmr and radial current factor fcr; thermodynamic effects incorporated, Duration sets as the transit time of small disturbances across the pinched plasma column
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
10
The Lee Model Code - Five Phases (7/8) Expanded Column Phase ◦ Snowplow model:
two coupled equations as in the axial phase; the column attains the radius of the anode, use the expanded column inductance to compute the current. This phase is not considered important as it occurs after the focus pinch.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
11
The Lee Model Code (8/8) Institute for Plasma Focus Studies ◦ http://www.plasmafocus.net/
Internet Workshop on Plasma Focus Numerical Experiments (IPFS-IBC1) 14 April-19 May 2008
◦ http://www.plasmafocus.net/IPFS/Papers/IWPCAkeynot e2ResultsofInternet-basedWorkshop.doc
• Lee S
Radiative Dense Plasma Focus Computation Package: RADPF o
o
http://www.intimal.edu.my/school/fas/UFLF/File1RADPF. htm http://www.plasmafocus.net/IPFS/modelpackage/File1R ADPF.htm International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
12
Computation of Neutron yield (1/2) Adapted from Beam-target neutron generating mechanism (ref Gribkov et al) • A beam of fast deuteron ions close to the anode • Interacts with the hot dense plasma of the focus pinch column • Produces the fusion neutrons Given by: Yb-t= Cn niIpinch2zp2(ln(b/rp))σ /U0.5 where ni = ion density b = cathode radius, rp = radius of the plasma pinch column with length zp, σ = cross-section of the D-D fusion reaction, n- branch, U= beam energy, and Cn = calibration constant NOTE International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
13
Computation of Neutron yield (2/2) Note: •
The D-D cross-section is sensitive to the beam energy in the range 15-150 kV; so it is necessary to use the appropriate range of beam energy to compute σ.
•
The code computes induced voltages (due to current motion inductive effects) Vmax of the order of only 15-50 kV. However it is known, from experiments that the ion energy responsible for the beam-target neutrons is in the range 50-150keV, and for smaller lower-voltage machines the relevant energy could be lower at 3060keV.
•
In line with experimental observations the D-D cross section σ is reasonably obtained by using U= 3Vmax.
•
The model uses a value of Cn =2.7x107 obtained by calibrating the yield at an experimental point of 0.5 MA.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
14
Computation of Neon SXR yield (1/2) Neon SXR energy generated YSXR = Neon line radiation QL dQL 2 QL calculated from: 4.6 x10 31 ni ZZ n4 rp2 z f / T dt where : Zn = atomic number, ni = number density , Z = effective charge number, rp = pinch radius, zf = pinch length and T = temperature QL is obtained by integrating over the pinch duration.
NOTE International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
15
Computation of Neon SXR yield (2/2) Note:
•
The SXR yield is the reduced quantity of generated energy after plasma self-absorption which depends primarily on density and temperature
•
The model computes the volumetric plasma self-absorption factor A derived from the photonic excitation number M which is a function of the Zn,ni, Z and T.
•
In our range of operation the numerical experiments show that the self absorption is not significant.
•
Liu Mahe (1999) first pointed out that a temperature around 300 eV is optimum for SXR production. Shan Bing‟s (2000) subsequent work and our experience through numerical experiments suggest that around 2x106 K (below 200 eV) or even a little lower could be better.
•
Hence for SXR scaling there is an optimum small range of temperatures (T window) to operate.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
16
Numerical Experiments (1/2) The Lee code is configured to work as any plasma focus: Input:
o bank parameters: L0, C0 and stray circuit resistance r0; o tube parameters: b, a and z0 o operational parameters: V0 and P0 and the fill gas.
Standard practice: fit the computed total current waveform to an experimentally measured total current waveform using four model parameters ◦ For the axial phase: mass swept-up factor fm; plasma current factor fr;; ◦ For the radial phases radial mass swept-up factor, fmr; and radial plasma current factor fcr
Important information apparent from the current trace: ◦ Axial and radial phase dynamics ◦ Crucial energy transfer into the focus pinch
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
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Numerical Experiments (2/2) The exact time profile of the total current trace: ◦ depends on the bank parameters, tube parameters, operational parameters, fraction of mass swept-up, fraction of sheath current in the axial and radial phases. ◦ determines the axial and radial speeds which in turn affect the profile and magnitudes of the discharge current. ◦ reflects the Joule heating and radiative yields. ◦ reflects the sudden transition of the current flow from a constricted pinch to a large column flow (at the end of pinch phase). ◦ powers all dynamic, electrodynamic, thermodynamic and radiation processes in the various phases of the plasma focus. ◦ contains information on all the dynamic, electrodynamic, thermodynamic and radiation processes that occur in the various phases of the plasma focus.
This explains the importance attached to matching the computed current trace to the measured current trace in the procedure adopted by the Lee model code. International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
18
Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (1/4)
To study the neutrons emitted by PF1000-like bank energies from 10kJ to 25 MJ. 1) Apply the Lee model code to fit a measured current trace of the PF1000: C0 = 1332 μF,V0 = 27 kV, P0 = 3.5 torr D2; b = 16 cm, a = 11.55 cm or c=1.39; z0 = 60 cm; external (or static) inductance L0= 33.5 nH and; damping factor RESF= 1.22 (or stray resistance r0=6.1 mΩ). 2) Apply the Lee model code to the PF1000 like bank energies for a range of C0 ranging from 14 µF (10 kJ) to 39960 µF (25 MJ): Voltage, V0 = 35 kV; P0 = 10 torr deuterium; RESF = 1.22; ratio c=b/a is 1.39. For each C0, anode length z0 is varied to find the optimum z0. For each z0, anode radius a0 is varied to get end axial speed of 10 cm/µs. International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
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Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (3/5)
Fitted model parameters : fm = 0.13, fc = 0.7, fmr = 0.35 and fcr=0.65.
Computed current trace agrees very well with measured trace through all the phases: axial and radial, right down to the bottom of the current dip indicating the end of the pinch phase as shown below. PF1000: C0 = 1332 μF; V0 = 27 kV; P0 = 3.5 Torr D2; b = 16 cm; a = 11.55 cm; z0 = 60 cm; L0= 33.5 nH; r0 = 6.1 mΩ or RESF=1.22.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
20
Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (4/5)
Voltage,V0 = 35 kV; P0 = 10 torr deuterium; RESF = 1.22; ratio c=b/a is 1.39. Numerical experiments: C0 ranging from 14 µF(8.5 kJ) to 39960 µF (24 MJ) For each C0, anode length z0 is varied to find the optimum z0. For each z0, anode radius a0 is varied to get end axial speed of 10 cm/µs. Yn scaling changes: • Yn~E02.0 at tens of kJ • Yn~E00.84 at the highest energies (up to 25MJ)
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
21
Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (5/5) Scaling of Yn with Ipeak and Ipinch:
Yn vs Ipinch (higher line), Yn vs Ipeak (lower line)
Log Yn, Yn in 10 10
10000.0
Yn=3.2x1011 Ipinch4.5
y = 10-12x4.5
and
100.0
y = 7x10-12x3.8
Yn=1.8x1010 Ipeak3.8 where Ipeak = (0.3-5.7)MA
1.0 100
1000
10000
and
Ipinch = (0.2 -2.4)MA.
0.0
Log I, I in kA
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
22
Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (1/4) To study the neon SXR emitted by a modern fast bank energies from 0.2 kJ to 1 MJ. Apply the Lee model code to a proposed modern fast plasma focus machine: 1) With optimised values: c=b/a =1.5 V0 = 20 kV L0= 30 nH RESF = 0.1 Model parameters : fm=0.06, fc=0.7, fmr=0.16, fcr=0.7. 2)
For C0 varying from 1 μF (0.2 kJ) to 5000 μF (1MJ): For each C0, vary P0, z0, and a0 to find the optimum Ysxr International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
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Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (2/4)
Computed Total Current versus Time For L0 = 30nH; V0 = 20 kV; C0 = 30 uF; RESF = 0.1; c=1.5 Model parameters : fm = 0.06, fc = 0.7, fmr =0.16, fcr = 0.7 Optimised a=2.285cm; b=3.43 cm and z0=5.2 cm.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
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Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (3/4)
Ysxr scales as:
•E01.6 at low energies in the 0.2 to several kJ region. •E00.76 at high energies towards 1MJ.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
25
Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (4/4)
• Ysxr~Ipeak3.2 (0.1–2.4 MA) and • Ysxr~Ipinch3.6 (0.07-1.3 MA) • Black data points with fixed parameters RESF=0.1; c=1.5; L0=30nH; V0=20 kV and model parameters fm=0.06, fc=0.7, fmr=0.16, fcr=0.7. • White data points are for specific machines with different values for the parameters :c, L0, V0 etc.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
26
Conclusion (1/2)
The scaling laws obtained (at optimized condition) for Neutrons:
Yn~E02.0 at tens of kJ to Yn~E00.84 at the highest energies (up to 25MJ)
Yn =3.2x1011Ipinch4.5 (0.2-2.4 MA) Yn=1.8x1010Ipeak3.8 (0.3-5.7MA)
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
27
Conclusion (2/2)
The scaling laws obtained (at optimized condition) for neon SXR:
Ysxr~E01.6 at low energies Ysxr~E00.8 towards 1 MJ
Ysxr~Ipeak3.2 (0.1–2.4 MA) and Ysxr~Ipinch3.6 (0.07-1.3 MA)
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
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Acknowledgement The authors acknowledge the contributions of: • Paul Lee and • Rajdeep Singh Rawat
to various parts of this paper.
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
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Papers from Lee model code
S Lee and S H Saw, “Pinch current limitation effect in plasma focus,” Appl. Phys. Lett. 92, 2008, 021503. S Lee and S H Saw, “Neutron scaling laws from numerical experiments,” J Fusion Energy 27, 2008, pp. 292-295. S Lee, P Lee, S H Saw and R S Rawat, “Numerical experiments on plasma focus pinch current limitation,” Plasma Phys. Control. Fusion 50, 2008, 065012 (8pp). S Lee, S H Saw, P C K Lee, R S Rawat and H Schmidt, “Computing plasma focus pinch current from total current measurement,” Appl. Phys. Lett. 92 , 2008, 111501. S Lee, “Current and neutron scaling for megajoule plasma focus machine,” Plasma Phys. Control. Fusion 50, 2008, 105005, (14pp). S Lee and S H Saw, “Response to “Comments on “Pinch current limitation effect in plasma focus,” Appl. Phys. Lett.94, 2009, 076102. S Lee, S H Saw, L Soto, S V Springham and S P Moo, “Numerical experiments on plasma focus neutron yield versus pressure compared with laboratory experiments,” Plasma Phys. Control. Fusion 51, 2009, 075006 (11 pp). S H Saw, P C K Lee, R S Rawat and S Lee, “Optimizing UNU/ICTP PFF Plasma Focus for Neon Soft X-ray Operation,” accepted for publication in IEEE Trans. on Plasma Science. Lee S, Rawat R S, Lee P and Saw S H. “Soft x-ray yield from NX2 plasma focuscorrelation with plasma pinch parameters” accepted for publication in JAP. S Lee, S H Saw, P Lee and R S Rawat, “Numerical experiments on plasma focus neon soft x-ray scaling”, (to be published).
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
30
THANK YOU Email: [email protected] Website: 1) http://www.intimal.edu.my/school/fas/uflf/ 2) http://www.plasmafocus.net/ 3) http://iwpca2008.intimal.edu.my
International Workshop on Plasma Diagnostics & Applications Scaling Laws for Plasma Focus Machines from Numerical Experiments -S H Saw & S Lee
2nd – 3rd July 2009
IPFS
The Plasma Dynamics in Focus
Axial Accelaration Phase Inverse Pinch Phase HV
30 mF, 15 kV
Computation & experiment
a
experiment Pinch dissembles
Radial inward shock phase
RS phase
Expanded column phase
rp rs tcomp
tf
radius rmin t Fig 3. Schematic of radial phases
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