1. liquid wall ablation under ife photon energy deposition 2
TRANSCRIPT
May 5-6, 2003/ARR 1
1. Liquid Wall Ablation under IFE Photon EnergyDeposition
2. Flibe Properties from EoS
A. René Raffray and Mofreh ZaghloulUniversity of California, San Diego
ARIES Town Meeting on Liquid Wall Chamber DynamicsHilton Garden Inn, Livermore, CA
May 5-6, 2003
May 5-6, 2003/ARR 2
Physical Processes in X-Ray Ablation
EnergyDeposition &
Transient HeatTransport
InducedThermal- Spikes
MechanicalResponse
PhaseTransitions
•Stresses and Strainsand HydrodynamicMotion•Fractures and Spall
• Surface Vaporization•Heterogeneous Nucleation•Homogeneous Nucleation(Phase Explosion)
Ablation Processes
Expansion,Cooling and
Condensation
SurfaceVaporization
Phase ExplosionLiquid/Vapor
Mixture
Spall Fractures
Liquid
FilmX-Rays
Fast Ions
Slow Ions
Impulse
Impulse
sy
sx
sz
May 5-6, 2003/ARR 3
High Photon Heating Rate Could Lead to Explosive Boiling
Photon-likeheating rate
Ion-likeheatingrate
• Effect of free surface vaporization isreduced for very high for heating rate(photon-like)
• Vaporization into heterogeneous nucleiis also very low for high heating rate
From K.Song and X.Xu, AppliedSurfaceScience 127-129 (1998)111-116
• Rapid boiling involving homogeneousnucleation leads to superheating to ametastable liquid state
• The metastable liquid has an excessfree energy, so it decomposesexplosively into liquid and vaporphases.
- As T/Ttc increases past 0.9, Becker-Döhring theory of nucleation indicate anavalanche-like and explosive growth of nucleation rate (by 20-30 orders of magnitude)
May 5-6, 2003/ARR 4
Analysis Based on Photon Spectra for 458 MJ IndirectDrive Target
• High photon energy for indirect drive target case (25% compared to 6% ion energy))• More details on target spectra available on ARIES Web site:
http://aries.ucsd.edu/ARIES/
(25%)
(1%)
May 5-6, 2003/ARR 5
Photon Energy Deposition Density Profile in Flibe Filmand Explosive Boiling Region (for Rchamber=6.5m)
1x107
1x108
1x109
1x1010
1x1011
1x1012
0 5 10 15
Ener
gy d
epos
ition
(J/m
3 )
Penetration depth (micron)
Cohesion energy (total evaporation energy)
2.5
Evap.region
10.4
2-phase region
Sensible energy (energy to reach saturation)
Sensibleenergy basedon uniformvaporpressurefollowingphotonpassage inchamber andincludingevaporatedFlibe fromfilm
0.9 Tcritical
4.1
Explo.boil.region
Bounding estimates of thermally-induced aerosol source term from photon energy deposition:(1)Upper bound: the whole 2-phase region; (2)Lower bound: explosive boiling region
May 5-6, 2003/ARR 6
1x106
1x107
1x108
1x109
1x1010
1x1011
1x1012
1x1013
0 1 2 3 4 5
Ener
gy d
epos
ition
(J/m
3 )
Penetration depth (micron)
Photon Energy Deposition Density Profile in Pb Film and Explosive Boiling Region (for Rchamber=6.5m)
Cohesion energy (total evaporation energy)
Sensible energy (energy to reach saturation)
0.9 Tcritical
1.9 2.5 3.9
2-phase region
Explo.boil.region
Evaporated region
May 5-6, 2003/ARR 7
Explosive Boiling Results for Rchamber=3.5 m
Explosive Boiling Ablated Thickness for:Flibe Pb
Rchamber= 6.5 m 4.1 mm 2.5 mm Rchamber= 3.5 m 10.9 mm 3.83 mm
May 5-6, 2003/ARR 8
Mechanical Response to Induced Shock
• Rapid increase in internal energy due to x-ray energy deposition andablation impulse creates high pressure within the material
- The induced shock wave propagates to the wall, gets reflected back to the freesurface where a rarefaction wave is produced (creating tensile stresses) andpropagates back through the material. The process is repeated.
- If the magnitude of the rarefaction wave is greater than the tensile strength ofthe material, fracture or spall will occur establishing a new surface.
• Evolution of spall in a body subject to transient stresses is complex
- Material dependent: brittle, ductile or liquid
- Small perturbations can lead to opening of voids and initiation of spall process
- A reasonable prediction of the dynamic spall strength, time to failure, and some measure of the nominal fragment size created in the spall event are needed to characterize the spall process
- An upper bound theoretical spall strength can be derived from intermolecularpotential
May 5-6, 2003/ARR 9
Theoretical (Maximum) Spall Strength Provides an UpperBound Estimate in the Absence of Appropriate Spall Data
˙˚
˘ÍÎ
Ș¯
ˆÁË
Ê ---˜
¯
ˆÁË
Ê --=
a)vv(
exp2a
)vv(2expU)v(U 00
coh
UUcoh= Specific cohesive energy
v = 1/r = Specific volume
v0 = Specific volume at zero pressure
a=(2v0 Ucoh / B0 )1/2
B0 = Bulk modulus
˙˚
˘ÍÎ
Ș¯
ˆÁË
Ê ---˜
¯
ˆÁË
Ê --=-=
a)vv(
expa
)vv(2exp
aU2
dvdU
)v(P 00coh
- The cold pressure is given by:- The cold pressure is given by:
TheoreticalTheoretical spall spall strength,strength, P Pthth,given by minimum of P(v):
†
Pth =Ucoh B0
8 v0
• Based on intermolecular potential reflecting dependence on cohesive energy and bulk modulus with inherent energy balance
- Using a three-parameter potential such as the Morse potential
May 5-6, 2003/ARR 10
Spall Strength is Highly Temperature Dependent
• Using the Soft Sphere EOS to Estimate Temperature-Dependent Spall Strength
˙˙˚
˘
ÍÍÎ
Ș̃¯
ˆÁÁË
Ê-˜̃
¯
ˆÁÁË
Ê++˜̃
¯
ˆÁÁË
Ê++=
TKTKQ)4n(
61
TKC
23
TKNU)T,v(UB
m
3/1
B
9/n
B
3/nnBcoh
er
er
er
˙˙˚
˘
ÍÍÎ
Ș̃¯
ˆÁÁË
Ê-˜̃
¯
ˆÁÁË
Ê++˜̃
¯
ˆÁÁË
Ê+=
TKm
TKQ)4n(n
181
TKCn
31
1V
TKN)T,v(P
B
m
3/1
B
9/n
B
3/nn
B er
er
er
• Theoretical spall strength is then calculated from:
.0vd
)T,v(Pd=
n, m, Q, e, and s are adjustable parametersto satisfy the available experimental data
N: Number of molecules,
V: Specific volume,
r= N s3 / (21/2) V,
s : the sphere diameter,
Cn: FCC Madelung constant.
May 5-6, 2003/ARR 11
Temperature-Dependent Spall Strengths of Example Materials
-0.0657Gas-0.22353749
-0.2814-0.1010-0.52212999-0.6848-0.4267-0.89812250-1.4212-0.8950-1.40981450-2.4914-1.4401-2.0014750
Flibe (GPa)Li (GPa)Pb (GPa)T (K)
May 5-6, 2003/ARR 12
Parameters of Different IFE Reactor Design Studies &Pressure Pulse Profile
17.3
0.008
6.5
23.31
0.04
3.5
15.2
0.026
6.5
115
Pb/SiC
3.553.55Closest distance from target(m)
45.77
0.022
115
Flibe/SS
Present Study
6059.022Reactive impulse** (Pa s)
0.02340.02780.0125Vaporized mass (kg/m2)
95129*31X-ray yield (MJ)
Pb-17Li/SiCFlibe/CPb/SiCLiquid/Structure
HiballOsirusPrometheus-LParameters
* X-ray and debris
• The pressure wave is steady (nochange in shape)
• Parameters of different IFEreactor design studies and thepresent study are comparable
• Osirus profile scaled according tothe relative impulses and used forthe present analysis.
** Ablated material velocity ~ sonic velocity ~ 586/2094 m/s for Pb/Flibe at Tcrit (~5100/4500 K)
May 5-6, 2003/ARR 13
Illustration of Spalling Following a Pressure WavePropagation in the Flibe Layer on a Perfectly Stiff Wall
r @ 2000 kg/m3, Cs @ 3300 m/s, Tin = 885.7 K, Pth = -1.887 GPa
Spalled Thickness @ 2.1 µm & Spall Time (t3 – t1) @ 16.9 ns
Spall time from the beginning of the pressure pulse = 2 L/Cs+ (t3-t1)@ 200 ns for a 0.3 mm flibe layer
1. For a perfectly stiff wall,the pressure wave isreflected from the wall andreturns to the free surfaceas a pressure pulse
2. Pfree-surface= Pchamber and thepressure pulse arriving atthe free boundary must bereflected back as a tensilewave
3. If the net tensile stress >the spall strength of thematerial, rupture occursestablishing a newsurface
May 5-6, 2003/ARR 14
Summary for the Case of Perfectly Stiff Wall
3.66.2R=6.5 m
5.615.0R=3.5 mTotal fragmentedthickness
(µm)
1.12.1R=6.5 m
1.84.1R=3.5 mSpallation thickness(µm)
2.54.1R=6.5 m
3.810.9R=3.5 mExplosive boilingthickness
(µm)
PbFlibe
May 5-6, 2003/ARR 15
BUT No Wall is Perfectly Stiff !
†
P = G P0 =(Zwall - Z film )(Zwall + Z film )
P0
6.09C46.7SS5.75SiC
19.31Pb6.84Flibe
Z (Mkg/m2 s)MaterialA pressure pulse with initial pressure P0 will be reflected atthe mismatched interface with a new pressure, P where:
With the acoustic impedance Z = r Cs
In principle, combination ofliquid and structural materialcan be chosen to eliminate orminimize spalling
E.g. For Pb/SiC and 3.5 m Cavity Radius fi G= -0.541• Pressure wave is reflected as a reduced tensile wave.• The max. pres., Pmax, of the reflected wave = -6.67 GPa• Spall tensile strength, Pth, = -1.875 GPa
• ÁPmaxÁ>ÁPthÁ fi Spall occurs (at 0.94 µm from the wall)
May 5-6, 2003/ARR 16
1.784
3.585
6.8 > |Pth|
18.2 > |Pth|
0.745(reflected as a
reduced pressurepulse)
0.745(reflected as a
reduced pressurepulse )
Flibe/SS
-0.541(reflected as areduced tensile
pulse)
-0.058(reflected as areduced tensile
pulse)
R = 3.5m
G-0.541
(reflected as areduced tensile
pulse)
-0.058(reflected as areduced tensile
pulse)
R = 6.5m
L-0.240.0R = 6.5m
L-0.960.0R = 3.5mSpallation
thickness(µm)
4.35 > |Pth|0.53< |Pth|R = 6.5m
6.67 > |Pth|1.4 < |Pth|R = 3.5m
Magnitude ofthe maximum
reflectedpressure
Pmax(GPa)
Pb/SiCFlibe/C
Spallation Analysis for Different ProposedLiquid/Wall Combinations
May 5-6, 2003/ARR 17
Summary of Study of Wall Ablation Mechanisms asAerosol Source Term
• Integrated effect of liquid wall thermal and mechanical responsesto X-ray energy deposition to provide bounding estimates ofablation as source term for aerosol analysis- First principle consideration
- Ablation depths of liquid wall from explosive boiling and spalling
- Spall strength of materials as compared to anticipated IFE shocks
- Acoustic impedances of structural material and liquid can be chosen to avoid or minimize spalling
• Some key remaining questions- Form (vapor conditions, size and distribution of liquid droplets) of the
removed material?
- What happens to the spalled material (not enough energy to push it out…where does it go?)
- Spall time scale
May 5-6, 2003/ARR 18
Flibe Properties from EoS
M. ZaghloulUniversity of California, San Diego
ARIES Town Meeting on Liquid Wall Chamber DynamicsHilton Garden Inn, Livermore, CA
May 5-6, 2003
May 5-6, 2003/ARR 19
Flibe Thermodynamic Properties
Liquid+
Vapor
Vaporchemical
decomposition of molecules
Plasmamultiple
ionization
TTcritcrit ~ 4500 K~ 4500 K
T ~ 1 T ~ 1 eVeV
Tem
pera
ture
Te m
pera
ture
ß The soft-sphere equation of state for liquid flibetakes into account the dependence on r and T(*).
ß Vapor pressure correlations from ORNL & Olanderet al., Fusion Technology,41, 141 (2002).
ß Chemical equilibrium codes that use data fromJANAF tables are used (e.g. the computer codeSTANJAN, Stanford Univ.)
ß Fitted equations of state for flibe gas(**).
ß Inaccuracies include
- Use of Post-Jensen ionization data calculated from theCorona equilibrium (valid only for extremely low densities and very high temperatures-optically thin plasma-and show no dependence on density).
- Post-Jensen data give no information on the individualpopulations needed for the internal energy computationand so further approximations have to be made.
- No Coulomb corrections.
- No excitation energies included in the computations ofinternal energies.
(*)(*) Xiang Xiang M. Chen M. Chen , Virgil E. Schrock, and Per F. Peterson ““ The soft-sphere equation of state for liquid The soft-sphere equation of state for liquid Flibe Flibe,,”” Fusion Tech. 21, 1525 (1992). Fusion Tech. 21, 1525 (1992).
(**) Xiang M. Chen, Virgil E. Schrock, and Per F. Peterson, “Fitted Equation of State for Flibe Gas,” Fusion Technology 26, 912 (1994).
Improved Modeling andComputations in the High-Temperature Region are Needed
(Revisited)
May 5-6, 2003/ARR 20
Flibe Vapor Pressure and Stability of LiBeF3(*)
(*) Mofreh Zaghloul, D. K. Sze, A. R. Raffray, “Thermo-physical Properties and Equilibrium VaporComposition of Lithium-Fluoride Beryllium-Fluoride (LiF/BeF2) Molten Salt ,” 15th Topical Meeting on the
Technology of Fusion Energy, TOFE, Washington, DC, Nov. 2002.
• Based on previous experimental results by Buchler and Stauffer(***), Olanderet al(**) assumed a LiBeF3 contribution of ~10% to flibe vapor, and estimatedthe LiBeF3 vapor pressure based on a negative energy of formation (DG0 < 0)
• However, from Knack et al. and using NIST-JANAF tables, DG0 for LiBeF3 >0 for T > 555 K (Meta-stable)
• Accordingly, Olander et al. overestimated the vapor pressure of LiBeF3 by~ 30 orders of magnitude
• However the effect on their estimation of the overall vapor pressure of flibe ismuch smaller since LiBeF3 contribution was assumed to be ~10%
(**) Olander et al., Fusion Technology,41, 141 (2002).
(***) O. KNACKE, O. KUBASCHEWSKI and K. HESSELMANN (Eds.), Thermochemical Properties of Inorganic Substances I, second edition, Springer-Verlag, Berlin, Germany (1991).
)g(3LiBeF)g(2BeF)g(LiF =+
May 5-6, 2003/ARR 21
Proposed Correlation for Flibe Vapor PressureOver a Wide Range of Temperature
One can combine the valuescomputed for flibe vaporpressure using thermo-chemical data from NIST-JANAF tables in the lowtemperature region toORNL measurements(adequately accurate for T>1000 C) and use a best fitto obtain a correlation validover a wide range of temp.
)K(T/208.11026424.9Plog Torr10 -=
May 5-6, 2003/ARR 22
AssumptionsI- Fully-Dissociated Flibe-Gas (T> 1 eV) (Mixture of Monatomic Gases);
(**) T. Fujimoto et al, Phys. Rev. A 42, 6588 (1990).
(***) W. Ebeling et al, Theory of Bound States and Ionization Equilibrium in Plasmas and Solids (Akademic-Verlag, 1976)
Ionization Equilibrium and Thermodynamic Functions(*)(*)Mofreh R. Zaghloul, “A consistent model for the equilibrium thermodynamic functions of partially
ionized Flibe plasma with Coulomb Corrections,” Physics of Plasmas, 10, No. 2 (2003) 527-538.
˜̃¯
ˆÁÁË
Ê-= Â
£
= TK
EexpgU
B
n,j,rIE
1nn,j,rj,r
effj,rn,j,r
Terminated Partition Function
j,rj,reff
j,r III D-=
Continuum Lowering(***)
V- Coulomb Corrections
F,Be,Lij,ZZi3
1jav
3
1jj,e
max,Z
1ij,i
j
===Â ÂÂ= ==
aIV- Electro-neutrality.
III- No Nuclear Reactions (Constant Number of Heavy Particles ) nH= Const.;
5.1nuc
a
67nuc
e2nuc
183c
)Z/49.0(55.0a
,10Z
TZ105.1)cm(n
-=
˜¯
ˆÁË
Ê¥=-
II- Local Thermodynamic Equilibrium LTE(Validity Criterion by Fujimoto(**))
May 5-6, 2003/ARR 23
Thermodynamic Relations
CoulavBH
NNT
PZTKnV
FP
ke
D++=˜̃¯
ˆÁÁË
Ê
∂
∂-= )1(
},{,
˜˜¯
ˆÁÁË
Ê+˜̃
¯
ˆÁÁË
Ê+-+-+=
˜¯
ˆÁË
涶
-=⋅+==
  Â= = =
Coul
J
1j
Z
1i
i
1zi,jzziH0BavH
}N,N{,V
2
EW)II(n)TT(K)Z1(n231
TF
TT
1)STF(
1E
1e
jmax,
ke
DDar
rrr
1- Pressure
2- Internal energy
3- Contribution ofexcitation energies ofion i of element j tointernal energy ( ) ( )ÂÂ
-£-£
--=
∂∂
=
i,ji,j,i,ji,ji,j,i,j II
B,i,j,i,j
II
B,i,j,i,j,i,j
i,j2
Bi,j
TKexpgTKexpg
UlnT
TKW
Dw
zzz
Dw
zzzz
zz
www
4- Enthalpy )Pe(1
he += rr
5- Adiabatic exponent ,C/C vP=g
2/1
T
2/1
S
s
PPv ˙
˚
˘ÍÎ
Ș̃¯
ˆÁÁË
Ê
∂∂
=˜̃¯
ˆÁÁË
Ê
∂∂
=r
gr6- Sound Speed
The most rigorous expressions for thermodynamic functions have been used
May 5-6, 2003/ARR 24
Results of Detailed Composition & Average Ionization State
• As the temperature increases, the molarfractions of neutral species (Li, Be, and F)decrease monotonically as a result of theprogressive ionization.
• With further increase of the temperature,higher ionized ionic species appear on theexpense of lower-fold ionized species.
• The lower the density the lowerrecombination rate, and the higher <Z>.
• The lower the density the more prominent isthe step-like structure of the <Z>.
• At very high T, the value of <Z> approachesits expected value of a fully-dissociated,fully-ionized flibe-plasma , <Z> = 6.57.
May 5-6, 2003/ARR 25
Results for Flibe Pressure & Internal Energy
• Substantial deviation from the ideal-gas linear behavior.• Ionization and Coulomb interactions among charged particles are responsible
for the deviation in the pressure curves, while for internal energy anothercontribution comes from excitation.
May 5-6, 2003/ARR 26
Results for Flibe Specific Heat, Adiabatic Exponent & Sound Speed
• Non-linearity characteristics of hightemperature plasma are most prominent.
• At very high T, Cp increases to its expectedvalue of a fully-dissociated, fully-ionized flibe-plasma (Cp = 1.1138¥104 J/kg K) and so for theother properties.
May 5-6, 2003/ARR 27
ResultsValidity of LTE Assumption & Effect of Coulomb Corrections
• LTE assumption is applicable fordensities ≥ 0.001 kg/m3 for thetemperature range shown.
• <Z> is the most affected parameter atlow temperatures.
• Percentage relative differences ofother parameters at the same densityare bounded within ± 15 %.
May 5-6, 2003/ARR 28
Comparison with Chen’s et al Results (*).
(*) Xiang M. Chen, Virgil E. Schrock, and Per F. Peterson, Fusion Technology 26, 912 (1994).
• Differences of up to ~ 65% • Differences of up to ~ 150%
May 5-6, 2003/ARR 29
Optical Characteristics of Flibe Multi-Frequency AbsorptionCoefficient
In the evaluation of the multi-frequency absorption coefficient, kn
• Molecular absorption is neglected (the plasma temperature is considered to besufficiently high > 1 eV that the plasma is fully dissociated);
• Photon energies are low enough (< 1MeV) that high-energy absorption processes suchas pair production and nuclear photoabsorption can be ignored;
• Only electronic transitions in the field of the ions are considered (bound-bound, bound-free, and free-free) as these constitute the major absorption processes for mostconditions of interest.