July 1, 2002/ARR1
Scoping Study of FLiBe Evaporation and Condensation
A. R. Raffray and M. Zaghloul
University of California, San Diego
ARIES-IFE Meeting
General Atomics
San Diego
July 1-2, 2002
July 1, 2002/ARR2
Outline
• FLiBe properties used in analysis
- Vapor pressure as a function of temperature
- Other properties
• Condensation rates and characteristic time for FLiBe
• Aerosol source term - Photon energy deposition and explosive boiling
- Estimate of amount of FLiBe expulsed from surface
• End-goal: example parameter window plot
July 1, 2002/ARR3
FLiBe Vapor Pressure
-3
-2
-1
0
1
2
3
4
5
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
Log P (Pa), ORNL
Log P (Pa), Olander
Log P (Pa), Zaghloul fit
1000/T (1/K)
FLiBe Vapor Pressure as a function of Temperature
ORNL Olander
• e-mail communications among UCSD, ORNL and Berkeley
• From Olander’s calculated values: log10 (P(Torr)) = 9.55-11109.56/T(K); T= 773 K - 973 K
• From ORNL’s measured values: log10 (P(Torr))=9.009-10444.11/T(k); T= 1223 K - 1554 K
• Expression used for the analysis (fit from above two):
- log10 (P(Torr))=9.3806-10965.26/T(K); T = 773 K - 1554 K
July 1, 2002/ARR4
Physical Properties for Film Materials
= 4.9810-8 TT2 2 - 2.05 T + T +
38.01
= 8.965106 - 2347 T T (* *)
(= 5.3x106 at 1687 K)
= 46.61 - 0.003 T T +4.7710-7 TT22
(=8.6x106 at 1893 K))hfg (J/kg)
519.73 – 0.01 T2413 - 0.488 T11291 – 1.1647 TDensity (kg/m3)
452732.2600.5Tmelting (K)
LiFLiBe Pb Property
Log10 (Ptorr) =
7.764 - 7877.9/T
Log10 (Ptorr) =
9.38 - 10965.3/TLog10 (Ptorr) = 7.91 - 9923/TVapor Pressure
32234498.8 (* )4836 Tcritical (K)
2347
1687
4227.57 - 0.0733 T =183.6 -0.07 T -1.6x106 T-2
+ 3.5x10-5 T2+ 5x10-9 T3
CP (J/kg-K)
15901893Tboiling, 1 atm (K)
All temperatures, T, in KAll temperatures, T, in K
(*) Xiang M. Chen, Virgil E. Schrock and Per F. Peterson, “The Soft-Sphere Equation of State for Liquid FLiBe,” (*) Xiang M. Chen, Virgil E. Schrock and Per F. Peterson, “The Soft-Sphere Equation of State for Liquid FLiBe,” Fusion Technology, Vol. 21, 1992.Fusion Technology, Vol. 21, 1992.
(**) Derived from Cp and the Cohesive energy @ 1atm.(**) Derived from Cp and the Cohesive energy @ 1atm.
July 1, 2002/ARR5
Condensation Flux and Characteristic Time to Clear Chamber as a Function of FLiBe Vapor and Film Conditions
• Characteristic time to clear chamber, tchar, based on condensation rates and FLiBe inventory for given conditions
jcond
jevap
TfPg
Tg
jnet=MR2π
⎛ ⎝ ⎜ ⎞
⎠ ⎟
0.5Γσc
Pg
Tg0.5
−σePf
T f0.5
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
1x100 1x101 1x102 1x103 1x104
Vapor Pressure (Pa)
FLiBe:Film temperature = 800KFilm Psat = 0.0063 Pa
Vapor velocity = 0
Vapor Temp. (K)
1000
6000
3000
2000
• For ~0.1 s between shots Pvap prior to next shot could be up to ~10 x Psat
• Not likely to be a major problem
• Of more concern is aerosol generation and behavior
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æ
æ
æ æ æ æ æ æ æ æ æ
ø
ø
ø ø ø ø ø ø ø ø ø
”
”
”
” ” ” ” ” ” ” ” ”
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
1x10-2 1x10-1 1x100 1x101 1x102 1x103 1x104
Vapor pressure (Pa)
ƒ
æ
ø
”
FLiBe film temperature = 800KFilm Psat = 0.0063 Pa
Vapor velocity = 0Chamber radius = 5 m
Vapor Temp.
6000 K
3000 K
2000 K
1000 K
• For higher Pvap (>0.1 Pa for assumed conditions), tchar is independent of
• For lower Pvap as condensation slows down, tchar increases substantially
July 1, 2002/ARR6
X-ray Spectra and Cold Opacities Used in Aerosol Source Term Estimate
• X-ray spectra much softer for indirect drive target (90% of total energy associated with < 5 keV photons
• Cold opacity calculated from cross section data available from LLNL (EPDL97)
July 1, 2002/ARR7
Photon Energy Deposition in Vapor and Film
• Example Vapor Pressure = 1 mTorr - Corresponding to TPb= 910 K; TFLiBe= 886 K; TLi= 732 K
• Photon energy deposited in vapor for R=6.5 m: - ~1% in Pb vapor - < 0.1% in FLiBe vapor
• For film, most of photon energy deposited:- within order of 1 m for Pb film - within order of 10 m for FLiBe film- important for aerosol source term
July 1, 2002/ARR8
Processes Leading to Aerosol Formation following High Energy Deposition Over Short Time Scale
Energy Deposition &
Transient Heat Transport
Induced Thermal- Spikes
Mechanical Response
Phase Transitions
•Stresses and Strains and Hydrodynamic Motion•Fractures and Spall
• Surface Vaporization•Heterogeneous Nucleation•Homogeneous Nucleation (Phase Explosion)
Material Removal Processes
Expansion, Cooling and
Condensation
Surface Vaporization
Phase Explosion Liquid/Vapor
Mixture
Spall Fractures
Liquid
FilmX-Rays
Fast Ions
Slow Ions
Impulse
Impulse
y
x
z
July 1, 2002/ARR9
Vaporization from Free Surface• Occurs continuously at liquid surface
• Governed by the Hertz-Knudsen equation for flux of atoms
j =1
2πmkαe
PsTf
−αcPvTv
⎛ ⎝ ⎜
⎞ ⎠ ⎟
e = vaporization coefficient,
c = condensation coefficient,
m = mass of evaporating atom,k = Boltzmann’s constant,
• Liquid-vapor phase boundary recedes with velocity:
drdt
=jmρ
γ =dTdt
dr =α(Ps−Pv)
γρm
2πkT
⎛ ⎝ ⎜ )
12
dT
• For constant heating rate, , and expression for Ps =f(T), the following equation can be integrated to estimate fractional mass evaporated over the temperature rise.
Ps = saturation pressure
Pv = pressure of vaporTf = film temperature
Tv = vapor temperature
• Free surface vaporization is very high for heating rate corresponding to ion energy deposition
• For much higher heating rate (photon-like) free surface vaporization does not have the time to occur and its effect is much reduced
Photon-like heating rate
Ion-like heating rate
Example results for Pb
July 1, 2002/ARR10
Vaporization into Heterogeneous Nuclei• Occurs at or somewhat above boiling
temperature, T0
• Vapor phase appears at perturbations in the liquid (impurities etc.)
• From Matynyuk, the mass vaporized into heterogeneous nuclei per unit time is given by:
dM
dt= αem
M2/3 2πmkT36ρv
2
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
1/3hfgT0
(T−T0)
• The equation can be integrated over temperature for a given heating rate, , and following some simplifying assumptions (Fucke and Seydel).
v = density of vapor in the nucleus,
hfg = enthalpy of vaporization per unit mass,
0 = density of saturated vapor at normal boiling
temperature (T0)
P0 is the external static pressure • Heterogeneous nucleation is dependent on the number of nuclei per unit mass but is very low for heating rate corresponding to ion energy deposition and even lower for photon-like energy deposition
Photon-like heating rate
Ion-like heating rate
Example results for Pb
July 1, 2002/ARR11
Phase Explosion (Explosive Boiling)
•• RapidRapid boiling involving homogeneous nucleation
• High heating rate
- Pvapor does not build up as fast and thus falls below Psat @ Tsurface
- superheating to a metastable liquid state
- limit of superheating is the limit of thermodynamic phase stability, the spinode (defined by P/v)T = 0)
•• A given metastable state can be achieved in two ways:
- increasing T over BP while keeping P < Psat (e.g. high heating rate)
- reducing P from Psat while keeping T >T sat (e.g. rarefaction wave)
• A metastable liquid has an excess free energy, so it decomposes explosively into liquid and vapor phases.
- As T/Ttc > 0.9, Becker-Döhring theory of nucleation indicates avalanche-like and explosive growth of nucleation rate (by 20-30 orders of magnitude)
dNdt
=Aexp(−ΔGc
kT) ΔGc =
16πσ 3
3(ρohfgβ)2;
July 1, 2002/ARR12
Photon Energy Deposition Density Profile in FLiBe Film and Explosive Boiling Region
Cohesion energy (total evaporation energy)
Evaporated thickness
Explosive boiling region
2-phase region
Sensible energy (energy to reach saturation)
0.9 Tcritical
3.7 5.52 12.24
Explosive boiling region as lower bound estimate for aerosol source term , ~5.52 m for FLiBe
Sensible energy based on uniform vapor pressure following photon passage in chamber and including evaporated FLiBe from film
July 1, 2002/ARR13
Summary of Results for Different Film Materials under the Indirect Drive Photon Threat Spectra
•• TTsatsat estimated from P estimated from Pvapor,interfacevapor,interface initial vapor pressure (P initial vapor pressure (P00=1 mtorr) heated by photon =1 mtorr) heated by photon
passage plus additional pressure due to evaporation from film based on chamber volumepassage plus additional pressure due to evaporation from film based on chamber volume
•• Adding the vapor component from the 2-phase region remaining after explosive boiling Adding the vapor component from the 2-phase region remaining after explosive boiling only slightly increases total expulsed mass (monly slightly increases total expulsed mass (mtottot vs. m vs. mexplosive,boil.explosive,boil.))
•• mmtottot would be lower-bound source term for chamber aerosol analysis would be lower-bound source term for chamber aerosol analysis
Li vapor
(1 mtorr 732 K)
FLiBe vapor
(1 mtorr 886 K)
Pb vapor
(1 mtorr 910 K)
9.0010-20.180.15 Vapor quality in the remaining 2-phase region
11.5110.079.14Cohesive energy, Et (GJ/m3)
5.08
5.59
1.92105
1.2113.91mtot(kg)
3.392.46explosive boil. (m)
6.071042.44105Pvapor,interface / P0
4.27 0.8512.89mexplosive boil. (kg)
July 1, 2002/ARR14
Comparison of Simple Explosive Boiling Estimates with ABLATOR Results
• ABLATOR graciously provided by LLNL
• ABLATOR is an integrated code modeling energy deposition and armor thermal response including melting, evaporation, boiling and spalling
• In the interest of time comparison done for a case already available in ABLATOR input file
• Results for Al armor under given X-ray spectra and fluence 32.5 J/cm2 assuming asquare pulse over 3 ns• Melting depth
- ABLATOR: 10.6 m- Simple volumetric model: 10.1 m
• Vaporization depth:- ABLATOR: 1.863 m- Simple volumetric model: 2.25 m
• Explosive boiling depth:- ABLATOR (assumed as depth where nucleation rate > 1024 s-1): 4.32 m- Simple volumetric model (T~ 0.9 Tcrit): 4.14 m
• Results from simple model reasonably close to ABLATOR results suggesting that the simple model could be used for scoping analysis
July 1, 2002/ARR15
• Use explosive boiling results as input for aerosol calculations
• Perform aerosol analysis to obtain droplet concentration and sizes prior to next shot (NOT DONE YET)
• Apply target and driver constraints (e.g. from R. Petzoldt)
10 3
10 5
10 7
10 9
10 11
10 13
10 15
0.1 1 10
anp_reg1_kal_data
100 µs
500 µs
1000 µs
5000 µs
10000 µs
50000 µs
100000 µs
Number Concentration (#/m
3)
Particle Diameter (µm)
• Need aerosol analysis for explosive boiling source case for Pb and FLiBe
• Need target tracking constraints for FLiBe
• Need to finalize driver constraints on aerosol size and distribution
From P. Sharpe’s preliminary calculations for Pb
Example Aerosol Operating Parameter Window
100 ID: 580 mDD: 0.05 m
Tracking (only as example)