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

ƒ

ƒ

ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ

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)