Modeling Effort on Chamber Clearingfor IFE Liquid Chambers at UCLA

Presented by:

P. Calderoni

Town Meeting on IFE Liquid Wall Chamber DynamicsLivermore, CAMay 5-6, 2003

Outline

This presentation will address twocomponents of our modeling efforts onchamber clearing:

- Vapor Condensation

- Droplet clearing in a pressure decay field

Vapor Condensation Modeling - ApproachCouple UCB model for condensation at a liquid / vapor

interface (based on Schrage kinetic theory) withTsunami calculations in 2-D volume

Apply “enhanced” Tsunami to simulate flibe vaporcondensation experiments, maintaining all assumptions

implicit in Tsunami and condensation model

Compare with experimentsEvaluate effect of measured interface conditions:

traces of non condensable gasesvapor density dropping into transitions regime

accumulation of less volatile BeF2

Generalize liquid / vapor interface model:add diffusional layer at the interface for non condensable gases

add velocity and temperature slip at the interfaceadd diffusional layer at the interface for BeF2

Gas dynamics modeling in IFE liquid chambers

nd 221π

λ =

Gas dynamics regime characterized by Knudsen number:

Considering:Hard-sphere modeldiameter d = 4 ALowest density inHYLIFE chamber =3X1013 #/cm3

Upper limit ofmean free path

in HYLIFE:4.7 cm

LKn λ

= mean free path:

characteristic length

Molecular collisions and collision with system boundaries are equallyimportant

Kn < 0.01 Continuum Regime

0.01 < Kn < 0.1 Slip regime

Mean free path small compared to system - molecularcollision dominate - gas approximated as continuosmedium

Gas approximated as continuos several mean free paths away fromadjoining medium - Kinetic theory near interfaces to account forboth molecular collisions and collision with system boundaries

0.1 < Kn < 3 Transition regime

Kn > 3 Free Molecules RegimeMolecular collisions infrequent - rarefied gas kinetic theory applies

Condensation model assumptions

2-D gas dynamics calculation assumptions:

Two dimensional geometry

Gas phase is a continuum

Gas state changes are isoentropic everywhere exceptat shock waves, which are treated as discontinuities

Liquid structures are rigid - liquid inertia in the time scales of interestprevents structures from moving or deforming - no work is transferredfrom gas to liquid

Gas viscosity is negligible - viscous time scale L2/ν >> dynamic time scale L/c

Inside the volume gas is adiabatic - conduction time scale L2/α >>dynamic time scale L/c - radiative losses assumed to be negligible

Flibe is an ideal gas law with constant γ - fitted EOS corrected fordissociation and ionization not effective in the considered range

Condensation model assumptions

1-D liquid / vapor interface assumptions:

Vapor condenses only on liquid surfaces present as initial condition -no droplet nucleation in the volume

Liquid layers are semi-infinite slabs - thermal diffusion length (α ∆t)1/2

<< surface curvature - initial layer T is uniform - liquid T away from theinterface remains constant - droplet spray cooling not considered

Heat and mass transfer at the interface only in the normal direction - interfacevelocity due to mass addition is neglected because of mass continuity

Heat transfer in the liquid layer by conduction in the normal direction -a convection term due to condensing flux introduced in the energy eq

Liquid surface is always in thermodynamic equilibrium with the vapor - highmass transfer rates during initial transient neglected - continuum assumption

Recombination and chemical diffusion effects for flibe are fast - vaporchemical composition is fixed by initial conditions

Vapor composition is uniform in the volume and at the interface

Interface kinetic theory accommodation coefficients (sticking andevaporation) are assumed to be = 1

Interface condensation: Schrage theory

( )[ ] dwdvduwvuuTkm

Tkm

NdN

bb

wvu

++−

−

= 222

0

2/3,,

2exp

2π

The effect of condensation on the molecular motion is to impose anet flux in the direction normal to the interface:

Integrating over v, w and positive (toward) u:

+

+

−

=+

2/1

0

2/1

020

2/1

0 21

22exp

2 Tkmuerf

Tkmu

Tkmu

mTkn

bbb

bN π

πφ

Re-writing:

( )[ ] serfseTR

pG s ++

= −+ 1

22

2/1

0

ππµ where ( )

2/12/1

2/10

2/1

00 222

=

=

=

γγγ

µ MRTu

TRus

RTp ρ=Finally, the net flux across theinterface:

2/12/1

21

21

−

Γ=

lslse

vsvsck RT

pfRT

pfGππ

Modeling the interface and liquid layer

Energy equation

with bc

0),( ≥=∞ ∞ tTtT l

Two equations to couple liquid and vapor properties at the interface:

mass balance 000

≥=== +− ==tGuuG

xlslsvsvsxρρ

energy balance conduction in the vaporneglected for short diffusionlength - neglect radiation

Vapor stagnationenthalpy:

Where for flibe

[ ]refvsatplpgreffgvsatfg TpTccThpTh −−+= )()()())((

−

×= Tsatp

10054407.910

760101300

0),(

0

0 ≥∂

∂−=

=

txtxTkhG

xlvs

[ ] [ ]2

)()(2

0 vsvssatfgvssatvspgvs

upThpTTch ++−=

0),(

0int ≥

∂∂

−=′′ txtxTkq l

In the liquid layer:

∞≤≤∂

∂=

∂∂

+∂

∂ xxtxT

xtxTu

ttxT

ls 0),(),(),(2

2

α

Convection term added toaccount for condensing massacross the interface - ulsevaluated from G

Initial non-equilibrium conditionsIn early stages of condensation the contact of highly superheatedvapor with the cold surface causes high mass transfer rates atthe interface - the effect (suction) is to increase the vaporvelocity that is evaluated by Tsunami

Schrage theory fails to account for high mass transfer ratesbecause of the surface equilibrium assumption - velocityassociated with mass flux predicted by the Schrage eq can behigher than physical limitations associated with super sonicchoking effect

Gas dynamics limiting flux:

cuMcMMcG vv <

−

++−

−−

−

++−

−

+=−+

− 11

12

max 12

11

12

12

11

12 γ

γγ

γγγ

γρ

γγγ

γρ

cuuG vvv >= ρmax

),(min maxvk GGG =Correction:

Numerical iteration scheme

( )

xkcG

TxkTchG

Tl

plk

ll

lplvsk

ls

∆+

∆++

=∞

0

2/12/1

21

21

−

Γ=

lslse

vsvsck RT

pfRT

pfGππ

Vapor (p, T) at the interface are given by Tsunami (as well as the gas dynamic limiting flux)

Equilibrium assumption reduces unknown liquid properties to one, surface temperature:

Condensation gives the second eq to solve for T and G:

)( 1−= nls

n TGG )( nls

nls GTT = ε≤

−−

−

1

1

nls

nls

nls

TTT

))((

1

nlowlsls

nhighls

nl

nlowls

TGTT

TT

−−

−−

=

=

2)( n

lowlsnlsnewn

ls

nls

newnhighls

TTT

TT

−−

−−

+=

=

[ ] [ ]11111

1112

1 2)(

−−

−−−

−−+

− −∆∆

−+−∆∆

+= ni

ni

l

nni

ni

ni

ni

ni TT

xtGTTT

xtTT

ρα

1

2)(2

−

∆

+∆

<∆x

Gx

tl

n

ρα

In the liquid layer, using an upwind scheme for the condensation case:

Iteration step:

Newton-Raphson averaging method:

or

2)( n

highlsnlsnewn

ls

nls

newnlowls

TTT

TT

−−

−−

−=

=

Introducing condensation effect in Tsunami

The condensation module evaluates G at each step - the mass flux condition must now beused at Tsunami boundary cells interface instead of the usual adiabatic condition adoptedat cells interfaces in the volume

Tsunami numerical scheme requires computation of mass (continuity), momentum (Riemannsolver across the discontinuity) and energy (adiabatic assumption) fluxes at the edge ofeach cell

Mass flux is G

Energy flux from same interface balance -written in Tsunami terms:

+

−=

2)1(

2vs

vs

vs upGfluxEnergyργ

γ

( )( )TvssvsvsR

TvsvsvsL

puuU

puU

,,

,,

+−=

=

ρ

ρ ( )TpuU **** ,,ρ= **uG ρ= ερ<

−=∆

max

**

vnorm G

GuG

vss

Guρ20 =

0)1(

0)1(11

11

<∆+=

>∆−=−−

−−

GifGuu

GifGuuknorm

ks

ks

knorm

ks

ks

Momentum flux determined by mirror node, introducing suction velocity:

Iteration scheme:

Numerical domaingeometry

Uniform grid: 10 x 100 cells - 0.5 x 0.5 cm each

4.1

0331.0

==

=

v

p

CC

molkgm

γ

0

00 TR

Wp⋅⋅

=ρ10

0 −⋅

=γTRe

Background gas is flibe, considered as anideal gas with:

Initial background conditions specified as:

Injected gas considered by Tsunami as DEBRIS,initially available in a 3 x 3 cell volume V0

Initial superheated vapor conditions specified astotal injected mass [kg] and total initial energy [J]

Code runs - parameters case study

Tsunami BCrecovered:

us = 0

us = 2 uvs

Top and bottom boundaryare impermeable

Boundary conditions:

Open interface

Impermeable surface

Condensation

Parametric study for:

Sticking coefficient at the interface

Liquid layer thickness

Initial liquid temperature (constant at solidwall interface)

Reference case:

fc = 1

0.5 mm

600 C

Liquid initial temperature

Liquid layer thickness

Liquid initial temperature

Temperature distribution in the 2-D axi-symmetrical numerical domain as a function

of time for the reference case

Temperature distribution - middle cells

Gas temperatures evaluatedby Tsunami fall below theimposed initial backgroundtemperature whenremaining mass is low

Temperature distribution - Top

In the boundary cells atthe top and bottom ofthe chambers thevapor interface T ishigher then Tsunamievaluated temperaturein the inner cells

Heat conduction in the liquid layer

Pressure distribution in the 2-D axi-symmetrical numerical domain as a

function of time for the reference case

Liquid surface equilibriumassumptions not valid fortransient condensation

Ohno fitted equation for flibe:

−

×= Tsatp

10054407.910

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Pressure distribution - middle cells

Ideal gas assumption for flibeoverestimates vapor pressureduring high temperaturesinitial transient

Density distribution in the 2-D axi-symmetrical numerical domain as a

function of time for the reference case

Free molecular regime

Gas dynamics modeling in IFE liquid chambers

01.0≅=L

Kn λ

Direct Simulation MonteCarlo method

HIBALL - Wisconsin (1989)KOYO - Osaka Un. (2002)

solving Boltzmann equation

dvdggtxvftxcftxvftxcfcfF

xnfc

tnf

Ω×−∫∫=

=∂∂⋅+

∂∂⋅+

∂∂

),()],,(),,(),,'(),,'([

)()(

χσ

by decoupling molecularmotion from collisions:

),(*)1(),,()0,,()1(),(*xcFtJtxcF

xcFtDxcF∆+=∆

∆−=

Continuum fluid regime

Hydrodynamic code TsunamiHYLIFE - UCB (1979-2003)

NIF - LLNL, UCB (1994-1996)

solving Euler equation

2

2

21),,(

),,(

0

ueEpuEpuuF

EuUxF

tU

T

T

ρ

ρρρ

ρρρ

+=

++=

=

=∂∂

+∂∂

with the eq of state

),( Epp ρ=

Proposed extension to gas dynamics slip regimeKinetic theory of gas dynamics on a diffused surface in dilute gasconditions:• references include older theoretical studies of Cuette flow conditionsand newer numerical studies with DSMC methods (1988 - 2001)• models are based on imposing a velocity and temperature slip to thegas near the surface to compensate for the difference in the velocitydistribution of the particles approaching and leaving the surface• DSMC simulation show model is valid for Kn < 0.1

Proposed extension for Tsunami is based on Harvie and Fletcher study(2001) that explicitly include the mass flux in the velocity and temperatureslip formulation:

xvUvvi ∂∂

−= λ0

+

−

=ΦGG

xv

fffffUvvcmc

cml ∂

∂

−+Φ−−+

=−)1()1)(1(1

0 λxvv∂∂

Φ−=λ

0

xTUTTi ∂∂′−= λ0

459 −=′ γU

mcct ffff )1( −+=

xT

ffffUTT

ct

tcl ∂

∂

−Φ−+

−Φ−−′=−)1)(1(

)1(20 γ

λ

xTTTl ∂∂

Φ−Φ

=−4.04.1

35.110 λ

Droplet Clearing in a Pressure DecayField

Problem Definition: Droplet clearingrepresents another aspect of thechamber clearing issue. Dropletsproduced from the blast should becleared away before the next shot.

Approach: Start with thedevelopment of an incompressiblecode for analyzing droplet heat andmass transfer with respect to aknown pressure decay

Goal: Ultimately to couple theTsunani code with the developedincompressible free surface heatand mass transfer code for chamberclearing evaluation

Movie: A hot droplet reacting to thecold surrounding environment

Droplet Heat Transfer and Phase Change withTruchas: Preliminary Evaluation

Truchas is a software programdeveloped at LANL to simulatesolidification manufacturingprocesses, most notably metalcasting and welding operations.Include models and algorithmsfor:• Interfacial motion and heat

transfer• Properties varying with

temperature• Phase change• finite volume method• Simulations are fully 3

dimensional onunstructured grids.

Movie: A hot droplet falling down through a coldenvironment

Internal circulation caused by the temperaturedifference is now employed in Truchas code

through Boussinesq approximation

Tgg o∆=∆ αρρBuoyancy force

Melting temp=723KBottom temp=723 kTop temp=523 kInitial temp inside droplet=735 kinitial temp outside droplet=723 k

x

y

0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

4

Assumptions & Near Term Goal

The mass evaporated from the droplet intothe surrounding pressure field will bediscarded

Incorporate a time dependent temperatureboundary condition (set at Tsatcorresponding to saturated temperate as afunction of known pressure decay)

Await approval from LLNL to modifyTruchas code