a pyrolysis and ablation toolbox based on...

Feb 3, 2009 Multiscale modeling of pyrolysis and ablation 1October 7-9, 2008

A pyrolysis and ablation toolbox based on OpenFOAM- with application to material response under high-enthalpy environments

Jean Lachaud* and Nagi N. Mansour+

* NASA Postdoctoral Fellow at Ames, [email protected]+ NASA Ames Research Center, [email protected]

Sponsored by NASA’s Fundamental Aeronautics Program - Hypersonics Project

5th OpenFOAM Workshop, Chalmers, Gothenburg, Sweden, June 21-24, 2010

2

. Context and application. Context and applicationAtmospheric entry of space vehicles – in the news: HAYABUSA (JAXA)

News : The Japanese capsule "HAYABUSA“ re-entered into the earth atmosphere on June 13th, 2010 at 12.2 km/s.

TrajectoryCapsule

Hayabusa is an unmanned space mission led by the Japan Aerospace Exploration Agency (JAXA) to return a sample of material from a small near-Earth asteroid named 25143 Itokawa to Earth for further analysis.Hayabusa was launched in May 2003.

Pyrolysis and ablation of the heat-shield

during re-entry

3

. Context and application. Context and applicationAtmospheric entry of space vehicles – in the news: HAYABUSA (JAXA)

June 14, 2010 - Credit: JAXA

HAYABUSA (JAXA) after successful landing in Australia.

. Context and application. Context and applicationSimilar mission : Stardust (NASA) capsule re-entered in 2006, was recovered, and analyzed

1 - Virgin PICA

SEM micrographs (1)

1

3 - Charred PICA

3

2 - Partially charred2

4 - Partially ablated

4

Stardust, Jan. 15, 2006

Imag

e cr

edit

: NAS

A

[1] M. Stackpoole et al., Post-Flight Evaluation of Stardust Sample Return Capsule Forebody Heatshield Material, AIAA 2008-1202

(1)

4

Stardust is an interplanetary mission, whose primary purpose was to collect cosmic dust from the comet Wild 2 and to return it to Earth. Stardust re-entered the Earth atmosphere at a record velocity of 12.7 km/s in January 2006. Amino acids have been found.

Current state-of-the-art models based on equilibrium chemistry are conservative.It has been estimated that they over predicted recession by about 60% at the stagnation point [1].

5

. Objective & Motivation. Objective & Motivation

Objective• Develop high-fidelity material response models for fibrous phenolic impregnated materials.

Motivations• Widely used class of materials (Huygens with AQ60, Stardust with PICA, Hayabusa with carbon/phenolic, MSL with PICA, Dragon with PICA-X, Marco Polo maybe ).• Reduce design uncertainties.• Provide guidelines for material development.

Mars Science Laboratory (MSL) heatshield

Reduce uncertainties and improve uncertainty quantification

Dragon capsule, SpaceX

6

. Outline. Outline

1. Presentation of the approach– Porous medium model– Implementation strategy

2. PAT-0: first step, reach the state-of-the-art– Model– Code-to-code comparison and application

3. PAT-1 : PAT-0 + Darcy– Model– 3D applications

4. Conclusion & perspectives

Pyrolysis and Ablation Toolbox based on OpenFOAM (PATO)

7

1. Presentation of the approach1. Presentation of the approachComplete pyrolysis-ablation model -- upgrades to the state-of-the-art models are in blue

, Cix i i i i i i

i

Mp v v with v y v y x and J X vM

-1K

12

( ) ( ) ( )a g ma p t g p x x x i g i i m i t m i

i

iig i p t i i i

i S i S

m m t m f f

S S

t f

iy c V p h

h h K v v

c T c v T k T h h h y

,

_

,1

( ) 1 expi i

n lawn A

m v v i p i i

s

t i i ii

Ewith A

RT

t f f f i ii S

s

Species and mass transport

Energy balance

Pyrolysis laws

In depth ablation laws (sublimation and heterogeneous chemistry)

t m m m i ii S

s

Fibers : Carbonized matrix :

Species conservation x i ii iit X X v

( ) C Di i i i i i iJ X v X v V J J

Momentum balance in porous media (Brinkman’s model)

Finite-rate chemistry

' '''' ' rj rjf b

i t i ri ri r j r jr R j S j S

d X k X k X

i i

pX xRT

( )i f i

Multicomponent mass transfer in porous media (Dusty Gas Model: Stefan-Maxwell + Bosanquet)

,

D D Dj i i j i

i B Kj i j i

J x J x JXD D

Di i iJ X V

1. Presentation of the approach1. Presentation of the approachCode development strategy

We wish to implement the complete model in a code that is:

Modular :

• PAT-0: State-of-the-art model (Kendall, 1965)• PAT-1: PAT-0 + Darcy• PAT-2: PAT-1 + Finite-rate chemistry• PAT-3: PAT-1 + Brinkman• etc

3D, parallelizable

High level implementation for fast developments (object oriented)

The idea is to do more physics, less implementation. Rely on community developments and innovations in CFD.

We decided to use OpenFOAM open source CFD toolbox as a framework.

8

9

. Outline. Outline






10

2. PAT2. PAT--0 : first step; reach the state0 : first step; reach the state--ofof--thethe--art model [1]art model [1]Description of the main features (1/2)

[1] Moyer, C. B., and Rindal, R. A., “An analysis of the coupled chemically reacting boundary layer and the charring ablator”, NASA CR-1061, 1968.

• Continuity equation: immediate transfer of pyrolysis gases through the char layer• Energy balance equation (written in OpenFOAM format)Solve(rhoCp*fvm::ddt(T) - fvm::div(fvc::interpolate(rhoCp)*mesh.phi(),T)- fvm::laplacian(k, T)+(h_bar-hg)*fvc::ddt(rho_m) - fvc::div(fvc::interpolate(h_bar-hg)*mesh.phi(),rho_m)+ (fvc::grad(hgCenter) & vectorY)*mDotG);

• n Pyrolysis laws written in advancement (Xsi) of each reactionfvm::ddt(Xsi_i) - pow((1-Xsi_i),m_i)*A_i*pow(T,n_i)*exp(-E_i/(R*T))

-fvm::div(mesh.phi(), Xsi_i)

rho_m=rho_mv*((F_1*(1-Xsi_1))+(F_2*(1-Xsi_2))+(F_3*(1-Xsi_3))+(F_4*(1-Xsi_4))

• Surface Energy Balance

Written asgrad(T)=1/k[ … ]

fixedGradient BCoverwritten from the mainat each time step

1111

2. PAT2. PAT--0 : state0 : state--ofof--thethe--art model [1]art model [1]Description of the main features (2/2)

[1] Moyer, C. B., Rindal, R. A., “An analysis of the coupled chemically reacting boundary layer and the charring ablator”, NASA CR-1061, 1968.[2] F.S. Milos and Y.-K. Chen. Comprehensive model for multicomponent ablation thermochemistry. AIAA 1997-0141.

• Evaluation of the surface recession velocity (ablation)- Equilibrium chemistry is assumed in a control volume close to the wall (gas-gas & gas-solid)- Mass transport in the boundary layer and element conservation dictate the ablation rate- This problem can be solved a priori, knowingthe elemental composition of the pyrolysis gas (Yk,pg), the pyroloysis gas flux(mDotpg), the wall temperature (Tw) and pressure (pw) mDotca

pw B’g B’c Tw hw

1 10 10 3000 3e6

1 10 1 2000 2e6

1 10 0.1 1000 1e6

1 0.1 10 3300 3.4e6

1 0.1 1 2200 2.4e6

1 0.1 0.1 1100 1.3e6

0.1 0.1 0.1 2800 2.5e6

• “B’ tables “ may be generated when Yk,pg are constant. In the following we will use tables generated using MAT [2].By definition: B’= mDot / (rho_e u_e CM), where rho_e and u_e are the boundary layer edge density and velocity, and CM is the mass Stanton number.• A 3D class has been written in OpenFOAM to read and interpolate in B’ tables.

• Very inconvenient when Yk,pg are not constant. Solution: Integration of Mutation - the equilibrium chemistry solver of Pr. T. Magin (von Karman Institute)- in OpenFOAM.

e.g.pw = 1B’g = 10Tw = 1500

0.1 < B’c < 1

12

2. PAT2. PAT--0 : state0 : state--ofof--thethe--art modelart modelVerification against FIAT (NASA Ames state-of-the-art code [1] )

[1] Chen, Y.-K., and Milos, F. S., “Fully Implicit Ablation and Thermal Analysis Program (FIAT),” Fourth International Conference on Composites Engineering (ICCE/4), New Orleans, LA, 1997, pp. 661–662.

0 20 40 60 80 100 1200

200

400

600

800

1000

1200

1400

Imposed surface temperature

time (s)

T (K

)

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.03500

200

400

600

800

1000

1200

1400

Temperature profiles

x (mm)

T (K

)

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350200

205

210

215

220

225

230

Density profiles

x (mm)

dens

ity (k

g/m

³)

28s

32s

50s

100s

28s32s

50s100s

TopBottom

MaterialPICA old version

T=f (time)

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2. PAT2. PAT--0 : state0 : state--ofof--thethe--art modelart modelApplication: Stardust re-entry

• 1D simulation at the stagnation point• Environment data provided by the TPS-ADP / Stardust post-flight analysis team(Enthalpy, Stanton number, pressure, radiative heat flux)• Recession computed using B’ table interpolation : Pw, Tw, B’g B’c

2. 2. PATPAT--0 in 3D0 in 3DSimulation of an X-Jet experiment on a Teflon cylinder

Avco model for Teflon [Avco Research Report 226, August 1965]• Surface energy balance accounting for the heat of ablation

• Ablation rate:

• Heat transfer in the material (no pyrolysis in the case of Teflon)

Experiment• NASA Ames Xjet at atmospheric pressure, 0.85 MW/m², 30 seconds• Teflon cylinder : D= 63 mm, h= 25 mm• Test off-centered with a Gaussian flux repartition : 3D problem

31 2 4exp( / )w w ws T T T

Sample after test:Recession of 4mm in the middle of the concavity(the grid has been drawn for the clarity of the photograph)

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2. 2. PATPAT--0 in 3D0 in 3DSimulation of an X-Jet experiment on a Teflon cylinder

Ablation of a Teflon cylinder - NASA Ames X-Jet 0.85 MW/m², 30 seconds(the movie is accelerated: x10)

15

• Satisfactory qualitative agreement of the ablation profiles.• Total recession at the center of the ablation zone:

sample: 4 mm +/- 0.5 mm, simulation: 4.1 mm

16

. Outline. Outline






3. PAT3. PAT--1 : PAT1 : PAT--0 + Darcy0 + DarcyDescription of the main features

17

• Everything that PAT-0 has.

• Anisotropic (3D) heat transfer

• Darcy’s law (Averaging of the momentum equation in porous media) – substituted in the continuity eq.

v = - 1/eps * 1/mu * (K & fvc::grad(p)) // K (permeability) is a 2nd order tensor// eps (porosity), mu (viscosity)

• Continuity equation (written in OpenFOAM format)

eps/T*fvm::ddt(p) + p * eps * fvc::ddt(1/T) // gas dilatation, explicit

- fvm::laplacian(K*p/(mu*T), p) // Divergence becomes Laplacian// with Darcy substituted inside

+ R/M * fvc::ddt(rho_m) // Pyrolysis gas source

- fvm::div(fvc::interpolate(eps/T)*mesh.phi(), p) // Mesh- fvc::div(fvc::interpolate(p*eps)*mesh.phi(), 1/T) // Motion- fvc::div(fvc::interpolate(R/M)*mesh.phi(), rho_m) // Correction

3. PAT3. PAT--1 : PAT1 : PAT--0 + Darcy0 + DarcySame X-Jet experiment on a PICA cylinder

PAT-1 model Chen-Tran material model for PICA [NASA TM 110440, 1997] Ablation rate: B’ table (equilibrium chemistry) [NASA TM 110440, 1997]

Negligible ablation, resulting in a higher surface temperature than for Teflon

18

3. PAT3. PAT--1 : PAT1 : PAT--0 + Darcy0 + DarcySame X-Jet experiment on a PICA cylinder

Pressure map and pyrolysis gas velocity vectorsAt t=30s

19

2020

Complete pyrolysis-ablation model

, Cii i i i i i

ix

Mv with v y v y x and J X vp vM

-1K

12


i

iig i p t i i i

i S i S

m m t m f f

S S

t f

iy c V p h

h h K v v

c T c v T Tk h h h y

,

_

,1

( ) 1 expi i

n lawn A

m v v i p i i

s

t i i ii

Ewith A

RT

t f f f i ii S

s


Energy balance

Pyrolysis laws

In depth ablation laws (sublimation and heterogeneous chemistry)

t m m m i ii S

s


Species conservation x i ii iit X X v


Momentum balance in porous media (Brinkman’s model)

Finite-rate chemistry

' '''' ' rj rjf b


d X k X k X

i i

pX xRT

( )i f i


,

D D Dj i i j i

i B Kj i j i

J x J x JXD D

Di i iJ X V

black: State-of-the art (completed) red: first improvements completed blue: being/to be implemented

. Conclusion and perspectives. Conclusion and perspectives

• OpenFOAM is an ideal platform to develop multiphysics models.

• Two pyrolysis-ablation modules implemented- PAT-0: Pyrolysis-ablation module reproducing NASA’s state-of-the-art model + 3D.- PAT-1: PAT-0 + Darcy. We can go further an analyze pyrolysis gas transport in 3D and

pressure increase.

• No limit: the following modules will have an increasing degree of complexity relyingon community developments in CFD

• Current developments

- PAT-3: PAT-2 with a volume of fluid approach to model ablation in depth.

• Future developments

- Coupling with a hypersonic solver

. Conclusion and perspectives. Conclusion and perspectives

certain pyrolysis-ablation modeli

PATi

- PAT-2: PAT-1 + Finite-rate chemistry. Finite-rate chemistry solver available in OpenFoam.

21

22

A- Microscopic Analysis of the TPS of Stardust.

B- Pyrolysis-ablation modelfor low-density fibrous media: illustrations.

C- Equilibrium chemistry

AppendicesAppendices

Problem studied: Isothermal oxidation of the char layer (pyrolysis gases are assumed frozen)Model

Starting point : differential recession of aheterogeneous surface by gasification

Microscopic ablation model: Transport, Reaction, and Local Surface Recession

i = {fiber , matrix};iJ v n

i i = {fiber , matrix}J k X ;

( ) 0X D X Xt

gv

q

xz

J

v

X(x,y,z=0,t)=XoVertical mass transfer of oxygen

X(x,y,z,t)

CARBONIZEDMATRIX

Local recession velocity

Local ablation flux per surface unit

Oxygen transport

Localconcentration

Diffusion Convection(if any)

n

n

NomenclatureX = Oxygen concentration (mol m-3)D = Diffusion coefficient (m2 s-1)k = Reactivity (m s-1)n = Normal to the surface ( - )vg = Pyrolysis gas velocity (m s-1)Ω = Solid molar volume (m3 mol-1)

vg

In the followingvg = 0

Can be solved analytically for simple geometries, but for this study a simulation tool is required.

AA-- Microscopic Analysis of the TPS of Stardust.Microscopic Analysis of the TPS of Stardust.

23

Simulation tool: Random Walk for diffusion & Triangle Marching Cube for front tracking

h

[1] J. Lachaud, G.L. Vignoles. A Brownian motion technique to simulate gasification and its application to C/C composite ablation. Computational Materials Science, 44(4), 2009, pp. 1034-1041. doi:10.1016/j.commatsci.2008.07.015

3D simulation tool : AMA[1] (Ablation Marche Aléatoire)(“Random Walk Ablation”)


24

[1] J. Lachaud, I. Cozmuta, N. N. Mansour. Multiscale approach to ablation modeling of phenolic impregnated carbon ablators. Journal of Spacecraft and Rockets, to appear..

Flow regime in the pores of the material : from Knudsen to continuum

2

5 ( )3.2 1

(

0

))

( )

(p

Knudsen number in the poresmean free pathKn

Mean free path of O in airT in Kp in Pa

d pore diameter

Just as an illustration, Knudsen number in the pores along the Stardust trajectory[1]

Conditions given at the stagnation point

Path of a molecule in a periodic cell in slip regime (code AMA)


25

[1] J. Lachaud, I. Cozmuta, N. N. Mansour. Multiscale approach to ablation modeling of phenolic impregnated carbon ablators. Journal of Spacecraft and Rockets, accepted for publication, to appear.

Analytical analysis of a fiber preform in steady-state (diffusion time << ablation time)

0

cosh ( / 1)( )

coshsz L

X z X

• The concentration field is a function of the Thiele number [1]

Ls: material depth (m)Deff: effective diffusivity (m²/s)kf: fiber reactivity (m/s)s : specific surface (m2/m3)

Thiele number

s

eff

f

LDsk

Ls

surface

bottom

z

The Thiele number is the key dimensionless number for the ablation of fibrous preforms:

• Large Thiele number : small ablation zone; can be seen as a surface ablation.• Smaller Thiele number : larger ablation zone.

X(z=0)/X0=1

X(z)/X0


26

Hypotheses• The material is fully charred

• No pyrolysis gas blowing • Air

• Φ = 40, e. g. • T = 3360 K (isothermal)• P = 0.26 Atm• ρf = 32 ρm

• oxidation by O2

km = 10 kf = 13.7 m/s[Drawin 1992, Lachaud 2007]

AMA simulation – Cartesian grid: 300x100x100– Computational time : 2 days on a

single core.

Oxidation at moderate Thiele Number, Φ = 40

Oxidation by the top

Periodic on the sides

Physical time: 1.2 s


27

Oxidation at moderate Thiele number (Φ = 40): 3 stages

t = 0.0 s 0.4 s 0.8 s 1.2 s

1) Matrix is removed first

2) Fiber diameterdecreases

3) Overall surface Recession (a steady-state is reached)


28

Oxidation at very high Thiele Number, Φ > 1000

Example: oxidation by O (atomic) at 3360 K, 0.26 atmSticking probability β ≈ 0.45<1 [Park 1990], i.e. k ≈ 100 m/s. Hypothesis: km= kf ; ρf = 32 ρm

50 µm

Transient regime: as described for moderate Thiele numberSteady-state:- Small ablation zone (50 µm)- Could be seen as surface ablation with development of a specific surface roughness

0.75

mm

29


30

Ablation : surface phenomenon at microscopic scale, but volumetric at macroscopic scale

( , )2 ( ) ( , )f

s f

d z tk T X z t

t

2

, 0_0

ff f t

f

dd

– Volume averaging : fiber diameter mean porosity (ε)

– Mean fiber diameter evolution

Illustration of the averaging on a carbon fiber preform : micro macro

rf

Ablation fluxLocal approach

Fiber


( ( , , ) ) geff i

XD T P X v X

t

– Matrix (similar relation)

– Oxygen transport (volume averaged)

Diffusion Convection Consumption (heterogeneous reaction)Local concentration

mm m ms k X

t

sm : specific surface (in m2/m3)of the porous matrix

31

Macroscopic model derived from the microscopic approach : Pure oxidation of the char layer

Den

sity

(kg/

m3 )

Depth from initial surface

3.5 mmsurfa

ce

botto

m

Loosely coupled with traditional tools:

• data from FIAT inside the material- Temperature gradient- Pyrolysis gas velocity

• data from DPLR at the wall- Oxygen partial pressure - Total pressure

Data provided by Ioana Cozmuta

Carbonized matrix

Fiber preform

End of the ablative part of Stardust trajectory : 90s (Twall= 1350 K < Tsublim) to 130s (Twall=850 K <Toxi)


surface

32

Char layer

Top of the ablation zone

AblationZone

Ability of the model to fit flight data


33

BB-- PyrolysisPyrolysis--ablation model for lowablation model for low--density fibrous media: illustrationsdensity fibrous media: illustrationsIllustration in equations

, Cix i i i i i i

i

Mp v v with v y v y x and J X vM

-1K

12


i

iig i p t i i i

i S i S

m m t m f f

S i

f

S

t

y c V k pc T c v T T h h h y h

h h K v v

,

_

,1

( ) 1 expi i

n lawn A

m v v i p i i

s

t i i ii

Ewith A

RT

t f f f i ii S

s


Energy balance

Pyrolysis laws

Ablation laws (sublimation and heterogeneous chemistry)

t m m m i ii S

s


Species conservation

x i ii iit X X v


Momentum balance in a porous medium (Brinkman model in continuous regime)

Finite-rate chemistry in the pores

' '''' ' rj rjf b


d X k X k X

i i

pX xRT

( )i f i


,

D D Dj i i j i

i B Kj i j i

J x J x JXD D

Di i iJ X V

Finite-rate chemistry : SchematicNB: This is just an illustration -- incomplete mechanism!

Virgin

Pyrolysis zone

CharNot ablated

OH

Phenolic formaldehyde resin

CH4 + CO + 2 C2H2

H2 +

~ 200°C

~ 1200°C

Boundary layer

CH4 + 2O2 = CO2 + 2H2O

2 O

O2

2 C (solid) + O2 = 2 CO

O + O = O2

2 O

Material code :- ablation + pyrolysisfinite rate chemistry(homogeneous and heterogeneous)Coll. with Prof. G. Blanquart, Caltech- pyrolysis : multi-laws- mass transport

CFD :- finite rate chemistry(homogeneous only)- mass transportCollaboration withProf. T Magin, VKIAblation zone

Shock

34

C (sublimation)

C + O = CO

O

CO2 + C (solid) = 2CO

BB-- PyrolysisPyrolysis--ablation model for lowablation model for low--density fibrous media: illustrationsdensity fibrous media: illustrations

Phenolic Pyrolysis model development

35


0.00E+00

4.00E-05

8.00E-05

1.20E-04

1.60E-04

2.00E-04

0.00E+00 4.00E+02 8.00E+02 1.20E+03 1.60E+03

Spec

ies

Prod

ucti

on R

ate

Temperature

H2O

C6H6

C6H6O

C7H8

C7H8O

0.00E+00

2.00E-05

4.00E-05

6.00E-05

8.00E-05

0.00E+00 4.00E+02 8.00E+02 1.20E+03 1.60E+03

Spec

ies

Prod

ucti

on R

ate

Temperature

CO

CO2

CH4

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

0.00E+00 4.00E+02 8.00E+02 1.20E+03 1.60E+03Sp

ecie

s Pr

oduc

tion

Rat

e

Temperature

H2

1+2 3 4

Phenolic Pyrolysis model development

36


Finite-rate chemistry of the pyrolysis gases is needed to go beyond current SoA

[April69] April, G. C., “Energy Transfer in the char Zone of a Charring Ablator”, Ph. D. Dissertation, Department of Chemical engineering, Louisiana State University, 1969.

B

Sootproduction

Fiber/Gas reactions

Added to ShutoffReaction when H2 is not present

37


Illustration: finite-rate chemistry of the pyrolysis gases at 1500K, 1atm

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Mas

s Fr

acti

on

time (s)

CH4

H2

C2H6

C2H4

C2H2

C

A1OH

A1

H2O

CO

CO2

Starting from an estimated composition of the pyrolysis gases, and using a mechanism adapted from the mechanism of April [April69],

[April69] April, G. C., “Energy Transfer in the char Zone of a Charring Ablator”, Ph. D. Dissertation, Department of Chemical engineering, Louisiana State University, 1969.

U= O(1 m/s) [Expected velocity of pyrolyzedgases]L= O(0.01 m) [expected depth of pyrolysis front]t = O(0.01 s) [expected residence time scale] same order as

Finite rate Chemistry regime

38


Knudsen effects : Diffusion coefficient in a capillary (Bosanquet model)

Fibers randomly oriented : tortuosity effects non negligible in Kn regime

Effective diffusion coefficient

eff refD D

1 13 3

1 1 1 1 1

ref B K pD D D v vd (harmonic average)

Tortuosity has to be obtained by Monte Carlo simulation inside the porous media (not available yet in the literature fornon-overlapping fibers)

1 1 1

39


1) Displacement of 10000 walkersfollowed during (chosen for convergence)

2) Einstein relation on diffusion process :

Monte Carlo Simulation :• Random Walk rules

• (T,P) fixed = (λ,D) fixed

• λ : Maxwell-Boltzmann distribution

• constant velocity norm

(D = 1/3 v λ) with 3D random direction drawing

Tortuosity as a function of Kn for the fibrous material of this study

Determination of the effective tortuosity

2

2j

e jD

Illustration : path of a walker in a periodic cell

40


41

Determination of the permeability, K, by volume averaging

SEM observation : 2D

1) Microscopic scale analysis

Tomography : 3D

3) Comparison with macroscopic scale measurements (permeability)

Ref. : Marshall, J. Thermophysics 13(3), 1998.

Kexperiment ( virgin)= 1.3 10-11 m² Kexperiment (pyrolyzed)= 3.8 10-11 m²e.g. PICA

K ( virgin)= 1.6 10-11 m² K (pyrolyzed)= 2.0 10-11 m²

2) DNS : up-scaling (here, we solve the Navier-Stokes momentum equation in the fibrous medium)

gp v -1K 1.1

Origin of Forchheimer term- separation- recirculation (still laminar)Presented case : vg = 60 m/s

For low densities: collaboration with Alexandre Martin & Iain Boyd DSMC and Lattice-Boltzmann for 0.1 < Kn < 10 (where Navier-Stokes is not valid) We are analyzing Klinkenberg effects (slippage)


4242

Velocity map computed with FEMLAB (2D)

Velocity (m/s)

Gap, drilling, holes.. at macroscopic scale : Brinkman’s model

PICA (K=3.10-11 m² Darcy)

Hole (K >> 1 Stokes)

PICA (K=3.10-11 m² Darcy)

vp v -1K Darcy + Stokes”

P2 =

0.0

1 A

tm

P1 = 1 Atm

5 cm

2 cm

Modeling of an Arc Jet sample with a hole (e.g. pressure-sensor passage)

“Brinkman ≈


Tkqeff .

),,(000000

kji

eff

kk

kk

i

j

k

1 10.034k W m K

1 1(1 ) 1.52air fiberk k k W m K

1 10.025 W m Kairk

1 110 W m Kfiberk

85.0

Modeling heat transfer by conduction; illustration: regular array of parallel fibers


Regular array of parallel fibers : tilted

k axis tilt (45°)

i or j axis tilt (45°)

Verification for rotations

PkPkkji

effzyx

eff),,(

1

),,(

kjizyx uPu ,,,, .

1000cossin0sincos

P

After projection (on x) :

2 2cos . sin .effk k k

x

y

k

x

y

k


97.04 random fibers : d=10µm ; L=150µm

1.

1._1.0_

KmWsimulationeffk

1.

1._025.0

KmWairk

1.

1._10

KmWfiberk

Modeling heat transfer by conduction; illustration: 3D random


4646

Radiative heat transfer in porous media: Collaboration with Tom van Eekelen, Samcef

T < 700K : conduction (radiation is negligible) T > 700 K : conduction + radiation

Semi-analytical expression for the equivalent radiative conductivity (under Rosseland-Diessler hypotheses)

The conductivity is a function of- Δx : sample size- T : average temperature- εσ : emissivity- σext : extinction coefficient

Objectives- verify this model- determine σext


47

Elementary cell: 252 fibers (configuration 3)

L = 2 mm

L configuration N = 2N x 0.25 (in mm)

Result

Consistent with literature data

Radiation in porous media: Collaboration with Tom van Eekelen, Samcef

Perspectives:-Extension to 3D analysis (3D images obtained from computed micro-tomography)


48

Computation of a “B’ table” with Mutation, Julien de Mûelenaere and Thierry Magin

CC-- Equilibrium chemistryEquilibrium chemistry

p=1atm

a pyrolysis and ablation toolbox based on...

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