Particlemethodsfortortuosityfactorsin

porousmedia

AblationWS,2017Bozeman,MT

JosephC.Ferguson1ArnaudBorner 1

FrancescoPanerai 2Nagi N.Mansour3

1.ScienceandTechnologyCorp.atNASAAmesResearchCenter2.AnalyticalMechanicalAssociatesInc.atNASAAmesResearchCenter3.AdvancedSupercomputingDivision,NASAAmesResearchCenter

https://ntrs.nasa.gov/search.jsp?R=20170009457 2020-05-22T14:11:34+00:00Z

Ablative Thermal Protection Systems

2

+

FiberForm® Resin

PICAArtist rendering of MSL entry

http://mars.nasa.gov/mer/gallery/artwork/entry_br.html

Material Design and Modeling

3

X-ray micro-tomography

• AdvancedLightSource(ALS)attheLawrenceBerkeleyNatl.Laboratory

• Synchrotronelectronacceleratorusedtoproduce14KeVX-rays

• Usedformanyresearchareas,includingoptics,chemicalreactiondynamics,biologicalimaging,andX-raymicro-tomography.

BerkeleyLabs,Flickr

http://www2.lbl.gov/MicroWorlds/ALSTool

4

X-ray micro-tomographyCollectX-rayimagesofthesampleasyourotate

itthrough180°Usethisseriesofimagesto“reconstruct”the3Dobject

MultipleanglesPenetratingpower Courtesy of D. Parkinson (ALS)

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X-ray micro-tomography

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VisualizationofFiberForm inPuMA atmultiplescales

Porous Materials Analysis (PuMA) TechnicalSpecifications

• WritteninC++• GUIbuiltonQT• Visualizationmodulebasedon

OpenGL• ParallelizedusingOpenMP for

sharedmemorysystems

7

DomainGeneration

Visualization MaterialProperties MaterialResponse

Artificial Material

Generator

Micro-tomography Import, Processing, and Thresholding

Marching Cubes

OpenGL Surface

Rendering

Porosity

Specific Surface Area

Effective Thermal Conductivity

Effective Electrical Conductivity

Diffusivity / Tortuosity(Bulk and Knudsen)

Representative Elementary Volume

Oxidation Simulations

Transient Heat Transfer *

*Under Development

Tortuosity Factors• Quantifiesamaterialsresistancetoa

diffusiveflux• Importantinmodelingdiffusion/reaction

systems– suchasablativeTPSresponse

• 𝜂 = tortuosityfactors• 𝜀 = porosity• 𝐷%&' = referencediffusioncoefficient• 𝐷&'' = effectivediffusioncoefficient

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𝜂 = 𝜀𝐷%&'𝐷&''

SurfacerenderingofFiberForm tomographyinPuMAV2.1.Visualizationcontains≈500milliontriangles.

• Non-dimensionalnumberwhichdefinesthediffusionregime

• Continuum:Kn <<1• Transitional:Kn ≈ 1• Rarified:Kn >>1

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LowKnudsen HighKnudsen

2DdiffusivitysimulationsusingarandomwalkmethodinPuMA.Particlepathsarevisualizedinred.

Kn = �̅�𝑙/=

MeanFreePathCharacteristicLength

Knudsen Number

Tortuosity Factors• Quantifiesamaterialsresistancetoa

diffusiveflux• Importantinmodelingdiffusion/reaction

systems– suchasablativeTPSresponse

• 𝜂 = tortuosityfactors• 𝜀 = porosity• 𝐷%&' = referencediffusioncoefficient• 𝐷&'' = effectivediffusioncoefficient

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𝜂 = 𝜀𝐷%&'𝐷&''

SurfacerenderingofFiberForm tomographyinPuMAV2.1.Visualizationcontains≈500milliontriangles.

Reference Diffusion Coefficient• 𝐷%&' = referencediffusioncoefficient

• 𝐷>?@A = BC�̅��̅�,whichisundefinedasthe

meanfreepathapproachesinfinity• 𝐷%&' thereforemustbebasedonalength

scale.Inthiscase,theDiffusioncoefficientthroughacapillaryofdiameter𝑙/

�̅� =meanthermalvelocity�̅� =meanfreepath

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Continuum FreeMolecular

𝐷%&' = 𝐷>?@A 𝐷>?@A doesnotexist

Bosanquet Approximation• Usedtoapproximate𝐷%&' basedonknown

valuesfor𝐷> and𝐷A.[1]

• Rewrittenforsinglespeciesdiffusioninacapillary,𝐷%&' becomes[2]

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1𝐷%&'

= 1𝐷>

+1𝐷A

𝐷%&' = 13�̅�

�̅�𝑙/�̅� + 𝑙/

[1]Tomadakis,1998[2]Pollard,1948

Choice of Length Scale

1. Define𝑙/ basedonanapproximategeometriclengthscaleforthematerial.TypicallyHI

Jormean

interceptlength.(Tomadakis,Lachaud,Geodict)

2. Define𝑙/ afterthesimulationsarecompleteasthevaluewhichmakesthetortuosityfactorvs.Knudsennumberplotconvergetoasinglevalue.(Zalc)

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Poresizedistribution,computedinGeoDict,ofFiberForm.

Length Scale Option #1Define𝑙/ basedonanapproximategeometriclengthscaleforthematerial.TypicallyHI

Jormean

interceptlength.(Tomadakis,Lachaud,Geodict,PuMA)

• Mostoftenusedintheliteratureandsoftware

• Requiresvaluesof𝜂>, 𝜂A and𝑙/ inordertoapplytheBosanquet approximation

• 𝜂 isnolongerapurelygeometricalproperty,asitisnowafunctionoftheKnudsennumber

• Since𝜂> hadnophysicalmeaningwithout𝑙/,thiscanproduceconfusingresultsof𝜂A<1

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FigurefromTomadakis,1993

Length Scale Option #2Define𝑙/ afterthesimulationsarecompleteasthevaluewhichmakesthetortuosityvs.Knudsennumberplotconvergetoasinglevalue.(Zalc,PuMA)

• Requiresonlyonevalueof𝜂 andacomputedlengthscale,𝑙/, inordertoapplytheBosanquetapproximation

• 𝜂 isnowlongerapurelygeometricalproperty,nolongerafunctionofKn

• Easiertounderstandandimplement

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TortuosityfactorvsKnudsenNumberfor1Dfibers,computedinPuMA,showingtheparallelandperpendiculartortuosityfactorsforOption#1andOption#2

Choice of Length Scale

1. Define𝑙/ basedonanapproximategeometriclengthscaleforthematerial.TypicallyHI

Jormean

interceptlength.(Tomadakis,Lachaud,Geodict)

2. Define𝒍𝑫 afterthesimulationsarecompleteasthevaluewhichmakesthetortuosityfactorvs.Knudsennumberplotconvergetoasinglevalue.(Zalc)

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Poresizedistribution,computedinGeoDict,ofFiberForm.

Applying Tortuosity Factors• Usedtocompute𝐷&'' withinaporous

media,withknowntortuosityfactor,𝜂,knownlengthscale,𝑙/,andknowngasproperties.

• UsingBosanquet approximationtoapproximate𝐷%&',theequationbecomes

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𝐷&'' = 𝜀𝐷%&'𝜂

SurfacerenderingofFiberForm tomographyinPuMAV2.1.Visualizationcontains≈500milliontriangles.

𝐷&'' = 𝜀3𝜂 �̅�

�̅�𝑙/�̅� + 𝑙/

Numerical Methods

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Continuum Rarified

• Canbesolvedusingtypicalnumericalmethodssuchasfinitevolumeandfinitedifference

1. Geodict - ExplicitJumpSolver2. PuMA – ExplicitJumpSolver3. TauFactor – FiniteVolumesolver

• MustbesolvedusingparticlemethodstoaccountforKnudseneffects

1. PuMA – Randomwalksolver2. Geodict – Randomwalksolver

(Knudsenregime)3. SPARTA– DirectSimulationMonte

Carlo

Random Walk Solver• Particlemethodforsolvingdiffusion• Velocityandmeanpathforeachparticlebasedon

exponentialdistribution• Diffusereflectionsareusedforsurfacecollisions• Symmetricboundaryconditions

• 𝜉O isthemeansquaredisplacementoftheparticles

• Meanthermalvelocity,�̅�,andmeanfreepath,�̅�, areimposedtosimulatethedesiredgasspeciesandconditions.

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𝐷&''P =𝜉O

2𝑡

HighKnudsenLow

Knudsen

Wall Collisions• Diffusereflectionsusedforsurfacecollisions• Collisiondetectioncanbebasedonisosurface or

cuberille grid

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HighKnudsenLow

Knudsen

Isosurface (a)andcuberille (b)approximationsofacylinderwithradius3voxels.

(a) (b)

Wall Collisions• Diffusereflectionsusedforsurfacecollisions• Collisiondetectioncanbebasedonisosurface or

cuberille grid

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HighKnudsenLow

Knudsen

Percentdifference(isosurface vscuberille)vsKnudsennumberforthreedifferentidealgeometries

Comparison to Literature

The 5% error is likely due to the limitations of computing in 1993.Simulations by Tomadakis were using only 200 particles andlikely on a small dataset. The PuMA simulations were run on200,000 particles for a total walk length of 10,000 times thedomain length 22

TestCase#1

• 3DFibers,5123• Intersecting,isotropic• 0.6porosity

Comparison to Literature

The 5% error is likely due to the limitations of computing in 1993.Simulations by Tomadakis were using only 200 particles andlikely on a small dataset. The PuMA simulations were run on200,000 particles for a total walk length of 10,000 times thedomain length 23

TestCase#2

• 1DFibers,512x512x256• Nonintersecting• 0.7porosity

Bosanquet Analysis

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TestCase#1

• 3DFibers,5123• Intersecting,isotropic• 0.6porosity

Bosanquet Analysis

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TestCase#2

• FiberForm,0.8mm3

• Transverseisotropic• 0.89porosity

Direct Simulation Monte Carlo• DSMCisaparticlemethodtosimulate

transitionalandrarifiedflowswithhighfidelity

• Verycomputationallyexpensive,preventinglargeorfrequentsimulations

• DSMCdiffusionsimulationsconductedinSPARTA,developedatSandiaNationalLabs.

• Pressurevariedtochangethemeanfreepath,andthereforetheKnudsennumber

• Usedasaverificationcasefortherandomwalkmethod

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DSMCsimulationoftransitionalflowovertheSpaceShuttle.Sparta.sandia.gov

Direct Simulation Monte Carlo

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TestCase#1

• 3DFibers,5123• Intersecting,isotropic• 0.6porosity

Conclusion and Outlook

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DomainGeneration

Visualization MaterialProperties MaterialResponse

Artificial Material

Generator

Micro-tomography Import, Processing, and Thresholding

Marching Cubes

OpenGL Surface

Rendering

Porosity

Specific Surface Area

Effective Thermal Conductivity

Effective Electrical Conductivity

Diffusivity / Tortuosity(Bulk and Knudsen)

Representative Elementary Volume

Oxidation Simulations

Transient Heat Transfer *

*Under Development

• Implemented finite difference and random walk tortuosity factor solvers into PuMA V2.1

• Demonstrated the necessity of using an isosurface collision detection for complex 3d media, a capability which currently only exists in PuMA

• Verified random walk model for tortuosity factors against Direct Simulation Monte Carlo (DSMC) simulations.

• Recommend changing current definitions of tortuosity factor to restore the value as a purely geometrical property.

Acknowledgements

• This work was supported by the Entry System Modeling project (M.J. Wright project manager) of the NASA Game Changing Development program.

• T. Sandstrom, C. Henze, D. Ellsworth, and B. Nelson for useful discussions during the development of PuMA and the parallelization of the oxidation model.

• A.A. MacDowell, H.S. Barnard, D.Y. Parkinson are acknowledged for their assistance with tomography measurements.

• The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

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Questions?

AblationWS,2017Bozeman,MT

JosephC.Ferguson1ArnaudBorner 1

FrancescoPanerai 2Nagi N.Mansour3

1.ScienceandTechnologyCorp.atNASAAmesResearchCenter2.AnalyticalMechanicalAssociatesInc.atNASAAmesResearchCenter3.AdvancedSupercomputingDivision,NASAAmesResearchCenter