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
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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