Extension of the bifurcation method for diffusion coefficients to porous medium transport

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Extension of the bifurcation method for diffusion

coefficients to porous medium transport

Cédric Descamps, Gerard L. Vignoles

To cite this version:

Cédric Descamps, Gerard L. Vignoles. Extension of the bifurcation method for diffusion coefficients to

porous medium transport. Comptes rendus de l’Académie des sciences. Série IIb, Mécanique, Elsevier,

2000, 328 (6), pp.465-470. �10.1016/S1620-7742(00)00038-6�. �hal-00327610�

Extension of the bifurcation method for diffusion coefficients to porous

medium transport

C. DESCAMPS

Snecma Propulsion Solide

Les 5 chemins,

F33187 LE HAILLAN Cedex, France

Cedric.descamps@snecma.fr

and

G. L. VIGNOLES

LCTS

Laboratoire des Composites ThermoStructuraux

3,Allée La Boëtie – Université Bordeaux 1

F33600 PESSAC, France

vinhola@lcts.u-bordeaux1.fr

Published in :

diusion oeÆ ients to porous medium

transport

Cedri Des amps and Gerard L. Vignoles

Laboratoiredes CompositesThermoStru turaux (UMR 5801 CNRS-SNECMA-CEA-UB1) 3, Allee La Boetie,Domaine Universitaire,

F33600PESSAC, Fran e Fax: (00 33) 5 56 8412 25 E-mail: des ampsl ts.u-bordeaux.fr

Abstra t

WepresentanextensionofBartlett'sbifur ationmethod(Bartlettetal.(1968))for

theapproximate omputationof multi omponentdiusion oeÆ ientsina gaseous

mixturetodiusioninporous media. Onbehalf ofthe remarkthatthebifur ation

oeÆ ients F i

aremerely proportional tothe square rootof the molar massesM i

,

we statethat Knudsendiusionmayalso be representedthrough somebifur ation

fa torF K

.Thisapproximationistestedinavarietyof ases,displayinggoodresults

ex eptforvery light gasspe ies.

Extension de la methode de bifur ation pour les

oeÆ- ients de diusion au transport en milieu poreux

Resume|Nouspresentonsuneextensiondelamethodedebifur ationdeBartlett

que les oeÆ ients de bifur ation F i

sont approximativement proportionnels a la

ra ine arreedes massesmolaires M i

, il estpose quela diusionde Knudsenpeut

aussi ^etre representee par un fa teur de bifur ation F K

. Cette approximation est

testee pour dierents as, et montre de bons resultats sauf pour des espe es tres

legeres.

Mots- les: Diusionmulti omposants; Milieuxporeux; Methode d'approximation

Key words: Multi omponent diusion; Porous Media;Approximationmethod

Version fran aise abregee

Pour le traitement dela diusion gazeuse multi omposants, qui requiert une

inversion des relations de Stefan-Maxwell (eq. 1), Bartlett et al. (1968) ont

propose une methode dite \de bifur ation", basee sur la onstatation que

haque oeÆ ient de diusion binaire pouvait ^etre ree rit suivant l'equation

(4).Ce iestjustieenfaitparla orrelationassezetroiteentrelesfa teursde

bifur ation F i

etl'inverse dela ra ine arreedes masses molairesM 1=2 i

. Une

expression expli ite pour les ux massiques est alors obtenueen fon tion des

gradients (eqs. 5,6). Les oeÆ ients de bifur ation doivent ^etre pre- al ules

par une methodedeminimisation d'erreur, omme lesimplexe.

Nous proposons d'etendre ette pro edure aux equations de gaz poussiereux

de Mason et Malinauskas (1983) (eqs. 10), qui ontiennent en parti ulier la

ontributionsupplementairedeladiusiondeKnudsen(eq.12).Comme

elle- i fait aussi intervenir une proportionnalite a M 1=2 i

i K

poreux (eq.13).Desrelationsexpli itessontalorsobtenuespour les ux

mas-siquesdiusifsdes gaz (eqs. 21,22),etenprenant orre tementen ompteles

dependan esdierentesdes diusionsbinaireetdeKnudsenalatemperature

eta la pression,ondoit orriger F K

suivant l'expression (24).

Lamethodeoriginalementproposeepresenteunepre isiondel'ordredequelques

pour- ents,sauf pour desespe es treslegeres outreslourdes.C'estegalement

le as del'extension proposeei i.

1 Introdu tion

In many modeling problems involving gas mixtures (distillation, ombustion,

atalysis, et ...), an important question is how to treat multi omponent

diusion. Con eptual diÆ ulties have been over ome sin e a long time, but

numeri al issues are still of a tuality, be ause a hieving simultaneously

a - ura y, onvergen e and omputational rapidity is not a simple task. In this

ontext, the bifur ation method appeared as a fairly good ompromise. On

the other hand, a popular treatment of diusive uxes in porous media,

fea-turingKnudsendiusion(also alledeusion),istheDusty-GasModel,whi h

requiresasmu h omputationaleortasforthedeterminationoffree-medium

diusive uxes. We address here an extension of the bifur ation prin iple to

tothe Dusty-GasModel framewill bepresented and tested.

2 The Bifur ation Method

Letus rst re all brie y the ontext of Bartlett's approximation.

2.1 Multi omponentdiusion relations

Molar diusive uxes J i

are relatedtothe respe tivefor es rP i =RT through Maxwell-Stefan relations: rP i RT = n X j6=i x j J i x i J j D ij (1)

In a mass balan e equation, the divergen e of the uxes are needed : this

means that eq. (1) has to be solvedfor the uxes. This system also presents

also a singularity be ause (i) all mass uxes have to sum up to zero and (ii)

all gradients have to sum up to rP tot

=RT, whi h is determined elsewhere.

Removing these singularities and solving for the uxes may lead to heavy

omputations,soadire t,expli itformulawouldbeofgreatpra ti alinterest

inthe aseoflargesystemse:g: ombustionorpyrolysisproblems(Giovangigli

(1996)).

ij D ij = 3 16 p 4RT  q M ij  2 ij D ij (2) where M ij

is the redu ed mass of the pair (i;j) :

M ij = 1 2 M i M j M i +M j (3)

2.2 Prin iple of the bifur ation method

Theprin iple ofthebifur ation methodistostatethat ea hmulti omponent

diusion oeÆ ient may be expressedas :

D ij = D ref F i F j (4) where D ref

is some referen e diusion oeÆ ient and F i

a bifur ation fa tor

proper to ea h spe ies i. If n spe ies are present, then only n fa tors F i

are

to be determined instead of n(n 1)=2 oeÆ ients D ij

. In addition to this,

a dire t, expli it expression for the diusive mass uxes as a fun tion of the

gradients isobtained : j i =M i J i = n X j=1 D ij rP j (5)

8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : D ij = ! i M j RTF j D ref n X l=1 x l F l si i6=j D ii =(1 ! i ) M i RTF i D ref n X l=1 x l F l si i=j (6)

The reader is referred to the original referen e (Bartlett et al. (1968)) for

the detailed derivation ofthe latterexpressions.Using this relationspeeds up

strongly the resolution of mass transport problems, while keeping the main

physi al onstraint that the sum of the mass uxes is zero.

Comparing (4) to the theoreti al expression (2), one sees that this

approxi-mationwill bevalid assoonas :

q M i +M j  2 ij D ij = st=G (7)

The quantityG inthisrelationhas been omputed forasetof 69gasspe ies,

that is, for 2346 spe ies pairs. Fig. 1 is an histogram of G, showing that

this quantity possesses a modest standard deviation. The most important

deviationswere observedforvery heavy,poorly volatile spe ies su hasHgI 2

.

In order to determine the F i

oeÆ ient set, the simplex method is used. It is

hosen to minimize the following error fun tion :

E = n X i=1 n X j=i+1 (F i F j D ij D ref ) 2 (8)

exa tlyequal totheanalyti alpredi tion,sin einthis asetheapproximation

is indeedexa t.

Bartlett et al. (1968) report from their tests a mean error of 5% onthe

mul-ti omponent diusion oeÆ ients. We also performed a test on a set of 14

hydro arbons plus mole ular hydrogen. The mean error is 2.03%. The

maxi-maldeviationis 25.75%fortheH 2

CH 4

pair.Thiswastobeexpe tedsin e

they are the lightest spe ies presentin the system : their G fa toris well

be-low the mean G fa tor of all spe ies pairs. Forthe 69-spe ies system used to

he k validity of relation (7), the mean error was 3.44%, showing reasonable

usefulness of the approximation s heme tolarge systems.

2.3 In uen e of temperature

As the binary diusion oeÆ ients vary with temperature in amoderatebut

ompli ated way (from T 3=2

to T 2

), one should a priori take into a ount a

thermal variation of the F i

fa tors. Fig. 2 shows that when T in reases, the

F i

on erning the lightest spe ies tend to in rease while the heaviest spe ies

fa tors tend tode rease. However,one noti es that the overall thermal

varia-tion in a broad temperature range (300-1500K) for the F i

is small ompared

to the initial errors due tothe method. A ordingly, one may safely onsider

that these fa tors do not vary independently with temperature, the global

dependen e of diusion onT being in orporated inside D ref

Bartlettet al.(1968) alsohas shown fromexperimentaldataof Svehla (1962)

that the F i

maybeestimated fromthe empiri al orrelation :

F i /M i 0:461 (9)

This fa t may be expe ted from formulas (2,3), where it is readily seen that

D ij is proportional to (M i M j ) 1=2

if the G fa tor in eq. (1) does not vary

appre iably. Fig. 4 shows a plot of the F i

omputed in the 69-spe ies test

ase versus M 1=2 i

, in whi h dire t proportionality appears as a fairly good

approximation.

3 Appli ation to diusion in porous media

3.1 The Dusty-Gas Model equations

A popular extension of the Stefan-Maxwell relations to porous media is the

Dusty-GasModel(MasonandMalinauskas(1983)).Knudsendiusionisviewed

as the diusion of a spe ial gas pair, namely a true gas and the solid phase

des ribed asa olle tion of giant gasmole ules (\dust")whi hare held xed

(n+1) spe ies. Relations(1) be ome then : n X j6=i D 1 ij;e  x i ~ J D j x j ~ J D i  D 1 iK;e ~ J D i = rP i RT (10)

wherethe gas pairdiusion oeÆ ientshavebeen repla ed by ee tive

quan-tities D ij;e

whi h take intoa ount the in uen e ofthe porous medium:

D ij;e = 1 b D ij (11)

and where the Knudsen diusion oeÆ ientsD iK;e read : D iK;e = 1 K D iK = 1 K 1 3 s 8k B T M i d h (12) The quantities , d h ,  b , and  K

refer respe tivelytothe porosity,mean pore

diameter,binary diusiontortuosity,and Knudsendiusiontortuosityforthe

onsidered porous medium. All of them may be oordinate-dependent if the

medium is non-homogeneous, and the last two are se ond-order symmetri

tensors inthe ase of an anisotropi medium(Ofori and Sotir hos (1997)).

Ineq. 12we see learlythat the proportionalitytoM 1=2 i

ispreservedforthe

true gas i. A ordingly, the same kind of F i

that is used for multi omponent

Relation(12) mayberewritten : D iK = D ref F i F K (13) where F i et D ref

are the diusion fa tors and the referen e diusivity

om-putedfromthebinary(ordinary)diusiondata.F K

in orporatesa ontribution

arising fromthe solid phase. As a matterof fa t, F K

should be onsidered as

a eld whenit is non-homogeneous.

Now dire t expli it relations for the uxes may be attained, as in the ase of

standard multi omponent diusion. Summing up eqs. (10) on all gas spe ies

i, one has : n X i=1 0  n X j6=i D 1 ij;eff (x i ~ J D j x j ~ J D i ) D 1 iK ~ J D i 1 A = n X i=1 rP i RT (14)

and, aftersimpli ation :

n X i=1 D 1 iK ~ J D i = rP RT (15)

This relation is known as Graham'slaw of eusion. Introdu ing the

bifur a-tion, eqs. (10) and (15) are rewritten :

n X j6=i  b  x i ~ J D j x j ~ J D i  F i F j  K ~ J D i F i F K =D ref rP i RT (16) n X i=1 ~ J D i F i F K =  1 K D ref rP RT (17)

 b  x i ~ J D i F i F i x i ~ J D i F i F i  (18)

Eq. (16) nowreads :

 b 0  x i F i n X j=1 ~ J D j F j ~ J D i F i n X j=1 x j F j 1 A  K ~ J D i F i F K =D ref rP i RT (19)

Combiningthis last equationwith eq. (17) yields :

D ref RT  rP i  b  1 K x i F i F K rP  = 0   b n X j=1 x j F j + K F K 1 A ~ J D i F i (20)

Using mass uxes insteadof mole uxes, one obtainsnally a Fi kianform :

~ j D i = n X j=1 D ij rP j (21)

where the porous-mediummulti omponent diusion oeÆ ientsD ij read : 8 > > > > > > > > > < > > > > > > > > > : D ij = D ref M i RTF K 0  n X j=1  b x j F j + K F K 1 A 1  b  1 K x i if i6=j D ii = D ref M i RTF i F K 0  n X j=1  b x j F j + K F K 1 A 1   b  1 K x i F i +F K Id  if i=j (22)

3.3 Determination of the Knudsen diusion fa tor

The Knudsen diusion fa torF K

may be dire tly determined fromthe

inver-sion of (13) : F K = D ref F i D K i (23)

K

approximately the ase, sin e the F i

fa tors are only approximately

propor-tionaltoM 1=2 i

.Fig.5isaplotofF K

fromeq.(23)forthepre eding15-spe ies

set. Itis learlyseen fromthis gurethat the dis repan yfromthe averageis

largerfor thelightest gaseousspe ies, and mainlyfor hydrogen.Alternatively

to the hoi e of a simple average , the F K

fa tor may be determined by the

aforementioned simplex method, altogetherwith the F i

fa tors.

Knudsendiusiondependsontemperatureandpressureinadistin twaythan

ordinarydiusion:theformervariesasT 1=2

whilethelattervariesasT 3=2 P 1 ormore(a tually,asT 1:75 P 1

a ordingtoFuller'smodel forbinary diusion

oeÆ ients (Reid et al. (1987))). A onsequen e of this is that if D ref arries the usual T 1:75 P 1

dependen e, then the Knudsen bifur ation fa tor F K

has

tobe anexpli it fun tionof temperatureandpressure, whilethe F i are not : F K =F 0 K  T T 0  1:25 P 0 P ! (24) 3.4 Computational a ura y

Themulti omponentdiusionmatri esobtainedthroughthisbifur ationmethod

havebeen omparedtothestandardDusty-GasModelvaluesinthe15-spe ies

test ase. Verya eptablemean errorsareobtained (about 1%),but the

max-imal error is important for the lightest spe ies (up to 30%). This ould be

is that the mean error islowerin the porous medium ase.

Figs. 7 and 6 are plots of the mean error as a fun tion of temperature and

pore diameter. Again, itappears thatthe temperaturehas onlyasmall ee t

onthe results. On the other hand,the pore diameter has astrongerin uen e

:the larger the pores, thelarger the meanerror, but the smallerthe maximal

error. Tointerpretthis fa t, re allthat the maximal error is due tothe

light-estgasspe ies, forwhi hana priori evaluation of F K

wouldbe substantially

largerthantheaverage(g.5).Hen e,thiserror(duetotheextrahypothesis)

be omes more and more visible asthe pore radius de reases.

4 Con lusion

An extension of the bifur ation method presented by Bartlett et al. (1968)

tothe Dusty-GasModel(Mason and Malinauskas(1983))has been presented

andtested.Itprovidesanexpli itexpressionforthediusivemulti omponent

uxes inporous mediawhi hretainsGraham'slawofeusionasa onstraint.

Theapproximated oeÆ ientsexhibitthe samedegreeof pre isionthan when

noporous medium ispresent,that is :

 The mean error is of the same order of magnitude, and even somewhat

lower.

 Tests performed in the absen e of very light gas spe ies have proved very

satisfa tory a ura y onall spe ies.

Thelimitsof ombiningthebifur ation methodandtheDusty-GasModelare

only the independent limits of both models ; so, inside them, it appears as

a valuable omputational speed-up for problems dealingwith mass transport

of large hemi al systems in porous media like har ombustion, pyro arbon

inltration, et ...

5 A knowledgements

The authors are indebted to SEP, divison de SNECMA,for a Ph.D. grant to

C.D.. They also wish to thank Ph.Le Helley (SEP-SNECMA) and G. Dua

Bartlett, E. P., Kendall, R. M., and Rindal, R., An analysis of the

oupled hemi ally rea ting boundary layer and harring ablator. Part IV

: A unied approximation for mixture transport properties for

multi om-ponent boundary layer appli ations, Te hni al Report CR-1063, NASA,

(1968).

Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport

Phe-nomena, John Wiley, New York City,NY, USA, 1960.

Giovangigli, V.,ModelisationdelaCombustion,CNRS, Paris, Fran e,

Im-ages des Mathematiques, 1996.

Mason, E. A. and Malinauskas, A. P., Gastransport in porous media:

the Dusty-Gasmodel,Chemi alengineeringmonographs,Elsevier,

Amster-dam, The Netherlands, 1983.

Ofori, J.Y.andSotir hos,S.V.,Multidimensionalmodelingof hemi al

vapor inltration: Appli ation to isobari CVI, Ind. Eng. Chem. Res. 36

(1997), 357-367.

Reid, R. C., Prausnitz, J. M., and Poling, B. E., The properties of

gases andliquids, M GrawHillBookCompany,NewYorkCity,NY, USA,

4 th

edition,1987.

Svehla, R. A., Estimatedvis osities andthermal ondu tivitiesof gasesat

0

1

2

3

4

5

6

7

8

0.315

0.615

0.915

1.215

1.515

%

Function G

Fig.1. Histogramof G(see eq. (7)in text)

0

0.5

1

1.5

2

2.5

3

200

400

600

800 1000 1200 1400 1600

H

₂

CH

₄

C

₃

H

₈

C

₆

H

₆

C

₆

H

₁₂

F

_i

T(K)

Fig. 2. Bifur ation oeÆ ients F i

vs: temperature. Chemi al system of 5 C- and

0

0.5

1

1.5

2

0

2

4

6

8

10

F

i

F

i

M

i

1/2

Fig. 4.Plot ofdiusionfa tors vs: thesquare rootofthemolar mass.

Fig.5.Plotof theKnudsendiusionfa torfrom eq. (23)vs: thesquarerootofthe

thelargestdiusion oeÆ ient.

Fig.7.Meanerrorin%onthemulti omponentdiusionmatrixwithrespe ttothe