Chemical Engineering Science 68 (2012) 392–400

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Chemical Engineering Science

0009-25

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May the Knudsen equation be legitimately, or at least usefully, applied todilute adsorbable gas flow in mesoporous media?

J.H. Petropoulos n, K.G. Papadokostaki

National Center for Scientific Research ‘‘Demokritos’’, Institute of Physical Chemistry, 15310 Ag. Paraskevi Attikis, Athens, Greece

a r t i c l e i n f o

Article history:

Received 16 June 2011

Received in revised form

2 September 2011

Accepted 25 September 2011Available online 1 October 2011

Keywords:

Gas adsorption

Porous media

Modeling of transport processes

Diffusion in membranes

Knudsen flow

Surface flow

09/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ces.2011.09.046

esponding author. Tel.: þ30 210650 3787, þ

0 2106511766.

ail address: petrop@chem.demokritos.gr (J.H.

a b s t r a c t

The question of applicability of Knudsen’s equation to dilute adsorbable gas flow in mesoporous media

was the subject of a recent debate between Ruthven et al. (2009), Ruthven (2010) and Bhatia and

Nicholson (2010, 2011). Here, we present a critique of the said debate and introduce new information

of critical importance, which has nevertheless been disregarded by both sides thereof. It is pointed out

that von Smolukowski’s derivation leaves no doubt that Knudsen’s equation is meant to apply to the

limiting case of zero gas adsorption. In practice, this limit may be expected to be approached at high

temperature/low gas adsorbability but the crux of the matter is (i) choice of the right transport

parameter (in this case the ‘‘reduced permeability’’) to provide a useful criterion of approach to the

Knudsen limit and (ii) thorough study of the course followed by the said parameter in its approach to

the said limit. In this light, we begin our review of the aforesaid new information with conventional

surface flow theory, which can deal satisfactorily with higher and moderate, but not with the case of

weak, adsorption (because the reduced permeability does not converge smoothly to the Knudsen limit).

Even so, we show that it can provide a good answer to a significant question raised in the aforesaid

debate. We continue with the more advanced (but still analytical) surface flow approach of Nicholson

and Petropoulos (1973), which has provided a plausible physical mechanism for the experimentally

observed passage of the reduced permeability through a minimum with rising temperature, in the

weak adsorption region. It is shown here that the said experimental data constitute the first

observations (including the ‘‘fine structure’’) of what we have called ‘‘apparent or quasiKnudsen’’ (to

avoid confusion with proper Knudsen) flow behavior, which is the central theme of the debate in

question. We then pass onto a discussion on the confirmation of the above interpretation of

quasiKnudsen-flow ‘‘fine structure’’, as well as the identification of conditions favorable to the

appearance of such flow in practice, afforded by the results of ab initio evaluation of dilute adsorbable

gas flow, under suitable adsorption potentials, in model pores (and model pore networks) by and

Petropoulos (1981, 1985) and Petropoulos and Petrou (1991).

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

There is little doubt that when, over a century ago, Knudsenpublished his well known equation, he could hardly imagine thatits use would spread far and wide to an extent that can probablybe matched only by the extent of its misuse! A recent debate onthe question of applicability of the Knudsen equation to diluteadsorbable gas flow in mesoporous media is a case in point.

In our view, even if Knudsen’s derivation of his equation (basedon arguments of momentum balance) had left any doubt about therange of its applicability, such doubt was put to rest, very shortly

ll rights reserved.

30 210650 3661;

Petropoulos).

thereafter, by Smolukowski’s more rigorous (see e.g. Kennard, 1938)molecular path tracing (MPT) derivation, based on linear gasmolecular trajectories within the model pore and hence clearlyrestricted to nonadsorbable gas flow. This fact constitutes a strongcaveat against indiscriminate invocation of the said equation inpractice, except as a limiting case of sufficiently low gas adsorptionenergy (En-0) and/or sufficiently high temperature (T-N). On theother hand, the possibility of Knudsen-like behavior under othersuitable conditions (but still within the dilute-gas Knudsen flow

regime) is not expressly precluded; but should be qualified asapparent or quasiKnudsen behavior to avoid confusion with theabove limiting case.

In fact, it is the latter possibility that has provided fertileground for the aforementioned debate, one side of which(Ruthven et al., 2009; Ruthven, 2010), focused on actualobservations of such quasiKnudsen behavior, accepting them

J.H. Petropoulos, K.G. Papadokostaki / Chemical Engineering Science 68 (2012) 392–400 393

(somewhat incautiously, in view of the aforementionedcaveat) as evidence of genuine applicability of the Knudsenequation. The other side, namely the view that adsorbabilityeffects should always be kept in mind and duly accounted forin modeling, was presented by Bhatia and Nicholson (2010,2011); but this was done in a way that calls for criticalreexamination (see Section 2).

In particular, Ruthven et al. (2009) referred to their study ofdilute gas permeation through a thin mesoporous silica mem-brane deposited on a porous support of much larger pore size.The data were suitably pretreated to separate out the effect of theporous support on measured permeability, on the assumptionthat gas flow in the said support occurred in the viscous regime.The resulting ‘‘corrected data’’ were found to vary with tempera-ture (T) and gas molecular weight (M) as predicted by theKnudsen equation and to yield Knudsen diffusivities Dexp

K in goodaccord with the corresponding Knudsen ‘‘benchmark’’ theoreticalones DK, if the ‘‘tortuosity factor’’ of the silica membrane was setat t¼3.2 (regarded by these authors as an eminently reasonablevalue). These (in conjunction with some other pertinent) findingsled Ruthven et al. (2009) to criticize the results of simulations ofdilute adsorbable gas diffusion in single model pores, reportedby the Bhatia group (Bhatia and Nicholson, 2003; Bhatia et al.,2004; Jepps et al., 2003, 2004), for returning Knudsen diffusivityvalues ðDexp

K Þ which (i) were typically substantially short of thosederived from the Knudsen equation (DK) and/or (ii) exhibitedvariability with temperature in excess of the Knudsen predictedT1/2 dependence.

The former criticism was rebutted by Bhatia and Nicholson(2010), who justified the discrepancy in question in terms of theeffect of gas–solid interactions, invoking in particular theobserved conformity of the said simulation results with pertinentpredictions of their so called ‘‘oscillator model’’ (N.B. not toconfuse this with models of molecular oscillators).

On the other hand, Bhatia and Nicholson (2010) rejected the(corrected) data of Ruthven et al. (2009), on the grounds that thecorrection procedure was at fault, arguing that (i) the meaneffective pore radius of the relevant porous support (rpE50 nm)was more consistent with a dominant Knudsen, rather than withthe assumed viscous, gas flow regime therein and (ii) adoptionof the viscous flow assumption led to material overestimationof Dexp

K ; so that the correct result for t¼3.2 should be DexpK oDK,

in line with the effect of gas adsorbability invoked above. Ofcourse, one might just as well choose to readjust t to another‘‘reasonable value’’; as was (inevitably) done subsequently inthe response of Ruthven (2010), who calculated that, under theconditions specified by the Bhatia side, Dexp

K ¼DK could berestored for tE4! However, the Ruthven side was also ableto adduce more compelling arguments in rebuttal of Bhatiaet al.’s criticism, principal among which was the fact that theporous support in question exhibited the kind of bimodal poresize distribution that (in practice) is strongly indicative of abidisperse micro- or meso-pore network permeated by amacropore one; with the latter network typically occupying arelatively small fraction of the total pore volume but carryingthe lion’s share of the gas flux (see, e.g. Petropoulos and Petrou,1992). Thus, Ruthven et al.’s assumption of viscous flow in theirporous support (which was characterized by effective macro-pore size of the order of micrometers) is, to all appearances,justified; hence their claim that their permeability data wereproperly corrected is acceptable.

In a more recent paper, Bhatia and Nicholson (2011) come(somewhat belatedly; see Section 3) to recognize the reality ofquasiKnudsen flow and to obtain some insight into its ‘‘finestructure’’, but still insist on denying the validity of the afore-mentioned Ruthven et al. (2009) corrected data. It is also good

that they come to appreciate the substantial role played by thesorption coefficient in determining flow behavior. On the otherhand, they steadfastly and unaccountably remain oblivious ofsome very striking gaps in their references to pertinent literature,which happen, in our view, to be of crucial importance for thematter at issue here.

Accordingly, we proceed below to review briefly the missingliterature and to demonstrate its importance; but also to drawattention to some specific points (especially in connection withthe definition and usage of diffusivity parameters), which are alsoliable to mislead or confuse the interested reader. For thispurpose, we begin by defining our terms as fully and preciselyas possible.

2. Review of missing literature

2.1. Definition of diffusivity and permeability

According to standard diffusion theory, any solid-penetrantsystem obeying Fick’s and Henry’s laws is characterized byconstant sorption (S), diffusion (D), and hence permeability(P¼SD), coefficients, as defined in Eq. (1) et seq. below. Measure-ment of the steady-state permeation flux J yields P (or Q¼P/RT;see Section 2.2) and the main point to note (not very consistentlykept in mind in the above debate) is that D cannot be properlydeduced therefrom without knowing (or making an explicitwell founded assumption about) the corresponding value of S

(or Sp¼S/RT; see Section 2.2).

2.2. Conventional theory of dilute gas flow in mesoporous media

Consider steady-state permeation flux (in the Knudsen regime)Jg through a porous disk of porosity e and specific internal surfacearea A (per unit volume of the porous medium), cross-sectionalarea Ac and thickness l, under an imposed external gaseousconcentration difference Dcg (equal to a gas pressure differenceDp/RT). The permeability P (¼RTQ, where Q is the permeabilityformulated in terms of Dp) is defined by

�Jg ¼ AcPDcg=l¼ AcSDDcg=l ð1Þ

with S¼(C/cg)equil¼Sp/RT [Sp¼(C/p)equil], where C is the concen-tration of gas sorbed per unit volume of porous medium and cg isthe external gas concentration.

A nonadsorbed gas is taken up by simple occlusion in (i.e. bysimple filling of) the pores; hence C ¼ Co

g ¼ ecg and S¼ Sog ¼ e.

The occluded molecules are expected to exhibit Knudsen diffu-sivity represented by D¼Do

g. The salient features of the physics offlow behavior may be assumed to be practically independent ofthe structural details of real disordered porous media of interesthere. Accordingly, one may define a single ‘‘structure factor’’kg (according to the usage followed by e.g. Barrer and Gabor,1959; Barrer, 1963; Petropoulos and Petrou, 1992) linking Do

g toan easily calculable theoretical Knudsen benchmark value DK. Thebenchmark normally chosen is a uniform long circular cylindricalmodel pore, with well defined diffusely reflecting walls of radiusrp¼2e/A sufficiently small to ensure lg5lG (where lg, lG denotethe gas molecular mean free path in the intrapore and externalgas phases respectively) and Knudsen’s equation yields lg¼2rp

(see, e.g. Present, 1959; Barrer, 1963). Thus:

Dog ¼ kgDK ¼ ð4=3Þkgrpv1 ¼ ð8=3Þkgv1e=A; v1 ¼ ðRT=2pMÞ1=2

ð2a;bÞ

where the unidimensional mean gas molecular speed v1 embodiesthe requisite physics and kg represents the cumulative structuraleffect of (i) real single-pore geometry, such as short pore length

Table 1Diffusion coefficients (dimensionless) for 2D slit (of diameter dp¼3ho and length

lp¼30ho, with a triangular adsorption potential well of depth U0 and width ho),

calculated (a) analytically: D¼DA; (b) by MPT: D¼DMPT; (c) by MC (transmission

probability): D¼DMC1 (value in parentheses corrected for edge effects for compar-

ison with DMPT); (d) by MC (random walk): D¼DMC2. DMC results were based on

samples of 103 or (in some cases) 104 molecules (see text and Nicholson et al.,

1979).

U0/RT 0 0.5 2.0

DA 3.36

DMPT 3.36 1.84 0.94

DMC1 (N¼103) 2.86 1.76 1.00

DMC1 (N¼104) 2.92 (3.52)

DMC2 (N¼103) 3.10 1.72 0.89

DMC2 (N¼104) 3.07

J.H. Petropoulos, K.G. Papadokostaki / Chemical Engineering Science 68 (2012) 392–400394

and cross-sectional shape other than circular (which lead to kgo1or kg41 respectively); (ii) ‘‘heteroporosity’’ (i.e. variability of realpore structural features, such as pore size distribution) in theporous solid and pore network connectivity; (iii) macroscopicanisotropy or structural inhomogeneity of the porous medium(such as preferred pore orientation or non-uniform porosity dis-tribution therein; see, e.g. Petropoulos and Petrou, 1992;Galiatsatou et al., 2006; Petropoulos and Papadokostaki, 2011).

An adsorbable gas is taken up by a porous medium to anextent exceeding that due to simple occlusion; thus C ¼ Co

gþCs

and S¼ SogþSs. Following Gibbs, this ‘‘excess sorption’’ coefficient

Ss¼Aks (where ks is the Henry-law adsorption coefficient of thegas) is conventionally considered to represent adsorption on theinternal surfaces of the porous medium. The adsorbed molecules(admolecules) are, for simplicity, pictured as forming a dilutemonolayer on the pore surfaces, wherein surface transport,governed by diffusivity Ds, is considered to occur by an activatedjump mechanism with activation energy E# (e.g. Barrer, 1963;Hwang and Kammermeyer, 1966a,b; Lee and O’Connell, 1975;Okazaki et al., 1981; Uhlhorn et al., 1989); thus giving rise to acorresponding ‘‘surface permeability’’ term Ps, as shown below:

S¼ SogþSs ¼ eþAks; P¼ PgþPs ¼ So

gDgþSsDs ¼ eDgþAksDs

ð3a;bÞ

Ds ¼D#s k#s =ks ¼ ðk

#s =ksÞkskalsv1 ð4Þ

where superscript # denotes parameters pertaining to activated(i.e. free to move along x) admolecules; ls is the typical jumplength along the surface; ka is a numerical factor characteristic ofthe geometry of the lattice of surface adsorption sites and ks is astructure factor, originally thought to represent the effect ofstructural deviation of real pore surfaces from ideal surfacelattices (Barrer, 1963; Barrer and Gabor 1960); but subsequentlyshown by Nicholson and Petropoulos (1968) to be a function ofboth gas adsorbability and temperature and to account primarilyfor structural effects on surface flow in porous media as comparedwith surface flow in single model pores. For this purpose, theseauthors (e.g. Nicholson and Petropoulos, 1971, 1975a; Nicholsonet al., 1988; Petropoulos and Petrou, 1991, 1992; Petropoulos andPapadokostaki, 2011) studied theoretically the behavior of kg

and ks as a function of salient structural model parameters atboth the mesoscopic (using the heteroporous network modelintroduced in this field by Nicholson and Petropoulos, 1971)and macroscopic (using models of axial and radial spatial varia-tion of porosity; e.g. Nicholson and Petropoulos, 1982) levels.

A basic postulate of conventional surface flow theory (by analogyto the above Gibbs definition of adsorption) is to set Dg �Do

g, i.e. toidentify Dg of a given gas in Eq. (3b) with the Knudsen diffusivity ofthe corresponding (fictitious) nonadsorbed gas. This enables experi-mental determination of kg in Eq. (2a) (by the use of He (ksE0) as a‘‘calibrating gas’’), and hence evaluation of Dg ¼Do

g in Eq. (3b)(making use of Eq. (2b)) and ultimately of Ds (given Ss) (e.g. Barrerand Gabor, 1959; Barrer, 1963; Lee and O’Connell, 1975).

The above conventional surface-flow theory has the virtue ofmaking an, admittedly gross (but amenable to subsequent refine-ment; see Section 2.3), distinction between relatively faster (Dg)and slower (Ds) intrapore gas diffusion mechanisms; which isirretrievably lost in the ‘‘overall diffusion coefficient’’ D, thusobscuring insight into the physics of adsorbable gas transport.This difficulty is, not infrequently, compounded by indiscriminateuse of the terms diffusivity and permeability. Statements illus-trating the deplorable consequences of this state of affairs, may bequoted from both sides of the debate, notably: ‘‘y a surfacediffusion contribution y would increase rather than decrease thediffusivity’’ (Ruthven et al., 2009) or ‘‘If one insists on using the

Knudsen model, while still considering adsorption, then the fluxpredicted by Eq. (2) must be enhanced by a factor equal to theHenry law constant y’’ (Bhatia and Nicholson, 2010). The exactmeaning of the Bhatia Henry-law constant K and accompanyingdiffusivity Do may be ascertained by reference to Eq. (1) of Bhatiaand Nicholson (2011); comparison with S and D in Eq. (1) here,then shows that Ke�S and Do/t�D. Hence the aforesaid quotationfrom Bhatia and Nicholson (2010) is obviously based on themistaken presumption (not in line with their own theory!) that,in the systems under consideration here, D remains constantwhen S is raised; while the preceding quotation from Ruthvenet al. (2009) suggests lack of clearcut distinction between diffu-sivity and permeability. On the other hand, Eqs. (1) and (3) hereyield

D¼ P=S¼Dg½1þðAks=eÞðDs=DgÞ�=ð1þAks=eÞ ð5Þ

Bearing in mind that, at least for disordered porous media ofinterest here, one normally finds DsoDg, and the discrepancywidens with rising ks (e.g. Barrer and Gabor, 1959; Barrer, 1963),Eq. (5) yields DoDg and the discrepancy is similarly accentuatedby rising ks. Thus, as far as correct prediction of the effect of gasadsorbability on D is concerned, the (more than a half-centuryold) conventional surface-flow theory does (in its simple butinformative way) as good a job as the ensuing, more elaborate,non-explicit models of Nicholson et al. (1979) and Nicholson andPetropoulos (1981) (see Table 1) or the ‘‘oscillator’’ model of Jeppset al. (2003, 2004) (invoked by Bhatia and Nicholson, 2010)!

The label ‘‘tortuosity’’ attached to structure factor t associatedwith D harks back to the simplistic way of modeling mesoporousmedia as bundles of tortuous homoporous capillaries; however t(or rather 1/t) as defined here cumulates the roles of the structurefactors kg and ks defined above (accordingly, Bhatia et al.’s findingthat here t is not a purely geometrical constant is certainly notnews), which are themselves not simple entities. It is thusdifficult to see how any chosen (without precise justification)value(s) of the said t can usefully serve as grounds for basing anyserious theoretical argument or data analysis thereon (as hasbeen done more than once in the above debate). As far as we cansee, checking carefully the constancy of the reduced permeability(defined in Section 2.3) with high quality experimental data,remains the best possible criterion of conformity to genuineKnudsen behavior at our disposal.

2.3. Advanced analytical surface transport theory

For the study of the effect of temperature and adsorbabilityon P, the combined Eqs. (3b) and (4) may be convenientlyrewritten as

P=v1 ¼ ð8=3Þkge2=AþkskalsAk#s ¼ BgþBsk#s ð6Þ

J.H. Petropoulos, K.G. Papadokostaki / Chemical Engineering Science 68 (2012) 392–400 395

where Bg is independent, and Bs only weakly dependent, on gasadsorbability and temperature. Thus, the physical behavior ofthe reduced permeability, expressed as P/v1, as P(M/T)1/2, asQ(MT)1/2 or as j¼ P=Po

g (if an acceptable Pog value is available),

should be governed primarily by that of k#s . Given the fact thatthe adsorption properties of normal and activated admoleculesdiffer primarily by the magnitude of pertinent adsorptionenergy (En for the former and En

�E# for the latter), the behaviorof k#s may reasonably be identified with that of ks for amore weakly adsorbed gas. Thus, in accordance with theobserved behavior of ks over substantial ranges of temperatureand adsorbability (e.g. Barrer, 1963; Barrer and Gabor, 1959,1960), k#s (and hence P/v1) should decline with rising tempera-ture and/or diminishing adsorbability, as described by classicaladsorption theory; which may be suitably adapted here to read(e.g. Hwang and Kammermeyer, 1966a,b; Nicholson andPetropoulos, 1973):

k#s ¼ C#s =Cog ¼ ðf

#s =f gÞ expððEn

2E#Þ=RTÞ ¼ ðf #s =f gÞ expðU0=RTÞ ð7Þ

In Eq. (7) the f’s are partition functions per unit volume:f #s ¼ f syf sz=L; fg¼L�3 (denoting by L¼h/(2pmkT)1/2 the transla-tional partition function and by y and z the space coordinates(normal to x) along and across the surface, respectively), and theexponential term is clearly dominant, as long as U0/RT exceedsunity substantially.

Intuitively, one might expect the aforesaid decline of P/v1 withrising T to continue smoothly in the low adsorption region ofU0/RTE1 or o1 up to the limit P/v1-Bg as T-N. However,extensive measurements of gas permeabilities through mesopor-ous ‘‘Vycor’’ glass by Hwang and Kammermeyer (1966a), over avery wide temperature range, showed that reduced permeabilitytypically passes through a minimum in the high T (low adsorp-tion) region, thus leaving the location of the T-N (genuineKnudsen) limit uncertain (see Fig. 1; as well as additional similardata in Fig. 2 of Hwang and Kammermeyer, 1966b). These authorsnoted that such behavior could be accounted for by Eqs. (6)and (7), in conjunction with the simple harmonic oscillator

0 100 200 300 400 500 600

4.0

4.5

5.0

5.5

6.0

6.5

7.0

H2HeNeN2O2CO2

Q (M

T)1/

2 x 1

04

T (K)

A

0

4

5

6

Ne

Q (M

T)1/

2 x 10

4

T (K)

B

C

A

200 400 600

Fig. 1. Temperature dependence of the reduced permeability of a series of dilute

gases permeating through ‘‘Vycor’’ mesoporous glass (rp¼4 nm) measured by

Hwang and Kammermeyer (1966a); points: experimental data; broken lines:

curve-fit by authors based on Eq. (8); line A: Knudsen limit according to the said

curve-fit. Inset: typical observed flow behavior illustrated by the Ne data; line A:

as above; line B: Knudsen limit consistent with the surface transport theory

proposed by Nicholson and Petropoulos (1973); line C: median line for the low

adsorption region. In the ordinate. Q is given in cm3 (STP) cm cm�2 s�1 cm Hg�1.

(S.H.O.) approximations f #sy ¼ kT=hn#y and f #sy ¼ kT=hn#z . The result-ing Eq. (8) is shown below as given by them, expressingthe reduced permeability in terms of the lumped constants B1,B2 and D:

Q ðMTÞ1=2¼ B1þB2T expðD=TÞ ð8Þ

where the theoretical meaning of the said constants follows easilyfrom the detailed derivation of Eq. (8); but their values for eachgas were determined empirically by curve fitting the reducedpermeability data, under the condition that B1, which representsthe Knudsen limit (B1¼Bg/2pR), should be the same for all gases.

Although the data were thus fitted reasonably well, theresulting value of B1 lay well below all experimental curves(see line A in Fig. 1), thus implying unnaturally large reducedsurface permeabilities (especially for He), which moreover tendedto increase with T. Nevertheless, as shown more clearly in theinset of Fig. 1, there was also evidence of a progressivelydiminishing rate of increase near the high end of the experimentalT range (subsequently confirmed by the more extensive high T

range covered by Shindo et al., 1983); which was interpreted byHwang and Kammermeyer (1966a) as presaging passage of therelevant Q(MT)1/2 curves through a maximum with subsequentdecline towards B1 (represented by line A in Fig. 1 and in theinset), thus resulting in a physically very curious behavior.

However, closer examination by Nicholson and Petropoulos(1973) revealed that the S.H.O. approximations introduced inEq. (8) were not sustainable in the region of the reduced perme-ability minimum. In particular, model calculations of js ¼ Ps=Po

g

by these authors, using more accurate expressions for f #sz and f #sy

in Eq. (7) (see Eq. (9) below), showed js varying monotonicallythroughout the T range of interest, under conditions where Eq. (8)produced a spurious prominent minimum in the weak adsorptionregion (see Fig. 2 of Nicholson and Petropoulos, 1973); thusinvalidating the curve-fitting exercise of Hwang andKammermeyer (1966a) and strongly suggesting that the realKnudsen limit (Bg/2pR) might more plausibly be identified withline B in the inset of Fig. 1. This, in turn, obviously signals failureof the conventional theory of surface transport in the weakadsorption region, because it leads to the unphysical resultPso0 in this case. (N.B. however, that ensuing detailed theoreticalanalysis by Petropoulos and Havredaki (1986) has shown that thebasic concepts of the said conventional surface flow theoryremain formally valid, to a first approximation, in the strongeradsorption region, when applied to sufficiently narrow meso-pores; thus conventional surface flow theory retains much of itsimportance as a readily usable tool for adsorbable gas-flow dataanalysis (e.g. Choi et al., 2001; Markovic et al., 2009)).

On the other hand, analogous studies on silica, alumina andgraphite mesoporous media (Havredaki and Petropoulos, 1983;Petropoulos and Havredaki, 1986) demonstrated that the unusualdilute gas flow behavior observed by Hwang and Kammermeyer(1966a,b) and by Shindo et al. (1983) was not confined topermeation in mesoporous ‘‘Vycor’’ glass but was a more generalphenomenon. Two points of particular interest emphasized by theHavredaki and Petropoulos studies are (i) the importance ofsupporting (but lacking in the above debate) equilibrium sorptiondata and (ii) the possibility, in cases of relatively simple dispersivegas–solid interactions, to represent reasonably well (with the helpof the said S data) the combined effects of temperature and gasadsorbability on reduced permeability, by a single curve(see Fig. 2); in contrast to the scattered data (but note that thegeneral picture of passage through a minimum is still preserved)displayed in the corresponding plot for a mesoporous colloidalgraphite compact in Fig. 9b of Petropoulos and Havredaki (1986).

The above findings emphasized the need for more sophisti-cated modeling. Accordingly, a comprehensive statistical

J.H. Petropoulos, K.G. Papadokostaki / Chemical Engineering Science 68 (2012) 392–400396

thermodynamic approach applicable to all mobile (in the x

direction) intrapore molecules and, in particular, bereft of thearbitrary a priori assumptions Cg ¼ Co

g, Sg ¼ Sog, Dg ¼Do

g of conven-tional theory was adopted and applied to a slit-shaped pore ofwidth dp, endowed with a 9:3 UP(z) adsorption potential (shownschematically in Fig. 3a) for the calculation of f #sz (supplementedby a cosine periodic barrier for evaluation of f #sy, which actuallyplays a minor role and may be eliminated without significant lossby putting LfsyE1).

The division into gas-phase and adsorbed-phase intraporemobile molecules was retained, but their respective concentra-tions (Cg ¼ Co

g and C#s ) were redefined as C0g and C#0s , assigned,respectively, to molecules energetic enough to overcome thedesorption potential energy barrier in the middle of the slit(U0P–UMP) or not (see Fig. 3). Accordingly (bearing in mind thathere e¼1), we have:

S#0s ¼ Ak#0s ¼ C#0s =cg ¼L2f #0syf #0sz expðU0P=RTÞ � Isz expðUMP=RTÞ ð9Þ

S0g ¼ kg ¼ C0g=cg ¼L2f 0gyf 0gz expðU0P=RTÞ � Igz expðUMP=RTÞ ð10Þ

A' B'

zmzmin

UP= U0P

UP= UMP

AB

U=UP=0

z0

U

z

Fig. 3. Pictorial representation of the adsorption potential UP(z) used for mesoporous sl

surface atoms and (b) model triangular potential (for further details see Nicholson and

1010.10.9

1.0

1.1

1.2

1.3

Al2O3 HeN2KrC2H6CO2

P (M

/ T

)1/2

x 10

3

S

SiO2

C

Fig. 2. Experimental study by Havredaki and Petropoulos (1983) of the reduced

permeability of dilute gases through mesoporous silica (rp¼4 nm) and alumina

(rp¼5 nm) media, as a function of gas adsorbability and temperature: 298 K (plain

points), 348 K (barred points). (line C represents the median horizontal line drawn

through the silica data). Adapted from Havredaki and Petropoulos (1983).

The derivation of integrals Isz, Igz was based on the Hamiltonian

Hz

kT¼

U0PþUPðzÞ

RTþ

pz2

2mkT

under conditions �poz opzopo

z for Isz (adsorbed phase) and �poz 4

pz4poz for Igz (gas phase), where pz denotes molecular momentum

along z, and poz is the ‘‘escape momentum’’ given by

poz

ð2mkTÞ1=2¼ �

UPðzÞþUMP

RT

� �1=2

resulting in

Isz ¼ d�1p

Z zb

za

exp�UPðzÞ�UMP

RT

� �erf�UPðzÞ�UMP

RT

� �1=2

dz ð11Þ

Igz ¼ d�1p

Z zb

za

exp�UPðzÞ�UMP

RT

� �erfc

�UPðzÞ�UMP

RT

� �1=2

dz ð12Þ

where za¼zmin and zb¼2zm–zmin.The integrands (shown in Eqs. (11) and (12)) of Isz, Igz

appearing on the right hand side of Eqs. (9) and (10), describethe distribution of intrapore mobile molecules across the slit.As illustrated in Fig. 4, Cs

#0ðzÞ, at equilibrium with a given cg in a

typical mesopore, declines from a maximum value near thepore wall (at z¼zmin) to zero at z¼zm, and C0g z¼ zminð ÞoCo

g ¼ cg,

zz =ho

U = U

U

U =U =0

z =0

dp

z

its with structureless walls: (a) 9:3 potential; where z¼0, 2zm at the centers of the

Petropoulos, 1973; Nicholson et al., 1979).

00

1

2

0

1

2

z/zo

C#s'

Cog

Cg'Co

g

5 10 15 20 25 30 35 40 45 50

Fig. 4. Examples of calculated distribution of the local intrapore concentration of gas-

phase, C0g zð Þ, and activated admolecule, C#0s zð Þ, concentrations, across a mesoporous

slit subject to a 9:3 adsorption potential: U0P/RT¼0.5 (full lines), 2 (broken lines).

surface flux

gas-phase fluxRed

uced

per

mea

bilit

y

Temperature

observed total flux

Knudsen limit

Fig. 5. Illustration of the mechanism, proposed by Nicholson and Petropoulos

(1973) and Shindo et al. (1983), for the passage of the reduced permeability

through a minimum with rising temperature.

J.H. Petropoulos, K.G. Papadokostaki / Chemical Engineering Science 68 (2012) 392–400 397

correspondingly rises to a maximum as z-zmð-Cog, if the slit is

sufficiently wide to ensure UMP-0; see Fig. 4). Hence, in termsof the overall molecular concentrations in the slit C0g, C#0s (seeEqs. (9) and (10)), we normally have kgo1; and, as T-N,C#0s -0 (hence ks

#0-0) and C0g-Co

g (hence kg-1).Accordingly, as illustrated schematically in Fig. 5, the passage

of the reduced permeability through a minimum with rising T,can in principle be interpreted in terms of a continuous decline ofks#0, which is dominant at lower T but is eventually overridden, at

high enough T, by the concomitant continuous rise of kg (N.B. thatthe latter parameter does not appear in the conventional theory,because it reduces to unity throughout).

In more quantitative terms, with k01=k1 allowing for the effectof the adsorption field on lg, the diffusivities in the slit, fornonadsorded and adsorbable gas molecules respectively, may bewritten (with the aid of structure factors k01 , k1, k2) as

Dog ¼ k1v1dp ¼ Po

g ; D0g ¼ k01v1dp; D#0s ¼ k2v1l

0

s ð13Þ

The permeability P of the slit, normalized with respect to thenonadsorbed gas, is then

j¼ ðP0gþP0sÞ=Pog ¼ ðS

0

gD0gþS#0s D#0s Þ=Po

g ¼jgþjs ¼ ðk0

1=k1Þkgþðk2l0

s=k1dpÞk#0g

ð14Þ

and setting, to a first approximation, B0g ¼ k01=k1 � const� 1,B0s ¼ k2l

0

s=k1dp � const

j¼jgþjs ¼ ðB0gIgzþB0sIszÞ expðUMP=RTÞ ð15Þ

Suitable model calculations with Eq. (15) then confirmed thatthe mechanism proposed in Fig. 5 for generation of the j(T)minimum was physically realistic (Nicholson and Petropoulos,1973; see Tables I and II therein).

The above model approach has the advantage of rationalizingconventional surface flow theory and of including seamlesslytherein the aforementioned, previously unsuspected, low-adsorp-tion j(T) behavior, while preserving analytical mathematicaltreatment. However, on the quantitative side, Eq. (15) is obviouslysubject to the inherent limitations of its derivation and applica-tion mentioned above. (For a similar, but much simpler, approach,the reader is referred to Shindo et al., 1983.)

2.4. Ab initio MPT approach to dilute adsorbable gas flow

To eliminate the aforementioned limitations, a more rigorousab initio approach to the calculation of the total gas flux through

the model pore, was developed based on von Smolukowski’sMPT method of deriving Knudsen’s equation (see Section 1).This methodology was suitably modified to account for the effectof the adsorption field on both local gas molecular concentrationand molecular trajectories (at the same time rendering thedistinction between intrapore gas-phase and activated adsorbedmolecules unnecessary), and used to (i) verify the generalconclusions drawn from the above analytical approach and(ii) undertake a detailed study of dilute adsorbable gas transportin mesopores, as described by Nicholson and Petropoulos (1975b,1981, 1985), Nicholson et al. (1979) and confirmed and extendedby Petropoulos and Petrou (1991).

Principally for reasons of computational economy, a modelpore in the form of a two-dimensional (2D) slit, with structurelesswalls (implying that all intrapore gas molecules are free to movealong the surfaces, i.e. E#

¼0) and diffuse molecule reflection atthe wall, was chosen as the main basis for the above studies.Bearing in mind that real pore networks are made up of pores offinite length lp, the said model pore has the unique advantage ofyielding (at all lp/dp) an analytical expression for the diffusioncoefficient of a nonadsorbed gas D(U0¼0)¼D1; and may thusserve both as a reliable benchmark for corresponding calculatedMPT values (DMPT) at U0¼0 or U040 and as a close analog of3D-slit physical behavior (accordingly, the aforementioned 9:3adsorption potential was retained). The relevant calculationsinvolved evaluation of the flux through the cross section at themiddle of the model pore (assuming a constant concentrationgradient along the pore) by tracing the (nonlinear) moleculartrajectories between the said cross section and the pore wallsor the cross sections at the ends (x¼0 or x¼ lp) of the pore(for details see Nicholson et al., 1979; Nicholson and Petropoulos,1981). The final MPT expressions for evaluation of the reducedpermeability j of the 2D slit are given by Nicholson andPetropoulos (1981) and Petropoulos and Petrou (1991).

2.4.1. Verification of methodology

For the purpose of checking the aforementioned MPT metho-dology, a simplified model triangular adsorption potential (seeFig. 3b), allowing for fast analytical evaluation of molecular flighttimes, was adopted (Nicholson et al., 1979). As illustrated by theexample of calculated results given in Table 1, excellent agree-ment between DMPT for U0¼0 and the corresponding analyticalD1¼DA values, was obtained. For U040, MPT was tested againstMonte Carlo (MC) calculations, where gas molecules were intro-duced on the upstream side (x¼0) of the model pore and theirpaths therein followed up to their exit at either the upstream ordownstream (x¼ lp) side. Two methods were used to calculate theresulting diffusion coefficients: DMC1 derived from transmissionprobabilities (¼Nn/N; where Nn denotes number of moleculesexiting at x¼ lp out of a total number N of molecules introduced)and DMC2 from the molecular displacements along the model poretreated as a one-dimensional random walk. Table 1 shows goodagreement between DMC1 and DMC2, which improves with risingN. Good agreement is also found between DMC and DMPT at higherU0. Some discrepancies, which became significant at U0-0, wereshown to be due to the development of significant ‘‘edge effects’’,under these conditions, in the MC study; which led to reductionof the concentration gradient within the pore substantiallybelow that pertinent to the MPT calculation. After due correction(see Appendix of Nicholson et al., 1979), reasonable agreementbetween MPT and MC was obtained for U0¼0 also (see Table 1).

Needless to say, the results shown in Table 1 are fully in linewith the expected drop of D with rising gas adsorbabilitypreviously demonstrated using Eq. (5).

J.H. Petropoulos, K.G. Papadokostaki / Chemical Engineering Science 68 (2012) 392–400398

2.4.2. MPT model investigation of the physics of flow in mesopores

For the purpose of making use of the MPT approach for asystematic study of the physics of permeation (Nicholson andPetropoulos, 1981), the 9:3 adsorption potential used in the previousanalytical study was retained. In order to ensure exact correspon-dence of the slit width dp felt by adsorbable and nonadsorbed gasmolecules (to enable reliable calculation of j¼ P=Po

g), hard wallswere inserted (see lines A, A0 in Fig. 3a) at z¼zmin and at z¼2zm�zmin

yielding dp¼2zm�2zmin (but the alternative choice of lines B, B0 inFig. 3a yielding dp¼2zm�2zo made no material difference to thegeneral calculated behavior; cf. Petropoulos and Petrou, 1991). Theplots of reduced permeability, thus evaluated, as a function of U0/RT

duly confirmed the occurrence of the minimum predicted by theprevious analytical treatment, in slits of a wide range of dp/zo values(shown on the curves of Fig. 6) and length lp¼10dp. As also confirmedby the more sophisticated model approach of Petropoulos and Petrou(1991) (see esp. Fig. 4 therein), the minimum in question is mostpronounced in pores of moderate width and tends to deepen withincreasing length. It tends to shift to higher U0/RT and to becomebroader and shallower (and hence more indistinct) in wider pores. Innarrow pores, it tends to move to lower U0/RT, becoming shallowerand narrower up to complete extinction (as illustrated by thebehavior of microporous and mesoporous colloidal graphite com-pacts; see Fig. 9a and b of Petropoulos and Havredaki, 1986).

Furthermore, a parallel study based on cylindrical model pores(also subject to the appropriate 9:3 potential; see Nicholson andPetropoulos, 1981), as well as the results obtained from the 2Dslit with triangular potential, confirmed that the general featuresof the reduced permeability behavior, displayed in Fig. 6, wereinsensitive to changes of the functional form of the adsorptionpotential or of the cross-sectional shape or length of the modelpore. This was also shown (Nicholson and Petropoulos, 1985;Petropoulos and Petrou, 1991) to be true for simple heteroporousmodel media consisting of random combinations (in variousproportions) of model pores of two different widths, in the formof networks of various connectivities (including the limiting casesof parallel and serial arrays); thus providing a link between theabove single-pore results and real porous media.

As confirmed by the experimental data of Fig. 2, the relativelysimple theoretical picture presented by Fig. 6, wherein thecombined effect of varying gas adsorbability (represented by

Fig. 6. Results of ab initio MPT calculations of the reduced permeability j¼ P=Pog

of 2D slits, subject to a 9:3 adsorption potential, of width dp/zo¼2(zm�zmin)/zo

(values shown on the curves) and length lp¼10dp (see text and Nicholson and

Petropoulos, 1981); N.B. also that the range of meaningful calculated j41values,

in practice, will depend on the applicability of the requisite preconditions (such as

Knudsen flow regime, Fick’s and Henry’s laws) to the particular real systems under

consideration. From Nicholson and Petropoulos (1981).

changing U0 keeping zo constant) and temperature can bedescribed (to an acceptable degree of approximation) by a singleparameter (U0/RT), probably suffices to explain the behavior ofmesoporous media–gas systems subject to nonspecific molecularinteractions.

However, the effect of increasing gas molecular size (repre-sented by increasing zo and zmin in the above analytical and MPTmodel approaches), which normally accompanies enhanced gasadsorbability, becomes increasingly important as the pore dia-meter decreases and ultimately becomes dominant in porediameters approaching molecular dimensions.

2.4.3. Refined ab initio MPT modeling approach

To confirm and amplify the preceding mesopore results(as discussed above), as well as to model seamlessly the transitionfrom mesopores to molecular sieving micropores (see below),a more advanced version of the aforementioned MPT approach wasdeveloped (Petropoulos and Petrou, 1991) and applied to a 2D-slitembedded in a square lattice of carbon atoms, with a series ofmonatomic gases (representative of He to Xe) as permeants. Theadsorption potential therein was, in each case, built up by summa-tion of the relevant binary (gas–solid atom) Lennard–Jones interac-tions, and MPT was applied using a sufficiently dense grid of UP(z, x)values, thus accounting fully for the effect of the accompanyingchange in gas-molecular size (for further details see above refer-ence). Calculated reduced permeabilities demonstrating the effect ofadsorbability-cum-molecular size and of temperature in a wider anda narrower model micropore, are plotted in Fig. 7. The resultingbehavior in the wider micropore resembles that noted in Fig. 6,except for the fact that the coincidence of the plots representing theaforementioned effects in Fig. 6 has now broken down. In thenarrower micropore, the effect of temperature remains effectivelythe same as in the wider one but what was formerly an adsorb-ability effect has now been completely supplanted by molecularsieving.

Fig. 7. Results of refined model MTP computations of reduced gas permeability

(normalized to slit width 2zm; hence the resulting j values have only relative

significance), demonstrating the effect of (a) adsorbability-cum-gas molecular size

(akin to what might be expected for the series He, Ne, Ar, Kr, Xe at 300 K;

full lines) and (b) temperature (representative calculations for Ar in the range

200–800 K; broken lines), in wider (2zm¼1.19 nm; open points) and narrower,

molecular sieving (2zm¼0.71 nm; filled points) 2D-micropores with atomically

structured walls. The respective effective pore widths [namely the distances

across the slit, at its narrowest points x¼xo, over which the designated penetrant

molecule could move under an attractive potential UP(z, xo)o0], were: 0.60 (He)

– 0.49 (Xe) nm (wider pore) and 0.14 (He) – 0.025 (Xe) nm (narrower pore). Uo12

measures the strength of the gas–solid Lennard–Jones atomic interactions, which

are cumulated to build up the adsorption potential in the model pore-gas system

(for further details, see text and Petropoulos and Petrou, 1991). Adapted from

Petropoulos and Petrou (1991).

0.000

2

4

6

8

10

12

HeCH4N2ArC3H8

Red

uced

mod

ified

per

mea

ne x

104

M/T (g mol-1K-1)

0.06 0.12 0.18

Fig. 9. Plot of the corrected Ruthven et al. (2009) data (kindly supplied to us in the

form of their ‘‘modified reduced permeance’’ (eDo/tl)(M/T)1/2 proportional to our

P(M/T)1/2) in a manner akin to that used in Fig. 2. Remarkable similarity to the

silica plot of Fig. 2 is discernible, as is confirmed by the least-squares fit of a

quadratic curve (full line) yielding correlation factor R2¼0.426, as opposed to the

linear median fit (broken line), which yields R2¼0.026.

J.H. Petropoulos, K.G. Papadokostaki / Chemical Engineering Science 68 (2012) 392–400 399

3. Relevance of the above experimental and modeling work tothe Bhatia–Ruthven debate

The rather modest deviations from a horizontal median line,exhibited by the generally broad and shallow minima of theexperimental reduced permeability curves of Figs. 1 and 2 (seeline C therein; cf. also the corresponding theoretical curves ofFig. 6), strongly suggest that the said experimental data qualify asthe earliest cases of observed quasiKnudsen behavior!

Conformity to such behavior is normally assessed by the cognos-centi, using as criterion the linearity of the plot of measuredpermeability P (or ‘‘effective diffusivity’’; which, given the usual lackof supporting S data, should differ only by a constant factor frommeasured permeability; see Section 2.1 above) versus the Knudsenbenchmark diffusivity (or permeability, because here S¼1) which isproportional to (T/M)1/2. Indeed, replotting the data of Fig. 2, on thisbasis, yields good straight lines (see Fig. 8). We have also found this tobe true of the pertinent (weak adsorption region) porous ‘‘Vycor’’glass data of Hwang and Kammermeyer (1966a,b).

Conversely, replotting the (corrected; see Section 1) experi-mental data of Ruthven et al. (2009) (which have been advancedas a typical case of quasiKnudsen behavior; cf. Fig. 3 therein ), in amanner akin to that adopted in Fig. 2 (except for the fact that M

was the only correlate of gas adsorbability available in this case)yields a clearly discernible (despite the substantial experimentalscatter) broad and shallow minimum (see Fig. 9), quite similar tothat displayed by the more precise experimental data of Fig. 2.

It should undoubtedly prove very instructive to reexamineother quasiKnudsen experimental (as well as simulated) data inthe above manner.

For the time being, we may say in conclusion that (i) all the aboveolder and recent data are examples of the same phenomenon, whichhas been designated here as ‘‘quasiKnudsen flow’’ in an attempt toeradicate confusion with genuine Knudsen flow behavior; (ii) thepassage of the reduced permeability through a minimum in the weakadsorption (low adsorbability/high temperature) region, before thetrue Knudsen limit is approached (at Uo-0, T-N), first reported in1966, constitutes what we have here called for convenience the ‘‘finestructure’’ of quasiKnudsen flow; (iii) no need to coin terminologyakin to the foregoing one proved necessary in connection with theolder data, thanks to the fact the said data were sufficiently preciseand their analysis sufficiently refined, to show up right awaythe aforementioned ‘‘fine structure’’, thus nipping in the bud thepossibility of confusion with genuine Knudsen flow; while (iv) the

00

4

8

12

Al2O3

HeN2KrC2H6CO2

P x

103

(cm

2 s-1

)

(T/M)1/2 (g-1mol K)1/2

SiO2

2 4 6 8 10

Fig. 8. Replot of the Havredaki and Petropoulos (1983) data of Fig. 2, in the

manner usually adopted for the detection of quasiKnudsen behavior, confirming

good conformity to the said behavior.

ensuing theoretical modeling proved sufficiently perceptive first toshow that quasiKnudsen flow was a natural extension of conven-tional surface flow (Sections 2.2 and 2.3), and then obtain detailedphysical insight (Section 2.4) into the variability of its form and of theconditions of its occurrence, extending well beyond the limits of thedebate examined here.

Nomenclature

a activity of sorbed gas (units as for C)ag activity of gas in external gas phase (units as for cg)A specific internal surface area per unit volume of the

porous medium (m2/m3)Bg, Bs parameters defined in Eq. (6)B1, B2, D lumped constants used in Eq. (8)C concentration of sorbed gas per unit volume of porous

medium (mol/m3)cg gas concentration in external gas phase (mol/m3)dp pore diameter (m)D diffusion coefficient of sorbed gas (m2/s)DK Knudsen diffusion coefficient benchmark value (m2/s)En adsorption energy (J/mol)E# activation energy of adsorbed-phase (surface) diffusion

(J/mol)f standard statistical thermodynamic partition functionho parameter of model triangular potential (m)Igz, Isz integrals defined in Eqs. (11) and (12)ks Henry-law gas adsorption coefficient: Ss/A (m)k1, k01, k2 structure factors defined in Eq. (13)l thickness of porous solid (m)lp pore length (m)M gas molecular weight (g/mol)p gas pressure in external gas phase (Pa)pz intrapore gas molecular momentum along z (kg m/s)P permeability coefficient formulated in terms of the

imposed Dcg (m2/s); reduced permeability P(M/T)1/2

or P/v1

Q permeability coefficient formulated in terms of theimposed Dp (Q¼P/RT); reduced permeability Q(MT)1/2

J.H. Petropoulos, K.G. Papadokostaki / Chemical Engineering Science 68 (2012) 392–400400

rp mean effective pore radius of porous medium (¼2e/A)equal to radius of benchmark pore

S sorption coefficient expressed as C/cg

Sp sorption coefficient expressed as C/pUP(z) adsorption potential energy field across a model pore

(J/mol)U0 defined in Eq. (7) (¼En

�E#)U0P, UMP adsorption potential energy parameters defined in Fig. 3

(J/mol)v1 unidimensional mean gas molecular speed (m/s)x, y, z single pore spatial coordinates in the direction of flow,

normal to flow along the surface and normal to thesurface, respectively (m)

zo, zm, zmin adsorption potential distance parameters definedin Fig. 3

e porosity of the porous solidkg structure factor linking from Dg to DK

ks structure factor applicable to adsorbed gas (surface) flowlg, lG gas molecular mean free path in the intrapore and

external gas phase, respectively (m)ls length of activated diffusion jump along pore surfaceL translational partition functiont tortuosity factor applicable to intrapore gas flow as

defined in the debatef reduced permeability expressed as P=Po

g

Subscripts

g refers to intrapore gas-phase parameters (except for cg

which refers to the external phase)s refers to intrapore adsorbed phase (surface) parameters

Superscripts

o denotes intrapore nonadsorbed gas parameters# denotes parameters pertaining to activated admolecules0 denotes parameters pertaining to advanced surface

flow theory

Acknowledgment

The senior author wishes to record his appreciation of thestrong interest shown, and sustained encouragement offered, bythe late Prof. R.M. Barrer F.R.S., and to reiterate his gratitude toDr. D. Nicholson for his unfailing colleagueship and close colla-boration, during the pursuit of the work reviewed succinctly inSection 2 above.

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