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

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524.5

Mass Transfer Across Selectively Permeable Membranes

787

Here the subscripts

and

refer to conditions within the membrane and in the external

solution, respectively. The activity

is to be calculated at the composition of the mem-

brane phase but at the pressure of the external solution, and

In practice, the solutes are normally considered incompressible and

to be constant

across the interface.

Often conditions inside the membrane are very difficult, or even impossible, to de-

termine, and Eq.

24.5-6

is primarily useful under these circumstances to obtain a qualita-

tive understanding of membrane behavior. Partly for this reason complete descriptions

of membrane transport are rare (see, however, Scattergood and Lightfoot'). Highly sim-

plified, but often directly useful, introductions to membrane transport are available in a

variety of One venerable approximation, found especially useful by biolo-

gists, is that of Kedem and Katchalsky."

However, rapid progress is being made in obtaining fundamental data, and much of

this is reported in the

Journal of Membrane

Science.

One important area is that of micro-

porous

membrane^.^

One can also expect advances in modeling behavior. It has long been

known7 that the generalized Lorentz reciprocal theorem for creeping flows8 provides a

sound basis for extending hydrodynamic diffusion theory of

517.4

to multicomponent

diffusion in microporous membranes. Recently developed computational techniques9

should make the necessary computations tractable enough to provide real predictive

power. These techniques can also be used to develop self-assembling str~ctures,'~ which

offer new possibilities for highly selective membranes.

This field offers an extremely wide variety of membrane types and of mass transfer

processes taking place in them. One can distinguish between biological" and synthetic12

membranes, but there are very wide ranges of composition and behavior within each of

these categories. Among the synthetic group there are "homogeneous" membranes, in

which the matrix acts as a true solvent for permeating species, and "microporous" mem-

branes, in which the permeating species are confined to matrix-free regions, as well as

mixtures of the two types. These factors are important from a materials standpoint, but

the formalisms needed to describe their transport behavior are much the same for all.

Cussler,

Diffusion: Mass Transfer in Fluid Systems,

2nd edition, Cambridge University Press

(1997),

580.

Deen,

Analysis of Transport Phenomena,

Oxford University Press (1998), p. 597.

Seader and

Henley,

Separation Process Principles,

Wiley, New York (1998).

Kedem and

Katchalsky,

Biochem. Biophys. Acta,

27,229 (1958).

K. Kaneko,

Membrane Sci.,

96,5949 (1994); K. Sakai,

Membrane Sci.,

96,91-130 (1994); S. Nakao,

Membrane Sci.,

96,181-165 (1994).

N. Lightfoot,

Bassingthwaighte, and

F. Grabowski,

Ann. Biomed. Eng.,

4,78-90 (1976).

J. Happel and

Brenner,

Low Reynolds Number Hydrodynamics,

Prentice-Hall(1965), Martinus

Nijhoff (1983),

62,

85.

Kim

and

Karrila,

Microhydrodynamics: Principles and Selected Applications,

Butterworth-

Heinemann, Boston (1991).

Mustakis,

C. Clear, P. F. Nealey, and

Kim,

ASME Fluids Engineering Division Summer

Meeting,

FEDSM, June 22-26 (1997).

"B.

Alberts et al.,

The

Molecular Biology of the Cell,

Garland, New York (19991, Chapters 10 and 11.

S. W.

and

K. Sirkar,

Membrane Handbook,

Van Nostrand Reinhold, New York (19921,

p. 954; R.

Noble and S.

Stern,

Membrane Separations Technology,

Membrane Science and Technology

Series,

Elsevier (Amsterdam),

718;

van Reis and

Zydney, "Protein Ultrafiltration" in

Encyclopedia of Bioprocess Technology

(M.

C. Flickinger and

W. Drew, eds.), Wiley, New York (1999),

pp.

2197-2214;

Zeman and

Zydney,

Microfiltration and Ultrafiltration,

Marcel Dekker, New York

(1996).

788

Chapter 24 Other Mechanisms for Mass Transport

There are also a wide variety of process conditions in widespread use. Here we consider

only a few examples to illustrate commonly encountered situations.

EXAMPLE

24.5-1

Concentration

Diffusion Between

Preexisting

Bulk

Phases

Consider "solute"

diffusing through a membrane placed between binary solutions of solute

A in solvent

under the influence of concentration gradients alone.

This is a commonly encoun-

tered situation, including dialysis, blood oxygenation, and many gas-separation systems.I3

There are many variants, including

facilitated difision

(see Problem 24C.8). Hemodialysis is a

special case, where pressure differences are used to drive water across the membrane, but

concentration diffusion of solutes is of primary interest from the present standpoint. Assume

for the moment that the flux of solvent, NB, is already known. Develop an analog to Fick's

first law for this system.

SOLUTION

We are primarily concerned here with solute A, and the Maxwell-Stefan equation for it takes

the form

This may be rearranged to give

This is reminiscent of Fick's law, with the first term on the right corresponding to the Fickian

diffusive flux and the second to the convective term. However, the effective diffusivity now

contains a membrane contribution, and the convective term is now weighted by

ratio of dif-

fusivities. This situation corresponds to the situation pictured in Fig. 24.5-1. The arrow point-

ing to the right represents the diffusion relative to the solvent, modified by the interaction

with the matrix, while the arrow pointing to the left represents the "drag" of the membrane,

which tends to reduce the transport relative to the convection occurring in the absence of the

membrane matrix-namely, xA(NA

NB). Note that, in general, there are mass transfer

boundary layers on both sides of the membrane.

There are several limiting situations of interest. If the membrane interacts only very

weakly with the solute,

BAB.

Then

Diffusion relative to

solvent, and interaction

with

membrane

(dialysis processes)

Fig.

24.5-1.

Intramembrane

mass transport.

'%.

Koros and

K. Fleming,

Membrane

Sci.,

83,l-80

(1993).

324.5 Mass Transfer Across Selectively Permeable Membranes

789

which is exactly Fick's law. However, it must be remembered that both molar concentration

and diffusivity are those in the membrane phase. We next look at a limiting situation, where

x*(NA

NB)

(24.5-1 1)

and the distribution between membrane and solution is linear, so that

K~c~m

(25.5-12)

where the subscripts

and

refer to the external solution and membrane phases, respec-

tively, and KD is the distribution coefficient for the two phases. We may then write

P(c/.eO

cAe&

(24.5-13)

where

is known as the

membrane permeability,

and

D~,eff

(a;,

The subscripts

AeO

and

refer to the solute concentrations at the "upstreamr' and "down-

stream" sides of the membrane.

EXAMPLE

24.5-2

Ultrafiltration

and

Now consider the filtration processes, in which it is desired to remove a solvent selectively

relative to a solute by pressure-driven flow across a solute-rejecting membrane. Applications

include ultrafiltration and reverse osmosis, the former dealing with macromolecular and the

Reverse

Osmosis

latter with small solutes.14 Microfiltration and nanofiltration are formally similar, but the par-

ticulate nature of the entities being removed presents additional complications we do not

wish to consider here.I2 Develop

framework for describing solvent flow rate and filtrate

composition as functions of driving pressure.

SOLUTION

Inevitably some solute moves through the membrane along with the solvent, as indicated in

Fig. 24.5-1, and it will now be necessary to consider the Maxwell-Stefan equations for both

species. However, membrane filtration is a complex process requiring a great deal of informa-

tion to obtain a complete a priori description, and we therefore begin with an overview of

characteristic behavior using Fig. 24.5-2 as a point of departure. Here both the flow through

the membrane and the composition of the filtrate are shown schematically as functions of the

Transmembrane pressure drop

Fig.

24.5-2.

Ultrafiltration: flow

and solute rejection.

Petersen,

Membrane

Sci.,

83,81-150

(1993).

790

Chapter

Other Mechanisms for Mass Transport

transmembrane pressure drop. First we note that the flow increases with pressure drop,

slowly at first, but approaching a linear relation asymptotically, and the asymptote crosses

the line of zero velocity at a finite pressure drop, The ratio of the filtrate solute concen-

tration to the feed concentration drops with increasing pressure drop, from unity toward an

asymptote,15 which is normally much lower than unity. Our main concern in this brief intro-

duction is to explain this characteristic behavior in terms of key thermodynamic and trans-

port behavior.

This situation differs fundamentally from that just described, in that pressure diffusion

now comes into play, and in that the downstream solution is produced by the transmem-

brane mass transfer. Hence the downstream ratio of solute to solvent is the same as the ratio

of the corresponding mass transfer rates. There is, therefore, no boundary layer on the down-

stream side of the membrane, and it is an almost universal practice to use a composite struc-

ture. Such composite membranes consist of a very thin selective layer on the upstream face,

and a comparatively thick, highly porous, nonselective backing that provides mechanical

strength. This backing can be ignored in the present example.

We begin by focusing on the intramembrane behavior for which the Maxwell-Stefan

equations, modified from

Eq.

24.5-4, assume the forms

Here the subscripts

and

refer to the partially rejected solute and solvent (usually water)

respectively. The terms

x,~,

have been replaced by c,v,/c, to make the presence of the vol-

ume fractions

(24.5-18)

explicit. In addition, the first term on the right has been rewritten as a reminder that the de-

rivative represents the gradient of the partial molar free energy

with composition, temperature, and pressure being held constant.

We begin by examining flow behavior, and to do this we add Eqs. 24.5-16 and

to get

relation between species transport rates and

intramembrane

pressure gradient

The subscript

on pressure is a reminder that we have so far calculated only the pressure

drop inside the membrane. Here advantage has been taken of the Gibbs-Duhem equation

and the fact that the volume fractions sum to unity.

To obtain the directly measurable difference between upstream and downstream solu-

tion pressures, we must go back to

Eq.

24.5-6, which takes the form

'%ctually there is a continuing, very slow drop of filtrate solute concentration even at very high

pressure drops, presumably resulting from the compression of the membrane. However, we shall not

attempt

deal with this small effect here.

s24.5 Mass Transfer Across Selectively Permeable Membranes

791

Looking at the upstream side of the membrane, for example,

Eq.

24.5-22 states that a finite

pressure drop across the membrane interface is required to drive the solute against an in-

crease in the thermodynamic activity. Then the measurable transmembrane pressure drop is

where

is the membrane thickness, and

where the subscripts

and

refer to the upstream and downstream sides of the membrane,

respectively. The intramembrane osmotic pressures are seldom known, but they are substan-

tially smaller than the corresponding solution values (see Problems 24C.7 and

8).

At the pres-

ent state of understanding,

Eq.

24.5-24 explains why there is a finite intercept to the

asymptotic flow behavior, and an elimination of the membrane contributions provides an

upper limit to it. It also provides some insight into intramembrane behavior from experimen-

tal observations, but not an a priori prediction of the intercept.

Next we eliminate the pressure gradient from

Eq.

24.5-16 with the aid of

Eq.

24.5-23

This expression can be integrated to obtain the solute concentration profile (see, for example,

Problems 24C.7 and

8).

In general the concentration profile shows an increasingly negative

slope in the flow direction, and this feature becomes more pronounced as the flow rate

through the membrane increases-that is, as the transmembrane pressure drop becomes

larger.

At very low flow rates,

and

are relatively small and diffusion is relatively fast.

There is only a small drop in solute concentration across the membrane, and the result is the

poor rejection seen for low pressure drops in Fig. 24.5-2. This behavior is suggested by the

zero-flow concentration profile in Fig. 24.5-1.

At very high flow rates, on the other hand, concentration gradients are large and diffu-

sion is weak, except very close to the downstream boundary of the membrane, where a very

large negative concentration gradient develops. Near the upstream boundary, the two mass-

flux terms are large compared to their difference, and one may neglect concentration gradi-

ents in calculating the mass-flux ratio:

The bases of solute exclusion now become clear:

Thermodynamic exclusion, defined by the ratio

xs/x,

Frictional differentiation, defined by differences in the interaction terms with the

membrane

(Vs/Vw)(3kM/Bh,).

Both effects are used in practice and are illustrated in the problems.

Consider now membranes containing immobilized charges consisting of polyelectrolyte gels.

Such gels contain repeated covalently bound ionic groups, as pictured in Fig. 24.5-3. The

Charged Membranes

membrane interior may then be viewed as a solution containing spatially bound

fixed

charges,

and

lbm~~

~xclusion'~

mobile counter-ions, invading electrolyte, and water. For simplicity assume that the fixed

Strathmann, "Electrodialysis," Section

Membrane Handbook

(W.

S. Ho and

Sirkar,

eds.), Van Nostrand Reinhold, New

York

(1992).

792

Chapter 24 Other Mechanisms for Mass Transport

SOLUTION

Fig.

24.5-3. A

sulfonic acid based ion-

exchange membrane.

Membrane

Molecular

structure

charges are anions, written as X-, and the counter-ions are cations, written as M'. The exter-

nal solution is aqueous M+X-, and it provides the invading electrolyte M'X-. Show how the

presence of fixed charges produces an exclusion of invading electrolyte.

This system is dominated by the behavior at the membrane boundary, and we therefore return

to Eq. 24.5-6, written for the water and for the salts

or MY. The expression for water is

where the subscripts

and

refer to the external solution and the membrane, respectively.

Since the intramembrane electrolyte concentration is always higher than the external, thus re-

sulting

a lower internal chemical activity for water, the membrane interior is at a higher

pressure than the external solution (see, for example, Problem 248.4).

The corresponding equation for the salt

yields

It follows that

X~'eX~-e

Y5e

exp(%

An)

X~-mX~

Y~wi

and therefore that the concentration of salt in the membrane phase is less than that in the so-

lution. This suppression of invading electrolyte by the presence of fixed charges is known as

Donnan exclusion

(see Problem 248.3).

The preponderance of counter-ions, here

M+,

inside the membrane, tends to cause them

to diffuse out to the external solution, whereas the co-ions, here X-, tend to diffuse into the

membrane. The result is the development of an electrical potential difference between the

membrane and external solution. This is normaIly estimated, by negIecting osmotic effects

and assuming activity coefficients of unity, as

Equations 24.5-27 to

also apply to the relations between solutions on opposite sides of

membrane containing a partiaIly excluded solute on one side, which now corresponds to the

membrane phase of the above development. Equation 24.5-31 is often used for lack of knowl-

edge

the neglected effects.

524.6 Mass Transport in Porous Media

793

Thus Eq. 24.5-31 is particularly widely used

biologists to explain the origin of the

ubiquitous potentials observed across biological membranes.'' However, the means by which

biological membranes can produce and control ion selectivity are extremely sophisticated and

are only beginning to be understood.17

524.6

MASS TRANSPORT

POROUS

MEDIA

Porous media are important in many mass transfer applications, some of which, such as

catalysis1 have already been touched on in this text

(§18.7),

and they exhibit a very wide

variety of morph~logies.~,~ Adsorptive processes, such as chromatography, usually take

place in granular beds and the absorbent particles themselves are often porous solids. Sec-

ondary recovery of crude petroleum typically involves mass transfer in porous rock, and

freeze drying, or lyophilization, of foods and pharmaceuticals4 depends on the transport of

water vapor through a porous layer of dried solids. Related transport processes occur

throughout the large field of particle technology,5 and, as already indicated in

S24.5,

some

membranes may be considered as microporous structures. Microporous structures abound

in living organisms and contribute importantly to both water and solute distribution3

Discussion of porous solids also brings us full circle, back to the discussions of mo-

mentum transfer with which this text began. Many of the models used to describe mass

transfer in porous media are hydrodynamic in origin, and sometimes the concepts of

mass and momentum transfer become blurred.

Predicting the transport of liquids and gases in porous media is a difficult and chal-

lenging problem, and no completely satisfactory theory is available. Mass is transported

in a porous medium by a variety of mechanisms: (i) by ordinary diffusion, described by

the MaxwellStefan equations; (ii) by Knudsen diffusion; (iii) by viscous flow according

to the Hagen-Poiseuille equation;

(iv)

by surface diffusion-that is, the creeping of ad-

sorbed molecules along the surfaces of the pores; (v) by thermal transpiration, which is

the thermal analog of viscous slip; and (vi) by thermal diffusion. In this discussion, we

neglect the last three of these mechanisms.

This problem has been attacked by many investigators: and summarized by othem7

We give here the principal results of their work. Available models are based either on

B. Hill,

Ionic Channels of Excitable Membranes,

Sinauer Associates, Sunderland, Mass. (1992);

M. Ashcroft,

Ion Channels and Disease: Channelopathies, Academic Press,

New York (1999); D.

Aidley,

The Physiology of Excitable Cells,

Cambridge University Press (1998).

(a) R. Aris,

The Mathematical Tkeoy of Diffusion and Reaction in Permeable Catalysts,

Vols.

and

Oxford

University Press (1975);

(b)

0. Levenspiel,

Chemical Reaction Engineering,

3rd edition, Wiley, New York (1999).

Sahimi,

Flow and Transport in Porous Media and Fractured Rock,

Verlagsgesellschaft, Weinheim,

Germany (1995); V. StanPk, Fixed Bed Operations, Ellis Horwood, Chichester, England (1994).

Curry, R.

Adamson, Bing-Mei Fu, and S. Weinbaum,

Bioengineering Conference

(Sun River,

Oregon), ASME, New York (1997).

(a) L. Rey and

C. May,

"Freeze-Drying/Lyophilization

of Pharmaceutical and Biological

Products" in

Drugs and the Pharmaceutical Sciences

(J.

Swarbrick, ed.), Marcel Dekker, New York (1999);

(b)

Sheehan and

Liabis,

Biotech. and Bioeng.,

60,712-728 (1998).

M. Rhodes,

Introduction to Particle Technology,

Wiley, New York (1998).

Hoogschagen,

Chem. Phys.,

21,2096 (1953),

Ind. Eng. Chem.,

47,906-913 (1955);

S. Scott

and

A. L. Dullien,

AIChE Journal,

8,113-117 (1962);

B. Rothfeld,

AIChE Journal,

9,19-24 (1963);

Silveston,

AIChE Journal,

10,132-133 (1964); R. D. Gunn and C.

King,

AIChE Journal,

15,507-514

(1969); C. Feng and W.

Stewart,

Ind. Eng. Chem. Fund.,

12,143-147 (1973); C.

Feng, V. V. Kostrov,

and

E. Stewart,

Ind. Eng. Chem. Fund.,

13,5-9 (1974).

E. A. Mason and R. B. Evans,

II1,J. Chem. Ed.,

46,358-364 (1969);

B. Evans

111,

D. Love, and

A. Mason,

Chem. Ed.,

46,423427 (1969); R. Jackson,

Transport in Porous Catalysts,

Elsevier, Amsterdam

(1977); R. E. Cunningham and R.

J. J.

Williams,

Diffusion in Gases and Porous Media,

Plenum Press, New

York (1980); Chapter 6 of this book gives a summary of the history of the subject of diffusion.

794

Chapter 24 Other Mechanisms for Mass Transport

cylindrical channels or aggregates of spheroidal particles, and we shall review a few rep-

resentative examples here. We shall also restrict the discussion to two limiting situations

within the pores of the solid matrix:

(i)

Free-molecule flow of

gases,

in which the molecular diameters are short and mean

free paths are long relative to the characteristic dimensions of the pores. Under

these conditions, there is no significant interaction between the intrapore

species.

(ii)

Continuum flow of

gases

liquids, in which both the diameters and the spacing

of the intrapore molecules are short compared to the pore dimensions. Here the

intrapore fluid can be described by the generalized hydrodynamic theory,%nd

the generalized Maxwell-Stefan equations for multicomponent diffusion can

be used.

There are also phenomena for gas transport, known as the

slip-flow

phenomena, in which

the mean free paths are comparable to the pore dimensions: but we shall not discuss

these here.

Free-Molecule Transport

Transport of rarefied gases is an example of Knudsen flow, already presented in Prob-

lem 2B.9. For a long capillary tube of radius

the Knudsen formula takes the form

Here

is the partial pressure of species

in any mixture. Note that

Eq.

24.6-1 states that

the transport of any individual species under these limiting conditions is unaffected by

the presence of others. Thus the total molar flow rate

tube is proportional to the

cube of the tube radius and to the inverse square root of the molecular weight. This de-

pendence on molecular weight is known as

Graham's law.

Equation 24.6-1 can be rewritten as

which defines the "Knudsen diffusivity"

DAK.

However, this must be considered as a bi-

nary diffusivity for species

relative to the porous medium that is

not

consistent with

Fick's law, because the molar flux contains no convective term. As a result

DAK

is not a

state property, containing as it does, the tube radius

To allow for the tortuous nature

of the channels in a porous medium and the limited cross-sectional area available for

flow, the flux expression must be further modified by writing

where

(&/7)DAK

and

(NA)

is the molar flux based on the total cross section of the porous medium. In this

expression

is the fractional void space in the porous material, and

is a tortuosity fac-

Lightfoot,

B. Bassingthwaighte, and

Grabowski,

Ann.

Biomed. Eng.,

4,78-90 (1976).

Jackson,

Transport in Porous

Catalysts,

Elsevier, Amsterdam (1977);

Cunningham and

Williams,

Diffxsion in Gases and Porous Media,

Plenum Press, New

York

(1980).

s24.6 Mass Transport in Porous Media

795

tor. Although models existla,'"for estimating the magnitude of

it must normally be de-

termined experimentally."

As an alternative to

Eq.

24.6-4 for the effective Knudsen diffusivity, one may treat the

aggregate as a collection of large immobile spheres (or "giant gas molecules"), and use

the Chapman-Enskog kinetic theory.'' Problem 24B.6 shows that this approach yields

predictions very similar to those of

Eq.

24.6-4. There is remarkable model insensitivity.

EXAMPLE

24.6-1

Knudsen

Diffusion

Two large well-stirred reservoirs, each of volume

are joined by a short duct of cross-

sectional area

and length

filled with a porous solid as indicated in Figure 24.6-1. Initially

reservoir

is filled with hydrogen at uniform pressure

and reservoir

with nitrogen, also

at p,. The entire system is maintained at a constant temperature. At time

a small valve in

the duct is opened, and the two reservoirs are allowed to equilibrate with each other. Develop

an expression for the total pressure in each reservoir as a function of time, assuming that the

flow of each gas through the connecting duct follows Eq. 24.6-1, and that the ideal gas law

holds throughout the system.

SOLUTION

We begin by assuming quasi-steady-state behavior in the duct so that, for either gas, the rate

of transfer from reservoir 1 to reservoir 2 is given

where

W,,

is the molar rate of flow of species

(either nitrogen or hydrogen) and

is the ef-

fective radius of the pores in the plug joining the two reservoirs. Now a macroscopic mass

balance for reservoir 2 gives

Now

mass balance over the whole system yields

The initial conditions are that at time

These initial conditions complete the specification of the system behavior, and we see that the

distributions of the two gases are independent of each other.

For

nitrogen

we can define the dimensionless variables

t,b

pN/po and

(RTKN/V)t.

Then we may write Eq. 24.6-7 for nitrogen in compartment 2

with the initial condition $hN2(0)

The solution to this problem is then

Stewart and

Johnson,

Catalysis,

4,248-252

(1965).

Butt,

Reaction Kinetics and Reactor Design,

2nd edition, Marcel Dekker, New York (19991,

500, Table

7.4.

Evans

111,

Watson, and E.

Mason,

Chem.

Phys.,

35,2076-2083

(1961).

796

Chapter 24 Other Mechanisms for Mass Transport

Connecting

tube filled with

a porous solid

Two interconnected

reservoirs each

with volume

2 3

Dimensionless time,

Fig.

24.6-1.

Knudsen flow.

For

hydrogen

we note that

a~~

3.74K,.

Hence the differential equation for hy-

drogen is

with the initial condition

$,,(O)

The solution to the differentia1 equation is then

The results are plotted in Fig. 24.6-1.

The ratio

NA/NB

of molar fluxes obtained here was first observed

Gra-

ham13 in 1833 and rediscovered by ~oo~scha~en' in

1953.

Though derived here for Knudsen

flow, this relation is valid also for isobaric diffusion well outside the Knudsen region.

has

been derived from kinetic theory by several investigators and verified experimentally in

tubes and porous media6,7,'3,14 up to very large ratios of passage width to mean free path. Two

sets of confirmatory data are shown in Table 24.6-1. In both sets of e~~eriments,'~.'~ an appa-

ratus similar to that in Fig. 24.6-1 was used, and various tests gases were used against air. The

flux ratios

N,,,/N,,,

were initial values, when each reservoir contained only air or the test gas.

Continuum Transport

To date, fluid mechanical modeling of intrapore transport has been limited to binary so-

lutions in which molecules of the minor constituent (solute) are large compared to those

of the solvent. Models for this situation are based on hydrodynamic diffusion theory ex-

tended to porous

structure^.^

Descriptions are obtained by solving the creeping flow

equations of motion for spheres (representing the solute) through a continuum (repre-

senting the solvent) in closed channels.15 Important effects include partial exclusion of

solute at the channel entrance and selective interaction with the channel wall. Result to

date are limited to single solutes, but the rapid development of computational tech-

nique~'~ should permit extension to more complex systems. Hydrodynamic diffusion

'q.

Graham,

Phil. Mag.,

2,175,269,351 (1833).

Thomas

Graham

(1805-1869), son of a prosperous

manufacturer, attended the University of Clasgow from 1819 to 1826; in 1837, he was named professor of

chemistry, University College, London, became a Fellow of the Royal Society in 1834, and in the same

year was named Master of the Mint.

A. Mason and B. Kronstadt, IMP-ARO(D)-12, University of Maryland, Institute for Molecular

Physics, March 20,1967.

Z.-Y

Yan,

Weinbaum, and

Pfeffer,

Fluid Mech.,

162,415438 (1986).

S. Kim and

Karrila,

Microhydvodynamics: Principles and Selected Applications,

Butterworth-

Heinemann, Boston (1991).