0 Copyright 1990 American Chemical Society
DECEMBER 1990 VOLUME 6, NUMBER 12
T h e Kendall Award Address Macrotransport Processes? Howard Brenner Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received May 23, 1990 Macrotransport proceases (generalizedTaylor dispersion phenomena) constitute coarsegraineddescriptions of comparable convective-diffusive-reactive microtransport processes, the latter supposed governed by microscale linear constitutive equations and boundary conditions but characterized by spatially nonuniform phenomenological coefficients. Following a brief introduction, we focus-by way of background information-upon the original (and now classical) Taylol-Aris problem, involvingthe combined convective and molecular diffusive transport of a point-size Brownian solute “molecule”(tracer) suspended in a Poiseuille solvent flow within a circular tube. A series of elementary generalizations of this prototype problem to chromatograhic-like solute transport processes in tubes is used to illustrate some novel statisticalphysical results and hence to suggest the potential implicit in future applications of the theory. These examplesemphasizethe fact that a solute molecule may, on average, move axially down the tube at a different mean velocity (either larger or smaller) than that of a solvent molecule;solvent molecules collectively create the Poiseuille flow. Moreover, a solute molecule may suffer axial dispersion about the mean velocity at a rate greatly exceeding that attributable to its axial molecular diffusion alone. Such “chromatographic anomalies”represent novel macroscale nonlinearities originating from physicochemicalinteractions between the spatially inhomogeneous convective-diffusive-reactive microtransport processes. 1. Introduction
The research described herein builds upon a scheme initiated more than 30 years ago by TaylorlJ and firmly established by Aris,3 originally relating to the convective flow and dispersion of a soluble solute dissolved in a solvent undergoing Poiseuille flow in a circular tube. It is wellknown4 that such a transport process is governed at the microscale by the convective diffusion equation, involving radial, azimuthal, an axial transport of the solute. Taylor set himself the problem of coarse graining this equation, so as to describe only the mean, axial transport of solute Expanded version of a lecture delivered on the occasion of receiving the 1988 American Chemical Society Award in Colloid and Surface Chemistry (sponsored by The Kendall Company), 62nd Colloid and Surface Science Symposium, T h e Pennsylvania State University, June 21, 1988. (1)Taylor, G. I. h o c . R. SOC.London 1953, A219,186. ( 2 ) Taylor, G. I. h o c . R. SOC.London 1954, A225,473. (3) Aris, R. h o c . R. SOC.London 1956, A235,67. (4) Bird, R. B.; Stewart, W. E.; Lightfoot,E. N. Transport Phenomena; Wiley: New York, 1960. f
0743-7463/90/2406-1715$02.50/0
down the tube, which is normally the only auantitv of physical interest in applications. -In bringing this qkest to fruition, he paved the way for an entirely new fieldnamely, generalized Taylor dispersion theory or, as it has more recently come to be called (in a somewhat broader context), macrotransport processes. This exciting and growing field has many useful applications, extending both geometrically and conceptually greatly beyond the original tube flow problems analyzed by Taylor1t2and A r k 3 Many researchers have contributed to its subsequent development. Indeed, macrotransport -Processes has now reached an advanced state of scientific and technoloeical maturitv. one that is described by a well-defined paradigmwenablingthe systematic graining of a rich variety of detailed mi‘rotransport pr0cesses94 governed by linear constitutive equations. The heterogeneous nature of the audience for whom this presentation is intended excludes entering into the details of the paradigm- (although see ( 5 ) Brenner, H. PCH, PhysicoChem. Hydrodyn. 1980, 1,91.
0 1990 American Chemical Society
1716 Langmuir, Val. 6, No. 12, 1990
The Kendall Award Address
section 2.3). However, in the limited time (and space) available, we can do little more than enter into a few elementary examples illuminating the useful (and interesting) results obtained to date. In this context, we have chosen to focus exclusively upon the role of macrotransport processes in rationalizing a variety of chromatographic separation concepts. These involve calculations of the mean velocity (U*)and dispenivity (D*) of a convected Brownian solute through capillary tubes (and, concomitantly,capillary-tube models of packed beds), including the case of chemically reactive solutes undergoing first-order irreversible reactions-for which a third macrotransport coefficient, namely the solute’s volumetric reactivity K*,is required. On a personal note, my own initiation into the field of Taylor dispersion phenomena occurred at the capable hands of the late F. J. M. (“Fritz”)Horn during his 196970 sabbatical year spent in the Chemical Engineering Department at Carnegie-Mellon University, where I was resident a t the time. This ultimately led to my first publication in the field? coauthored with my student Larry Gaydos (who, to the everlasting confusion of bibliophiles, shortly thereafter changed his name back to the original family spelling of Gajdos’O just prior to completing his dissertation.)*l Since that time, I have had the enormous satisfaction of introducing others to the subject of macrotransport processes and of seeing many of them, especially my own research students and collaborators, acquire an enthusiasm for the subject no less than my own. It is largely due to the efforts of the latter that I have been honored with the American Chemical Society (“Kendall”) Award in Colloid and Surface Chemistry. (Other researchers, of course, beyond those with whom I have collaborated, have contributed importantly to the development of the subject. Constraints imposed by the personalized format of the presentation did not easily allow their citation here). Finally, I wish to thank the American Chemical Society for the 1988 Colloid and Surface Chemistry Award leading to this lecture. Equally do I thank the Kendall Co. for their sponsorship of this award. 2. T h e Taylor-Aris Problem 1. Convection and Diffusion of a Solute in a Poiseuille Flow. Microtransport Equations. Referring to Figure 1, consider the Poiseuille flow
u(r) = 2 P [ 1 - ( r / r o ) 2 ]
(2.1)
of an incompressible viscous fluid (the “solvent”) at mean velocity (2.2)
through an infinitely long circular capillary tube of radius ro. Imagine that a t time t = 0 a mass m of solventdissolved solute is introduced into the system such that the initial solute concentration distribution (mass per unit volume) at point (r,&z)of the circular cylindricalcoordinate system (0 5 r < ro, 0 I4 < 2 ~-a, < z < m) at time t = 0 is (6) Brenner, H. PCH, PhysicoChem. Hydrodyn. 1982,3,139. ( 7 ) Shapiro, M.; Brenner, H. Chem. Eng. Sci. 1986,41,1417. (8) Shapiro, M.; Brenner, H. AZChE J. 1987,33, 1155. (9) Brenner, H.; Gaydos, L. J. J.Colloid Interface Sci. 1977,58,312. (10) Gajdos, L. J.; Brenner, H. Sep. Sci. Technol. 1978, 13, 215. (11) Gajdos, L. J. Boundary Effects in the Transport of Arbitrarily-
Shaped Brownian Particles. Ph.D. Dissertation, Carnegie-Mellon University: Pittsburgh, 1977.
t ’0
1
c+o
I
ac = o
m
Figure I. Poiseuille flow of a viscous solvent at mean velocity V through an infinitely long circular capillary tube of radius r,. At time t = 0, a mass m of solvent-dissolved solute (depicted by the closed shaded domain) is introduced into the flowing fluid without disturbing the latter. c = c(r,d,z,O) (2.3) where c c(r,&z,t) represents the solute concentration field at any time t 2 0. Following its initial introduction into the system, the solute distribution evolves in space and time in accordance with the well-known microscale convective diffusion equation4
(2.4)
with D the molecular diffusivity of the solute. Assuming the tube wall to be impenetrable to solute (and solvent) requires imposing the no-flux boundary condition aclar = 0 at r = ro
(2.5)
Moreover, we also require satisfaction of the far-field boundary condition aslzl- m (2.6) where the spatial attenuation of the solute concentration field toward zero is faster than algebraic. Given these boundary conditions, it is readily demonstrated that the solute mass m originally introduced into the system is conserved for all time; explicitly c -0
l:S,’”S,”c(r,$,z,t)r
dr d 4 dz = m = const ( t IO) (2.7)
The initial- and boundary-value problem (2.3H2.6) posed for the solute’s microconcentration field c(r,$,z,t) possesses a unique solution. Because of the nonconstancy of the phenomenological function u(r) appearing in the convectiveterm of the solute conservation continuity equation (2.4), this solution must be obtained numerically rather than analytically. Given the computational capacities and speed of modern computers, this would not be a difficult task-though the numerical solution would be specific to the initial condition (2.3); that is, a different numerical calculation would be required for each different choice of initial condition. Mean Concentration Field E(z,t). In any event, such exhaustively detailed information as would be embodied in the microsolution c(r,&z,t)-numerically or otherwiseis rarely, if ever, required in practice. Rather, the quantity of usual interest in applications is the mean (areaaverage) solute concentration field, namely
Obviously, E can be computed numerically from comparable
Langmuir, Vol. 6, No. 12, 1990 1717
The Kendall Award Address
pi-))-q INPUT DATA
c ( I , 4, z , 0 )
E - 0 asIzI-0)
OUTPUT
FIRST PRINCIPLES
(2.13)
and macroscale initial condition (cf. (2.8) and (2.3))
TAYLOR-ARIS PARADIGM
c(z,O) = & c t c ( r , r j , z , O ) r
dr drj
From (2.7) and (2.8)) we note that
Figure 2. Two (equally rigorous) alternative schemes for proceeding from the specified microscale parameters and initial data to the area average solute concentration E .
numerical knowledge of c, although again the result would be specific to the choice of initial solute distribution, c(r,rj,z,O).
To the extent that knowledge of (7. is all that is required in practice, one is completely done with the physical problem a t this stage. Unfortunately, such strictly numerical knowledge-while perhaps useful in parametric engineering design studies-rarely furnishes useful conceptual insights into the overall, macroscale physical phenomena arising from interactions existing between microscale convective and diffusive solute transport mechanisms. Moreover, the computation o f t via this “first-principles” scheme is very wasteful of computer resources in the sense that having expended considerable effort to first compute the microfield c(r,@,z,t),much of this detailed information is effectively discarded when the latter is coarse grained during the integration step (2.8) leading to ~ ( z , t ) . 2. Macrotransport Processes. Given the preceding remarks, it is natural to inquire as to whether a more insightful and computer resource effective scheme or paradigm exists for (rigorously) proceeding form the specified microscale parameters and input data depicted in Figure 2 to the desired coarse-grained field E(z,t). This important question was answered in the affirmative by Taylor112more than 30 years ago, with a major conceptual assist from Aris? leading to what is today called Taylor dispersion theory. In particular, Taylor was able to show that if one is interested only in the asymptotic, longtime macroscale solute distribution E = c(z,t) for times t satisfying the nondimensional inequality Dtlr;
>> 1
(2.9)
then E itself obeys a (one-dimensional, macroscale) convective “dispersion” equation, namely (2.10)
possessing constant macroscale phenomenological coefficients U* and D*. The latter are respectively given by
U* = p
(2.11) (2.12)
and are thus exp_ressed in terms of the prescribed input parameters ro, V, and D (cf. Figure 2) governing the comparable microtransport problem defining c. Not only are these two coefficients independent of spatial position z and time t , but equally important they are independent of the initial (microscale) solute distribution (2.3). To obtain the mean field E , one has thus only to solve the constant-coefficient macrotransport equation (2.10) subject to the far-field boundary condition (cf. (2.8) and (2.6))
(2.14)
XTO
(2.15)
Insofar as the effective use of computer resources is concerned, the one-dimensional Taylor-Aris macroscale equation (2.10) for computing E(z,t)-albeit valid only for the long times t indicated in (2.9)-is obviously far superior to the ab initio scheme embodied in the numerical solution of the three-dimensional microscale convective-diffusive equation (2.4) (leading to E(z,t) via (2.8)). Moreover, the existence, functional dependence upon microscale parameters in Figure 2, and physical interpretation of the macrotransport coefficients U* and D* as representing the mean axial solute velocity and dispersivity about the mean obviously furnish useful physical insights into the global aspects of the transport process. Comparable insight would clearly be absent from the purely numerical “first principles” solution of the microscale equation (2.4) leading to E for some specific initial solute distribution (2.3). 3. Taylor-Aris Paradigm. The Taylor-Aris paradigm ultimately resulting in the expressions (2.11) and (2.12) for the macrotransport coefficients U* and D* derives from asymptotically equating (for long times (2.9) and for each integer value m = 0, 1, 2, ...) the axial moments
M,
tfJ:&”“S,”(z
- z’),cr
dr drj dz
(2.16)
of the microtransport equation (2.4) governing c with the comparable axial moments (2.17)
of the macrotransport equation (2.10) governing E . Mere explicitly, the theory involves asymptotically matching M, with M , insofar as dominant-order algebraic terms in t are concerned. 3. Conditional Probability Density 1. Reformulation of the Microtransport Equations in Terms of the Conditional Probability Density P. a. Green’s Function. Rather than considering the combined behavior of the many (noninteracting) individual solute “molecules” comprising the solute microconcentration field c(r,$,z,t), it proves both convenient and conceptually useful to consider the (mean or Brownianlevel statistical) behavior of a single such “molecular”entity. In particular, owing to the linearity of the system of microtransport equations (2.3H2.6) defining the microscale field c, the principle of superposition may be invoked by introducing Green’s function
P = P(r,rj,z,tlr’,4’,z’) (3.1) for this system of equations via the defining equation
c(f,rj’,z’,O)r’ dr’ drj’ dz’ (3.2) Thereby, the Green function plays the mathematical role of an integral “proportionality function” operator, serving to convert the initial solute concentration distribution
1718 Langmuir, Vol. 6, No. 12, 1990
The Kendall Award Address
H v, = 0: (I,,
I?
Stochastic ‘trajectory’
Figure 3. Stochastic ’trajectory” of a Brownian tracer or ’molecule’ initially introduced at time t = 0 into the system at the point (~‘.d’.z’)and currently present at timet at the point (r,wl.
c(r,o,z,O)prevailing at time t = 0 into the current solute concentration distribution c(r.@,Z,tJprevailing at time 1 . Upon substitution of (3.2) into the system (2.3)-(2.6) of microtransport equations governing c, it readily follows that P represents the solution of the following microscale, initial- and boundary-value, convect ive-diffusive problem: (3.3) (3.4a)
’
P - 0 asIz-z’I-together with the initial condition
P = 6(z-z’) ( t = 0) = 0 (t < 0)
(3.5)
with 6 the Dirac 6 function. This system of equations is essentially identical with the comparable system (2.3)(2.6) governing c, except that the original generic initial condition (2.3) has now been formally replaced by the instantaneous “source“ term (3.4).the latter representing the instantaneous introdurtion at time t = 0 of a spatially concentrated unit ma3s of soluteat the point (r‘,&,z’) into the otherwise solute-free (flowing)solvent. In this context, the analogue of (2.7) is lIS,”s,”Prdr
d@dz = 1 (t t 0) = 0 ( t > ro2/D,it can be showns that the second central axial moment (3.15) grows linearly with time; explicitly (2
- 2)2 = 2D*t
(3.18)
where D* is a constant, independent of t (as well as of r‘ and $’), given explicitly by Taylor’s second relation, (2.12). Analogous to Einstein’s’3 - celebrated Brownian motion relation, namely, ( A z ) ~= 2Dt for the mean-square unidirectional displacement of a solute molecule diffusing (with molecular diffusivity D) through the quiescent solvent, (3.18) gives the dispersion about the mean solute position 2 (RZ’ U*t) at time t in the flowing solvent. However, in contrast with Einstein’s mean-square displacement relation, which is exact for all times t , (3.18) is only asymptotically valid after the solute molecule has had an opportunity to sample (many times) all of the Poiseuille velocities u(r) in the tube cross section. The Einstein-like Lagrangian interpretation (3.18) of the dispersivity D* contrasts with the classical Eulerian Fick’s law total axial flux interpretation of D* implicit in the macroscale conservation law (3.9) (or, equivalently, (2.10)). Referring to Figure 3, this dispersion arises from the fact that, due to the stochastic nature of the molecular Brownian motion, not all solute molecules originally introduced at the inlet plane z’ will cross the distant exit plane z (=z’ + L ) at the same time (even if they are introduced simultaneously a t the same cross sectional location (r‘,@)).Rather, there will exist a distribution of arrival times at the exit plane about the mean time t = L/U*. In particular, the leading term of (2.12) represents the contribution to this dispersion resulting from the solute’s axial molecular diffusion, whereas the second (generally more dominant) term arises from the solute’s transverse molecular diffusion. The latter contribution is highly nonlinear, resulting from microscale interaction between lateral Brownian motion and axial convection. Taylor’ himself explains why this contribution to the dispersion is greater the smaller the (transverse) molecular diffusivity. 4. Generalized Taylor Dispersion Theory. Macrotransport Processes. The same single-particle Lagrangian approach embodied in the archetypal tube flow problem can be applied to large classes of microscale convective-diffusive-reactive transport phenomena governed by linear constitutive equations, leading to generalized Taylor dispersion theory. More recently, this subject has been termed macrotransport processes. A representative sampling of illustrative problems is addressed in subsequent sections. For simplicity of presentation, these have all been selected as constituting simple variants of the original Taylor-Aris problem.
+
4. Finite-Size Sphere. Hydrodynamic
Chromatography The preceding Taylor-Aris tube flow analysis has been extended9J4from an effectively point-size solute molecule to a finite-size colloidal Brownian sphere (radius = a). Hydrodynamic and steric wall effects resulting from the nonzero value of the parameter X = a/r,
(4.1) represent the unique features of this problem, leading (13) Einstein, A. Investigations on the Theory of the Brownian Mouement; Dover: New York, 1956. (14)Mavrovouniotis, G. M.; Brenner, H. J . Colloid Interface Sci. 1988, 124. 269.
1720 Langmuir, Vol. 6, No. 12, 1990
The Kendall Award Address
ultimately to a "steric exclusion" chromatographic scheme pioneered by Smalll5 for separating colloidal spherical particles of different sizes via Poiseuille flow in capillary tubes. Imagine that the Brownian sphere undergoing transport is composed of a transparent material possessing the same refractive index as the solvent in which it is suspended, thereby rendering the sphere invisible to an observer. Suppose now that a point-size black dot is placed at the center of the sphere, so that this point constitutes the only object within the solvent visible to the observer. The stochastic trajectory of this body-fixed point (termed the particle "locator point") will then represent the entire sphere's motion through the solvent. Being a point (rather than a finite-size body), to which classical continuummechanical principles can be applied, one may now calculate the conditional probability density P(r,@,z,tlr',@',z') that the sphere center is situated at the point (r,@,z)at time t , given that it was initially situated at the position (r',$',z') at time t = 0. The microscale conservation equation governing transport of the finite-size sphere (center) through the circular cylinder differs in the values of its phenomenological functions from the comparable point-size convective diffusion equation (3.3) owing to the existence of wall effects arising from the nonzero value of A. In particular, (3.3) is here replaced by + V ( rap) z= V.[D(r)VP] (4.2)
1.15
1.10 \
I& 1.05 1.oo
0
0.05
0.10
0.20
0.15
L = a/ r,, Figure 5. Mean axial velocity o* of a neutrally buoyant Brownian sphere of radius a transported by a Poiseuille flow (mean velocity = V) through a circular capillary tube of radius r,.
i I
at
where U ( r )is the axial velocity with which a neutrally buoyant force-free and couple-free sphere of radius a (whose center is situated at r) would move when suspended in the (undisturbed) Poiseuille flow (2.1) of the mean solvent velocity P. Due to wall effects, the translational molecular diffusivity D(r) of the sphere is no longer a position-independent scalar as in (3.3) but is now rather a second-rank tensor whose value depends upon the radial position r of the sphere center within the tube. Assuming the pertinent phenomenological functions to be known, equation (4.2) may, in principle, be solved subject to the respective initial and boundary conditions (3.4) and (3.6), and with (3.5) here replaced by the noflux condition
aP/ar = 0 at r = r, - a
-=-
(A
V This expression is valid only for the case
> a), have traveled the same axial distance L down the tube.
0.75
1.00
Figure 6. Hindered mean axial molecular diffusivity DM of a Brownian sphere of radiw a through a quiescent viscous solvent confined within a circular capillary tube of radius roe X S 0.20 (approximate)
(4.7)
owing to the current unavailability of the phenomenological functions required in (4.2) for larger values of A. Equation 4.6 is plotted in Figure 5. Similar calculations performed for the dispersivity may be expressed in the form (4.8)
D*=DM+DC
where the "hindered" diffusivity DM (cf. Figure 6) of the Brownian sphere through the quiescent solvent is9
DM 1 --N
D,
In (Ae1) - 1.539h + O(X2) (1- X)2
,
(A V. The effect of the dispersion about these peaks, quantified by the respective dispersivities D * 2 and D*I, is to decrease the sharpness of the separation between the two species. Knowledge of the functional dependencies of U*i and D*i ( i = 1,2, ...) upon the fundamental parameters of the system (namely, ro, P, ai, Di) for each of the solute species i being separated permits a complete engineering design (17) That is, using (2.1)
0 1
Z’
-
Figure 7. Chromatographic separation (in a circular capillary tube) of an initially uniform mixture introduced into the flowing fluid at the plane z = z’ at time t = 0. At time t , the two species, which respectively move at the different mean axial velocities U*1 and U*2, have separated into two diffuse “bands”. The diffuseness embodied in these bands, which limits the sharpneas of the separation, arises from the respective dispersivities D*1 and D*2 of the two species.
of the chromatographic separation system. In practice, of course, one often uses (monodisperse)packed beds rather than simple capillary tubes to effect this separation. One must then resort to additional geometrical modeling, involving-in the simplest case-use of an “effectiveradius” ro for the characteristic size of the packed-bed interstitial pore space; more realistic geometrical models of packed beds-typically spatially periodic mode1slgm-have also been the subject of generalized Taylor dispersion theory analyses.
5. Solute Adsorption on the Walls. Affinity Chromatography Returning to the original Taylor-Aris-type analysis of section 3, involving a point-size Brownian solute molecule suspended in a Poiseuille flow within a circular capillary tube, consider the case where the solute molecule experiences adsorption forces tending to deposit it on the tube In the simplest possible case, the previous Taylor-Aris microscale no-flux boundary condition (3.5) is here replaced by the linear Henry’s law adsorption isotherm
P = kPIr=ro
(5.1)
with k the Henry’s law constant and p the surfaceexcess solute “concentration” (probability density) on the wall, measured in amount of solute per unit area. In its adsorbed state, the solute may diffuse along the wall with a surface diffusivity D8 in addition to undergoing both diffusion in the bulk fluid (with molecular diffusivity D) and Poiseuille flow convection (with mean velocity Vj. (18)Brenner, H.; Adler, P. M. Philos. Trans. R. Soc. London 1982, A307, 149. (19)Nitxhe, L. C.; Brenner, H. AIChE J. 1990,36,1403. (20) (a) Adler, P. M.; Brenner, H. PCH, PhysicoChem. Hydrodyn. 1984, 5,269. (b)Dungan, S. R.; Shapiro, M.; Brenner, H. h o c . R. SOC.London 1990. ~. .. A429.639. (21) Golay, M. J. E. In Gas Chromatography; Deaty, D. H., Ed.;Butterworths: London, 1968; p 36. (22) Dill, L. H.; Brenner, H. J. Colloid Interface Sci. 1982, 85, 101. ~~~~~
That no weighting factor appears in the integrand of the latter stems from the fact that, due to the transverse Brownian motion of the sphere, the sphere center samples all points equally in the circular domain (4.5).
Distance z
~
1722 Langmuir, Vol. 6, No. 12, 1990
The Kendall Award Address
With23 K = 2k/r, (5.2) a nondimensional Henry’s law constant, application of the generalized Taylor dispersion theory paradigm derived from the Lagrangian equations (3.16) and (3.18) eventually yieldsz2
u*
-=.=-
V
1 l+K
for the mean axial solute velocity
(5.3)
u*and
D* = D M + D C
(5.4) for the total axial solute dispersivity required in (3.9). Here
+
= D KDs (5.5) l+K is the purely molecular contribution (corresponding to the quiescent fluid case) and DM
the Taylor or convective contribution. Since K > 0, (5.3) reveals that, on average, the solute molecule moves through the tube more slowly than does a comparable solvent molecule, i.e.
Equation_5.5 for the macroscale or effective molecular diffusivity DMembodies respective contributions from both the volumetric ( D ) and surface (Ds)molecular diffusion processes. This corresponds physically to these processes operating in parallel. Whenever Ds > D,the overall molecular diffusion process through the quiescent solvent occurs more rapidly than would otherwise occur for pure bulk transport, and conversely. 6.. Chemically Reactive Solute Consider the case where the point-size formerly passive solute of sections 2 and 3 now undergoes a first-order irreversible chemical reaction at the tube wall (e.g., a catalyst is present there). In this situation, the TaylorAris microscale no-flux condition (3.5) at the tube wall is replaced by the linear boundary condition
-D-ap = kP at r = r,
(6.1) dr with k the reaction velocity constant. In such circumstances, solute is no longer conserved (as a chemical species). Rather, upon reaching the wall (via Brownian motion) a solute molecule may be permanently removed from the bulk fluid as a result of undergoing chemical reaction. The previous Eulerian macrotransport equation (3.9) governinq, the long-time transport of the macroscale solute density P (3.8) is replaced in present circumstances by the expression7*8.24
u*< v
(5.7) This phenomenon arises here because the (short-range) adsorptive forces acting upon the solute molecule when in proximity to the tube wall cause it to spend a disproportionately large fraction of its time in that neighborhood, where the slower moving Poiseuille flow streamlines prevail. In other words, whereas a solvent molecule is free to sample, without bias, all points in the tube cross section, the solute molecule is biased in favor of the slower moving streamlines proximate to the wall. By favoring these streamlines, a solute molecule thus moves, on average, more slowly in the axial direction than does a solvent molecule. Equation 5.3 points up the possibility of chromatographically separating two (or more) solute species possessing different adsorptivities ki (i = 1,2, ...). Thus, if K1 and K2 are the dimensionless adsorptivities (5.2) of solutes 1 and 2, their respective mean velocities down the tube will be in the ratio
u*l 1 + K2 -- r, + 2k2
-=-
U*2 1 + K ,
r,+2kl
(5.8)
This yields the inequality U*1 > U * 2 whenever k l < kz, so that the least strongly bound species (i.e., species “1”) is the faster moving of the two. Equation 5.8 provides the basis for understanding affinity chromatography, whereby the separation of solutes possessing different affinities k is accomplished by virtue of their different mean velocities. The ensuing chromatographic separation of the two species is similar to that depicted schematically in Figure 7. (23) The inverse length factor 2/r, appearing in (5.2) represents the so-called specific surface of the capillary tube; that is, for an arbitrary axial length 1 of capillary, it represents the ratio
corresponding to the wetted surface area per unit volume. This factor arises naturally in adsorption problems when the areal results are expressed on a volumetric macroscale basis, as in (3.9).
The microscale first-order irreversible surface chemical reaction (6.1) occurring at the wall thus manifests itself at the macroscale as a first-order irreversible volumetric chemical reaction (characterized by the bulk reaction velocity constant R*). Surprisingly, the mean velocity u* with which the reactive solute moves axially is not the same as the mean solvent velocity V. As such, the solute is not simply convected on average with the inert carrier; rather, the solute moves faster than does the fluid; explicitly u* > v (6.3) This pheomenon stems from the fact that only those solute molecules “smart enough” to stay away from the wall region survive their trip downstream, thus enabling them to accomplish the transit from tube entrance to exit; conversely, those molecules “foolish” enough to meander (by lateral Brownian motion) over to the tube wall are destroyed by the reaction. As such, those solute molecules that exit the.tube (and are hence counted in the entrance/ exit scheme depicted in Figure 4) have not sampled the slower moving streamlines of the Poiseuille flow existing near to the wall. Stated alternatively, those molecules surviving the trip downstream, and hence exiting the tube, have on average sampled only the faster moving streamlines distant from the tube wall. In contrast, the (inert) solvent molecules (all of which survive the trip downstream) sample all cross-sectionalpoints and hence all streamlines in the tube equally, in effect “wasting” a portion of their time in the slowest moving stceamlines. The exact amount by which U*exceeds V depends upon the Damkohler number25 Da = kr,/D (6.4) In a similar vein, D* = D M + Dc, where D M = D, as before; however, Dc does not possess the same classical (24) Sankarasubramanian, R.; Gill, W. N. R o c . R. SOC.London 1973,
A333, 115.
The Kendall Award Address Taylor value (2.12) for a passive solute as it does for a reactive solute; rather, the ratio Dc/Doc (where the denominator DOCdenotes the classical Taylor value ro2P/ 4 8 0 ) will also depend upon the Damkohler number (6.4), the rather complex functional dependence being given explicitly by Shapiro and Brenner.' Of course, the effective macroscale volumetric rfaction velocity constant K* (in the nondimensional form K*ro/k) is also functionally dependent upon the Damkohler number. For small values of the latter, one finds that
K* = 2k/ro, (Da > 1) (6.6) (see ref 25 for the source of the 5.893 coefficient.) Here, the effective reaction rate is limited by the ability of the solute to diffuse to the wall. Finally, we observe that despiie the fgct that the phenomenological coefficients U*, D*, and K * appearing in the reactive macrotransport equation (6.2) are independent of the initial microscale data (i.e., of the initial at which the solute molecule cross-sectional position (r',@') is introduced into the system at time? = 0), the macroscale conditional probability density P, defined by (3.8), will nevertheless depend upon this initial conditionexplicitly upon r'. The reason for this derives from the fact that the macrotransport equation (6.2) governing P , rather than being exactly true for all times t , is only valid asymptotically for times t satisfying the transverse sampling-time criterion (2.9). And, for earlier times, namely, t I O(ro2/D),preceding that for which the macroscale description (6.2)becomes valid, a solute molecule initially introduced too near the reactive wall will have a higher probability of being destroyed by chemical reaction than one introduced further from the wall. Thus, not all solute molecules initially introduced into the system survive long enough for the macrotransport equation (6.2) to henceforth become a valid descriptor of their fate. As a result of these facts, in order that the value (3.8) of P (derived by "first-principles" quadrature from the mi-
K* = 5.783D/r:,
(25) The exact relation is7
iJ* 1 + (&pa-' - 219,-')2 _ -I+ 3[(~@a-')* + 11
where 00 1 Bo(Da) is the smallest positive root of the transcendental equation#'
-
8J,(B) DaJ,(B) = 0 with Jn the Bessel function of order n. In the limit of small and large Da, the respective asymptotic roots of the latter equation are,respectively Bo
2Da (Da > 1) the 2.4048 coefficientbeing the smalleat positive root of the equation Jo(p) = 0. For later reference, we also note that, in general
fi* = B;D/r: for arbitrary Da. (26) Carslaw, H. S.;Jaeger, J. C. Conduction of Heat in Solids, 2nd ed.; Oxford London, 1959; p 493.
Langmuir, Vol. 6, No. 12, 1990 1723 croscale solution P of eqs (3.31, (3.4), (3.6),and (6.1)) asymptotically match the comparable solution P of the macroscale equation (6.2) (subject to the far-field boundary condition (3.10)),one must subject the solution of the latter to a fictitious initial condition, say &,O) at t = 0, in place of the true initial condition (3.11). The scheme for deriving this fictitious initial condition from the given microscale phenomenological data, namely, u(r), U,ro, D , and k (as well as r'), is discussed by Shapiro and Brenner.7t8*27 Practical packed-bed reactor design principles may need to be reexamined in view of the significant differences observed here between macroscale descriptions of passive vs reactive solute transport processes. In particular, residence time distribution-type analyses (especially of fmtorder irreversible chemical reactions) appear to be conceptually inappropriate in situations for which the reaction velocity constant is nonuniform across the tube cross section: such as is the case here when it is nonzero only at the tube wall, r = ro. 7. Aerosol Deposition As a final example, we consider the case of Brownian aerosol or hydrosol solute particles being deposited (by molecular diffusion) from a dilute suspension upon the walls of the capillary tube within which a Poiseuille flow ocurs at mean velocity U.2a Assuming that these submicron particles, upon reaching the wall, are held there permanently by (short-range) attractive forces, the TaylorAris microscale no-flux condition (3.5) at the tube wall is here replaced by the zero solute "concentration" boundary condition29
P=O atr=ro
(7.1)
The wall thus functions as a sink for any solute particle reaching it (by lateral Brownian motion). As a result of its removal from the bulk fluid, solute is not conserved (as a chemical species) at the macroscale. The situation here is entirely analogous to the microscale first-orddr irreversible surface chemical reaction case of section 6 in the limiting situation where the reaction velocity constant k 00. As a result of this analogy, the basic aerosol macrotransport equation is given by (6.2), in which the effective volumetric reactivity coefficient is (cf. (6.6))
-
K* = 5.783D/r;
(7.2)
This macroscale phenomenological coefficient quantifies the rate of solute disappearance from the bulk fluid and hence quantifies the corresponding rate of solute deposition upon the tube wall. Thus, whereas no real chemical reaction occurs at the microscale it nevertheless appears at the macroscale, as if a first-order irreversible bulkphase reaction is, in fact, occurring. On the basis of the microscale surface chemical reaction analogy with the results of section 6 , it follows that, on average, an aerosol particle will move axially through the cube at a mean velocity U* exceeding the mean velocity V of the carrier fluid. Explicitly, we find that3O
o*/P= 1.564
(7.3)
The dispersivity D* can analogously be obtained as a limiting case of the results of section 6 as Da--. (27) Shapiro, M.; Brenner, H. Chem. Eng. Sci. 1988,43,551. (28) Shapiro, M.; Brenner, H. J. Aerosol Sci. 1990,21,97. (29) Shapiro, M.; Brenner, H.; Guell, D. C. J . Colloid Interface Sci. 1990,136,552. (30)This derives from the U*/V formula appearing in ref 25, upon setting BO = 2.4048 and letting Da m.
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1724 Langmuir, Vol. 6, No. 12, 1990
Of course, to obtain the correct solution P of the aerosol macrotransport equation (6.2) one must again introduce the idea of a fictitious initial condition, as discussed in the preceding section. 8. Closure Given the implicit restrictions imposed by the nature of this presentation, we have only barely touched upon the power of macrotransport analyses in the macroscale modeling and physical interpretation of large classes of microscale convective-diffusive-reactive transport processes. A comprehensive list of applications involving further examples of the use of the generalized Taylor dispersion theory paradigm718to effect the coarse graining of complex microscale phenomena is given by Frankel and B1-enner.3~Also contained therein will be found references to the works of many early researchers who pioneered the subject, as well as to contemporary researchers who have significantly expanded its boundaries.
Acknowledgment. Many of our research activities pertaining to macrotransport processes were supported by the National Science Foundation, the Office of Basic (31) Frankel, I.; Brenner, H. J.Fluid Mech. 1989,204, 97.
The Kendall Award Address Energy Sciences of the Department of Energy, the U.S. Army Research Office, and the Microgravity Science and Applications Division of the National Aeronautics and Space Administration, to all of whom I am grateful for their support and encouragement. A portion of this manuscript was written while on sabbatical leave as Chevron Visiting Professor of Chemical Engineering at the California Institute of Technology. Their support and hospitality were much appreciated. During this period I was also the recipient of a Guggenheim Memorial Foundation Fellowship. Finally, I would like to express my gratitude to the many friends, colleagues, and former and current research students who participated in the “Kendall Award” Symposium held in my honor at Penn State during the period June 20-22,1988, under the sponsorship of the 62nd Colloid and Surface Science Symposium. I am especially happy to acknowledge my former student, Peter M. Bungay of the National Institutes of Health, who organized and coordinated this symposium. My one regret is that Stanley G. Mason, a friend and collaborator whose work and style greatly influenced my own career in colloid science, and himself a former Kendall awardee, did not live to see how successfully his mentoring had come to fruition.
