Influence of crossflow hydrodynamics on retention ratio in flow field-flow fractionation

Analytica Chimicu Actu, 246 (1991) 161-169 Elsevier Science Publishers B.V., Amsterdam

161

Influence of crossflow hydrodynamics on retention ratio in flow field-flow fractionation Joe M. Davis Department

of Chemistry and Biochemistry,

Southern Illinois University, Curbondule, IL 62901 (U.S.A.)

(Received

8th June 1990)

Abstract The axial velocity profile in flow field-flow fractionation (flow FFF) differs from that in most forms of FFF, because analytes are polarized near the accumulation wall by a crossflow of carrier instead of a gradient in chemical potential. In accordance with the Navier-Stokes equations, this crossflow alters the axial velocity profile in an open parallel plane channel from a parabola to one which is skewed toward the accumulation wall. Because of this skewing, the commonly used expression for the retention ratio in open parallel plate channels, which is based on a parabolic profile, is incorrect for flow FFF. An alternative expression for retention ratio R, which accounts for this skewing, is derived. The expression depends on the reduced zone thickness, h, and on the Reynolds number, Re = 1U, 1w/2v, where )U, ) is the magnitude of the crossflow velocity, w is the channel width and Y is the kinematic viscosity. In the limiting but important case in which R varies linearly with h, the error between this expression for R and that based on the parabolic profile is less than l%, as long as Re < 0.030. Experimental values of Re in open parallel plate channels are shown to be at least one order of magnitude smaller than this threshold value. Consequently, the expression for the retention ratio based on the parabolic profile is adequate in most instances. In the most restrictive case considered, it is shown that crossflow hydrodynamics should not perturb transport rates provided that the analytes elute within 10 void volumes. Keywords:

Field-flow

fractionation;

Crossflow

hydrodynamics;

This paper addresses the influence of crossflow hydrodynamics on analyte transport in flow fieldflow fractionation (flow FFF). FFF is an important family of separation methods that are particularly well suited to the fractionation and characterization of macromolecular and colloidal mixtures [1,2]. In general, separation by FFF is realized by the localization of analytes into polarization layers (zones) of various thicknesses at the walls of a thin channel, through which a carrier is slowly pumped. The various layers move through the channel at speeds proportional to their thicknesses; tightly focused zones move slower than less focused zones. In most modes of FFF (e.g., thermal, electrical and sedimentation), analytes are polarized at the 0003-2670/91/$03.50

0 1991 - Elsevier Science

Publishers

Retention

ratio

channel walls by a gradient in chemical potential, which is physically generated by applying an external force perpendicular to the channel walls. For example, in sedimentation FFF, the channel is coiled, placed in a centrifuge basket and rotated to induce a sedimentation force. In flow FFF, however, this gradient is replaced by a crossflow of carrier, which leaves the channel through a semipermeable membrane defining the accumulation wall. For example, crossflow is generated across the open parallel plate channel shown in Fig. 1 by pumping carrier through the two semipermeable membranes defining the channel walls [3]. Other recently developed (or revived) modes of flow FFF, e.g., asymmetric flow FFF [4-61 and hollow-fiber flow FFF [7-91, rely on the converB.V.

J.M. DAVIS

162

Fig. 1. Schematic diagram of OPPC for flow FFF, defines the spatial orientations of x, z, U, and u.

which

sion of a fraction of axial flow into crossflow. Regardless of its mode of generation, the crossflow carriers all analytes by convection toward a single accumulation wall. Separation is affected by differences in analyte diffusion coefficient [3], which alone controls the thickness of the polarization layer. A particularly attractive feature of flow FFF is the universality of this polarization mechanism: all analyte types and classes are affected. In contrast, gradients in chemical potential are selective and may not affect some analytes at all. The universality of flow FFF is reflected by its application to the separation and characterization of many mixture types, including polystyrene latices [3,8,10], viruses and bacteriophages [6,10,11], proteins [3,5,6,10,12], plasmids [5,6], hydrophilic [13,14] and hydrophobic [15] polymers, silica sols [16], fulvic and humic acids [17], chromatographic supports [18] and ground minerals [19]. The spatial and temporal characteristics of the channel geometries, the various chemical potential gradients or crossflows and the axial velocity profiles which are important in FFF can often be described by fairly simple mathematics. This simplicity has facilitated the derivation of several sets of equations for the transport rate and axial dispersion of analyte zones in FFF channels [20-221. Such equations are useful in the design, construction and optimization of these channels. More

important, one can regress against these equations various experimental measurements of analyte transport (e.g., retention ratio and plate height) to estimate various analyte properties (e.g., mass, density, electrical charge, molecular weight and diffusion coefficient). Thus, FFF can be used both to characterize and to separate analyte mixtures. The accuracy of the estimated analyte properties clearly depends on the use of a good model, which requires (among other things) a correct mathematical description of the physical system. In particular, the axial velocity profile must be correctly described, because it determines the speed with which analytes are swept downstream. In general, this profile is not affected by a gradient in chemical potential. As shown below, however, the axial profile is affected by the crossflow through a flow FFF channel, even though these flows are orthogonal. The result is that the axial velocity profile is asymmetric and is skewed toward the accumulation wall. The effect of this asymmetric flow on the axial transport rate of analyte zones, as measured by the retention ratio, is the subject of this paper. The purpose of this work was not to refine a theory that already successfully describes the transport rates of the high-molecular-weight species to which flow FFF largely has been applied. Rather, it was to assess the applicability of this theory to low-molecular-weight species (i.e., 50-1000 dalton). Because separation by flow FFF is based on differences in diffusion coefficient, the resolution of such mixtures by chromatography is often more selective and efficient. Some highly polydisperse materials (e.g., humic and fulvic acids of ca. 1000 dalton) nevertheless are perhaps best addressed by flow FFF [17]. In addition, very fine colloids (e.g., inorganic and metallic ~01s) can be resolved only poorly by means of steric exclusion. The catalytic, electrocatalytic and electrochemical properties of such sols are typically dependent on the sol size. Because sols of different size have different diffusion coefficients, flow FFF could prove useful in the future high-resolution fractionation of such mixtures. In this instance, it is important to establish that the large crossflow velocities required to form thin polarization layers from such species of high diffusivity do not affect

INFLUENCE

OF CROSSFLOW

HYDRODYNAMICS

significantly the axial velocity quently, axial transport rates.

profile

ON RETENTION

RATIO

and, conse-

THEORY

The axial transport nels is experimentally ratio, R [21]:

rate of zones in FFF chanquantified by the retention

R = t,/t, = V,/V,

(1)

where t, and t, (V, and V,) are the times (eluent volumes) required to elute a non-retained zone and the zone of interest. This ratio also can be expressed theoretically by the equation [21] R = (c*u)/(c*)(u)

(2)

concentration where c* IS . the quasi-equilibrium formed at the accumulation wall by the crossflow or chemical potential gradient, u is the axial velocity profile and the brackets represent averages of the enclosed functions over the channel cross-section. With appropriate expressions for c* and u, one can derive equations which connect theory (as expressed by Eqn. 2) to experimental observables (as expressed by Eqn. 1). Axial velocity profile The axial velocity u in any FFF channel is determined as an appropriate solution to the Navier-Stokes equations. Various such solutions have been proposed, principally because channel geometries and fluid properties can differ in different cases. In this paper, only the commonly used rectangular conduit, or open parallel plate channel (OPPC), shown in Fig. 1 will be considered. For the channel breadth-to-width ratios that are characteristic of OPPCs (e.g., ca. loo), the channel walls effectively can be represented by two infinitely parallel plates [23]. In this instance the flow in the breadth direction is zero. For an incompressible carrier, the steady axial velocity u is governed by the differential equation [24]

ap

-Jy+JQ

a

a~

( 1=o Pa,

(3)

where ap/az is the axial pressure gradient causing the flow, p is the carrier viscosity, x is the lateral

IN FLOW

FFF

163

coordinate perpendicular to the accumulation wall and z is the axial coordinate, which parallels the flow direction (see Fig. 1). Although the axial flow-rate can be programmed [25], it is usually held constant. In this common case, the pressure gradient, ap/az, is constant. Further, the viscosity p can often be approximated as constant and factored out of the parenthetical grouping in Eqn. 3 (this approximation is not valid in thermal FFF, because p varies with x; the solution to u for this case is given elsewhere [26]). Under these constraints, the solution to Eqn. 3, subject to the non-slip boundary conditions

u(x=O)=O u(x=w)=O

(4

is u=6[5-(;)l](u) where w-‘j,,“‘u For Stokes between properly

w is the channel width and (u) = dx is th e average carrier velocity. an ideal flow FFF channel, the Navierequation for the steady axial velocity u two infinitely parallel plates is more written for an incompressible carrier as

PWI

(6) where p is the carrier density and U is the crossflow linear velocity, which parallels x. The term pU au/ax now replaces the zero on the right-hand side of Eqn. 3, and the term a(p &Y/az)/ax is now included on the left-hand side of this equation. Clearly, the solution to Eqn. 6 must differ from Eqn. 5. In effect, previous theories for flow FFF based on Eqn. 3 are valid only in the limit of very small crossflow velocities, i.e., as ] U 1 + 0. To solve Eqn. 6, some simplifying assumptions will be made. First, it will be assumed that the crossflow velocity U is equal to the constant Q,. This assumption is consistent with the infinitely parallel plate model, in which the channel contents are unbounded in the breadth direction. A constant crossflow can be realized experimentally if the semipermeable membranes are uniform, the

J.M. DAVIS

164

crossflow is fed equally through all portions of the membranes and the crossflow rate is independent of time. Because the crossflow velocity U, (of magnitude 1U, I.) is opposite to coordinate x, the term pU au/ax in Eqn. 6 can be expressed as -p 1U, 1au/tlx. One can also equate the term XJ,Gz in Eqn. 6 to zero, because the constant 7-J, is independent of the axial coordinate. As before, it will be assumed that the axial flow-rate is constant and that the carrier has a constant temperature; these assumptions imply that ap/az, p and ~1 are also constants. Under these constraints, the boundary conditions in Eqn. 4 determine the solution to Eqn. 6 as [27] 2 “=Re

eR’(l -2;

+

_ e-=Ww) sinh( Re)

1 V

max

Re

-2;+

x{

eR’(l

_ e-2W’w) sinh( Re)

IU, ]w/2v

iwe-‘/‘dx/o”[

-2;

+ sinLke) -1

)I i

By introducing

(10)

dx

the variable

[20] (11)

1 (7b)

Eqn.

10 in the dimensionless

(v>

x

(1 _

e-2ReAS

d

)I X

4

(Jro”-’ e-l dS/oh-‘[

-2X{

+ sini&e)

-1

(8)

Quasi-equilibrium concentration If analyte does not leak through the semipermeable accumulation wall, the quasi-equilibrium concentration c* is [3] = c e-XIuOI/D = c e-x/’ 0 0

i

{=x/l

is a Reynolds number defined by the magnitude of the crossflow velocity I U, 1, channel width w, and kinematic viscosity v = p/p. By expanding Eqns. 7a and b with Eqns. Al and A2 in the Appendix, one can show that the former equations appropriately reduce to Eqn. 5 as Re -+ 0.

c*

dx

(1 _ e-2Rex/w

x

is the maximum where v,,, = -w2(8p/8z)/8~ carrier velocity expressed by Eqn. 5 (equal to 3(v)/2 and found at x/w = l/2), (v) = w - ‘~Owv dx as before and Re=

e

smh( Re )

)I

one can re-express form

Re Recoth(Re)-1

.

(1 _ e-2Rex/w

x

(7a)

or ‘=

Retention ratio In accordance with Eqn. 2, one can combine Eqns. 7 and 9 to express the flow FFF retention ratio R as

~(1

e-2ReU

d

)I

5)

(12)

where x=1/w

(13)

is the reduced zone thickness. As in other forms of FFF, X is the theoretical parameter of primary importance. Equation 12 can be evaluated analytically as e-X-’

R=2Re

(2XRe+

-X+

1))’

(1 - e-‘-l) e-Am’ hRe eRe sinh( Re) - (1 _ e-Am’)

(9)

where co is the concentration at the wall, D is the analyte diffusion coefficient and I= D/l U, I is the characteristic zone thickness, as shown in Fig. 1.

_

X

(Re coth( Re) - 1))’

(14) I

INFLUENCE

OF CROSSFLOW

HYDRODYNAMICS

ON RETENTION

RATIO

IN FLOW

which has several interesting limits. The appropriate expansions prerequisite to the evaluation of these limits are reported in the Appendix. As X approaches infinity, 1U, 1 approaches zero and R should approach one, because analyte molecules occupy all axial streamlines with equal probability. By expanding Eqn. 14 with Eqn. A3 in the Appendix, one can indeed show that R

=l

as expected. Also, as Re approaches zero, one should recover the classical equation for the retention ratio based on the parabolic velocity profile, Eqn. 5. By expanding Eqn. 14 with Eqns. A2, A4 and A5 in the Appendix, one can show that R ,im Redo =6X(coth[(2h)-‘]

(16)

-2X)

as expected. Other interesting limits merit examination. large values of Re, Eqn. A6 holds and

R

Re-m

For

[ (1 - eeA-‘)-’

- X]

(17)

which cannot be simplified further without specifying X. For Xs such that ReX B l/2 (i.e., 2ReX B l), Eqn. 17 reduces to = 2[ (1 - e_x-‘)-1

- A]

lim Re+ 00 lim Reh >3 l/2

(18)

Finally, as X approaches zero, Eqn. A4 applies (the expansion is the same whether Re or A approaches zero) and R

= 2XRe

lim h+O

Re eR’/sinh( Re) - 1 Re coth(Re) - 1 I

(19)

One observes that R varies linearly with A in this limit, as it does in other forms of FFF [21].

RESULTS

Re =

0

AND

1

I

DISCUSSION

Figure 2 is a plot of Eqn. 7a, the reduced x/w velocity u/emax, vs. the reduced coordinate for various Reynolds numbers Re. Equation 7a is plotted instead of Eqn. 7b because the dependence of u on Re for a fixed axial pressure gradient is more clearly indicated by the former equation

0.2

0.4

0.6

0.8

1

x/w Fig. 2. Plot of reduced velocity, u/u,,,,,, nate, x/w, for various Reynolds numbers,

than by the latter. 5, which applies ity clearly shifts Re increases. In duces to U=

4ReX = 2Reh+l

R

’

(15)

lim X-m

lim

1.2

165

FFF

vs. reduced Re.

coordi-

The bold curve is a plot of Eqn. as Re + 0. The maximum veloctoward the accumulation wall as the limit Re --, 00, Eqn. 7a re-

(1 - vm4nax

; lim Re+

m

and u varies linearly with x. Interestingly, Eqn. 20 is identical with the velocity profile expected in an infinitely parallel plate channel, whose contents are not subject to an axial pressure gradient but whose accumulation wall translates in the axial direction with a velocity equal to 4u,,,,/Re [27]. The physical origin of the distortion of v from a parabola is the convective component, U, au/ax, of the substantial derivative [24,28]. This term represents the convective acceleration of a fluid element in the axial direction; fluid (and entrained analyte) is decelerated near the accumulation wall and accelerated near the other wall. This representation becomes obvious, if one expresses the crossflow velocity U, as dx/dr and expresses the substantial derivative with the chain rule as au u0,x=,,,x=D,

dx

au

Dv (21)

where Du/Dt represents the substantial derivative in the axial direction [28]. Equation 7a shows that u varies inversely with Re for a fixed axial pressure gradient. The axial velocity decreases with increasing Re, because the shear stress and the friction factor near the accu-

166

J.M. DAVIS

-3 -4 -4

-3

-2

-1 log

Fig. Re. and the

0

1

3L

3. Plot of log R vs. log h for various Reynolds numbers The lower and upper bold curves are graphs of Eqns. 16 18, respectively. Other solid curves are graphs of Eqn. 14; dashed lines are graphs of Eqn. 19.

mulation wall increase with increasing crossflow [27]. One can qualitatively observe this increase in Fig. 2, which shows that the rate at which u varies with x near the accumulation wall is most rapid for large Re. Various workers have suggested that the distortion of the axial velocity profile in flow FFF can be avoided experimentally by using an average axial velocity which is much greater than the crossflow velocity. Presumably the reasoning underlying this assertion is that the substantial derivative becomes negligible in this limit. This assertion, however, is incorrect. As shown by Eqn. 7b, the spatial variation of u with x is independent of the average axial velocity (u), which simply scales u. The physical origin of this behavior is that the convective acceleration, U, au/ax, and the gradient in shear stress, /.La’u/~x’, in Eqn. 6 are both proportional to (u). One can reduce the convective acceleration, relative to the gradient in shear stress, only by reducing Re. Figure 3 is a plot of log R vs. log h for several values of Re. The two bold curves define the lower and upper bounds on R, which correspond to Eqns. 16 and 18. The remaining solid curves are graphs of Eqn. 14; the dashed lines are graphs of the linear approximation, Eqn. 19. For any value of A, R increases with increasing Re, because analyte zones increasingly are localized in a region of relatively rapid flow. For intermediate values of h (e.g., X = O.l), R actually rises above unity, i.e.,

zones move faster than the carrier. This behavior also has been predicted for the retention of analyte zones near the inner walls of annular channels, in which the velocity profiles also are skewed toward the inner wall [29,30]. As Re approaches infinity, R is always greater than one for the indicated Xs, because the analyte is polarized in the region of the most rapid flow. To determine if this flow asymmetry affects the axial transport rate of zones in flow FFF, one can calculate the percentage error between the two retention ratios, Eqns. 14 and 16. In general, this error is a function of A and Re. If one considers the limiting but important case in which R varies linearly with h, then the percentage error, PE, between these equations is

lim X+0 PE

and depends only on Re. Figure 4 is a plot of this error, which becomes substantial for Re > 1. The error is negative, because zones are moved more rapidly by the asymmetric profile than by the parabolic profile for equal A values. One can readily show, by computing the percentage error between Eqns. 14 and 16 for various Res and Xs, that Eqn. 22 represents the upper bound on this error and that the error decreases with increasing X. The threshold above which one decides that this error is unacceptably large must be chosen somewhat arbitrarily. Here, a threshold equal to 1% is

-20..

PE -4o--6O--80.-

-3

-2

-1 log

R:

1

2

Fig. 4. Plot of Eqn. 22, the percentage error PE between 14 and 16, vs. log Re. Eqn. 22 applies as h + 0.

Eqns.

INFLUENCE

OF CROSSFLOW

HYDRODYNAMICS

ON RETENTION

RATIO

adopted. The numerical solution of Eqn. 22 shows that Re must be less than about 0.030 to satisfy this criterion. It is instructive to reinterpret with this theory the published results of flow FFF experiments in OPPCs. As suggested in the Introduction, the physical properties (e.g., size, molecular weight and diffusion coefficient) of the high-molecularweight species fractionated by flow FFF in OPPCs closely agree with the predictions of Eqn. 16, which is based on the parabolic velocity profile. In other words, the theory developed here is not necessary to estimate them correctly (this is perhaps unsurprising; otherwise, the theory would have been developed before now!). A review of this literature indicated that the Res characteristic of these experiments are much smaller than the threshold value, 0.030, above which axial transport is affected. To interpret the literature, one expresses Eqn. 8 as VW2

Re=

IU, Iw/2v=

(23)

%

where vc is the volumetric crossflow rate and V, is the channel void volume. Table 1 gives values of I$, V, and w selected from several applications of flow FFF in OPPCs and the Res calculated from them with Eqn. 23. In each instance, the largest value of fc in the indicated reference is reported.

TABLE I Values of c,

V, and w for flow FFF in OPPCs

< (ml s-‘)

vo (mu

km)

0.024 0.0085 0.017 0.035 0.011 0.0034 0.067 0.056 0.027 0.056 0.032

1.8 1.84 1.68 4.2 3.90 1.85 3.88 1.38 1.28 2.63 2.44

0.0254 0.038 0.038 0.047 0.053 0.041 0.0508 0.0254 0.025 0.051 0.047

Re a

Ref.

4.3 x 10-4 3.3 x 10-4 7.3 x 10-4 9.2~10-~ 5.0x10-4 1.5 x 10-4 2.2x10-3 1.3x10-3 6.6x10-4 2.8 x 1O-3 1.4x10-3

3, 10 11 12 14 15 b 16 17 18 19 31 32

a Re was calculated from Eqn. 23. Y = 0.01 cm2 s-’ unless noted otherwise. b Y f: 0.0080 cm2 s-’ (ethylbenzene).

IN FLOW

167

FFF

(For studies based on more than one channel, the data corresponding to the largest Re are reported). Unless noted otherwise, the value of v used in these calculations is that of water, 0.01 cm* s-l (20° C). The Res so calculated only vary from 1.5 X lop4 to 2.8 X 10m3. Even the largest Re is about ten times smaller than the threshold value, and the error between Eqns. 14 and 16 is only ca. 0.1%. It is unsurprising, therefore, that these data are well explained by Eqn. 16. It is perhaps worth noting that these axial transport rates are unaffected by crossflow hydrodynamics, not only because ( U, 1 (or tic) is small, but because the channel width w is small. As shown by Eqns. 8 and 23, Re depends on both I U, I and w. In general, w is kept small in all forms of FFF to avoid many undesirable effects, including convection, zone dilution and large non-equilibrium plate heights. Clearly, another reason for using small ws in flow FFF is to reduce Re. The same volumetric crossflow rates reported in Table 1 easily could cause anomalous transport in thicker channels. Under what conditions does crossflow hydrodynamics affect axial transport in flow FFF? To answer this question, Eqn. 8 is now reintepreted as Re=

IU,Iw/2v=

IU,IwD/2Dv=D/2Xv

(24)

This equation defines the ratio, D/X, for any value of Re. In physical terms, Eqn. 24 states that the attainment of a specific value of X for an analyte with a large (small) D requires the use of a large (small) crossflow, which is proportional to Re. For v = 0.01 cm* s-r, Eqn. 24 predicts that Re will be less than 0.030 (and Eqns. 14 and 16 will differ by less than 1%) provided that the ratio D/X is less than 6 X lop4 cm* s-r. Figure 5 is a plot of log X vs. log D for this threshold ratio (in this plot, D is implicitly ratioed against a reference diffusion coefficient equal to one to obtain a dimensionless result). The domain in which anomalous transport is expected is indicated by the shaded triangle in the lower portion of the figure. From this plot, one can infer that X must be unusually small before crossflow hydrodynamics significantly affects R. In fact, one can set fairly rigorous limits to A and R, above which axial transport will not be

J.M.

168

log

; -4 -5 -6 -7 -10

-9

-6 log

-7

-6

-5

D

Fig. 5. Plot of log h vs. log D for D/X = 6.0~10~~

cm2 s-‘.

affected by crossflow. By doing so, one can determine conditions under which the simpler expression for R (Eqn. 16) applies, regardless of the molecular weight of the analyte. One can impose the most restrictive (and hence conservative) constraint on low-molecular-weight analytes with large Ds, which require high crossflows to form thin polarization layers. An upper limit to D in liquids is about 1.0 X lop5 cm2 s-l, which corresponds to an equivalent Stokes radius of 2.1 A in water at 20 “C. If this value is assigned to D and v is assigned the value 0.01 cm* s-l, then Eqn. 24 predicts that Re will be less than 0.030 only if h is greater than 0.017. From this lower limit to A, one can now estimate a lower limit to R. Because Eqns. 14 and 16 differ by less than 18, when Re < 0.030, one can approximate R by the latter equation, which predicts that R > 0.098 = 0.10 for X values greater than 0.017. In accordance with Eqn. 1, the analyte retention volume V, consequently must be smaller than V,/R = lOI& In other words, zones which elute within about ten void volumes should not be subject to anomalous transport. A survey of the literature on flow FFF in OPPCs indicates that virtually all species elute within this constraint. Indeed, many practical problems (e.g., irreversible adsorption and immobilization by crossflow) arise if one attempts to exceed this elution range [11,14]. Conclusions One can conclude that anomalous transport in flow FFF is not to be expected for species which

DAVIS

elute from OPPCs within ten void volumes. Within this elution range, the molecular weight (or size) of the analyte is immaterial, and the concern about the axial transport rates of low-molecularweight analytes stated in the Introduction is unfounded. To induce deliberately anomalous transport in channels of conventional widths, one must use volumetric flow-rates which are ten or more times greater than those in common use today. Although this study suggests that lateral crossflow does not affect axial transport in ideal flow FFF channels, anomalous transport certainly may be observed in practice. This behavior may be found, for example, in channels with very small ws, whose uniform construction challenges present technology. The word “ideal” perhaps should be emphasized, because a non-ideal (e.g., non-planar) accumulation wall could convert a fraction of the crossflow into axial flow, i.e., U could vary with z. In this more complicated case, the term XJ/az could not be dropped from Eqn. 6, and u would vary with both x and z, as it does in asymmetric and hollow-fiber flow FFF [4,8]. Perhaps the most conservative conclusion one can draw from this work is that anomalous transport in a well designed OPPC will not originate from the convective acceleration of axial flow. The author thanks J. Calvin Giddings of the University of Utah for bringing ref. 27 to his attention.

REFERENCES 1 K.D. Caldwell, Anal. Chem., 60 (1988) 959A. 2 J.C. Giddings, Chem. Eng. News, 66 (1988) 34. 3 J.C. Giddings, F.J. Yang and M.N. Myers, Anal. Chem., 48 (1976) 1126. 4 K.-G. Wahlund and J.C. Giddings, Anal. Chem., 59 (1987) 1332. 5 K.-G. Wahlund and A. Lit&n, J. Chromatogr., 461 (1989) 73. 6 A. Lit&n and K.-G. Wahlund, J. Chromatogr., 476 (1989) 413. 7 H.-L. Lee, J.F.G. Reis, J. Dohner and E.N. Lightfoot, AIChE J., 20 (1974) 776. 8 J.A. Jbsson and A. Carlshaf, Anal. Chem., 61 (1989) 11. 9 A. Carlshaf and J.A. Jijnsson, J. Chromatogr., 461 (1988) 89.

INFLUENCE

OF CROSSFLOW

HYDRODYNAMICS

ON RETENTION

10 J.C. Giddings, F.J. Yang and M.N. Myers, Science, 193 (1976) 1244. F.J. Yang and M.N. Myers, J. Virol., 21 11 J.C. Giddings, (1977) 131. 12 J.C. Giddings, F.J. Yang and M.N. Myers, Anal. Biochem., 81 (1977) 395. 13 J.C. Giddings, G.C. Lin and M.N. Myers, J. Liq. Chromatogr., 1 (1978) 1. 14 K.G. Wahlund, H.S. Winegamer, K.D. Caldwell and J.C. Giddings, Anal. Chem., 58 (1986) 573. 15 S.L. Brimhall, M.N. Myers, K.D. Caldwell and J.C. Giddings, J. Polym. Sci., Polym. Lett. Ed., 22 (1984) 339. 16 J.C. Giddings, G.C. Lin and M.N. Myers, J. Colloid Interface Sci., 65 (1978) 67. 17 R. Beckett, 2. Jue and J.C. Giddings, Environ. Sci. Technol., 21 (1987) 289. 18 SK. Ratanathanawongs and J.C. Giddings, J. Chromatogr., 467 (1989) 341. 19 B.N. Barman, M.N. Myers and J.C. Giddings, Powder Technol., 59 (1989) 53. 20 J.C. Giddings, J. Chem. Phys., 49 (1968) 81. 21 M.E. Hovingh, G.E. Thompson and J.C. Giddings, Anal. Chem., 42 (1970) 195. 22 J.C. Giddings, Y.H. Yoon, K.D. Caldwell, M.N. Myers and M.E. Hovingh, Sep. Sci., 10 (1975) 447. 23 J.C. Giddings and M.R. Schure, Chem. Eng. Sci., 42 (1987) 1471. 24 R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960, p. 84. 25 J.C. Giddings, K.D. Caldwell, J.F. Moellmer, T.H. Dickinson, M.N. Myers and M. Martin, Anal. Chem., 51 (1979) 30. 26 J.J. Gunderson, K.D. Caldwell and J.C. Giddings, Sep. Sci. Technol., 19 (1984) 667. 27 F.M. White, Viscous Fluid Flow, McGraw-Hill, New York, 1974, pp. 151-152. Introduction to Fluid 28 R.W. Fox and A.T. McDonald, Mechanics, Wiley, New York, 2nd edn., 1973. 29 J.M. Davis and J.C. Giddings, J. Phys. Chem., 89 (1985) 3398. 30 J.M. Davis, Anal. Chem., 58 (1986) 161.

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IN FLOW

FFF

169

31 J.C. Giddings, X. Chen, K.-G. Wahlund and M.N. Myers, Anal. Chem., 59 (1987) 1957. 32 X. Chen, K.-G. Wahlund, and J.C. Giddings, Anal. Chem., 60 (1988) 362.

APPENDIX

The following series expansions and approximations are useful in the evaluation of several limits of velocity u and retention ratio R: _ e-2Re~/w)

eR’(l

-2c+2Re[G-($)2]

sinh( Re) lim Re-6

Re2)

+ 0(

Re coth( Re) - 1 = Re2/3

+ 0( Re3)

lim Re-0

(Al) (‘42)

e-x-’

-x-l/2+0(x-‘)

(A3)

(1 - e-‘-‘) lim X+m

(2Reh

= 1 - 2ReX + 4Re2A2

+ l)-’

liy,;ei*

Ooor

(W

+ O[(ReX)3] Re eRp sinh( Re )

= 1 + Re + Re2/3

+ 0( Re3)

(W

lim Re-0

ReeRe

=2Re

(Ah)

sinh( Re) lim Re-cc

where O(a) (Y.

represents

a truncation

error of order