Relative Velocity Profile and Flow-Rate in Sedimentation Field-Flow Fractionation Michel Martin Ecole SupCrieure de Physique et Chimie Industrielles, Laboratoire de Physique et MCcanique des Milieux HBtCrogbnes (URA CNRS 857), 10, rue Vauquelin. 75231 Paris Cedex 05, France

Key Words:

Sedimentation field-flow fractionation Curved channels Curved Poiseuille flow Flow-rate Velocity profile

Summary Although the classical retention theory is used for interpreting data or optimizing separations in sedimentation field-flow fractionation (SedFFF), as in most other field-flow fractionation techniques, the assumption of a parabolic flow profile on which this theory is based is not rigorously correct in SedFFF because of the curvature of the channel walls. In order to examine quantitatively the influence of this effect, the relative velocity profile in SedFFF is obtained by solving the Navier-Stokes equation in cylindrical coordinates. Dis- crepancies found in the literature about the definition of the mean velocity in such channels are discussed. Relationships between mean velocity, flow-rate and pressure gradient are given. Approximating the velocity profile by a third-degree polynomial of the radial coordinate which provides the same slope as the exact profile at a reference wall, for small values of 6, the curvature ratio (ratio of the channel thickness to the mean curvature radius), shows that the adjustable parameter of the approximate profile, v, is equal to f 6/3, the sign depending on whether the reference wall is the inner or outer wall. The curvature ratio appears to be a good indicator of the error made on retention when using the straight channel approximation in retention theory. The error is quite small for typical SedFFF channels. It may have to be taken into account for precise determinations if thicker channels and/or miniaturized systems are used.

1 Introduction

Among the various field-flow fractionation (FFF) te.chniques currently implemented, sedimentation field-flow fractionation (SedFFF) appears to be especially well suited for the selective separation of a large variety of colloidal materials. The SedFFF channel is enclosed in the gap between two coaxial cylindrical walls rotating at the same velocity around their symmetry axis. A liquid flow carries the species along the channel in the azimuthal direction (i. e. in the direction perpendicular to planes containing the symmetry axis). The centrifugal force resulting from the rotation leads to a non-uniform distribution of the concentration of a colloidal species within the channel cross-section. The combination of the non-uniformities of the concentration profile and of the flow velocity profile leads to the retention of the species, and eventually to its separation from other differently retained species [l].

Dedicated to Professor C.A. Cramers on the Occasion of his 60th Birthday

The optimization of the separation or the characterization of the sample species from their retention time in the FFF channel requires a precise knowledge of both the concentration and flow distributions. Although the classical retention theory, established for straight channels, is frequently used for this purpose, in principle, it does not strictly apply to SedFFF because of the curvature of the channel walls. In such channels indeed, even in steady operating conditions, the velocity vector of a fluid element is never constant because its direction is changing with time and the velocity profile in a cross-section differs from that in a straight channel. Obviously, when the channel thickness is much smaller than the curvature radius of the walls, the effect of the curvature will marginally be perceived and the flow behavior in a SedFFF channel will be very close to that in a straight channel. Never- theless, it is important to know in which conditions the straight channel approximation can be used in SedFFF and to have a quantitative estimate of the error resulting from this approxima- tion. This is the purpose of the present study.

Furthermore, in order to compute the degree of retention by means of a dimensionless factor, such as the retention ratio, the relative velocity profile, i.e. the ratio of the velocity profile to the mean carrier velocity, is neededrather than the absolute velocity profile. However, divergences are found in the literature about the defi- nition of the mean velocity in flow configurations corresponding to that of SedFFF. The basis for the definition of the mean velocity is discussed and the relationships between mean velocity, flow- rate and pressure gradient in SedFFF are also addressed in the following.

2 Theory

2.1 Carrier Velocity Profile

Let Ri and R, be the radii of, respectively, the inner and outer walls of the SedFFF channel, r the radial coordinate (distance from the axis of symmetry). In every plane containing the axis of symmetry, the cross-section of the SedFFF channel is rectan- gular with dimensions 6, the channel breadth (in the direction parallel to the axis of rotation), and w = R, - Ri, the channel thickness. One considers the situation in which b >> w, which prevails in SedFFF (blw is generally larger than 75). If the flow-rate is lower than a certain threshold, there is no secondary

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Relative Velocity Profile and Flow-Rate in Sedimentation FFF

b

Y

axis

P W

Figure 1. Sketch of the SedFFF channel with the cylindrical coordinate system (r , 8, y). The channel is rotating around the axis of symmetry and the carrier flows in the azimuthal (6) direction.

flow [2] , except in a small region near the small edges of the channel [3]. In typical operating conditions, this threshold is not reached in SedFFF [4]. Therefore, apart from this small region, the flow velocity vectors are everywhere perpendicular to the channel cross-section. There is flow only in the azimuthal direc- tion, 0 (see Figure 1).

In these conditions, the steady isothermal flow of an incompress- ible Newtonian fluid is given by the following form of the Navier-Stokes equation in cylindrical coordinates [5]:

where vg is the flow velocity along the azimuthal direction, q the camer viscosity, and p the pressure. By double integration of equation (1) with the assumption of a constant azimuthal pressure gradient, dp/d0, one gets [6]:

1 aP B ve = --[rlnr + A r + -1 2 q ae r

The constants A and B are determined using the boundary con- ditions, vg = 0 for r = R i and r = R,:

(3) R:lnRo - R f l n R i R ~ R : R, A = - andB = In-

R: - Rf R o - Rj R i

These equations have been used or re-derived by several authors working with azimuthal flow in cylindrical channels although they may appear different forms because different variables were used to describe the geometry of the system [see, for instance, refs. 7-91.

2.2 Basic Definitions of the Flow Rate and of the Mean Carrier Velocity

However, if a general agreement is found about the velocity profile, vg(r), discrepancies between various authors are found as concerns the expression of the mean azimuthal velocity, <vg>, and the related expression of the flow rate, Q. Yet, in order to compute the retention ratio in SedFFF, a correct expression of the relative velocity profile, vg(r)/< vg>, is needed. It seems that this arises from a confusion between situations in which the flow streamlines are perpendicular to the channel symmetry axis, as in SedFFF, and situations in which the flow is parallel to the symmetry axis, as has been proposed for FFF in annular channels (annular FFF) [lo]. In both cases, the flow-rate is obtained by integrating the volume element of fluid flushed per unit time, i.e. r d0 d r dy / dt where y is the coordinate in the direction parallel to the channel axis, over the two directions of the cross-section perpendicular to the channel flow. For flow in they direction, as in annular FFF, this double integration is thus performed over the transversal coordinates rand 0, which gives:

Q (annular FFF) = 5 I F d r dY d0 dt (0) ( r )

i.e., identifying dyldt as vy :

Q (annular FFF) = 271 r vy r d r Ri

(4)

In SedFFF, the fluid flows, as noted above, in the 0 direction, and the integration must be performed over the transversal coordi- nates r and y, giving:

d0 Q(SedFFF)=J r -d rdy

(Y) ( r ) dt

or, noting that r d0ldt = vg and that vg does not depend on y under the above assumptions:

Q (SedFFF) = b r vg dr Ri

(7)

The mean flow velocity is then defined as the flow-rate divided by the cross-sectional area perpendicular to the flow streamlines, i.e. for annular FFF:

R, ) r d r

. I ,

- R, (8) - 2 2 R, - Ri

<vy> (annular FFF) = 5 r d r d 0 (e) ( r )

and, for SedFFF:

The discrepancies mentioned above arise from the incorrect use of v(r) r dr as the integrand in the above equations in the case of a purely azimuthal flow instead of v(r) dr. Most of the authors dealing with azimuthal flow between two coaxial cylinders used

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Relative Velocity Profile and Flow-Rate in Sedimentation FFF

the correct definition of the mean velocity [6, 9, 11-151. How- ever, some authors applied, for azimuthal flow, the definition corresponding to axial flow.

2.3 Flow Rate and Pressure Gradient in Sedimentation Field- Flow Fructionation

By combining equations (2) and (9) and integrating, one gets:

which, in combination with equation (7), provides the relationship between the flow-rate and the pressure gradient:

I, ~ ."- _I

2 q 1 (Rt - Rf)2

(1 1) It is easily verified that, as the gap, w = Ro - Ri, becomes much smaller than the radius of curvature, Rc = (Ro + Ri)/2, equation (10) becomes:

which, as expected, is the mean velocity of a Poiseuille flow between two infinite parallel plane walls, when Re at3 is identified with az, the differential displacement length along the flow streamlines, and when w/Rc vanishes.

It is worth noting that, in the SedFFF channel configuration, a constant flow-rate corresponds to a constant angular pressure gradient, dp/dt3, rather than to a constant azimuthal pressure gradient, apl(rdt3), since r varies from the inner to the outer wall. Accordingly, a streamline near the inner wall is shorter and undergoes a larger azimuthal pressure gradient than a streamline near the outer wall. This is why the azimuthal velocity is larger near the inner wall than near the outer wall, at equal distances from the corresponding wall. In other words, the position of the maximum velocity, which is on the centerline in straight FFF channels, is shifted toward the inner wall in SedFFF channels. This can be easily verified. Indeed, one finds, by derivation of equation (2), that the sign of the velocity gradient, dvgldr, at the centerline ( r = Re), is the same as that at r = Ro and opposite to that at r = Ri.

It is also interesting to note that the cross-sectional area of the outer half of the SedFFF channel, i.e. the cross-sectional area between the cylinders of radii Re and Ro and of height b, is the same as the cross-sectional area of the inner half (between cyl- inders of radii Ri and Re) and equal to bw/2. This implies that, if the azimuthal velocity, vg, were constant, i.e. independent of r instead of being given by equation 2, the flow-rate would be the same in these two halves and equal to vgbw/2. Still, the volume of the outer half is larger than that of the inner half. This means that, in this hypothetical constant- vg situation, it would take a longer time to flush the outer half than the inner half, or to travel along the channel near the outer wall than near in the inner wall. This situation is not paradoxical. It is similar to that of racers

running along the curved track of a stadium. Nevertheless, this effect has an influence on the pertinent definition of the retention ratio, as will be discussed in a forthcoming publication [ 161.

2.4 Relative (Azimuthal) Velocity Profile in Sedimentation Field- Flow Fractionation

The relative velocity profile, along the azimuthal direction, which is required for the computation of the retention ratio in SedFFF, is obtained from combination of equations (2) and (10):

R2lnRi - Ri lnR, 1 Rf Ri Ro 2 ( r l n r + r + -

v0 R i - Rf r R; - R~ - -

One expects that, as the channel thickness, w, becomes small in comparison to the average radius of curvature, Rc, this relative velocity will closely approach the parabolic profile obtained in a straight channel.

Writing the relative profile of equation (13) in terms of the transversal coordinate in the field direction, x = Ro - r, and of the curvature ratio, 6, defined as :

and taking the polynomial development of the resulting expres- sion for small values of 6 and of x/ Rc, limited to the first degree in 6, one obtains after lengthy but straightforward transfonna- tions:

v0 <vg> W W

This result can be shown to be identical to that previously obtained using a different coordinate system [17].

In the past, it has been found useful to characterize a velocity profile slightly deviating from the ideal parabolic shape by a third-degree polynomial in x/w containing an adjustable parame- ter, v, defined in such a way that the slope of the approximate profile equals that of the true profile for x = 0 [ 181. Then, this slope is (1 + v) times that of the parabolic profile, for the same flow-rate. In this case, the approximate velocity profile is given by:

Such an equation is conveniently used in thermal FFF to account for the distortion of the flow profile due to the temperature dependence of the carrier viscosity [19]. The comparison of

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Relative Velocitv Profile and Flow-Rate in Sedimentation FFF

equations 15 and 16 indicates that, in the case of SedFFF with channels of small curvature ratios, the v parameter is given by:

6 l w v(SedFFF) = -- = -- 3 3 Rc

Obviously, when 6 = 0, one retrieves the classical parabolic velocity profile for a two-dimensional Poiseuille flow. As ex- pected, with the origin of the x axis selected at the outer wall, v is negative, which indicates, as mentioned above, that the slope of the relative velocity profile at the outer wall is lower than that it would have in a straight channel. It is easy to show that if the origin of the x axis were selected at the inner wall, v would be positive and equal to 613. This is consistent with the fact that the position, x, of the maximum of the azimuthal velocity profile is shifted from the centerline toward the inner wall by an amount which was shown to be approximately equal to 6 ~ 1 1 2 [20].

It must be clear that this deviation of the flow profile from the parabolic profile in SedFFFis not due to the rotation of the channel but only to its curvature. Indeed, the profiles given by equations (1 3) or (1 5 ) do not depend on the rotation velocity of the channel. The assumption used to determine the integration constants A and B in equation (2) is that the carrier velocity at the two walls is zero. Since the no-slip condition dictates that the fluid in contact with the walls moves at the same velocity as the walls, this assumption implies, first, that the two walls rotate at the same velocity (as is the case for SedFFF experiments, but not for Taylor-Couette experiments), and, second, that the reference azimuthal velocity is the velocity of the walls.

3 Conclusion

Although there is not a unique way to compare an exact relative velocity profile, such as the profile given by equation (13) in SedFFF, with a reference profile, such as the parabolic profile obtained for flow in straight channels, and to describe their relative difference by a number, the adjustable parameter v, arising in equation (1 6) can be considered as a very convenient one. Indeed, from its definition, it is equal to the relative difference of the slopes of the velocity profiles near one wall (the wall at x = 0). Since the migration velocity of a species highly compressed near this wall by the applied centrifugal force is, with a good approximation, proportional to the slope of the velocity profile at the wall, the adjustable parameter v, which can be called the flow distortion parameter, reflects the relative variation of the zone velocity of such a species and, hence, of the time it requires to travel a certain distance. In the case of SedFFF, the v parameter is seen to equal to f 613, the sign depending on whether the reference (accumulation) wall is the inner or outer wall. The curvature ratio, 6, appears to be a good indicator of the error resulting from the straight channel approximation in SedFFF.

In typical SedFFF channels (w = 0.25 mm, R, = 15 cm, for instance), 6 is of the order of lop3. This small correction can, for most purposes, be neglected. However, the present analysis in-

dicates that it may not be so if the curvature ratio is significantly increased, i.e. if the radius of curvature is reduced for the sake of miniaturization of the ‘SedFFF system andor if the channel thickness is increased in order either to increase the sample throughput or to set up the various spacer layers contained in a centrifugal SPLITT (split-flow thin) system [21,22].

Acknowledgments

Fruitful discussions with Joe M. Davis (Southern Illinois University, Carbondale, IL, USA), J. Eduardo Wesfreid (Ecole SupCrieure de Physique et de Chimie Industrielles, Paris, France) and Innocent Mutabazi (University of Le Havre, France) are grateful acknowledged.

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