Splitter Imperfections in Annular Split-Flow Thin Separation Channels

Oct 28, 2003 - P. Stephen Williams,*,† Keith Decker,‡ Masayuki Nakamura,‡,§ Jeffrey J. Chalmers,‡. Lee R. Moore,† and Maciej Zborowski†. ...
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Anal. Chem. 2003, 75, 6687-6695

Splitter Imperfections in Annular Split-Flow Thin Separation Channels: Experimental Study of Nonspecific Crossover P. Stephen Williams,*,† Keith Decker,‡ Masayuki Nakamura,‡,§ Jeffrey J. Chalmers,‡ Lee R. Moore,† and Maciej Zborowski†

Department of Biomedical Engineering, The Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, Ohio 44195, and Department of Chemical Engineering, The Ohio State University, 140 West 19th Avenue, Columbus, Ohio 43210

The separation performance of a split-flow thin (SPLITT) separation device depends on uniformity of channel thickness and the precise placement of the flow splitters at fixed distances between the channel walls. The observation of nonspecific crossover, that is, the transport of sample materials across the channel thickness without the influence of an applied field, has routinely been taken to indicate the presence of irregularities in splitter shape or placement. Computational fluid dynamics software may be used to predict the influence of splitter imperfections on nonspecific crossover, where it is assumed that sample transport is by convection alone. A previous study has shown how small inlet splitter imperfections can account for the relatively low levels of nonspecific crossover observed with typical annular SPLITT devices. This study, however, could not distinguish between the possible sources of nonspecific crossover; hydrodynamic lift or shear-induced diffusion could have contributed. To confirm the validity of the computational approach, a series of experiments has been carried out on a channel having a deliberately and severely bent splitter. Nonspecific crossover was measured for a range of inlet and outlet flow rate ratios, with the bent splitter placed at both the channel inlet and outlet. The severity of the splitter distortion was sufficient to produce significant nonspecific crossover over a wide range of flow conditions. Good agreement was found between experiment and prediction based on computational fluid dynamics, with experiment generally showing only slightly higher crossover than prediction. The quantitative agreement for this extreme case suggests that the contribution to nonspecific crossover due to geometrical imperfections can be well described using computational fluid dynamics. The quadrupole magnetic flow sorter (QMS) has been developed for the enrichment, depletion, or isolation of various biological cell types for applications in cellular therapy and * Corresponding author. Fax: 216-444-9198. E-mail: [email protected]. † The Cleveland Clinic Foundation. ‡ The Ohio State University. § Present address: 3M Corporate Technology Center, 3M Center, Building 218-1S-05, St. Paul, MN 55144-1000. 10.1021/ac030152n CCC: $25.00 Published on Web 10/28/2003

© 2003 American Chemical Society

biotechnology.1-10 The mechanism of sorting follows the principles of split-flow thin (SPLITT) separation, developed by Giddings and co-workers (see, for example, refs 11-26). In these systems, separation of materials is achieved within a laminar flow of fluid in a thin enclosed channel of high cross-sectional aspect ratio. In the common transport mode of operation, the sample suspension is introduced to the channel as a thin lamina close to one of the walls. A field is maintained across the channel thickness, perpen(1) Zborowski, M.; Moore, L. R.; Sun, L.; Chalmers, J. J. In Scientific and Clinical Applications of Magnetic Carriers: An Overview, Ha¨feli, U., Schu ¨tt, W., Teller, J., Zborowski, M., Eds.; Plenum Press: New York, 1997; pp 247-260. (2) Zborowski, M.; Williams, P. S.; Sun, L.; Moore, L. R.; Chalmers, J. J. J. Liq. Chromatogr., Relat. Technol. 1997, 20, 2887-2905. (3) Chalmers, J. J.; Zborowski, M.; Sun, L.; Moore, L. R. Biotechnol. Prog. 1998, 14, 141-148. (4) Sun, L.; Zborowski, M.; Moore, L. R.; Chalmers, J. J. Cytometry 1998, 33, 469-475. (5) Williams, P. S.; Zborowski, M.; Chalmers, J. J. Anal. Chem. 1999, 71, 37993807. (6) Zborowski, M.; Sun, L.; Moore, L. R.; Williams, P. S.; Chalmers, J. J. J. Magn. Magn. Mater. 1999, 194, 224-230. (7) Zborowski, M.; Sun, L.; Moore, L. R.; Chalmers, J. J. ASAIO J. 1999, 45, 127-130. (8) Hoyos, M.; Moore, L. R.; McCloskey, K. E.; Margel, S.; Zuberi, M.; Chalmers, J. J.; Zborowski, M. J. Chromatogr., A 2000, 903, 99-116. (9) Moore, L. R.; Rodriguez, A. R.; Williams, P. S.; McCloskey, K.; Bolwell, M.; Nakamura, M.; Chalmers, J. J.; Zborowski, M. J. Magn. Magn. Mater. 2001, 225, 277-284. (10) Hoyos, M.; McCloskey, K. E.; Moore, L. R.; Nakamura, M.; Bolwell, B. J.; Chalmers, J. J.; Zborowski, M. Sep. Sci. Technol. 2002, 37, 745-767. (11) Giddings, J. C. Sep. Sci. Technol. 1985, 20, 749-768. (12) Springston, S. R.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1987, 59, 344-350. (13) Giddings, J. C. Sep. Sci. Technol. 1988, 23, 119-131. (14) Levin, S.; Myers, M. N.; Giddings, J. C. Sep. Sci. Technol. 1989, 24, 12451259. (15) Levin, S.; Giddings, J. C. J. Chem. Technol. Biotechnol. 1991, 50, 43-56. (16) Gao, Y.; Myers, M. N.; Barman, B. N.; Giddings, J. C. Part. Sci. Technol. 1991, 9, 105-118. (17) Williams, P. S.; Levin, S.; Lenczycki, T.; Giddings, J. C. Ind. Eng. Chem. Res. 1992, 31, 2172-2181. (18) Giddings, J. C. Sep. Sci. Technol. 1992, 27, 1489-1504. (19) Fuh, C. B.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1992, 64, 31253132. (20) Williams, P. S. Sep. Sci. Technol. 1994, 29, 11-45. (21) Fuh, C. B.; Myers, M. N.; Giddings, J. C. Ind. Eng. Chem. Res. 1994, 33, 355-362. (22) Zhang, J.; Williams, P. S.; Myers, M. N.; Giddings, J. C. Sep. Sci. Technol. 1994, 29, 2493-2522. (23) Fuh, C. B.; Giddings, J. C. Biotechnol. Prog. 1995, 11, 14-20. (24) Fuh, C. B.; Chen, S. Y. J. Chromatogr., A 1998, 813, 313-324. (25) Fuh, C. B.; Chen, S. Y. J. Chromatogr., A 1999, 857, 193-204. (26) Fuh, C. B. Anal. Chem. 2000, 72, 266A-271A.

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Figure 1. (a) Schematic showing cross-sectional view of a SPLITT separation channel. ISS and OSS represent the virtual inlet and outlet splitting surfaces described in the text. (b) Schematic of an annular SPLITT channel in which the field acts to drive sample materials radially outward.

dicular to the direction of flow. As the sample suspension passes along the channel, the suspended materials are driven across the channel thickness due to their interaction with this applied field. When some of the suspended components interact more strongly with the field than others, they migrate more quickly across the channel thickness. The sample components thereby become selectively distributed across the channel thickness as they are carried along by the fluid flow. The separation of suspended sample components is completed by dividing the fluid into two fractions at the outlet end of the channel, the division being made at some predetermined optimal distance between the channel walls. The optimal distance is determined by the relative rates of cross-channel migration of the components to be separated. The faster migrating components are thereby isolated into one fraction and the slower components into the other. A thin physical flow splitter mounted parallel to the channel walls divides the fluid, and the two resulting streams exit at outlets on each side of this splitter. It was mentioned above that the sample suspension must be initially confined to a thin lamina region close to one of the walls. This is achieved using a similar flow splitter, separating two inlets to the channel. The thin sample lamina is obtained by introducing unbalanced fluid flow rates at the two inlets. The sample suspension flow rate is kept relatively low, and a higher flow rate of pure suspending medium is introduced to the other inlet on the opposite side of the inlet splitter. The inlet splitter serves to smoothly merge the two inlet streams. When conditions have been optimized, a separation may be carried out in continuous mode (referred to as continuous SPLITT fractionation, or CSF), with the sample being introduced steadily at one inlet and the two fractions being collected at the outlets. 6688

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The principles of operation are illustrated by the schematic shown in Figure 1a. The sample suspension is input at inlet a′ adjacent to wall A, and the higher flow of pure suspending medium at inlet b′ adjacent to wall B. The merging flows lie to each side of a virtual surface, known as the inlet splitting surface (ISS), within the overall channel flow. In the figure, this is seen to bend from the downstream edge of the inlet splitter toward wall A in the region where the two inlet flows merge. It remains at a constant distance from the wall along the remainder of the channel where the flow is fully developed. The sample suspension is initially confined to the thin laminar region between the ISS and the wall by the convective flow. A second virtual surface known as the outlet splitting surface (OSS) extends back from the outlet splitter, dividing the fluid elements that pass to each side of the splitter. The region between the ISS and OSS is known as the transport lamina. By definition, fluid streamlines do not cross either of the virtual surfaces. Leaving aside the possible influence of particle-particle interaction, a suspended particle can cross the ISS or OSS only by its interaction with the applied field, giving rise to a lateral force on the particle. All those suspended particles that migrate quickly enough that they cross the transport lamina and the OSS before reaching the outlet splitter are carried to outlet b. Those that interact to a lesser extent, such that their migration is insufficient to carry them beyond the OSS, exit at outlet a. A separation of a bimodal mixture is shown in Figure 1a, where one component migrates very slowly and exits at outlet a, while the other migrates more quickly and exits at outlet b. Figure 1b shows a schematic of an annular SPLITT channel in which the field interacts selectively with sample components, driving them radially outward as they are carried downward through the channel.

In the case of the QMS, the biological cells of interest are labeled with immunospecific colloidal magnetic particles. Nanometer-sized magnetite or maghemite particles coated with dextran, and conjugated to specific antibodies, are commonly used. Labeled cells, being paramagnetic, migrate radially outward in a quadrupole magnetic field. To exploit this radial migration pattern, an annular separation channel is mounted coaxially within the quadrupole field. The inlet and outlet flow splitters are cylindrical and are mounted inside the annular channel, coaxial with the channel and magnet. Specially constructed manifolds carry the fluids to and from the pairs of inlets and outlets. The currently used manifolds employ rubber O-rings to hold the cylindrical flow splitters in place and to maintain a seal between the fluid streams. We recently presented a study27 of the influence of small imperfections in the inlet flow splitter of an annular SPLITT channel on nonspecific crossover. Nonspecific crossover refers to the transport of suspended material across the OSS to exit at outlet b, without any interaction with the applied field. Nonspecific crossover therefore interferes with, and degrades, the selective specific crossover that is caused by differing strengths of interaction of sample components with the field. There are several possible contributing mechanisms for nonspecific crossover, as discussed in this previous article.27 Some are exacerbated by high suspension concentrations and may be classified as overloading effects. These effects may be reduced simply by reducing the sample suspension concentration. They include shear-induced diffusion28,29 and viscous interaction between nonmigrating and migrating particles. This is not the case for crossover caused by physical imperfections in flow splitters, which have a direct effect on the flow pattern and therefore on the convective transport of entrained particles. Lift forces may be expected to have negligible influence on nonspecific crossover, with the possible exception of very dense particles in an upward fluid flow.20 This is because lift forces generally tend to have significant influence on particles that are in close proximity to bounding walls and would have very little influence on particle transport across splitting surfaces further away from the walls. Great care is taken in the construction and assembly of the components of the annular SPLITT channels, yet small deviations from the ideal geometry do occur. The significant wear, or break, of a rubber O-ring, or the partial impedance to flow on one side or other of a flow splitter, can result in very poor performance that is easily detected by the resulting gross nonspecific crossover. The inadvertent bending of a splitter, or the improper seating and alignment of a splitter in an inlet or outlet manifold, would also be signaled by high nonspecific crossover. This has led to the practice of measuring nonspecific crossover as a routine check of channel integrity. In the case of QMS, measurement is carried out using biological cell-sized polystyrene (PS) particle standards. The channel is removed from the quadrupole magnet to eliminate forces arising out of the slight magnetic property of the PS beads relative to the aqueous suspending fluid. The PS particle standards are used rather than biological cells because cells may aggregate or adhere to channel walls and thereby interfere with the measurement of nonspecific crossover. (27) Williams, P. S.; Moore, L. R.; Chalmers, J. J.; Zborowski, M. Anal. Chem. 2003, 75, 1365-1373. (28) Leighton, D.; Acrivos, A. J. Fluid Mech. 1987, 177, 100-131. (29) Leighton, D.; Acrivos, A. J. Fluid Mech. 1987, 181, 415-439.

As mentioned above, this initial study27 was carried out for relatively small distortions of the inlet splitter; specifically, a 100µm offset of the splitter from the channel axis without any additional distortion and an elliptical distortion of the splitter such that the difference between major and minor axes was 200 µm without any offset from the channel axis. Computational fluid dynamics (CFD) software was used to predict the fluid flow pattern throughout the system. The nonspecific crossover was predicted by establishing the shape and position of the distorted ISS within the annular channel at a point where the flow was fully developed. (The midway point along the length of the channel was arbitrarily selected.) As mentioned above, fluid streamlines fall to each side of the ISS and do not cross it. Entrained particles that do not interact with an applied field are therefore assumed not to cross the ISS. However, if the distorted ISS falls to the far side of the OSS on any fraction of the annular circumference, then entrained particles will be carried into the region of channel flow that is directed to outlet b. The fully developed laminar fluid velocity profile for a Newtonian fluid in an annulus is known (see ref 30). Assuming a perfect outlet splitter, this allows the calculation of the position of the OSS for any outlet flow rate ratio.5,27 The volumetric crossover flow rate V˙ cross of fluid originating at inlet a′, along with its entrained particles, is then easily calculated for any outlet flow rate ratio. This is obtained by integrating over the fraction of the fluid velocity profile that lies between the OSS and the ISS where the radial distance of the ISS from the channel axis, rISS, exceeds that of the OSS, rOSS. The fractional nonspecific crossover is given by the ratio of V˙ cross to the sample flow rate V˙ (a′). The degree of nonspecific crossover exhibited by typical wellconstructed annular SPLITT channels is found to be similar to the predictions for the small distortions considered. This fact alone indicates the good quality of channel construction. However, the agreement does not indicate the specific factors contributing to the observed crossover. It is possible that the channels have a construction that is better (i.e., closer to perfect) than those considered for the simulations, and there is some contribution from other mechanisms. This uncertainty takes on significance in view of the fact that a new one-piece design for the channel inlet and outlet manifolds is under development. The one-piece design incorporates the fluid ports and flow splitter and mates precisely with the inner core rod and the outer tubular channel wall. It eliminates the problem area of rubber O-rings and seated cylindrical flow splitters. The acceptable tolerances for the fabrication of this new manifold design need to be established. It is therefore necessary to confirm by experiment the predicted degree of nonspecific crossover for an imperfect system. To ensure that the dominant cause for crossover was the geometrical imperfection, the experimental measurements were carried out using an existing channel where one of the flow splitters was deliberately and severely bent. This was expected to give rise to an easily measurable nonspecific crossover that could be compared to predictions based on CFD modeling of the flow in the channel. If a good agreement could be demonstrated for such a severe distortion, then it could be assumed that CFD predictions would be equally good for slight imperfections. This would allow (30) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960; pp 51-54.

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Figure 2. Geometry of the distorted splitter, with thickness plotted to scale, on the channel cross section. The seated end remains circular while the distorted end projects into the channel.

the establishment of fabrication tolerances for the one-piece manifolds. Fabrication of manifolds to the established tolerances would reduce the nonspecific crossover due to geometrical imperfections to an acceptably low level. The level may be so low that nonspecific crossover due to other mechanisms would become evident. These contributions must be reduced by other means, such as reducing sample concentration where overloading is the underlying cause.27 EXPERIMENTAL MEASUREMENTS The channel used for the experiments was the Mark IV, according to our in-house classification. The length of the channel between the splitters was 9.32 cm, the radius of the inner core ri was 0.238 cm, and the radius of the outer wall ro was 0.453 cm. The flow splitters had a thickness of 0.015 cm and when perfectly cylindrical had an inner radius of 0.347 cm and an outer radius of 0.362 cm. One of the splitters was measured and confirmed to be cylindrical. The other was bent into an approximate elliptical shape at one end; the other end, which had to be seated in the flow manifold, remained circular. The distortion of the splitter was observed to increase linearly along its free length (the length protruding into the channel) of 0.50 cm. The splitter was digitally photographed within the manifold and a computer imaging program used to obtain coordinate points around its circumference relative to the manifold. These points were corrected to account for slight distortion of the image and converted to polar coordinates relative to the channel axis at the origin. The distorted end of the splitter is plotted in Figure 2 together with the undistorted circular cross section. The thickness of the lines is drawn to the scale of the splitter thickness. The nonspecific crossover experiments were performed with the channel mounted vertically in the absence of a magnetic field, with fluid flowing in a downward direction. Inlet flow rates were provided by a dual-drive Harvard 33 syringe pump (Harvard Instruments, Inc., Holliston, MA). A second pump controlled one 6690

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of the outlet flow rates. The aqueous carrier fluid contained 0.05% Triton X-100 (Sigma-Aldrich) and 0.02% sodium azide as a bactericide. The sample consisted of 8-µm-diameter polystyrene standard particles (Duke Scientific Co., Palo Alto, CA) and was introduced into the a′ inlet flow stream using a Rheodyne sample valve (model 7725i, Altech Associates, Inc., Deerfield, IL) fitted with a 500-µL sample loop. Sample concentration was ∼107 beads/ mL. Fixed-wavelength (254 nm) UV detectors (model VUV-10, HyperQuan, Inc., Colorado Springs, CO) were connected to the two outlets. A two-channel analog-to-digital converter (model DI190, Dataq Instruments, Akron, OH) was used to gather data from the detectors, and these were stored and manipulated using WinDaq Lite software (Dataq Instruments) on a PC. All experiments were carried out at a total flow rate V˙ of 10.00 mL/min. Series of experiments were carried out at fractional sample flow rates V˙ (a′)/V˙ of 0.10 and 0.20, with nonspecific crossover being determined over a wide range of outlet flow rate ratios. These flow conditions are typical of those used for cell separation using the QMS. The PS beads are similar in size to biological cells and are routinely used to check the channel and splitter integrity of the systems. Experiments were performed with the channel oriented in each direction, such that the distorted splitter served at the inlet and at the outlet. The nonspecific crossover was calculated by determining peak areas for the two detectors, taking into account the outlet flow rates, the detector gains, and the relative detector response. The latter was determined by direct injection of samples into both detectors connected in series and measuring the resulting peak areas. COMPUTER SIMULATIONS The approach for predicting fractional nonspecific crossover as a function of outlet flow rate ratio for the case of a distorted inlet flow splitter was described previously.27 The approach taken for a distorted outlet splitter differs from this, as discussed below. The channel was modeled over the 0.50-cm free lengths of the flow splitters and the 9.32-cm channel length between the splitters. A fine mesh was generated over the model, ensuring the mesh was most fine and uniform around the regions of merging flows at the splitter edges. The mesh had 48 elements across the annular channel thickness, with 8 uniform elements across the thickness of the splitter and 20 across the regions to each side of the splitters. On the inner side, a two-way bias in size was applied such that the element at the center was 3 times the size of the elements at the wall and close to the splitter. On the outer side, a two-way bias with ratio of 2:1 applied. There were 100 elements along the channel length between the splitters (a two-way bias was used with 100:1 ratio, so that elements got progressively smaller close to the splitters) and 30 elements along the length of each splitter (again with two-way bias at a 5:1 ratio). Around the full circumference, there were 120 equal sized elements. There was no plane of symmetry for the deliberately bent splitter, so the complete channel had to be modeled. Also, the more severe distortion modeled in this work required a greater number of mesh elements around the circumference than for the previous study27 (120 elements for the full circumference compared to 18 for the half-circumference in the earlier study). The computational fluid dynamics software package, CFX-4 (CFX International, AEA Technology plc, Harwell, Didcot, U.K.) was used to solve the Navier-Stokes equations, which describe

the flow of a Newtonian fluid through the channel. A temperature of 298 K was assumed, with fluid properties corresponding to those of water at this temperature (density F of 0.9966 g/mL and viscosity η of 0.009 00 P), giving a channel Reynolds number Re () 2(ro - ri) F/η, where is the mean fluid velocity) of 17.0 at a total flow rate of 10.0 mL/min. This is well below the transition to turbulent flow, which takes place at Re of ∼2000.30 In every case, the initial conditions were set as uniform fluid velocities normal to the plane of the channel inlets and “mass flow boundaries” at the outlets, the latter being equivalent to fully developed flow. In the case of a cylindrical, axially symmetric inlet splitter, the fluid quickly attains a fully developed flow profile in the annular regions on the inner and outer sides of the splitter. This occurs without any circumferential component to flow and is effectively complete well within the 0.50-cm splitter length. However, in the case of a distorted inlet splitter, the flow space varies continuously along the length of the splitter on both the inside and outside of the splitter. This causes circumferential components to flow as fluid tends to leave regions of narrowing space and approach regions of increasing space. Circumferential migration of fluid takes place in opposite directions on the inner and outer sides of the splitter. This redistribution of fluid occurs along the full length of the splitter. With the merging of the two flow streams into the uniform annular channel, there is a further circumferential redistribution of flow. In the simulated conditions, the higher flow rate enters the b′ inlet, and much of this flow is diverted to regions around the 30° and 200° points on the circumference as shown in Figure 2. It is around these regions that the annular thickness increases on the outer side of the splitter, corresponding to where the splitter dips inward toward the core rod. These regions of increasing thickness provide reduced resistance to flow and increased capacity for higher volumetric flow. This increased flow entering the channel from inlet b′ around the 30° and 200° points must be redistributed because, downstream of the splitter, the flow approaches a uniform annular flow in the undistorted channel. For the modeled system, the circumferential components to flow dissipate over a distance of around 1.0 or 1.5 cm, depending on the inlet flow rate ratio. The uniform annular flow is fully developed, and negligible circumferential components remain at the channel midpoint, 4.66 cm downstream from the splitter. In the case of a distorted inlet splitter, there is a further complication with regard to the initial inlet flow conditions considered for CFD solution. The overall resistance to flow varies around the circumference of both the a′ and b′ inlet regions. Strictly, this should be taken into account in imposing the normal flow velocities in the plane of the inlets. This was not done for the work presented. The result of this would be a slightly increased circumferential redistribution of flow in the region of the splitter, but it is not expected that it would greatly influence the overall result. PREDICTION OF NONSPECIFIC CROSSOVER Distorted Inlet Splitter. As mentioned earlier, the method for predicting nonspecific crossover as a function of outlet flow rate ratio for a distorted inlet splitter has been described.27 Only a brief description is therefore given here. The fluid flow pattern is obtained using CFD for a given inlet flow rate ratio and total flow rate. The outlet flow rate ratio is unimportant because

calculated nonspecific crossover is determined for conditions existing at the midpoint along channel length. The position of the ISS is determined at the channel midpoint by considering the distribution of a slowly diffusing solute that is considered to enter with the fluid at inlet a′. A diffusion coefficient of 10-6 cm2/s was assumed. Radial concentration profiles are extracted at 5° intervals at the channel midpoint. The low diffusion coefficient ensures the maintenance of the step functional form of the concentration profile along the channel length, although a small degree of smearing occurs, largely due to the numerical solution process. The position of the ISS is assumed to correspond to points where concentration is interpolated to half the initial concentration (i.e., c/c0 ) 0.5). Examination of radial profiles of axial velocity confirmed the circumferentially uniform velocity profile predicted from theory (see ref 30), given as a function of reduced radius F () r/ro) by

v(F) )

2 (1 - F2 + A2 ln F) A1

(1)

where is the mean fluid velocity along the length of the annular channel and A1 and A2 are functions of the ratio of inner to outer radii, ri/ro ) Fi of the annulus:

A1 ) (1 + Fi2 - A2)

(2)

A2 ) (1 - Fi2)/ln(1/Fi)

(3)

The position of the OSS for a perfect outlet splitter may therefore be calculated for any outlet flow rate ratio. Integration of flow velocity over the cross section of the channel from the inner wall at Fi to the OSS at FOSS yields the flow rate V˙ (a) at outlet a. The ratio of V˙ (a) to total flow rate V˙ may be shown2,5,27 to be given by

I2[Fi,FOSS] V˙ (a) ) V˙ A (1 - F 2) 1

(4)

i

in which I2[Fi,FOSS] represents the result of an integration and is given by

I2[Fi,FOSS] ) [f2(F)]FiFOSS ) [2F2 - F4 + 2A2F2 ln F A2F2]FiFOSS (5) Note that the function f2(F) is also defined in eq 5. Equation 4 may be solved numerically for FOSS, for any given V˙ (a)/V˙ . The nonspecific crossover is obtained by integration of fluid velocity over the region of channel cross section where FISS(θ) > FOSS. This follows from the fact that all fluid that flows on the outside of the OSS exits the channel at outlet b. The integration of fluid velocity is represented by

∫ ∫

V˙ cross ) ro2

2π

0

FISS(θ)

FOSS

v(F)F dF dθ

for

FISS(θ) > FOSS (6)

In discrete summation form, the volumetric crossover of fluid with Analytical Chemistry, Vol. 75, No. 23, December 1, 2003

6691

entrained material is given by

V˙ cross )

πro2

n

nA1

k)1

∑(f (F 2

ISS(2kπ/n))

- f2(FOSS))

for

FISS(2kπ/n) > FOSS (7) in which the angle θ is here taken at n equally spaced intervals around the circumference. The volumetric flow rate at inlet a′ may be similarly represented:

V˙ (a′) )

πro2

n

nA1

k)1

∑(f (F 2

ISS(2kπ/n))

- f2(Fi))

(8)

The fractional nonspecific crossover is therefore given by n

V˙ cross V˙ (a′)

∑(f (F

ISS(2kπ/n))

2

)

2π

0

FISS(θ)

Fbi

vz(F,θ)F dF dθ

(10)

in which Fbi is the reduced radius of the inner wall of the b outlet (equal to the outer radius of the undistorted splitter) and vz(F,θ) is the local fluid velocity component in the z-direction (parallel to the channel axis). The integration up to FISS(θ) is achieved by considering only regions where c/c0 > 0.5. As a discrete summation, this takes the form

2πro2(1 - Fbi)

n

m

∑∑v (F ,θ )F z

mn

j

k

(11)

j

k)1 j)1

for

n

∑(f (F 2

ISS(2kπ/n))

- f2(Fi))

k)1

FISS(2kπ/n) > FOSS (9) For the results presented, n was equal to 72, corresponding to the 5° intervals around the circumference. There is an advantage in taking the summation given by eq 8 for V˙ (a′) rather than the nominal V˙ (a′). It was mentioned that the initial conditions for the CFD solver required uniform normal fluid velocities at the inlets, which resulted in small discretization errors in volumetric flow rates. The quotient of summations in eq 9 gave a partial correction for this error. Distorted Outlet Splitter. The approach taken to calculation of nonspecific crossover for a distorted outlet splitter necessarily differs from that taken for a distorted inlet splitter. This is because both the radial location and the shape of the OSS in the main body of the channel depend on the outlet flow rate ratio. It is the ISS that, in this case, is cylindrical through the main body of the channel, with its radial position predictable from theory. Only as the distorted outlet splitter is approached does the ISS become distorted, as fluid is not drawn equally to each side of the splitter. For each considered inlet flow rate ratio, it is therefore necessary to obtain CFD solutions to the fluid flow pattern for a range of discrete outlet flow rate ratios. There are then two possible approaches to determination of nonspecific crossover for a given combination of inlet and outlet flow rate ratios. In the first, the position and shape of the OSS may be determined at the channel midpoint by back-projection of streamlines from the outlet splitter. The volumetric crossover V˙ cross may then be determined by an integration similar to that of eq 6, from FOSS(θ) to a constant FISS over the full circumference for all FOSS(θ) < FISS, with fractional nonspecific crossover given by an equation similar to eq 9. At the time this work was carried out, software capable of providing backprojection calculations along streamlines was not available to us. The alternative approach was necessarily followed, where the convection of the slowly diffusing solute provided the means to obtaining nonspecific crossover. This approach therefore relies 6692

∫ ∫

V˙ cross ) ro2

V˙ cross )

- f2(FOSS))

k)1

on the same assumptions made in the approach followed for the distorted inlet splitter. The determination of V˙ cross was made by integration of local fluid velocity in the plane of the outlets over that region of outlet b where c/c0 > 0.5. The integration is therefore carried out over the region of outlet b that is assumed to lie inside the distorted ISS. The integration yielding V˙ cross may be written as

Analytical Chemistry, Vol. 75, No. 23, December 1, 2003

where the summations are carried out for all (j,k) for which c(Fj,θk)/c0 > 0.5. When the outlet splitter is distorted in the manner considered, fully developed flow is not attained. Also, the mean fluid velocity in the outlet plane varies around the circumference. It is therefore necessary to extract both radial concentration profiles and fluid velocity profiles at intervals around the circumference. Conservation of mass requires all of the suspension that does not exit at the b outlet must exit at the a outlet. It follows that

V˙ (a′) - V˙ cross )

2πro2(Fao - Fi) mn

n

m

∑∑v (F ,θ )F z

j

k

j

(12)

k)1 j)1

in which Fao is the reduced radius of the outer wall of the a outlet (equal to the reduced inner radius of the undistorted splitter) and the discrete Fj here cover the thickness of the a outlet annulus in the outlet plane, and again the summations are carried out for all (j,k) for which c(Fj,θk)/c0 > 0.5. Evaluating the summations of eqs 11 and 12 gives a check on the internal consistency of the approach. Consistency was observed in every case, with deviation of V˙ (a′) thereby obtained from the nominal V˙ (a′) having a magnitude attributable to discretization error. RESULTS AND DISCUSSION Distorted Inlet Splitter. Figure 3 shows the locations of two inlet splitting surfaces within the channel annulus at the channel midpoint. The splitting surfaces are the result of inlet flow rate ratios V˙ (a′)/V˙ of 0.10 (inner surface) and 0.20 (outer surface). In each case, fully developed flow was predicted to be established in the annular channel within ∼1.5 cm of the downstream edge of the inlet splitter. For the case of V˙ (a)/V˙ ) 0.10, the disturbance to fully developed flow was predicted to occur within a distance of ∼0.25 cm of the leading edge of the outlet splitter. This latter distance is reduced further as the radial position of the OSS in the annular channel approaches that of the outlet splitter. The distance necessary to establish uniform flow downstream of the distorted inlet splitter is so much greater because the fluid must attain a uniform distribution around the circumference of the

Figure 3. Positions of the ISS at the midpoint of the channel length for V˙ (a)/V˙ of 0.10 (inner curve) and 0.20 (outer curve), determined by interpolation of concentration profiles at 5° intervals to half of the sample concentration. The distorted end of the inlet splitter is included.

annular channel. The channel midpoint is well-removed from the regions where flow deviates from uniform annular flow, as required by the approach taken to calculate nonspecific crossover. The locations of the points defining the curves were obtained by interpolation to c/c0 ) 0.5 on radial concentration profiles extracted at 5° intervals around the circumference. Included in the figure is the position of the distorted downstream edge of the inlet splitter. It was mentioned earlier that much of the fluid entering at the b′ inlet is diverted to regions centered at 30° and 200° for the orientation presented in the figure. The suspension entering at inlet a′ is diverted to regions centered around 110° and 310°. The merging of the two flow streams results in the suspension being largely confined to relatively narrow regions, with the regions becoming narrower with increasing ratio of V˙ (b′) to V˙ (a′). The predicted distortion of the ISS is seen to be more severe than that of the splitter for the two inlet flow rate ratios considered. Figure 4a shows the distribution in suspension concentration at the channel midpoint for V˙ (a′)/V˙ ) 0.20. The color spectrum corresponds to red at c/c0 ) 1 to blue at c/c0 ) 0. The narrow region of transition from red to blue confirms the retention of the step function concentration profile. The spatial distribution is consistent with the ISS shown in Figure 3. The distribution in fluid velocity component vz at the channel midpoint is shown in Figure 4b. The color spectrum corresponds to red at vz ) 1.4 cm/s to blue at vz ) 0.0 cm/s. The same color scheme is used for all figures, and this range is sufficient for all conditions shown. The velocity profile in Figure 4b is consistent with fully developed annular flow with a maximum velocity of 0.536 cm/s. Figure 4c shows the distribution of the product of local concentration and local velocity vz (red, (c/c0)vz ) 1.4 cm/s, to blue, (c/c0)vz ) 0.0 cm/s). This shows the variation in the local rate of transport of suspended material across the channel cross section. As explained in the previous section, the calculation of predicted nonspecific crossover is carried out using these data at the channel midpoint. Calculations involving a distorted outlet splitter require data to be gathered at the channel outlets.

Figure 4. (a-c) Cross-sectional slices at the channel midpoint showing local concentration, fluid velocity component vz, and product of concentration and fluid velocity, respectively, for V˙ (a′)/V˙ ) 0.20. (d-f) Corresponding slices taken in the plane of the channel outlets for V˙ (a′)/V˙ ) V˙ (a)/V˙ ) 0.20. Color map for (a) and (d): red, c/c0 ) 1, to blue, c/c0 ) 0. Color map for (b, c, e, f): red, 1.4 cm/s, to blue, 0.0 cm/s.

However, it is interesting to compare predicted suspension distribution at the outlets for the two cases. We therefore include Figure 4d-f, which show the same quantitative data as Figure 4a-c, but in the plane of the channel outlets for the case of V˙ (a)/V˙ ) V˙ (a′)/V˙ ) 0.20. The higher flow rate exiting outlet b compared to outlet a is evident in Figure 4e. It may also be seen that the flow is uniform around the circumference of each outlet, and examination of the velocity profiles indicates that they are also fully developed. Under the conditions considered, FOSS ) 0.6724 at the channel midpoint. This corresponds to a position for the OSS that is 0.30 of the annular thickness from the inner wall. A substantial fraction of the suspension is therefore carried to outlet b, in fact, all of the suspension finding itself in the outer 70% of the annular thickness at the channel midpoint. This is, to some extent, evident in Figure 4d, but it is Figure 4f that shows the result most clearly. The higher flow velocity at outlet b transports relatively more of the suspension than the flow at outlet a. Figure 5 shows the experimental observations, plotted as points, of nonspecific crossover as a function of outlet flow rate ratio V˙ (a)/V˙ . The smooth curve was obtained numerically from the results of CFD simulation as explained in the previous section. Also included in the figure is the linear decrease in relative nonspecific crossover V˙ cross/V(a′) from unity to zero as V˙ (a)/V˙ increases from 0.0 to 0.2 corresponding to ideal behavior with perfect splitters. It should be noted that 25 data points are plotted in the figure, with 4 points at V˙ (a)/V˙ ) 0.1, 0.3, 0.5, and 0.7, 3 points at 0.2, and 2 points at 0.4, 0.6, and 0.8. Reproducibility of results was therefore good. The numerical predictions indicate an expected V˙ cross/V(a′) of 0.534 at V˙ (a)/V˙ ) 0.20, supporting the qualitative remarks given above regarding Figure 4f. Figure 5 shows a generally good agreement between experiment and prediction, with experiment showing a slightly higher nonspecific crossover than the prediction from simulation. The experimental results for V˙ (a′)/V˙ ) 0.10 are compared with theory in Figure 6. Again, agreement is good, with a tendency for experiment to indicate a slightly higher crossover than prediction. In this case, Analytical Chemistry, Vol. 75, No. 23, December 1, 2003

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Figure 5. Fractional nonspecific crossover V˙ cross/V(a′) as a function of outlet flow rate ratio V˙ (a)/V˙ , for distorted inlet splitter and inlet flow rate ratio V˙ (a′)/V˙ ) 0.20. The points correspond to experimental measurements, the curve was predicted using CFD as described in the text, and the straight-line dependence is predicted for an ideal system.

Figure 7. Suspension concentration distribution in the plane of the channel outlets for a distorted outlet splitter, V˙ (a′)/V˙ ) 0.20, and outlet flow rate ratios V˙ (a)/V˙ of (a) 0.1, (b) 0.2, (c) 0.3, (d) 0.4, (e) 0.5, (f) 0.6, (g) 0.7, and (h) 0.8. Color map as for Figure 4a and d.

Figure 8. As for Figure 7 but showing the fluid velocity component vz parallel to the channel axis in the plane of the channel outlets. Color map as for Figure 4b and e.

Figure 6. As for Figure 5, but with V˙ (a′)/V˙ ) 0.10.

the figure includes 19 data points, with 4 points at V˙ (a)/V˙ ) 0.3, 3 points at 0.1 and 0.7, 2 points at 0.05, 0.2, 0.4, and 0.5, and 1 point at 0.6. Again, there was considerable reproducibility in repeated experiments. Distorted Outlet Splitter. As explained in the previous section, calculations of predicted nonspecific crossover make use of conditions predicted in the plane of the channel outlets. Figure 7 shows the suspension concentration in the plane of the outlets for V˙ (a′)/V˙ ) 0.2 and V˙ (a)/V˙ of (a) 0.1, (b) 0.2, (c) 0.3, (d) 0.4, (e) 0.5, (f) 0.6, (g) 0.7, and (h) 0.8. The orientation of the distorted outlet splitter corresponds to that shown in Figures 2 and 3. The inlet splitter is assumed to be perfectly circular and concentric with the channel, and the ISS is therefore also circular along most of the channel length. As the fluid approaches the distorted outlet splitter, the ISS is itself distorted as fluid is drawn nonuniformly into the outlets. For V˙ (a)/V˙ ) 0.1, the distortion is predicted to occur within a distance of ∼1.0 cm from the leading edge of the distorted outlet splitter, and for V˙ (a)/V˙ ) 0.2 within a distance of ∼0.5 cm. In the case of Figure 7a, V˙ (a)/V˙ ) 0.1 and so 90% of the fluid exits at outlet b. The resistance to flow is least and capacity for flow highest in the b outlet in the regions centered around 200° and 30°. In these regions, the OSS dips close to the inner channel wall (below the ISS) and draws suspended material into 6694 Analytical Chemistry, Vol. 75, No. 23, December 1, 2003

the b outlet. The opposite effect occurs around 310° and 110° where the flow to outlet b is restricted and the flow to outlet a enhanced. The net result is that the OSS is kept well above the ISS, and pure carrier fluid that entered the channel at inlet b′ is, in these regions, drawn into outlet a. It should be pointed out that, had both the inlet and outlet splitters been perfect, then we would expect 50% crossover for these flow conditions. No pure carrier fluid would have been drawn into outlet a, and the concentration of the suspension exiting outlet a would be equal to the initial concentration. The circular OSS would lie just within the circular ISS, skimming off a uniform fraction of the suspension, which would go on to occupy a thin uniform region in outlet b adjacent to the splitter. Parts b-h of Figure 7 show that as V˙ (a)/V˙ increases then more and more of the suspension is drawn to the a outlet. When V˙ (a)/V˙ ) 0.8 (see Figure 7h), the suspension is apparently excluded from outlet b. As explained earlier, the local fluid velocity component vz must also be taken into account to calculate predicted crossover. Parts a-h of Figure 8 show the fluid velocity component vz in the plane of the channel outlets for the same discrete outlet flow rate ratios of Figure 7a-h. Regions of higher and lower flow velocity are seen around the circumference. Examination of Figure 8a in conjunction with Figure 7a indicates that the regions of outlet b which draw in the crossover suspension also correspond to regions of high velocity. This is as expected, and the effects act in concert to exacerbate nonspecific crossover. This is clearly apparent in Figure 9a, which shows the product of local concentration and fluid velocity in the plane of the outlets. On the other hand, when V˙ (a)/V˙ is increased to 0.7 (as shown in

Figure 9. As for Figure 7 but showing the product of local suspension concentration and velocity component vz in the plane of the channel outlets. Color map as for Figure 4c and f. Figure 11. As for Figure 10, but with V˙ (a′)/V˙ ) 0.10.

slightly elevated nonspecific crossover compared to prediction. This is the same tendency observed for the experiments carried out with the distorted inlet splitter. Figure 10 includes 23 data points, with 4 points at V˙ /V˙ ) 0.1, 3 points at 0.15, 2 points at 0.175, 0.25, 0.3, 0.35, 0.4, 0.45, and 0.5, and 1 point at 0.2 and 0.225. Figure 11 includes 32 data points with 6 points at V˙ (a)/V˙ ) 0.08, 5 points at 0.06 and 0.12, 4 points at 0.04 and 0.1, 2 points at 0.3, 0.4, and 0.45, and 1 point at 0.2 and 0.5.

Figure 10. Fractional nonspecific crossover V˙ cross/V(a′)as a function of outlet flow rate ratio V˙ (a)/V˙ , for distorted outlet splitter and inlet flow rate ratio V˙ (a′)/V˙ ) 0.20. The points correspond to experimental measurements, the curve was predicted using CFD as described in the text, and the straight-line dependence is predicted for an ideal system.

Figures 7g, 8g, and 9g), the reduced flow velocity in outlet b has the effect of reducing the crossover. The apparently significant crossover in Figure 7g (showing local concentration) becomes less significant when local velocity is taken into account, as shown in Figure 9g. The predicted relative nonspecific crossover was calculated to be only 0.010 for these conditions. It was mentioned earlier that it is of interest to compare suspension distribution in the plane of the outlets for a distorted inlet splitter and a distorted outlet splitter under the same flow conditions. Figures 4d and 7b both correspond to V˙ (a)/V˙ ) V˙ (a′)/V˙ ) 0.2, but the distributions are seen to be markedly different. The distribution at the channel midpoint resulting from a distorted inlet splitter (shown in Figure 4a) is essentially retained as the fluid is drawn uniformly into concentric outlets. The generation of the very different distribution at a distorted outlet was described above. The spreading of the suspension around the circumference of the b outlet occurs with the redistribution of fluid within the distorted outlet regions. Figures 10 and 11 show the experimentally measured crossovers for V˙ (a′)/V˙ ) 0.2 and 0.1, respectively. The smooth curves in each figure were obtained by numerical calculation as explained in the previous section. The straight-line dependencies for ideal behavior are also included. Agreement between experiment and prediction is generally excellent, with experiment tending to show

CONCLUSIONS For the severely distorted flow splitter considered in this work, the quantitative agreement between experimental measurements of nonspecific crossover and prediction based on CFD modeling of the fluid flow was shown to be generally very good. For all flow conditions, the experimentally observed crossover was found to be just a little higher than predicted. Certainly, the departure from ideal nonspecific crossover was almost fully accounted for by the convective flow pattern predicted by CFD. This may be due to a small modeling error in the distortion of the splitter. Some other mechanism, such as shear-induced diffusion, could also contribute to particle transport across streamlines. However, the agreement for this severe splitter distortion is considered sufficiently good that we can have confidence in the convective crossover predictions for only slightly imperfect splitters. This will allow us to proceed with determining the influence of small splitter imperfections on resolution of migrating species. These calculations will establish acceptable tolerances for the fabrication of inlet and outlet manifolds. ACKNOWLEDGMENT This work was supported by Grants R01 CA62349 (M.Z.) and R33 CA81662 (J.J.C.) from the National Institutes of Health, and Grants BES-9731059 (J.J.C. and M.Z.) and CTS-0125657 (P.S.W., J.J.C., and M.Z.) from the National Science Foundation. The authors also acknowledge expert technical help in channel construction by J. Proudfit (Prototype Laboratory, Lerner Research Institute, The Cleveland Clinic Foundation).

Received for review April 21, 2003. Accepted September 10, 2003. AC030152N Analytical Chemistry, Vol. 75, No. 23, December 1, 2003

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