optimizing the rotor design for controlled-shear affinity filtration using computational fluid...

Optimizing the Rotor Design forControlled-Shear Affinity FiltrationUsing Computational Fluid Dynamics

Patrick Francis,1,2 D. Mark Martinez,2 Fariborz Taghipour,2

Bruce D. Bowen,2 Charles A. Haynes1,2

1Michael Smith Laboratories, Rm. 301, 2185 East Mall,Vancouver BC V6T 1Z3, Canada; telephone: (604) 822-5136;fax: (604) 822-2114; e-mail: [email protected] of Chemical and Biological Engineering,University of British Columbia, Vancouver BC V6T 1Z3, Canada

Received 9 November 2005; accepted 15 June 2006

Published online 25 August 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/bit.21090

Abstract: Controlled shear affinity filtration (CSAF) is anovel integrated processing technology that positions arotor directly above an affinity membrane chromatogra-phy column to permit protein capture and purificationdirectly from cell culture. The conical rotor is intended toprovide a uniform and tunable shear stress at themembrane surface that inhibits membrane fouling andcell cake formation by providing a hydrodynamic forceaway from and a drag force parallel to the mem-brane surface. Computational fluid dynamics (CFD)simulations are used to show that the rotor in the originalCSAF device (Vogel et al., 2002) does not provide uniformshear stress at the membrane surface. This results in theneed to operate the system at unnecessarily high rotorspeeds to reacha requiredshear stressof at least 0.17Paatevery radial position of the membrane surface, compro-mising the scale-up of the technology. Results from CFDsimulations are compared with particle image velocime-try (PIV) experiments and a numerical solution for lowReynolds number conditions to confirm that our CFDmodel accurately describes the hydrodynamics in therotor chamber of the CSAF device over a range of rotorvelocities, filtrate fluxes, and (both laminar and turbulent)retentate flows. CFD simulations were then carried out incombination with a root-finding method to optimize theshape of the CSAF rotor. The optimized rotor geometryproduces a nearly constant shear stress of 0.17 Pa at arotational velocity of 250 rpm, 60% lower than the originalCSAF design. This permits the optimized CSAF device tobe scaledup toamaximumrotordiameter 2.5 times largerthan is permissible in the original device, thereby provid-ingmore thana sixfold increase in volumetric throughput.� 2006 Wiley Periodicals, Inc.

Keywords: computational fluid dynamics; rotating diskfilter; cell separation; membrane filtration; moduledesign; process optimization

INTRODUCTION

Improvements in strain engineering, as well as in clone

selection, media composition, and culturing conditions have

pushed typical recombinant product titers in both fed-batch

and perfusion cultures above 1 g/L (Low, 2005). As a result,

the cost of goods (i.e., total manufacturing costs) for cell-

culture derived products is now dominated by the relatively

high cost of product capture, purification, and formulation.

Downstream processing of complex recombinant proteins

typically requires a large number of sequential unit

operations, each of which results in a loss of product. One

obvious method to improve product yields and thereby

reduce cost of goods is to reduce the number of required

operations through rational downstream process integration

and optimization. For proteins produced and secreted by

mammalian cells, optimization of the cell separation and

initial product capture steps is especially important, as

process volumes and volume reduction tend to be largest in

these stages. Integration of the cell-retention/removal step

with product capture could potentially increase the overall

yield by 10–20% and could also reduce processing times.

Vogel et al. (2002) recently proposed a novel technique for

the integration of cell separation and product capture.

Constant shear affinity filtration (CSAF) technology com-

bines a specially designed rotating disk filter with an affinity

membrane chromatography column to capture and purify

proteins directly from cell culture (Fig. 1). The CSAF

technology has been shown to be an effective method of

capturing human tissue-type plasminogen activator (t-PA)

directly from a recombinant Chinese hamster ovary (CHO)

cell culture, producing a 100% cell-free eluatewith a product

yield of 86%, and a purification factor of 16.7 (Vogel et al.,

2002). Inspired by the hydrodynamic properties of the cone-

and-plate viscometer, the rotor within the original CSAF

device is conical in shape in an attempt to produce a constant

and tunable shear stress at the membrane surface that inhibits

membrane fouling and clogging by providing both a uniform

hydrodynamic force away from (Saffman, 1964) and a drag

force parallel to (Belfort et al., 1994) the membrane surface.

The shear stress is controlled by the rotational speed of the

rotor, allowing the transmembrane pressure (TMP) to be

�2006 Wiley Periodicals, Inc.

Correspondence to: C.A. Haynes

Canada Research Chair in interfacial biotechnology.

largely decoupled from the shear stress over the whole

membrane area. This creates a homogeneous filtrate flow,

which in turn leads to optimal dynamic capacities and

reduced broadening of the breakthrough curve within the

associated membrane chromatography column.

However, at high rotor speeds, centrifugal forces are

known to cause an outward radial flow at the rotating surface

and an inward radial flow at the plate surface of a cone-and-

plate rotor geometry (Savins and Metzner, 1970). These

‘‘secondary’’ flows are significant when the Reynolds

number, Re, defined as Rrot2 O�2/n (Ellenberger and Fortuin,

1985), where Rrot is the radius of the cone, O is the angular

velocity of the rotor, � is the angle between the cone and

plate, and � is the kinematic viscosity of the fluid, is greater

than 1. In the case of the original CSAF device, these radial

flow effects are estimated to begin at rotational velocities

above 2 rpm and increase with increasing Re. As a result, the

original CSAF device, although effective at the 1-L scale,

does not provide uniform shear stress at the membrane

surface, creating a need for higher than necessary rotor

speeds (700 rpm) to achieve the desired performance.

Regrettably, high rotor speeds restrict device scale-up, as

mammalian cells are compromised by regions of high shear

stress, and cell lysis will lead to the release of intracellular

impurities.

The aim of this work is to utilize computational fluid

dynamics (CFD) to model the effects of retentate-fluid-

chamber and rotor geometries on hydrodynamics in the

CSAF device. CFD is finding increasing use in the modeling

of biological processes, including bioreactors (Davidson

et al., 2003; Williams et al., 2002) and filtration units (Rainer

et al., 2002; Taha and Cui, 2002). Recently, Castilho and

Anspach (2003) used CFD to model a dynamic filter for cell

harvesting and recycle that utilizes a rotor assembly in the

fluid chamber. Their filtration device differs fromours in both

design and purpose. As a result, their study did not account

for filtrate flux, the presence of an affinity membrane stack,

and the need for uniform solvent velocity through the

membrane stack. Nevertheless, it demonstrated the potential

for using CFD to visualize and quantify hydrodynamics

underneath a rotor and near a membrane surface. Here, we

extend the CFD model of Castilho and Anspach (2003) to

accurately model hydrodynamics within the fluid chamber

and membrane of the CSAF device under conditions where

solvent flux through the membrane stack is non-zero. Shear

stress profiles at the membrane surface of the CSAF device

are computed for a range of rotor geometries constructed in

silico to establish an optimal rotor shape. CFD model results

are validated and verified through comparison with particle

imaging velocimetry (PIV) data and results from a simplified

numerical solution of the Navier-Stokes equations for

conditions of low Re.

MATERIALS AND METHODS

Computational Fluid Dynamics

The geometry of the CSAF fluid chamber and affinity

membrane stack is shown in Figure 1, with important

dimensions and physical properties of the system provided in

Table I. The partial differential equations describing mass

andmomentum conservation within the CSAF fluid chamber

are as follows:

r � ð~vÞ ¼ 0 ð1Þ

�ð~v � rÞ~v ¼ �rpþ �r2~v ð2Þ

where r is the fluid density, p is the pressure, m is the fluid

viscosity, and ~v is the three-dimensional flow field in the

rotor chamber. Flow through the porous medium (i.e., the

affinity membrane stack) is described using the continuity

equation and the Brinkman-extended Darcy equation

(Brinkman, 1942):

r � ð~uÞ ¼ 0 ð3Þ

�

"

D~u

Dt¼ �"rpþ �r2~u� "�

K~u ð4Þ

where " and K are the porosity and permeability of the

membranes, respectively and~u is the three-dimensional flow

field in the membrane stack.

Figure 1. Schematic of CSAF prototype. A, B, and C refer to the feed,

retentate, and filtrate, respectively. D represents the affinity membrane stack

and E the rotor.

Table I. Characteristics of the CSAF device used in CFD model.

Rotor chamber height (Hcham) 20 mm

Rotor chamber radius (Rcham) 43.75 mm

Inlet and retentate radius (Rin/ret) 3.175 mm

Radial position of inlet/retentate (Rpos) 22.9 mm

Rotor radius (Rrot) 35 mm

Rotor gap height (�) 0.2 mm

Membrane stack radius (Rmem) 30 mm

Membrane stack thickness (Ho) 150 mmMembrane permeability (K) 6.6� 10�15 m2

Membrane porosity (") 0.58

1208 Biotechnology and Bioengineering, Vol. 95, No. 6, December 20, 2006

DOI 10.1002/bit

Equations (1–4) were solved using a three-dimensional,

double-precision segregated solver (Fluent 6.1.22 (Fluent,

Inc., 1998)) in which discretization, pressure–velocity

coupling, and pressure interpolation were performed using

a second-order upwind scheme, the SIMPLE algorithm, and

the program PRESTO!, respectively (Fluent, Inc., 1998;

Patankar, 1980). This solution strategy was chosen by

considering the geometry of the system, the rotating flow

patterns within it, computing time, numerical accuracy, and

ease of convergence (Fluent, Inc., 1998; Patankar, 1980).

The working fluid was assumed to be Newtonian,

isothermal, and incompressible. No-slip boundary conditions

were applied to all CSAF device walls excluding the

membrane surface. For those CFD simulations intended for

direct comparison with PIVexperiments, where there was no

flux through the membrane stack, the fluid chamber was

modeled as a closed systemwith no-slip boundary conditions

applied at the membrane surface (retentate side). In all other

simulations, a 0 Pa outlet pressure was applied at the outlet-

flow surface of the membrane stack, and the pressures at the

feed inlet and retentate outlet ports were adjusted to achieve

the desired filtrate flux. The stacked-membrane affinity

columnwasmodeled as a continuous porousmediumwith an

axial viscous resistance two orders of magnitude less than

that in the radial and tangential directions. Membrane

permeability and porosity were calculated from data

provided by the manufacturer (Pall, Inc., East Hills, NY).

All simulations were calculated using a steady-state

assumption in which the membrane permeability remained

constant; therefore, membrane fouling was neglected. The

realizable k-" model was used to describe turbulent flows

within the device as this model has been shown to be accurate

for similar geometries (Serra and Wiesner, 2000; Williams

et al., 1991). To ensure the solution method was accurate in

fluid regions near moving and non-moving surfaces, an

enhanced wall-treatment including pressure gradient effects

was implemented. Iterations continued until all scaled

residuals were below 10�4 and velocities remained constant

at representative points below the rotor.

The computational grids were built from structured

hexahedral elements using Gambit 2.1.6 (Fluent, Inc.,

Lebanon, NH); approximately 700,000 mesh elements were

used for each simulation. The grid distribution was densest

below the rotor, reaching a maximum just above and within

the surface of the membrane (porous region). In the area

above the rotor, mesh size was increased with no loss of

accuracy.Mesh independencewas achievedby increasing the

number of nodes until no further significant changes were

seen in the flow field; increasing the number of mesh

elements from 560,000 to 750,000 caused a maximum

change in velocity of less than 1%.

Particle Image Velocimetry

Particle image velocimetry (PIV) measurements were made

using a Flow-map 2D system (Dantec Dynamics, Mahwah,

NJ) in conjunction with a dual-head Nd:YAG laser (New

Wave Research, Fremont, CA) having a wavelength of

532 nm (green) and energies of up to 52 mJ. The laser sheet

was 1 mm thick. A CCD camera (Hamamatsu Photonics,

Bridgewater, NJ) with a resolution of 1,344� 1,024 pixels

and a 12-bit dynamic range was used to capture the images.

ANikonAFMicro-Nikkor lens (60/2.8), fitted with a 514 nm

line filter, was used for focusing. The camera and laser were

synchronized using the software package Flow Manager

(Dantec Dynamics); the time interval between two pulses

was set to 450 ms. Each velocity field was calculated from the

adaptive cross-correlation of 100 image sets based on

interrogation areas of 32� 32 pixels with a 50% overlap.

Polyamid seeding particles (Dantec Dynamics) were used as

tracers. These particles had a mean diameter of 20 mm and

a density of 1.03 g/cm3. The seeding concentration was

adjusted so as to achieve between 5 and 10 particles per

interrogation area (Khopkar et al., 2003).

The prototype CSAF device used in the PIV experiments

consisted only of the fluid chamber with no membrane stack

or associated filtrate flux. It wasmade of a plexiglass cylinder

with an inner diameter of 87.5 mm and a height of 20 mm.

The rotor, also made of plexiglass, had a diameter of 70 mm,

an angle of 48, a maximum thickness at the apex of 7.45 mm,

and a gap height from the bottom surface of 0.2 mm; it was

driven by a variable speed motor. The rotor’s shaft was 5 mm

in diameter and was made of stainless steel. In order to

decrease distortion of the image caused by refraction of light

at curved surfaces, the entire unit was placed in a rectangular

plexiglass box filled with glycerine. The laser sheet was

orientated vertically to capture tangential and axial velocities

along the center-line at a given radial position, or horizontally

to capture the associated radial velocity at a given axial

position.

NUMERICAL SOLUTION FOR LOWREYNOLDS NUMBERS

In order to verify the CFD model, tangential fluid flow

profiles within both the fluid chamber and the membrane

stack were obtained from an approximate analytical solution

of the mass and momentum conservation equations for low

Reynolds number flows within a simplified rotor chamber in

which the rotor is modeled as having an infinite radius

(Fig. A.1). The simplified CSAF-like system, therefore,

consists of a fluid bounded below by a non-deformable

porous membrane and above by an infinite rigid cone of fixed

angle of incidence � rotating at an angular velocity O. Thegap between the cone and the membrane surface varies

with radial position and is defined as h(r). The assumptions

made for this model are that the fluid is Newtonian and

incompressible, and the flow is laminar; body forces are

neglected and the solution is time-independent and axisym-

metric. Inertial terms in the equation of motion are retained,

as the curvature of the streamlines may be large near the

lower boundary. Flow within the membrane stack (porous

domain) positioned beneath the rotor was modeled using the

Brinkman-extended Darcy equation (Brinkman, 1942). The

Francis et al.: Optimizing the Rotor Design for CSAF 1209

Biotechnology and Bioengineering. DOI 10.1002/bit

full derivation of this model and its analytical solution for

tangential flows can be found in the Appendix. Analytical

expressions for axial and radial flow components, within both

the rotor chamber and the porousmedium,were not obtained.

Results from themodel in the limit of no axial flow through

the isotropic porous medium (when tangential and radial

velocities within the porous medium will be proportionately

highest) indicate that, in the case of the CSAF device,

tangential flows are negligible at all axial positions of the

porous medium. This is confirmed in Figure 3, which shows

that there are no tangential flows within the porous medium

of the simplified CSAF device if the value of Da/" (the Darcynumber Da¼K/H2, where K is the membrane permeability

and H is the total height of the rotor, H¼Hoþ h(r), divided

by the membrane porosity ") is 10�5 or less and fluid flow in

the rotor chamber is laminar. Da/" values for the membranes

used in our CSAF device are on the order of 1� 10�9.

Furthermore, because the dominant flow in the rotor chamber

is in the tangential direction, we can safely assume that radial

flows in the porous medium are also negligible. CFD

simulations performed both in the presence and absence of

filtrate flux through the porous medium also show that

tangential and radial components of flow are negligible in the

porous medium of the CSAF device. This allowed us to

further simplify the model derived in the Appendix by

replacing the porous medium with a fixed wall allowing a

uniform axial filtrate velocity (Vm) across its entire surface

(Fig. 2).

The numerical solution to this simplifiedmodel for all flow

components was found by assuming the forms of the velocity

components. Following the procedure outlined by von

Karman (1921), we postulate that the velocity components

may be written as:

vr ¼ � 1

2

Vm

hðrÞ r@

@zg

z

hðrÞ

� �� ð5Þ

vy ¼ rO�z

hðrÞ

� �ð6Þ

vz ¼ Vm gz

hðrÞ

� �� 1

2

r2z ddr½hðrÞ� @

@r g zhðrÞ

� �h ihðrÞ2

0@

1A ð7Þ

where g and � are dimensionless functions, to be determined

subsequently, that satisfy the continuity equation (Eq. 1).

These equations, after substitution into the momentum

equations, elimination of the pressure terms through cross-

differentiation of the z- and r-momentum equations and non-

dimensionalization, yield the following system of two

ordinary differential equations:

�hðxÞRrot

�ð�Þ d

d�gð�Þ

� �� gð�Þ d

d��ð�Þ

� ��

� 22þ d2

d�2�ð�Þ ¼ 0

ð8Þ

Figure A.1. Physical configuration of the two-domain similarity solution

for CSAF hydrodyamics. A conical rotor with an angle of incidence � and a

localized height, h(r), above the porousmedium of thicknessHo, is rotated at

a velocity O. The axial velocity at z¼ 0 is specified as Vm.

Figure 2. Physical configuration of the one-domain lowReynolds number

numerical solution of the CSAF hydrodynamics. A conical rotor with angle

of incidence� and a localized height above themembrane of h(r) is rotated at

a velocityO. Fluid flows out through themembrane at a constant velocityVm.

Figure 3. The tangential component of velocity profiles �1 and �2estimated in the limit of �¼ 0. �1 and �2 represent the tangential velocity

components in the porous and fluid domains, respectively. Four curves are

shown: (a) Da/"¼ 1� 10�5, (b) Da/"¼ 1� 10�3, (c) Da/"¼ 1� 10�2 and

(d) Da/"¼ 1� 105. These estimates were made with B¼ 0.15, ¼ 0.25,

H¼ 2� 10�3 m, and "¼ 0.58.


DOI 10.1002/bit

� �hðxÞRrot

gð�Þ d3

d�3gð�Þ

� �� 4hðxÞ32

�R3rot

�ð�Þ d

d��ð�Þ

� �

þ d4

d�4gð�Þ

� �¼ 0

ð9Þ

subject to no-slip and no-penetration boundary conditions at

the rotor surface, and no radial or tangential slip at the

membrane surface:

gð1Þ ¼ g0ð1Þ ¼ 0 �ð1Þ ¼ 1

g0ð0Þ ¼ �ð0Þ ¼ 0gð0Þ ¼ �1ð10Þ

when terms of order h0(�) and smaller have been neglected.

Hence this formulation is valid for the case of a nearly

constant gap size over small radial distances. The variables xand � are the non-dimensional coordinates:

� ¼ r

Rrot

ð11Þ

� ¼ z

hðrÞ ð12Þ

and \alpha and \beta are the Reynolds numbers for flow

through the membrane and within the rotor chamber,

respectively:

� ¼ VmRrot

�ð13Þ

¼ OR2rot

�ð14Þ

After specifying a value of x, the system of two equations

was solved using a 4th-order Runge-Kutta shooting method

(Holland and Liapis, 1983). Trial values of �0(0), g00(0), andg000(0) were assumed and Equations (8) and (9) were solved

as an initial value problem over the domain 0� �� 1. A

root-finding procedure was then used to update the values of

�0(0), g00(0), and g000(0) until the boundary conditions at

�¼ 1 were satisfied to within a tolerance of 0.1%.

RESULTS AND DISCUSSION

Validation of CFD Simulations

The meshing scheme and solution strategy used to model the

CSAF device by CFD were verified and validated by

comparing CFD simulations to results from the numerical

model and PIV experiments, respectively. The numerical

model is applicable to low rotational velocities correspond-

ing to laminar flow. Past studies have shown for a fixed-angle

cone and plate viscometer that turbulence occurs at

Re¼Rrot2 O�2/�� 48 (Sdougos et al., 1984). Note that the

Re defined by Sdougos et al. includes the square of the rotor

angle, while in our model does not so that it can be

generalized to handle non-conical rotor shapes. For a rotor

with a fixed incidence angle of 48, the onset of turbulence

would therefore occur at � 9,800. For ¼ 1,285 and

x¼ 0.71, where x is the dimensionless rotor radius r/Rrot,

Figure 4 shows that velocity profiles given by the CFD

Figure 4. CFD computed (^) dimensionless (a) tangential (�), (b) radial(g0), and (c) axial (g) velocity profiles under the rotor at a radial distance fromthe rotor center-point of x¼ 0.71, compared to those calculated by the one-

domain similarity solution (solid line). Rrot¼ 35 mm, O¼ 10 rpm, and

Vm¼ 3.65� 10�5 m/s.



simulation closely match those of the numerical model.

Indeed, good agreement between velocity profiles is

observed at all values of x less than 0.75 under laminar flow

conditions. Reported shear stress values represent the square

root of the sum of the squares of the two shear stress

components present at themembrane surface (tzy and tzr) andwere calculated by first fitting the near-wall velocity data to a

third-order polynomial using the Levenberg–Marquardt

method and then differentiating. The shear stress profiles

calculated from the CFD simulation show good agreement

with the numerical model (Fig. 5) at all x up to ca. 0.65, but

begin to diverge (as do velocity profiles) as the rotor edge is

approached. This divergence is expected since the numerical

model applies to a rotor of infinite radius and therefore does

not take into account rotor edge effects, while the CFD

solution has a finite rotor bounded by a cylindrical wall, a

small distance away (Fig. 1).

In order to validate the CFD model at higher rotational

velocities, PIV experiments were performed on a prototype

CSAF unit. PIV is a powerful non-intrusive technique to

obtain fluid velocities and has been used extensively as a

stand-alone method for fluid flow studies (Hill et al., 2000;

Hopkins et al., 2000; Pruvost et al., 2000; Shafiqul Islam

et al., 2002; Xiong et al., 2003) as well as in conjunction with

CFD models (Armenante et al., 1997; Khopkar et al., 2003;

Ranade, 1997). These studies include using CFD and PIV to

investigate hydrodynamics in bioreactors (Haut et al., 2003;

Vial et al., 2002), highlighting the potential use of CFD in

modeling and optimizing bioprocessing equipment. As noted

before, the prototype CSAF unit consisted of only the

retentate-fluid-chamber and did not include associated fluxes

through the membrane cartridge. CFD simulations were

therefore carried out for the CSAF device in the absence of

filtrate flux, allowing the results to be directly compared with

PIV data.

In Figure 6, representative tangential velocity profiles

determined by CFD simulation are compared with PIV data

for two different radial positions beneath the rotor. PIV data

are restricted to regions away from the rotor and membrane

surface due to the inherent difficulty in capturing velocities

near reflective surfaces. Good agreement between the

calculated and experimental data sets is observed, which is

representative of the level of agreement observed at other

radial positions and rotational velocities. Thus, the CFD

results are seen to accurately represent both the experimental

andmodeling data, indicating that themeshing geometry and

solution scheme were effective and that the realizable k-"turbulence model was appropriate over the range of

permissible CSAF operating conditions.

The CFD calculated tangential velocity profiles indicate

that there are two boundary layers in the gap region: one near

the rotor surface and the second adjacent to the membrane

(stator) surface. In the intermediate region between the two

surfaces, the fluid rotates at a nearly constant velocity XOr,where 0<X< 1. This type of velocity profile has been

reported previously for flow geometries similar to that

beneath the CSAF rotor (Daily and Nece, 1960; Ketola and

Figure A.1. 5. CFD computed (^) shear profile along the membrane

surface compared to that calculated by the one-domain similarity solution

(solid line) when Rrot¼ 35 mm, O¼ 10 rpm, and Vm¼ 3.65� 10�5 m/s.

Figure 6. Tangential velocity profiles determined by PIV measurement

(&) compared to CFD results (~). Experiments/simulations were carried

outwith an angled rotor at a rotational velocity of 700 rpm for radial positions

of (a) x¼ 0.57, (b) x¼ 0.86 when �¼ 0. The error bars represent the

standard deviation.


DOI 10.1002/bit

McGrew, 1968). In addition, although much weaker than

tangential flows, a radial outflow of fluid is observed along

the rotor surface and an inflow at the stator surface.

CFD Modeling of Original CSAF Device

Once validated, the CFD model was used to investigate the

shear stress profile at the membrane surface of the original

CSAF device, which utilizes a rotor having a fixed angle of 48and a gap height of 0.2 mm between the rotor tip and the

membrane surface. When the system is operated at a filtrate

flux of 125 L/m2�h and its optimal rotor speed of 700 rpm

(Vogel et al., 2002), CFD simulations show that the shear

stress at the membrane surface has a complex dependence on

radial position, increasing to a local maximum near x¼ 0.1

and then declining before increasing abruptly at x above ca.

0.55 (Fig. 7). The presence of filtrate flux lessens the

formation of the boundary layers (Fig. 8) compared to what

was observed in absence of filtrate flux (Fig. 6). As a result,

smooth velocity profiles are observed under a larger portion

of the rotor, with the onset of boundary layer formation at the

membrane and rotor surfaces occurring at x ca. 0.55. This

emergence of distinct boundary layers, and the associated

steepness in the tangential velocity profiles at x� 0.55,

coincides with the abrupt increase in shear stress (Fig. 7),

indicating that boundary layer formation must be prevented

or at least minimized to achieve constant shear stress across

the entire membrane surface.

In the original CSAF device, Rmem¼ 0.85 �Rrot; that is, the

radius of the membrane Rmem is less than that of the rotor Rrot

to minimize distortions in shear stress profiles at the

Figure 7. Shear profile along the membrane surface for the original CSAF rotor at O¼ 700 rpm (&) and for the CFD optimized rotor at O¼ 250 rpm (~)

when Rrot¼ 35 mm and filtrate flux is 125 L/m2�h.

Figure 8. Tangential velocity profiles under the original CSAF rotor at various radial positions: (a) x¼ 0.29, (b) x¼ 0.57 (c) x¼ 0.71, (d) x¼ 0.86 when

O¼ 700 rpm, Rrot¼ 35 mm and the filtrate flux is 125 L/m2�h.



membrane surface due to rotor edge effects. Although

constant shear stress at the membrane surface was not

achieved, the original CSAF devicewas shown to be effective

at both cell removal and product capture. When combined

with our CFD results, this indicates that a threshold shear

stress value exists for a given filtrate flux above which cells

are effectively swept off of the membrane. At a rotor speed of

700 rpm and a filtrate flux of 125 L/m2�h, the shear stress fallsto a local minimum of ca. 0.17 Pa near x¼ 0.5, suggesting

that the satisfactory performance of the original CSAF

technology is due to meeting or exceeding this threshold

shear stress at all radial positions while at the same time

creating a constant pressure at the membrane surface to

ensure a homogeneous filtrate flux. The CFD results show

that the latter condition is met, as the variations in pressure at

themembrane surface aremore than two orders ofmagnitude

less than the transmembrane pressure.

As the high rotor velocities required to achieve a threshold

shear stress of 0.17 Pa adversely affect scaleability of the

original CSAF system, we explored the use of in silico CFD

simulations to reshape the rotor such that constant shear

stress at or above 0.17 Pa was observed across the membrane

surface at significantly reduced rotor speeds.

Optimization of CSAF Rotor Geometry

The geometry of the CSAF rotor was optimized by adjusting

the shape of the original rotor based on the calculated shear

stress profile at the membrane surface and a secant-type root-

finding method based on the approximate simplifying

assumption that the local shear stress at a given radial

position r decreases linearly with h(r), the distance between

the rotor and membrane at r. The rotor height was thereby

decreased by a calculated weighting factor at radial positions

characterized by low shear stress at the membrane surface,

and vice versa. Throughout the optimization process, the

solvent flux through the membrane was set at 125 L/m2�h,consistent with the axial velocity through the membrane

during steady-state operation of the original device. The

hydrodynamics of the newly designed rotor were then

determined by CFD simulation, and the process repeated

(seven iterations in total) until a design was achieved that

produced a constant shear stress profile everywhere except

near r¼ 0; Or is zero at the center-point of the rotor

irrespective ofO. As a result, the shear stress tends to zero asr approaches zero.

The shape of the CFD optimized rotor is compared to the

original rotor design in Figure 9. Due to its variable angle of

incidence with the membrane, the new rotor creates a

significantly more desirable shear stress profile at the

membrane surface (Fig. 7). Furthermore, because of the

reduced rotational velocity and the refinement in rotor shape,

the onset of boundary layer formation in the tangential

velocity profiles (Fig. 10) occurs at x ca. 0.86; that is, at a

radial position beyond the membrane. As a result, no abrupt

changes in shear stress are observed. Instead, uniform shear

stress is achieved at all radial positions except directly

beneath the rotor center-point and at the outer edge of the

membrane stack where shear stress increases. More impor-

tantly, the threshold shear stress value of 0.17 Pa is achieved

across virtually the entire membrane surface at a rotor speed

of ca. 250 rpm, 60% less than the rotor speed required in the

original CSAF device. Consequently, the new rotor should

greatly improve our ability to scale-up the technology. In

addition, with this change in rotor geometry, the pressure

remains essentially constant across the entire membrane

surface, varying by less than 1% of the transmembrane

pressure and thereby providing uniform filtrate flow through

the affinity membrane stack.

It is important to note that our objective function for rotor

optimization sought to achieve a uniform surface shear stress

of 0.17 Pa at the lowest possible rotor velocity and at a filtrate

flux of 125 L/m2�h. As different host cells or culture

conditions could require the CSAF to be operated at different

Figure 9. Schematic of the CFD optimized rotor shape (A) compared to the original CSAF rotor having a fixed angle of 48 (B).

Figure 10. Tangential velocity profiles under the CFD optimized rotor at

various radial positions: (a) x¼ 0.29, (b) x¼ 0.57, (c) x¼ 0.71, (d) x¼ 0.86

when O¼ 250 rpm, Rrot¼ 35 mm, and the filtrate flux is 125 L/m2�h.


DOI 10.1002/bit

rotor speeds or filtrate fluxes, we investigated the perfor-

mance sensitivity of the optimized rotor to these two process

variables. Increases in filtrate flux up to 50 times that used in

the original CSAF studies (Vogel et al., 2002) had no effect on

the shear stress profile at themembrane surface due to the fact

that Vm remains much smaller than the tangential velocities

in the rotor chamber. The effect of the rotational velocity on

the performance of the optimized rotor is shown in Figure 11.

Rotor velocities between 200 and 500 rpm yield essentially

uniform shear stress at the membrane surface, with the value

of the surface shear stress increasing with increasing rotor

speed.At 700 rpm, the shear stress at themembrane surface is

less uniform, showing a small local maximum followed by a

shallow local minimum, due to the increased effect of

turbulent secondary flows. Thus, the new rotor produces the

desired shear stress profile for a wide range of rotational

velocities. However, if the rotational velocity required to

achieve sufficient surface shear stress to clear retained cells

from the membrane lies below 100 rpm or above 500 rpm,

different rotor geometries may be required. The CFD

simulations reported here provide an efficient and effective

means for designing such rotors.

CONCLUSIONS

Close agreement of velocity profiles determined by CFD

simulations with those determined by PIV experiments and

an appropriate numerical model indicate the capability of

CFD for modeling the hydrodynamics within the rotor

chamber of the CSAF technology. CFD simulations of the

original CSAF design show that unnecessarily high rotor

velocities are required to achieve sufficient shear stress to

clear cells from the membrane surface and ensure a uniform

filtrate flux and pressure over the entiremembrane surface.At

the bench-top scale, such high rotor speeds are not

problematic; however, they become disadvantageous during

scale-up as they will limit rotor size: the larger the rotor, the

higher the rotor tip velocity, resulting in regions of high shear

stress that promote cell lysis. CFD simulationswere therefore

used to redesign the rotor to attain comparable and uniform

surface shear stresses at lower rotational velocities. The CFD

optimized rotor provides uniform surface shear stress,

reaching the required threshold value of 0.17 Pa at a

rotational velocity that is 60% lower than that required in

the original device. The reduced rotational velocity will

permit the maximum allowable rotor diameter to be

increased by a factor of 2.5 over that possible in the original

design. This will result in over a sixfold increase in the

effective filtration area and, thus, the volumetric throughput

of the device.

Once the maximum size of the device has been reached,

further scale-up of CSAF technology can be attained through

scaling by number. A novel approach to accomplish this is to

stack CSAF units one atop another and use a central shaft to

drive all rotors with a single motor or mag-driven gearbox. A

small cylinder at the center of the affinity membrane stack

must be removed. However, as the shear stress near the center

of the membrane approaches zero, the removal of this small

amount of membrane has a relatively small influence on

throughput, yet allows constant shear stress at the membrane

surface and uniform filtrate flux to be achieved across the

entire device. This in turn should lead to optimal dynamic

capacities and increase the sharpness of the product break-

through curve from the associated membrane chromatogra-

phy column.

NOTATION

B shear stress proportionality constant (—)

Da Darcy number (—)

g1 assumed axial velocity function in domain 1 (—)

g2 assumed axial velocity function in domain 2 (—)

h gap height between rotor and porous domain (m)

Hcham rotor chamber height (m)

Ho thickness of the membrane stack (m)

K permeability (m2)

p pressure (Pa)

r radial position (m)

Figure 11. Shear profiles along the membrane surface for the CFD optimized rotor at different rotor speeds: 200 rpm (&), 300 rpm (~), 400 rpm (^),

500 rpm (þ), and 700 rpm (�). Rrot¼ 35 mm and filtrate flux is 125 L/m2�h.



Rcham rotor chamber radius (m)

Rin inlet line radius (m)

Rmem membrane stack radius (m)

Rpos radial position of inlet and retentate lines (m)

Rret retentate line radius (m)

Rrot rotor radius (m)

u fluid velocity in porous domain (m/s)

Vm flux through bottom membrane (m/s)

v fluid velocity in rotor chamber (m/s)

z axial position (m)

� Reynolds number of flow through membrane (—)

rotational Reynolds number (—)

" porosity (—)

z non-dimensionalized interface height (—)

� non-dimensionalized axial position (—)

� tangential position (deg)

�1 assumed tangential velocity function in domain 1 (—)

�2 assumed tangential velocity function in domain 2 (—)

� viscosity (kg/m � s)� kinematic viscosity (m2/s)

� non-dimensionalized radial position (—)

� density (kg/m3)

� rotor gap height (m)

shear stress (Pa)

� angle of incidence between rotor and membrane (rad)

O rotational speed (rad/s)

APPENDIX

The physical configuration of the simplifiedCSAF system for

which a two-domain model could be derived, and an

analytical solution obtained for tangential fluid velocities

within the rotor chamber and membrane stack is shown in

FigureA.1. Fluid domainD2, inwhich is placed a fixed-angle

rotor of infinite radius, is bounded below by a second domain

(D1) comprised of an isotropic porous medium (the

membrane stack) with porosity " and permeability K. The

principal equations from which the two-domain model is

derived are provided in Section ‘‘Numerical Solution for Low

Reynolds Numbers,’’ and the reader is encouraged to read

that section before proceeding.

Thevelocity components are given byEquations (5) through

(7). Substitution of these expressions into the Brinkman-

extendedDarcy equation (Eq. 4) and using the solution strategy

described in Section ‘‘Numerical Solution for Low Reynolds

Numbers’’ for the simplified version of this model yields the

following set of ordinary differential equations:

�HðxÞ"Rrot

�1ð�Þd

d�g1ð�Þ

� �� g1ð�Þ

d

d��1ð�Þ

� ��

þ d2

d�2�1ð�Þ �

"HðxÞ2

K�1ð�Þ ¼ 0 ðA:1Þ

�HðxÞRrot

��2ð�Þ

�d

d�g2ð�Þ

�� g2ð�Þ

�d

d��2ð�Þ

��

þ d2

d�2�2ð�Þ ¼ 0 ðA:2Þ

d4

d�4g1ð�Þ

� �� 42HðxÞ3

�"R3rot

�1�d

d��1ð�Þ

� �

� �HðxÞ"Rrot

g1ð�Þd3

d�3g1ð�Þ

� �� "HðxÞ2

K

d2

d�2g1ð�Þ

� �¼ 0

ðA:3Þ

d4

d�4g2ð�Þ

� �� 42HðxÞ3

�R3rot

�2ð�Þd

d��2ð�Þ

� �

� �HðxÞRrot

g2ð�Þd3

d�3g2ð�Þ

� �¼ 0 ðA:4Þ

where H(x)¼Hoþ h(x). Domain 1 is defined as:

(x,\eta)¼ [0,1],[0,z] and Domain 2 as: (x,\eta)¼ [0,1],[z,1]where z is the ratio Ho/H and represents the interface

between the fluid and porous domains.

In the limit where�! 0, Equations (A.2) and (A.3) reduce

to:

d2

d�2�1ð�Þ �

"HðxÞ2

K�1ð�Þ ¼ 0 ðA:5Þ

d2

d�2�2ð�Þ ¼ 0 ðA:6Þ

The boundary conditions are: no slip at the rotating wall and

lower boundary of the porous medium; equivalent fluid

velocities at the interface of the two domains and a

discontinuity in the shear stress at the interface which, as

proposed by Ochoa-Tapia and Whitaker (1995a,b), is

inversely proportional to the permeability of the porous

medium:

�1ð0Þ ¼ 0 ðA:7Þ

�2ð1Þ ¼ 1 ðA:8Þ

�1ðzÞ ¼ �2ðzÞ ðA:9Þ

�01 zð Þ ¼ "�0

2ðzÞ þ "BK

HðxÞ2

!�1=2

�1ðzÞ ð10Þ

where B is an empirical constant determined, in this case, to

be 0.15 through comparison to CFD results.

These equations, coupled with the boundary conditions,

can be solved analytically at a specified value of � to yield thefollowing expressions for the tangential velocities in the

porous and fluid domains:

�1ð�Þ ¼ C1e

ffiffi"

pffiffiffiDa

p �

� �þ C2e

�ffiffi"

pffiffiffiDa

p �

� �ðA:11Þ


DOI 10.1002/bit

and where Da is the Darcy number, Da¼K/H2.

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�2ð�Þ ¼ C3� þ C4 ðA:12Þ

where,

C1 ¼�"

ffiffiffiffiffiffiDa

pe

�ffiffi"

pffiffiffiDa

p z

� �

e�2ffiffi"

pffiffiffiDa

p z

� �½ðz� 1Þð

ffiffiffi"

pþ "BÞ� þ ðz� 1Þð

ffiffiffi"

p� "BÞ � "


pðA:13Þ

C2 ¼"ffiffiffiffiffiffiDa

pe

�ffiffi"

pffiffiffiDa

p z

� �

e�2ffiffi"

pffiffiffiDa

p z

� �ðz� 1Þð ffiffiffi

"p þ "BÞ½ � þ ðz� 1Þð ffiffiffi

"p � "BÞ � "


pðA:14Þ



optimizing the rotor design for controlled-shear affinity filtration using computational fluid...

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