fractionation of non-brownian rod-like particle...

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Chemical Engineering Science

0009-25

doi:10.1

� CorrE-m

PleasChem

journal homepage: www.elsevier.com/locate/ces

Fractionation of non-Brownian rod-like particle suspensions in aviscoplastic fluid

A. Madani a, S. Storey a, J.A. Olson a, I.A. Frigaard a, J. Salmela b, D.M. Martinez c,�

a Department of Mechanical Engineering, University of British Columbia, 2054-6250 Applied Science Lane, Vancouver, BC, Canada V6T 1Z4b VTT Process, P.O. Box 1603, FIN-40101 Jyväskylä, Finlandc Chemical and Biological Engineering, University of British Columbia, 2360 East Mall, Vancouver, Canada V6T 1Z3

a r t i c l e i n f o

Article history:

Received 2 April 2009

Received in revised form

12 November 2009

Accepted 13 November 2009

Keywords:

Non-Newtonian fluid mechanics

Viscoplastic fluid

Sedimentation

Papermaking suspensions

09/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ces.2009.11.017

esponding author.

ail address: [email protected] (D.M. Mart

e cite this article as: Madani, A., eical Engineering Science (2009), do

a b s t r a c t

In this work we explore a fractionation technique for non-Brownian rod-like particle suspensions based

upon the control of the threshold for motion in a yield stress fluid. The principle is demonstrated by

observing the motion of particles under the influence of a centrifugal force in a weak gel. Here we

develop calibration curves of the force required to initiate motion in a gel under numerous

configurations of the particles. Demonstration separations of bidisperse suspensions are reported.

Here we achieve complete separation of dilute suspensions based upon length, diameter, or density. The

method is then applied to an industrially important suspension, that is a polydispersed papermaking

fibre suspension, in which a length-based fractionation is reported.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The focus of the present work is the study of separation orfractionation of different classes of non-Brownian particles in ayield stress fluid. The motivation for the present work stems froman interest in one particular particle suspension, i.e. papermakingfibre suspensions. Papermakers fractionate to allow targetedprocessing of the low quality fraction to improve productperformance while minimizing the amount of processing andsubsequent energy, chemicals and capital required. The pulpfractions can also be used to produce papers with radicallydifferent properties. For example, fractionating the short fibrefraction from the long fibre fraction can create a short fibred paperthat has excellent printing characteristics and a long fibred paperwith high tensile strength. Further, the fibre fractions can beblended to create a spectrum of products designed for each of themany paper applications.

There are essentially two industrial methods to fractionatepulp fibres in use today: pressure screens and hydrocyclones. Inpressure screen, the fibre suspension is fed into the annular gapbetween a rotor and an outer cylinder containing small apertures.The screen apertures are either narrow slots (as small as 0.1 mmwide) or small diameter holes (as small as 0.8 mm in diameter).The surface of the cylinder can either be smooth or be contoured.

ll rights reserved.

inez).

t al., Fractionation of non-i:10.1016/j.ces.2009.11.017

Separation here is based upon length as the small particles passthrough the apertures. Several studies have been completed tounderstand how the operation and design of screens and screen-ing systems affect fibre fractionation efficiency and generalreviews of the topic are given by Sloane (2000) and Julien SaintAmand and Perrin (1999).

With regards to the second fractionation method, there are anumber of experimental studies that have shown that hydro-cyclones are able to separate fibres based on its specific surface(fibre surface area per gram). This results in a separation ofearlywood, large diameter thin walled fibres from latewood, smalldiameter, thick walled fibres, e.g. Paavilainen (1992). In addition,hardwood vessel elements have been shown to be effectivelyremoved by hydrocylones. In a particularly interesting study byVomhoff and Grundström (2003) a chemical pulp was separatedusing a system of five hydrocyclone stages and the resulting twodistinct fractions were refined separately and the qualitydetermined. The one fraction contained 89% earlywood fibresand the other 46% earlywood fibres. The unrefined fine fibrefraction was shown to have a tensile strength 3.5 times that of thecoarse fibre fraction. After refining, the coarse fibre fractiontensile increased to that of the fine fibre fraction but requiredsignificantly more refining.

The efficiency for fractionation in both hydrocyclones andpressure screens is known to be relatively low. We feel that theinherent separation potential is not realized because of thecomplexity of the flow within theses devices. A stochastic elementis introduced into the system through turbulence, the presence of

Brownian rod-like particle suspensions in a viscoplastic fluid.

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boundaries, flocculation, and long range hydrodynamic interac-tions and these effects diminish the efficiency of the separationprocess. In this work we propose a new principle for fractionationand present a series of experiments demonstrating its utility. Herewe show that fractionation can indeed proceed based on eitherlength, diameter or density in a mechanistic fashion. In Section2.1, the literature review, we present the background materialleading up to this novel principle. Here we demonstrate first whyseparation with a Newtonian fluid is difficult (but not impossible)and then proceed to develop and justify the novel principle. InSections 3 and 4, simple demonstration experiments highlightingthis principle are described and the results discussed. Theforeseeable limitations of this process are discussed in Section 5.

2. Prior work

2.1. The underlying fractionation mechanism

Understanding the relative motion of non-Brownian particlesin a flowing fibre suspension undergoing fractionation is difficult,if not impossible. Insight into this mechanism can be gained byexamining the simplest case of the motion of an isolated rod, in anunbounded fluid, settling slowly under the influence of gravity.This work was performed originally by Batchelor (1972), andsubsequently extended by Mackaplow and Shaqfeh (1998), andindicates that the terminal velocity u is given by a function of theform

u¼Drd2f16m ½ðln 2rþ0:193þOðln 2rÞ

�1Þg

þðln 2r�1:807þOðln 2rÞ�1Þðp � gÞp� ð1Þ

where r is the aspect ratio of the fibre defined by L=D; m is theviscosity of the fluid; and p is a unit vector that indicates fibreorientation. The bold font indicates a vector quantity. Thisequation implies that unlike spheres, fibres can have significantmotion perpendicular to gravity; the drift velocity is stronglydependent upon its orientation. Jayaweera and Mason (1965)indicate that with Reo0:01, cylinders do not tend to rotate duringsettling—they tended to fall in the same attitude with which theywere released. The key finding here is that for cylindrical particles,the terminal velocity is not a unique value; it is also related to theaspect ratio and orientation of the cylinder. With these findings,one can argue that a mechanistic means of separating particlesexists. Under more realistic conditions, i.e. at elevated concentra-tions and Reynolds numbers, disturbances must be introducedinto the suspension which would result in a distribution in theterminal velocity. We shall consider two classes of disturbancesbelow and characterize the resulting distributions in the terminalvelocity.

In the first case, category 1, we consider disturbances in thesuspension created by the presence of other particles. Theconcentration of the fibre suspension need not be large and werestrict our argument to the influence of the ‘wake’ of one particlemoving in the vicinity of another. This phenomenon is not new.With swarms of settling particles Happel and Brenner (1965)explain that the physics becomes more complex as eachindividual fibre settles and rotates under the influence of the‘wakes’ or ‘long-range hydrodynamic disturbances’ of the othersettling particles. This leads to inhomogeneous settling rates andlocal floc formation. With monodisperse glass fibre suspensions,floc formation has been observed by Kumar and Ramarao (1991)who noted that as the fibre concentration increased, the numberof flocs increased and caused greater hindrance effects. Herzhaft

Please cite this article as: Madani, A., et al., Fractionation of non-Chemical Engineering Science (2009), doi:10.1016/j.ces.2009.11.017

and Guazzelli (1999) found that in the dilute regime, theensemble-averaged settling velocity actually increases with con-centration and may exceed the velocity of an isolated particle.They also observed that under these conditions most of the fibreswere aligned in the direction of gravity. In the semi-dilute regime,however, the sedimentation velocity decreased according to theRichardson and Zaki (1954) correlation. There are only a fewnumerical simulations available in the literature which attempt todescribe this process under these conditions (Mackaplow andShaqfeh, 1998; Butler and Shaqfeh, 1989; Koch and Shaqfeh,1989). In general these works were conducted in the limit ofRe-0, using slender body theory, and suggest that the settlingsuspension should segregate into particle clumps. The simulationswere in good agreement with experimental results of Herzhaftand Guazzelli (1999).

We continue this discussion by considering the secondcategory of disturbances which are created at elevated Reynoldsnumbers. Our attention is focussed on this regime as this Re rangerepresents the range relevant for papermaking fibres settling inwater (Marton and Robie, 1969). It has been shown that, unlike inthe Stokes’ regime, at Reynolds numbers Re�Oð1Þ isolatednon-spherical particles, in the dilute limit tend to exhibitpreferential orientation during settling (Jayaweera and Mason,1965; Feng et al., 1998; Jianzhong et al., 2003). The torqueinduced on thin cylinders causes the body to rotate into a stableposition with its symmetry axis aligned horizontally. At elevatedconcentrations, the long range hydrodynamic interactions perturbthe flow field to the point where recirculation or swirling likestructures are apparent (Holm et al., 2004). In later work Salmelaet al. (2007) completed a more comprehensive study using asimilar index-of refraction matching technique and studied themotion of settling particles as a function of concentration, aspectratio, fluid viscosity, and fibre length for both monodisperse andbidisperse suspensions. What sets this work apart is that themotion of tracer fibres could be tracked three-dimensionally.Hence fibre fractionation could be studied in this equipment. LikeHerzhaft and Guazzelli (1999), Salmela et al. (2007) reported non-monotonic behaviour in the average settling velocity as a functionof concentration with the maximum velocity occurring at avolume fraction of f¼ 0:05%. With papermaking fibres, non-monotonic behaviour has also been observed with the maximumin the initial settling speed occurring at a crowding number ofN� 16, see Holm et al. (2004). Salmela et al. (2007) also indicatedthat the orientation state of the suspension also displayedcomplex behaviour under these conditions. They reported thatthe suspension was preferentially oriented in the horizontal stateat low concentrations and then adopted an increasing proportionof fibres in the vertical state as concentration increased.

We now turn to what we consider the key finding fromunpublished results of Salmela and his coworkers, which shedslight onto the mechanistic understanding of the ability tofractionate based upon settling. Although not reported in Salmelaet al. (2007), the results stem from this study. Here they examinethe motion of a bidispersed suspension composed of glass rods oftwo different aspect ratios r¼ 23 and 50 and measured theensemble-averaged settling velocity of each class of particle.These authors measured this as a function of the initialsuspension concentration, see Fig. 1. Here the results have beenmade dimensionless by scaling with the average of the isolatedvelocity of the two different particles settling in the verticalposition; these values were determined by experiment. What isclear from this figure is that under dilute conditions each fibrefraction settles at statistically different velocities. In this regimefractionation may be possible. With increasing concentration, thedifferences between the terminal velocities diminish until theirdifference is indistinguishable.


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10−2 10−1 1000.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Dim

ensi

onle

ss V

eloc

ity

Volume Concentration φ [%]

Fig. 1. Dimensionless mean settling velocity versus volume concentration f for abidispersed suspension composed of equal masses of particles with aspect ratios of

r¼ 50% (&) and r¼ 23% (�). The uncertainty in the estimate is reported as the95% confidence interval. The Reynolds number in this case based upon the average

fibre length was estimated to be Re� 0:12. This result is an unpublished resultfrom the study given in Salmela et al. (2007).

A. Madani et al. / Chemical Engineering Science ] (]]]]) ]]]–]]] 3

To summarize the existing literature, we find that for isolatedparticles in an unbounded fluid the terminal velocity is related tothe density, diameter, aspect ratio and orientation of the cylinder.Under creeping flow conditions with extremely dilute suspen-sions, we can draw the conclusion that we can indeed separatefibres based upon the terminal velocity, if and only if, theorientation distribution remains constant during descent. We findthat at elevated concentrations, for all Re above the creeping flowconditions, long range hydrodynamic disturbances lead to flocformation and swirling chaotic structures. Salmela et al. (2007)have shown that there is no statistical difference between theterminal velocity at any reasonable concentration. This presentstudy is motivated from these findings.

2.2. Towards a novel fractionation principle

From the above review we see that the motion of a particlesuspension at any industrially relevant concentration andReynolds numbers is extremely complex. Chaotic behaviour isevident due to long-range hydrodynamic interactions. Since theseobservations were made under ideal settling conditions, weanticipate an even richer behaviour to occur in the complexthree-dimensional, time-dependent, turbulent flow fields encoun-tered in industrial hydrocyclones Ko (2005), Narashima et al.(2007), Nowakowski and Dyakowski (2003), Bergstrom andVomhoff (2007), Hsieh and Rajamani (1991), Sevilla and Branion(1997), Boysan et al. (1982) and Ma et al. (2000). As a result,instead of trying to improve an already complex coupled fluid–fibre interaction problem, we attempt to devise a new separationprinciple which is not hindered by long-range disturbances. Wefeel that this may be achieved through use of a yield stress fluid.1

To introduce our novel fractionation principle we consideronce again the illustrative problem of an isolated particle settlingin a yield stress fluid. With yield stress fluids, the settling problemis more complex: the presence of a yield stress implies that

1 Everyday examples of yield stress fluids are Ketchup, Mayonnaise, and salad

dressing. With these fluids, the yield stress is the applied stress that must exceed

in order to make the fluid flow. Ketchup has a yield stress of approximately 15 Pa.


settling can only occur if the net applied force Fa is greater thanthe resistive force due to the yield stress of the material. Despitethe simplicity of the problem, the mechanism by which theparticle settles is poorly understood. Clearly a critical force isrequired to initiate the motion of the particle. To highlight this,consider an example a particle immersed in a material having ayield stress ty. If Fa is not sufficient to overcome the yield stressthen the object will be suspended indefinitely. Motion willcommence when

Fa4tyAe ð2Þ

where Ae is the surface area over which the force due to the yieldstress is applied. The exact value for Ae however remains an openquestion. The above relationship can be made dimensionless byscaling each side by a characteristic area of the particle. In thiscase if we divide each side of the equation by D2, we can define adimensionless force ratio F which represents the criteria formotion

F ¼ FatyD2

4AeD2

ð3Þ

The bound where motion begins is defined as the critical forceratio Fc .

The key to the separation is the determination of Ae. A numberof researchers have attempted to determine Ae through simula-tion and we divide the methods into four groups. In the firstcategory are estimates made through an assumption regardingthe shape of Ae. For example, through a simple force balanceAndres (1961) argues that for spherical particles Ae is related tothe projected area of the particles and reports that motion shouldcommence when Fc ¼ 4:8. In the second category are regulariza-tion models in which a continuous function is used to approx-imate the rheological behaviour of the yield stress fluid (Blackeryand Mitsoulis, 1997; Liu et al., 2002; Beaulne and Mitsoulis, 1997;Jie and Zhu, 2006). In essence this approach avoids the singularityinherent with strain rate decreasing to zero in unyielded regions.A detailed review of this is given in Frigaard and Nouar (2005).These authors advance the argument that these regularizationmethods converge to the correct flow field, but cannot recover theeffective area properly. The third category of methods involvesdomain mapping. In this category falls the work of Beris et al.(1985), which we consider to be the benchmark paper in this area.Combining a regularized model with an intricate mapping of theyield surfaces onto a standard domain, they were able to calculatethe position of the yield surface of a settling sphere veryaccurately for one class of yield stress fluids, that is a Binghamplastic.2 By doing so they argue that two yield surfaces areevident: a kidney shaped surface in the far field and twosomewhat triangular-cusps attached to the leading and trailingedges of sphere (Fig. 2). With this, these authors indicate thatparticle motion would commence when

Fc ¼VDrgtyD2

� 7p2� 11 ð4Þ

This expression is approximately 3 times greater than thatgiven by Andres (1961) and are closer in magnitude to theexperimental measurements of Tabuteau et al. (2007), Attapatuet al. (1995), Jossic and Magnin (2001), and Laxton and Berg(2005), i.e. 16oFc o25. In the fourth category are simulationsconducted using augmented Lagrangian schemes (Glowiniski andLe Tallek, 1987; Fortin and Glowiniski, 1983). Roquet and

2 A Bingham plastic is a viscoplastic material that behaves as a rigid body at

low stresses but flows as a viscous fluid at high stress. When flowing, there is a

linear relationship between the applied stress and the strain rate.


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U

(1)

(2)

-g

Fig. 2. Schematic illustrating the yielded (region (1)—white) and unyielded flowregions (region (2)—blue), according to the numerical results by Beris et al. (1985).

This figure was reproduced from Putz et al. (2008). (For interpretation of the

references to colour in this figure legend, the reader is referred to the web version

of this article.)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.356

7

8

9

10

11

12

13

14

15

shear rate (1/s)

shea

r stre

ss (P

a)

0 5 10 150

2000

4000

6000

8000

10000

12000

14000

Shear Stress (Pa)

Vis

cosi

ty (P

a.s)

Fig. 3. (a) The local viscosity versus shear stress for a 0.16% Carbopol solution. Theyield stress is given at the maximum in this curve (b). A typical rheogram of shear

stress versus shear rate.


Saramito (1983) combined the augmented Lagrangian methodwith anisotropic grid refinement to obtain very accuratenumerical results and have applied this method successfully tothe flow around settling cylinders.

We now turn our attention to the experimental literature. Anextensive summary of settling and sedimentation experiments indifferent media, including viscoplastic fluids, can be found in thebook by Chhabra (2007). This review was published in 2007 andfocuses generally on the calculation of the drag coefficient and theterminal velocity, i.e. engineering properties useful for designpurposes. Recently there has been renewed interest in thisproblem due to the availability of (digital) particle image analysis(PIV). Gueslin et al. (2006), for example, used this technique tomeasure the flow field of a spherical particle settling in Laponite,an extremely thixotropic yield stress fluid. The objective of thiswork was to study the aging properties of this fluid. Putz et al.(2008) experimentally estimated Ae through flow visualizationexperiments for two different size spheres settling in Carbopolsolutions. Their results indicate that the shape of the yield surfaceapproximates that of an ovoid spheroid with its major axisapproximately 5 times greater than the radius of the particle.

To summarize, what is clear from this body of work is thatduring settling the flow is confined in the vicinity of the particlewithin an envelope the size of which is related to the yield stressof the material. For particles to settle, a critical force must beapplied to overcome the resistance created by the yield stress.Relatively little is known about the shape of this surface and themagnitude of the applied force to create motion. What can be saidis that the unyielded envelope is larger than the body itself but itsshape has yet to be determined rigorously.

What is key in this description is that the differences in staticstability of particles in a yield stress fluid, under the influence ofan applied force, may represent a novel criterion for separation.We expand upon this idea to highlight the principle. In this study,we consider two different particles of equal dimensions but ofdifferent densities. Each particle is spun in a centrifuge at exactlythe same angular velocity and at equal radial positions from theaxis of rotation. In other words they experience the sameacceleration field. The particle with greater density, however,experiences a larger centrifugal force as it has a larger mass. If weapply Eq. (2), we see that separation of these two particles willoccur if Fa1 4tyAe4Fa2 , where Fa1 and Fa2 are the centrifugalforces applied to each particle. This simple example leads us tothe question: Can we choose an applied force and a yield stress inwhich separation of particles is achieved? In this work weattempt to address this question and devise simple experimentsto illustrate this principle.


3. Experimental details

In this work we measure the critical force required to initiatemotion of a particle in a yield stress fluid under the action of acentrifugal force. Two different centrifuges were used dependingupon the force required. For low rotational speed experiments anin-house centrifuge was built. In this case, the centrifuge, whichwas made from plexiglass, was 250 cm in diameter, and 5 cmthick and was driven by a motor and controller, with a precisionof 5 rpm, up to a maximum of rotational rate of 300 rpm. Highspeed experiments were conducted in a Universal Model UVcentrifuge at rotational rates of up to 3000 rpm

The main fluid used in this study was Carbopol-940 obtainedfrom Noveon. In most of the work a 0.16% wt/wt solution wasprepared using DI water and then neutralized using a dilute NaOHsolution. The density of the resulting solution was that of water.The resulting gels were degassed and then allowed to restovernight. Rheological characterizations were performed at roomtemperature in a cone and plate flow geometry on a Bohlinrheometer. Shear sweeps were performed in controlled strainmode typically in the range of 1� 10�221 s�1 in order todetermine the yield stress, a typical rheogram is shown in Fig. 3


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Table 1The experimental conditions for determination of Fc for spherical particles.

D (mm) ty (Pa) N Fc

2.4 0.9 70 29ð70:7Þ2.4 3.1 200 24ð71Þ3.2 0.9 150 23ð71:3Þ3.2 3.3 150 20:3ð70:4Þ4.0 3.1 75 17:4ð70:3Þ4.0 3.3 170 17:3ð70:3Þ4.8 3.1 150 14:8ð70:3Þ4.8 3.3 170 14:4ð70:2Þ5.6 3.1 70 13:6ð70:5Þ5.6 3.3 130 13:2ð70:3Þ

The spheres in this case all had a density of 7800 kg=m3. N represents the number

of replicates and the uncertainty in Fc is reported at the 95% confidence interval.

Table 2The experimental conditions for determination of Fc for cylindrical rods with their

axes oriented normal to the direction of force.

ty (Pa) r (kg=m3) D (mm) L=D N Fc

3.3 7800 1.6 6.0 90 190ð73Þ3.3 7800 1.6 12.0 50 372ð79Þ3.3 7800 1.6 16.0 55 517ð717Þ3.3 7800 1.6 2.6 110 83ð71Þ3.3 7800 1.6 4.0 100 127ð72Þ3.3 7800 3.2 2.0 90 50ð71Þ3.3 7800 3.2 3.0 120 90ð72Þ3.3 7800 3.2 4.0 95 118ð73Þ4.0 8300 1.6 4.0 80 115ð72Þ4.0 8300 1.6 6.0 75 179ð74Þ4.0 8300 1.6 8.0 75 237ð75Þ4.0 8300 3.2 3.0 60 81ð72Þ4.0 8300 3.2 4.0 75 124ð73Þ4.0 8300 3.2 5.0 75 145ð74Þ4.0 8300 3.2 6.0 90 174ð75Þ4.0 8300 4.8 2.6 85 77ð72Þ4.0 8300 4.8 3.3 65 107ð73Þ4.0 8300 4.8 4.0 70 114ð74Þ

N represents the number of replicates and the uncertainty in Fc is reported at the

95% confidence interval.

Table 3The experimental conditions for determination of Fc for cylindrical rods with their

axes oriented parallel to the direction of force.

ty (Pa) r (kg=m3) D (mm) L=D N Fc

4.0 2700 1.6 16.0 50 183ð75Þ4.0 2700 1.6 20.0 50 169ð76Þ4.0 2700 1.6 28.0 30 214ð75Þ4.0 7800 1.6 4.0 85 49ð72Þ4.0 7800 1.6 6.0 75 59ð72Þ4.0 7800 1.6 8.0 80 67ð73Þ4.0 7800 1.6 12.0 90 96ð73Þ4.0 7800 1.6 16.0 70 128ð76Þ4.0 7800 1.6 20.0 90 170ð712Þ4.0 7800 1.6 25.0 55 165ð77Þ4.0 7800 2.4 2.6 70 37ð71Þ4.0 7800 2.4 4.0 80 52ð72Þ4.0 7800 2.4 6.6 95 80ð73Þ4.0 7800 2.4 8.0 95 111ð74Þ4.0 7800 2.4 10.6 65 115ð72Þ4.0 7800 3.2 2.0 100 28ð71Þ4.0 7800 3.2 3.0 120 43ð72Þ4.0 7800 3.2 4.0 100 51ð73Þ4.0 7800 3.2 6.0 80 79ð73Þ4.0 7800 3.2 8.0 65 103ð74Þ4.0 7800 4.8 2.0 80 33ð71Þ4.0 7800 4.8 2.6 80 41ð72Þ4.0 7800 4.8 3.3 70 51ð72Þ4.0 7800 4.8 4.0 65 57ð72Þ4.0 8300 1.6 4.0 90 60ð72Þ4.0 8300 1.6 6.0 100 83ð72Þ4.0 8300 1.6 8.0 100 114ð73Þ4.0 8300 3.2 3.0 80 47ð71Þ4.0 8300 3.2 4.0 80 66ð72Þ4.0 8300 3.2 5.0 100 70ð72Þ4.0 8300 3.2 6.0 90 75ð74Þ4.0 8300 4.8 2.6 90 45ð72Þ4.0 8300 4.8 3.3 80 65ð72Þ4.0 8300 4.8 4.0 75 75ð72Þ

N represents the number of replicates and the uncertainty in Fc is reported at the

95% confidence interval.

Table 4The experimental conditions for determination of Fc for bent cylindrical rods with

their axes oriented normal to the direction of force, see Fig. 4.

L=D S=L N Fc

10 0.51 20 183ð78Þ10 0.70 20 183ð714Þ10 0.89 20 233ð714Þ16 0.59 30 244ð737Þ16 0.71 30 290ð717Þ16 0.83 30 319ð712Þ20 0.50 60 267ð712Þ20 0.63 60 363ð714Þ20 0.82 60 423ð723Þ

In this case the density of the cylinders were 7800 kg=m3, ty ¼ 5:3 Pa, andD¼ 1:6 mm. N represents the number of replicates and the uncertainty in Fc isreported at the 95% confidence interval.


and the yield stress was estimated at the maximum of thecurve shown in Fig. 3(b). Here the uncertainty in the estimate islarge. This is typical of most reported estimates for thisparameter.

A number of different studies were performed in whichdensity, diameter, aspect ratio, shape (degree of curl) ofcylindrical rods were employed. Both rigid and flexible rods wereconsidered. The particles were made from aluminum, brass andstainless steel and were manufactured by McMaster Carr. In ourinitial study, spherical particles were under the conditionsoutlined in Table 1. Here particles were placed in the low speedcentrifuge at a radial distance R from the axis of rotation and thenspun at a fixed angular velocity o for approximately 5 min. At theend of the experiment, the position of the sphere was inspectedand if unchanged (to within a prescribed tolerance), the test wasrepeated by increasing the speed of the centrifuge. The procedurecontinued until motion was induced. With this the critical forceratio to induce motion was estimated using the followingexpression:

Fc ¼DrVRo2

tyD2ð5Þ

It should be noted that the Carbopol solution was pretreatedbefore data collection by allowing a number of spheres to settle in


the solution. Both Attapatu et al. (1995) and Putz et al. (2008)indicate that reproducible settling experiments can be obtained ifapproximately 4–5 spheres are allowed to settle in the gel.

In the second series of experiments we examined the stabilityof cylinders. In Tables 2–5 a large series of experiments wereconducted to measure Fc as a function of the density and aspectratio L=D of the rod, the angle of orientation (relative to thedirection of motion), and its shape, that is the degree of kinkreported as S=L. S=L is defined as the end-to-end length S divided


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Table 5The experimental conditions for determination of Fc for bent cylindrical rods with

their axes oriented parallel to the direction of force, see Fig. 4.

L=D S=L N Fc

10 0.51 60 139ð75Þ10 0.63 60 117ð74Þ10 0.82 60 87ð74Þ16 0.59 50 213ð75Þ16 0.71 50 190ð76Þ16 0.83 50 129ð74Þ20 0.57 50 247ð75Þ20 0.07 55 238ð76Þ20 0.89 40 220ð75Þ

In this case the density of the cylinders were 7800 kg=m3, ty ¼ 5:3 Pa, andD¼ 1:6 mm. N represents the number of replicates and the uncertainty in Fc isreported at the 95% confidence interval.

Fig. 4. Schematic illustration of the orientation of bent rods used in the device. Wedefine ‘parallel’ to indicate the direction of the chord between the ends of the fibre

make in relation to the direction of motion. In (a), we see that this is parallel to the

direction of motion. In (b), the orientation is perpendicular to the direction of the

applied force.

60 80 100 120 140 160 1800

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Pro

babi

lity

Dis

tribu

tion

Fc (dimensionless)

Fig. 5. The histogram (a) and the cumulative probability distribution (b) for Fc mea7800 kg=m3 with a fluid having a yield stress of 4 Pa. Approximately 70 observations w



by the length of the rod. In these studies, the orientation isdefined by the direction of the chord connecting the two ends ofthe rod relative to the direction that the force is being applied. Forthe bent fibres as this is difficult to visualize so a schematic of thisis shown in Fig. 4.

4. Results and discussion

4.1. Spheres

Before proceeding to present the main findings of this section,it is instructive to examine the histograms of one test. As shownin Tables 1–5, a large number of tests were conducted for eachexperimental condition from which we could develop both ahistogram and a cumulative probability distribution. An exampleof this is shown in Fig. 5 where the histogram is smooth andclustered around a definite value. The cumulative distributionrepresents the fraction of particles which have remained unstable.In this case the histogram was skewed towards the lower valuesof Fc and this finding is representative of all cases tested. Thecritical values for each experiment are given in these tables inwhich the uncertainty in the mean is reported at the 95%confidence interval. What is evident is that motion occurs overa range of Fc and variation (standard deviation) in this example is21% of the mean. It should be noted that the example chosenrepresents a data set with one of the largest standard deviations.Generally the results were much better. The error in this casestems from the initial positioning of the particle in the centrifuge.A small error is introduced as it is difficult to repositionthe particle, with the correct orientation, between trials.As a result, a large number of replicates for each test wereconducted to increase our confidence in our estimate of the mean.

At this point we turn our attention to the estimates of Fcmeasured as a function of particle diameter, see Fig. 6. The firstobservation that can be made from this figure is that Fc is a strongfunction of D. We consider this finding quite surprising as there is

80 100 120 140 160 180−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Cum

. Pro

babi

lity

Dis

tribu

tion

Fc (dimensionless)

sured on a 1.6 mm diameter and 25.4 mm length cylinder having a density of

ere made. Cylinder axis of symmetry is parallel to the direction of motion.


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Fig. 7. A demonstration of the fractionation of a bidisperse suspension of sphericalparticles. In (a) an image of the suspension is given before the commencement of

the centrifuge. (b) is the state of the suspension after the application of the

centrifugal force. It should be noted that most of the darker particles are on the

periphery of the centrifuge.

2 2.5 3 3.5 4 4.5 5 5.5 6x 10−3

5

10

15

20

25

D (m)

F c

[27]

[33]

[35, 36, 42, 43]

Fig. 6. Estimate of the critical force required to initiate motion for stainless steelspheres with diameters in the range 2:4 mmrDr5:6 mm. The dotted linerepresents linear regression of the data with a second order polynomial. The

arrows represent literature values for theoretical estimates, i.e. Andres (1961) and

Beris et al. (1985), or experimental works, Attapatu et al. (1995), Tabuteau et al.

(2007), Jossic and Magnin (2001), and Laxton and Berg (2005). The uncertainty in

the estimates are reported at the 95% confidence interval.

300

400

500

600

F c

Parallel

Perpendicular

1.6 mm2.4 mm3.2 mm4.8 mm


no indication of this in the literature. When we re-examine thepertinent works in the literature, i.e. Attapatu et al. (1995) andJossic and Magnin (2001), we conclude that previously reportedresults were obtained from indirect methods; Fc was determinedby extrapolation from measurements of a slowly translatingsphere. The question of ‘slip’ on the surface of the particle was notaddressed in this work whilst making this extrapolation. Inaddition, the uncertainty in the estimates given by Attapatu et al.(1995) and Jossic and Magnin (2001) are within the range ofvariation given in Fig. 6. From this we conclude that the methodsused in the previous literature studies were not sensitive enoughto observe this functional relationship. Like Putz et al. (2008) wetoo must draw the conclusion that the unyielded envelope is notspherical in shape (otherwise it would be independent of D). Ourinterpretation assumes that Carbopol chains do not adsorb on theparticles and do not affect their DLVO interactions or theirdistance of maximum approach.

Next we turn to the main point of this section and we attemptto demonstrate the fractionation principle using one specificexample. Here we create a separation based upon the differencesin densities of two spheres having equal diameter. The spheres areplaced in a centrifuge at equal radial distances from the axis ofrotation and the angular velocity is set. From this we can estimatethe force ratio F1 and F2 acting on each particle during theexperiment. Three different scenarios can be constructed:

200

1.

100

PC

Both F1 and F2 are less than the critical force ratio Fcdetermined in Fig. 6. In this case both particles remain stablytrapped.

2.

0 5 10 15 20 25 300

Both F1 and F2 are greater than the critical force ratio Fc. Heremotion begins for both particles.

3.

L/D

If F14Fc 4F2, particle 2 remains stably trapped while particle1 moves. Under this case separation has occurred. This is theseparation principle

Fig. 8. Measurement of the critical force ratio Fc to cause motion in isolatedcylinders of various aspect ratios. The data in this figure are also shown in Tables 2

and 3 where the number of replicates and the confidence interval for each data

point is given. Two sets of data are given: for cylinders oriented parallel to the

direction of the force and cylinders oriented perpendicularly to the direction of the

applied centrifugal field.

To highlight this last scenario we examine Fig. 7(a) where we havecreated a bidisperse suspension of spheres with similar diametersbut vastly different densities. The sphere with the larger density is

lease cite this article as: Madani, A., et al., Fractionation of non-hemical Engineering Science (2009), doi:10.1016/j.ces.2009.11.017

shown in black and that with the smaller density in red. An equalnumber of each type of sphere is distributed throughout thecentrifuge. If the centrifugal force is chosen correctly, as shown inFig. 7(b) separation can indeed occur. Here we see that the heavierspheres migrated towards the periphery whilst the lighter oneswere held by the gel.

4.2. Cylinders

From the previous subsection, we demonstrated that thesegels could be used to create a separation of a bidispersesuspension of spheres. As a proof of principle, we showed anearly perfect separation of a bidisperse suspension of spheresbased upon particle density. In this section we continue to buildon our understanding of this separation and report on the motionof isolated cylinders in a centrifuge. Cylinders are a morecomplicated geometry and we suspect that orientation couldaffect the shape of the unyielded envelope. We present the resultsin this subsection in a similar manner as to those presented in the


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Fig. 9. A demonstration of the fractionation of a bidisperse suspension of cylindrical particles. In (a) an image of the suspension is given before the commencement of thecentrifuge. (b) is the state of the suspension after the application of the centrifugal force.

3 Denier is unit of measurement of linear density of textile fibre mass—calcu-

lated as 1 g per 9000 m. It is related to diameter of the fibre.


previous subsection. First the results for isolated particles aregiven followed by a number of fractionation demonstrationexperiments.

To begin, like spheres we measured Fc for a large number ofcylinders of different aspect ratio, diameter, density, degreeof curvature, and orientation (measured relative to the applicationof force). The results for rigid cylinders are shown in Fig. 8. Thefirst observation that can be made is that with increasing aspectratio, a larger force is required to initiate motion for the cylindersoriented perpendicularly to the direction of force than thoseoriented in the parallel sense. We report these two bounds as wefeel that any orientation state between these two limits should liesomewhere between these two curves. The second observationthat we can see is that for cylinders oriented perpendicular to thedirection of the force, Fc was found to vary nearly linearly withthe aspect ratio of the particle. The third observation that can bemade from Fig. 8 is that for particles oriented parallel to thedirection of the force, at larger aspect ratios, Fc seeminglyapproaches a constant value. This latter finding, however,should be confirmed with larger aspect ratio particles.

In the next stage we attempt to demonstrate this principle bycreating separations based upon either diameter or density. InFig. 9 the cylinders in this case have equal diameter and lengthbut have two different densities. The lighter coloured rod has thelarger density. In Fig. 9(a) an image of the particles is given beforethe commencement of the centrifuge. In Fig. 9(b) the state of thesuspension after the application of the centrifugal force is shown.It should be noted that all of the denser particles moved to theperiphery while the less dense ones remained trapped. In asimilar manner we demonstrate a fractionation based upondiameter. In Fig. 10, the cylinders in this case have equaldensity and length but have two different diameters. The largerdiameter rods are located in the upper left quadrant of the imagein (a). In (a) an image of the particles is given before thecommencement of the centrifuge. (b) is the state of thesuspension after the application of the centrifugal force. It


should be noted that all of the larger diameter particles movedto the periphery while the smaller diameter ones remainedtrapped.

With the principle of separation shown for isolated particles,we attempt to perform a demonstration of this principle withparticles better representing papermaking fibres. We do so in twoseparate studies with nylon fibre suspensions of various lengthand denier.3 As the density of these particles are so small incomparison to the test particles used, the separation wasconducted in the higher speed centrifuge. In the first series oftests, see Fig. 11, nylon fibre suspensions of various denier weresuspended in a Carbopol gel and rotated at increasing rpm. Theportion of the fibres which were stably trapped were recoveredand the number of fibres determined using a Fibre QualityAnalyser (www.optest.com). The orientation of the fibres was notcontrolled in this experiment and should be considered to berandomly distributed. The results indicate that the thresholdrequired to cause motion increased with increasing denier of theparticle. What is evident in this figure is that a completeseparation of say a 1.5 denier particle from a 12 denier particlecould be achieved at 980 rpm. In the second set of tests, asuspension of equal denier nylon fibres but with different lengthsare separated in a centrifuge, see Fig. 12. Here nearly completeseparation was achieved at 1200 rpm.

Following the purpose of fractionating papermaking fibres,having a proof on fractionation using Nylon fibres, we triedseparating wood fibre in the Carbopol under centrifugal force. Todo so, we repeated the same process of Nylon fibres and chosesemi-bleached kraft pulp and made the pulp solution by addingthe fibres into Carbopol. The result presented here is for the fibreconcentration of 0.16% and fluid yield stress of approximately 1 Pa(Fig. 13).


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Fig. 10. A demonstration of the fractionation of a bidisperse suspension of cylindrical particles. In (a) an image of the suspension is given before the commencement of thecentrifuge. (b) is the state of the suspension after the application of the centrifugal force.

Fig. 11. The fraction of nylon particles remaining stably trapped after theapplication of a centrifugal force. The fibres were randomly oriented initially. The

nylon particle were of equal length. Denier is a representation of the diameter of

the fibre.


From Fig. 13 one can see how the length distribution changesafter applying the centrifugal force. The length weighted average,Lw of the initial sample was 2.56 mm with the maximumdistribution of fibres around 3 mm, while after being exposed tocentrifugal force the Lw has dropped to the value of 1.44 mm withmaximum distribution around 1 mm.

5. Limitations—applications to real systems

At this point we are trying to build an understanding of thelimitations of the proposed separation principle. Thus far we have


studied ideal systems both in terms of the definition of theproperties of the particles and in terms of the concentrationregimes tested, i.e. dilute. A small discussion of each is givenbelow.

We begin to test the assumption regarding the shapeof papermaking fibres by including the effects of kink or curlinto the particle. To do so we modified our system by bendingthe cylindrical rods and then measuring Fc in the mannerdescribed above, see Fig. 14. Similar to the rigid bars, the forceis a function of L=D. What is evident is that Fc is quite sensitiveto the bend in the rod. This will need to be addressed infurther work.

In addition, the size of the unyielded envelope places an upperlimit on the concentration of the suspension which can be used.Here, the separation principle will be diminished if the unyieldedzones of adjacent particles overlap. Just like with particle flowwith Newtonian fluids, this represents a long-range hydrody-namic disturbance. At this point we attempt to estimate the sizeof the unyielded zone based upon our experimental results. To doso we conducted a series of tests where spheres of a knowndiameter which were placed at equal R in the centrifuge but atdifferent interparticle spacings. The isolated particle results couldbe obtained when the interparticle spacing was greater than oneparticle diameter.

At this point it is difficult to say if this principle will bemore efficient than existing fractionation technologies,i.e. hydrocyclones or pressure screens, as the work was conductedbatchwise, at low concentrations and with primarily idealizedparticles. Further work is required to ascertain properly theefficiency and limitations of this principle when scaled-upto a continuous operation and compared to existing technologies.The results of this study are not limited to the use ofCarbopol-940. We have used this yield stress fluid convenient aswe have experience with it. Other yield stress fluids could beemployed.


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0 1 2 3 4 5 60

5

10

15

20

L (mm)

Freq

uenc

y (%

)

0 1 2 3 4 5 60

5

10

15

20

L (mm)

Freq

uenc

y (%

)

Fig. 12. The separation of nylon fibres of equal diameters with different lengths. (a) The fibre length distribution of the sample initially. (b) The fibre length distribution ofthe stably trapped fibres after treatment with the centrifuge at 1200 rpm.

0 1 2 3 4 5 6 70

1

2

3

4

5

L (mm)

Freq

uenc

y (%

)

0 1 2 3 4 50

1

2

3

4

5

6

L (mm)

Freq

uenc

y (%

)

Fig. 13. The separation of a softwood bleached Kraft pulp based upon length. (a) The fibre length distribution of the sample initially. (b) The fibre length distribution of thestably trapped fibres after treatment with the centrifuge at 2900 rpm.

−0.2 0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

20

25

30

35

40

S/L (dimensionless)

F cD

/L (

dim

ensi

onle

ss)

L/D = 20L/D = 16L/D = 10

Fig. 14. Measurement of the critical force ratio Fc to cause motion in isolated bentcylinders. The measurements were conducted both as a function of aspect ratio

and degree of kink S=L. Particles oriented perpendicular to the direction of force as

shown as filled symbols. Particles oriented parallel to the direction of force are

unfilled symbols. The uncertainty in the estimates are reported at the 95%

confidence interval.



6. Summary and conclusions

In this work we advanced the argument of a novel separationprinciple. Here we propose that the separation principle is basedupon controlling the threshold for motion of individual classes ofparticles. If the threshold can be controlled to a high enoughresolution then in principle any separation could be achieved. Inthis work we set our threshold by the balance between thecentrifugal force and the yield stress of a gel. Other systems couldbe used to control this to a finer scale.

This principle is strikingly different than that used in presentindustrial or even laboratory devices. The separation is basedupon the difference in the terminal velocities of the particle whichis related to parameters such as density and specific surface. Theefficiency of this separation is diminished due to long-rangehydrodynamic interactions.

We have demonstrated under very ideal conditions thatseparation can indeed be achieved with these types of gels. Wefound out that with rods the separation is based upon the massper unit area of the particle and orientation has a large effect onstability. It was shown that when the cylinders were orientedparallel to the direction of force, the threshold is independent oflength for large L=D. Simple test cases were performed with nylonfibre suspensions in which separation was achieved based uponeither length or diameter.


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Finally, we argued that there are limitations to this proposedsystem before its reduction into practice. First, this system too islimited by the long range hydrodynamic interactions. However,the interactions are on a smaller scale and limited to theinteraction length created by the unyielded envelope. In addition,we see that the separation is based upon the mass per unit area ofthe unyielded envelope. The size and shape of the unyieldenvelope is still an open question in the literature. Understandingthis will allow for an estimate of the upper limit of the efficiencyof this proposed separation technique in terms of concentrationand classes of particles which can be separated from each other.

Acknowledgments

We gratefully acknowledge financial support of the NaturalSciences and Engineering Research Council of Canada through theCollaborative Research and Development program and throughthe support of our partners BC Hydro, Paprican, Catalyst Papers,Howe Sound Pulp and Paper, West Fraser Quesnel River Pulp,Canfor, Andritz, Arkema, Honeywell, WestCan Engineering, Ad-vanced Fiber Technologies, Ontario Power Authority and CEATIinternational. In addition D.M. Martinez would like to thank DrDaniel Ouellet and Mr John Senger for many fruitful discussions inthe initial part of this work.

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dx.doi.org/10.1016/j.ces.2009.11.017

Fractionation of non-Brownian rod-like particle suspensions in a viscoplastic fluidIntroductionPrior workThe underlying fractionation mechanismTowards a novel fractionation principle

Experimental detailsResults and discussionSpheresCylinders

Limitations--applications to real systemsSummary and conclusionsAcknowledgmentsReferences

fractionation of non-brownian rod-like particle...

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