8 (2007) 17–21www.elsevier.com/locate/powtec

Powder Technology 17

Settling velocity of cubes in Newtonian and power law liquids

Neha Agarwal, R.P. Chhabra ⁎

Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur, 208016, India

Received 10 January 2007; received in revised form 1 April 2007; accepted 4 April 2007Available online 12 April 2007

Abstract

New extensive data on the free settling velocity of thirty cubes of various densities and sizes falling in scores of Newtonian and Power lawliquids are reported herein to supplement the existing data, for there is very little prior data on cubes in power law liquids. The new data embracethe range of conditions as follows: sphericity of 0.805; power law index, 0.61 to 1 and consistency index, 0.0078–15.31 Pa sn; Reynolds number,0.0013 to 860. The new results are shown to be consistent with an existing drag correlation which has been tested extensively using the literaturedata for spherical and non-spherical particles falling in Newtonian and power law liquids with acceptable levels of accuracy.© 2007 Elsevier B.V. All rights reserved.

Keywords: Cubes; Settling velocity; Drag coefficient; Power law; Sphericity

1. Introduction

Owing to the frequent occurrence of liquid–solid systems ina wide range of industrial settings, considerable research effortshas been devoted to the development of reliable expressions topredict the free settling velocity of spherical and non-sphericalparticles in Newtonian [1–5] and in power law type non-Newtonian liquids [6–11]. It is customary to express thisinformation using the standard dimensionless parameters,namely, drag coefficient, Reynolds number and sphericity inNewtonian media, with the further inclusion of the power lawindex for settling in power law liquids. Since most of thepertinent studies have been thoroughly reviewed recently[12,13], only their salient features are recapitulated here. Forspherical particles, based on a combination of numerical andexperimental results, satisfactory schemes are now available fora priori estimation of the settling velocity of a sphere inNewtonian media of known physical characteristics [12–14]under most conditions of interest. In contrast, much lessinformation is available on spheres falling in power law fluids[6,9,13,15,16]. Reliable numerical predictions of sphere drag inpower law liquids are now available up to the Reynolds number

⁎ Corresponding author. Tel.: +91 512 259 7393; fax: +91 512 259 0104.E-mail address: chhabra@iitk.ac.in (R.P. Chhabra).

0032-5910/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.powtec.2007.04.002

values of 500 and of the power law index ∼0.5 to 1.8 [16].These ranges are supplemented by experimental results up tothe Reynolds number values of ∼1500 [13,16,17]. Owing totheir generally high viscosity levels, in most applications,settling of particles in power law liquids usually occurs at low tomoderate Reynolds numbers and therefore the upper limit ofRe=1500 of currently available results is not as serious as itmight appear.

In contrast, even less is known about the settling velocity ofnon-spherical particles both in Newtonian and in power lawfluids. Extensive evaluations of the available data against thepredictive correlations have been reported amongst others byChhabra et al. [1], Tang et al. [3] and Yow et al. [4] for regularshaped particles falling in Newtonian media and by Rajithaet al. [8] in power law liquids whereas the prediction of drag ofirregular shaped particles in Newtonian fluids has been dealtwith by Tran-Cong et al. [2]. During the course of theseextensive evaluations, it became clear that very few experi-mental results are available for cubes falling freely even inNewtonian liquids, let alone in power law liquids. For instance,particles of this shape are frequently encountered in foodprocessing applications, e.g. carrot slices are of almost cubicalshape. Aside from such practical applications, there is anintrinsic interest in studying cubes as these constitute regularshaped particles with a constant value of sphericity and thusallow the delineation of the effects arising from the shear-

Table 1Properties of test liquids

(w/w %) Fluid T (K) n (–) m (Pa sn) ρ (kg/m3)

19.5 Glucose 297 1 0.0078 109923 Glucose 298 1 0.0081 111323.8 Glucose 298 1 0.0092 115956 Glucose 305 1 0.417 135467 Glucose 306 1 2.15 137175.9 Glucose 296 1 4.04 138577 Glucose 295 1 4.10 138990.6 Glucose 307 1 8.46 13903.2 CMC 299 0.61 15.31 10182.5 CMC 299.4 0.79 8.27 10171.8 CMC 300.4 0.81 5.08 10071.2 CMC 300.2 0.79 4.25 10090.90 CMC 301 0.79 3.79 1016

18 N. Agarwal, R.P. Chhabra / Powder Technology 178 (2007) 17–21

thinning characteristics of the liquids and from the non-spherical shape of the particles. Hence, such studies contributeto the overall understanding of the settling behavior of non-spherical particles in non-Newtonian fluids. This study aims tofill this gap in the current body of information.

New extensive experimental results are reported in this workon the settling velocity of thirty cubes made of different size andof six different materials falling in several Newtonian andpower law liquids, thereby encompassing wide ranges of theparticle Reynolds number and of the power law index. Detailedanalysis revealed that these results are consistent with thepredictions of an existing expression which has already beentested extensively for non-spherical particles settling in New-tonian (using 1900 individual data points) and power lawliquids using 1437 individual data points [6].

2. Experimental

Thirty cubes made of brass (8403 kg/m3), of stainless steel(7741 kg/m3), of aluminium (2684 kg/m3), of teflon (2123 kg/m3), of acrylic (1173 kg/m3) and of nylon (1137 kg/m3) rangingin size from 6 mm to ∼25 mm have been used in this studythereby covering wide ranges of particle size and density. Thesphericity of cubes is, of course, fixed at ψ=0.805. In addition,a few spheres of teflon and steel were also used to ascertain thereliability and accuracy of the experimental protocols therebyestablishing the extent of uncertainty for the results reported inthis work.

Eight glucose-in-water solutions of varying concentrationswere used as the model Newtonian experimental test fluids.Five aqueous Carboxymethyl Cellulose (CMC of mediumviscosity grade, marketed by Loba chemicals, Mumbai, India)solutions of varying concentrations (0.9–3.2% w/w) wereemployed as model power law fluids. The polymer solutionswere prepared by slowly adding the required amount of drypolymer powder to water in an agitated tank. In order tominimize the extent of bacterial degradation, polymer solutionswere seeded with a few ppm concentration of Formalin.

The density of each test fluid was measured using a constantvolume density bottle whereas the steady shear data of eachsolution was obtained using the concentric cylinder configura-tion in a Bohlin CVO-100 rheometer at the same temperature asthat encountered in the settling experiments. The rheologicalcharacterization of CMC polymer solutions was carried out overthe shear rate range of 0.1 to 200 s−1 and the correspondingestimates of the shear rate values produced by settling cubes,approximated as (2V/ds), are well within this range. Further-more, no measurable viscoelasticity (storage modulus inoscillatory shear) was observed in the highest concentrationCMC solution used in this work and therefore the polymersolutions used here are believed to be inelastic shear-thinningliquids.

A clear glass tube (of 100 mm inside diameter and 1500 mmlong) was used to perform the settling experiments. Each testliquid was loaded into the fall tube for a long period of time forthe air bubbles to escape and for the thermal equilibrium to bereached before performing the particle sedimentation tests.

Similarly, the test particles were also soaked in the test liquidsprior to being released in the liquid in order to eliminate thepossibility of any air entrainment. Test particles were dropped asclose to the centre of the fall tube as possible and their descentwas timed using an electronic stop watch reading up to 10 ms.In most cases, cubes retained their original orientation, except afew falling at high velocities in water. The descent of a particlewas timed over two test sections, both of which were locatedsufficiently away from the two ends for the end effects to benegligible [13,14] and from each other to ascertain that theparticle had attained its terminal falling velocity. Only tests withconcordant fall times over the two test sections were finallyaccepted. Furthermore, each particle was dropped at least twiceand the reproducibility of the results is of the order of 1.5%. Theterminal settling velocity measured in this work ranged from∼0.76 mm/s to 1 m/s. Altogether 159 individual data points (70in glucose solutions and 89 in CMC solutions) have beengleaned in this study to elucidate the dependence of the freesettling velocity of cube-shaped particles on fluid properties.

3. Results and discussion

3.1. Physical properties of test liquids

As expected, aqueous glucose solutions exhibited constantshear viscosity thereby confirming their Newtonian flowbehavior. Table 1 summarizes the values of their density andviscosity; the latter is seen to vary by a factor of ∼1000. On theother hand, as expected, all polymer solutions used heredisplayed varying levels of shear-thinning behavior whichcould be adequately approximated by the usual two parameterpower law model, i.e.;

s ¼ mðdgÞn: ð1Þ

Using the steady shear data, the best values of theconsistency index, m and of the flow behavior index, n wereestablished using the non-linear regression. These values arealso summarized in Table 1. While the values of m show a fourfold variation, the test fluids show a moderate degree of shear-

Fig. 1. Comparison between the present results and the predictions of Eq. (4) forNewtonian liquids for cubes.

19N. Agarwal, R.P. Chhabra / Powder Technology 178 (2007) 17–21

thinning. The value of m can also be interpreted as the effectiveviscosity of a power law fluid at a shear rate of 1 s−1.

3.2. Calibration experiments with spheres

The settling velocity is known to be strongly influenced bythe confining walls [18]. For spherical particles, this effect isstrongly dependent on the value of sphere-to-tube diameter ratioand the Reynolds number. Thus, the experimentally measuredsettling data was corrected for wall effects by using the methodsavailable in the literature for Newtonian [18] and for power lawliquids [13] for both spheres and cubes. It is worthwhile to addhere that the severity of wall effects gradually diminishes withthe increasing Reynolds number. In the extreme case, thiscorrection entails an uncertainty of the order of 7%. After thisstep, the raw experimental data was converted into their di-mensionless form using the following definitions:

Drag Coefficient : CD ¼ 43gdV 2

qs � qq

� �ð2Þ

Reynolds Number : Re ¼ qV 2�ndn

m: ð3Þ

Dimensional arguments suggest the drag coefficient, CD, tobe a function of the Reynolds Number, Re, and the power lawindex, n. In this work, the value of the Reynolds number forspheres varied from 0.006 to ∼4.2 in Newtonian fluids (n=1)and from ∼0.06 to ∼2.88 in power law fluids. Theexperimental values were within ±5.5% of the expected valuesof the drag coefficient [13,14]. Though, identical experimentalprotocols were followed for tests with cubes, the results for non-spherical particles and/or in power law fluids entail additionalexperimental error due to the uncertainty of the orientation ofthe settling particle and in the measurement and fitting of theirrheological properties. Based on the analysis of repeatmeasurements, it is believed that the new results presentedherein for cubes are probably accurate to within ∼6–7%.

3.3. Results for cubes

Extensive evaluation of the literature data for regular non-spherical shaped particles falling in Newtonian and power lawliquids showed the following expression to yield overallminimum mean deviation for scores of particle shapes[7,8,13].

CD ¼ CD0 þAc

APCDlðCD0Þ2bk

6Xb6Xbþ CD0

� �b

þ CDl

6Xb6Xbþ 128CD0

� �11=12ð4Þ

where

CD0 ¼24XRe

; CDl ¼ 0:44 ð5aÞ

X ¼ 6ðn�1Þ=2 3n2 þ nþ 1

� �nþ1

ð5bÞ

a ¼ 3n2 þ nþ 1

ð5cÞ

b ¼ expf3ða� ln6Þg ð5dÞ

k ¼ 3� a6a

exp3� a2a

� �ln3

� �ð5eÞ

b ¼ 1148

ffiffiffi6

p1� exp

3� a2a

� �2

ln

ffiffiffi6

p � 1ffiffiffi6

p� �( )" #

: ð5f Þ

It needs to be emphasized here that Eq. (4) was initiallydeveloped for spheres falling in power law liquids. Eq. (5b) isbased on the numerical results of Tripathi et al. [19] over the rangeof conditions as 0.4≤n≤1.8 in the creeping flow regime. Thefactor, α, given by Eq. (5c) is linked to the numerical value of Xthrough the rate of energy dissipation considerations [5]. Thevalue ofCD∞

is assumed to be identical to the limiting value of thedrag coefficient for a sphere in Newtonian fluids. While there isno justification for this assumption, but the currently availablenumerical results for a sphere in power law liquids are limited to amaximum Reynolds number of 500 [16] and experimental resultsgo up to Re ∼1500 [13]. Under these conditions, this term isunlikely to contribute significantly to the overall value of the dragcoefficient, as shown in references [7,8,13]. It is particularlyjustified here as the maximum value of the Reynolds number inthe present study is only 864. Therefore, the Newtonian value of0.44 corresponding to a sphere is retained here as the firstapproximation, the remaining three constants, namely, α, β and κare evaluated using the numerical drag value up to Re=100 [19].Indeed, Eq. (4) reproduced most of the literature data (670 datapoints) on sphere drag in power law fluidswith an average error of21% [7].

Fig. 2. Comparison between the present results and the predictions of Eq. (4) forpower law liquids for cubes. (Numbers in parentheses correspond to CMCconcentrations).

20 N. Agarwal, R.P. Chhabra / Powder Technology 178 (2007) 17–21

Subsequently, Rajitha et al. [8] demonstrated that Eq. (4)reproduced drag data for non-spherical particles (includingcones, cylinders, needles, disks, prisms, etc.) in Newtonian andpower law liquids rather well, provided the sphere diameter d isreplaced by an equal volume sphere diameter, ds. It is useful torecall here that the ratio (Ac/AP) denotes the ratio of the surfacearea of the cube to the projected area normal to the direction ofsettling. Thus, this ratio will be equal to 4 for a sphere.Furthermore, it can be easily shown that

Ac

AP¼ 4

Wdsdn

� �2

: ð6Þ

This ratio thus accounts for the orientation of the particleduring its descent and for deviation from spherical shape via thesphericity Ψ. For about 1500 individual data points encompass-ing wide range of conditions (10−5≤Re≤300; 0.31≤n≤1;0.62≤Ψ≤0.86), Rajitha et al. [8] reported the overall meandeviations to be 25% and 30% for Newtonian and power lawfluids, respectively. Given the diversity of the particle shapes andorientations, these errors are regarded to be reasonable andacceptable, at least from process engineering calculationsstandpoint.

At the outset, the new results obtained in this work for thesettling of cubes will therefore be compared with the predictionsof Eq. (4). Fig. 1 shows a parity plot for experimental andcalculated using Eq. (4) values of drag coefficient forNewtonian (n=1) glucose solutions encompassing the rangeof conditions as 0.012≤Re≤864. The average deviationbetween the experimental and predicted values is ∼14.5%with about 10 data points showing deviations N25%. Clearly,these deviations are well within the error bands of Eq. (4).

Fig. 2 shows the corresponding comparison for power lawliquids which extended over the range of conditions as1.3×10−3≤Re≤∼7. The average deviation in this case is alsoabout∼15%which is again well within the error bands of Eq. (4).

In summary, therefore, the new experimental data (159 datapoints) for drag on cubes in Newtonian and power law fluids isconsistent with an existing drag correlation which reconcilesdata for not only spherical and non-spherical particles, but alsofor Newtonian and power law liquids with acceptable levels ofaccuracy.

4. Concluding remarks

In this work, new data for the settling velocity of cubes inNewtonian and power law liquids have been obtained tosupplement the existing data base for drag on non-sphericalparticles. These results are well represented by an existing dragcorrelation which has been already tested thoroughly againstextensive data for the scores of particle shapes in Newtonianand power law liquids over wide ranges of the particle Reynoldsnumber and particle shapes.

NomenclatureAc Surface area of particle, m2

AP Projected area of particle, m2

CD Drag coefficient, dimensionlessCD0 Drag coefficient at low Reynolds number,

dimensionlessCD∞

Drag coefficient at high Reynolds numbers,dimensionless

d Sphere diameter, mdn Diameter of a circle with an area equal to the projected

area, mds Equal volume sphere diameter, mg Acceleration due to gravity, m/s2

k Constant, Eq. (5e), dimensionlessm Power law consistency index, Pa sn

n Power law index, dimensionlessRe Reynolds number, dimensionlessV Falling velocity, m/sX Correction factor, Eq. (5b)

Greek lettersα Constant, Eq. (5c), dimensionlessβ Constant, Eq. (5f), dimensionlessγ· Shear rate, s−1

ρ Fluid density, kg/m3

ρs Particle density, kg/m3

Ψ Sphericity, dimensionlessτ Shear stress, Pa

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