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Particuology 9 (2011) 298–305

Contents lists available at ScienceDirect

Particuology

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xperimental study of settling and drag on cuboids with square base

insheng Wang, Haiying Qi ∗, Junzong Zhuey Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, China

r t i c l e i n f o

rticle history:eceived 12 July 2010eceived in revised form 12 October 2010ccepted 22 November 2010

a b s t r a c t

The drag on non-spherical particles is an important basic parameter for multi-phase flows such as inbiomass combustion, chemical blending, and mineral processing. Though there is much experimentalresearch on such particles, there are few results for cuboids. This paper presents data for cuboids with asquare base in static glycerin–water solutions of various volume concentrations. Complex motions were

eywords:uboid particlesnsteady settling behaviorrchimedes numbererminal velocityrag coefficient

observed and characterized. A dimensionless expression is given for terminal velocity ut as a functionof Archimedes number Ar which is used to develop an accurate correlation for friction factor CD. Theaccuracy of the correlation is 7.9% compared to experimental data in the literature. For both square platesand square rods, the terminal velocity per unit mass, ut/mp, was used to characterize the influence ofparticle geometry on velocity, which was shown to be linear.

© 2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of

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. Introduction

Two-phase flow plays an important role in both technical andatural processes, such as combustion of pulverized coal, sus-ended fibre flows in paper forming, chemical blending, oil drillingnd mineral processing. To optimize processes and to design equip-ent, the terminal velocity and drag coefficient are significant

arameters. Early studies focused on spherical particles with accu-ate correlations for the drag coefficient and terminal velocityased on experimental data and theoretical studies (Clift, Grace &ebber, 1978). However, non-spherical particles have not been asell characterized even though they are common in nature and in

ndustrial processes. In addition, unsteady sedimentation of non-pherical particles involves complex motions, such as swinging andotation, for which drag coefficients for non-spherical particles areuite different from those for spherical particles.

Non-spherical particles could be either regular or irregular inhape. Although most particles in industry and in nature are irreg-lar, it is important first to study particles with regular shapes forrag of non-spherical particles. Based on research of regular parti-les, the shape description and drag model could then be extendedo irregular particles.

As non-spherical particles, cuboids with a square base provide

typical shape for which experimental data could be expediently

nalyzed. However, not much experimental data for particles withuch shapes are available in the literature as compared to disks

∗ Corresponding author. Tel.: +86 10 62796036; fax: +86 10 62796036.E-mail address: hyqi@mail.tsinghua.edu.cn (H. Qi).

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674-2001/$ – see front matter © 2011 Chinese Society of Particuology and Institute of Process Eoi:10.1016/j.partic.2010.11.002

Sciences. Published by Elsevier B.V. All rights reserved.

nd cylinders. Based on study of particles of typical shapes, it isossible first to develop suitable drag models for regular particles,hich may be improved for irregular particles.

The primary objective of this paper is to present a reliableorrelation to predict the terminal velocity and drag coefficientor cuboids with a square base. The sphericity of these particlesaries from 0.642 to 0.805, and aspect ratios from 0.3 to 5. Theelationship between terminal velocity and particle geometry isubsequently analyzed, in order to enrich experimental researchn non-spherical particles.

. Previous work

.1. Shape description

The drag coefficients for spheres were initially obtained exper-mentally with sphere diameter, dp, as the characteristic length.sually, studies on non-spherical particles use the volume-quivalent spherical diameter, dv, as the characteristic length.owever dv does not uniquely describe the particle shape. There-

ore shape description is a significant issue in two phase flowtudies with non-spherical particles.

Generally, the aspect ratio, E, was used to describe the axisym-etric particles, such as cylinders and spheroids. This factor was

efined as the ratio of the length projected on the axis of sym-etry to the maximum diameter normal to the axis (Clift et al.,

978; Gabitto & Tsouris, 2008). Apart from aspect ratio varioushape factors have been defined and used to describe shape effectf non-spherical particles, regular or irregular, including particlephericity, , particle circularity, ϕ, Corey shape factor, ˇ, and

ngineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

J. Wang et al. / Particuology 9 (2011) 298–305 299

Table 1Shape description formulas available in literature for non-spherical particles.

Shape description Symbol Definition Source

Aspect ratio E Ratio of particle length projected on symmetryaxis to maximum diameter normal to the axis

Clift et al. (1978) and Gabitto and Tsouris (2008)

Sphericity = AvAp

Wadell (1933)

Circularity ϕ ϕ = PA/Pp =�dA/Pp Tran-Cong, Gay, and Michaelides (2004)Corey shape factor ˇ ˇ = c/(ab)1/2 Tran-Cong et al. (2004)Stokes shape factor ϕStokes ϕstokes = 18·�f ·ut

(�p−�f )·g·d2v

Pettyjohn and Christiansen (1948), Xie and Zhang (2001),Ganser (1993), and Bouwman, Bosma, Vonk, Wesselingh,and Frijlink (2004)

Av, surface area of volume-equivalent sphere; Ap, particle surface area; PA, projected perimeter of area-equivalent sphere; Pp, projected particle perimeter; a, b and c,the largest, intermediate and shortest particle axes (a > b > c); �p and �f , densities of particle and fluid; g, gravitational acceleration; �f , fluid viscosity; dv, diameter ofvolume-equivalent sphere.

Table 2Values of ai in Eq. (1).

a0 a1 a2 a3

K2 2.3288 −6.458 2.4486 0.0K3 4.905 −13.8944 18.42222 −10.2599K4 1.4681 12.2584 −20.7322 15.8855

Table 3Particle geometries (mm).

No. Height (h) Base length (a)

P1 3 5 8 10 15 20 25 30 10

Sw

sek

2

s2MA1f(cF

se

wgn

vp

Ppmfw(evt

to cubes or square cylinders, but, to the best of authors’ knowledge,no correlations applicable to drags for cuboid particles both platesand rods have yet been reported in the literature.

P2 10P3 15P4 20

tokes shape factor,ϕStokes, which are summarized in Table 1 alongith the available reference sources.

However, these methods to describe the particle shape are onlyuitable for one or several fixed shaped particles. Thus, a more pow-rful shape factor is needed for a general drag model handling manyinds of non-spherical particles.

.2. Drag coefficient for non-spherical particles

There are many studies of drag coefficients for fixedly, regularlyhaped particles reported in the literature (Fan, Yang, Yu, & Mao,003; Guo, Lin, & Nie, in press; Heiss & Coull, 1952; Ku & Lin, 2008;archildon, Clamen, & Gauvin, 1964; Militzer, Kan, Hamdullpur,

myotte, & Al-Taweel, 1989; Tripathi, Chhabra, & Sundararajan,994; Zhu, Lin, & Shao, 2000). These drag models are accurate onlyor the selected shapes and orientations. Holzer and Sommerfeld2008) summarized the drag coefficients as a function of parti-le Reynolds number for various shaped particles, as shown inig. 1.

The correlation proposed by Haider and Levenspiel (1989) forpherical and non-spherical particles based on more than 500xperimental data points is presented as follows:

CD = 24Re

(1 + K2 ReK1 ) + K3

1 + K4/Re,

K1 = 0.0964 + 0.5565 , Kj = exp

(3∑i=0

ai i

), (j = 2,3,4)

(1)

ith the values of ai given in Table 2. Although this expressionreatly improves predictions of particle sedimentation, it still can-

ot be applied in practice because of its poor accuracy.

For cuboids, Albertson (1952), Alger and Simons (1968) pro-ided data for settling of cubes in fluid. Agarwal and Chhabra (2007)resented extensive data of cubes in Power law liquids. Dutta,

Fs

4 6 8 10 12 15 18 204 6 8 10 12 15 18 204 6 8 10 12 15 18 20

anigrahi, and Muralidhar (2008) studied experimentally the flowast a square cylinder with different orientations, showing a mini-um drag coefficient at an orientation of 22.5◦. Ku and Lin (2008)

ound that cubic particles rotated faster than cylindrical particlesith the same size in pipe flow. In addition, Delidis and Stamatoudis

2009) compared the velocities of cubes and spheres in the accel-rating region for very high Reynolds number, showing that theelocity of spheres was always greater than that for the cubes ofhe equal volume.

In the above mentioned studies, the particle shapes were limited

ig. 1. Drag coefficients as a function of particle Reynolds number for differenthaped particles (Holzer & Sommerfeld, 2008).

3 uology 9 (2011) 298–305

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.3. Terminal velocity of non-spherical particles

The terminal velocity, ut, can be calculated based on the forcealance on particles:

t =√

2mpg(�p − �f)�p�fApCD

, (2)

here mp is the particle mass and g the gravitational acceleration.ince CD is a function of terminal velocity, the determination of ut

equires iteration.Terminal velocity ut can also be calculated in terms of other

imensionless numbers such as the Archimedes number defineds:

r = d3v�f(�p − �f)g

�2f

. (3)

Terminal velocities for various minerals have been correlated inhis way (Ganguly, 1990; Tsakalakis & Stamboltzis, 2001).

. Experimental

The present experiments used a rectangular parallelepipedest vessel, 1500 mm × 300 mm × 300 mm, with 87 cuboids madef aluminum (2703.2 kg/m3), PVC (1510.3 kg/m3) and PMMA1170.8 kg/m3) having different base lengths, a, and heights, h, asisted in Table 3 and shown in Fig. 2. The particles are divided intoour categories marked P1–P4, each with either a or h varied once

time. Six glycerin–water solutions of different concentrationsere used as the test fluids as shown in Table 4. During experi-ents, temperature fluctuated from 18 to 21 ◦C, small enough to

e considered negligible.

When a particle reaches its terminal velocity, the gravitational

orce, buoyancy and drag are balanced as:

D = Fg − Fb = �

6d3

v g(�p − �f), (4)

rae(

able 4roperties of glycerin aqueous solution at 20 ◦C.

No. F0 F20

Concentration C (vol%) 0 20Density �f (kg/m3) 971 1028.8Dynamic viscosity �f (10−3 Pa s) 1.523 3.064

ig. 3. Typical unstable settling patterns. (a) Too close to the wall (a = 4 mm, h = 10 mm, �p

a = 10 mm, h = 3 mm, �p = 1510.3 kg/m3, F40); (c) periodic movements with a small swing

Fig. 2. Schematic geometry of the particles to be studied.

here dv, �f, and �p are chosen from Tables 3 and 4, so that CD cane calculated from Eq. (4) once the terminal velocity, ut, is known.he terminal velocity was measured with a high-speed camera inhe experiments, which recorded the time lapse for the particlealling a certain distance.

The terminal velocities were measured in two adjacent regions,oth located sufficiently far from the end for neglecting end effects.he two measurements are considered acceptable only when theifference between the two was less than 5%, and ut is the averagef the two values.

Wall effect was eliminated by choosing only the data of particlesith settling oscillation amplitude less than 10% of the cross-

ectional dimension. Each particle was dropped three times withepeatability of results of about 2%. The system was calibrated with

3

glass sphere (� = 2539.8 kg/m ) of 16.17 mm in diameter with thexperimental drag coefficients within ±2.3% of the expected valuesClift et al., 1978).

F40 F60 F80 F100

40 60 80 1001086.6 1144.4 1198.4 1260

5.203 14.87 82.77 872.8

= 2703.2 kg/m3, F0); (b) periodic movements with a small swing for a square plate.for a square rod (a = 10 mm, h = 15 mm, �p = 1510.3 kg/m3, F40).

J. Wang et al. / Particuology 9 (2011) 298–305 301

Fig. 4. Motion stability diagram. (“\” means no sedimentation occurred because the particle density was less than fluid density.).

Fig. 5. Influence of particle geometry on ut for F60.

Fig. 6. Relationship between Rep and Ar for all cases. Fig. 7. Relationship between CD and Rep for all the small-oscillation cases as com-pared with that for spheres.

302 J. Wang et al. / Particuology

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ig. 8. Comparison between the present model and data in literatures (Albertson,952; Alger & Simons, 1968).

. Results and discussion

.1. Unstable settling patterns

During experiment, various patterns of settling behavior werebserved as shown in Fig. 3. Once released and within a feweconds, the particles tended to adjust themselves with their maxi-

sflt

Fig. 9. Influence of particle geo

9 (2011) 298–305

um projected areas normal to the direction of motion. In general,ost of the particles did not fall along the center line of the settling

olumn while swinging and rotating, especially in low-viscosityuids. In some extreme cases, the particles even ran up againsthe wall. Such behaviors were found to be independent of initialarticle orientation.

In low-viscosity fluids such as F0, F20, F40 and F60, par-icles of different shapes behaved differently at the terminalelocity. Movements with small oscillations were periodic: thequare plates spiraled down with a sharp angle between the nor-al to the particle base and the center line of the column as

hown in Fig. 3(b). Square rods zigzagged down with the princi-al axis perpendicular to the vertical as shown in Fig. 3(c). Cubesotated almost continually following a complex helical path. How-ver, in high-viscosity fluids F80 and F100, the high viscosityaused all particles to settle in a stable manner with little oscil-ations.

Motion of particle settling is classified into three categoriesn the basis of the extent of swing, as stable settling, oscilla-ions less than 30 mm and oscillations larger than 30 mm, ashown in Fig. 4. The repeatability of the data in the first twoategories is well, and they were referred to as “limited oscilla-ion”.

The diagram shows an obvious demarcation between cases of

mall and large oscillation. The oscillations were dependent onuid viscosity and particle density, size and shape. Large oscilla-ion occurred for low viscosity fluids and large particles with plates

metry on ut/mp for F60.

uology 9 (2011) 298–305 303

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J. Wang et al. / Partic

ere more stable than rods and cubes. This diagram distinguishesll the cases and provides guidance for further studies. The mech-nism for the different behavior is an important topic for furthertudy.

.2. Terminal velocity

The terminal velocity was calculated for the small oscillationases. The influence of geometry on the terminal velocity was stud-ed for the cases using F60 as examples. Fig. 5 uses either a or h ashe abscissa to show two zones for plates and for rods respectively,ivided in between by the cubes of h/a = 1.

The data in Fig. 5 for F60 show that ut is larger for the largerarticle densities, but as the particle size increases, ut first increasesnd then remains almost constant. The turning point occurs at theube geometry. One reason is that both the projected area and theass change with the geometry, but they have opposite effects on

he velocity. Another reason is that the spiral path is longer thanhe zigzag path, so the velocity in the gravitational direction alonghe spiral path is less than the one along the zigzag path.

Since both fluid and particle characteristics affect the terminalelocity, using one expression to describe all conditions is diffi-ult. The particle Reynolds number, Rep, defined by Eq. (5), is theatio of the inertia force to the viscous force which can represent aimensionless velocity.

ep = �fdvut

�f. (5)

The Archimedes number, Ar, represents the ratio of the buoyantorce to the viscosity force, which includes particle geometry andensity as well as fluid density and viscosity as defined in Eq. (3).

The plot of Rep versus Ar in Fig. 6 shows that all data are wellepresented by a single curve. As a result, the data can be well corre-ated by Eq. (6) with R2 = 0.997. The coefficients in Eq. (6) are listedn Table 5.

ep = exp(k1Ar−1/k2 + k3Ar

−1/k4 + k5) (6)

.3. Drag coefficients

Fig. 7 shows the relationship between CD and Rep for all themall-oscillation cases. The relationship is similar to but slightlybove the curve for spheres. The data for CD deviate from thator spheres for large Rep, because the effect of the particle shapen the flow field increases and oscillation also increases with Rep.or Rep > 100, CD remains almost constant at 2 as the differencerom the spherical data increases. There are also larger differencesompared with the spherical correlation in the region of smallep. For Rep < 0.2, the difference gradually increases, different fromhe reported results in the literature. Additional experiments withpherical particles indicated that the measured CD for spheres wasnly 2.3% higher than that reported in the literature. This implieshat the differences for the non-spherical particles resulted fromhe shape characteristics of the rectangular prisms.

A force balance can relate CD with Rep and Ar as:

D = 43dv(�p − �f )g

�f u2t

= 43Ar

Re2p

. (7)

Thus, CD can be correlated with Rep through the relationship ofr and Rep. The data in Fig. 8 can be used to express Ar as a function

f Rep with R2 = 0.994:

r = exp(A1 Re−1/B1p + A2 Re

−1/B2p + C), (8)

ith the coefficients A1, A2, B1, B2 and C listed in Table 6.

�ttd

Fig. 10. Relationship between ut/mp and 1/a or 1/(ah) for F60.

Combining Eqs. (7) and (8) then gives the drag coefficient as:

D = 43

exp(A1 Re−1/B1p + A2 Re−1/B2p + C)

Re2p

. (9)

Thus, Eq. (9) can be used to represent the drag coefficient foruboids with square base for Rep from 0.05 to 6300.

Other experimental data in the literature (Albertson, 1952;lger & Simons, 1968) were used to verify the applicability of therag correlation in the present study, as shown in Fig. 8. The meanelative error is 7.9%, indicating that the drag correlation is suitableor cuboids beyond their current experimental range.

As shown in Fig. 5, particle mass and particle volume have oppo-ite effects on terminal velocity. To eliminate the effect of particleass, the terminal velocity per unit mass, ut/mp, was used to corre-

ate the data with the particle geometry as shown in Fig. 9 for F60,ndicating that ut/mp decreases as particle size increases yet withgradually decreasing gradient.

The curves for the different heights in Fig. 9(d) are approxi-ately parallel. Thus, when ut/mp is plotted against 1/a as shown

n Fig. 10(a), ut/mp is found to be approximately inversely propor-ional to a. Fig. 11(b) shows a nearly linear relationship betweent/mp and 1/(ah), with the slope increasing with particle density,p. This relationship is suitable for both plates and rods. Thus,

here is essentially no difference between the terminal veloci-ies for the different shapes when the shape varies in any oneimension.

304 J. Wang et al. / Particuology 9 (2011) 298–305

Table 5Coefficients in Eq. (6).

k1 k2 k3 k4 k5

0.03568 −5.043 −15.32 10.72 10.51

Table 6Coefficients in Eq. (8).

A1 B1 A2 B2 C

−0.00136 0.5656 −2.938

Ft

4

hr(fspd

5

c

(

(

(

(

(

A

N

R

A

A

A

B

C

D

D

F

Gcylindrical shape. Powder Technology, 183, 314–322.

ig. 11. The same relationship between Rep and Ar for plates and rods when havinghe same base.

.4. Universal drag model for non-spherical particles

The above discussion indicates that particles with a square baseave the same relationship between Ar and Rep for both plates andods. Additional experimental data and the data from literatureGogus, Ipekci, & Kokpinar, 2001) showed that this rule is suitableor particles having a different base such as circular or triangular ashown in Fig. 11, thus, promising drag coefficient correlations forarticles with the corresponding bases to lead to correspondingrag models. Such results will be discussed in a further study.

. Conclusions

An experimental study of the sedimentation characteristics ofuboids with a square base showed that:

G

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12.32 4.308

1) Once released into a fluid, and within a few seconds, parti-cles adjusted themselves with their maximum projected areasoriented normal to the descent direction. Various complexmotions occurred during sedimentation, including oscillation,rotation and twirling. Square plate particles tended to follow aspiral path while square rod particles followed a zigzag path.These motions were independent of initial particle orientation.

2) The extent of particle oscillation was plotted to classify the dif-ferent behaviors. Such regime diagram serves to identify thevarious motion regimes to provide guidance for further studies.

3) The Archimedes number, Ar, was found to be related expo-nentially to the Reynolds number, Rep. A dimensionlessrelationship was developed to predict ut. A correlation betweenCD and Rep was developed from the relationship between Arand Rep. This correlation fits the experimental data of otherresearchers with mean relative error of 7.9%.

4) Even at low Rep, there is still obvious difference in CD for cuboidsand for spheres, indicating that shape characteristics led tolarger pressure drag for cuboids than for spheres.

5) The parameter, terminal velocity per unit mass, ut/mp, high-lights the effect of particle geometry. Analysis resulted in alinear relationship between ut/mp and particle geometry, indi-cating essentially no difference in terminal velocity betweendifferent shapes while any one dimension is varied.

cknowledgement

This work was supported by the Major Program of the Nationalatural Science Foundation of China with Grant No. 10632070.

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