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Powder Technology 139 (2004) 21–32

Drag coefficients of irregularly shaped particles

Sabine Tran-Cong, Michael Gay, Efstathios E. Michaelides*

School of Engineering and Center for Bioenvironmental Research, Tulane University, New Orleans, LA 70118, USA

Received 22 April 2003; received in revised form 16 September 2003; accepted 10 October 2003

Abstract

The steady-state free-fall conditions of isolated groups of ordered packed spheres moving through Newtonian fluids have been studied

experimentally. Measurements of the drag coefficients are reported in this paper for six different geometrical shapes, including isometric,

axisymmetric, orthotropic, plane and elongated conglomerates of spheres. From these measurements, a new and accurate empirical

correlation for the drag coefficient, CD, of variously shaped particles has been developed. This correlation has been formulated in terms of the

Reynolds number based on the particle nominal diameter, Re, the ratio of the surface-equivalent-sphere to the nominal diameters, dA/dn, and

the particle circularity, c. The predictions have been tested against both the experimental data for CD collected in this study and the ones

reported in previous works for cubes, rectangular parallelepipeds, tetrahedrons, cylinders and other shapes. A good agreement has been

observed for the variously shaped agglomerates of spheres as well as for the regularly shape particles, over the ranges 0.15 <Re < 1500,

0.80 < dA/dn < 1.50 and 0.4 < c < 1.0.

D 2003 Elsevier B.V. All rights reserved.

Keywords: Drag; Particles; Irregular; Correlations

1. Introduction

The settling behavior for variously shaped particles is of

fundamental importance since natural and artificial solid

particles occur in almost any shape ranging from roughly

spherical pollen and fly ash through cylindrical asbestos

fibers to irregular mineral particles. Irregularly shaped

particles are met in many applications, such as sedimenta-

tion and flocculation of aggregates of fine particles in rivers

and lakes, chemical blending, mineral processing, powder

sintering, manufacturing with phase change and solidifica-

tion processes. In most of these applications, the determi-

nation of the falling velocity of the particle is of interest for

the design and optimization of processes and equipment.

Since the falling velocity of a particle depends greatly on its

drag coefficient, reliable correlations for the drag coefficient

of these irregular particles are required for the understanding

of the processes and the design of equipment.

Early investigations focused on the drag force experi-

enced by spheres moving through a fluid. Extensive sets of

0032-5910/$ - see front matter D 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.powtec.2003.10.002

* Corresponding author. Tel.: +1-504-865-5764; fax: +1-504-862-

8747.

E-mail address: [email protected] (E.E. Michaelides).

data were collected and combined with theoretical work.

The sets of data resulted in several empirical correlations for

the drag coefficient, CD, in terms of the Reynolds number,

Re. These correlations for spherical bodies were reviewed

subsequently by many authors in treatises and review

articles by Clift et al. [4] and Khan and Richardson [11].

A comparison between most of these correlations for

spheres by Hartman and Yates [6] showed relatively low

deviations.

The studies on the drag of irregular particles mainly

addressed a limited number of solid shapes and, in most

cases, formulated the particle free-falling velocity and drag

with respect to a well defined particle shape. A number of

empirical correlations were proposed for regular polyhe-

drons by Pettyjohn and Christiansen [20] and Haider and

Levenspiel [5]; for cylinders by Marchildon et al. [16],

McKay et al. [18] and Unnikrishan and Chhabra, [28]; for

thin disks by Squires and Squires [25] and Willmarth et al.

[30]; for parallelepipeds by Heiss and Coull [8]; for cones

by Jayaweera and Mason [10]; and for flat annular rings by

Roger and Hussey [22]. In comparison to these correlations

for specific particle shapes, very few studies appeared for

irregularly shaped particles. Notable among them are the

ones by Hottovy and Sylvester [9], Baba and Komar [1],

Lasso and Weidman [15] and Hartman et al. [7].

S. Tran-Cong et al. / Powder Technology 139 (2004) 21–3222

It is generally recognized and experiments have shown

that the departure of a particle from the spherical shape

causes a decrease of its terminal velocity in a fluid. This

implies a higher CD for the non-spherical particle as

observed by Pettyjohn and Christiansen [20] and Komar

[13]. Although there are many sets of data and empirical

methods for the determination of the drag coefficient for

irregularly shaped particles, there is not an unambiguous

method that yields this drag coefficient for a more than one

particular shape. Most of the previous studies on the subject

have been limited in collecting a large number of data on

variously shaped particles as a necessary step in an effort to

better understand and formulate the particle shape effects on

the settling behavior.

The primary objective of this study is to provide a

reliable correlation for the drag coefficient that covers as

many shapes as possible of the infinite set of the shapes of

the irregularly shaped particles. The irregular particles are

made of ordered arrangements of several smaller spheres

joined together. Thus, isometric, axisymmetric, orthotropic,

plane and longitudinal agglomerates of spheres were exam-

ined. The terminal velocities of these irregular particles were

measured by an optical method in glycerin–water solutions

of different concentration and viscosity, and, hence, their

steady-state drag coefficients were determined. From the

results obtained, we modified the well-known steady-state

correlation for the drag coefficient of spheres in an infinite

fluid [23] and we derived a simple and accurate correlation

of the drag coefficients of irregular spheres in terms of the

Reynolds number based on the nominal diameter, Re, the

ratio of the surface-equivalent-sphere to nominal diameters,

dA/dn, and the particle circularity, c. The predictions were

then compared with experimental data for the CD within

the ranges 0.05 < Re < 1500, 0.80 < dA/dn < 1.50 and

0.4 < c < 1.0. This data included our measurements for

variously shaped conglomerates of spheres and data

obtained by Pettyjohn and Christiansen [20], Heiss and

Coull [8], and Lasso and Weidman [15].

2. Previous studies

2.1. Drag coefficient for spheres

The drag force, FD, experienced by a particle settling

with a uniform velocity, ws, in a quiescent fluid is defined in

terms of a drag coefficient, CD, by the following expression:

FD ¼ CDAp

qfw2s

2; ð1Þ

where Ap is the projected surface area of the particle normal

to the direction of its motion and qf is the density of the

fluid. In the case of a freely falling sphere, the balance

between the drag force and the force due to the gravity leads

to the following expression, which was used in the exper-

imental determination of the drag coefficient:

CD ¼ 4

3dðqs � qf Þ

qf

g

w2s

: ð2Þ

In Eq. (2), d and qs are the diameter and the density of

the solid sphere respectively, and g is the gravitational

acceleration. For spheres falling in infinite fluids, the drag

coefficient has been correlated in terms of the dimensionless

parameter Re = qsdws/lf, where lf is the dynamic viscosity

of the fluid. Several correlations for CD were proposed over

a wide range of Re. One of the most widely used was the

empirical equation of Schiller and Naumann [23], which is

simple and accurate in the range 0.1 <Re < 800 [4,11].

Based on this expression, Clift et al. [4] developed an

improved and more accurate expression which is valid for

higher Re, up to 3� 105 and reads as follows:

CD ¼ 24

Reð1þ 0:15 Re0:687Þ þ 0:42

1þ 4:25� 104 Re�1:16: ð3Þ

In this study, we will use this expression for the drag

coefficient for spheres.

2.2. Shape factors and pertinent geometric parameters

For non-spherical particles in an infinite medium, it was

generally recognized that CD must be expressed in terms of

Re as well as one or more shape factors. Several methods

were suggested for obtaining these shape factors and for

classifying the non-spherical particles by size or shape

[4,27]. One of the most important measures was defined

by Wadell [29] as the volume-equivalent-sphere diameter or

nominal diameter, dn, which is defined as follows:

dn ¼ffiffiffiffiffiffiffiffiffiffiffi6V=p3

p; ð4Þ

where V is the particle volume. This size parameter has been

used extensively in particulate as well as bubbly flows to

define the Reynolds number of the particle [4].

A similar parameter may be defined in terms of the

projected area of the sphere, Ap. The surface equivalent

sphere diameter, dA, thus defined is equal to:

dA ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi4Ap=p

q: ð5Þ

The ratio dn/dA was used as a pertinent and useful

dimensionless number in many studies on the drag coef-

ficients [8,21,24,28].

In similar manner and in order to account for the particle

elongation, axisymmetrical shapes were conveniently de-

scribed by their aspect ratio, E, which is defined as the

particle length along the symmetrical axis over the largest

diameter of the cross section. Relevant information on the

preferential settling orientation of cylinders with respect to

this shape factor was reported by Marchildon et al. [16] and

S. Tran-Cong et al. / Powder Technology 139 (2004) 21–32 23

Jayaweera and Mason [10]. It must be pointed out that the

aspect ratio is related to the two diameters defined above

and, that is often possible to express it analytically in terms

of these diameters. For example, the aspect ratio of a

cylinder falling with its flat surface at right angles to the

direction of motion, E = 4(dn/dA)3/p.

Another dimensionless number that is useful in studies of

spheroids and ellipsoids is the particle sphericity, w. Thisshape factor was first defined by Wadell [29] as the ratio of

the surface area of the equivalent-volume-sphere to the

actual surface area of the particle, A. Thus, w was given by:

w ¼ pd2n=A: ð6Þ

The sphericity is generally recognized to be an appro-

priate single dimensionless number measure for character-

izing the shape of isometric non-spherical particles [5,6,20].

From a practical standpoint, however, w is difficult to

determine for strongly irregular particles because it requires

a measure of the surface area, which it is not easy to

accomplish in all cases. For this reason, other shape factors

have been used, which are easier to determine for irregular

particles. Among these is the particle circularity (also called

surface sphericity), c, which is defined as follows:

c ¼ pdA=Pp; ð7Þ

where Pp is the projected perimeter of the particle in its

direction of motion. This is an easier parameter to mea-

sure. One of the difficulties of using the circularity, c, is

that occasionally it yields the same value for three-dimen-

sional and two-dimensional objects. For example, spheres

and disks that fall on their flat sides have the same

circularity.

Another approach for characterizing the shape of three-

dimensional irregular particles, while bypassing the diffi-

culty in determining their surface area, is to define dimen-

sionless numbers based on the largest, intermediate and

shortest particle axes. This approach was adopted by

McNown and Malaika [19], Komar and Reimers [14], Baba

and Komar [1] and others, who characterized irregular

particle shapes by using the so-called ‘‘Corey shape factor,’’

Co. This is defined as the ratio of the shortest particle axis to

the square root of the product of the other two axes.

One important feature of E and Co is that, for an ellipsoid

settling with its largest axis perpendicular to its motion, both

of these factors reduced to terms that are proportional to (dn/

dA)3. For a cylinder falling on its side, Co is equal to E� 1/2.

Accordingly, the Corey shape factor seems to be appropriate

to characterize the flatness of particles that exhibit a com-

pact shape. One of the disadvantages of using this shape

factor is that Co = 1 for spheres as well as for non-compact

particles having three perpendicular axes of same length,

such as star-shaped particles. For this reason, Co does not

give any information on the deviation of the shape of the

particle from the spherical shape.

It appears that a combination of several shape factors

may be necessary to properly describe the effect of the shape

of a particle on the hydrodynamic drag coefficient. After

studying the effect of all the above shape factors on the drag

coefficient, we found out that the particle volume, projected

area, flatness and circularity are well-characterized by the

nominal diameter, dn, the surface-equivalent-sphere diame-

ter, dA, the ratio dn/dA and the particle circularity, c.

2.3. The drag coefficients for non-spherical particles

Two approaches for the determination of the steady-state

drag force on a particle may be found in the literature. The

first and most widely used is based on the diameter and the

projected area of an equivalent-volume-sphere. Eq. (2) is the

expression that results from this approach, where the parti-

cle nominal diameter, dn plays the most important role.

The second approach is to calculate the drag force by

considering the actual projected surface area of the irregular

particle [10,16,17]. This drag coefficient, denoted by CDA,

is defined as follows:

CDA ¼ 4

3

d3nd2A

ðqs � qf Þqf

g

w2s

: ð8Þ

The ratio CD/CDA is equal to (dA/dn)2. Because the drag

force is a function of the projected area in the direction of

the flow, we used this approach for the determination of a

suitable correlation function. An a posteriori test of the data

and the derived correlation with both methods showed that

this was a very good choice for the reduction of the

experimental data.

3. Experimental facility and procedure

We conducted experiments for the determination of the

terminal velocity of irregular particles and the calculation of

the drag force exerted on them. Experiments were con-

ducted in a 1.85-m-long, 12.7-cm inner diameter, Plexiglas,

cylindrical tank with the lower end sealed. Water–glycerin

solutions of different concentrations and viscosities were

used as the fluid. The high-dynamic viscosities were mea-

sured by means of a Cannon–Fenske viscometer with

distilled water as the reference fluid, while the intermediate

and low viscosities were determined by using the falling ball

method. Comparisons with the results on the viscosity of the

fluid and published tables of the viscosity as a function of

the composition of the water–glycerin mixture showed

excellent agreement. The fluid was loaded into the tube at

least 24 h prior to the experiments, thereby allowing any air

bubbles to exit the liquid and for thermal equilibrium to be

Table 1

Shape factor values with respect to the particle shapes

Particle shape Surface-equivalent-sphere

to nominal diameter

ratio, dA/dn

Particle

circularity, c

Close-to-sphere

particles

ffiffiffi2

p

131=32þ 3

ffiffiffi3

p

p

� �1=2

ffiffiffi2

p

32þ 3

ffiffiffi3

p

p

� �1=2

Pyramids ffiffiffi2

p

141=35þ 16

p

� �1=21

55þ 16

p

� �1=2

Stars ffiffiffi5

p

71=3

1ffiffiffi5

p

H-shaped particles 71/6 1ffiffiffi7

p

Crosses 71/6 1ffiffiffi7

p


reached. Experiments were performed at atmospheric pres-

sure and at the controlled laboratory temperature. One of the

challenges was to ensure uniformity of temperature in the

long tube, because the viscosity of the fluid is a strong

function of the temperature. This uniformity of temperature

was assured by having a fan blowing in the direction of the

tube, thus ensuring that there is no stratification of the air in

the laboratory. Regular measurements of the temperature

and the viscosity assured that the properties of the fluid were

uniform during the experiments.

The particles used in the experiments were made from an

ordered assembly of several identical smooth glass spheres

glued together. Six different geometrical shapes were ex-

amined: spherical, pyramidal, star-shaped, H-shaped, cross-

shaped and cylindrical (bar). These shapes are shown in Fig.

1. The particle dimensions and weights were measured

using a Vernier caliper with an accuracy of F 1 Am and a

Fig. 1. Shapes of the agglomerates of spheres investigated.

Bars 71/6 1ffiffiffi7

p

scale accurate to F 0.0005 g. The nominal diameters were

calculated from the total volume of the solid phase that

constituted the various shapes. The surface-equivalent-

sphere diameters were calculated from the actual projected

areas of the particles while in their settling position. The

shape factors of all the particles used in this study are shown

in Table 1.

Since some of the particle shapes, such as the pyramidal

and the spherical, were constructed by putting together

small spheres, the solid particles formed are essentially

porous. We performed an analytical test by using the

permeability function developed by Brinkman [3] to find

out the effect of the porosity on the flow field: For

aggregates composed of primary spheres of radius a, a

Re

CD

Fig. 2. Measured drag coefficient, CD, versus Reynolds number, Re.


number of investigations concluded that the dimensionless

permeability j/a2 depends only on the solids fraction of the

medium [12]. The solids fraction, /, varies with the struc-

ture of the medium. In this study, we used two different

packing arrangements to construct the spherical and pyra-

midal shapes, which are shown in Fig. 1a and b, respec-

tively. For both arrangements, the solids fraction, /, wasequal to 0.7405, which corresponds to the closest packing of

spheres. Hence, the porosity was at its minimum value of

0.2595. The dimensionless permeability defined by Brink-

man [3] as,

ja2

¼ 2

9/1þ 3

4/ 1�

ffiffiffiffiffiffiffiffiffiffiffiffi8

/� 3

s ! !; ð9Þ

was very low (0.0012). This test showed that there was no

significant flow through these agglomerate particles, be-

Fig. 3. Correlation between the measured drag coefficients based on the particle p

equivalent-sphere diameter, ReA.

cause they exhibited a very high packing density so that

their porosity was very low. This is consistent with previous

observations in the studies by Sutherland and Tan [26] and

Lasso and Weidman [15]. In addition, a microscopic exam-

ination showed that the glue used to attach the smaller

particles together blocked several of the passages between

the smaller particles, thus, further reducing the porosity and

inhibiting any flow through the agglomerated particles.

The agglomerated composite particles were let free into

the liquid at the center of the cross-section and beneath the

free surface with the help of a pair of tweezers. In the case

of the asymmetric shapes, an initial series of settling

experiments were performed in order to determine the

vertical orientation of the agglomerates in the settling

process. These tests showed that, for Re>0.1, all the

irregular particles tend to align themselves with their

maximum cross section normal to the direction of the

rojected surface area, CDA, and the Reynolds number based on the surface-


settling motion. The settling at this preferred orientation

was maintained up to Re = 100. At higher Re, the particles

settled in a steady-state orientation only after they were

released with their maximum cross section parallel to the

horizontal axis. For any other initial orientation, the

elongated particles show a tendency to wobble around a

horizontal axis, while the fuller bluff bodies tend to rotate

and, hence, they follow a spiral motion rather than a

straight vertical path. During the experiments, we ensured

that the irregular particles quickly reached their steady-

state settling by releasing them in the orientation they

settle steadily.

The terminal velocities were measured using a high-

speed video camera of high resolution with the CCD

captor sensor discerning 256 grey levels. The camera

takes 500 frames/s, with an exposure time of 1/500 s.

Whenever we found that the agglomerated particles

Predic

Re

Fig. 4. Comparison between our experimenta

exhibited unsteady oscillatory motion (at very high Re),

hit the wall of the container or went too close to the wall

(2–3 diameters) to be influenced by the presence of the

wall the measurements were excluded from the analysis.

The measured values of terminal velocity were used to

calculate CD from Eq. (2). Considering all the experi-

ments performed, the Reynolds numbers extended from

0.15 to 2000. The experimental uncertainties were calcu-

lated to be 5% in the range 0.15 <Re < 1, 2% for Re

from 1 to 100, and less than 4% for 100 <Re < 2000. The

higher uncertainty in the last range is due to the

unsteadiness of the settling because of the wakes formed

behind the agglomerate particles. It must be pointed out,

that because of the high-speed camera we used, the

uncertainties observed in the current set of data are lower

than the ones reported for the determination of the drag

coefficient of spheres [4].

ted CD

l drag coefficients and the correlation.

Table 2

Statistical averages for the correlation of the experimental data

Sphere Pyramid Star H-shaped Cross Bar

Number of

data pts.

40 45 42 46 44 48

Mean absolute

percent

error (%)

7.97 9.02 7.24 5.97 7.00 4.71

Root mean

square error

0.55 1.08 0.97 0.59 0.68 1.18

R2 value 0.99944 0.99835 0.99958 0.99990 0.99978 0.99958

R-value 0.99972 0.99918 0.99979 0.99995 0.99989 0.99979


4. Experimental results and data reduction

The experimental data of the drag coefficient, CD, as a

function of the Reynolds number, Re, are plotted in Fig. 2.

At a given Reynolds number, the values of the drag

coefficient for all the irregular particles were higher than

the ones for spheres, regardless of the shape of the agglom-

erate. This difference increased with both the Reynolds

Fig. 5. Measured and predicted drag coefficients, CD, vers

numbers and the departure of the particle shape from an

isometric compact body. Three distinct trends were ob-

served during the experiments:

� First, agglomerates that have a near-spherical shape

showed the lowest drag coefficients and their drag curves

were always close to those of the spheres. The external

shape of these particles was close-to-sphere with a

corrugated outer surface.� Second, at a given Re, the highest values of CD were the

data for the agglomerates made with non-circular planar

constructions, such as the cylindrical bar, the cross-

shaped, and the H-shaped particles. All of these

agglomerates were characterized by the same projected

surface area and the same nominal diameter. Therefore,

they have the same ratio dA/dn. As shown in Fig. 2, the

data sets related to these planar shapes were very close to

each other, regardless of the actual particle elongation.

This leads to the conclusion that the ratio dA/dn is an

important parameter that characterizes the settling of

us Reynolds number, Re, for each shape examined.


irregular particles. Using this parameter as the character-

istic of particles brings together the drag coefficient data

for diverse groups and shapes of particles.� Third, the experimental data for the cylinder and star-

shaped agglomerates exhibited drag coefficients between

those of spheres and the shapes mentioned in the

previous paragraph. The only exception to this was

observed at Re>400, where the measured CD for the

pyramidal conglomerates were higher than the ones for

star-shaped particles. This is due to the fact that pyramids

at Re>400 shed unsteady vortices that slow the motion of

the particle itself.

Following the trends observed in Fig. 2, we correlated the

experimental data in terms of dimensionless numbers based

on the projected surface area of the irregular particles.We first

used the equivalent-surface-area for calculating a drag coef-

ficient. This resulted in a coefficient CDA as defined by Eq.

(8). This procedure brought the curves of CDA vs. Re for all

Fig. 6. Comparisons between the experimental drag coefficients for

isometric particles given by previous works [8,20] and the predictions

from both Eq. (10b) and Haider and Levenspiel’s [5] correlation.

the isometric shapes closer to each other up to Re = 100.

However, the curves of CDA vs. Re for particles made with

planar construction still deviated considerably from the

curves of CDA associated with spherical agglomerates. For

this reason, we selected the ratio dA/dn, which characterizes

the flatness of a particle. This ratio was also recognized to be

appropriate for this purpose in previous studies by Heiss and

Coull [8], Singh and Roychowdhury [24], Unnikrishan and

Chhabra [28], and Rodrigue et al. [21].

In order to express the deviation of the projected area of

the particle from that of a disk, we included the parameter of

circularity, c, in our analysis. Based on our experimental

observations and after examining several graphical ways to

reduce the whole set of experimental data, we decided that the

most general and accurate correlation would be obtained by

modifying Eq. (3) for the drag coefficient of spheres in an

infinite fluid. Thus, we derived a simple but accurate corre-

lation, which includes the parameters of circularity, c, and the

flatness ratio dA/dn. The final result is the following expres-

sion for the drag coefficient:

CDA ¼ 24

dA

dnRe

� � 1þ 0:15ffiffiffic

p dA

dnRe

� �0:687" #

þ 0:42

ffiffiffic

p1þ 4:25� 104

dA

dnRe

� ��1:16" # ; ð10aÞ

5. General observations

The corresponding set of data for each conglomerate

shapes examined is shown in Fig. 3. The various curves of

CDA vs. (dA/dn)*Re practically coincided in the range

0.2 < (dA/dn)Re < 10, which means that the data for all the

shapes are well correlated in this range. For higher Reynolds

numbers, the introduction of c was required to more accu-

rately correlate the data, as shown in Fig. 3b. It is observed in

this figure that the proposed correlation fits well the data up to

(dA/dn)*Re of the order of 1500.

From Eq. (10a), it is very easy to obtain an equation

for the actual drag coefficient, CD, which is defined in

terms of the nominal diameter by Eq. (2) and is the most

widely used definition for the drag coefficient of irregular

particles. The final correlation function obtained is as

follows:

CD ¼ 24

Re

dA

dn1þ 0:15ffiffiffi

cp dA

dnRe

� �0:687" #

þ0:42

dA

dn

� �2

ffiffiffic

p1þ 4:25� 104

dA

dnRe

� ��1:16" # : ð10bÞ


Eq. (10b) was derived from data in the ranges 0.15 <

Re < 1500, 0.80 < dA/dn < 1.50 and 0.4 < c < 1.0. These ranges

cover most of the irregularly shaped particles in engineering

applications. It must be pointed out that the functional form of

Eqs. (10a) and (10b) was decided upon only after several

functional forms and data reduction methods were tested. The

criterion used in the determination of the functional form to

be used was the minimization of the standard deviation

(standard error) of the predicted data from the actual data.

Fig. 7. Comparisons between the experimental drag coefficients for cylinders from

from both Eq. (10b) and Heiss and Coull’s [8] correlation.

Fig. 4a depicts the measured values of CD vs. the

predicted values of CD, over the range 0.15 <Re < 1500. It

is observed that all the points are very close to a line at 45j(shown by the dashed line). This signifies that there is very

good agreement between experimental data and the corre-

lation function. Fig. 4b depicts the all the fractional differ-

ences between the measured and the predicted values for

CD, as a function of Re. These individual experimental

deviations are less than 5% over the range 0.15 <Re < 3,

both Lasso and Weidman [15] and Heiss and Coull [8] and the predictions


less than 12% in the range 3 <Re < 200, where the wake

behind the particle is formed and less than 20% for

200 < Re < 1500, where the wakes behind the particles

become unstable and vortices are shed in the flow. The

average data absolute errors are less than 10%. It appears in

Fig. 4b that even up to Re = 1550, which is shown by the

dot-dashed line, the correlation and the data agree fairly

well. Above this value of Re, the fractional differences

Fig. 8. Comparisons between the predicted drag coefficients for rectangular pa

increased substantially and the correlations obtained are

not to be used. Table 2 gives a summary of the statistics

that led to the correlation shown in Eq. (10b).

Within the range of validity of Eq. (10b), the agglom-

erate shape that corresponds to the highest fractional

differences was the star. These particles were initially

released with one axis parallel to the vertical and the

calculations were made by assuming that this orientation

rallelepipeds from both Eq. (10b) and Heiss and Coull’s [8] correlation.

Table 3

Comparison between the experimental drag coefficients for cylinders given

by previous works and predictions from Eq. (10b)

E Re dA/dn c Measured CD

from Lasso

and Weidman

Predicted CD

from Eq. (10b)

Absolute

fractional

difference (%)

1.00 0.634 0.8736 1.0 41.59 36.37 12.54

1.00 0.100 0.8736 1.0 261.5 215.55 17.57

0.50 0.326 1.1006 1.0 95.50 87.03 8.86

0.50 0.050 1.1006 1.0 592.6 539.09 9.03

0.75 0.164 1.3867 1.0 253.4 213.93 15.56

0.75 0.024 1.3867 1.0 1614 1406.7 12.87


is stable throughout the settling process. However, at

intermediate and high Re, the stars showed an occasional

tendency to align themselves with two axes at 45j with

respect to the horizontal. In these cases, the actual surface

projected was higher than the one considered in the

calculations and this resulted in an under-prediction of

the drag coefficient, which is depicted in Fig. 5b.

The measured and predicted drag coefficients vs. the

Reynolds number are plotted in Fig. 5a, b and c. For the

close-to-sphere agglomerates, our measurements at low

Reynolds were consistent with the measured value of CD

from Lasso and Weidman [15] and with their correlation,

which is valid for Re up to 1. Our predictions for the

agglomerates made with both spherical and planar construc-

tions agreed very well with the measurements over the range

0.15 <Re < 1500, as shown in Fig. 5a and c. It must be

pointed out that the predictions for the cross-shaped, bar-

shaped and H-shaped agglomerates of spheres form a single

straight line, because these particles were characterized by

the same flatness ratio, dA/dn, and the same circularity, c.

The calculated drag coefficients for the pyramidal shape

slightly under-predicted the experimental data at Re higher

than 300. A similar trend was observed for the star-shaped

agglomerates over the range 20 <Re < 1000. However, these

discrepancies remained relatively low as shown in Fig. 4b.

We have also tested the derived correlation (Eq. (10b))

with experimental data and correlations given in previous

studies, for particles of regular shapes. Both isometric and

axisymmetric particles were considered. Comparisons for

isometric polyhedrons are shown in Fig. 6. Two shapes of

polyhedrons were examined: cubes and tetrahedrons. The

corresponding shape factors dA/dn and c for these shapes are

well within the range of validity of Eq. (10b). It is observed in

Fig. 6 that for low and intermediate Re, the predictions from

Eq. (10b) are always in very good agreement with both the

measured values from Pettyjohn and Christiansen [20] and

the correlation derived by Haider and Levenspiel [5]. The

individual fractional differences, which were higher for cubes

than for tetrahedrons, did not exceed the value of 10%, which

is close to the experimental uncertainty of previous methods.

This good agreement between the data and the correlations

is in the range 0.01 < Re < 200 for cubes and from

0.03 <Re < 50 for tetrahedrons. At higher values of Re, the

calculatedCD deviated from the experimental set of data. The

deviations abruptly increase and eventually become constant.

This is due to the formation of a strong wake at the aft of the

polyhedron, which tends to induce a wobbling or spinning

motion. This was also observed by Pettyjohn and Christian-

sen [20]. In this case, the two polyhedrons teetered or

wobbled and eventually spun or rolled on a horizontal axis

and followed a spiral path rather than a straight vertical path

in their descent. This difference in the settling process

explains the scattering observed in Fig. 6a.

It must be pointed out that, in our experiments with

agglomerates, this unstable settling process was observed

occasionally for 50 <Re < 1000, and systematically for Re

higher than 1600. Also that, even though the Haider and

Levenspiel [5] expression showed a good agreement with the

measuredCD for the isometric particles, it is unsuitable for the

predictions of the drag for agglomerates that are close to the

spherical shape. Because of the presence of an exponential

function in its formulation, a very small change in the

sphericity results in very large effects on the value for CD.

We also compared the drag coefficients obtained by Eq.

(10b) with experimental values for rectangular parallelepi-

peds and cylinders from previous studies. We considered the

data reported by Heiss and Coull [8] and by Lasso and

Weidman [15]. Both focused on very low Reynolds numb-

ers, Re < 1. We selected the data corresponding to particles

falling with their maximum cross section normal to the

motion of the particle and tested our correlation with the

experimental data. Then, we calculated the corresponding

CD using Eq. (10b). The results of this comparison are

shown in Figs. 7 and 8 as well as in Table 3. A glance at the

figures and the table proves that Eq. (10b) predicts fairly

well the experimental data for CD for axisymmetric par-

ticles, as long as they are settling with their axis either

horizontal or vertical. However, it should be mentioned that

the predictions for both cylinders and rectangles, which

were initially released with the maximum cross section

parallel to the motion, did not agree well with the measured

values of CD that are reported by Heiss and Coull [8] for

0.05 <Re < 0.1. The same observation was made by testing

the predicted CD values from Eq. (10b) with the data

reported by Lasso and Weidman [15] for solid cylinders in

the range 0.05 <Re < 0.6. This discrepancy is due to the fact

that these particles did not maintain their vertical orientation

as pointed out by Pettyjohn and Christiansen [20], Becker

[2] Marchildon et al. [16] and Jayaweera and Mason [10].

6. Conclusions

Laboratory measurements were conducted to determine

the terminal velocity of irregularly shaped agglomerates of

spheres. The drag coefficients were subsequently calculated

and reported. The experimental error of the measurements

was considerably lower than that of previous studies. From

these measurements and several optimization tests with


different functional forms, we derived a simple but accurate

correlation for the drag coefficients of irregular particles,

which is shown in Eq. (10b). A combination of several

shape factors was necessary to properly describe the effect

of the shape of a particle on the hydrodynamic drag

coefficient. We found out that the Reynolds number based

on the particle nominal diameter, Re, the ratio of the surface-

equivalent-sphere to the nominal diameters, dA/dn, and the

particle circularity, c are appropriate dimensionless number

measures for characterizing the deviation of the shape of the

particle from the spherical shape.

The proposed correlation is valid in the ranges of varia-

bles 0.15 <Re < 1500, 0.80 < dA/dn < 1.50 and 0.4 < c < 1.0.

These ranges cover most of the irregularly shaped particles in

engineering applications. Measurements and predictions are

in very good agreement for the sphere agglomerates, as well

as for more regularly shape particles that are reported in

previous studies provided that the particles were released

with their maximum cross section parallel to the motion.

Furthermore, in the case of spheres, the present correlation

for the drag coefficient reduces to a well-known expression

for spheres [4].

Acknowledgements

This research effort was partly supported by a grant by

the ONR to the Tulane/Xavier Center for Bioenvironmental

Research, for which the authors are thankful.

References

[1] J. Baba, P.D. Komar, Settling velocities of irregular grains at low

Reynolds number, J. Sediment. Petrol. 51 (1) (1981) 121–127.

[2] H.A. Becker, The effects of shape and Reynolds number on drag in

the motion of a freely oriented body in an infinite field, Can. J. Chem.

Eng. 37 (1959) 85–91.

[3] H.C. Brinkman, A calculation of the viscous force exerted by a flow-

ing fluid on a dense swarm of particles, Appl. Sci. Res., Sect. A,

Mech., Heat 1 (1947) 27–34.

[4] R. Clift, J.R. Grace, M.E. Weber, Bubbles, Drops and Particles, Aca-

demic Press, New York, 1978.

[5] A.M. Haider, O. Levenspiel, Drag coefficient and terminal velocity

of spherical and nonspherical particles, Powder Technol. 58 (1989)

63–70.

[6] M. Hartman, J.G. Yates, Free-fall of solid particles through fluids,

Collect. Czechoslov. Chem. Commun. 58 (5) (1993) 961–982.

[7] M. Hartman, O. Trnka, K. Svoboda, Free settling of nonspherical

particles, Ind. Eng. Chem. Res. 33 (1994) 1979–1983.

[8] J.F. Heiss, J. Coull, The effect of orientation and shape on the settling

velocity of non-isometric particles in a viscous medium, Chem. Eng.

Prog. 48 (3) (1952) 133–140.

[9] J.D. Hottovy, N.D. Sylvester, Drag coefficients for irregularly

shaped particles, Ind. Eng. Chem. Process Des. Dev. 18 (3)

(1979) 433–436.

[10] K.O.L.F. Jayaweera, B.J. Mason, The behavior of freely falling cylin-

ders and cones in viscous fluid, J. Fluid Mech. 22 (4) (1965) 709–720.

[11] A.R. Khan, J.F. Richardson, The resistance to motion of a solid sphere

in a fluid, Chem. Eng. Commun. 62 (1987) 135–150.

[12] A.S. Kim, K.D. Stolzenbach, The permeability of synthetic fractal

aggregates with realistic three-dimensional structure, J. Colloid Inter-

face Sci. 253 (2002) 315–328.

[13] P.D. Komar, Settling velocities of circular cylinders at low Reynolds

number, J. Geol. 48 (1980) 327–336.

[14] P.D. Komar, C.E. Reimers, Grain shape effects on settling rates,

J. Geol. 86 (1978) 193–209.

[15] I.A. Lasso, P.D. Weidman, Stokes drag on hollow cylinders and con-

glomerates, Phys. Fluids 29 (12) (1986) 3921–3934.

[16] E.K. Marchildon, A. Clamen, W.H. Gauvin, Drag and oscillatory

motion of freely falling cylindrical particles, Can. J. Chem. Eng. 42

(1964) 178–182.

[17] J.H. Masliyah, N. Epstein, Numerical study of steady flow past sphe-

roids, J. Fluid Mech. 44 (3) (1970) 493–512.

[18] G. McKay, W.R. Murphy, M. Hillis, Settling characteristics of discs

and cylinders, Chem. Eng. Res. Des. 66 (1988) 107–112.

[19] J.S. McNown, J. Malaika, Effects of particle shape on settling velocity

at low Reynolds’ numbers, Trans. - Am. Geophys. Union 31 (1950)

74–82.

[20] E.S. Pettyjohn, E.R. Christiansen, Effect of particle shape on free-

settling rates of isometric particles, Chem. Eng. Prog. 44 (1948)

157–172.

[21] D. Rodrigue, D. DeKee, R.P. Chhabra, Drag on non-spherical particles

in non-Newtonian fluids, Can. J. Chem. Eng. 72 (1994) 588–593.

[22] R.P. Roger, R.G. Hussey, Stokes drag on a flat annular ring, Phys.

Fluids 25 (6) (1982) 915–922.

[23] L. Schiller, A. Naumann, Uber die grundlegende Berechnung bei der

Schwekraftaufbereitung, Ver. Deutch Ing. 77 (1933) 318–320.

[24] A.N. Singh, K.C. Roychowdhury, Study of the effects of orientation

and shape on the settling velocity of non-isometric particles, Chem.

Eng. Sci. 24 (1969) 1185–1186.

[25] L. Squires, W. Squires, The sedimentation of thin discs, Trans. Am.

Inst. Chem. Eng. 33 (1937) 1–11.

[26] D.N. Sutherland, C.T. Tan, Sedimentation of a porous sphere, Chem.

Eng. Sci. 25 (1970) 1950–1951.

[27] L.B. Torobin, W.H. Gauvin, Fundamental aspects of solid–gas flow:

Part IV. The effects of particle rotation, roughness and shape, Can. J.

Chem. Eng. 38 (1960) 142–153.

[28] A. Unnikrishan, R.P. Chhabra, An experimental study of motion of

cylinders in Newtonian fluids: wall effects and drag coefficient, Can.

J. Chem. Eng. 69 (1991) 729–735.

[29] H. Wadell, Sphericity and roundness of rock particles, J. Geol. 41

(1933) 310–331.

[30] W.W. Willmarth, N.E. Hawks, R.L. Harvey, Steady and unsteady

motions and wakes of freely disks, Phys. Fluids 7 (1964) 197–208.

drag coefficients of irregularly shaped particles -...

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