ISSN 00405795, Theoretical Foundations of Chemical Engineering, 2011, Vol. 45, No. 3, pp. 248–266. © Pleiades Publishing, Ltd., 2011.
Original Russian Text © G.I. Kelbaliyev, 2011, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2011, Vol. 45, No. 3, pp. 264–283.
Drag Coefficients of Variously Shaped Solid Particles,
Drops, and Bubbles
G. I. Kelbaliyev
Institute of Chemical Problems, Academy of Sciences of Azerbaijan, H. Javid ave. 29, Baku, AZ1143 Azerbaijan
email: Kelbaliev@yahoo.com
Received June 9, 2009; in final form, February 24, 2010
Abstract—This article, which is largely a review, deals with the drag force and drag coefficient for rigid spher
ical and deformable particles in ordinary and nonNewtonian fluids. The most important theoretical formu
las for small Reynolds numbers of R e Ⰶ 1 and semiempirical formulas for the drag coefficient in a wide Re
range up to 106 are presented. The deformation of drops and bubbles and its effect on the drag coefficient are
considered.
DOI: 10.1134/S0040579511020084
INTRODUCTION
One of the significant parameters governing the
migration of dispersed particles in a flow is the drag
force arising from the interaction between the particle
surface and the flow past the particle. This force plays
the key role in mass, heat, and momentum transfer.
The drag coefficient of a spherical particle in the clas
sical sense is given by the formula
CD =
FS
( ρ U 2) s
C
2
∞
,
M
where U ∞ is the flow velocity far from the particle. The
drag FS depends on the shape of the surface past which
the fluid moves, on the hydrodynamic characteristics
of the flow, on the size and properties of the particles,
and on the properties of the medium. Numerous
experimental studies aimed at determining the drag
coefficient resulted in the construction of the so
called standard Rayleigh curve for a single solid sphere
moving with a constant velocity in a quiescent isother
mal liquid. For complexshaped bodies, theoretical
determination of the drag coefficient involves serious
difficulties and, in most cases, this quantity is esti
mated using experimental data and empirical correla
tions. This is particularly true for the drag coefficient
of deformable particles (drops and bubbles) at large Re
values with the corresponding specific features of the
flow (boundary layer breakaway, hydrodynamic wake,
drag crisis, etc.) and for nonspherical single particles.
In chemical technology there are many processes
involving solid particles whose size changes in the flow
from zero to some limit (crystallization, condensa
tion, swelling, granulation) and many processes in
which particles can disappear entirely as a result of
physicochemical transformations (dissolution, subli
mation, evaporation). Along with these processes,
there are mass transfer processes in which the particle
size varies with time in a wide range (agglomeration,
coalescence, unsteadystate particle growth or dimi
nution). As the drop grows in size, the flow past it
begins to differ from the flow past a solid particle
because, along with the above physical phenomena,
other effects come into play. These are drop shape and
surface pulsations (because of the mobility of the
interface and the uniform pressure distribution on it),
internal liquid circulation, shape variations, fragmen
tation, etc. Because of this, various expressions have
been suggested in the literature for calculating the drag
coefficient for solid particles, drops, and bubbles both
for small Re values and for Re varying in a fairly wide
range [1–13]. For gas bubbles and liquid drops, the
determination of the drag coefficient at large Re values
is complicated by shape distortion depending on the
Weber number We, Morton number Mo, Acrivos
number Ac, and Bond number Bo. Table 1 presents
the definitions of, and the relationships between, these
numbers. In the general case, the drag experienced by
a liquid drop moving in a medium can be expressed as
2 + 3γ
FSK = 1
FS [1 + ζ ( γ, Ac, Re)] ,
3 1+ γ
where FSK is the drag force acting on the drop. In view
of We = Ac Re 2, we obtain ζ ( γ, Ac, Re) =
ζ ( γ, We, Re) , although, as will be demonstrated
below, this equality is not valid in the general case.
The variation of the character of the C D (Re)
dependence over a wide Re range is due to the hydro
dynamic changes in the flow past the particle. Assum
ing that the drag coefficient in isothermal flow
depends on the flow velocity and viscosity and on the
particle shape, introducing some equivalent diameter
d, and using dimensional and similarity methods, we
248
DRAG COEFFICIENTS OF VARIOUSLY SHAPED SOLID PARTICLES
249
Table 1. Similarity parameters and their interrelation
Number
Definition
1
Archimedes
Ar =
gd 3 Δρ
ν 2c ρc
2
Morton
Mo =
gη4c Δρ
ρ c σ3 ρ c
3
Weber
We =
ρ cU d
σ
4
Bond or Eötvös
Bo =
gd 2
Δρ
ρ
5
Acrivos
Ac =
6
Froude
Fr = U
dg
2
Relation
Ar = BoAc–1
Ar = M oAc − 3
4/3
1/3
We = Re d Mo Fr
2
(1)
= A f (Re) .
Re
As was demonstrated by theoretical calculations
and experimental studies, the drag coefficient for
spherical particles (solid particles, liquid drops, gas
bubbles) at Re Ⰶ 1 (Stokes flow) obeys the following
equation [1, 8, 12]:
−k
(2)
C D = 24 .
Re
For somewhat larger Re values, Eq. (1) can be rewrit
ten as
(3)
C D = 24 (1 + ζ (Re)) ,
Re
where ζ (Re) is, in most cases, an empirical function of
Re derived from experimental data. For drops and
bubbles and large Reynolds numbers, Eq. (1) is repre
sented as C D = A f (Re, Mo, We, Ac) .
Re
The purpose of this work is to analyze the state of
the art in the determination of the drag coefficients of
rigid and deformable particles in ordinary and non
Newtonian fluids.
DRAG COEFFICIENT
OF A SOLID SPHERICAL PARTICLE
Theoretical and experimental studies of the drag
coefficient for a solid spherical particle have been car
ried out by many researchers [1–6, 12, 13]. The com
plete list of the relevant works would make up an
−2
−1/3
= Ac Re d
2
−4
= BoWe Re d
We = Re d (MoBo)
2
2
−1/2
−2
Ac = We Re d
2
C D = A Re
−4
Bo = ArAc = WeFr −1
ρcν c
σd
obtain C D = AU nd mν ck . With the dimensions of these
parameters taken into account, n = –k and m = –k. In
view of the arbitrariness of the coefficient A, the
expression for the drag coefficient appears as
3
Mo = We Re d Fr
Fr = WeBo
− 1 Δρ
ρc
extensive bibliography. Note that the Stokes solution
(2) is not the exact solution of the flowpastparticle
problem with inertial terms. Calculations demon
strated that, for distances larger than the particle size,
the neglected terms carry a significant correction.
Ossen suggested linearizing the Navier–Stokes equa
tion by replacing the convective term (U ∇) U with
(U ∞∇)U ∞, where U ∞ is the velocity of unperturbed flow
at infinity [4, 5, 14]. This leads to the following expres
sion:
(4)
ζ (Re) = 3 Re + .....
16
An analysis carried out by Proudman and Pearson [4,
5] by the matched asymptotic expansions method led
to the relationship
(5)
ζ (Re) = 3 Re + 9 Re 2 ln Re .
16
160
2
Quantitatively, expansion (5) is inappropriate because
the ln Re term makes the result dependent on the method
of determining the Reynolds number. A threeterm
expansion was obtained by Chester and Breach [7]:
(
2
ζ (Re) = 3 Re + 9 Re ln Re + A
16
160
2
⎛ Re 2 Re ⎞
+ O⎜
ln ⎟ ,
2⎠
⎝ 4
A = γ 1 + 5 ln 2 − 323 ≈ 0.835,
3
360
)
(6)
where γ 1 is the Euler constant, whose value is 0.577.
Note that there is still no substantiated method for
improving asymptotic series with logarithmic terms.
Voloshchuk and Sedunov [14] reported that the three
term expansion (6) effectively refines Eq. (4) only at
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
No. 3
2011
250
KELBALIYEV
Re ≤ 0.44, and augmented Eq. (6) to make it better up
to Re ≤ 5. Thus, the solutions of the hydrodynamic
problem of flow past a spherical particle demonstrate
that the Stokes equation for the drag force is only the
first term of the series in powers of Re.
The above approximate theoretical solutions are
appropriate for very small Reynolds numbers of
Re Ⰶ 1, i.e., for laminar flow. For large Re values, the
theoretical solutions of the hydrodynamic problem are
complicated by the appearance of convective terms in
the Navier–Stokes equation and by various hydrody
namic phenomena (boundary layer breakaway, turbu
lence, wake formation, etc.). Because of this, numer
ous formulas fitted to the experimental Rayleigh curve
have been reported for different Re ranges.
Wake formation is associated with the specific flow
pattern around the spherical body and with the
appearance of convective forces. At Re ≈ 10, the iner
tial terms begin to distort the symmetric flow pattern
and the laminar boundary layer begins to break away
from the rear part of the sphere surface [4]; at Re ≈ 20,
according to experimental data of Taneda [9], an
annular vortex forms behind the sphere. Van Dyke [15]
analyzed the liquid velocity field and discovered that,
even at Re ≥ 8 , there is a wake as an annular vortex
behind the sphere. At Re ≈ 100 , the flow at the rear
part of the sphere becomes unsteady and the system of
vortices spreads downstream over a distance approxi
mately equal to the sphere diameter [4]. At Re ≈ 500,
the flow becomes irregular and a system of vortices
separates from the rear part of the sphere with a fre
quency increasing with Re to from a wake. As Re
increases, annular vortices form and separate from the
surface constantly, thus causing periodic changes in
the flow velocity field and in the drug. At fairly large
Re values, part of the laminar boundary layer on the
sphere surface before the breakaway line becomes
unsteady and turbulent. This turbulization of the
boundary layer appreciably shifts the breakaway line
downstream, the turbulent wake behind the sphere
narrows, and a drag crisis occurs at Re ≈ 3 × 10. This
causes a dramatic decrease in the drag down to
C D ≈ 0.1.
Note that, although the drag curve is differentiable
and continuous, its derivatives are not monotonic
functions in wide Re ranges. In particular, at Re values
corresponding to the drag crisis, the derivatives show a
discontinuity of the first kind: as the discontinuity is
approached from the left Re → +3 × 105, C D ≈ 0.51;
as it is approached from the right Re → −3 × 105,
C D ≈ 0.1. In drag coefficient calculations, the data
pertaining to this region are smoothed to some extent.
For very small Re values, an analysis of calculated and
experimental data leads to the conclusion that ζ (Re) is
a monotonically increasing function satisfying the fol
lowing conditions [14]:
d ζ (Re) 3
≤ , ζ (Re) ~ Re 3 .
d Re
16
Levich [8] estimates the region of flow breakaway
from the surface of drops and bubbles as
2
(7)
θ≈ 1 ,
Re
that is, the flow almost does not break away from the
bubble. Michaelides [6] reported the following empir
ical correlation for the wake of the flow past a solid
spherical particle:
( )
θ ≈ 180 − 42.5 ⎡ln Re ⎤
⎣⎢ 20 ⎦⎥
0.483
.
Using experimental data of other authors [6, 9], we
obtained, for the θ ≤ 180 range, the following simpler
formula with a correlation factor of r 2 = 0.995:
θ = 276
.
1/7
Re
(8)
It follows from (7) and (8) that the flow breakaway
angle for a drop or bubble is much smaller than for a
solid particle. Owing to the mobility of the surface of
the drop and bubble, the flow breakaway from the sur
face occurs in the very small region in which flow tur
bulization at large Re values takes place. Accordingly,
the flow breakaway point for the drop or bubble, as
distinct from the solid particle, is situated so low that
the turbulent wake covers only an insignificant part of
the surface. Thus, the total drag on a solid particle can
be considered as the drag force acting on the surface
upstream of the flow breakaway point and having a vis
cous character plus the drag force acting in the break
away region. As was noted by Levich [8], the drag
force is dissipative in character and can be expressed in
terms of energy dissipation, and it is equal to the
Stokes force at small Re values. This flow pattern past
the spherical body poses serious limitations on the
description of the Rayleigh curve in a wide Re range.
An analysis of the existing formulas for calculating the
drag coefficient for solid spherical particles demon
strated that, for Re ≤ 500 , most of the equations
reduce to the following forms:
(
)
(9)
C D = 24 1 + A Re ,
Re
where A = 0.17 and n = 2/3—Klyachko–Mazin for
mula [10, 11]; A = 0.189 and n = 0.632—Prup
pacher–Steinberger formula [16]; A = 0.15 and n =
0.687—Schiller–Neumann formula [1, 17, 18]; A =
0.1538 and n = 2/3—Kelbaliev–Ceylan formula [19];
A = 0.1935, n = 0.6305, and 20 ≤ Re ≤ 260 —Clift–
n
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
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2011
DRAG COEFFICIENTS OF VARIOUSLY SHAPED SOLID PARTICLES
Gauvin formula [20]; A = 3/16 and n = 2/3—Stuck
enbruck formula [6, 11];
(
)
n
(10)
C D = 24 1 + A Re m ,
Re
where A = 0.0665, m = 2/3, n = 3/2, and 0.5 ≤ Re d ≤
2 × 103 [3]; A = 0.27, m = 1, and n = 0.43 [3, 21]. At
Re ≤ 500 , the Klyachko–Mazin and Schiller–Neu
mann equations coincide within acceptably low accu
racy:
C D = 24 + ζ (Re) ,
Re
(11)
where ζ (Re) = 6.48Re −0.573 + + 0.36, 20 ≤ Re ≤ 500
[3, 10] , ζ (Re) = 0.03276 × × Re −0.18−0.05 lg Re,
0.01 ≤ R e ≤ 20
[20] , ζ (Re) = 6 + 0.4,
1 + Re
5
1 < Re < 10 [6] ;
(12)
CD = A n ,
Re
where the coefficients A and n take the values listed in
Table 2 [10]. This table can be supplemented with the
following data: for Re = 04–7 × 104, A = 0.16 and n =
–0.102. A number of approximations for the drag
coefficients as Eq. (12) are presented in [1–6; 11, 19,
22–31]. In order to widen the Re range, the above for
mulas were brought into the following forms:
(
)
C D = 24 1 + a Re m + ψ (Re) ,
Re
n
C D = 24 (1 + k Re) + ψ1 (Re) ,
Re
(
where ψ (Re) = 0.42 1 + 42300 Re −1
(13)
(14)
) , a = 0.15,
−1
m = 0.687, Re < 2 × 10 5 [1, 6] ;
(
ψ (Re) = 0.42 1 + 1.902 × 10 4 Re −1.18
) ,
−1
a = 0.241, m = 0.687, Re < 1.5 × 105 [11];
(
ψ(Re) = 0.407 1 + 8710 Re −1
) ,
−1
a = 0.15, m = 0.684, Re < 2 × 10 5 [22] ;
(
ψ (Re) = 0.413 1 + 16300 Re
) ,
−1.09 −1
a = 0.173, m = 0.657, Re < 2 × 105 [25];
(
ψ (Re) = 0.4251 1 + 6880.9 Re −1
) ,
−1
a = 0.1806, m = 0.6459, Re < 2 × 10 4 [28];
(
)
ψ (Re) = 0.47 ⎡⎣1 − exp −0.04 Re 0.38 ⎤⎦ ,
k = 0.27, n = 0.43, Re < 2 × 10 [27].
5
251
Table 2. Determining the particle drag coefficient in differ
ent Re ranges
Re
A
n
Re
A
n
<0.1
0.1–1.0
1.0–10
24
26.9
26.5
1.0
0.95
0.80
10–100
100–800
800–1000
16.8
6.1
5.8
0.60
0.38
0.37
For wide Re ranges, the formulas for the drag coeffi
cient of a spherical particle are more complicated and
cumbersome (Table 3). These formulas are valid for
comparatively large Re values and allow the drug coef
ficient to be calculated with a relative error of below
5–10%. Note that Eq. (17) was obtained by the New
ton method [18], which enables one to combine sepa
rate piecewise approximations into a single formula
for a wide Re range. The Clift–Gauvin equation (15)
describes the entire drag curve for 0.01 ≤ Re ≤ 10 6 as
a piecewise approximation for particular Re ranges.
Equations (13), (14), and (17) presume the observance
of the Newton law in the 10 4 ≤ Re ≤ 105, range, which
means that the friction force is proportional to the
square of flow velocity (C D ≈ 0.44) . This leads to a sig
nificant deviation of the calculated data from the
actual experimental curve in this Re range. Equation
(21) fits the drag curve only in the drag crisis region. As
follows from the data presented in Table 3, use of series
or sums of functions is the only way to solve the prob
lem of approximating the drag curve for a spherical
particle in a wide Re range. By using the drag coeffi
cient in the form of Eq. (3), it is possible in most cases
to reduce the problem of fitting the drag curve to
experimental data for spherical particles to appropri
ately choosing a ζ (Re) . function. However, it should
be remembered that the highest computational effi
ciency is attained with formulas having the minimum
possible number of terms and parameters to be deter
mined.
A satisfactory fit to the standard drag curve in the
6
0.1 ≤ Re ≤ 10 range is provided by the following
series [31]:
∞
CD =
∑ A Re
m(n)
n
n =1
(
)
exp −bn Re ( ) .
k n
(23)
Here, A, b, m, and k are the coefficients of the series
depending on the term number. This equation, con
sisting of similar functions, is very flexible and can
describe the smallest changes in the drag over the
widest Re range. In addition to the formulas listed in
Table 3, there are other correlations for the spherical
particle drag coefficient in particular Re ranges [1, 6,
11, 30–34]. Here, we suggest one more equation for
calculating the drag coefficient of a solid spherical
particle over the 0.01 ≤ Re ≤ 10 6 range:
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
No. 3
2011
252
KELBALIYEV
4
taking into account the compressibility of the liquid
was reported for the sphere drag coefficient:
СD
10
C D = 24
Re
103
)(
(
)
⎡ 1 + 0.15Re 0.687 1 + exp ⎡− 0.427 − 3 ⎤ ⎤
⎢
⎣⎢ M 4.63 Re 0.88 ⎦⎥ ⎥ , (26)
×⎢
⎥
1 + M ⎡3.82 + 1.28 exp − 1.25Re ⎤
⎢
⎥
⎢
⎥
Re ⎣
M ⎦
⎣
⎦
102
10
2
1
1
0.1
0.01 0.1
3
1
102
10
103
104
105
106 Re
Fig. 1. Drag coefficient of a solid spherical particle:
24
(1) C D ≈ 0.44; (2) C D =
; (3) Schiller–Neumann equa
Re
tion.
2
⎡
⎤
C D = 24 ⎢1 + Re 3 ζ (Re)⎥
Re ⎣
⎦
−24
+ 0.22 1 − exp − 1.2 × 10 Re 4 ,
(
ζ (Re) =
+
))
(
Re
5 + 6.5Re 2
Re
2
3
3 × 10 + 36 Re + 2.8 × 10
4
(24)
2
−13
Re
10
3
.
For 0.01 ≤ Re < 800 , Eq. (24) reduces to
8
⎛
⎞
Re 3
⎟
C D ≈ 24 ⎜⎜1 +
2⎟
Re ⎜ 5 + 6.5Re ⎟
⎝
⎠
(25)
which is a modification of Eq. (9). Figure 1 plots the
Rayleigh curve and the data calculated via Eq. (24).
This equation provides a satisfactory fit with accept
able accuracy of 8–10% and a correlation factor of
r 2 ≈ 0.9431. The error of approximation arises from
the low accuracy of the data in the drag crisis region.
Note that a considerable effect of the compressibil
ity of the liquid is observed at high velocities of the
flow past the particle, when the drag crisis takes place.
This effect is taken into account in terms of the Mach
number M = U , where c is the speed of sound in the
c
given medium [35]. At M Ⰶ 1, the liquid can be con
sidered to be incompressible. The following equation
)
(
Re < 3 × 105.
At Re < 500, this equation coincides with Schiller and
Neumann’s formula. Experimental studies demon
strated that an increase in M causes an increase in the
Re value at which the turbulization of the boundary
layer occurs.
DRAG COEFFICIENT
OF NONSPHERICAL PARTICLES
The spherical shape of a particle is an idealization,
and real particles deviate from the strict spherical
shape, appearing as an oval, ellipse, disc, cylinder, or
an irregularly shaped body. Irregularly shaped parti
cles are characterized by a sphericity factor, which is
the ratio of the surface area of the sphere having the
same volume as the irregularly shaped particle (s) to
the actual surface area of the particle ( s d ): ϕ = s . For
sd
cubic particles, ϕ = 0.806 ; for cylindrical particles,
ϕ = 0.69 ; for a disc, − ϕ = 0.32. Numerous expres
sions and experimental data have been reported for the
drag coefficient of irregularly shaped particles [28,
36–41] (Table 4). Formula (31) in Table 4 is applicable
to a round disc with thickness H and diameter d. the
drag coefficient of a cylindrical body with 5 < L < 50
d
and Re < 1 is given by the following expression[6]:
4
(32)
CD =
.
Re ⎡ln L − 0.1197⎤
⎢⎣ d
⎥⎦
The expression for the drag coefficient of a cylindrical
body in lateral flow can be derived from experimental
data [32] in the following form (Re = Ud , d is the cyl
νc
inder diameter):
()
CD
=
1.875
−3
2.55
⎞
10 ⎛1 + Re + 0.368 × 10 Re
(33)
0.778 ⎜
1.15
3.95 ⎟
−14
Re
⎝ 60 + 6.8Re + 0.4 × 10 Re ⎠
(
+ 0.36 1 − exp(− 1.2 × 10
−24
4
)
Re ) .
Figure 2 presents a comparison between the data cal
culated via Eq. (33) and experimental drag coefficient
data for a cylindrical particle for 0.1 ≤ Re ≤ 10 6. Equa
tion (33) has the following particular solutions:
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
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2011
DRAG COEFFICIENTS OF VARIOUSLY SHAPED SOLID PARTICLES
253
Table 3. Drag coefficients of spherical particles over a wide Re range
En
try
⎧ 24 ,
⎪Re
⎪ 24
0.82 − 0.05 l o g Re
1 + 0.1315 Re
,
⎪
⎪ Re
⎪ 24 1 + 0.1935 Re 0.6305 ,
⎪ Re
⎪ 1.6425−1.1242 l o g Re + 0.1558 l o g 2 Re
,
⎪10
⎪⎪ −2.4571 + 2.5558 l o g Re − 0.9295 l o g 2 Re + 0.1049 l o g3 Re
C D = ⎨10
,
⎪ −1.9181 + 0.637 l o g Re − 0.0636 l o g 2 Re
,
⎪10
⎪ −4.339 + 1.5809 l o g Re − 0.1546 l o g 2 Re
,
⎪10
⎪29.78 − 5.3 l o g Re ,
⎪
⎪0.19 l o g Re − 0.49,
⎪
4
⎪0.19 − 8 × 10 ,
⎪⎩
Re
( )(
( )(
1
2
For Refer
mula ence
Expression
⎫
⎪
⎪
0.01 < Re ≤ 20
⎪
⎪
⎪
20 < Re ≤ 260
⎪
⎪
260 < Re ≤ 1500
⎪
⎪
4 ⎪
1500 < Re ≤ 1.2 × 10 ⎬
4
4 ⎪
1.2 × 10 < Re ≤ 4.4 × 10 ⎪
4
5⎪
4.4 × 10 < Re ≤ 3.38 × 10 ⎪
5
5
⎪
3.38 × 10 < Re ≤ 4 × 10
⎪
5
6
4 × 10 < Re ≤ 10
⎪
⎪
6
⎪
Re > 10
⎪⎭
Re < 0.01
)
)
⎡
⎤
1
CD = ⎢
+
ϕ
4
⎥
−1
−1
⎣( ϕ1 + ϕ2 ) + ϕ3
⎦
1/10
ϕ1 =
,
( )
24
Re
10
10
⎛1.57 × 10 8 ⎞
6
1
ϕ
=
, ϕ4 =
, Re < 10
ϕ2 =
,
3
⎜
− 10
1.625 ⎟
17
2.63
10
−
−
⎡
⎤
⎡ 0.148Re 0.11 −10 + 0.5 −10 ⎤
⎝ Re
⎠
× 10 Re
+ 0.2
⎣⎢
⎦⎥
⎣⎢
⎦⎥
CD =
CD =
4
5
CD =
(
)
3
( )
3.6
24 ⎡
Re
1 + 18.5 Re +
Re ⎢⎣
2
24 m(Re)
10
,
Re
(
)
m (Re) = 0.261Re
⎤
⎥
⎦
0.369
4 Re
,
9 330 + Re 4/5
− 0.105 Re
0.431
−
0.1 ≤ Re ≤ 5 × 10
0.124
,
2
1 + l o g Re
4
Re < 3 × 10
b = 0.4 + 0.77 exp ( − 0.04 Re ) , Re < 3 × 10
⎧28.12 − 5.3 l o g Re,
⎪
C D = ⎨0.1 log Re − 0.46,
⎪0.19 − 4 × 10 4 Re −1,
⎩
(
[19]
5
(18)
[24]
(19)
[26]
(20)
[29]
(21)
[11]
(22)
[30]
2
a = 0.09 + 0.077 exp(− 0.4 Re),
5
1.7 × 10 ≤ Re ≤ 2 × 10 ,⎫
5
5
5
5 ⎪
2 × 10 < Re ≤ 5 × 10 ,⎬
Re > 5 × 10
5
⎪
⎭
)
m
n
24
3
A + B Re + C Re , A = 1, B = ,
m = 1, C = 0, Re < 1,
Re
16
A = 1, B = 0.1935, m = 0.6305, C = 0, 1 < Re < 285,
A = 1, B = 0.015, m = 1, C = 0.2283, n = 0.424, 285 < Re < 2000,
5
0.44
, m = 1, C = 0, 2000 < Re < 3.5 × 10
A = 0, B =
24
CD =
(17)
4/5
+
⎡1 + 0.545 Re + 0.1Re1/2 (1 − 0.03 Re )⎤
C D = 24 ⎢
⎥,
Re ⎢⎣
1 + a Re b
⎥⎦
7
[21]
)
2
−5
3
−10
4
0.2841
9.04
1+
λ (Re) , λ (Re) = 0.96208 Re + 2.73646 × 10 Re − 3.93861 × 10 Re
2
Re
Re
−15
5
−21
6
−27
7
6
+ 2.47686 × 10 Re − 7.15934 × 10 Re + 7.43723 × 10 Re , Re < 3 × 10
6
8
11 1/30
(16)
+⎜
1
(
[1]
10
⎛ 21 ⎞
⎛ 4 ⎞
10
+ ⎜ 0.33 ⎟ + 0.4 ,
0.67 ⎟
⎝ Re ⎠
⎝ Re ⎠
10
(15)
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
No. 3
2011
254
KELBALIYEV
Table 4. Drag coefficients of irregularly particles
En
try
For
mula
Expression
CD =
1
CD =
2
(
30
67.289
+
,
Re exp (5.03ϕ)
)
0.2 ≤ ϕ ≤ 1.0,
(
0.1 ≤ Re ≤ 20
Refer
ence
(27) [35, 36]
)
b Re
b
2
24
1 + b1 Re 2 + 3
, b1 = exp 2.3288 − 6.4581ϕ + 2.488ϕ , b2 = 0.0964 + 0.5565ϕ,
b4 + Re
Re
2
3
b3 = exp(4.905 − 13.8944ϕ + +18.4222ϕ − 10.2599ϕ ),
(
2
3
(28) [28, 38]
)
b4 = exp 1.4681 + 12.258ϕ − 20.7322ϕ + 15.8855ϕ .
CD =
3
73.69 Re exp ( − 5.748ϕ)
24
,
[1 + 8.171exp(−4.0655ϕ)] Re 0.0964 + 0.5565ϕ +
Re
Re + 5.378 exp (6.2122ϕ)
0.7 ≤ ϕ < 1, 1 < Re ≤ 20
(29)
[28]
0.6567
24 ⎡
⎤ + 0.4305 ,
1 + 0.118 ( K1K 2 Re)
⎣
⎦
3305
K 1K 2 Re
1+
K1K 2 Re
−0.5741
1
2
K1 = +
, log K 2 = 1.8148 ( − l o g ϕ)
3 3 ϕ
(30)
[39]
(31)
[1, 6]
CD =
4
CD =
5
(
)
( )
⎡ Re 2 Re 2
Ud
64
Re
Re ⎤
1+
, Re ≤ 0.01, Re = e , C D = 64 ⎢1 +
+
ln
Re < 1
⎥,
2
π Re
2π
νc
π Re ⎣
2π
2 ⎦
5π
m
2
64
CD =
1 + 10 , m = −0.883 + 0.906l o gRe − 0.025l o g Re,
π Re
1/3
2
0.01 < Re < 1.5, d e = 6d H
, C D = 64 1 + 0.138 Re 0.792 , 1.5 ≤ Re < 133,
π Re
H—disk thickness, d—disc diameter
(
)
(
)
10 , Re ≤ 1
Re 0.778
1.875
⎛
⎞
Re
,
[32] ,C D ≈ 100.778 ⎜1 +
1.156 ⎟
Re
⎝ 60 + 6.8 Re
⎠
Re ≤ 6.103,
(
DRAG COEFFICIENT
OF DROPS AND BUBBLES
CD ≈
CD ≈
(
(34)
)
10 1 + 0.1076 Re 0.778 , 0.1 ≤ Re ≤ 6.10 3.
Re 0.778
These relationships for a cylindrical body can be
approximated as
C D ≈ 1.0, 6.10 2 ≤ Re ≤ 6.103
CD ≈ 1.1–1.4,
)
8.10 3 < Re < 2.10 5.
As follows from Fig. 3, the drag crisis associated
with the shift of the boundary layer breakaway zone to
the downflow part of the cylinder because of turbuliza
tion takes place at Re ≥ 3 × 105.
The motion of a liquid drop or gas bubble differs
from the motion of a solid spherical particle of the
same volume and mass. The liquid flowing past a drop
initiates, through friction, the circulation of the inter
nal liquid of the drop. This effect depends on the
η
dimensionless parameter γ = d , which is the ratio of
ηc
the dynamic viscosity of the internal liquid to that of
the external liquid and characterizes the degree of
mobility of the drop surface. In addition, as the exter
nal liquid flows past the drop, forces arising from pres
sure nonuniformity act on the drop surface to distort
the spherical shape of the drop. These forces are
opposed by surface tension, which tends to conserve
the spherical shape. The presence of adsorbed matter
on the drop or bubble surface can exert a significant
effect on the surface forces, on the drag coefficient,
and on the drop (bubble) settling or floating velocity.
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
No. 3
2011
DRAG COEFFICIENTS OF VARIOUSLY SHAPED SOLID PARTICLES
Taking into account these specific features of flow
past liquid drops and using the matched asymptotic
expansions method in the determination of the drop
drag coefficient, Taylor and Acrivos carried out a the
oretical analysis and arrived at the following formula
for Re Ⰶ 1:
255
СD
100
10
CD K = 8
Re
⎡2 + 3γ Re 2 + 3γ 1 ⎛ 2 + 3γ ⎞ 2 Re 2 Re⎤ (35)
×⎢
+
+ ⎜
ln ⎥ .
⎟
16 1 + γ 40 ⎝ 1 + γ ⎠ 2
2 ⎦⎥
⎣⎢ 1 + γ
1
Chester and Breach [7] and Ockendon and Evans [43]
modified the formula for relatively large Reynolds
numbers of Re < 6. For large Re values, numerical
methods were suggested for solving the Navier–Stokes
equation (Re < 6) in view of the appearance of convec
tive terms [44, 45]. There have been theoretical and
experimental studies, including numerical solutions,
on the drag coefficient, deformation, and settling
(floating) velocity of drops and bubbles for moderate
and large Re values [1, 13, 46–53]. Table 5 presents
some formulas for calculating the drag coefficients of
drops and bubbles. Unfortunately, most of the formu
las for bubbles are valid only for small and moderate
Re and We values (Re < 50, We < 1). The exceptions
are formulas (39), (45), and (46). Equations (39) and
(40) for the drag coefficient of gas bubbles was derived
from experimental data of Raymond and Rozant [54]
by the Newton method. Equation (39) is valid for
Re ≤ 100, M o ≤ 7 ; for Re ≤ 1.385 , it reduces to
Eq. (40), which is independent of Mo. As was demon
strated by numerous experimental data on the behav
ior of drops and bubbles, their drag coefficient and
degree of distortion depend on Re, Mo, and We. In
particular, the Morton number depends on the prop
erties of the medium and drops and takes the following
values for different liquids at normal temperature:
mineral oil, Mo = 1.45 × 10 −2 ; water + 42% glycerol,
water
+
13%
ethanol,
Mo = 1.75 × 10 −4 ;
−8
Mo = 1.17 × 10 ; methanol, Mo = 0.89 × 10 −10 ; dis
tilled water, Mo = 3.1 × 10–11. The second term in
Eq. (39) characterizes the dependence of the drag
coefficient of a bubble on its deformation, Re, and
Mo. As follows from Eq. (40), at Re < 0.4 spherical
drops and bubbles behave as solid particles and their
drag coefficient is given by formula (2), although for
0.4 ≤ Re ≤ 1.385 − C D = 16 . Thus, the following
Re
formula can be suggested for the 0.01 ≤ Re < 0.4
range [19]:
0.1
10–1
( )
⎡
⎤
8
8
1
⎢
⎥ , (49)
C DK = ζ (Re) =
1+
Re
Re ⎢ 1 − 0.5 1 + 250Re5 −2 ⎥
⎣
⎦
(
)
1
2
100
101
102
103
104
105
106 Re
Fig. 2. Drag coefficient of a cylinder in a lateral flow:
(1) C D ≈ 1.1; and (2) CD = 10Re
−0.78
.
with ζ (Re) → 2,for Re > 0.4 and ζ (Re) → 3, for
Re ≤ 0.01. Figure 3 compares the drag coefficients of
bubbles calculated via formula (39) to experimental
data [54].
Experimental data for the minimum in the drag
1
curve suggest the relationship Re Mo 6 ≈ 7 [55], and, if
Re Mo1/6 > 7, the drop will undergo deformation. In
another work [52], this condition is expressed as
We Re 0.85 > 165. It was reported [29] that the drop will
not be deformed if Mo < 1.2 × 10–7 Bo8.15 for Bo < 5 and
if Mo < 0.2 × 10–7 Bo2.83 for Bo > 5. Based on the data
of Harmathy [56], Ceylan et al. [31] suggested the for
mula
C DK
≈ 6 [1 − exp ( −0.126Bo)] .
CD
(50)
Here, CD is the drag coefficient of the solid particle.
For Bo < 20, the following formula was obtained using
experimental data reported by Harmathy [56]:
C DK
≈ 1.275 Bo.
CD
Based on formula (50), the steadystate value of the
drag coefficient for drops and bubbles for large Re
Re CD ≈ 4 and Bo can be found as C D∞ ≈ 6C D = 8 . If
3
9
the drag coefficients for a solid spherical particle and
for a gas bubble are known, the drag coefficient for a
liquid drop can be determined as follows [3, 11]:
(
)
C DK =
γC D + C DG
,
1+ γ
where CDK , CD , and C DG are the drag coefficients of the
drop, solid particle, and gas bubble, respectively. This
formula is precise at Re Ⰶ 1 and, as was demonstrated
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
No. 3
2011
256
KELBALIYEV
Table 5. Drop and bubble drag coefficients
En
try
Expression
For
mula
Refer
ence
1
⎛2 3 + γ⎞
C DK = 24 ⎜
⎟ , Re Ⰶ 1
Re ⎝ 1 + γ ⎠
(36)
[8]
(37)
[1]
48 ⎛1 + 12Mo ⎞
Mo
+ 0.9
⎜
1/3 ⎟
1/3
3/2
Re ⎝1 + 36Mo ⎠
1.4 1 + 30Mo
+ Bo
(38)
[46]
( )
(39)
⎧ 1.68 ,
⎪⎪Re1/4
C DK = ⎨
⎪ 14.9
,
0.78
⎩⎪Re
2
C DG =
3
4
9
10
1/3
3/2
(
)
( )
C DK =
1 ⎡ ⎛ 24
4 ⎞ 14.9 ⎤
γ⎜ +
, 0.1 ≤ Re < 10
⎟+
1 + γ ⎢⎣ ⎝ Re Re1/3 ⎠ Re 0.78 ⎥⎦
C DK =
6
8
⎪⎪
⎬
1 ≤ Re ≤ 10 ⎪
⎭⎪
4/3
1/3
12 1/55
⎧16ζ 0 (Re) ⎡
⎫
Re Mo
⎤
Re
8
1
+
+
, Re < 100 ⎪
⎪
−
1/3
⎢
⎥
4/3
1/3
3 24 (1 + Mo1/3 )
+ Re Mo
⎪⎪ Re ⎣ 1.385 ⎦
⎪⎪
C DG = ⎨
⎬
12 1/55
⎪16ζ 0 (Re) ⎛
⎪
⎞
Re
1
⎪ Re ⎜⎝1 + 1.385 ⎟⎠ , Re < 10, ζ 0 (Re) = 1 +
5 −2 ⎪
1 − 0.5 (1 + 250 Re ) ⎪
⎪⎩
⎭
5
7
0.1 ≤ Re ≤ 0.5⎫
C DK =
⎡ H ( χ )⎤
48
G ( χ) ⎢1 + 1/2 ⎥ ,
Re
⎣ Re ⎦
Re < 20
3γ + 2
8 3γ + 2 ⎡
3λ + 2 ⎤
1 + 0.05
Re − 0.01
Re ln Re,
⎢
⎥
⎦
Re 1 + γ ⎣
1+λ
1+ γ
(
)
2
CDK = 1.87 783γ + 2142γ + 1080 Re
⎧⎡ ⎛ 24
C DK = ⎨⎢γ ⎜
⎩⎣ ⎝ Re
+
−0.74
1 < Re < 5,
0<γ<∞
[(60 + 20γ)(4 + 3γ)]−1 , 2 < Re < 50
( )
⎫
3γ + 2
2 −1
4 ⎞ 14.9 ⎤
+ 0.78 ⎥ + 40
+ 15γ + 10⎬ ⎡⎣(1 + γ ) 5 + Re ⎤⎦ ,
1/3 ⎟
Re
Re ⎠ Re ⎦
⎭
(
⎧16 ,
⎪Re
⎪ 14.9
⎪⎪ 0.78 ,
C DK = ⎨Re
⎪49.9
−8
2.615
2.21
1 − 1/2 + 1.17 × 10 Re
,
⎪
Re
Re
⎪
⎪⎩2.61,
(
)
Re ≤ 400
⎫
⎪
⎪
1.5 < Re ≤ 80 ⎪
⎪
⎬
⎪
80 < Re ≤ 1530⎪
⎪
Re > 1530 ⎪
⎭
[19]
(40)
(41)
[47]
(42)
[48]
(43)
[51]
(44)
[11]
(45)
[51]
(46)
[52]
Re < 1.5
)
26.5 ⎡(1.3 + γ) − 0.5⎤
⎥ , Re ≤ 5,
0.78 ⎢
Re ⎣(1.3 + γ )( 2 + γ )⎦
γ ≤ 1.4
(47)
[53]
{
Re < 50
(48)
[47]
No. 3
2011
2
11
12
C DK =
C DK =
(
)
−0.5 ⎤
16
8 1
1 + ⎡ + 1 + 3.31Re
⎣⎢Re 2
⎦⎥
Re
}
−1
,
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
DRAG COEFFICIENTS OF VARIOUSLY SHAPED SOLID PARTICLES
by calculations, is accurate to within 5% at
Re ≤ 100. The correlations between the drag coeffi
cients of the solid spherical particle, drop, and bubble
for 5 < Re < 1000 are presented below [6]:
2−γ
4γ
C D (Re, 0) +
C D (Re, 2) ,
2
6+γ
0≤γ<2
γ−2
C D (Re, γ ) = 4 C D (Re, 2) +
C D (Re, ∞) ,
2+γ
γ+2
2 ≤ γ ≤ ∞,
C D (Re, γ ) =
where C D (Re, 0) , C D (Re, 2), and C D (Re, ∞) are the
drag coefficients for a bubble, a drop with γ = 2 , and a
solid particle, respectively. The drag coefficient is sig
nificantly affected by variations of the shape of drops
and bubbles as a result of their deformation. It was
found that the behaviors of drops and bubbles are
qualitatively the same and differ markedly from the
behavior of solid particles. The shape of a drop or bub
ble is not fixed. It forms as the drop or bubble moves
and is determined by the current balance of the pres
sure force acting on the surface of the deformable par
ticle from the side of the medium, which tends to com
press the particle in the motion direction, and the sur
face tension, which opposes this compression. The
pressure force is proportional to the velocity head,
2
ρU
FD ~ c ∞ , and the surface tension is proportional to
2
F
the capillary pressure, Fσ = 2σ . If We ~ D Ⰶ 1 at
Fσ
dr
Re Ⰶ 1, the drops and bubbles will be strictly spherical.
When FD ≥ Fσ , the drop surface loses stability and the
drop undergoes deformation, taking on the shape of an
oblate ellipsoid of revolution and, with a further
increase in Re and We, various configurations, includ
ing that of a thread. Here, the deformation factor is
defined as the ratio of the minor semiaxis of the ellipse
a
(a0 ) to the major semiaxis (b0,): χ = 0 . Mypa [57] sug
b0
gested the following estimator for the deformation fac
tor at large Re values:
( χ + χ − 2)
× ⎡χ arc sec ( χ) − ( χ − 1) ⎤ ( χ − 1) .
⎣⎢
⎦⎥
We = 4χ
2
−4/3
3
2
1/2
2
−3
Wellek et al. [58] reported the empirical formula
Y =
b0
.
= 1 + 0.09
a0
We 0.95
Taylor and Acrivos [42] suggested the following for
mula for estimating the deformation of drops:
χ = 1 − 9 We.
16
257
СD
+
100
3
10
1
а
b
c
d
e
f
1
3
2
0.1
1
10
1000 Re
100
Fig. 3. Bubble drag coefficient as a function of Re for Mo =
(a) 7.0, (b) 1.4, (c) 0.023, (d) 9 × 10–4, (e) 1.1 × 10–5, and
(f) 9 × 10–7.
A still simpler formula was reported by Raymond
and Rozant [54]:
χ ≈ 1 − We .
9
Taylor and Acrivos [42], assuming that the radius of the
deformed drop is r = R [1 + ζ ( cos θ)] , theoretically
obtained the following expansion by asymptotic
matching of the external and internal expansions for
the function ζ ( cos θ) :
ζ ( cos θ) = λV Ac Re 2 P2 ( cos θ)
11 + 10γ 2 3
− 3 λV
Ac Re P3 ( cos θ) + ...,
70
1+ γ
λV =
(
1
81 + 57 γ + 103 γ 2 + 3 γ 3
3
40
4
4 (1 + γ ) 80 20
−
γ Δρ 1
,
12 ρC (1 + γ ) 2
2
)
(51)
3 + 2γ
FSK = 1
FS
3 1+ γ
2
⎡ 1 3 + 2γ
⎤
⎛ 3 + 2γ ⎞
2
1
× ⎢1 +
Re + ⎜
⎟ Re ln Re + ...⎥ ,
40 ⎝ 1 + γ ⎠
⎢⎣ 8 1 + γ
⎥⎦
where P2 ( cos θ) and P3 ( cos θ) are second and third
order Legendre polynomials. For gas bubbles in a liq
uid medium, γ → 0 and λV → 5 . Thus, drop defor
48
mation exerts no effect on the drag up to the terms of
the order of Re ln Re, inclusive. Unfortunately, lack of
the necessary physical data concerning the drag on
drops at medium Re values encumbers the indepth
analysis of this expansion.
According to Sherman [59], the variation of the
shape of a drop at small deformations is described by
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
No. 3
2011
258
KELBALIYEV
χ
+
1.0
+
+
+
+
++
+
+
0.8
+
а
b
c
d
e
+
+
+ +
0.6
0.4
0.2
where β = π κ S . Experimental data of Raymond and
24
Rozant [54] suggest that β = 1.475 × 10 −4 Mo −1/5 and
1
χ S = 1 Mo8. Figure 4 compares the experimental and
2
calculated gas bubble deformation factors for different
Mo values. As Mo decreases, the deformation factor
falls sharply and, accordingly, the shape of the bubble
(drop) changes significantly.
A somewhat different approximation for the defor
mation factor can be derived from Eq. (51) by retain
ing only its first term and assuming that We = Ac Re 2d
and P2 ( cos θ) = 0.5 3 cos 2 θ − 1 :
(
0
2
4
6
8
10
12 d, mm
Fig. 4. Bubble deformation factor as a function of the
equivalent size for Mo = (a) 7.0, (b) 1.4, (c) 0.023,
(d) × 10–4, and (e) 1.1 × 10–5.
)
λ
r = R (1 − ζ ( cos θ)) = R ⎡1 − V We 3 cos 2 θ − 1 ⎤ . (54)
⎢⎣
⎥⎦
2
(
For θ = 0 and r = a0, Eq. (54) reduces to
r = a0 = R (1 − λV We) .
the following relationship (n the notation used in this
work):
()
1
1
ρ c ⎞ 4 19γ + 16
1 − χ Gd g 2 ⎛
=
,
⎜ Mo ⎟
Δρ ⎠ 16γ + 16
1+ χ
2 K ⎝
1
(52)
where υ is the volume of a drop, κ S is the elasticity
coefficient, and χS is the steadystate value of the
deformation factor. By passing to the Δd → 0, limit,
we obtain
Δχ ∂χ
=
= −κ S ( χ − χ S ) υ.
∂d
Δd →0 Δ d
lim
For spherical drops, assuming that ϑd = πd , we can
6
bring the above equation to
3
3
dχ
= − πd κ S ( χ − χ S )
6
dd
χ ( d → 0) = 1.
(53)
The solution of Eq. (52) will then appear as follows
[19, 60]:
(
)
χ = χ S + (1 − χ S ) exp −β d ,
4
(
)
For θ = π, we have r = b0 = R 1 + λV We and the
2
2
deformation factor will appear as
a0 1 − λV We
(55)
=
.
b0 1 + λV We
2
Using experimental data of Raymond and Rozant
[54], λV in the first approximation can be represented
as follows [19, 60]:
χ=
⎛ ⎞2
where K = ⎜ σ ⎟ is the capillary constant and G is the
⎝ dg ⎠
shear modulus.
In terms of elasticity theory, drop deformation is
described by the following equation [19, 60]:
1 Δχ = −κ Δd,
S
υχ − χ S
)
(
) (
)
λV = 1 1 − 3 We = 1 1 − 3 Ca .
12
25 Re
12
25
There has been a comparison between the observed
and calculated (Eq. (55)) bubble deformation factors
for various We and Re values. It follows from these data
that the rate of decrease of the deformation factor of a
drop increases with increasing We and Re. As Mo
increases, the deformation factor increases progres
sively more rapidly even at small drop sizes. Similar
experimental data on the drag coefficient and defor
mation of air bubbles in various liquid media were pre
sented by Haberman and Morton [61].
The deformation of drops and bubbles exerts a sig
nificant effect on their drag coefficient. Under the
assumption that the volume of the liquid in a drop is
shapeindependent, the volume of an ellipsoidal drop
a
is υ = 4 πa02b0. Taking into account that χ = 0 and that
3
b0
3
4
the volume of a spherical drop is υ = πR , we obtain
3
1
⎛ ⎞3
1
−2
⎜ χυ ⎟
a0 = ⎜ ⎟ = Rχ 3 and b0 = υ = Rχ 3. The drag
4
4 πa 2
⎜ π⎟
0
⎝3 ⎠
3
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
No. 3
2011
DRAG COEFFICIENTS OF VARIOUSLY SHAPED SOLID PARTICLES
coefficient of an ellipsoidal drop in a lateral flow for
small and moderate Re values will then appear as
1
3
2
(56)
C DE ≈ C DC R = χ C DC ,
a0b0
This expression interrelates the drag coefficients of an
ellipsoidal particle ( CDE ) and a spherical particle
( CDC ). The variation of the shape of drops and bubbles
and their deformation at a constant volume is associ
ated with the variation of their surface area,
8πa0b0
(57)
.
1+ χ
Equation (57) describes the dependence of the surface
area of a deformed drop or bubble on the deformation
factor. It follows from this equation that, with increas
ing deformation (decreasing deformation factor), the
surface area of the drop increases. When there is no
deformation, χ = 1, a0 = b0 = R, and S = 4πR 2,; that is,
the drop is spherical and its surface area is equal to the
surface area of the sphere. The deformation of drops
depends mainly on the capillary number Ca and vis
cosity ratio γ. There has been an experimental study of
drop deformation at γ= 1.0, 3.0, and 5.0 and various
Re values [62], with the deformation factor defined as
the ratio of the length of the deformed drop (L) to the
thickness of the drop: Y = L/a. The following drop
deformation equation was derived from experimental
data in that study:
S =
Y =
1
,
Y s + (1 − Y s ) exp −β0d 2
(
259
Y
7
а
b
c
6
5
4
3
2
1
0
0.05
0.10
0.15
0.20
0.25
0.30 d, mm
Fig. 5. Drop deformation factor for γ = (a) 1.0, (b) 3.0, and
(c) 5.0.
the liquid shows pseudoplastic properties; at n > 1, the
liquid is dilatant. This fact imposes special constraints
on the hydrodynamic description of nonNewtonian
fluids and modifies the description of the drag coeffi
cient in these liquids.
Theoretical and experimental hydrodynamic stud
ies and drag coefficient data for solid particles, drops,
and bubbles in nonNewtonian fluids are presented in
[3, 51, 67–75]. The following formula was obtained by
asymptotic methods for the drag coefficient of a bub
ble in a nonNewtonian fluid (n < 1) for small Rey
nolds numbers of Re < 1 [3, 76]:
(58)
)
C D = n16 X ( n) ,
2 Re
1
where Ys is the steadystate value: Y s = 1 γ 2, and
7
−1
4
β 0 = 1.08 × 10 −3 Re 2 γ . Figure 5 presents a compari
son between the calculated and experimental drag
coefficients of a drop [62] for different γ values.
The deformation of drops and bubbles is associated
with changes in their shape and is accompanied by
their breakup into smaller particles [60, 63]. The
changes in the particle size have a significant effect on
the drag coefficient. At the same time, the minimum
and maximum sizes of drops and bubbles depend on
the drag coefficient. The effects of external factors
(physical phenomena on the surface) on the drag coef
ficient of drops and bubbles are considered in other
works [64–66].
DRAG COEFFICIENT OF PARTICLES
IN A NONNEWTONIAN FLUID
NonNewtonian fluids differ from ordinary liquids
in that their viscosity changes with an increasing shear
rate. The extent to which a liquid is nonNewtonian is
characterized by the shear rate exponent n. At n < 1,
X ( n) = 2
n −1
3
n−3
2
2 13 + 4n − 8n
(59)
(2n + 1)( n + 2)
,
where X ( n) is the exponent n dependent parameter
characterizing the rheological properties of the flow.
Calculations using formula (59) demonstrate that, for
pseudoplastic and dilatant liquids, the drag coefficient
is, respectively, larger and smaller than the drag coeffi
cient of a bubble in nonNewtonian flow [3]. In addi
tion to Eq. (59), there have been experimental X data
and X ( n) dependences reported by Chabra [51]:
n−3
X ( n) = 2 3 2 [1 − 3.83 ( n − 1)] ,
0.7 ≤ n ≤ 1;
n
( )
X ( n) = 2 3γ
3
X ( n)
2
n−3
2
2 13 + 4n − 8n ,
(2n + 1)( n + 2)
n+3
⎡2 2n + 1
=3 2 ⎢
2
(
−
⎣
γ > 10;
)(2 − n)⎤ 2
⎥⎦ ;
n
n−3
X ( n) = 2 3 2 1 + 7n − 5n , n < 1.
n ( n + 2)
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
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KELBALIYEV
a number of formulas characterizing the deformation
of bubbles in a nonNewtonian fluid:
X (n)
1.8
b0
= 0.0628δ 0.46, 20 ≤ δ ≤ 100,
a0
δ ≤ 4,
Y K = 1.4,
YK =
1.2
−
Y K = 6.17δ 1.07,
, Mo = We Fr 2 − 3n Re t−4,
0.64 ≤ n ≤ 0.9
Table 6 presents a variety of formulas for calculat
ing the drag coefficient of solid spherical particles,
drops, and bubbles in nonNewtonian fluids. Formu
las (62), (65), and (67) in Table 6 refer to the drag
coefficient of a solid spherical particle; formulas (63),
(64), and (66), to irregularly shaped particles with dif
ferent shape factors; formula (68), to gas bubbles.
According to Chabra [51], the drag coefficient of a
bubble in a nonNewtonian fluid in the 5 ≤ Re ≤ 25
range is given by the formula
δ = Re t Mo
0.6
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 n
Fig. 6. X ( n) as a function of n
K (n)
50
4 ≤ δ ≤ 20,
n+2
0.078
C DG =
45
K ( n)
.
Re t
Here, K ( n) is an ndependent coefficient. Based on
experimental data of Chabra [51], we suggest the fol
lowing formula:
45
45
25
K ( n) = 30.6
25
0.2
0.4
0.6
0.8
1.0 n
Fig. 7. K (n) as a function of n.
The second formula in Eq. (60) characterizes the
behavior of drops in a nonNewtonian fluid. These
theoretical expressions do no not provide a general
solution for the problem.
A satisfactory approximation for X ( n) over the
0.1 ≤ n ≤ 1.8 range for solid particles in a nonNewto
nian fluid is provided by the following relationship based
on experimental data of Chabra and Richardson [69]:
3
⎡ 3⎛
⎞⎤
2
7
n
2⎜
⎢
X ( n) =
(5n + 9) exp −n 1 + ⎟⎥ .
⎢ ⎜ 10 ⎟⎥
450
⎣ ⎝
⎠⎦
2
(61)
It follows from (61) that, at X ( n) ≈ 1. Figure 6 illus
trates the fit between the calculated and experimental
X ( n) data. Chabra and Richardson [69] also reported
exp(2.53n)
⎛ 67
⎞
⎜ n + 1⎟
⎝
⎠
3
.
As is clear from Fig. 7, the coefficient K ( n) varies sig
nificantly in the 0 ≤ n ≤ 1 range and, passing through a
minimum at n = 1, tends to K ( n) ≈ 48,, which charac
terizes the drag coefficient of a bubble in a Newtonian
fluid suggested by Levich [8] for small Re values.
By analogy with Eq. (24), we suggest another equa
tion for the drag coefficient of solid particles in a non
Newtonian fluid:
CD =
2
⎤
24 X ( n, Re t ) ⎡
3
1
Re
+
t ξ ( Re t )⎥
⎢
Re t
⎣
⎦
(
))
(
+ 0.3 1 − exp −0.2 × 10 −19 Re t4 ,
3
⎡
⎤
n −1
1
Re t2
5
6
⋅
⎢
⎥,
X ( n, Re t ) =
1 + (1 − n) 3
3⎥
1
1 + 4n ⎢
1 + (1 − n) 3 Re t2 ⎥⎦ (70)
⎢⎣
0.8 < n ≤ 1, 0.1 ≤ Re t ≤ 105,
Re t2
5 + 8Re t2
Re t
ξ (Re t ) =
+
6.8 × 10 Re
6
−5
4
2
10
+ 32Re t3 + 952.8 × 10 −13 Re t3
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DRAG COEFFICIENTS OF VARIOUSLY SHAPED SOLID PARTICLES
261
Table 6. Drag coefficients of particles in nonNewtonian fluids
En
try
Expression
CD =
1
0.654
24 ⎡
⎤⎦ + 0.4305k2 , Re t = k1 Re, k1−1 = 0.33 + 0.67ϕ −1/2,
1 + 0.1118 ( k2 Re t )
⎣
3305
Re t
1+
k2 Re t
l o g k2 = 1.815 ( − l o g ϕ)
2
CD =
(
(
CD =
CD =
(
)
0.2
32.5
1 + 2.5Re t ,
Re t
4
(
( n ) ≤ Ret ≤ 100 (2 n )
Re t < 150, 0.77 ≤ n < 1,
X * ( n ) = X ( n) +
4n
2
(
)
4
,
10
−3
2
[64]
(64)
[67]
(65)
[38]
(66)
[68]
(67)
[69]
(68)
[72]
10
(69)
[70]
−5
−n+3
, Re t < 10
3n
3
−5
3
≤ Re t < 10 ,
For Re t ≤ 1000 , this equation reduces to
3
24 X ( n, Re t ) ⎛
−5
2⎞
CD ≈
⎜1 + 0.125Re t + 2.51 × 10 Re t ⎟ .
Re t
⎝
⎠
0.35 ≤ ϕ < 0.7
Re t < 1000, 0.16 ≤ n < 1
2n − 3 n
3
1−n
lg 10 Re t ,
3n + 1
n−3
24 Re t 3
⎪
⎬
⎪
⎪
⎭
24 2
X * ( n) = X ( n) = 3
X * ( n) = X ( n ) +
(63)
⎫
)
24
X * ( n) ,
Re t
[47]
( n ) ≤ Ret ≤ 24 (2n ) 0.38 ≤ n ≤ 1⎪
0.2 2
0.657
16
0.431
1 + 0.173Re t
+
,
−1.09
Re t
1 + 16300 Re t
CD =
8
Re t < 2.5 × 10
73.69 Re t exp ( −5.075ϕ)
0.0964 + 0.557ϕ
24 ⎡
⎤⎦ +
1 + 8.1723exp ( −4.066ϕ) Re t
⎣
Re t
Re t + 5.378exp (6.212ϕ)
5
(62)
5
)
⎧35.2 ( 2 n )
⎡ 20.9 ( 2 n )⎤
,
⎪ 1.03 + n ⎢1 −
1.11 ⎥
Re t
⎢
⎥
⎪ Re t
⎣
⎦
CD = ⎨
n
⎪37 ( 2 )
⎪ 1.1 + 0.25 + 0.36n,
⎩ Re t
Refer
ence
≤ Re d ≤ 10 , 0.1 < ϕ < 1
1 + 0.0000238 Re t
0.5
24
1 + 0.24 Re t +
,
−1
Re t
1 + 370 Re t
CD =
7
−3
)
4
6
, 10
0.657
24
0.413
1 + 0.173Re t
+
, Re t < 1600, 0.16 ≤ n ≤ 1
−1.09
Re t
1 + 16300 Re t
CD =
3
0.574
Formula
≤ Re t < 10
−3
0.56 ≤ n ≤ 0.89
EFFECTS OF VARIOUS PHENOMENA
ON THE DRAG COEFFICIENT OF PARTICLES
2
Figure 8 illustrates the fit between the drag coefficient
data calculated via Eq. (70) and experimental data
[72–75]. It follows from Fig. 8 and calculations using
Eq. (70) that, as n decreases, the drag coefficient of
solid particles in viscoplastic liquids become smaller
than in Newtonian fluids.
Apart from the above parameters, the following
factors produce a significant effect on the friction
forces and the drag coefficient of particles: flow turbu
lence, roughness of the particle surface, bulk porosity
of the particles, and mass transfer processes on the
drop and bubble surface (dissolution of gas bubbles in
the liquid, evaporation of drops, etc.). These factors
cause unsteady changes in the particle size and many
other features.
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
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KELBALIYEV
particle. This can markedly change the friction forces
and drag coefficient. Particles freely moving in the
flow can spin under the action of the Magnus force,
change their orientation, and execute complicated
nonlinear motion. Depending on its size, the particle
can be entrained by a pulsating turbulent flow to some
extent, and the Reynolds number in this case is esti
U − Vd d
mated as Re =
. The smaller the particle, the
νc
larger the extent to which it is entrained by the turbu
lent pulses; that is, the extent of entrainment of parti
cles tends to unity as the particle size increases. For
drops and bubbles, an increase in Re raises the proba
СD
3
10
а
b
c
+ d
e
102
10
1
0.1
0.1
1
10
10
2
10
3
104
5 Re
10
Fig. 8. The drag coefficient of a solid particle in a non
Newtonian fluid as a function of Re according to the data
of different authors: (a) n = 0.84–0.86 [64], (b) n = 0.75–
0.90 [69], (c) n = 0.75–0.92 [69], (d) n = 0.56–0.75 [70],
and (e) n = 0.73 [71]. The dashed line represents the drag
coefficient of the particle in a Newtonian fluid.
There has been a qualitative analysis of the effect of
the turbulence of the external hydrodynamic field on
the drag on a solid spherical particle in the case of the
particle size being of the same order as the internal
scale of turbulence or larger [4, 18, 77–79]. At low tur
bulence intensity in the external flow and small Re val
ues, turbulence slightly raises the drag by enhancing
the dissipation in the wake. When the particle is in tur
bulent flow, the drag crisis occurs at smaller Re values
[18]. It was noted [79] that, as the turbulence intensity
increases from 0.5 to 2.5%, the critical Re value
decreases from2.7 × 10 5 to 1.25 × 10 5. At a turbulence
intensity of 30% and above, the drag crisis can occur at
much smaller Re values. The effect of turbulence on
the drag coefficient of solid particles is estimated as
FTP
.
πd ρ d
2
(U )
8
With isotropic turbulence taken into account [78], the
drag coefficient of solid particles appears as
C D ( Re ) =
2
⎡
⎞ ⎤
−4 ⎛
C DT = C D ⎢1 + 8.76 × 10 ⎜ d ⎟ ⎥ ,
⎝ λ 0 ⎠ ⎥⎦
⎢⎣
3
where λ 0 is the Kolmogorov scale of turbulence and
C DT is the drag coefficient of the particle in turbulent
flow.
Note the difference between a fixed particle in a
flowing medium and a particle moving in the flow. For
the fixed particle, the flow pattern around it may
depend on the degree of turbulence and highintensity
turbulence can alter the structure of the boundary
layer and break down the turbulent wake behind the
(
)
bility of their ellipsoidal distortion We ~ Re 2 and
their breakup frequency [60, 63]. Polyanin et al. [11]
suggested formulas for calculating the drag coefficient
of ellipsoidal particles in lateral and longitudinal flows.
Tropea et al. [79] reported experimental data on the
effect of surface roughness on the drag crisis. As the
roughness of the particle surface increases, the laminar
boundary layer is turbulized progressively more rapidly
and the drag crisis occurs at smaller Re values. Various
expressions taking into account surface roughness have
been suggested for estimating the drag and turbulent dif
fusion coefficients of particles, the particle settling veloc
ity, and heat transfer [77, 78, 80, 81].
The drag force on a permeable sphere is equivalent
to the drag force of an impermeable sphere with a
−1
smaller diameter of dk = d × ⎛⎜1 + 2k2 ⎞⎟ [4], where k is
⎝ d ⎠
the permeability coefficient. Expressions for the drag
coefficient of porous cylindrical particles were pre
sented by Tropea et al. [79].
A marked effect on the drag on drops is exerted by
mass transfer processes, including evaporation from
the drop surface, which reduces the drag coefficient:
C DM = 27 Re −0.84, Re < 1. The following formula was
suggested for the drag coefficient of a spherical drop
with an evaporating surface [65, 66]:
1 + 0.545Re + 0.1Re (1 − 0.03Re)
C DK = 24
,
B
Re
1 + A Re P
(71)
10 < Re < 200, 1 < Re P < 20,
A = 0.09 + 0.077 exp ( −0.4 Re) ,
B = 0.4 + 0.77 exp ( −0.04 Re) ,
0.5
VP d
is the Reynolds number defined in
νc
terms of the evaporation rate and V P is the rate of evap
oration from the drop surface. With the Stefan flow
taken into account, the drag coefficient of an evapo
where Re P =
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
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DRAG COEFFICIENTS OF VARIOUSLY SHAPED SOLID PARTICLES
rating spherical drop for small Re d values appears as
follows [51, 70]:
CD =
24
,
Re (1 + Bm )
Bm =
Y s − Y∞
,
1 − Ys
where Y s, and Y ∞ are the mole fractions of vapor on the
drop surface and far from the drop. Ackovic [82]
reported calculation of the drag coefficient for
unsteadystate bubble growth in a binary mixture.
Other factors in the friction force and drag coeffi
cient are the wall effect and interaction with other par
ticles due to the superposition of hydrodynamic fields.
The wall effect is taken into account by introducing
the coefficient f = F ( ϑ, Re) , ϑ = d (D is the channel
D
diameter) [1, 5, 31, 51]. In the particular case of Re >
100, the coefficient accounting for the wall effect is
(
)
3
written as f = 1 − ϑ2 2 . [1]. The drag coefficient of
drops and bubbles depends strongly on the presence of
adsorbed surfactants on the surface. The presence of
traces of adsorbable admixtures in the solution
strengthens the interface, making the drops to behave
as solid particles. ChiaShun Yih [64] presented a
number formulas taking into account the effect of the
concentration of adsorbed surfactants on the drag
coefficient of drops and bubbles at small Re values.
CONCLUSIONS
The above expressions for the drag coefficient of
particles are empirical and, for very small Re values,
semiempirical or theoretical. Aside from the above
formulas for the drag coefficient of solid particles,
drops, and bubbles in Newtonian and nonNewtonian
fluids, there are many relationships valid in various
ranges of Re, We, Mo, and Bo [1, 6, 25, 26, 38, 47, 51,
64, 65].
It is natural that serious problems are encountered
at large Re values, which arise from the changes in the
flow pattern around the particle. This leads to unpre
dictable variation of the derivative of the drag coeffi
cient with respect to Re. An analysis of numerous
expressions and approximations for the drag coeffi
cient of solid particles, drops, and bubbles demon
strates that, for obtaining a good result, a researcher
should make the right choice among the empirical
ξ (Re) functions, guided by their experience and intu
ition. The ξ (Re) function should be continuously dif
ferentiable over the entire Re range, and it is desirable
that it be fairly simple and contain the smallest possi
ble number of unknown coefficients.
Note that experimental drag coefficient data for
solid particles over the 2 × 105 ≤ Re ≤ 10 7 range vary
widely [1, 4, 6, 32, 64, 65]. A similar situation takes
−3
place at very small Reynolds numbers of Re < 10 . It
follows from the formulas presented here that, in the
263
hypothetical case of Re → 0, the drag coefficient tends
to infinity: C D → ∞,; that is, we have a singular prob
lem in which the particle size is zero and the drag on
the particle is infinitely high. In order to eliminate this
singularity, it is likely necessary to set CD = 0 at
Re → 0. It is likely that the drag curve will pass
through a maximum at some Re → Re m value. Obvi
ously, the boundary of the singularity determined the
lower limit of the validity of the Stokes law.
If the particle size is small compared to the free
path of the molecules of the liquid (λ,), implying that
the Knudsen number is Kn = λ Ⰶ 1,then molecular
a
slip will take place, which will reduce the drag coeffi
cient (Kn ~ M/Re). In this case, it is possible to apply
the Cunningham correction factor [1, 4, 10], with
which the drag coefficient can be expressed in the
form of the following semiempirical relationship:
C D ≈ 7.09Kn [ξ (1 − Kn ) + Kn]
where ξ is the slip factor,
(
U
2RT
),
−1
( 2URT ) is the ratio of the
flow velocity to the molecular motion velocity, T is
temperature, and R is the gas constant.
Since there are a wide variety of formulas for calcu
lating the drag coefficient of solid particles, drops, and
bubbles, it is essential to find the best formula. How
ever, here it is difficult to give a clue to researchers
because the right choice will be determined by the fol
lowing factors:
(1) Flow pattern of the dispersion. When the parti
cle size in the flow varies continuously because of par
ticle coagulation or breakup and some other physical
processes (evaporation, dissolution, condensation,
etc.), it is appropriate to choose formulas valid in a
wide Re range. In particular, this is true for the turbu
lent flow of liquid–gas and gas–liquid dispersions in
which the drop or bubble size varies from 1 µm to
200 µm and above, which covers a wide Re range and,
if the particles undergo deformation, wide ranges of
Mo and We. Obviously, simple formulas should be
used in CD calculations for a narrow Re range. In par
ticular case of solid spherical particles and Re ≤ 500 ,
the best choice ensuring an acceptable degree of accu
racy is relationship (9), the Klyacko–Mazin equation,
the Schiller–Neumann equation, or relationship (25).
For a narrow Re range, an acceptable choice is
Eq. (12).
(2) Re, Mo, and We ranges and the accuracy of the
formula for calculating the drag coefficient. The rela
tive error of most of the above formulas in a given Re
range is <10%. However, although the formulas are
simple and do not exceed the acceptable error in their
validity range, they lead to appreciable errors even
when the Re value is slightly outside this range.
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 45
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KELBALIYEV
(3) Particle shape and size. Note that serious diffi
culties are encountered in the determination of the
drag coefficient of deformable drops and bubbles at
Re Ⰷ 1, We > 1, and Mo Ⰶ 1 in both Newtonian and
nonNewtonian fluids. Under these conditions, drops
and bubbles can take a shape that cannot be analyzed
experimentally or described theoretically.
NOTATION
CD —drag coefficient of a solid particle;
C DK —drag coefficient of a drop;
C DG —drag coefficient of a bubble;
c —speed of sound, m/s;
d —diameter of a spherical or cylindrical particle, m;
FS —friction force, N;
f —wall effect;
G —shear modulus, s/m;
L —length, m;
n—shear rate exponent for nonNewtonian fluids;
R—radius, m;
r 2—correlation factor;
s M —midsection area of the particle, m2;
S—surface area of a sphere, m2;
S d —surface area of an irregularly shaped particle, m2;
U—flow velocity, m/s;
Vd —particle velocity, m/s;
Y—drop deformation factor;
γ —ratio of the viscosity of the particle to the vis
cosity of the medium;
λ 0 —Kolmogorov scale of turbulence;
λ —free path length of the molecules of the liquid;
ν c —kinematic viscosity of the medium, m2/s;
ρ d —density of the particle, kg/m3;
ρ c —density of the medium, kg/m3;
σ—surface tension, N/m;
χ —deformation factor of drops and bubbles;
ς —nonNewtonian viscosity coefficient;
Ac—Acrivos number;
Bo—Bond number;
Ca—capillary number;
Fr—Froude number;
Kn—Knudsen number;
Mo—Morton number;
M—Mach number;
U − Vp d
—Reynolds number for a particle;
Re =
νc
Re t =
ρ cU
tonian fluids.
2−n
ς
d
—Reynolds number for nonNew
SUBSCRIPTS AND SUPERSCRIPTS
c—continuous medium;
d—dispersed phase.
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