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Wall Effects

10.1 INTRODUCTION

The steady motion of rigid and fluid particles through quiescent unbounded
(or infinite expanse of) fluids has been a subject of theoretical study for many
years, dating back to the pioneering studies of Newton (1687), Hadamard and
Rybczynski (Clift et al., 1978), and Stokes (1851) to mention a few. Owing
to the finite size of vessels and tubes employed in such experimental studies,
it is clearly not possible to realize the unbounded flow conditions. In practice
therefore, the confining walls exert an extra retardation effect on a particle,
whether settling freely in a quiescent fluid, or being suspended in an upward
flowing stream of fluid; this effect has also been studied for centuries (Newton,
1687; Munroe, 1888; Ladenburg, 1907). The effect is caused by the upward flux
of the fluid displaced by the particle; the smaller the gap between the particle and
the boundary, more severe is the effect. A knowledge of this so-called wall effect
for both rigid and fluid particles is necessary for a rational understanding and
interpretation of experimental data in a number of situations of overwhelming
pragmatic significance. For a rigid sphere, typical examples include falling ball
viscometry, hydrodynamic chromatography, membrane transport, hydraulic
and pneumatic transport of coarse particles in pipes, etc. Furthermore, in recent
years, the problem of flow around solid particles in a tube has received further
impetus from the use of electric fields to achieve enhanced rates of transport
phenomena and of separations in multiphase systems. On the other hand, in the
case of fluid particles, not only is their velocity influenced by the presence of
boundaries, but their shapes are also greatly altered due to the extra dissipation
at the rigid walls. Conversely, their free surface enables them to negotiate
their way through the narrow throats in undulating tubes and in porous media,
as encountered in the enhanced oil recovery processes. It is thus much more
difficult to quantify the severity of wall effects for bubbles and drops than that
for rigid particles. In view of the significant differences in the velocity field
in the immediate vicinity of the particle, the rate of interphase heat and mass
transfer is also influenced (generally enhanced) due to the confining walls in
relation to the unconfined case. Evidently, the magnitude of the wall effect will
depend upon the size and the shape of the confining walls, that is, whether the
particle is moving axially or non-axially in circular or noncircular ducts (such as

521

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Bubbles, Drops, and Particles in Fluids

square, triangular, elliptic cross-sections, planar slit) or it is sedimenting toward
or parallel to a plane wall, etc.

Similarly, sphere-in-sphere geometry has also been used to gain some know-

ledge about the extent of wall effects in a confined domain (Happel and Brenner,
1965). There is no question that the case of a rigid or a fluid sphere falling at
the axis of a cylindrical tube represents the most commonly used and studied
configuration, both theoretically and experimentally and hence, this chapter is
mainly concerned with the extent of wall effects on a rigid or a fluid sphere
falling along the axis of a cylindrical tube. Additional complications arise for
nonspherical particles due to their preferred orientation which may get accen-
tuated in the presence of boundaries. The available body of information for
nonspherical particles in cylindrical tubes as well as that for spherical particles
in noncircular tubes is rather limited, as will be seen in this chapter.

10.2 DEFINITION

Due to the extra retardation effect exerted by the confining walls, the terminal
falling velocity of a rigid particle is lower in a confined geometry than that in
an unbounded fluid under otherwise identical conditions. Conversely, the drag
experienced by the particle in a confined medium is higher than that without
any walls being present otherwise under identical conditions. There are several
ways to quantify this effect; perhaps the simplest of all is by defining a wall
factor, f ,as the ratio of the two velocities as

= V/V

∞

(10.1)

where V is the terminal velocity of a sphere (of diameter d) falling on the
axis of a tube of diameter D, and V

∞

is the terminal velocity of the same

sphere in an unbounded medium. Obviously, the wall factor, f , as defined
here, will take on values between zero and unity. Other definitions of the wall
factor involving the ratio of drag forces in the absence and presence of the
confining walls, the reciprocal of f , the ratio of viscosities calculated using
the Stokes formula with and without walls (Bacon, 1936; Sutterby, 1973b;
Clift et al., 1978) etc. have also been used in the literature. Obviously all these
definitions are not mutually exclusive, though beyond the creeping flow region,
the interrelationships between them is far from being straightforward and in fact
is quite involved (Happel and Brenner, 1965; Clift et al., 1978; Kim and Karrila,
1991). The scaling of the field equations and the pertinent boundary conditions
for a spherical particle moving axially in an incompressible Newtonian medium
in a cylindrical tube reveals the wall factor f to be a function of the sphere
Reynolds number (Re), the sphere to tube diameter ratio,

λ(= d/D) and the

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viscosity ratio, X

, that is,

= φ(Re, λ, X

)

(10.2)

Additional dimensionless groups will emerge depending upon the choice of a
rheological model to approximate non-Newtonian flow behavior. For instance,
for the case of the simple power-law fluid model, the power-law flow beha-
vior index, n, or a Bingham number for Bingham plastic model fluids, or
the Weissenberg or the Deborah number for visco-elastic fluids, etc. will
appear in Equation 10.2. Likewise, other dimensionless groups will also emerge
for particles falling in off-center locations in cylindrical tubes or in noncircular
tubes. In this chapter, it is endeavored to present the current status of the progress
made in establishing the functional dependence depicted in Equation 10.2 for
Newtonian and non-Newtonian fluids. We begin with the case of rigid particles,
(i.e., X

→ ∞), followed by that of bubbles (X

→ 0) and finally that of

droplets falling/rising freely at the axis of cylindrical tubes so that no additional
complexities are involved due to eccentricity and/or noncircular cross-section
of the sedimentation vessels.

10.3 RIGID SPHERES

It is instructive to begin with the nature of wall effects on a solid sphere set-
tling in Newtonian fluids, and this, in turn, sets the stage for presenting the
analogous treatment for spheres falling in non-Newtonian fluids in the ensuing
section.

10.3.1 N

EWTONIAN

LUIDS

10.3.1.1 TheoreticalTreatments

From a theoretical standpoint, the effect of confining walls is to change the
boundary conditions for the equations of motion and continuity for the con-
tinuous phase. In place of the condition of a uniform flow faraway from the
sphere, confining walls impose conditions which must be satisfied at definite
boundaries. Further complications arise from the prevailing velocity profiles
(uniform or Poiseuille) in the tube. The available experimental and analyt-
ical or numerical results clearly indicate that the wall factor is a function of
λ only under both creeping and fully turbulent conditions, whereas it depends
on both

λ and Re in the intermediate Reynolds number range. It is thus con-

venient to present the available body of information separately for each flow
region.

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10.3.1.1.1 Creeping Flow Region
In the creeping flow region (Re

≪ 1), accurate analytical solutions have

been obtained by using the so-called method of reflections for the system of a
sphere falling in a cylindrical tube. Reflection solutions due to Faxen (1923),
Happel and Byrne (1954) and Wakiya (1957) give good predictions for the
wall factor which are restricted to small values of the sphere to tube diameter
ratios

λ(= d/D) 0.1. For instance, the celebrated result of Faxen (1923) is

given by

= 1 − 2.104λ + 2.09λ

− 0.95λ

+ · · ·

(10.3)

Subsequently, Bohlin (1960) has extended these results to higher values of

up to about

≈0.6, as follows:

= 1 − 2.10443λ + 2.08877λ

− 0.94813λ

− 1.372λ

+ 3.87λ

− 4.19λ

· · ·

(10.4)

The coefficients of the first three terms in Equation 10.3 and Equation 10.4 are
virtually identical. Subsequently, Bohlin (1960) also presented an expression
for the conditions when the Poiseuille flow is imposed on the sphere motion in
a cylinder.

Almost concurrently but independently, Haberman and Sayre (1958)

provided the following analytical expression for the wall factor f for a sphere
settling in a stationary liquid filled in a cylinder:

− 2.105λ + 2.0865λ

− 1.7068λ

+ 0.72603λ

− 0.75857λ

(10.5)

This expression was stated to be applicable in the range 0

≤ λ ≤ 0.8. Sub-

sequently, the accuracy of Equation 10.5 has been demonstrated by many
numerical studies (Paine and Scherr, 1975; Tullock et al., 1992; Bowen and
Sharif, 1994; Higdon and Muldowney, 1995; Wham et al., 1996) up to about
λ ≈ 0.9 thereby covering almost the complete range of interest 0 ≤ λ ≤∼ 1.
As will be seen later, Equation 10.5 is in excellent agreement with the experi-
mental results available in the literature (Happel and Brenner, 1965; Clift et al.,
1978; Chhabra et al., 2003).

10.3.1.1.2 Intermediate Reynolds Number Region
It is not clear beyond what value of the Reynolds number, the expressions
presented in the preceding section cease to apply. Broadly speaking, the larger

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the value of

λ, higher is the value of the particle Reynolds number marking the

end of the creeping flow region. This issue will be dealt with in more detail
later in this section. There are a few numerical studies available in the literature
in which the effect of the particle Reynolds number on the wall factor has been
investigated. Faxen (1923) applied the Oseen-type linearization to a sphere
moving axially in a tube, but the resulting predictions are no more reliable than
that for an unbounded fluid (Haberman and Sayre, 1958). Johansson (1974)
numerically solved the Navier–Stokes equations for the Poiseuille flow around
an axially fixed sphere in a cylinder. For

λ = 0.1, he calculated the values of

drag coefficient for Reynolds number (based on the mean velocity and the tube
diameter) up to about 150. The corresponding particle Reynolds number will
be one-tenth of this value. Based on limited comparisons with experimental
results, Johansson (1974) concluded that the wall effects become negligible for
Re

> 50 for λ = 0.1, which is also consistent with the subsequent findings.

Subsequently, Oh and Lee (1988) treated the same problem for

λ = 0.5 and 0.74

and for the values of the particle Reynolds number (based on sphere diameter)
up to about

∼200. However, it is virtually impossible to use the numerical

results from both these studies as sufficient details are not available enabling
the recalculation of their results in the form of the wall factor f , as defined here.
Xu and Michaelides (1996) studied the flow over ellipsoidal particles placed
axially in cylindrical tubes. For a sphere, they compared their predictions with
the analysis of Bohlin (1960). Unfortunately, however, neither the value of

nor the range of the Reynolds number employed by them is known. This indeed
severely limits the utility of their results. Wham et al. (1996) have investigated
the effect of cylindrical confining walls on freely falling spheres in quiescent
liquids as well as on spheres suspended in upward moving liquids. Their results
encompass the values of diameter ratio in the range 0.08

≤ λ ≤ 0.70 and

the sphere Reynolds number up to 200. After some rearrangement, their final
expression can be recast in the following form:

+ 0.03708(0.5 Re

∞

)

[1 + 0.03708 (0.5 f Re

∞

)

(10.6)

where

= 1.514 − 0.1016 ln(0.5 Re

∞

)

(10.7a)

= 1.514 − 0.1016 ln(0.5 f Re

∞

)

(10.7b)

− 0.75857 λ

− Kλ + 2.0865λ

− 1.7068λ

+ 0.72603λ

(10.7c)

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Bubbles, Drops, and Particles in Fluids

and

= 0.6628 + 1.458 exp(−0.028175 f Re

∞

)

(10.7d)

It should be noted that Equation 10.6 and Equation 10.7 are implicit in f and thus
an iterative procedure is needed to estimate the value of f for known values of
λ and Re

∞

(based on the sphere diameter and V

∞

). It is also appropriate to add

here that in the limit of vanishingly small Reynolds numbers, these expressions
do reduce to Equation 10.5. It can be easily shown from the predictions of
Equation 10.6 and Equation 10.7 that the wall factor indeed is independent of
the Reynolds number in the viscous region, though the critical value of Re

∞

below which the wall factor is independent of the Reynolds number is seen
to vary appreciably. Based on the criterion of 5% deviation from the constant
limiting value of f , Equation 10.6 and Equation 10.7 were used to establish
the critical values of Re

∞

below which the creeping flow approximation might

be applicable. Unfortunately, this approach predicts the critical value of Re

∞

to decrease with the increasing value of

λ which is exactly contrary to the

experimental observations, as will be seen in the next section. More recently,
Henschke et al. (2000) have also reported numerical results for the wall effects
on a sphere settling in cylindrical tubes in the range of conditions as 0.01

≤

λ ≤ ∼ 0.97 and 10

−3

≤ Re

∞

≤ 10

. However, smaller the value of

λ, wider

was the range of Reynolds number of numerical simulations. The agreement
between their predictions and experiments was reported to be good.

10.3.1.1.3 Turbulent Flow Region
Under these conditions, the flow in dominated by inertial forces, and Newton
(1687) (also see Barr, 1931) presented the following expression for the wall
factor for a sphere settling in a vessel of cross-sectional area, A:

= (1 − β)(1 − 0.5β)

0.5

(10.8)

where

β = (π d

/4A), and clearly for the special case of a sphere settling in a

cylindrical vessel,

β = λ

, and hence Equation 10.8 becomes:

= (1 − λ

)(1 − 0.5λ

)

0.5

(10.9)

Although virtually no analytical details are available for the reasoning behind
this expression, it will be seen in a later section that indeed Equation 10.8
provides excellent agreement with the experimental results available in the
literature for this flow regime (DiFelice et al., 1995; Chhabra et al., 1996).

From the foregoing discussion, it is thus abundantly clear that analytical and

numerical predictions of the wall factor are available for up to about

λ ≤ 0.97

and Re

∞

< 10

, though not for all values of

λ, the numerical results extend up

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to Re

= 10

. With the notable exception of Equation 10.8, all works on wall

effects at high Reynolds numbers are based on experimental data reported by
different investigators, as will be seen in the next section.

10.3.1.2 Experimental Results and Correlations

Detailed discussion on the experimental determination of the wall factor,
f , experimented uncertainty, etc. is available elsewhere (Chhabra, 2002b).
Table 10.1 provides a succinct summary of the experimental studies available
on this subject where it is clearly seen that indeed experimental results are now

TABLE 10.1

Summary of ExperimentalStudies on WallEffects

Investigator

Range of

Flow regime

Remarks

Munroe (1888–89)

0.11–0.83

Inertial region

Equation 10.17

Lunnon (1928)

<0.7

Inertial region

Equation 10.18

Francis (1933)

<0.97

Viscous

Equation 10.10

Bacon (1936)

<0.3155

Viscous

Emphasis on falling

ball viscometry

Lee (1947)

<0.168

Viscous

Empirical correlation

McPherson (1947)

<0.1571

< 317

Graphs

Engez (1948)

<0.9

< 1000

Empirical correlation

Fayon and

Happel (1960)

0.13–0.313

0.1

≤ Re ≤ 40

Empirical correlation

Fidleris and

Whitmore (1961)

<0.6

0.05

≤ Re ≤ 20, 000

Graphs

Sutterby (1973)

<0.13

< 4

Emphasis on falling

ball viscometry

Achenbach (1974)

0.5–0.92

× 10

≤ Re ≤ 2 × 10

Correlation

Iwaoka and

Ishii (1979)

<0.9

Viscous

Empirical correlation

Chhabra and

Uhlherr (1980c)

<0.5

Viscous and intermediate

No correlation

Lali et al. (1989)

0.05–0.78

0.002

≤ Re

∞

≤ 200

Empirical correlation

Humphrey and

0.442, 0.757,

Viscous

Vertical and inclined

Murata (1992)

0.882

tubes

Bougas and

Stamatoudis (1993)

≤0.7

13,500–70,000

Accelerating spheres

Uhlherr and

Chhabra (1995)

0.03–0.9

0.038–47,000

Graphs

Ataide et al. (1999)

<0.55

< 311

Empirical correlation

Kehlenbeck and

DiFelice (1999)

0.1–0.9

≤ Re

∞

≤ 185

Equation 10.13 and

Equation 10.14

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Bubbles, Drops, and Particles in Fluids

0.728

0.614

0.337

d /D

0.949

0.894

0.838

0.783

0.672

0.504

0.279

–1

–2

–1

Reynolds number, Re

ag coefficient,

FIGURE 10.1 Effect of

λ on drag coefficient — Reynolds number relationship in

Newtonian fluids for a sphere. (Modified from Uhlherr, P.H.T. and Chhabra, R.P., Can.
J. Chem. Eng., 73, 918, 1995.)

available for the complete range of

λ and up to about Re ≈ 2×10

. Figure 10.1

and

Figure 10.2,

taken from Uhlherr and Chhabra (1995), show representat-

ive results illustrating the nature of variation of the wall factor with

λ and the

Reynolds number in the three flow regimes, namely, viscous, transition and
fully turbulent (inertial), respectively. Figure 10.1 employs the usual drag coef-
ficient (C

), Reynolds number (Re) coordinates and illustrates the universal

form of relationship for a range of values of

λ. With increasing value of λ, the

viscous region seems to persist up to larger and larger values of the Reynolds
number (Re) and this is also accompanied by a rather late transition to the fully
turbulent conditions thereby resulting in a substantial intermediate transition
region. Suffice it to add here that in the overlapping range of conditions, there
is a good internal consistency of data culled from different sources. Figure 10.2,
on the other hand, confirms the expectation that the wall factor is a function of
λ only both in the viscous and in the inertial regions, as asserted earlier. It is
thus convenient to deal with each region separately.

10.3.1.2.1 Creeping Flow Region
At the outset, it is perhaps appropriate to address the issue of the limiting value
of the Reynolds number denoting the cessation of the creeping flow. Based on an
arbitrary 5% criterion, the critical Reynolds number is defined here as the value

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1 / f

Reynolds number, Re

–1

4
6

FIGURE 10.2 Variation of wall factor with Reynolds number and diameter ratio (curve
numbers 1 to 12 are respectively for

λ = 0.95, 0.89, 0.84, 0.78, 0.73, 0.67, 0.614, 0.50,

0.40, 0.36, 0.266, and 0.169). (Based on the results shown in

Figure 10.1.)

TABLE 10.2

Limiting

Values

Reynolds

Number

for

Creeping

Flow

Conditions

∞

0.1

0.027

0.021

0.2

0.04

0.023

0.3

0.05

0.020

0.4

0.083

0.023

0.5

0.18

0.035

0.6

0.52

0.069

0.7

2.1

0.18

0.8

8.4

0.42

0.9

25.17

0.56

at which the value of the wall factor is 5% higher than the corresponding value
given by Equation 10.5. Based on this criterion applied to the experimental
results available in the literature (McNown et al., 1948; Fidleris and Whitmore,
1961; Uhlherr and Chhabra, 1995; Kehlenbeck and DiFelice, 1999; DiFelice
and Kehlenbeck, 2000), the resulting limiting values of the sphere Reynolds
number are summarized in Table 10.2. It is clearly seen that the so-called
creeping flow persists up to higher value of the Reynolds number, Re

∞

as the

value of

λ increases.

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Now, the creeping flow region is assumed to occur for the values of Re or

∞

smaller than those listed in

Table 10.2.

In this region, perhaps the simplest

(and possibly also the most widely used) experimental correlation is that of
Francis (1933)

− λ

− 0.475 λ

(10.10)

This expression covers virtually the entire range of the diameter ratio up to
λ ≤ 0.97.

By using a fluid dynamic analogy between a single particle in a tube and

a multiparticle suspension, DiFelice (1996) was able to obtain satisfactory
predictions of the wall factor and his expression is given as

− λ

− 0.33 λ

(10.11)

where

α is obtained from the following relationship:

3.3

− α

α − 0.85

= 0.1 Re

∞

(10.12)

The numerical constants appearing in Equation 10.12 are based on the exper-
imental results of Fidleris and Whitmore (1961) and thus this expression is
limited to the range of conditions implicit in the study of Fidleris and Whitmore
(1961). In the viscous region, it predicts

α ≈ 3.3.

Figure 10.3

contrasts various predictions of the wall factor together with

representative experimental data taken from different sources in the creeping
flow region. Overall, good correspondence is seen to exist between various pre-
dictions themselves and also between the predictions and experimental results
in this region. Suffice it to add here that it is virtually impossible to discrimin-
ate between Equation 10.5, Equation 10.10 and Equation 10.11, as all perform
equally well over the entire range of

λ.

10.3.1.2.2 Intermediate Reynolds Number Region
As mentioned previously, only a few numerical studies are available on wall
effects beyond the viscous flow region and out of these, only the study of Wham
et al. (1996), Equation 10.6 to Equation 10.7, are convenient for calculating
the values of the wall factor, f as a function of

λ and Re (or Re

∞

). There

is a paucity of empirical correlations also in this flow region. Subsequent to
the paper by DiFelice (1996), Kehlenbeck and DiFelice (1999) revisited this
problem. Detailed comparisons between the predictions of Equation 10.11 and

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1.0

Lee (1947)
Fidleris and Whitmore (1961)
Iwaoka and Ishii (1979)
Chhabra (1980)
Uhlherr and Chhabra (1995)

Equation 10.5
Equation 10.10
Paine and Scherr (1975)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.2

0.4

0.6

Diameter ratio, l

all f

actor

0.8

1.0

FIGURE 10.3 Representative comparison between various predictions and experi-
mental results on wall factor in the creeping flow region.

Equation 10.12 with the new extensive experimental data of their own and those
of Barr (1931), Francis (1933), McNown et al. (1948), Fidleris and Whitmore
(1961), Okuda (1975) and Uhlherr and Chhabra (1995) revealed inadequacies of
their earlier equation, namely, Equation 10.11. Hence, Kehlenbeck and DiFelice
(1999) proposed the following new correlation:

− λ

+ (λ/λ

)

(10.13)

where both

and p are functions of Re

∞

− 0.283

1.2

− λ

= 0.041 Re

0.524

∞

(10.14)

For Re

∞

≤ 35,

= 1.44 + 0.5466 Re

0.434

∞

(10.15a)

and for Re

∞

≥ 35,

= 2.3 + 37.3 Re

0.434

∞

(10.15b)

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Bubbles, Drops, and Particles in Fluids

They also proposed a simpler version of Equation 10.13 by assigning a constant
value p

= 2.2 and then λ

is obtained from the following relation:

− 0.27

1.2

− λ

= 0.05 Re

0.65

∞

(10.16)

Based on detailed comparisons between the predictions of Equation 10.6 and
Equation 10.7, Equation 10.11 through Equation 10.15 and the experimental
results in the common range of conditions, the main findings can be summarized
as follows: Equation 10.6 and Equation 10.7 based on the numerical predictions
yield values of the wall factor which are within 5 to 6% of the corresponding
experimental values only for

λ ≤ 0.3 and Re

∞

≤ 200. As the value of λ

increases, the correspondence deteriorates rather rapidly. Thus, for instance,
Equation 10.6 and Equation 10.7 yield acceptable values of the wall factor only
for Re

∞

< 10 when λ = 0.4. Similarly, for λ ≥ 0.5 and for all values of

∞

(≤ 200), these equations result in values of the wall factor which deviate

from experimental values by as much as 200%. Furthermore, for

λ = 0.7 and

∞

> 100, these equations predict f > 1 which is clearly inadmissible. On the

other hand, Equation 10.13 through Equation 10.15 predict bulk of the literature
data with an accuracy of about 10% or so under most conditions (Chhabra et al.,
2003).

10.3.1.2.3 Fully Turbulent Region
As mentioned previously, and seen in

Figure 10.1

and

Figure 10.2

under fully

turbulent conditions (Re

→ ∞), the wall factor again becomes independent

of the Reynolds number. In general, the effect progressively diminishes as the
Reynolds number is increased. An inspection of

Table 10.1

shows that many

empirical correlations are available in the literature for the estimation of the
wall factor in this regime. Perhaps the simplest and also the most successful
of these are the Newton’s formula, Equation 10.8 or Equation 10.9 and that of
Munroe (1888–89) which is based on his own experimental results in the range
0.11

≤ λ ≤ 0.83. The Munroe’s equation is given by

= 1 − λ

1.5

(10.17)

The other commonly cited correlation in this flow region is that of Mott (1951)
which in fact is based on the experimental results of Lunnon (1928). Mott
(1951) correlated Lunnon’s limited data by the following two expressions;

For 0.15

≤ λ ≤ 0.5;

= (1 + 3.2λ

)

−0.5

(10.18a)

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533

and in the range 0.4

≤ λ ≤ 0.7

= (1 + 16λ

)

−0.5

(10.18b)

Finally, it can readily be seen that in the limit of Re

∞

→ ∞, the correlations

of Kehlenbeck and DiFelice (1999), Equation 10.13 through Equation 10.16,
respectively reduce to

− λ

2.3

+ (λ/1.2)

2.3

(10.19)

Similarly, in the limit of Re

∞

→ ∞, Equation 10.11 and Equation 10.12 yield

− λ

− 0.33λ

0.85

(10.20)

Figure 10.4 shows a comparison between various predictions, namely,
Equation 10.8, Equation 10.17, Equation 10.18, Equation 10.19, and

1.0

Lunnon (1928)
Fidleris and Whitmore (1961)
Achenbach (1974)
Awbi and Tan (1981)
Dudukovic and Koncar-Djurdjevic (1981)

Uhlherr and Chhabra (1995)

Munroe (1888) Bougas and Stamatoudis (1993)

Equation 10.8

Equation 10.17

Equation 10.18a

Equation 10.18b

Equation 10.19

Equation 10.20

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

Area ratio, b

all f

actor

0.8

1.0

FIGURE 10.4 Comparison between various predictions and experimental results on
wall factor under fully turbulent conditions

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534

Bubbles, Drops, and Particles in Fluids

Equation 10.20. In general, there appears to be an overall good agreement,
though Equation 10.8 and Equation 10.17 have the added advantage of their
particularly simple forms. Included in this figure are the available experimental
data culled from various sources including some results for wall effects in
square channels (Awbi and Tan, 1981). Furthermore, the data included in this
figure relates to the freely falling spheres (Uhlherr and Chhabra, 1995) as well
as to the spheres fixed in flow streams (Achenbach, 1974; Dudukovic and
Koncar-Djurdjevic, 1981); clearly, there are no discernable trends present in
this figure. This is so presumably owing to the generally turbulent conditions
in the tube and therefore the velocity profile of the oncoming stream is relat-
ively flat. Evidently, excellent correspondence is seen to exist between most
experimental results and the predictions of Equation 10.8, except for the results
of Dudukovic and Koncar-Djurdjevic (1981) which lie slightly below the other
results. This is so probably due to the fact that the value of the Reynolds number
is not sufficiently high for the fully turbulent conditions to exist in this work.

Finally, it would be useful to delineate the limiting values of the sphere

Reynolds number beyond which the fully turbulent conditions can be assumed
to prevail and therefore, Equation 10.8 or Equation 10.17 can be used to calcu-
late the value of the wall factor. As seen in

Figure 10.1

and

Figure 10.2,

for a

fixed value of

λ, the wall factor is a function of the Reynolds number and only

approaches the values predicted by Equation 10.8 or Equation 10.17 asymp-
totically. In view of the fact that the wall factor can only be estimated with an
accuracy of about 10% in this regime, it is appropriate to define the critical
Reynolds number as the value at which the value of the wall factor reaches the
95% of the value predicted by Equation 10.8 for a fixed value of

λ. Based on this

criterion, the critical values of the Reynolds number have been extracted from
the literature data (Fidleris and Whitmore, 1961; Uhlherr and Chhabra, 1995);
the resulting values are summarized in Table 10.3 (Chhabra et al., 1996). An
inspection of this table shows the strong dependence of the critical Reynolds
number on

λ. Owing to the limited experimental data in this region, it is not

possible to obtain the corresponding values for

λ > 0.85 (Chhabra et al., 1996).

However, the value of 60 for

λ = 0.1 is in good agreement with the value of 50

reported by Johansson (1974).

TABLE 10.3
Values of Reynolds Number for the Onset of Fully Turbulent

Conditions

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.85

110

200

500

2000

6700

1.25

× 10

1.5

× 10

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Wall Effects

535

In summary, it is clear from the foregoing description that based on a com-

bination of theoretical, numerical, and experimental results, it is possible to
estimate the value of the wall factor for spheres falling axially in cylindrical
tubes under most conditions of practical interest, that is, 0

≤ λ ≤ ∼ 1 and

∞

≤ Re

critical

. In the low and high Reynolds number regions, the wall factor

is a function of

λ only whereas it depends on both the diameter ratio (λ) and the

Reynolds number (Re or Re

∞

) in the intermediate Reynolds number region.

Apart from influencing the terminal fall velocity, the presence of boundar-

ies also alters the detailed structure of the flow field prevailing around a sphere
moving or fixed in a cylindrical tube. For instance, Coutanceau (1971) reported
that the formation of the wake is delayed due to the confining walls. Simil-
arly, Cliffe et al. (2000) have reported that the transition from axisymmetric
to asymmetrical type of flow is also delayed, thereby suggesting that the walls
tend to stabilize the flow. Notwithstanding the fact that the cylindrical tubes
represent the most commonly employed geometry, some analytical and numer-
ical results on the extent of wall effects (in the creeping region) on spheres
settling in ducts of square and triangular cross-sections, and in planar slits
are also available (Brenner, 1961; Cox and Brenner, 1967; Happel and Bart,
1974; Tullock et al., 1992). The corresponding experimental results have been
reported by Miyamura et al. (1981), Chow et al. (1989), Ilic et al. (1992) and
Balaramakrishna and Chhabra (1992). The effects of eccentricity (spheres fall-
ing off-center) on the falling velocity and drag of a sphere have been studied,
amongst others, by Tozeren (1983) and by Higdon and Muldowney (1995).
Under otherwise identical conditions, eccentricity often leads to an increased
drag on the sphere. Similarly, the wall effects on a sphere falling in the pres-
ence of curved boundaries such as in the annular gap formed by two concentric
cylinders have been studied analytically by Hasimoto (1976), Shinohara and
Hasimoto (1980), Alam et al. (1980), Fukumoto (1985) and experimentally by
Zheng et al. (1992). Preliminary comparisons between the observations and the
predictions for some of these configurations appear to be encouraging. Like-
wise, the case of a sphere settling towards or parallel to a plane wall has been
studied by Cooley and O’Neill (1969), Adamczyk et al. (1983), Ambari et al.
(1983, 1984b), Lecoq et al. (1993) and Gondret et al. (1999) amongst others.

10.3.2 I

NELASTIC

-N

EWTONIAN

IQUIDS

10.3.2.1 Theoreticaland NumericalTreatments

Very little theoretical and numerical work has been carried out on the effect
of containing walls on sphere motion in purely viscous fluids without a yield
stress. Kawase and Ulbrecht (1983c) assessed the significance of the wall effects

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536

Bubbles, Drops, and Particles in Fluids

TABLE 10.4
Values of the Drag Correction Factor (n,

λ)

Values of

0.5

0.25

0.125

0.10

0.05

0.02

1.0

5.944

1.979

1.354

1.267

1.119

1.046

1.004

0.9

5.093

1.925

1.402

1.330

1.214

1.163

1.137

0.8

4.366

1.869

1.443

1.387

1.301

1.269

1.257

0.7

3.740

1.809

1.477

1.434

1.376

1.360

1.355

0.6

3.201

1.745

1.500

1.469

1.435

1.428

1.429

0.5

2.740

1.678

1.510

1.493

1.475

1.473

1.476

0.4

2.340

1.607

1.507

1.495

1.489

1.490

1.492

Extrapolated.

Source: Missirlis, K.A., Assimacopoulos, D., Mitsoulis, E., and

Chhabra, R.P., J. Non-Newt. Fluid Mech. 96, 459 (2001).

by studying the creeping motion of a sphere in a spherical envelope of power-
law fluid and concluded that shear-thinning behavior suppresses the effect of
walls. Gu and Tanner (1985), on the other hand, numerically solved the field
equations for the creeping motion of power-law fluids both for a sphere-in-a-
cylinder and for a sphere-in-a-sphere configurations. Both studies concluded
that the wall effects are less severe in power-law fluids (n

< 1) than that

in Newtonian fluids otherwise under identical conditions, which is consistent
with the reported experimental findings, as will be seen later in this section. In
an extensive numerical, study Missirlis et al. (2001) studied the wall effects on
the settling velocity of a sphere falling freely at the axis of a cylindrical tube
in the creeping flow regime. Their results encompass wide ranges of conditions
as: 1

≥ n ≥ 0.1 and 0.50 ≥ λ ≥ 0.02. The wall effect is predicted to decrease

with the decreasing values of diameter ratio,

λ, and the power-law index, n.

Table 10.4 summarizes the numerical values of the drag correction factor

(n, λ), introduced in

Chapter 3,

including the extrapolated values for n

→ 0

and for

λ → 0. The wall factor, f , is related to the drag correction factor via

the following relationship:

(n, λ = 0)

(n, λ)

(10.21)

Obviously, Equation 10.21 becomes indeterminate in the limit of n

= 0.

Figure 10.5

shows the dependence of the theoretical estimates of the wall factor

on the power-law index and the diameter ratio

λ. The wall factor is seen to

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Wall Effects

537

1.0

0.8

0.6

0.4

0.2

0.0

0.1

0.2

0.3

all f

actor

Diameter ratio, l

0.4

0.5

n = 0.6
n = 0.8

n =1

n = 0.2

n = 0.4

FIGURE 10.5 Dependence of the wall factor on

λ and n in creeping flow regime.

(Based on the results of Missirlis, K.A., Assimacopoulos, D., Mitsoulis, E. and
Chhabra, R.P., J. Non-Newt. Fluid Mech., 96, 459, 2001.)

approach the limiting value of f

= 1 at finite values of the diameter ratio, λ as

the value of n is gradually reduced below unity. Also, for each value of n, the
wall factor shows almost linear dependence on the diameter ratio,

λ.

A typical comparison between these predictions and the experimental results

is shown in

Figure 10.6

where a satisfactory correspondence is seen to exist.

However, the match between the theory and experiments deteriorates rather
rapidly with the decreasing value of the power-law index, n. Furthermore,
while the theory predicts a strong influence of the power-law index n on the wall
factor, the available experimental results suggest it to be negligible (Chhabra
and Uhlherr, 1980c).

10.3.2.2 ExperimentalStudies

Amongst the numerous experimental studies available on the free settling of
spheres (see

Chapter 3),

very few investigators have addressed the question of

wall effects on the sphere motion falling in cylindrical tubes. Wall effects in
inelastic non-Newtonian liquids have been either ignored (Slattery and Bird,
1961) or the Newtonian expressions (Turian, 1967; Uhlherr et al., 1976;
Acharya et al., 1976) have been used for correcting the terminal falling velocity
data in power-law fluids without any justification. Over the years, consider-
able experimental and theoretical evidence has accumulated which suggests
that the wall effects are generally less severe in generalized Newtonian fluids
than those in Newtonian media (Sato et al., 1966; Turian, 1967; Chhabra et al.,
1977; Chhabra and Uhlherr, 1980c; Gu and Tanner, 1985; Missirlis et al.,

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538

Bubbles, Drops, and Particles in Fluids

Experimental

Predicted

0.0

0.1

0.2

all f

actor

0.3

0.4

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.5

Diameter ratio, l

n = 0.67

n = 0.86

FIGURE 10.6 Typical comparison between the predicted (Missirlis et al., 2001) and
experimental (Chhabra, 1980) values of the wall factor for power-law fluids in the
creeping flow regime.

2001). If power-law model is employed to depict the shear-thinning beha-
vior, the flow behavior index (n) seems to exert virtually no influence on the
extent of wall effects, which is in stark contrast to the numerical predictions
as discussed earlier. By analogy with the wall effects in Newtonian fluids, it is
generally recognized that the wall factor exhibits a functional dependence on
the Reynolds number and

λ which is qualitatively similar to that observed for

spheres in Newtonian fluids. For power-law fluids in the creeping flow regime,
the wall factor is independent of the sphere Reynolds number suitably modified
for power-law fluids and varies linearly with the diameter ratio as

= 1 − Aλ

(10.22)

Equation 10.22 implies that for a given sphere/fluid combination, the fall velo-
city decreases linearly with the sphere-to-tube diameter ratio. This expectation
is borne out by the representative results shown in

Figure 10.7.

Despite some

confusion (Turian, 1967) regarding the value A in Equation 10.22, most of the
literature data (Karino et al., 1972; Chhabra, 1980; Lali et al., 1989) covering
0

≤ λ ≤ 0.5; 0.52 ≤ n ≤ 0.95; Re

≤ ∼ 1 is well correlated (Chhabra

et al., 1977) with a single value of A

= 1.6. By comparing Equation 10.22 with

the corresponding expression for Newtonian fluids, the wall effects are seen to
be smaller in power-law fluids than those in Newtonian media, at least in the

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Wall Effects

539

PVC spheres
Sapphire spheres

d = 12.686 mm

d = 3.175 mm

Measured ter

minal v

elocity

(mm s

–

)

d = 2 mm

d = 8.749 mm

0.0

0.1

0.2

0.3

Diameter ratio, l

0.4

0.5

0.6

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

FIGURE 10.7 Typical variation of sphere fall velocity with diameter ratio in a 1%
aqueous methocel solution in the low Reynolds number regime. (Based on the data from
Chhabra, R.P., Ph.D thesis, Monash University, Melbourne, Australia, 1980.)

viscous region. This inference is qualitatively consistent with the findings of
Tanner (1964), Caswell (1970), Gu and Tanner (1985) and of Missirlis et al.
(2001).

Similarly, the value of the wall factor is believed to reach a constant value

∞

) at high values of the Reynolds number. However, the critical value of the

latter denoting the attainment of a constant value in the fully turbulent region
is strongly dependent upon the diameter ratio. For instance, for

λ = 0.1, this

transition seems to take place ca. Re

∼ 50 whereas the corresponding value

is about 1000 for

λ = 0.5. The fact that neither of these values is signific-

antly different from the corresponding transition values for Newtonian fluids

(Table 10.3)

seems to suggest that one could use the same values for power-

law fluids also, at least as a rough guide. Based on the meager amount of data
(

λ ≤ 0.5), Chhabra and Uhlherr (1980c) gave the following simple relation

for f

∞

= 1 − 3λ

3.5

(10.23)

In the intermediate transition regime, the wall factor depends upon both the
diameter ratio and the Reynolds number (i.e., the terminal fall velocity no
longer decreases linearly with diameter ratio as shown in

Figure 10.8).

There

is only limited data available in the literature demonstrating the presence of

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540

Bubbles, Drops, and Particles in Fluids

Measured terminal velocity,

(mm s

–

)

250

225

200

175

150

125

100

0.0

0.1

0.2

0.3

d = 6.33 mm

d = 7.92 mm

d = 9.51 mm

d = 6.39 mm

d = 12.69 mm

d = 8.75 mm

PVC spheres
Perspex spheres

0.4

0.5

Diameter ratio, l

0.6

FIGURE 10.8 Typical variation of sphere fall velocity with diameter ratio in a 0.5%
aqueous methocel solution in the intermediate Reynolds number region. (Based on the
data from Chhabra, R.P., Ph.D thesis, Monash University, Melbourne, Australia, 1980.)

all the three flow regions (Chhabra et al., 1977; Chhabra and Uhlherr, 1980c),
and these are reproduced in

Figure 10.9;

the corresponding Newtonian results

are also included in this figure for a qualitative comparison. The functional
dependence of the wall factor on the Reynolds number and the diameter ratio
is well approximated by the following analytical relationship (Chhabra and
Uhlherr, 1980c):

(1/f ) − (1/f

∞

)

(1/f

) − (1/f

∞

)

= {1 + 1.3Re

}

−0.33

(10.24)

where f

and f

∞

, are the limiting values of the wall factor in the low and high

Reynolds number regions estimated using Equation 10.22 and Equation 10.23
respectively. Equation 10.24 encompasses the following ranges of conditions:
10

−2

≤ Re

≤ 10

; 0

≤ λ ≤ 0.5; 0.53 ≤ n ≤ 0.95. The resulting overall

average deviation is of the order of 8%. Other similar expressions (Turian,
1967; Lali et al., 1989; Ataide et al., 1998, 1999) based on the power-law fluid
model are also available in the literature, but none of these have been tested as
rigorously as the ones presented here. For instance, based on their experimental
data over the ranges of conditions as 0.56

≤ n ≤ 0.84 and 1 ≤ Re

∞

≤ 200,

Lali et al. (1989) put forward the following empirical correlation for wall effects:

= (1 − λ)

∞

(10.25)

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Wall Effects

541

1.00

0.80

0.60

0.40

0.20

Reynolds number, RePL

–1

Data

all correction f

actor

Reynolds number, RePL

all correction f

actor

1.00

0.95

1.00

0.90

all correcton f

actor

0.80

0.70

0.60

0.50

0.40
0.35

0.90

0.85

d /D = 0.20

d /D = 0.10

d /D = 0.4

d /D = 0.5

d /D = 0.3

Data

0.80

(a)

(b)

FIGURE 10.9 Wall factor as a function of the Reynolds number and diameter ratio for
spheres falling in power-law liquids. The prediction of Equation 10.24 is shown as solid
line; The corresponding Newtonian results are shown as broken lines. (Based on the
data from Chhabra, R.P., Ph.D thesis, Monash University, Melbourne, Australia, 1980.)

RAJ: “dk3171_c010” — 2006/6/8 — 23:06 — page 542 — #22

542

Bubbles, Drops, and Particles in Fluids

where

1.8

+ 0.2(n − 1)

(10.26a)

0.1

+ 0.2(n − 1)

(10.26b)

Admittedly, Equation 10.25 has implicit in it the dependence of the wall factor
on the power-law index via Equation 10.26, in the overlapping range of con-
ditions, the predictions of Equation 10.24 and Equation 10.25 differ at most
by 7–8% which is well within the limits of experimental accuracy of the wall
factor, f . However, for a few combinations of

λ and Re

∞

, Equation 10.25

yields values of the wall factor which are greater than unity which are clearly
physically unrealistic.

Unfortunately, sufficient information is not yet available to infer the values

of the Reynolds number to delineate the boundaries of the flow regimes for
power-law liquids. In addition to the wall effects in cylindrical tubes, scant
results are also available on wall effects for a sphere falling in vessels of square
cross-section (Balaramakrishna and Chhabra, 1992; Machac and Lecjaks,
1995). The wall effects appear to be less severe in square ducts than in a
cylindrical tube for power-law liquids. Dewsbury et al. (2002b) have repor-
ted completely different type of wall effects on the (light) solid spheres rising
through viscous Newtonian and inelastic polymer solutions. Owing to the non-
vertical rise, it is much more difficult to quantify the wall effects in these
systems.

10.3.3 V

ISCO-

LASTIC

IQUIDS

Unlike in the case of Newtonian and pseudoplastic fluids, intuitively it appears
that a sphere moving in a visco-plastic medium will experience the effect of
the containing walls only if the boundary intersects with the sheared zone
moving with the sphere (Carreau et al., 1997). This seems to suggest that
for a given visco-plastic medium-sphere combination, there must be a crit-
ical value of the sphere-to-tube diameter ratio below which there would be
no wall effects, namely, the wall factor would be unity. This expectation is
borne out by the numerical study of Blackery and Mitsoulis (1997) and con-
firmed in

Figure 10.10,

where the values of wall factor are shown for three

steel spheres as reported by Atapattu et al. (1986). Based on their experimental
results encompassing the range 0.0091

≤ Y

≤ 0.053, Atapattu et al. (1986)

proposed the following predictive expression for wall effects in visco-plastic

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Wall Effects

543

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.0

0.1

0.2

0.3

Diameter ratio, d /D (-)

0.4

all f

actor

(-)

1.0

d = 3.00 mm
d = 4.76 mm
d = 6.00 mm

= 0.027

= 0.017

= 0.013

1.0

FIGURE 10.10 Wall factor as a function of diameter ratio and yield parameter in
visco-plastic media. (Modified from Atapattu, D.D., Chhabra, R.P. and Uhlherr, P.H.T.,
Proceedings of the International Conference on Hydraulic Transport., Bhubaneswar,
India, p. 342, 1986.)

media:

= 1

λ ≤ λ

crit

= 1 − 1.7(λ − λ

crit

)

λ ≥ λ

crit

(10.27)

where

crit

= 0.055 + 3.44Y

Note that this simple correlation is based solely on the yield stress of the

fluid. Finally, one should anticipate a relation between the critical sphere-to-
tube diameter ratio and the size of the sheared zone (

δ) moving along with the

sphere; this contention is examined in

Figure 10.11

where the predicted values

of (d

/δ) are contrasted with the experimentally determined values of (λ

crit

)

and the two seem to differ by almost a factor of two. Undoubtedly, one reason
for this discrepancy could be the difference between the ideal Bingham fluid
and the real fluid behavior. The other contributing factor could be the extent
of back flow of the displaced liquid which inevitably occurs with the descent

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544

Bubbles, Drops, and Particles in Fluids

Yoshioka et al. (maximum principle) (Yoshioka et al., 1971)

Yoshioka et al. (minimum principle)

Data of Atapattu et al. (1986)

Beris et al. (1985)

Equation 10.27

0.01

0.02

0.03

0.04

0.05

0.06

Yield parameter, Y

(-)

)

crit

or (

)

(-)

0.8

0.6

0.4

0.2

FIGURE 10.11 Relationship between the critical diameter ratio (

crit

) and (d

/δ) as a

function of the yield parameter.

of a sphere in a cylindrical fall tube, whereas no such effect is encountered in
theoretical predictions which assume the infinite extent of fluid. Subsequent
numerical work (Beaulne and Mitsoulis, 1997; Blackery and Mitsoulis, 1997)
suggests the wall effects to be more significant up to about Bi

≈ 10 beyond

which the drag results for 0.02

≤ λ ≤ 0.5 collapse on to a single drag curve.

This is qualitatively consistent with the form of Equation 10.27.

10.3.4 V

ISCO

-E

LASTIC

IQUIDS

As seen in

Chapter 5,

the available literature (limited to the creeping region) on

the sedimentation of a sphere in visco-elastic liquids relates either to the usual
polymer solutions which exhibit both shear-dependent viscosity and visco-
elasticity (finite primary normal stress difference) or to the so-called Boger
fluids. In spite of the fact that several studies are available on the drag of spheres
in shear-thinning visco-elastic polymer solutions, only Cho et al. (1980) and
Chhabra et al. (1981b) have presented detailed results on wall effects. Depend-
ing upon whether the primary normal stress difference data is available or not,
one of the following empirical expressions can be used for estimating the value
of the wall factor, f

= 1 − 1.3λ

0.94

()

−0.077

(10.28a)

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Wall Effects

545

= 1 − 0.94λ

0.80

(We)

−0.073

(10.28b)

These equations are valid for the following ranges of conditions:

λ ≤ 0.5;

0.9

≤ ≤ 300; 0.4 ≤ n ≤ 0.55; 0.02 ≤ We ≤ 11 and Re

< 1. The

dimensionless Carreau number (

) and the Weissenbeerg number (We) are

measures of visco-elastic effects, as discussed in

Chapter 5.

In the presence of

shear-thinning behavior, elastic effects further suppress the wall effects, though
the effect is very small (Tanner, 1964; Caswell, 1970).

10.3.4.1 Boger Fluids

The early limited experimental work on spheres in Boger fluids suggested
the Newtonian wall correction to be adequate for

λ ≤ ∼ 0.25 provided the

sphere motion occurred in the constant relaxation time range, that is, at
low Weissenberg numbers (Chhabra and Uhlherr, 1988b; Chmielewski et al.,
1990a; Tirtaatmadja et al., 1990). The wall effects were reported to diminish
progressively with the increasing values of the Weissenberg number, We. While
the extensive numerical results for

λ = 0.5 have already been presented in

Chapter 5, limited numerical and experimental results are also available for the
other values of

λ on both sides of 0.5. In most cases, the values of the drag

correction factor, Y , have been presented as functions of We and

λ. Indeed the

Y — We map is strongly influenced by the value of

λ and therefore extra-

polations from one value of

λ to another may not yield even qualitatively

correct results, as can be seen in

Figure 10.12

for three values of

λ. It is

clearly seen that the value of the drag correction factor, Y may be smaller
or greater than unity depending upon the value of

λ. This appears reason-

able because as the value of

λ increases, the stresses in the nip region must

rise steeply, thereby leading to significant drag enhancements for large val-
ues of

λ due to the strong extensional flow. This assertion is in line with the

scant experimental results (Oh and Lee, 1992; Degand and Walters, 1995)
and some of the numerical simulations for

λ ∼ 0.9 (Mitsoulis, 1998a). It is

difficult to quantify the severity of wall effects over the entire range of

in a simple manner; however, both experiments and predictions suggest the
wall effects to be negligible up to about

λ ≤ ∼ 0.15 and the wall effects

are further suppressed even with a moderate degree of shear-thinning beha-
vior. Such experiments, however, require special care to ensure that the sphere
settles along the axis of the cylinder and without undergoing rotation (Tanner,
1964).

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546

Bubbles, Drops, and Particles in Fluids

2.5

l = 0.2

l = 0.4

l = 0.5

2.0

1.5

1.0

0.5

1.0

Weissenberg number, We

1.5

2.0

(We)

FIGURE 10.12 Wall effects on a falling sphere in visco-elastic liquids in low Reynolds
number regime. (Based on the results of Jones, W.M., Price, A.H. and Walters, K.,
J. Non-Newt. Fluid Mech., 53, 175, 1994.)

10.4 NONSPHERICAL RIGID PARTICLES

Additional complications arise in quantifying the extent of wall effects for
nonspherical rigid particles owing to the inherent difficulties in an unambiguous
description of their size, shape, and orientation in the free settling motion. This
problem is further accentuated by the loss of symmetry with respect to the
shape of sedimentation vessels. Notwithstanding these intrinsic difficulties, the
current status of the available body of information is reviewed in the ensuing
sections, adhering to the same pattern as followed in the preceding section
for spherical particles. Other pertinent reviews are available in the books like
Happel and Brenner (1965), Clift et al. (1978), and Kim and Karrila (1991).

10.4.1 N

EWTONIAN

IQUIDS

In the absence of inertial effects (Re

→ 0), attempts have been made to par-

allel the Faxen-type analysis for an arbitrarily shaped and oriented particle
translating axially in long cylindrical tubes (Brenner, 1962; Rallison, 1978;
Hirschfeld et al., 1984; Kasper, 1987). This approach yields results that are
valid when the ratio of the particle size to that of the sedimentation vessel is
small (

≪1). As expected, for a fixed shape and orientation, the terminal set-

tling velocity decreases linearly with the size ratio. Similarly, the wall effects
on circular thin disks falling freely (oriented normal to the direction of grav-
ity) at the axis of cylindrical tubes have been studied analytically by Shail
and Norton (1969), Pulley et al. (1996), Wang (1996a), Zimmerman (2002)
and by Nitin and Chhabra (2005a). The theoretical and numerical results have

RAJ: “dk3171_c010” — 2006/6/8 — 23:06 — page 547 — #27

Wall Effects

547

been supplemented by experimental studies by Schmiedel (1928), Squires and
Squires (1937), Chhabra (1995c) and Rami et al. (2000). In the limit of creeping
flow and for

λ <∼ 0.08, the analytical results of Shail and Norton (1969) can

be rearranged as (Trahan et al., 1989)

= 1 − 1.7861λ + 1.278λ

− 0.3582

√

(10.29)

The predictions of Equation 10.29 are in line with the experimental results of
Squires and Squires (1937) and the calculations of Wang (1996a). The additional
effects arising from the ends of the tube have been studied by Trahan et al.
(1989). The sedimentation of a disk toward a plane boundary has been studied by
Trahan et al. (1987) and Davis (1990). Most of these and the other related
studies have been reviewed by Zimmerman (2002). More recently, Nitin and
Chhabra (2005a) have numerically studied wall effects on a circular disk moving
broadside at the axis of a cylindrical tube in the range 0.02

≤ λ ≤ 0.5 and

≤ Re ≤ 100. Qualitatively speaking, the wall effects diminish with the

increasing Reynolds number and the decreasing diameter ratio. Their results
can be recast in terms of the wall factor f as

= (1 + aλ + bλ

)

(10.30)

where the values of the two constants a and b are given in Table 10.5 as functions
of the Reynolds number.

Also, the wall effects on circular cylinders of finite aspect ratio settling

with their axis aligned with the direction of motion have been evaluated theor-
etically by Allan and Brown (1986), Trahan et al. (1989) and experimentally
by Huner and Hussey (1977), Ui et al. (1984), Unnikrishnan and Chhabra

TABLE 10.5
Values of a and b in

Equation 10.30

−a

−b

1.293

0.441

0.961

1.032

0.467

1.807

0.192

2.106

0.0546

2.147

0.0112

2.152

100

0.008

2.200

RAJ: “dk3171_c010” — 2006/6/8 — 23:06 — page 548 — #28

548

Bubbles, Drops, and Particles in Fluids

TABLE 10.6
Values of A in Equation 10.22 for Power-Law and

Newtonian Fluids

Values of A

Shape

Newtonian

Power-law

Range of

Discs and plates

1.7

1.52

≤0.32

Cylinders

Short

(L/d < 10)

1.33

1.2

≤0.19

Long

(L/d > 10)

10.58

3.0

Rectangular prisms

1.42

1.28

≤0.60

Cones

2.11

1.63

≤0.37

Hollow cylinders

—

1.13–1.25

≤ ∼0.5

(Ataide et al., 1998)

Source: From Chhabra, R.P., Transport Processes in Bubbles,

Drops and Particles, Dekee, D. and Chhabra, R.P., ed., p. 316,

Taylor & Francis, New York (2002b).

(1990, 1991), and Chhabra (1995c). Analogous results for cross-flow configur-
ation have been provided by Stalnaker and Hussey (1979) and by Chakraborty
et al. (2004). Recently, similar numerical results for ellipsoidal, prolate, and
oblate shaped particles have also been documented in the literature (Xu and
Michaelides, 1996; Shahcheraghi and Dwyer, 1998). On the other hand, a lim-
ited amount of experimental results on wall effects for cones, prisms, cubes,
needles, and cylinders sedimenting in cylindrical tubes are also available (Heiss
and Coull, 1952; Sharma and Chhabra, 1991; Venumadhav and Chhabra, 1995;
Chhabra, 1995c; Ataide et al., 1998) and these have been correlated empirically
(Chhabra, 1995c). Broadly speaking, the low Reynolds number predictions for
discs and cylinders are in line with experimental results. But clearly, all such
attempts are shape-specific and extrapolations beyond the range of conditions
are not recommended. Furthermore, though the presently available predict-
ive formulae have been tested only in a limited way, a summary is presented
in Table 10.6. Some results for wall effects on the straight chains of spheres
and conglomerates of spheres falling at the axis of cylindrical tubes are also
available (Kasper et al., 1985; Chhabra et al., 1995). Limited results on the
terminal falling velocities of porous particles (flocs) in cylindrical tubes have
been reported by Hsu and Hsieh (2004).

10.4.2 I

NELASTIC

-N

EWTONIAN

IQUIDS

Even less is known about the wall effects on nonspherical particles sediment-
ing in stagnant time-independent fluids. As far as we know, only Tanner (1993)

RAJ: “dk3171_c010” — 2006/6/8 — 23:06 — page 549 — #29

Wall Effects

549

has reported that the wall effects for power-law fluids normal to a cylinder are
less severe than those in a Newtonian fluid in the creeping flow region. More
recently, Nitin and Chhabra (2006) have reported the extent of wall effects on a
disk moving in power-law fluids in a cylindrical tube. The shear-thinning beha-
vior was found to suppress the wall effects with reference to that in Newtonian
fluids. This study embraces the range of conditions as: 0.02

≤ λ ≤ 0.5,

< 100 and 0.4 ≤ n ≤ 1.0. Likewise, only scant experimental results

for prisms, needles, cylinders, cones, and disks falling in cylindrical vessels
filled with power-law fluids are available in the literature (Unnikrishnan and
Chhabra, 1990; Sharma and Chhabra, 1991; Venumadhav and Chhabra, 1994;
Chhabra, 1996b; Ataide et al., 1998; Rami et al., 2000; Siman et al., 2002).
Based on data from all these sources, for a fixed shape and orientation, the
wall factor, f

in power-law fluids can be related linearly with

λ as given by

Equation 10.22 where

λ = d

/D; A is a constant that appears to be independent

of the power-law index, n, but it varies with the particle shape. The values
of A for the particle shapes studied thus far are summarized in

Table 10.6.

Also included are the corresponding values of A for Newtonian fluids (Chhabra
1995c). For each case, the wall effects are seen to be less severe in power-law
fluids than that in Newtonian media within the range of conditions (creeping
flow) studied so far. The value of A

= 1.7 for discs in Newtonian media is

remarkably close to the theoretical value of 1.6 (Shail and Norton, 1969; Wang,
1996a). The other scant results on wall effects for cubes and hollow cylinders
in power-law fluids suggest the values of A in the range 1.13 to 1.25 which are
comparable to those listed in Table 10.6. The limited results for linear chains
of spheres and conglomerates of spheres settling in power-law fluids also show
linear dependence of the settling velocity on the reciprocal of tube diameter
(Chhabra et al., 1995).

The effect of visco-elasticity on wall effects for a cylinder confined in a

planar slit has been studied by Huang and Feng (1995). They used the Oldroyd-B
fluid model to delineate the roles of weak inertia (Re

< 10) and of elasticity

(We

< ∼3) on the drag of the cylinder for a range of the blockage ratios (λ ≤

∼0.6). Both visco-elasticity and shear-thinning were shown to suppress the
wall effects, a finding which is in qualitative agreement with the experiments
of Dhahir and Walters (1989) for)

λ = 0.6.

10.5 DROPS AND BUBBLES

Much less is known about the wall effects on the terminal falling velocity of
a bubble or a droplet even when the continuous phase is Newtonian, let alone
for non-Newtonian media. It is generally believed that the sole effect of the
confining walls is to retard the particle as long as the drop-to-tube diameter
ratio,

λ ≤ ∼0.6 (Clift et al., 1978). Conversely, under these conditions, the

RAJ: “dk3171_c010” — 2006/6/8 — 23:06 — page 550 — #30

550

Bubbles, Drops, and Particles in Fluids

confining walls cause little deformation beyond that which may be present
in an unbounded medium. In some sense, therefore, the treatment for rigid
particles presented in the preceding section forms a good starting point for the
discussion on wall effects for fluid spheres for

λ ≤ ∼0.6, whereas owing to

the increasing degree of deformation due to walls, the case of

λ > 0.6 must

obviously be treated separately. However, since the results on wall effects for
fluid particles falling in non-Newtonian continuous phase are indeed scant and
the corresponding literature in Newtonian fluids has been extensively reviewed
by Clift et al. (1978), only a short discussion on wall effects for

λ ≤ ∼0.6

is included here. Furthermore, it is convenient to deal with the low Reynolds
number and high Reynolds number regimes separately. As usual we begin with
the behavior in Newtonian liquids.

10.5.1 N

EWTONIAN

ONTINUOUS

HASE

10.5.1.1 Low Reynolds Number Regime

It is useful to recall here that fluid particles at low Reynolds number in uncon-
fined media tend to deviate little from spherical shape, and furthermore, owing
to the presence of surfactants or large values of the viscosity ratio, X

, the inter-

face is generally immobilized (or stagnant). Indeed, only minor errors occur in
using the wall correction factors for rigid spheres up to about

λ ≈ 0.3. For a

fluid sphere with clean interface translating at the axis of a long cylindrical tube,
Haberman and Sayre (1958) derived the following approximate expression for
wall correction factor:

− a

λ + a

+ a

− a

+ a

(10.31)

where the constants a

to a

are given by

= 0.702

+ 3X

+ X

(10.32a)

= 2.087

+ X

(10.32b)

= 0.569

− 3X

+ X

(10.32c)

= 0.726

− X

+ X

(10.32d)

= 2.276

− X

+ 3X

(10.32e)

RAJ: “dk3171_c010” — 2006/6/8 — 23:06 — page 551 — #31

Wall Effects

551

Needless to say that Equation 10.31 does reduce to Equation 10.5 in the limit
of X

→ ∞. Likewise, subsequent numerical studies for spherical bubbles

(Hartholt et al., 1994; Higdon and Muldowney, 1995; Wham et al., 1997)
confirm the validity of Equation 10.31 in the limit of X

→ 0. Haberman

and Sayre (1958) also reported that the scant results for drops (of aqueous
glycerine solutions or silicone oil) falling in castor oil were consistent with the
predictions of Equation 10.31 for two values of X

≈ 0.005 and X

∼ 0.07 and

λ ≤ ∼0.5 and beyond this value of λ, they reported appreciable deformation
of fluid spheres due to the confining walls. The effect of slip condition on the
particle surface undergoing steady translation in a confined medium has also
been examined by Ramkissoon and Rahaman (2003).

10.5.1.2 High Reynolds Number Regime

Wham et al. (1997) extended the analysis of Haberman and Sayre (1958) to
finite values of Reynolds number (

≤100) but limited to 0.1 ≤ λ ≤ 0.2 and

≤ X

≤ 10. While the low values of λ signify spherical shape, the range

of values of the viscosity ratio does capture the drop-like behavior. Based
on their numerical results, Wham et al. (1997) presented a modified form of
Equation 10.31 which purports to encompass the full range of particle behavior.
Furthermore, Wham et al. (1997) also observed no wake formation at all for
X

≤ ∼2.75. Sherwood (2001), on the other hand, has invoked the irrotational

flow approximation to elucidate the role of weak wall effects (small values of

on the drag of a bubble rising through a liquid in a cylindrical tube and modified
the drag expression (Moore, 1959,1963) for an unconfined bubble. Similarly,
Bozzi et al. (1997) have numerically studied the extent of deformation due to
wall and inertial effects for

λ = 0.5, X

= 1, and Re ≤ ∼540.

While experimental studies on the drag on drops and bubbles reported in the

literature (see

Chapter 6)

have been carried out using vessels or containers of

finite size, very few workers have actually set out to glean data on wall effects by
using containers of different diameters. Admittedly, the presence of confining
walls influences shape, wake structures, and breakup behavior (Hetsroni et al.,
1970; Borhan and Pallinti, 1998, 1999), but most of the experimental literature
on wall effects is limited to establishing the influence of

λ on the terminal

velocity, that is, to delineate the relationship implied by Equation 10.2. The
studies of Strom and Kintner (1958), Harmathy (1960), Salami et al. (1965),
Eaton and Hoffer (1970), Mao et al. (1995), Bhavasar et al. (1996) and of
Chhabra and Bangun (1997) are representative of those in which wall effects
on falling drops have been studied. The corresponding representative studies
for bubbles are due to Uno and Kintner (1956), Harmathy (1960), Maneri and
Mendelson (1968), Tsuge and Hibino (1975) and Krishna et al. (1999b). Much
of the available data on bubbles and drops relates to the conditions of Re

> 1

RAJ: “dk3171_c010” — 2006/6/8 — 23:06 — page 552 — #32

552

Bubbles, Drops, and Particles in Fluids

and E

< 40, Clift et al. (1978) found that the experimental data for wall effects

on fluid particles were in line with that for rigid spheres, at least up to about

λ ≤

∼0.6. Furthermore, Clift et al. (1978) proposed that the value of the wall factor,
f , was greater than 0.98 (i.e., 2% reduction in velocity) under the following
conditions:

λ ≤ 0.06, Re ≤ 0.1

(10.33a)

λ ≤ 0.08 + 0.02 log Re, 0.1 ≤ Re ≤ 100

(10.33b)

λ ≤ 0.12, Re > 100

(10.33c)

On the other hand, for Re

> 100, the wall factor becomes independent of the

Reynolds number and Clift et al. correlated the literature data as:

= (1 − λ

)

1.5

(10.34)

The equation due to Strom and Kintner (1958) is similar to Equation 10.34
except for the value of the exponent being 1.43. There has not been much
activity in this field since the aforementioned early studies, except the scant
data of Mao et al. (1995) which is in line with Equation 10.34.

Beyond these conditions, the wall effects on spherical gas bubbles have been

discussed by Clift et al. (1978). Okhotskii (2001) has used the wave analogy
to develop a semiempirical expression for wall effects on bubbles. Based on
extensive experiments with air–water and air–Tellus oil (

µ = 75 mPa s

−1

)

systems, Krishna et al. (1999b) reported extensive results on wall effects on the
free rise velocity of bubbles up to about

λ ∼ 0.7 or so, and these data are also

consistent with Equation 10.33.

10.5.2 N

-N

EWTONIAN

ONTINUOUS

HASE

Little is known about the influence of non-Newtonian rheology of the continu-
ous phase on the free rise/fall of fluid particles. Chhabra and Bangun (1997)
reported preliminary results on the terminal velocity of droplets of distilled
water, ethylene glycol, chloroform, 1,2-dichloroethane, chlorobenzene, and
furfural falling in aqueous solutions of carboxymethyl cellulose and of poly-
acrylamide (0.52

≤ n ≤ 0.87) in the creeping flow regime (Reynolds number,

< 0.07) and λ ≤ ∼ 0.45. For a fixed diameter of drop, the terminal

falling velocity showed linear dependence on

λ. Since the maximum value of

the viscosity ratio, X

was only 0.0027, these results are tantamount to that for

bubbles. In these range of conditions, the wall factor was well approximated
by the expression, f

= 1.0 − 1.6λ. More recently, Dziubinski et al. (2001)

have studied wall effects on single bubbles rising through power-law liquids

RAJ: “dk3171_c010” — 2006/6/8 — 23:06 — page 553 — #33

Wall Effects

553

(0.43

≤ n ≤ 0.83; λ ≤ ∼ 0.84; 3.3 ≤ Re

≤ 1580; 0.6 ≤ E

≤ 10.8).

Qualitatively, the wall effects were reported to be less severe in power-law
fluids than that for Newtonian liquids, which is consistent with the trends seen
for solid spheres.

Coutanceau and Hajjam (1982) investigated the wall effects on a bubble

rising through visco-elastic and visco-inelastic liquids. For values of

λ ∼ 0.7

to 0.8 (defined using equal volume sphere diameter) and in creeping flow, they
reported significant departure from spherical shape. Indeed, bubbles of inver-
ted bottles-like shapes were reported. Similarly, scant data on slug velocities
of bubbles in non-Newtonian liquids in vertical and inclined tubes have been
reported by Johnson and White (1993), Carew et al. (1995) and Baca et al.
(2003), but no general predictive expressions are yet available for establishing
the role of confining walls on moving bubbles under these conditions. Harris
(1993, 1996) has investigated the behavior of bubbles in Newtonian media near
solid structures.

Before concluding this chapter, it is worthwhile to add here that limited

results are also available on heat and mass transfer from solid particles (mainly
spheres, cylinders, plates, cubes) to streaming non-Newtonian fluids in the pres-
ence of significant wall effects. Unfortunately, the influence of the confining
walls per se on Nusselt (or Sherwood) number has not been investigated sys-
tematically and each correlation relates to a single values of

λ without providing

any clues to allow quantitative predictions of the results in a new application
involving even a slightly different value of

λ. The pertinent literature, however,

has been presented in

Chapter 9.

10.6 CONCLUSIONS

In this chapter, the available body of knowledge on the severity of wall effects
on the rigid and fluid particles falling in Newtonian and non-Newtonian media
has been presented. Even for the simplest case of a solid sphere sediment-
ing at the axis of a cylindrical tube filled with a power-law liquid, the results
are nowhere near as extensive or conclusive as that for the Newtonian liquids.
Broadly speaking the shear-thinning suppresses wall effects for solid spher-
ical and non-spherical particles and for gas bubbles. Virtually no experimental
results are available for fluid spheres with finite values of the viscosity ratio.
Visco-elasticity further lowers the extent of wall effects, at least for falling
spheres. However, owing to the complex interplay between particle shape and
visco-elasticity, extrapolations outside the range of conditions must be treated
with reserve. Finally, it is now possible to estimate the effects of

λ on settling

velocity of a rigid sphere falling in power-law liquids in cylindrical tubes up to
λ ≤ 0.6 and Re

∼1000.

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554

Bubbles, Drops, and Particles in Fluids

NOMENCLATURE

Constant, Equation 10.22 (-)

Constant, Equation 10.6 (-)

Constant, Equation 10.25 (-)

Constant, Equation 10.30 (-)

–a

Constant, Equation 10.31 (-)

Constant, Equation 10.6 (-)

Bingham number

(-)

Constant, Equation 10.25 (-)

Constant, Equation 10.30 (-)

Constant, Equation 10.6 (-)

Drag coefficient (-)

Tube/vessel diameter (m)

Sphere diameter (m)

Equal volume sphere diameter (m)

Eotvos number (

=gρd

/σ) (-)

Wall factor (-)

Wall factor in the creeping flow region (-)

∞

Wall factor in the fully turbulent region (-)

Acceleration due to gravity (m s

−2

) (-)

Constant, Equation 10.6 (-)

Power-law consistency coefficient (Pa s

)

Power-law index (-)

First normal stress difference (Pa)

Constant, Equation 10.13 (-)

Reynolds number (

=ρVd/µ) (-)

Reynolds number (

=ρV

−n

/m) (-)

Terminal velocity in a confined medium (m s

−1

)

Weissenberg number (

=λ

/d) (-)

Viscosity of dispersed phase divided by viscosity of

continuous phase (-)

(n, λ)

Drag correction factor (

/24) (-)

yield parameter (

=τ

/gd(ρ

− ρ)) (-)

REEK

YMBOLS

Constant, Equation 10.11 (-)

Ratio of projected area of particle and

cross-sectional area of vessel (-)

RAJ: “dk3171_c010” — 2006/6/8 — 23:06 — page 555 — #35

Wall Effects

555

˙γ

Shear rate (s

−1

)

Size of yielded region (m)

Viscosity (Pa s)

Continuous phase density (kg m

−3

)

Particle density (kg m

−3

)

Carreau number (

=λ

/d) (-)

Particle to tube diameter ratio (-)

Carreau model parameter (s)

crit

Critical value of

λ, Equation 10.27

Fluid characteristic time (

/2τ ˙γ) (s)

Constant, Equation 10.13 (-)

Surface tension (N m

−1

)

Shear stress (Pa)

Yield stress (Pa)

UBSCRIPT

∞

Unconfined medium

Bingham model

RAJ: “dk3171_c010” — 2006/6/8 — 23:06 — page 556 — #36

Document Outline

Contents
Chapter 10 Wall Effects

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