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8.1 INTRODUCTION

When a liquid flows upward through a bed of particles, one can discern three
distinct flow regimes depending upon the flow rate of liquid. At sufficiently low
flow rate or velocity, it gives rise to a fixed bed; but if the velocity of liquid
is sufficiently high, the solid particles will be freely supported in the liquid to
give rise to what is known as a fluidized bed. At very high velocities, the solid
particles will be transported from the system. The bed in which the conditions
cease to exist as a fixed bed is described as the incipiently fluidized bed and
the value of the liquid velocity corresponding to this point is known as the
minimum fluidization velocity. When the flow rate of liquid is increased above
this value, the bed continues to expand so that the average distance between
the particles increases. The behavior of this kind is known as “particulate fluid-
ization.” It is now generally agreed that this type of fluidization occurs with most
solid–liquid systems (except when the solids are too heavy) and in gas–solid
systems over a limited range of conditions, especially with fine particles. The
ensuing discussion is, however, largely pertinent to particulate fluidization.

Fluidized beds are extensively used as heat/mass exchangers and chemical

reactors in chemical and processing industries. Significantly enhanced heat and
mass transfer rates can be achieved due to the vigorous mixing between the
liquid and solids in a fluidized bed. Existing and potential applications of this
mode of contacting involving Newtonian fluids have been dealt with by several
authors (Davidson and Harrison, 1971; Davidson et al., 1985; Fan, 1989; Kunii
and Levenspiel, 1990; Jamialahmadi and Müller-Steinhagen, 2000; Epstein,
2003) whereas the corresponding limited information for non-Newtonian sys-
tems has been discussed by Baker et al. (1981), Joshi (1983), Tonini (1987),
Chhabra (1993a, 1993b), Shilton and Niranjan (1993) and Chhabra et al. (2001).

In recent years, there has been a growing recognition of the fact that there

are numerous applications in biotechnology employing three-phase fluidization
(Shuler and Kargi, 1992). For instance, many industrially important biopro-
cesses frequently use agricultural wastes as cheap substrates and nutrients with
significant amounts of solids, which can lead to the clogging of packed beds.

395

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Conversely, the sedimentation of solids in bubble column reactors is also an
issue. Some of these difficulties can be obviated by fluidizing the solids with the
upward flow of liquid, and then bubbling the gas through it, thereby resulting
in an effective contacting of the three phases. This mode of operation is called
the three-phase fluidization. The specific examples of the practical applica-
tions of this technique have been summarized by Shuler and Kargi (1992) and
by Schugerl (1997). Airlift tower-loop and sparged reactors also share some
of the phenomena characterizing the hydrodynamic behavior of a three-phase
fluidized bed contactor.

It is readily acknowledged that the minimum fluidization velocity and the

extent of bed expansion as a function of the liquid velocity (beyond the point of
incipient fluidization) represent the two most important hydrodynamic design
parameters for the sizing of the two-phase and three-phase fluidized beds in an
envisaged application; consequently, these aspects have received the greatest
amount of attention in the literature. Other important aspects required for for-
mulating comprehensive models of fluidized beds include mixing and flow
patterns, heat and mass transfer characteristics, detailed particle trajectories,
residence time distribution, particle attrition, conversion and selectivity, etc.,
but these have been studied less extensively, even in the case of Newtonian
media. Excellent books and reviews are available on this subject (Davidson and
Harrison, 1971; Davidson et al., 1985; Fan, 1989; DiFelice, 1995; Kim and
Kang, 1997).

Other types of fluid–particle systems that are usually considered along

with the fluidized beds are packed beds and settling suspensions or hindered
settling in concentrated suspensions of noninteracting particles. While the pos-
sible link between a fixed and fluidized bed has already been alluded to in
the preceding section, the settling behavior of concentrated suspensions also
shows a great deal of similarity with a particulately fluidized bed. In this
chapter, this analogy would be extended to the case of inelastic non-Newtonian
media.

This chapter thus presents a critical evaluation of the available literature

on the hydrodynamic aspects of the two-phase and three-phase fluidized beds
and of the settling of concentrated suspensions of noninteracting particles,
especially when the liquid phase exhibits non-Newtonian behavior. In particu-
lar, consideration will be given to the prediction of the minimum fluidization
velocity, bed expansion characteristics, and the rate of sedimentation of concen-
trated suspensions under the influence of gravity. However, it appears desirable
and instructive to include a brief discussion on the current scene with regard
to the Newtonian fluids for each of these parameters that will not only lay the
stage for non-Newtonian fluids, but will also serve as a reference to draw qual-
itative inferences regarding the influence of nonlinear flow characteristics of
the liquid phase.

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8.2 TWO-PHASE FLUIDIZATION

8.2.1 M

INIMUM

LUIDIZATION

ELOCITY

8.2.1.1 Deﬁnition

If a fluid is passed upward through a bed of particles, the pressure drop across
the bed,

p will initially increase as the superficial velocity, V, of the liquid is

gradually increased as long as the bed behaves like a fixed bed (see Figure 8.1).
When the liquid velocity has reached such a value that the frictional pressure
drop (

p) is equal to the buoyant weight per unit area of the particles, any fur-

ther increase in the velocity results in the rearrangement of particles such that
the resistance to flow remains the same, that is, as the velocity increases, the
bed expands, but the pressure drop across the bed remains essentially constant
provided there is no severe channeling and the wall effects are negligible. This
is the point of incipient fluidization and the corresponding velocity is designated
as the minimum fluidization velocity, V

and the corresponding bed voidage

is denoted by

. Thus for V

> V

, the pressure drop across the bed remains

constant. If the velocity is gradually decreased, the pressure drop remains con-
stant up to the point of incipient fluidization, but the pressure drop values in the
fixed bed region turn out to be lower than that recorded while the velocity was
being increased. This difference is attributed to the slight change in the value of
porosity on account of “repacking” of the bed. In practice, however, departures
from the aforementioned ideal behavior are observed due mainly to interlocking
of particles, channeling, wall effects, etc. Besides, the transition from the fixed
to the fluidized bed conditions occurs gradually over a range of velocities rather

Increasing
velocity

Decreasing velocity

Fluidized region

Fixed bed region

Log

∆

Log V

FIGURE 8.1 Ideal pressure drop–velocity curve for flow through a bed of particles.

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Bubbles, Drops, and Particles in Fluids

than abruptly as shown in

Figure 8.1

(Richardson, 1971). As the minimum flu-

idization velocity has no absolute significance, the generally accepted standard
method for its determination from

p−V plots involves drawing separate lines

through the fixed and fluidized bed regions, and the point of intersection of these
two lines yields the value of V

, as shown in Figure 8.1. Over the years, consid-

erable research effort has been devoted to the development of reliable methods
for the estimation of the minimum fluidization velocity for a liquid–solid com-
bination. A brief account of the progress made thus far is presented in the next
section.

8.2.1.2 Prediction of V

Experimental determination of the minimum fluidization velocity is neither
always possible nor desirable. Hence the need for its prediction often arises
while performing process design calculations for fluidizing systems. For a
given liquid–solid combination, the value of the minimum fluidization velo-
city is influenced by a large number of variables including the particle size (and
distribution), shape and orientation, density and viscosity of the liquid medium,
type and design of distributor, and particle-to-column diameter ratio. Thus, most
advances in this field have been made by using dimensional considerations aided
by experimental observations.

Most attempts at developing predictive expressions for the estimation of

hinge upon the fact that at the point of incipient fluidization, the pressure

drop per unit length of the bed is given by its buoyant weight, which in turn is
equated to the value obtained by assuming the bed to behave like a fixed bed
with a mean voidage of

, that is,

= (1 − ε

)(ρ

− ρ)g

(8.1)

While an incipiently fluidized bed represents a slightly loosened bed, it is cus-
tomary to treat it as a fixed bed since the particles are still in contact with
each other, though this assumption has been questioned by Barnea and Mizrahi
(1973) and Barnea and Mednick (1975). Thus, one can use a suitable method
(as outlined in

Chapter 7)

for calculating the pressure drop across a fixed bed. For

instance, one can combine the Ergun equation (Equation 7.8) with Equation 8.1
to yield for an incipiently fluidized bed:

(1 − ε

)(ρ

− ρ)g =

150

(1 − ε

)

µV

1.75

ρV

(1 − ε

)

(8.2)

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399

It is customary to rewrite Equation 8.2 in a dimensionless form using the Galileo
number, Ga

= 150

− ε

+ 1.75

2
mf

(8.3)

where the Galileo number is defined as

− ρ

2
mf

(8.4)

and the Reynolds number,

ρV

(8.5)

Thus, for a given liquid–solid system, one can readily calculate the value
of Galileo number that in turn facilitates the calculation of Re

or V

via

Equation 8.3 provided the value of bed voidage is known at the point of incipi-
ent fluidization. A typical comparison between the predictions of Equation 8.3
and experimental values is shown in Figure 8.2; the average and maximum

Srinivas and Chhabra (1991)

Ciceron (2000)

Miura and Kawase (1998)

Experimental V

(mm s

–1

)

Predicted

(mm s

–1

)

FIGURE 8.2 Typical comparison between experimental and predicted values of V

using Equation 8.3 (Newtonian fluids).

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Bubbles, Drops, and Particles in Fluids

deviations are 11.8 and 20%, respectively, for 18 data points. The literature
abounds with numerous expressions for the estimation of V

for Newtonian

liquid systems. A selection of widely used correlations has been compiled by
Couderc (1985) and more recently by Jamialahmadi and Müller-Steinhagen
(2000) who have also concluded that it is now possible to predict the value
of V

for spherical particles with an accuracy of about 15 to 20% in a new

application, though somewhat larger errors are encountered for nonspherical
particles (Limas-Ballesteros et al., 1982; Flemmer et al., 1993).

8.2.1.3 Non-Newtonian Systems

Table 8.1

gives a succinct summary of the studies pertaining to the flow of

non-Newtonian liquids (primarily power-law fluids) in fluidized beds. Evid-
ently, only a few workers have dealt with the measurement and prediction of
the minimum fluidization velocity, and a listing of the correlations proposed
for power-law fluids is provided in

Table 8.2.

Suffice it to add here that except

the works of Kawase and Ulbrecht (1985b) and Machac et al. (1986), all other
expressions have been obtained by combining the Ergun equation in its modi-
fied form for power-law fluids (see

Section 7.4)

with Equation 8.1. Kawase and

Ulbrecht (1985b), on the other hand, modified the Stokes equation to include
the effects of non-Newtonian behavior and the bed voidage. They have obtained
an approximate analytical solution to the governing equations in conjunction
with the free surface cell model (Happel, 1958), whereas Machac et al. (1986)
have used purely empirical considerations to arrive at their expression in the
form of (V

∞

). While Jaiswal et al. (1992) improved upon the estimates of

the drag correction factor Y given by Kawase and Ulbrecht (1985b), Dhole et al.
(2004) have extended these calculations to high Reynolds number (up to

∼500)

and to shear-thickening fluid behavior (n

> 1). Aside from these, Miura and

Kawase (1998) extended the approach of Kawase and Ulbrecht (1985b) to high
Reynolds number flows, but it is limited to 0.74

≤ n ≤ 1. While Sabiri et al.

(1996a) extended their capillary model (Sabiri and Comiti, 1995) to predict the
minimum fluidization velocity and bed expansion behavior of beds of spher-
ical and nonspherical particles fluidized by inelastic carboxymethyl cellulose
(CMC) solutions. In a subsequent detailed study, the range of applicability of
this approach was delineated (Ciceron et al., 2002b). However, this approach
yields the functional relationship similar to that obtained using the power-law
version of the Ergun equation. Finally, with the exception of limited data of Yu
et al. (1968), Sharma and Chhabra (1992), and Sabiri et al. (1996a) all other
studies pertain to the beds of spherical particles.

The relative predictive performance of some of the formulae listed in

Table 8.2 has been evaluated (Chhabra, 1993c). Altogether, 70 independent
measurements culled from various sources (as indicated in

Figure 8.3)

and

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TABLE 8.1

Summary of Studies on Fluidization with Non-Newtonian Media

Investigator

Type of work

Details

Main results

Yu et al. (1968)

Experimental

= 100

= 2.5, 8.4, 4.8, 9

= 1, 050–2,450

Aqueous solutions of polyox

Ad hoc empirical modification of the Richardson–Zaki correlation

for predicting V

and V

− ε behavior. Reasonable agreement

between experiments and predictions in the range 0.81

≤ n ≤ 1

Wen and Fan (1973)

Experimental

= 50

= 0.12–1.43

= 1, 520–11,300; Aqueous

solutions of CMC

Axial dispersion coefficients almost identical to Newtonian values

(0.86

≤ n ≤ 1)

Mishra et al. (1975)

Experimental

= 80

= 4.3,6

= 1, 200, 2,500

PVA in water and grease/kerosene

mixtures

Weak non-Newtonian effects are observed

Brea et al. (1976)

Experimental

= 50

= 1.1–3.1

= 2, 500–8,890

Titanium dioxide slurries

Data on V

and bed expansion characteristics which are inline with

the modified correlation of Richardson and Zaki (1954)

Tonini et al. (1981)

Experimental

= 50

= 1.80

= 2, 940

Aqueous solutions of CMC

Mainly concerned with the mass transfer aspects in electrochemical

reactions. Only two moderately non-Newtonian test fluids were used

(Continued)

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Fluids

TABLE 8.1

Continued

Investigator

Type of work

Details

Main results

Kumar and Upadhyay

(1981)

Experimental

= 56, 80, 126

= 0.52–3.05

= 1, 300–2,500

One CMC solution (n

= 0.85)

No results on V

; limited results on V –

ε behavior

Kawase and Ulbrecht

(1985b)

Theoretical

—

Cell models are used to derive expressions for V

and V

–

ε and

agreement with the literature data was stated to be moderately good,

especially for weakly shear-thinning fluids

Briend et al. (1984)

Experimental

= 102

= 0.23–1.86

= 2, 480–11,350

Aqueous solutions of Carbopol and

Separan

Preliminary results on V

and V –

ε behavior which are in line with a

non-Newtonian form of the Blake–Kozeny equation

Machac et al. (1986,

1988)

Experimental

= 20, 40

= 1.46–3.98

= 2, 500–16,600

Aqueous solutions of Natrosol, CMC,

and Separan

Correlations for V

and V –

ε behavior for power-law and Carreau

model fluids

Lali et al. (1989)

Experimental

= 86

= 1.65, 3.1

= 2, 500

Aqueous solutions of CMC

Bed expansion behavior is in line with Newtonian results

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Srinivas and Chhabra

(1991)

Experimental

= 50, 100

= 1.28, 2.6, 3.58

= 2, 500

Aqueous solutions of CMC

Extensive results on V

and bed expansion characteristics. Detailed

comparisons with existing correlations

Jaiswal et al. (1992)

Theoretical

—

Cell model predictions; good agreement with the literature data for

V –

ε behavior and V

Sharma and Chhabra

(1992)

Experimental

= 50, 100

Nonspherical gravel chips fluidized

with aqueous CMC solutions

Extensive data on V

and V –

ε behavior

Dolejs et al. (1995,

1997)

Empirical

—

Hybrid model based on a combination of the submerged objects and

the capillary bundle approach

Sabiri et al. (1996a)

Experimental

= 90

Spheres and plates fluidized with

CMC solutions

Extensive data on V

and bed expansion

Machac et al. (1997,

1999)

Experimental

2-D fluidized beds

1.47

≤ d ≤ 4.12 mm

0.36

≤ n ≤ 1

Effects of shear-thinning and visco-elasticity on fluidization

Broniarz-Press et al.

(1999)

Experimental

= 90

1.8

≤ d ≤ 4.4

0.56

≤ n ≤ 1

Wall effects and non-Newtonian effects on V

Ciceron et al. (2002b)

Experimental

= 90

1.95

≤ d ≤ 6.87

CMC solutions

Extension of the capillary model to predict bed expansion data

Machac et al. (2003)

Experimental

= 90

1.9

≤ d ≤ 6.87

Effect of elasticity on fluidization

Note: D and d are in mm;

in kg m

−3

; CMC is carboxymethyl cellulose.

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Bubbles,

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TABLE 8.2

Expressions Available for Predicting V

in Power-Law Liquids

Investigator

Expression

Remarks

Yu et al. (1968)

= (α

)

(2−n)/n

For creeping flow only

12.5

[((9n + 3)/n)(1 − ε

)

]

Mishra et al.(1975)

7.143Ga

/2−n

= (α

)

/(2−n)

+ 85.714(α

)

/(2−n)

No upper limit on the value of Re

was

stated

β =

(n+2)/2
mf

(1 − ε

)

+ 1

√

−n

Brea et al. (1976)

3
mf

160

/(2−n)

+ 1.75Re

/(2−n)

No upper limit on value of Re

+ 1

(1 − ε

)

−1

(1 − ε

)

−1

Kumar and Upadhyay (1981)

Same as Brea et al., except substitute 150 for 160

No upper limit on value of Re

Kawase and Ulbrecht (1985b)

ρgd

18Ym

Creeping flow only

where Y

= f (n, ε) is available in the original publication

Machac et al. (1986)

= 0.019V

Creeping and transitional flow regimes

= 0.015

+ 0.73

Jaiswal et al. (1992) and Dhole et al.

(2004)

Same as Kawase and Ulbrecht (1985)

The drag correction factor Y

(n, ε) is

available up to Re

= 500

Notes: Re

ρV

(2−n)

;

Dmf

/2−n

It has been subsequently corrected by Kumar and Upadhyay (1981).

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405

–1

–2

–1

Experimental V

(mm s

–1

)

Yu et al. (1968)
Kumar and Upadhyay (1981)
Briend et al. (1984)
Machac et al. (1986)
Srinivas and Chhabra (1991)
Miura and Kawase (1998)
Ciceron (2000)

Predicted

(mm s

–

)

FIGURE 8.3 Typical comparison between the predicted and experimental values of
V

for power-law liquids. (Modified after Chhabra, R.P., Powder Technol., 76, 225,

1993.)

encompassing wide ranges of conditions (0.34

≤ n ≤ 1; 0.23 ≤ d ≤ 15.8 mm;

1, 050

≤ ρ

≤ 11, 350 kg m

−3

) were used. A comparative summary of the

results is shown in

Table 8.3.

Evidently, none of these methods seems to work

particularly well. However, in assessing the results presented in this table, it
should be borne in mind that the value of V

is extremely sensitive to the value

of bed voidage at the incipient fluidized conditions. Besides, the errors of the
order of 50 to 100% are not uncommon in the experimental determination of V

even with Newtonian liquids. Finally, a part of the discrepancy must be attribut-
able to the unaccounted possible visco-elastic effects and wall effects. In view of
this, the predictions of the method I of Machac et al. (1986) and of Jaiswal et al.
(1992) appear to entail the minimum average deviations. Figure 8.3 contrasts
the experimental and predicted values of V

for a wide ranges of conditions.

Preliminary results for nonspherical particles also appear to correlate with

similar levels of accuracy with the equations listed in

Table 8.2.

However, the

approaches of Sabiri et al. (1996a) and Ciceron et al. (2002b) are particularly
suited to nonspherical particles as it obviates the necessity of estimating any
shape factors and/or equivalent particle diameter. Finally, analogous develop-
ments for other fluid models, namely Bingham Plastic, Carreau model and Ellis
model fluids are also available in literature (Mishra et al., 1975; Kawase and
Ulbrecht, 1985b; Dolejs et al., 1995).

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TABLE 8.3

Average and Maximum Deviations in Pre-

dicting V

for Spherical Particles in

Power-Law Liquids

% Error

Investigator or method

Mean

Maximum

Yu et al.(1968)

146

Mishra et al. (1975)

Brea et al. (1976)

174

Kawase and Ulbrecht (1985b)

105

Kumar and Upadhyay (1981)

146

Machac et al. (1986)

Method I

Method II

111

Jaiswal et al. (1992)

Sabiri et al. (1996a)

103

% error

= 100 (experimental−predicted)/predicted.

In summary, the accuracy of the predictions of the minimum fluidization

velocity in non-Newtonian liquids is somewhat poorer than that in Newtonian
liquids. This is so partly due to the additional complications arising from pos-
sible visco-elastic behavior and due to shear-induced mechanical degradation
of macromolecules (Broniarz-Press et al., 1999).

8.2.2 B

XPANSION

EHAVIOR

As mentioned earlier, once the superficial velocity of the liquid exceeds the min-
imum fluidization velocity, the mean voidage of the bed gradually increases and
the frictional pressure drop across the bed remains constant at a value equal to
its buoyant weight. It has been tacitly assumed that this kind of behavior gener-
ally occurs and data are seldom reported to confirm this expectation.

Figure 8.4

illustrates this kind of behavior for the beds of 3 to 3.58 mm glass spheres
being fluidized by non-Newtonian polymer solutions, as reported by Srinivas
and Chhabra (1991) and by Ciceron (2000). This behavior is qualitatively sim-
ilar to that for Newtonian liquids and has been documented in the literature by
others also (Sabiri et al., 1996a; Miura et al., 2001a; Ciceron et al., 2002b).

An anomalous effect has been documented by Machac et al. (1986) who

found that, depending upon the value of the power-law index, n, the pressure
drop across the bed may drop to a value lower than the buoyant weight of the

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407

500

400

300

200

100

0.1

500

1000

1500

2000

2500

Srinivas and Chhabra
(1991)

Ciceron (2000)

Superficial velocity, V (mm s

–1

)

Pressure drop

,–

∆

FIGURE 8.4 Experimental pressure drop-velocity curve and determination of V

for

= 0.845.

•

— Srinivas and Chhabra (1991);

— Ciceron (2000).

0.4

0.5

1.2

1.0

0.8

0.6

0.4

0.2

0.6

Voidage,

0.7

∆

/(1

–

) (

–r

)

FIGURE 8.5 Anomalous pressure drop–velocity behavior observed in fluidization by
visco-elastic liquids, as reported by Machac et al. (1986).

bed (see Figure 8.5). This type of behavior, in turn, limits the maximum achiev-
able bed voidage (

max

) for a given liquid–solid system. The value of ε

max

was

found to decrease with the decreasing value of the power-law index, n. This
type of behavior appears to occur in visco-elastic liquids and is indicative of
segregation.

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Bubbles, Drops, and Particles in Fluids

1.0

0.5

0.2

0.1

0.05

0.01

0.1

0.2

0.5

Voidage,

1.0

0.2

0.5

1.0

n = 0.785
D = 101.6 mm
d =15.8 mm

n = 0.845
D = 101.6 mm
d = 3.57 mm

t⬁

FIGURE 8.6 Typical bed expansion data for n

= 0.785 and n = 0.845. (From

Srinivas, B. K. and Chhabra, R.P., Chem. Eng. Process., 29, 121, 1991.)

It is customary to depict the bed expansion behavior of fluidized beds by

plotting dimensionless velocity ratio (V

∞

) against bed voidage. Figure 8.6

and

Figure 8.7

show typical bed expansion results for beds of 3.57 and 15.8 mm

glass spheres being fluidized by non-Newtonian carboxymethyl cellulose solu-
tions of varying levels of pseudoplasticity. Similar results have been reported
by several other workers in this field (Brea et al., 1976; Lali et al., 1989). In the
case of fluidized beds, one is usually interested in the extent of bed expansion
rather than the value of pressure drop, and therefore, often such data have been
represented and correlated in terms of the following dimensionless variables
for Newtonian systems:

, Re

ε,

= 0

(8.6)

Many expressions of varying forms and complexity have been proposed to
depict the functional relationship embodied in Equation 8.6, the simplest of all
being

∞

= ε

(8.7)

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1.0

0.1

0.01

0.1

0.2

0.5

Voidage,

1.0

0.2

0.5

1.0

n = 0.835
D =50.8 mm
d =3.57 mm

n = 0.9
D = 101.6 mm
d = 3.57 mm

t⬁

FIGURE 8.7 Typical bed expansion data for n

= 0.835 and n = 0.9. (From

Srinivas, B.K. and Chhabra, R.P., Chem. Eng. Process., 29, 121, 1991.)

The literature abounds with numerous empirical expressions that purport to
predict the value of the index, Z, under most conditions of practical interest.
A thorough and critical review of the pertinent literature was presented by
Khan and Richardson (1989). In particular, two methods for the prediction of
the index, Z, in Equation 8.7 have gained wide acceptance and these will be
described briefly here. Based on a large body of experimental data, Richardson
and Zaki (1954) proposed a set of correlations for calculating the value of Z
that are given below in their modified form:

= 4.65 + 20

∞

< 0.2

(8.8a)

4.40

+ 18

−0.3

∞

0.2

≤ Re

∞

≤ 1

(8.8b)

4.40

+ 18

−0.1

∞

≤ Re

∞

≤ 200

(8.8c)

= 2.40

∞

> 200

(8.8d)

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Bubbles, Drops, and Particles in Fluids

The second method is due to Garside and Al-Dibouni (1977) who worked in
terms of the ratio (V

) rather than (V/V

∞

) as

= ε

(8.9)

where Z

(Re

) is given by

5.09

+ 0.2839Re

0.877

+ 0.104Re

0.87

−3

≤ Re

≤ 10

(8.10)

One can, however, rewrite Equation 8.10 in terms of (V

∞

) by introducing

a function of (d

/D), as proposed by Garside and Al-Dibouni (1977), as

∞

+ 2.35

−1

(8.11)

In essence, the new factor in front of the

term corrects the single sphere

terminal velocity for wall effects. Scores of other schemes to predict the value
of Z in Newtonian liquids are available in the literature and these have been
reviewed by DiFelice (1995) and Jamialahmadi and Müller-Steinhagen (2000).
The approach of Richardson and Zaki (1954) has been extended to multisize
sphere systems also, see Asif (1998).

8.2.2.1 Inelastic Non-Newtonian Systems

Owing to the qualitatively similar nature of the bed expansion curves observed
for Newtonian and inelastic non-Newtonian liquids, it is natural to explore the
possibility of extending the aforementioned two formulae to power-law fluids.
In this instance, one would expect the index Z to show a possible additional
dependence on non-Newtonian model parameters. For power-law type fluids,
the relevant definition of Re

becomes

ρV

−n

(8.12)

and Z

= Z(n, Re

, d

/D).

To delineate any possible dependence of the fluidization index Z on the

power-law index, n, in the first instance, the experimental values of Z culled
from different sources are compared with those calculated using the methods
of Richardson and Zaki (1954) and Garside and Al-Dibouni (1977) in

Table 8.4

and the resulting discrepancies, if any, can be unambiguously ascribed to the

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TABLE 8.4
Values of Z for Fluidization with Power-Law Liquids

Values of Z

(d /D)

∞

Experimental values

Equation 8.8

Equation 8.9–Equation 8.12

Equation 8.13

Data source

0.882

0.156

3.66

4.81

6.33

4.52

8.04

Srinivas and Chhabra (1991)

0.84

0.156

1.42

6.35

6.95

4.81

8.11

0.886

0.156

8.13

4.09

5.92

4.21

8.03

0.845

0.0351

0.33

4.93

5.20

5.01

5.54

0.835

0.0803

0.55

5.98

5.86

4.98

6.25

0.900

0.0351

1.03

4.73

5.03

4.22

5.46

0.941

0.0803

1.68

4.93

5.38

4.80

6.12

0.382

0.0351

0.58

8.30

5.11

4.96

6.98

0.382

0.0803

0.58

8.88

5.85

4.98

8.68

0.603

0.0351

5.85

5.22

4.22

4.38

6.01

0.603

0.0803

5.85

4.85

4.42

6.81

0.603

0.0256

2.06

6.05

4.52

4.83

5.82

0.603

0.0512

2.06

5.84

4.95

4.73

6.33

0.68

0.0523

0.33

6.10

5.52

5.02

6.18

Tonini et al. (1981)

0.88

0.0523

0.33

5.80

5.52

5.02

5.83

0.89

0.0408

22.8

3.58

3.86

3.68

5.59

Lali et al. (1989)

0.89

0.0218

5.0

3.98

4.08

4.41

5.21

0.82

0.0408

6.14

3.95

4.28

4.34

5.69

0.82

0.0217

1.58

5.02

4.58

4.89

5.31

0.696

0.0408

1.48

4.93

4.94

4.81

5.90

0.84

0.0218

0.61

4.80

4.86

4.95

5.44

0.85

0.0023

0.0134

4.90

4.80

5.09

5.03

Briend et al.

(1984)

0.86

0.0023

0.029

6.60

4.80

5.09

5.03

0.88

0.0125

16.08

5.10

3.49

3.83

5.20

0.64

0.0125

19.60

5.80

3.43

3.83

5.46

Strictly applicable for Re

∞

≤ ∼ 0.2

The value of V

∞

has been estimated using the numerical results of Gu and Tanner (1985).

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Bubbles, Drops, and Particles in Fluids

non-Newtonian behavior of liquids. An examination of

Table 8.4

suggests that

for large values of (d

/D) (see the first three entries), the experimental values of

Z are closer to the predictions of Garside and Al-Dibouni (1977) whereas for the
remaining cases, that is, small values of (d

/D), there is a good correspondence

with the predictions of Richardson and Zaki (1954). Attention is drawn to the
fact that exceptionally large values of Z have been documented in the literature
when the liquid exhibits visco-elastic behavior (Briend et al., 1984; Srinivas
and Chhabra, 1991) for which neither of the aforementioned schemes seem
to be applicable. Based on the comparison presented in Table 8.4, it is thus
reasonable to conclude that either of the abovementioned two methods may be
used to predict the value of Z for visco-inelastic power-law fluids. The limited
results available for nonspherical particles also provide further support to this
conclusion (Sharma and Chhabra, 1992). Brea et al. (1976), Tonini et al. (1981),
and Miura et al. (2001a), on the other hand, have presented the following or
similar ad hoc modifications of Equation 8.8a for power-law fluids:

= 4.65 + 20

(1 − n)

∞

≤ 0.2

(8.13)

Though in most cases the value of Re

∞

≤ 0.2, it is interesting to note that

the predictions of Equation 8.13, also included in Table 8.4, show good match
with the experimental values of Z in most cases, even for visco-elastic liquids
(Briend et al., 1984; Srinivas and Chhabra, 1991). This agreement is, however,
believed to be fortuitous.

Similarly, Machac et al. (1986, 1993) have proposed the explicit empir-

ical correlation for bed expansion in the low Reynolds number regime
(Re

∞

≤ 0.3) as

ε =

∞

0.218

−0.404(d/D)

−0.862 (1 − n)

∞

0.802

−1.35(d/D)

(8.14a)

In the intermediate Reynolds number range (0.3

≤ Re

∞

≤ 165), they

employed Equation 8.7, with relationship for Z as

4.7

+ 8.8

(Re

∞

)

−0.1

(8.14b)

Equation 8.14b has only slightly different values of the constants as compared
with Equation 8.8c

Subsequently, Machac et al. (1986) have extended the applicability of

Equation 8.14 to include Carreau model fluids.

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Fluidization and Hindered Settling

413

In the second approach, the drag coefficients based on the free surface cell

model (Happel, 1958), theoretical estimates of the drag force experienced by
an assemblage of spherical particles in relative motion with power-law liquids
have been used to predict the velocity-voidage or bed expansion behavior of
fluidized beds (Kawase and Ulbrecht, 1985b; Jaiswal et al., 1992). Using the
numerical values of drag, one can prepare the so-called fluidization charts by
introducing the normalized diameter (d

) and velocity (V

) as

(Re

∞

)

/(2+n)

= V

(3ρ/(4gρ))

/(2+n)

(8.15)

= (C

−n

(Re

∞

)

/(2+n)

= d

(2−n)/(2+n)

/(2+n)

(8.16)

Figure 8.8

through

Figure 8.10

show the results plotted in the form of V

vs. d

for three different values of n

= 1, 0.8, and 0.6. For a given liquid–solid system

(i.e., for known values of

ρ, ρ, d, n, m), the value of d

is known and Re

∞

can be estimated, thereby fixing a point on

ε = 1 curve. One can thus generate

∞

vs.

ε curves simply by drawing a line parallel to the y-axis and passing

through the point (d

, Re

∞

) on ε = 1 line. In the absence of wall effects,

the results so obtained are compared with the predictions of Equation 8.8 and
Equation 8.10 for a range of conditions in

Figure 8.11

and

Figure 8.12

for

Newtonian fluids (Jaiswal et al., 1991a) whereas

Figure 8.13

depicts a typical

comparison with experimental data for n

= 0.6. In both cases, the agreement

between theory and experiments is about as good as can be expected in this field.
Thus, this approach provides a theoretical vantage point for the prediction of
bed expansion behavior.

Finally, it is also possible to use the capillary bundle approach to predict the

bed expansion behavior. This approach is exemplified by the work of Mishra
et al. (1975). While all the aforementioned three approaches for predicting the
bed expansion behavior of a homogeneous fluidized bed have enjoyed varying
levels of success in correlating limited data, none has proved to be completely
satisfactory over the entire range of bed voidage and the Reynolds number. As
remarked in

Chapter 7,

intuitively it appears that the capillary bundle represent-

ation should be more relevant in low to moderate porosity system whereas the
drag theories or the submerged object approach is clearly more appropriate for
the beds of moderate to high values of voidage (Dolejs et al., 1995). Ciceron
et al. (2002b) have reported a comprehensive study based on a combination of
the capillary bundle approach of Comiti and Renaud (1989) and the submerged

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Bubbles, Drops, and Particles in Fluids

Normalized diameter, d

100

0.1

100

0.3

0.4

0.5

0.7

0.9

= 0.9999

= 1

0.1

00.1

n = 1

Normalized velocity,

FIGURE 8.8 V

– d

plot for n

= 1. (From Jaiswal, A.K., Sundararajan, T., and

Chhabra, R.P., Numerical Heat Transfer, 21 A, 275, 1992.)

object model of Mauret and Renaud (1997). For a power law fluid, one can
easily combine Equation 7.20, Equation 7.78, and Equation 8.1 to obtain the
dimensionless equation

= 6 (1 − 0.49 ln ε)

(1 − ε)

+ 3

/(2−n)

+ 0.581

− 0.49 ln ε

/(2−n)

(8.17)

where the Archimedes number, Ar

, is defined as

= d

(2+n)/(2−n)

/(2−n)

(ρ) gm

/(n−2)

(8.18)

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Fluidization and Hindered Settling

415

n = 0.8

0.1

–2

–1

Normalized diameter, d

Normalized velocity,

0.9999

0.9

0.7

0.5

0.4

0.3

FIGURE 8.9 V

–d

plot for n

= 0.8. (From Jaiswal, A.K., Sundararajan, T., and

Chhabra, R.P., Numerical Heat Transfer, 21A, 275, 1992.)

and the particle Reynolds number, Re

, is given by

ρV

−n

(8.19)

For a given liquid–solid system, d,

ρ, ρ, m, and n are all known and

therefore Equation 8.17 permits the calculation of bed voidage for a fixed
value of V

(>V

). It is appropriate to mention here that Sabiri et al. (1997a,

1997b) and Ciceron et al. (2002b) pointed out that it is seldom possible
to approximate the flow curve of a pseudoplastic liquid with a single set
of power-law constants, that is, m and n. It is therefore preferable to fit
a series of power-laws to the flow curve and one should thus use appro-
priate values of (m, n

) depending upon the shear rate at the pore wall. For

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416

Bubbles, Drops, and Particles in Fluids

–1

–2

Normalized diameter, d

0.001

0.1

ε = 0.9999

0.9 0.7

0.5

0.4 0.3

–3

n = 0.6

Normalized velocity,

Re =

5.75

Re = 20

FIGURE 8.10 V

–d

plot for n

= 0.6. (From Jaiswal, A.K., Sundararajan, T., and

Chhabra, R.P., Numerical Heat Transfer, 21A, 275, 1992.)

this purpose, the shear rate at the pore wall can be estimated using the
expression

˙γ

wall

+ 3

(1 − ε)

(8.20)

where T is the tortuosity factor and is given by Equation 7.22, that is,
(1

−0.49 ln ε ).

Figure 8.14

and

Figure 8.15

show typical comparisons between

the predictions of Equation 8.17 and the experimental results drawn from liter-
ature. Based on the extensive comparisons (Re

≤ 240) shown in Figure 8.14

and Figure 8.15 and others not reported herein, it was abundantly clear that

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Fluidization and Hindered Settling

417

–1

–2

–3

0.3

0.4

0.5

0.6

0.7

Voidage,

0.8

0.9

1.0

⬁

= 1

t⬁

′

FIGURE 8.11 Typical comparison between the calculated, shown as line and experi-
mental values of (V

∞

) for Re

∞

= 1 in Newtonian systems.

◦

— Richardson and

Zaki (1954);

•

— Garside and Al-Dibouni (1977); — cell model prediction (Jaiswal

el al., 1991a).

the predictions of Equation 8.17 began to diverge increasingly from the experi-
mental results, once the voidage exceeded a limiting value,

lim

≈ 0.65, which

itself may be a function of the Reynolds number.

For

ε ≥ ε

lim

conditions, Ciceron et al. (2002b) developed a submerged

object model, similar to that of Mauret and Renaud (1997). This approach
yields the relationship

ρ =

ρV

(8.21)

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418

Bubbles, Drops, and Particles in Fluids

–1

′

⬁

= 50

–2

–3

–2

–1

0.3

0.4

0.5

0.6

0.7

Voidage,

0.8

0.9

1.0

t⬁

⬁

= 100

′

FIGURE 8.12 Typical comparison between the calculated, shown as line and exper-
imental values of (V

∞

) for Re

∞

= 50 and 100 in Newtonian systems. Key to the

symbols same as in

Figure 8.11.

where

is the hydraulic tortuosity that is assumed to be a function of the

Reynolds number and porosity (Mauret and Renaud, 1997; Epstein, 1998). At
low Reynolds numbers, the hydraulic tortuosity is independent of the Reynolds
number. The values of

for a range of bed voidages and the Reynolds number

are calculated using the approach of Molerus (1980) for Newtonian fluids. In
the first instance, it is assumed that the same values can be used for power-
law fluids. While it might be a reasonable idea for inelastic liquids, it would
be difficult to justify this approximation for visco-elastic liquids. The other
unknown in Equation 8.21 is the drag coefficient C

for a sphere. Based on

the numerical results of Tripathi et al. (1994) and Tripathi and Chhabra (1995),
Darby (1996) put forward the correlation for C

√

(8.22)

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Fluidization and Hindered Settling

419

–1

–2

–3

⬁

= 5.75

n = 0.6

t⬁

0.2

0.4

0.6

Voidage,

0.8

1.0

′

FIGURE 8.13 Typical comparison between the calculated, shown as line (from Jaiswal
et al., 1992) and experimental values (

) of (V/V

∞

) in power-law liquids. (From

Srinivas, B.K. and Chhabra, R.P., Chem. Eng. Process., 29, 121, 1991.)

where

= 4.8

1.33

+ 0.37n

+ 0.7n

3.7

0.5

(8.23a)

1.82

+ 34

−1/8

(8.23b)

Equation 8.23 is valid for Re

≤ 100 and 0.4 ≤ n ≤ 1.6. The modified Reynolds

number Re

is given by Equation 8.19 except that the characteristic velocity is

given by (V

/ε) instead of V. One can again introduce the shear rate range-

dependent power-law constants by ascertaining the value of the mean shear rate

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Bubbles, Drops, and Particles in Fluids

0.4

0.6

0.8

Superficial v

elocity

(mm s

–

)

Bed voidage,

1.0

FIGURE 8.14 Comparison between the predictions of Equation 8.17 and experimental
data for 3 mm steel spheres fluidized by a TiO

slurry (Brea et al., 1976).

100

0.5

0.6

0.7

Bed voidage,

0.8

Superficial v

elocity

(mm s

–

)

0.9

FIGURE 8.15 Comparison between the predictions of Equation 8.17 and experimental
data for 2.46 mm glass spheres fluidized by Tylose solution (Machac, I., Balcar, M., and
Lecjaks, Z., Chem. Eng. Sci., 41, 591, 1986; Machac et al., 1993).

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Fluidization and Hindered Settling

421

Bed voidage,

Superficial v

elocity

(mm s

–

)

100

0.5

0.6

0.7

0.8

0.9

FIGURE 8.16 Comparison between the predictions of Equation 8.21 and experimental
results for 4.93 mm glass spheres fluidized by a CMC solutions. (Ciceron, D., Comiti, J.,
Chhabra, R.P., and Renaud, M., Chem. Engng. Sci., 57, 3225, 2002).

using the expression

˙γ

mean

(8.24)

Thus, for known values of the physical and rheological properties, this approach
is iterative in nature. Figure 8.16 and

Figure 8.17

show typical comparisons

between the predictions of Equation 8.21 and experimental results for a range
of conditions.

Based on extensive comparisons for bed-expansion data obtained with

Newtonian and with inelastic power-law liquids, the key findings of Ciceron
et al. (2002b) can be summarized as follows: the capillary model yields accept-
able predictions in the range of conditions as

≤ ε ≤ 0.96 for Re

> 5 and

0.4

≤ n ≤ 0.96 whereas the complementary submerged objects model works

well for

ε ≥ 0.6, n ≥ 0.4 and Re

≤ 100.

8.2.3 Effect of Visco-Elasticity

In some of the studies listed in

Table 8.1,

it is likely that some of the polymer

solutions used as test fluids may have exhibited some degree of visco-elastic
characteristics in addition to their pseudoplastic behavior, for example, the

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Bubbles, Drops, and Particles in Fluids

Bed voidage,

Superficial v

elocity

(mm s

–

)

1000

100

0.5

0.6

0.7

0.8

0.9

FIGURE 8.17 Comparison between the predictions of Equation 8.21 and experimental
data for 2.2 mm lead spheres fluidized by a TiO

slurry (Brea et al., 1976).

polyacrylamide solutions used by Briend et al. (1984) and Srinivas and Chhabra
(1991). While the effect of visco-elasticity on the minimum fluidization velocity
is far from clear, inhomogeneities and the formation of preferred flow channels
have been reported for fluidization with visco-elastic liquids (Machac et al.,
1986, 2003), similar to that seen in fixed beds (Muller et al., 1998). Therefore,
bed-expansion data for these systems does not yield a straight line on log–
log coordinates when the bed voidage is plotted against the superficial liquid
velocity. In some cases, even the maximum achievable bed voidage is well
below the theoretical limit of

ε = 1 or when the pressure drop across the bed

is graphed against bed voidage under fluidized conditions, the pressure drop
may drop below the value (1

− ε) (gρ) after reaching the critical value of

bed voidage, as shown in

Figure 8.5.

This effect has also been observed in

two-dimensional fluidized beds.

Figure 8.18

clearly shows the formation of

structures in the beds of particles being fluidized by highly elastic Boger fluids
(Machac et al., 2003).

Aside from the aforementioned studies on fluidization, Wen and Fan (1973)

have reported limited measurements on axial dispersion in beds of spherical
particles fluidized by non-Newtonian polymer solutions. They concluded that
the Newtonian formulae may be used as a first approximation for calculating
the value of axial dispersion coefficient for power-law liquids.

The foregoing treatment is obviously relevant when the density of the

particles is larger than that of the fluidizing medium. In many situations, it

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Fluidization and Hindered Settling

423

(a)

(b)

(c)

(d)

(e)

(f)

FIGURE 8.18 Formation of structures in sedimentation (e–f) (Allen, E. and
Uhlherr, P.H.T., J. Rheol., 33, 627, 1989) and in fluidization (a–d) (Machac, I., Comiti, J.,
Brokl, P., and Siska, B., Chem. Eng. Res. Des., 81A, 1217, 2003. With permission.)

is not so and the solids are lighter than the liquid. In such cases, the liquid is
introduced from the top and the velocity is gradually increased until the bed is
fluidized. This is called inverse fluidization (Fan, 1989). The hydrolysis of milk
protein and the bio-oxidation of ferrous iron by Thiobacillus is carried out in
inverse fluidized bed reactors, for instance. A reasonable body of knowledge
is available on the hydrodynamics of two-phase and three-phase inverse fluid-
ized beds when the liquid phase is Newtonian, for example, see Legile et al.
(1992), Yasser Ibrahim et al. (1996), Ulaganathan and Krishnaiah (1996), etc.
In contrast, little is known about the hydrodynamics of inverse fluidized beds
with a non-Newtonian liquid phase. Femin Bendict et al. (1998) have reported
scant results on the bed expansion and pressure drop characteristics for the flu-
idization of 6 mm LDPE and PP plastic particles by aqueous CMC solutions
(0.80

≤ n ≤ 0.86). Subsequently, Vijaya Lakshmi et al. (2000) have studied the

effect of particle size on the minimum fluidization velocity of plastic particles
by similar CMC solutions. The minimum fluidization velocity was shown to
decrease with the increasing particle density and the liquid viscosity, and with
the decreasing particle size.

8.3 THREE-PHASE FLUIDIZED BEDS

8.3.1 I

NTRODUCTION

If a gas is passed through a liquid–solid fluidized bed, it is possible to disperse
the gas in the form of small bubbles and thereby obtain good contact between
the liquid, the solid, and the gas. This mode of contacting is often referred to
as three-phase fluidized beds. Three-phase fluidized bed reactors have poten-
tial applications in the area of hydrogenation of coal slurries, Fischer–Tropsch
synthesis and the heterogeneous catalytic hydrodesulphurization and hydro-
cracking of oil fractions, and the removal of acidic components from dusty
gases. An important application is in a biological fluidized bed reactor in which

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424

Bubbles, Drops, and Particles in Fluids

oxygen transfer to the biomass takes place, first by its dissolution from air that
is bubbled through the bed, and then its subsequent transfer from the solution
to the biomass particles. Many other existing and potential applications of such
systems have been listed by Epstein (1981), Darton (1985), Muroyama and Fan
(1985), Fan (1989), and Fan and Yang (2003).

The hydrodynamic behavior of three-phase fluidized systems is much more

complex than that of a two-phase liquid–solid fluidized bed, even when the
liquid phase is Newtonian, and it is therefore not possible to present a detailed
treatment here. Attention is, however, drawn to the excellent books (Fan, 1989;
Yang, 2003) and review articles (Epstein, 1981, 2003; Darton, 1985; Muroyama
and Fan, 1985) available on this subject.

Notwithstanding the significance of the detailed bubble and wake phe-

nomena and flow patterns, it is customary to characterize these systems at a
macroscopic level in terms of the minimum fluidizing velocity, bed expansion
characteristic, gas-hold up, etc. While a sizeable body of knowledge is available
on this aspect when the liquid phase is Newtonian, very little is known about the
effect of non-Newtonian liquid behavior on these aspects (Burru and Briens,
1989, 1991; Zaidi et al., 1990a, 1990b; Miura and Kawase, 1997, 1998; Miura
et al., 2001a, 2001b). The available limited information is summarized here.

8.3.2 M

INIMUM

LUIDIZATION

ELOCITY

If a gas is introduced into a bed of solids fluidized by a liquid, it is generally
observed that the minimum fluidizing velocity of the liquid is reduced by the
presence of the gas stream. This is so in part due to the kinetic energy of the
gas that facilitates the fluidization of particles (Zhang et al., 1995). Owing to
the fluctuating nature of the flow, accurate measurements are rather difficult.
Though there exists some confusion about the precise definition of the minimum
fluidization velocity in three-phase fluidized systems, frequently the value of the
minimum fluidization velocity for a three-phase fluidized system is evaluated in
a manner similar to that in the absence of the gas.

Figure 8.19

shows the pressure

drop across the bed vs. superficial velocity of the liquid phase with and without
the introduction of the gas into the bed (of 5 mm glass spheres being fluidized
by a non-Newtonian CMC solution, Miura and Kawase, 1997). The value of
the minimum fluidization velocity is seen to decrease almost by 50% due to the
introduction of the gas.

Figure 8.20

and

Figure 8.21

show the representative

results on the combined effects of the particle size and the gas flow rate on the
minimum fluidization velocity in three-phase systems for water and for a power-
law liquid, respectively. Qualitatively similar trends are present in both these
figures. Following the initial drop, the minimum fluidization velocity seems to
level off to a constant value that is nearly independent of the gas flow rate. This
trend is clearly seen in Figure 8.20 and in Figure 8.21 for small particles, but

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Fluidization and Hindered Settling

425

350

300

250

200

150

100

Superficial liquid velocity, V

(mm s

–1

)

Pressure drop

, ∆

/ρ

g (mm)

= 0 mm s

–1

= 115 mm s

–1

Lmf

FIGURE 8.19 Experimental determination of the minimum fluidization velocity in
three-phase fluidized beds with a polymer solution, n

= 0.79. (Replotted from Miura, H.

and Kawase, Y., Chem. Eng. Sci., 52, 4095, 1997.)

d = 7 mm
d = 5 mm
d = 3 mm

100

150

Superficial gas velocity, V

(mm s

–1

)

Minimum fluidization velocity,

Lmf

(mm s

–

)

200

250

FIGURE 8.20 Effect of gas velocity and particle size on the minimum (liquid) fluidiz-
ation velocity for beds of glass beads being fluidized by water. (Modified after Miura, H.
and Kawase, Y., Powder Technol., 97, 124, 1998.)

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426

Bubbles, Drops, and Particles in Fluids

100

Superficial gas velocity, V

(mm s

–1

)

150

200

250

Minimum fluidization velocity,

Lmf

(mm s

–

)

n = 1(Water)
n = 0.75
n = 0.49

FIGURE 8.21 Effect of power-law rheology and gas flow rate on the minimum (liquid)
fluidization velocity for a bed of 5 mm glass spheres. (Modified after Miura, H. and
Kawase, Y., Chem. Eng. Sci., 52, 4095, 1997.)

is less evident for the larger glass beads. The limited data available thus far
seems to suggest that the reduction in the value of the minimum fluidization
velocity progressively diminishes with the decreasing value of the power-law
index and with the increasing viscosity of the liquid phase (Miura and Kawase,
1997, 1998). Finally, Miura and Kawase (1997, 1998) also concluded that their
data on the minimum fluidization velocity is well predicted by the approach of
Zhang et al. (1995) modified suitably for power-law liquids.

8.3.3 B

XPANSION

EHAVIOR

Depending upon the inertia of solids, the expansion of the bed may increase or
decrease when a gas is introduced into a liquid–solid fluidized bed. At relatively
low gas flow rates and with particles of large inertia (

∼5 to 6 mm glass beads

being fluidized by water), large gas bubbles are split up by the presence of the
solids to produce a dispersion of fine bubbles that present a high interfacial area
for mass transfer. Even smaller bubbles are produced if the surface tension of the
liquid is reduced. On the other hand, smaller particles are unable to overcome

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Fluidization and Hindered Settling

427

1.1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

Superficial liquid velocity, V

(mm s

–1

)

Bed v

oidage

, (-)

100

110 120

d = 5 mm

= 0 mm s

–1

= 46 mm s

–1

= 92 mm s

–1

= 140 mm s

–1

FIGURE 8.22 Effect of gas flow rate on bed expansion behavior of a bed of 5 mm glass
beads being fluidized by water (open symbols) and by a polymer solution (n

= 0.84,

filled symbols). (Modified from Miura, H. and Kawase, Y., Chem. Eng. Sci., 52, 4095,
1997.)

the surface tension forces and do not penetrate into the large gas bubbles. Fur-
thermore, large gas bubbles in fluidized beds have large wakes thereby drawing
liquid rapidly into their wake. This in turn reduces the flow of liquid in the
remainder of the bed, causing the bed to contract. However, while this mech-
anism of bed contraction is pertinent for the low viscosity Newtonian fluids,
such contraction is not observed with highly viscous non-Newtonian liquids.
Figure 8.22 and

Figure 8.23

show the effects of particle size and gas flow rate

on the bed voidage for water and for a non-Newtonian solution. Broadly speak-
ing, for a fixed particle size, the bed expansion behavior is not very sensitive
to the rheology of the liquid (Figure 8.22). Likewise, the effect of particle size
(Figure 8.23) is also qualitatively similar in water and non-Newtonian solutions.

8.3.4 G

OLDUP

Figure 8.24

and

Figure 8.25

show the variation of gas holdup with the increas-

ing gas flow rate for glass beads of different sizes being fluidized by water and

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428

Bubbles, Drops, and Particles in Fluids

1.1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

1.0

0.9

0.8

0.7

0.6

0.5

0.4

Superficial liquid velocity, V

(mm s

–1

)

Bed v

oidage

, (-)

100

110 120

= 140 mm s

–1

d = 3 mm
d = 5 mm
d = 7 mm

FIGURE 8.23 Effect of particle size on bed expansion behavior of a three-phase
fluidized system. Open symbols for water and filled symbols for a polymer solution
(n

= 0.72). (Modified from Miura et al., 2001b.)

by a non-Newtonian polymer solution. Once again, the gas holdup shows qual-
itatively similar dependence on particle size, gas, and liquid flow rates, thereby
suggesting negligible influence of the shear-thinning characteristics of the liquid
phase. In view of this, Miura et al. (2001a, 2001b) reported their results on gas
holdup for power-law fluids to be in line with the following expression due to
Begovich and Watson (1978) initially developed for Newtonian liquids (in SI
units)

= 1.61 V

0.72

0.168

−0.125

(8.25)

Note that this expression does not involve the viscosity of liquid.

Broadly speaking, in a three-phase fluidized system, the minimum fluid-

izing velocity of the liquid phase progressively decreases as the gas flow rate
is gradually increased. This can be explained by noting that the energy input
to the system in terms of the kinetic energy of the incoming gas phase facilitates
the transition from the fixed bed state to the fluidized bed conditions. Similarly,
the liquid holdup also decreases with the increasing gas flow rate. However, the
reduction in the value of the liquid fluidization velocity is strongly dependent

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Fluidization and Hindered Settling

429

100

125

150

0.4

0.3

0.2

0.1

0.3

0.2

0.1

0.0

Superficial gas velocity, V

(mm s

–1

)

d = 5 mm

= 0 mm s

–1

= 20 mm s

–1

= 46 mm s

–1

= 71 mm s

–1

Gas holdup

(-)

FIGURE 8.24 Effect of gas flow rate and liquid rheology on gas holdup in a fluidized
bed of 5 mm glass beads. Open symbols for water and filled symbols for a polymer
solution (n

= 0.72). (Modified from Miura et al., 2001b.)

on the particle size and it generally decreases with the decreasing particle size
and with the increasing degree of pseudoplasticity. Similarly, the bed void-
age progressively increases with the rising gas flow rate at a constant liquid
flow rate. However, the net increase in the bed voidage is also small for small
particles.

In summary, it is perhaps fair to conclude that the available scant data

on three-phase fluidized systems with shear-thinning liquids suggests the non-
Newtonian effects to be rather small. It is thus possible to predict the minimum
fluidization velocity, bed expansion and gas hold-up characteristics by suitable
modifications of the existing frameworks for Newtonian fluids.

8.4 SEDIMENTATION OR HINDERED SETTLING

Hindered settling or sedimentation under static and dynamic conditions

of nonflocculated homogeneous suspensions of uniform size spherical inert
particles represents an idealization of numerous industrially important pro-
cesses encountered in chemical and processing applications, especially relating

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430

Bubbles, Drops, and Particles in Fluids

0.32

0.24

0.16

0.08

0.00

100

125

Superficial gas velocity, V

(mm s

–1

)

150

Gas holdup

(-)

175

200

225

d = 3 mm

= 75.4 mm s

–1

n = 0.79

d = 5 mm
d = 7 mm

FIGURE 8.25 Effect of particle size on gas holdup in a three-phase fluidized system.
(Modified after Miura, H. and Kawase, Y., Chem. Eng. Sci., 52, 4095, 1997.)

to liquid–solid separation using hydrocyclones for slurries (Ortega-Rivas
and Svarovsky, 1998) and in the pipeline transportation of mineral slur-
ries (DeAngelis et al., 1993; DeAngelis and Mancini, 1997; Rosso, 2000).
Undoubtedly, the variable of central interest here is the rate of sedimentation in
the initial constant concentration zone. Reliable knowledge of the sedimentation
velocities is required for the design of equipment for handling suspensions and
slurries such as thickeners, etc. It is readily recognized that the hindered settling
velocity of a suspension of spheres is influenced by a large number of variables:
liquid viscosity and density, particle shape, particle size and size distribution,
density, particle to tube diameter ratio, and the fractional volume concentration
of particles. Further complications arise in the case of fine particles due to van
der Waals and other surface forces; these systems are not considered here but
have been discussed in detail by others, for example, see Russell (1980).

Based on heuristic and dimensional considerations, Richardson and Zaki

(1954) were probably the first to recognize that, inspite of the obvious differ-
ences in the detailed flow fields, a great deal of similarity exists at a macroscopic
level between the settling characteristics of a concentrated suspension and the
expansion behavior of a particulately fluidized bed of uniform size spheres.
Moreover, they asserted that their formulae, initially developed for fluidization

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Fluidization and Hindered Settling

431

0.5

0.2

0.1

0.05

0.1

0.2

0.5

0.1

0.2

0.5

t⬁

(1–c)

n = 1.00
d = 0.64 mm
d/D = 0.0083

n = 1.00
d =1.30 mm
d/D = 0.0167

FIGURE 8.26 Hindered settling data in Newtonian media. (From Chhabra et al.,
1992.)

(Equation 8.7 and Equation 8.8), were equally successful in representing the
hindered settling data also. It is thus customary to use similar type of expressions
for both fluidization and sedimentation.

Figure 8.26 shows typical hindered settling data for two sizes of glass beads

in a Newtonian castor oil. Attention is drawn particularly to the qualitatively
similar nature of these plots to those for fluidization (see Figure 8.6). Indeed the
values of Z calculated from Equation 8.8 are in excellent agreement with the
corresponding experimental values. Similar conclusions have been reached by
numerous investigators during the last 50 years or so (Zeidan et al., 2003). Good
reviews on the hindered settling of non colloidal particles in Newtonian media
at low Reynolds numbers are also available (Scott, 1984; Davis and Acrivos,
1985; Zeidan et al., 2003).

8.4.1 N

-N

EWTONIAN

TUDIES

Very little theoretical and experimental work is available on the sedimentation of
multiparticle particles and of clusters of particles in quiescent non-Newtonian
liquids. Caswell (1977) investigated theoretically the settling of two spheres
in a generalized visco-elastic model fluid. Depending upon the choice of
fluid parameters and the initial separation, the two spheres may converge
or diverge; subsequently, this behavior has been experimentally observed by

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432

Bubbles, Drops, and Particles in Fluids

Riddle et al. (1977). Subsequently, similar results for two-, three-, and four-
sphere systems in shear-thining fluids have also been reported by Zhu and Clark
(1998), Daugan et al. (2002a, 2002b), Zhu et al. (2003) and Huang et al. (2006).
All these studies seem to suggest a complex interplay between the rheology
and the initial configuration of the particles that may lead to the segregation
of particles. Qualitatively similar inhomogeneities and formation of structures
in concentrated suspensions in non-Newtonian suspending media have been
reported, both under static and dynamic conditions (Allen and Uhlherr, 1989;
Darcovich et al., 1996; Bobroff and Phillips, 1998; Siska et al., 1995, 1996;
Daugan et al., 2004), as can be seen in

Figure 8.18

for the hindered settling

of sand particles in visco-elastic polyacrylamide solutions. Note the striking
similarity between the heterogeneities observed in the fluidized beds and in
the hindered settling of concentrated suspensions of noninteracting particles.
Furthermore, the formation of structures may show additional dependence on
time, thereby resulting in enhancement or hindrance in settling (Bobroff and
Phillips, 1998; Daugan et al., 2004). While the exact reasons for such inhomo-
geneties are not known, either shear-thinning or visco-elasticity, or both are
often invoked to explain these phenomena qualitatively (Bobroff and Phillips,
1998).

On the other hand, Kawase and Ulbrecht (1981e) employed the Happel’s

free surface cell model for predicting the hindered settling velocity of
suspensions of spheres in power-law fluids. They have obtained a closed form
solution that is believed to be applicable for weakly non-Newtonian behavior,
that is, n not too different from unity. Unfortunately, their final expression for
n

= 1 shows deviations almost of the order of 100% from the well-established

results for Newtonian fluids and hence their predictions for non-Newtonian
fluid behavior must be treated with reserve.

There have been only a few experimental investigations relating to the

hindered settling of concentrated suspensions in non-Newtonian systems.
Balakrishna et al. (1971) reported preliminary results on the settling of fine sand
particles and concluded that there were no non-Newtonian effects present in
their results. Inspite of this, a cumbersome empirical expression for the velocity
ratio (V

∞

) was presented. Allen and Uhlherr (1989) have reported the rates

of sedimentation for glass beads of various sizes settling in aqueous solutions
of polyacrylamide. They observed regions of heterogeneities and segregation,
akin to that observed in gas–solid fluidization and in liquid–solid fluidization
with visco-elastic liquids (Machac et al., 2003) and in the shearing flow of sus-
pensions (Lyon et al., 2001). Furthermore, when their results are plotted in the
form of V

∞

vs.

ε on log–log coordinates, two distinct power law regions are

encountered. Though the exact reasons for this behavior are not immediately
obvious, Allen and Uhlherr (1989) have ascribed both these anomalous obser-
vations to the viscoelastic behavior of test fluids. Subsequently, these results

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Fluidization and Hindered Settling

433

0.5

0.2

0.1

0.05

0.02

0.1

0.2

0.5

1.0

0.1

0.2

0.5

(1–c)

t⬁

d = 3.3 mm
d /D = 0.0413
n = 0.797

d = 3.3 mm
d /D = 0.0703
n = 0.925

FIGURE 8.27 Hindered settling data in power-law liquids. (From Chhabra et al.,
1992.)

have been reconfirmed by Bobroff and Phillips (1998) and by Daugan et al.
(2004). In fact, the temporal variation of particle distribution may even lead
to the enhanced rather than hindered settling in such systems (Daugan et al.,
2004). Scant results on hindered settling of noninteracting spheres in mildly
shear-thinning liquids (Figure 8.27) do seem to conform to Equation 8.7 and
Equation 8.9 (Chhabra et al., 1992).

Finally, it is appropriate to report on some of the recent work on the sedi-

mentation of rod-like particles (fibers) in Newtonian suspending media. Typical
examples of the sedimentation of rod-like particles are found in paper pulp sus-
pensions, red blood cells, etc. (Kumar and Ramarao, 1991; Chen and Chen,
1997). The interparticle interactions in rod-like systems begin to manifest at
much lower volumetric concentrations of fibers than in particulate suspensions.
For instance, in a Newtonian liquid a 3% volume fraction of spheres, the
hindered settling velocity is reduced by 20% whereas for elongated bodies, this
order of hindrance occurs at much lower concentration of fibers (Turney et al.,
1995). Needless to say that with nonspherical particles, it is neither justified
to assume the homogeneity of the system nor is it necessary that the interfiber
interactions will impede their settling. Indeed, it is possible to obtain enhanced
rates of sedimentation in such systems (Herzhaft et al., 1996). However, the

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434

Bubbles, Drops, and Particles in Fluids

Monte-Carlo simulations of Mackaplow and Shaqfeh (1998) predict the degree
of hindrance to be linear in concentration provided the system is isotropic and
homogeneous. On the other hand, their dynamic simulations suggested spa-
tial and orientational distributions to become anisotropic and heterogeneous.
Obviously, the sedimentation of concentrated suspensions of rod-like particles
is much more complex than that of spherical particles. It is unlikely that the
simple expressions like Equation 8.7 or Equation 8.8 will ever suffice for such
systems.

8.5 CONCLUSIONS

In this chapter, the gross flow characteristics of settling suspensions and bed
expansion characteristics of a bed of spherical particles fluidized by power-
law fluids have been considered. In particular, different methods available to
predict the minimum fluidization velocity and velocity–voidage relationships
for both fluidization and sedimentation have been examined. Based on the
critical analysis of the literature data, it is safe to conclude that satisfactory
empirical and theoretical methods are available for estimating the value of the
minimum fluidization velocity for power-law liquids whereas velocity-voidage
behavior of fluidization involving power-law fluids is well predicted by suitable
modifications of the corresponding Newtonian formulae. It is obvious that the
capillary model works well up to a limiting value of bed voidage, beyond which
a submerged objects approach affords a somewhat better representation. The
limiting value of the bed voidage delineating the region of the applicability of
these two models is about 0.6 to 0.65, though it is somewhat dependent on the
Reynolds number and the power-law index.

The settling of multiple particles and of concentrated suspensions in non-

Newtonian liquids deviates significantly from their Newtonian analogs. For
instance, model studies involving two, three, or four spheres show a tendency
to form segregates even in the creeping flow region whereas this type of effect
is caused only by inertial effects in Newtonian fluids. The temporal variation of
particles in concentrated systems also leads to the formation of inhomogeneous
structures that may even be time-dependent. Thus, extreme caution should be
exercised in using the Newtonian formulae to predict the rate of sedimentation
when the suspending medium is non-Newtonian.

Very little is known about the role of visco-elasticity as well as about the

other facets of liquid–solid fluidization including mixing patterns, detailed velo-
city profiles, etc. all of which strongly influence the mass/heat transfer processes
and chemical reactions in a fluidized bed. The scant literature available on the
sedimentation and fluidization of fibrous materials suggest their behavior to be
significantly different from that of suspensions of spherical particles even at

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Fluidization and Hindered Settling

435

the gross macroscopic level. Finally, the limited available data for three-phase
fluidized beds seem to suggest the non-Newtonian effects to be no more serious
than that in the two-phase fluidized beds.

NOMENCLATURE

Constant, Equation 8.23 (-)

Archimedes number, Equation 8.18 (-)

Constant, Equation 8.23 (-)

Volume fraction of solids (

= 1 − ε) (-)

Drag coefficient (-)

Modified drag coefficient, Equation 8.22 (-)

Particle diameter (m)

Normalized particle diameter, Equation 8.16 (-)

Tube diameter (m)

Acceleration due to gravity (m s

−2

)

Galileo number, Equation 8.4 (-)

Height of packed bed (m)

Power-law consistency index (Pa s

)

Flow behavior index (-)

Pressure drop (Pa)

Reynolds number for Newtonian fluids (-)

Reynolds number for power-law fluid (-)

Particle Reynolds number, Equation 8.19 (-)

Reynolds number based on terminal falling velocity of
sphere (-)

Tortuosity factor (-)

Superficial velocity (m s

−1

)

Normalized superficial velocity, Equation 8.15 (-)

Free settling velocity (m s

−1

)

Drag correction factor (

= C

/24) (-)

Fluidization and sedimentation index, Equation 8.7 (-)

REEK

YMBOLS

˙γ

mean

Mean shear rate for a sphere, Equation 8.24 (s

−1

)

˙γ

wall

Pore wall shear rate, Equation 8.20 (s

−1

)

Mean voidage or holdup (-)

Newtonian viscosity (Pa s)

Fluid density (kg m

−3

)

Solid density (kg m

−3

)

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436

Bubbles, Drops, and Particles in Fluids

Density difference (kg m

−3

)

Hydraulic tortuosity, Equation 8.21 (-)

UBSCRIPTS

Gas

Liquid

Minimum fluidization

∞

Absence of wall effects

Document Outline

Contents
Chapter 8 Fluidization and Hindered Settling

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