SEDIMENTATION OF MULTISIZED PARTICLES
by
AKALANKKUMAR C. KOTHARI, B.E.
A THESIS
IN
CHEMICAL ENGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
MASTER OF SCIENCE
IN
CHEMICAL ENGINEERING
Approved
August 1981
I 7--
ACKNOl/LEDGMEflTS
The author wishes to express his deep aopreciation and thanks to
Dr. Sami Selim for his encouraqement and advice throughout this work.
The author wishes also to thank Dr. H. R. Heichelhei'^, Dr. Lizi
Mann, and Dr. R. W. Tock for serving on the thesis committee a^^d for
giving valuable suggestions.
Financial support provided by NSF grant - ENG78-27000 for carrving
out the literature survey and for ourchasing the glassbeat-^s is ar^^tefully acknowledged.
Financial suoport in the form of a teac'^in^ assis-
tantship from the Chemical Engineering Department at Texas Tech University is gratefully appreciated.
Special thanks is expressed to classmate David L. Roberts for
helping in the computer work and for reading the manuscript.
11
TABLE OF CONTENTS
PAGE
ACKNOWLEDGMENTS
ii
ABSTRACT
v
LIST OF TABLES
vi
LIST OF FIGURES
I.
II.
III.
viii
INTRODUCTION
1
SEDIMENTATION OF EQUISIZED PARTICLES
11
Dimensional Analysis
17
Published Correlations
23
SEDIMENTATION OF MULTISIZED PARTICLES
31
Sedimentation of Suspensions of Uniform Particles . . . .
35
Sedimentation of Particles in a Binary Suspension. . . .
36
Sedimentation of Suspensions of Multisized Particles. . .
39
Sedimentation of Suspension with Continuous
Particle Size Distribution
IV.
V.
47
PROCEDURE AND EXPERIMENTS
48
Glass Spheres
48
Coloring Glass Particles
50
Suspending Media
50
Experimental Setup
52
Experimental Procedure
52
RESULTS AND DISCUSSION
54
Comparison of Models for Binary Suspensions
55
Comparison of Models for Ternary Suspensions
101
Behavior of Models at High Reynolds Numbers
106
i11
PAGE
Comparison of Models for Suspensions with.
VI.
Continuous Size Distribution
108
CONCLUSIONS AND RECOMMENDATIONS
116
Conclusions
116
Recommendations for Further Work
117
NOMENCLATURE
119
BIBLIOGRAPHY
122
APPENDICES
A.
Experimental Data and Comparison with Predictions
from Proposed Model
B.
126
Computer Program for Prediction of Interface
Velocities in Multisized Particle Suspensions . . .
1 V
139
ABSTRACT
A new model is developed for the sedimentation of multisized particles.
Unlike previously published models, the present model takes into
account interparticle interactions by taking into consideration the
buoyancy effect induced by the smaller size particles on the terminal
falling velocities of large size particles.
The new and previously
published models are tested against published and newly collected data
on suspensions with discrete size distribution.
It is shown that the
present model is the most accurate and represents the experimental data
satisfactorily.
Moreover, the model satisfactorily represents the ex-
perimental data on binary countercurrent operations where the Reynolds
number is as high as 546.
It is also shown that the proposed model can
be used to predict the sedimentation velocities of suspensions with
continuous size distribution.
LIST OF TABLES
Table
2.1
Page
Published Correlations for the Solid-Fluid
24
Vertical Flow Operations
4.1
Properties of the Glass Spheres
49
4.2
5-1
Properties of Suspending Media
Comparison of the Proposed and Previously
Published Models with the Experimental Data
on Binary Suspensions
52
57
Comparison of the Proposed and Previously
Published Models with the Experimental Data
on Ternary Suspensions
102
Comparison of the Proposed and Previously
Published Models with the Experimental Data
of Lockett and Al-Habbooby on Countercurrent
Operations
107
Comparison of the Proposed Model with the
Experimental Data for Binary Suspension in
Ethylene Glycol
127
Comparison of the Proposed Model with the
Experimental Data for Binary Suspension in
Diethylene Glycol
129
Comparison of the Proposed Model with the
Experimental Data for Binary Suspension in
Aqueous Glycerol
130
Comparison of the Proposed Model with the
Experimental Data of Smith (1965) on Binary
Sedimentation
131
Comparison of the Proposed Model with the
Experimental Data of Mirza and Richardson
(1979) on Binary Sedimentation
132
Comparison of the Proposed Model with the
Experimental Data for Ternary Suspension in
Diethylene Glycol
133
5.2
5.3
A.l
A.2
A.3
A.4
A.5
A.6
vi
Table
A.7
A.8
A.9
Page
Comparison of the Proposed Model with the
Experimental Data of Lockett and Al-Habbooby
(1973) on Countercurrent Solid-Liquid Vertical
Flow
134
Comparison of the Proposed Model with the
Experimental Data for Suspensions with
Continuous Size Distribution
137
Comparison of the Proposed Model with the
Experimental Data for Suspensions with
Continuous Size Distribution
138
vn
LIST OF FIGURES
Figure
2.1
2.2
Page
Relation between rate of sedimentation and
voidage of suspension.
22
Slope n as a function of d/D for various
values of Re .
25
00
2.3
2.4
3.1
3.2
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
General correlation for the relative velocity
in solid-liquid systems, based on extended
definitions of Re. and Cp..
28
A growth curve showing relationship between
Ug/U^ and Re at several values of c.
29
Segregation and zone-formation during sedimentation of a binary suspension.
32
Formation of zones in sedimentation of multisized particle suspension.
40
Comparison of experimental results with model
for dL=0.0460 cm, d^= 0.0194 cm, and C^= 0.0696.
58
Comparison of experimental results with model
for dL= 0.0460 cm, d^= 0.0194 cm, and C^= 0.119.
59
Comparison of experimental results with model
for dL= 0.0460 cm, d^= 0.0194 cm, and C,_= 0.168.
60
Comparison of experimental results with the model
for d,= 0.0460 cm, d^= 0.0194 cm, and C^= 0.217
61
Comparison of experimental results with the model
for d,= 0.0460 cm, d^= 0.0137 cm, and 0^= 0.0595.
62
Comparison of experimental results with model
for d,= 0.0460 cm, d^= 0.0137 cm, and C^= 0.119.
63
Comparison of experimental results with model
for dL= 0.0460 cm, d^= 0.0137 cm, and C^= 0.168.
64
Comparison of experimental results with model
for dj_= 0.0460 cm, d^= 0.0137 cm, and C, = 0.217.
65
Comparison of experimental results with model
for d^= 0.0326 cm, d^= 0.0137 cm, and 0^= 0.0797.
66
vm
Figure
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
Page
Comparison of experimental results with model
f o r d^= 0.0326 cm, d^= 0.0137 cm, and 0^= 0.139.
67
Comparison of experimental results with model
f o r dj_= 0.0326 cm, d^= 0.0137 cm, and 0^= 0.126.
68
Comparison of experimental results with model
f o r d^= 0.326 cm, d^= 0.0137 cm, and C^= 0.200.
69
Comparison of experimental results with model
f o r d|_= 0.0326 cm, d^= 0.0081 cm, and C^= 0.0624.
70
Comparison of experimental results with model
f o r dj_= 0.0460 cm, d^= 0.0194 cm, and 0^= 0.0612.
71
Comparison of experimental result:- with model
f o r d(_= 0.0460 cm, d^= 0.0194 cm, and 0^= 0.122.
72
Comparison of experimental results with model
f o r d,= 0.0460 cm, d^= 0.0194 cm, and 0^= 0.173.
73
Comparison of experimental results w i t h model
f o r d^=0.0460 cm, d^= 0.0194 cm, and 0^= 0.225.
74
Comparison of experimental results with model
f o r d^=0.0460 cm, d^= 0.0137 cm, and 0^=0.0612.
75
Comparison of experimental results with model
f o r d,= 0.0460 cm, d^= 0.0137 cm, and 0^= 0.120.
76
Comparison of experimental results with model
f o r d,= 0.0460 cm, d^= 0.0137 cm, and C^= 0.188.
77
Comparison of experimental results with model
f o r d, = 0.0460 cm, d^= 0.0137 cm, and C^= 0.225.
78
Comparison of experimental results with model
f o r d,= 0.0326 cm, d^= 0.0137 cm, and C^= 0.0612.
79
Comparison of experimental results with model
f o r d.= 0.0326 cm, d^= 0.0137 cm, and C^= 0.122.
80
Comparison of experimental results with model
f o r d, = 0.0326 cm, d^= 0.0137 cm, and Cj_= 0.174.
81
Comparison of experimental results w i t h model
f o r d, = 0.0326 cm, d^^^ 0.0137 cm, and C^= 0.0615.
82
\x
Figure
5.26
5.27
5.28
5.29
5.30
5.31
5.32
5.33
5.34
5.35
5.36
5.37
5.38
5.39
5.40
Page
Comparison of experimental results with model
for d|_= 0.0326 cm, d^= 0.0137 cm, and 0^= 0.123.
83
Comparison of experimental results with model
for dj_= 0.0326 cm, d^= 0.0137 cm, and C^= 0.174.
84
Comparison of experimental results with model
for dj_= 0.0326 cm, d^= 0.0137 cm, and 0^= 0.225.
85
Comparison of experimental results with model
for dj_= 0.0326 cm, d^= 0.0081 cm, and C^= 0.0615.
86
Comparison of experimental results with model
for d^= 0.0326 cm, d-= 0.0081 cm, and C^= 0.123.
87
Comparison of the proposed model with experimental
data of Smith.
88
Comparison of the proposed model with experimental
data of Smith.
89
Comparison of the proposed model with experimental
data of Smith.
90
Comparison of the proposed model and MizraRichardson model with experimental data of Mizra
and Richardson.
91
Comparison of the proposed model and MizraRichardson model with experimental data of Mizra
and Richardson.
92
Comparison of the proposed model and MizraRichardson model with experimental data of Mizra
and Richardson.
93
Comparison of models with the new experimental
data for the lower interface.
94
Comparison of models with the new experimental
data for the lower interface.
95
Comparison of models with the new experimental
data for the lower interface.
96
Comparison of models with the new experimental
data for the lower interface.
97
Figure
Page
5.41
Comparison of models with the new experimental
data for the lower interface.
5.42
Comparison of models with the experimental
data for the upper interface.
99
Comparison of models with the experimental
data for the upper interface.
100
Comparison of the proposed model with the
experimental data on ternary suspensions.
103
Comparison of the proposed model with the
experimental data on ternary suspensions.
104
Comparison of the proposed model with the
experimental data on ternary suspensions.
105
Comparison of the proposed model with experimental data on suspensions with continuous
size distribution.
110
Comparison of the proposed model with experimental data on suspensions with continuous
size distribution.
Ill
Comparison of the proposed model with experimental data on suspensions with continuous
size distribution.
112
Comparison of the proposed model with experimental data on suspensions with continuous
size distribution.
113
Comparison of the proposed model with experimental data on suspensions with continuous
size distribution.
114
Comparison of the proposed model with experimental data on suspensions with continuous
size distribution.
115
5.43
5.44
5.45
5.46
5.47
5.48
5.49
5.50
5.51
5.52
XI
98
CHAPTER I
INTRODUCTION
Chemical engineers often come across systems in which relative
motion takes place between a fluid and suspended solids.
The last
fifty years of research have witnessed an increased importance of solidliquid operations and have resulted in the need for a greater understanding of their underlying characteristics.
Most of the investigations so
far have been confined to systems of equisized spherical particles moving
relative to a fluid.
Unfortunately, such idealized systems are rarely
found and attention has turned towards systems of multisized spherical
or non-spherical particles.
Sedimentation, fluidization, and co- or
counter-current solid-liquid operations are among the major processes
that fall under relative motion between solids and fluids.
Such pro-
cesses are widely employed in a large number of chemical and allied industries as listed below.
(1)
Various gravity settling equipment.
(a)
thickeners,
(b)
dust collectors,
(c)
gravity settlers,
(d)
spray dryers.
This includes:
(2)
Fluidized-bed and moving-bed processes.
(3)
Pneumatic transport of solids in vertical or inclined pipe-
lines using forced draft.
(4)
Settling and particle-growth in crystal 1izers.
(5)
Electrostatic separators.
1
(6)
Erythrocyte sedimentation, a standard clinical test.
(7)
In mining and paint industries where relative motion between
solid and fluid is usually found.
The first important theoretical study of the forces acting on an
immersed body moving relative to a viscous fluid was made by Stokes
(1851).
Although the theory can be used to predict terminal falling
velocities, it is confined to a spherical particle moving in an infinite
fluid medium at low relative velocity so that the inertia of the fluid
could be neglected.
Predicted terminal velocities are accurate within
1.5% for a particle Reynolds number (Re) less than 0.1.
It has been
observed that the container walls of the solid-fluid system oppose the
motion of a settling particle and so the terminal falling velocities
in a finite fluid medium are always lower than those in an infinite
medium.
Several theoretical and empirical correlations have been sug-
gested for the wall correction of terminal falling velocities.
Filderis
and Whitmore (1961) reported an extensive experimental study of the wall
effect for spheres over the Reynolds number range between 0.05 and
2 X 10 . Happel and Brenner (1965) discussed various theoretical equations describing the wall effect in the viscous region (Re < 0.2). For
Reynolds number less than 0.2, an equation developed by Francis (1933)
appears to be the most satisfactory.
A simple and quite accurate equa-
tion was reported recently by Garside and Al-Dibouni (1977) over the
Reynolds number range between 3 and 1200.
Stoke's Law overpredicts the terminal falling velocity by about
3.0"-^ when the particle Reynolds number is around 0.2.
A closer ap-
proximation was achieved by Oseen (1910) and by Goldstein (1929) up
to Reynolds number of 20. Terminal falling velocities at higher
Reynolds numbers can be calculated using an experimental plot of drag
coefficient versus Reynolds number.
This method involves a trial and
error computation as both coordinates contain the terminal velocity as
an unknown variable.
The terminal velocity can be eliminated from one
o
2
of the coordinates by using the Galileo number (d p^(p -P^)/LI.|: ). Zenz
(1957), Jottrand (1958) and Zenz and Othmer (1960) have constructed
"I/O
7
"[ I '\
plots of (ReCj^)^^ versus (Re Cpj)'^"^ to avoid trial and error
tions.
calcula-
Equations developed by Davies (1945) and by Turian et al. (1971)
can be used to predict terminal velocities with little loss in accuracy.
No satisfactory treatment has been developed to predict terminal
falling velocities of non-spherical particles but the problem has been
considered by several investigators.
The first important development
was reported by Heywood (1938), who published results covering the
range of particle Reynolds number between 0.01 and 1000.
Non-sphericity
was taken into account by introducing a volume coefficient.
Pettyjohn
and Christiansen (1948) made careful measurements of the settling rate
of a number of well defined isomeric bodies in the region of streamline
flow and showed that Stokes' Law could be extended to cover nonspherical particles.
Kunkel (1948), on the other hand, investigated
the magnitude of errors introduced by calculating the settling rate of
dust particles from Stokes' Law.
Gurel et al. (1955) carried out an
extensive investigation by gradually changing the shape of particles
from spherical to cubic and from cubic to cylindrical.
They claimed
that their equation predicts the terminal falling velocities within an
error of + 2%.
A theoretical work was reported by Becker (1959). who
claimed that his equation showed a minimum dependence on particle shape
and Reynolds number.
Hottovy and Sylvester (1979) conducted a study to
measure the terminal falling velocity of roundish but irregularly shaped
particles over the particle Reynolds number range between 7 and 3000.
Particles passing through a screen, but retained on the next size smaller screen, were assumed to have an average particle diameter equal to
the average of the opening width of the two screens.
Settling veloci-
ties agreed with those of spherical particles for Reynolds number less
than 100.
For Reynolds number ranging from 100 to 3000, the settling
velocities were lower than those of spherical particles and the error
introduced was as high as 50% at Reynolds number of 3000.
Sedimentation velocities of particles in concentrated suspensions
are considerably lower than their terminal falling velocities under
free settling conditions.
as hindered settling.
Such behavior of settling is primarily known
The decrease in rate of sedimentation of parti-
cles is due not only to the interference of neighboring particles but
also to the appreciable upward flow of displaced fluid.
An earlier study on sedimentation was carried out by Coe and Clevenger (1916), who concluded that a concentrated suspension may settle in
two different steps.
In the first, after an initial brief accelera-
tion period, the interface between the clear liquid and the suspension
moves downward at constant rate and a layer of sediment builds UD at
the bottom of the container.
When the interface is closer to the sedi-
ment, the rate of fall of the interface decreases rapidly until a direct interface between sediment and clear liauid is formed.
Further
sedimentation then results solely from a consolidation of the sediment,
with liquid being forced upwards around the solids which are then forming a loose bed with particles in contact with one another.
A number of attempts were made to predict the apparent settling
velocities of a concentrated suspension.
Robinson (1926) suggested a
modification of Stokes' Law and used the density and viscosity of the
suspension in place of the properties of the fluid.
Steinour (194^),
who studied the sedimentation of sm.all uniform particles, adopted a
similar approach, using the viscosity of the fluid, the density of the
suspension and a function of suspension voidage to account for the
shape and size of the flow spaces.
Hawksley (1950) also gave an ex-
pression for the rate of sedimentation, based on the assumption that
lateral forces produce a more or less uniform spacing in a horizontal
plane, and that the particles were presumed to arrange themselves in
such a way that they offered the minimum resistance to the upward flow
of displaced fluid.
Many investigations were reported in the last three decades and
several theoretical, semitheoretical, and empirical correlations were
suggested.
Most of the correlations could predict the sedimentation
velocities over a small range of Reynolds number and the predicted velocities showed fair agreement with experimental results.
Jottrand
(1952) developed a simple empirical correlation which could predict
sedimentation velocities for particle Reynolds number up to 0-4 but
showed significant lower velocities at all voidage.
Lewis and Bower-
man (1952) gave two simple empirical equations, one over the particle
Reynolds number range between 2 and 500 and the other for the particle
Reynolds number greater than 500.
Both equations showed ooor agreement
with experimental results.
Until the work of Richardson and Zaki (1954), no correlation was
available to predict sedimentation and fluidization velocities over an
entire range of Reynolds number.
They carried out an extensive investi-
gation on sedimentation and fluidization of various sizes of equisized
particles using several suspending media.
Unlike Robinson (1926),
Steinour (1944), and Hawksley (1950) who used the density and viscosity
of the suspension, Richardson and Zaki used the density and viscosity
of the suspending fluid in their correlation.
The latter consists of
six equations which cover the entire range of Reynolds number.
These
equations are perhaps the most widely used and predict the correct trend
in behavior for all flow regimes.
They do, however, predict signifi-
cantly higher velocities for all flow regimes and suffer from discontinuities at the transition Reynolds numbers.
Happel (1958) developed a mathematical treatment on the basis that
two concentric spheres can serve as a model for a random assemblage of
spheres moving relative to a fluid.
The inner sphere comprises one of
the particles in the assemblage and the outer sphere consists of a fluid
envelope with a "free surface".
Velocities predicted from this model
were substantially low at low particle concentrations.
A theoretical
relationship between the concentration and the sedimentation velocity
of a suspension of particles was developed independently by Maude and
Whitemore (1958) and by Zuber (1964).
Similar but somewhat nore comolex
equations for the low Reynolds number range were suggested by Loef^ler
and Ruth (1959) and by Oliver (1961).
Wen and Yu (1956) suggested a
correlation which could predict velocities over the Reynolds number
4
range between 0.01 and 10 . At high concentrations, predicted velocities were significantly high at all values of Reynolds number, and at
low concentrations, they were high at low Reynolds number and low at
high Reynolds number.
Lapidus and Elgin (1957), Struve et al. (1958)
and Price et al. (1959) carried out an extensive investigation on
countercurrent operations, cocurrent operations and fluidization.
They
showed that the data for cocurrent and countercurrent flow were in excellent agreement with the operating diagram determined from the holdupslip velocity (relative velocity between solid and fluid) relationship
obtained from the batch fluidization experiments.
They concluded that
all those systems in which relative motion takes place between fluid
and suspended solids showed similar operating behavior.
Barnea and Mizrahi (1973) collected published experim.ental data
from twelve different sources and developed a correlation for predicting
sedimentation velocities in all flow regimes.
Although it was complex
and difficult to use, the Barnea-Mizrahi correlation gave the lowest
error compared with all previous correlations.
Letan (1974) extended
the work of Zuber (1964) for higher Reynolds numbers and developed a
semitheoretical correlation which showed good agreement with the
Richardson and Zaki correlation.
The latter, as mentioned earlier,
predicts significantly higher velocities at all Reynolds number.
On the
other hand, the equation developed by Wen and Fan (1974) predicts consistently low values of velocities at all voidages.
Bedford and Hill (1976) provided a theoretical justification for
the use of the slip velocity in transient one-dimensional particulate
8
sedimentation.
This fact has not always been recognized in previous
sedimentation investigation and, even when recognized, has often been
included empirically.
Garside and Al-Dibouni (1977) collected a large
number of published data from various sources and developed a correlation based on a logistic curve, which, in their case, is a Graphical
presentation of slip velocity versus Reynolds number with voidage as a
parameter.
Their correlation showed a significant improvement over
the Richardson-Zaki equation and it is also easier to use than the
Barnea-Mizrahi equation.
Velocities predicted from the Garside-
Dibouni correlation showed an absolute average error of 8.5%, which is
the lowest among all correlations.
In practice, most solid-fluid operations involve solid particles
of various sizes.
In the limit, any suspension can be regarded as
being composed of a very large number of closely-sized fractions.
In
order to develop a sedimentation correlation for such systems, mixed
suspensions consisting of two or more distinct measurable particle
sizes rather than a spectrum of sizes could be used.
No systematic
study of such systems had apparently been made until the work of Hoffman et al. (1960), who studied the fluidization of binary and ternary
mixtures of glass beads.
They observed sharp size segregation in bi-
nary mixtures of size ratios from 1.58 to 2.23 and partial segregation
in mixtures of size ratio 1.24.
Smith (1965, 1966, 1967) presented a
theoretical analysis on sedimentation of binary particle mixtures in
very
slow flow.
He developed a physical model for the sedimentation
of binary mixtures by extending the spherical f^uid envelope model
develooed by Happel (1958) for the sedimentation of single size
particles.
The settling velocities predicted from this model were,
however, significantly lower compared with the experimental results.
Using glass spheres (regular shape) and quartz powder (irregular
shape), Davies (1968) provided an extensive set of experimental data on
the sedimentation of binary and ternary particle mixtures.
In an at-
tempt to reproduce Davies' experiments, Lockett and Al-Habbooby (1973)
found that it was not possible to achieve an initially uniform suspension due to the high settling rates encountered.
As a result, Davies'
experimental data on the sedimentation of binary and ternary particle
mixtures appear to be suspect.
Finkelstein et al. (1971) carried out
an experimental study on mixed sized particles in a fluidized continuous system.
Their studies illustrated the validity of the unique
characteristic holdup-slip velocity relationship formulated by Lapidus
and Elgin (1957) for systems of single-size particles.
Lockett and Al-Habbooby (1973) carried out an extensive experimental study on countercurrent opev^ations and sedim.entation of binary
particle mixtures of two distinct particle sizes.
Using the Richardson
and Zaki equation (1954) for single-size particles, they developed a
physical model for sedimentation of binary mixtures assuming that a
particle settles only according to the local voidage fraction areund
it, irrespective of whether its neighbors are particles of the same or
of another size and whether they are moving relative to it or not.
The model showed good agreement with the experimental results for the
initial settling velocities.
Since initial settling velocities are
considerably higher than average settling velocities, their correlation
cannot be used to predict average settling velocities.
Recently '*irza
10
Richardson (1979) carried out an experimental study on the sedimentation
of suspensions of two distinct particle sizes. Their experimental sedimentation velocities were compared with those predicted from the Lockett
and Al-Habbooby model.
It was found that the predicted velocities for
both interfaces were overpredicted in almost all cases by between 5 and
50%.
Introducing a correction factor into the Lockett and Al-Habbooby
equations, they were able to match their experimental and predicted results to within j;^ 10%. Clearly, such an approach is highly empirical
and cannot be applied over a wider range of conditions than those used
in their experimental work.
This being the case, Mirza and Richardson
suggested that a further investigation on the sedimentation of binary
suspensions is required.
The purpose of the present study is fourfold:
(1)
to develop a physical model for the sedimentation of binary
suspensions of two distinct particle sizes,
(2)
to compare the model with experimental results obtained here
and elsewhere,
(3)
to extend the model to suspensions of multisized particles
(suspensions with three or m.ore distinct particle sizes) and to test
the model developed against experimental results,
(4)
to extend the analysis to systems with a continuous particle
size distribution.
CHAPTER II
SEDIMENTATION OF EQUISIZED PARTICLES
There are wide discrepancies between the prediction of the many
published correlations for the velocity-voidage relationship observed
during fluidization and sedimentation in solid-liquid systems.
Barnea
and Mizrahi (1973) have discussed some of the difficulties involved
in describing the characteristics of multiparticulate systems and in
particular in consolidating into one correlation results from batch
and continuous sedimentation, fluidization and co- or counter-cur-rent
vertical two-phase flow.
In analyzing sedimentation operations, most
investigators have concentrated on developing expressions for the
settling velocity of the particles in terms of the properties of the
particles and fluid and the volumetric concentration of particles.
Not all, however, have recognized the following fundamental features:
Hydrostatic Effect:
Richardson and Meikle (1961) have shown ex-
perimentally that the effective buoyancy force acting on an individual
particle in a suspension is greate>^ than the buoyancy force exerted
by the fluid alone.
If p , p^ and p
represent the densities of
particles, fluid and suspension, respectively, the following expression for the driving force may be written
Driving force per unit mass of ^ /, _ ^ ^
particle in the fluid alone
-^^''p ' ' f^
(? ])
\ - i
Driving force per unit mass of
/
\
-c(o-c)
particle in the suspension
-^ p
s
= gc(o_. - p j
P
11
I
(2.2)
12
where p
can be expressed as
P3 = Pp(l-£) + p^z
(2.3)
£, the voidage, is the volume fraction of the fluid in the suspension.
It is clear then that the density of the suspension p , should be used
in the expression for the driving force in particulate sedimentation.
However, as Equation (2.2) indicates the final expression for the
driving force in a suspension is related to (p
- p^).
It is instruc-
tive to note that the dynamic pressure relevant to the inertial forces,
should still be related to the fluid density p^, since the particles
are moving all together relative to the fluid and not relative to the
suspension.
Thus p^ should be used in the expression of the Reynolds
number and the drag force.
Wall Hindrance:
While it is reasonable to neglect wall effects
in large-scale equipment, it might not be acceptable in laboratoryscale equipment.
Significant wall effects are detectable even when
a single particle is settling in a vessel whose diameter is larger
than the particle size by one or two orders of magnitude.
Since
the settling velocity of a particle in suspension is related to the
terminal free-falling velocity, wall effects in a sedimenting suspension can be taken into account by applying wall correction to the
terminal velocity of the free-settling particle.
Slip Velocity:
In particulate sedimentation, particle drag is
governed by the velocity of the particle relative to the ""luid, U , and
not by U , the velocity of the particle relative to the container.
While this effect may not be important in a very dilute suspension
13
due to the low velocity of the displaced fluid, it is highly significant in concentrated suspensions.
This fact has not always been re-
cognized in previous sedimentation research and, even when recognized,
has often been included empirically.
The extensive study of Meters
and Rhodes (1955) and the work of Lapidus and Elgin (1957) have shown
experimentally that only U^ can be used to compare and correlate data
in sol id-fluid operations.
A theoretical justification for the use
of the slip velocity U was eventually given by Bedford and Hill (1976)
Consider the sedimentation under gravity of solids particles of
uniform composition, size, and shape through a fluid held in a container of constant cross section.
The downward direction is indicated
by the positive x axis.
Regarding the particles and fluid as two superimposed continua,
in the absence of reaction, dissolution, or other mass transfer processes, each constituent must staisfy the usual one-dimensional conservation of mass equation
^'h%V-^
(2.4)
^-y^fV-°
(2.5)
where p
= mass of solids per unit volume of suspension
P
P^ = mass of fluid per unit volume of suspension
U
c
= velocity of particles relative to the container
U^ = velocity of fluid relative to the container
The partial densities p
and Cr can be expressed in terms of the den-
sities of the solid and fluid and the volumetric concentration as
14
where p
^p = CPp
(2.6)
Pf = (l-C)p^ = cp^
(2.7)
= density of the solids
p^ = density of the fluid
C = volumetric concentration of particles
e = voidage
Substitution of Equations (2.6) and (2.7) into Equations (2.4) and
(2.5) gives
h^%^' h^%V - °
(2.8)
|t(£Pf) + |^(£PfUf) = 0
(2.9)
Assuming solids and liquids to be incompressible. Equations (2.8) and
(2.9) can be written as
i^ll(%> = °
(2-10)
|| + |^(.V
(2.11)
=0
For a single particle moving in a suspension, the forces acting
on the particle in the x direction are the drag, gravity and buoyancy.
Let f. be the drag force which is a function of the relative velocity
between the particle and fluid U -U^, and the concentration of the
suspension C.
Then the net force F on the particle can be expressed
as
F = -fd(C, U^-U^) + p Vg - p^Vg,
(2.12)
15
where p
s
= density of suspension which is (p
p
+ 6^)
f
V = volume of a single p a r t i c l e
g = gravitational
acceleration
The force on the p a r t i c l e s per u n i t volume of mixture is ( Y ) F . Applying Newton's Second Law to p a r t i c l e s in a unit volume of suspension,
PpSp = - ( v ) f d ' C , U^-U^) + Cg(pp - p^)
(2.13)
where the particle acceleration a is given by
9U
3U
Equations (2.10), (2.11) and (2.13) with Equation (2.14) provide
three equations in the three variables C, U and U^.
E is not an independent variable as e = 1-C.
Here note that
Adding Equation (2.10)
and (2.11)
|^(C.e).|3^(CU^.cU,) = 0
(2.15)
Since C + e = 1, the first term on the left-hand side of Equation
(2.15) drops out giving
|-(CU + cU.) = 0
3x c
f
(2.16)
The term (CU + eUr) is the volume of the mixture passing a point x
per unit time per unit area.
For sedimentation, this volume is zero,
so that Equation (2.16) yields
' - - ^ ^
(2.17)
16
Using Equation (2.17) to eliminate U^ from Equation (2.13) and using
Equation (2.14) we get
3U^
5U
^ P ^ a r ^^cW-^
= -(v'^d'C' ^c 'iTc^c^
^ ^9(P. - 0^)
(2.18)
Simplifying using Equation (2.2), we obtain
9U
3U
^p^9t- ^^cW-^
,
= - Y ^d(C' T ^ ^^c' ^ 9(Pp - P^)(l-C)
(2.10)
Equations (2.10) and (2.19) have two unknowns U and C, Getting f ,
experimentally or empirically, these two equations can be solved tor
transient sedimentation problems.
3C
For steady sedimentation ^ becomes zero and Equation (2.10)
leads to
dU
dx ^ = 0
(2.20)
Substituting Equation (2.20) into Equation (2.19) and noting that
aU /8t = 0, we get
^df^' 1 ^ ^c' ^
'^9(PD
- P^)(l-C)
(2.21)
This merely indicates that the draa and buoyance ^orces are balanced
by the weight.
This equation clearly indicates that particle dnaq
in suspension is governed by the oarticle velocity relative ^.o the
fluid
U, - ^f-T^^
(2.22)
Use of this slip velocity is essential when various sol id-liquid operations are to be out on a common base.
17
Dimensional Analysis
Dimensional analysis is of prime importance in any transport operation.
It reduces a large number of independent variables to a few
number of dimensionless groups.
Dimensional analysis of sedimentation
was first studied by Richardson and Zaki (1954).
The following assump-
tions are common in the study of sedimentation of equisized spherical
particles:
(1)
The suspension consists of spherical particles, of common den-
sity, with a relatively narrow size distribution which can be reduced to
an average characteristic size.
(2)
There are no interactions of any kind between the particles,
except hydrodynamic effects through the fluid.
Flocculation and aggrega-
tion are assumed not to be present.
(3)
The relative positions of the particles in the suspension are
completely random, without any segregation.
This assumption comes closer
to reality with narrow size distributions and more concentrated suspensions.
The drag R. per unit projected area of a spherical particle settling
at its terminal falling velocity U^^ is a function of the density and
viscosity of the fluid, the diameter of the particle d, the terminal
falling velocity U. , and the ratio of the particle diameter d to the
container diameter D.
R^
=
Thus,
f(p^, y^, U^^, d, ^)
For the isolated spherical particle.
(2.23)
18
RjTr/4)d^
= (7T/6)d^(Pp - p^)g
(2.24)
For the sedimentation of a particle in a suspension, R. will alter to R|
because the upthrust is equal to the weight of displaced suspension.
Thus
R[(Tr/4)d2
= (TT/6)d^(Pp - p^)g
3
= (7T/6)d £(p - p^)g
(2.25)
From Equations (2.24) and (2.25) it follows that
R;
= cR^
(2.26)
R' is a function of the v e l o c i t y of p a r t i c l e r e l a t i v e to f l u i d U , and is
also a function of the voidage, e, which determines the flow patterns and
the area available f o r the flow of the displaced f l u i d .
K
f(p.f:> Ufy U^, d , £ , ^)
Thus,
(2.27)
Equations (2.23) and (2.27) may be rearranged to give the f o l l o w i n g exp l i c i t equations f o r U.
U
t^'
U^
=
and U
f(R^, P^> y.p» d, ^)
(2.28)
f ( R [ , p^, y^, d, e, -^)
(2.29)
Since R' = sR., Equation (2.29) may be written as
^s
^
^'^t' ^f
^f ^' ^' D^
Dividing Equation (2.30) by Equation ( 2 . 2 8 ) , we ootain
(2.30)
19
u,
u
(2.31)
f ( R ^ , P^, 1-1^, d , £ , -^)
t°°
Since U^/U^^ is dimensionless, the right-hand side of Equation (2.31)
should also be a function of dimensionless groups.
The quantities R. ,
P.p, y^ and d can be arranged to form the dimensionless group
2
2
R^cl p^/y^ . Therefore
U.
= f ( ^ y ^ , c, §)
U
(2.32)
t^'
Now,
R^d2p.
-)(^^-^)2
(
(2.33)
y^r
P-pU?
f t°o
y.
Here R./p^U.
U. d p - ^
R,
is a resistance c o e f f i c i e n t which is a unique function of
the Reynolds number U. dp^/y^ f o r a spherical p a r t i c l e .
U.
=
Ut«'
U^ cip.
,
f(- t"' f ' ^' d
D'
y.
Thus
(2.34)
The above derivation does not take into account the nature of the flow
past the particles in suspension.
Further simplification can be obtained
by considering various flow regimes.
The Viscous Regime:
In the Stokes' Law range the relative velocity is
sufficently low for inertial effects to be negligible so that the whole
resistance may be attributed to skin friction.
R:
t
oc y u d
^f s
Under this condition,
20
R'^ being independent of p^, the density of the fluid, Equation (2.31)
becomes
U^
= f(yf, d, R^, e, ^)
(2.35)
Here y^, d, and R^ cannot be arranged to form a dimensionless group
and therefore U^/U^^ must be independent of these variables under condition of streamline flow.
This leads to
i r - = f(e,~)
t°o
The Turbulent Regime:
(2.36)
At high velocities the effect of the inertia of
the fluid becomes important and the viscous forces are negligible.
Under this condition Newton's law is applicable and
R; - PfU^^
(2.37)
The resistance per unit area is thus independent of the viscosity of
the fluid and the diameter of the particle except insofar as the latter
influences the effect of the wall.
Equation (2.31) therefore becomes
^s
d
(/- = f(Pf, R^, e, §)
(2.38)
Again p^ and R. cannot be arranged to form a dimensionless group and
therefore
^s
U^=
d
f{e, §)
(2.39)
21
Thus, when either skin friction or form drag is predominant, U /U.
is a
function of e and -^ as shown in Equations (2.36) and (2.39), respectively
When both drag and skin resistances are of comparable magnitude, the
Reynolds group is significant in addition to e and -^ as indicated in
Equation (2.34).
In sedimentation experiments the velocity of the particles relative
to the container U is measured.
This velocity is related to the slip
velocity according to the expression
U^ = eU^
(2.40)
Substitution of U in place of U into Equation (2.34) gives
U,
U. dp.
t°o
.
f
Experimental results on sedimentation showed a linear relationsnip
between log U and log e. A typical plot is illustrated in Figure 2.1
which shows a straight line with a slope n and intercept U. corresponding to infinite dilution (£=1.0).
Thus
log U^ = n log £ + log U.
(2.42)
or
U^
= £"
(2.43)
^•
In Equation (2.43) U- is the sedimentation velocity of suspension in a
tube of diameter D at infinite dilution.
Thus U^ can be taken as a
terminal falling velocity of a particle in a tube.
Therefore,
—
1 1 :
O '
1
O7
i1 ' ' 1i
t
t
t
1
u
o
o
d 1
r
i
i
O1
y
1
1
1
•'!
1
i
'
1
'
i
^
X1
y^
:
'
'
i IX 1 : : 1 M
O "^
X
i ^ 1 !
1
1 i
'
^:•
:1
\
-O i* -O 22 -C 2 -OI8 -C i<3 -0"S -Oi2 -O O C Zs-OCb-
ZZ'*-'^Cl
Log £
'igure 2 . 1 .
Relation between rata of sedir';entation
and voidage of suspensions.
(2.ia)
f(§)
'J. 00
From Equations ( 2 . 4 1 ) , (2.43) and (2.44)
n
.>'t-^^f
_
d.
(2 ^S''
Noting that n is independent of £,
'^.-
11
00
n =
f(
d^
i..^D
•' D^
•f
In the viscous and turbulent "'lew regimes, U^/U^^^ 'n —uaticn i_.-:! ::ecomes indepencent of Reynolds number 'J,
(2.46) simplifies to
' - s/
-.
As a r e s u l t Ecuaficn
n
=
f(^)
(2.47)
This behavior has been observed by most investigators in t h e i r e x p e r i mental work.
For low and high Reynolds numbers, n was found to be func-
t i o n of -p as shown in Equation (2.47).
For intermediate Reynolds numbers
i t was found to be a function of Reynolds number and ^ as shown in Equation (2.46).
In some instances, however, i t was found that n varies i n -
s i g n i f i c a n t l y with -p so t h a t n becomes independent of ^ .
Sedimentation
v e l o c i t i e s may be predicted from Equation (2.43) in conjunction with
Equation (2.46) or (2-47).
The v e l o c i t y U. in Equation (2.43) may be
obtained from the terminal f a l l i n g v e l o c i t y U.
a f t e r applying awal1 cor-
r e c t i on.
Published Correlations
There are many correlations in the l i t e r a t u r e that aim to predict
the sedimentation or f l u i d i z a t i o n v e l o c i t i e s in s o l i d - f l u i d systems.
The
m a j o r i t y of these apply over a r e s t r i c t e d range of Reynolds number
although some have been suggested as being s a t i s f a c t o r y f o r a l l
regimes.
Table 2 . 1 .
flow
Most of the published correlations have been summarized in
Predicted v e l o c i t i e s by most of these correlations have been
compared by Garside and Al-Dibouni (1977).
The correlations suggested
by Richardson and Zaki (1954), Barnea and Mizrahi (1973), and Garside
and Al-Dibouni
(1977) are by f a r the most important.
Richardson and Zaki (1954) were the f i r s t to develop a c o r r e l a t i o n
which was able to predict sedimentation and f l u i d i z a t i o n v e l o c i t i e s in
a l l flow regimes.
They c a r r i e d out sedimentation and f l u i d i z a t i o n exper-
iments in various suspending media using d i f f e r e n t - d i a m e t e r tubes, and
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25
various spherical particles of different size and density.
Id.':-^' set
of data when plotted as logU^versus log £p qave a straight ''ne wit.'
slope n. Tne exper''mental values of n were plotted against ^ witn
Reynolds number as a parameter as shown in -igure 2.2.
A single
curve represents the data for all values of Reynolds numbers less
than 0.2.
The figure also shows another sinqie curve ~jr all values
of Reynolds numbers greater than 500.
"or in^^ermediate values 3^
Reynolds number, n was found to be a function of Re^.
is in agreement with Equations (2.46) and (2.^7).
This analysis
From analysis o*
t'^e curves, the following equations were obtained
n = 4.55 + 19.5 ^
0 < Re < 0.2
X)
0.2 < Re < 1.0
n = (4-35-17.5 ^)Re;'^-^^
(2.^9)
1.0 < Re < 200
n = (4-45 + 18 ^)Re;^-^
V- •0)
!-4
:oi
:c4
d/3
^icure 2.2.
S>pe n as a ^"unc'icn
r/1
26
200 < Re^ < 500
n = 4.45 Re^^'^^
(2.51)
Re^ > 500
n = 2.4
(2.52)
These values of n may be used in Equation (2-43) to predict the settling velocity.
This set of equations is perhaps the most widely used and predicts the correct trend in behavior for all flow regimes.
However,
predicted velocities are consistently high and show an absolute average deviation of about 20%. Also, the predicted relationship between velocities and Reynolds number shows discontinuities at Reynolds
numbers of 0.2, 1.0, 200 and 500.
Barnea and Mizrahi (1973) carried out an extensive investigation
on sedimentation and fluidization.
It was found that the transition
points between the laminar and intermediate and between the intermediate and turbulent regions were dependent not only on Reynolds number
but also on the concentration of the particles.
Poth effects were
taken into account by introducing a modified Reynolds number Re^ which
includes the slip velocity U and viscosity of the suspension y^. Experimental data were collected from twelve different sources and were
plotted as (Re^/Cp,^)^'^^ versus (Re^^C^^)^'^^, where
Re, = ^ _ 1 = ReJ-__^|^-^)
•(D
U3
and
4d(p
*
- p.)g
3y^ U.
.r
1+C
(2.53)
27
The plot is shown in Figure 2.3.
For convenience this relationship was
also made available in an algebraic form.
For a single particle moving
in a fluid C^ was found to be
Cn
^
=
(0.63 +-~±^)^
/ R ooe ^
(2.55)
By analogy, a similar equation was obtained from the curve for multiparticle systems.
This equation may be written as
Cnd, =
from which U^/U.
Re or U
(0-63 + - M ) ^
/R^
(2.56)
can be obtained algebraically as a function of C and
as a function of p^, u^, p , d and C.
This correlation predicts better results than the Richardson and Zaki
equations but it is complex and difficult to use.
regimes although the predicted values of U
It covers all flow
are consistently low for
Reynolds numbers less than about 300 and consistently high for Reynolds
number above this value.
This trend can be seen in Figure 2.3.
Garside and Al-Dibouni (1977) collected a large number of published
experimental data and plotted them as Reynolds number Re = (dU.po/iJ^)
against U /U. with the voidage £ as a parameter.
Here U. is the terminal
falling velocity of a single particle in a finite fluid medium.
The
curves obtained had the shape of a 'logistic' or growth curve (Wolfenden,
1942) as shown in Figure 2.4.
These curves may be described by the
equation
U
-^ - A
^t
-^—[j-
'•i
=
K Re-
(2.57)
23
•r1
.r
' 5
, ^ '
v^»^-v."
tr^
y
'J*
i7i
<m
o
/ '
^
W
O)
7
/'
I
.L
ZD
I
A
10
0.1
. >^V.y
"^ .^ — .'^
•aur?
J.
.^enerai cor^-eia'ion "c tne re i a t i ve ve ' oc: -.y
1igui a systems, basec on extended ce-'ni:^:cns
ana Cr>,.
29
.J
'
(/)
Re
"igure 2.4. A growth curve showing 'Relationship between 'J /'J
s' ^t
and Re at several values of £.
A and B are the asymtotic values of U /O^ at low and high /alues of Re,
r e s p e c t i v e l y , while the position and rata of increase of U^/0, in the in
termediate region were determined by the constants .< and : .
~ne oest
values of A, B, K and z were deterTiined for voidages of 3.5, 0.-:, 0 . ^ ,
C.3, 0.9 and 0.95.
a function of £.
Equations were then develooec for - , 3, K anc : ss
These equations iv.ay be l i s t s o as
""
3
=
^
4.14
~
.
J
^
0. 8£
9
n^
'?'=l
c o
f o r -: <_ 0.35
f o r £ > 0.35
('
\ -
55)
30
K = 0.06
(2.61)
z = £ + 0.2
(2.62)
Substituting K and z into Equation (2.57), we get
U.
r- - A
- ^ p
= 0.06 Re^"0-2
(2.63)
When compared with the experimental results, Equation (2.63) showed an absolute average deviation of 8.6%.
Since U^^/U^ = £
is perhaps the most widely used equation for
predicting veolcities of solids in fluidization and sedimentation, an
equation for predicting values of n as a function of Re was also
developed.
This equation may be written as
^'1 2 y = 0.1 Re°-^
(2.6^)
where 5.1 and 2.7 v/ere the asymtotic values of n at low and high
values of Re, respectively.
Although Equation (2.64) represents ex-
perimental results with an accuracy comparable to that achieved with
Equation (2.63), the behavior at high voidage (greater than about 0.9)
is not well represented.
Compared to all correlations. Equations (2.63) and {2.6^)
'vere
found to be the most accurate and are, therefore, recommended for
use in further investigation on sedimentation and fluidization.
CHAPTER III
SEDIMENTATION OF MULTISIZED PARTICLES
There seems a little prospect at the present time of dealing with
the hydrodynamics of binary particle-liquid suspensions in a fundamental way.
Even when the particles are all identical little progress
has been made; except for the very
slow flow regime, and for higher
values of Reynolds number we are dependent upon empirical correlations
such as that of Richardson and Zaki or of Garside and Al-Dibouni.
It
is therefore unrealistic to hope for an analytic solution to the problem for a binary particle mixture except perhaps for the very
slow
flow regime and this has been dealt with by Smith (1965, 1966, 1967)
with a little success.
The problem becomes even more complex when
more than two sizes of particles are sedimenting in a suspension.
Segregation of particles has been observed during the sedimentation of suspension of two or more sizes cf particles.
Distinct sedi-
menting zones are formed due to complete segregation of particles
when the size ratio of the two closest sizes of particles is greater
than 1.6.
Partial segregation without distinct zone formation has
been observed even at a particle size ratio of 1.19.
When a uniform
suspension of two distinct sizes of particles starts settling, the
segregation gives rise to tv;o sedimenting zones; a lower zone in which
particles of both sizes are settling and an upper zone in which only
smaller size particles are settling (see Figure 3.1). Sedim.entation
of a suspension of three sizes of particles gives rise to three distinct zones of sedimentation, the lowest where all the three sizes of
31
32
E
<v
"=i
-o
<v
oo
E
o
CO
E
CD
C
CO
E
c/1
CD OJ
C N
•r— •^
e (/)
sro
•1—
<X3r^
-M 1 — CO
E fO O)
O E p—
U CO (J
S_
-1-
^3 Z3
<V CT
C_J
03
+->
o
^vl
I—
O
C
Ol
<v 1>^-M
.sz E— SfC
E
o
C". c/^
E CD
•r— 1 —
o.
U
r
<zr> 1 —
cr.^ <D
<V
e
•^- fT3 r—
c E U
•r— in
n3 p ^
+J r— CO
C fC O)
-l-J
fO
+-> -o sE E fC
O <o Q .
•1^
T3
S- -rfT3 Z3
<V c r
r— "r—
O
I—
CO
N
•^
CO
c
c
•^-
1 —
u
1 —
u
OJ > ) 4-)
c;
-a
CD
CO
-a
• I —
o E
CO
O
rvi
+-> • 1 —
-(-> +J
CD sCO <T3
£2.
CD
4-> CD
<D CT)
1 —
i.
C 03
1
c —
O
O
03
-M
E
CD
e
o
s-
03
Q.
u
QJ CO
OJ ^-/. O)
E S- N
fO • 1 —
o
^sj p — CO
+j
4-)
E
CD
CO
CJ) CD
E
E
OJ
•r-
O
00
E
SO
M-
M
cr
E
c:
%C
4 I
CD
C
o
Ni
"O
E 4-)
C
E
• , - OJ
-M E
03
fO • - -
O
cr-T3
CD
CD
s- -c
en
CD
OO
E
O
s
03
cm
CD
S-
cr
CD
OO
'^
E
o o
r03
•rJ->
•-E
E
CD
(_)
E
O
CJ
33
particles are settling, the middle where medium and small sizes of
particles are settling, and the top where only small particles are
settling.
The samie physical phenomenon occurs in a suspension of m
sizes of particles, segregating and giving rise to m sedimenting zones
Interpretation of experimental behavior on sedimentation of binary particle mixtures was first suggested by Lockett and Al-Habbooby
(1973).
They developed a physical model which is based on the assump-
tion that a particle settles only according to the local voidage fraction around it, irrespective of whether its neighbors are particles
of the same or of other size and whether they are moving relative
to it or not.
In developing the model, they used the equation
U^ = U ^ d - O "
(3-1)
which was given by Richardson and Zaki (1954) for uniform spherical
particles.
Rather than using the equations developed by Richardson
and Zaki for predicting the exponent n in Equation (3.1), Lockett and
Al-Habbooby measured n experimentally for each size of particles.
Mirza and Richardson (1979) used the sam.e model for binary systems
and extended it to multisized particle systems.
They used Equation
(3.1) in their model with n values given by the Richardson and Zaki
equations.
Two remarks should be mentioned regarding this m.odel.
These
are:
(1)
Values of the exponent n in Equation (3.1) can be predicted
using generalized correlations.
Instead, Lockett and Al-Habbooby
measured n values experimentally as -^.64, 5.41, 5.14 and 5.07 for
34
four different sizes.
Corresponding values given by the Richardson and
Zaki equations are 4.75, 4-77, 4.82, and 4.92, respectively.
It has
been reported by several investigators that values for the exponent n
remain essentially constant rather than varying drastically as measured
by Lockett and Al-Habbooby.
In general, it is not recommended to mea-
sure n values by taking few experimental observations since reliable
correlations are available for their prediction.
Mirza and Richard-
son predicted n values using Richardson and Zaki (1954) equation which
predicts consistently lower values of n and thus considerably higher
values of settling velocities are obtained.
(2)
The assumption that the settling velocities of larger par-
ticles are unaffected by the sizes and velocities of surrounding
particles is physically unacceptable and results in higher settling
velocities of the larger particles.
It has been observed when a large
particle is settling in a suspension of smaller size particles that
it is displacing not only a fluid but also the smaller size particles.
Thus sedimentation velocities of large particles in a multisized particle system are lower compared to their velocities in a suspension
of their own kind at the same total concentration.
It is clear,
therefore, that inter-particle interactions cannot be ignored in
multisized particle systems.
These interactions may be accounted for
by taking into consideration the buoyancy effect of all particles
with sizes less than i on the terminal falling velocity of particles
with size i.
It is instructive to note that the use of Equation
(3.1) implies that the settling rates are not influenced by interparticle collisions which become increasingly important as the velocities of the particles increase.
35
Sedim.entation of Suspensions of Uniform Particles
The settling velocity of particles in a suspension of uniform
spherical particles is given by Equation (3.1).
The exponent n may
be predicted from the equation given by Garside and Al-Dibouni (1977)
~ ^ =
0.1 Re°-^
(3.2)
d U p.
where Re =
The upward velocity of the fluid U^ can be calculated by equating the
flow rate of solids and the flow rate of fluid at any cross section
of the settling zone. This gives
U^£ = U^C
(3.3)
U,C
U. = - ^
(3.4)
or
f
£
The r e l a t i v e v e l o c i t y between s o l i d and l i q u i d U^ is given as
U = U + U.
s e t
(3.5)
Substituting for U. from Equation (3.4) into Equation (3.5) and simplifying,
U^
U = -^
S
(3.6)
£
Combining Equations (3.1) and (3.6), we obtain
n-1
U^ = U^(l-C)'
= U/"^
(3.7)
36
Use of Equation (3.7) is required in the analysis of systems with two
or more sizes of particles.
Sedimentation of Particles in a Binary Suspension
In general, a suspension of two distinct sizes of particles will
give rise to four zones during the course of sedimentation (see Figure 3.1). From the top downwards these will consist of:
clear licuid,
suspension of smaller particles, suspension of particles of both sizes
with concentration equal to initial concentration, and finally, sediment.
Consider sedim.entation in the zone containing both sizes of particles.
Let suffixes S and L apply to the small and large particles
respectively, and suffixes 1 and 2 apply to upper and lower sedimenting zones.
In order to account for particle interactions in the
lower sedimenting zone, the terminal falling velocity for the larger
particles must be calculated as if they were settling in a suspension
of smaller size particles.
In the Stokes' Law range the terminal
falling velocity of a large particle U^^^j_^2 ""^ S^ven as
U, , , =-Ko^r
where o
^
(3-8)
is the density of a suspension consisting only of the small
s
size particles. The density p^ may be written as
37
The terminal falling velocity for the small size particles U^
c- ^
may be written as
U
_ ^S (Pp - Pf>9
t-,S,2
18 y.
(3.10)
Equations (3.8) and (3.10) may be corrected for wall effect using
equation given by Francis (1933).
t,i,2
^
t-,i,2
1-0.475 d./D
-4
where i = L,S
1 - d./D
(3.11)
Equation (3.7) may be written for the small and large size particles
in this zone to give
^s,L,2 " ^t,L,2 ^2
U
\X^
(3.12)
^,2-^
s,S,2
U. . o
t,S,2
(3.13)
£o
The n. ^ and n^^ ^ may be computed from Equation (3.2).
In sedimenta-
tion, the upward flow rate of liquid is equal to the downward flow
rate of particles at any cross section.
This gives
^f,2 ^2 " ^c,L,2 ^L,2 "^ ^c,S,2 ^S,2
(3.14)
Using Equation (3.5), Equation (3.14) may be written as
U f , 2 ^ 2 = (Us,L,2- U f , 2 ) ^ , 2 ^ (^^^3^2- Uf,2^S,2
(^l.S)
Equation (3.15) may be rearranged to give
^f,2
Now,
^s,L,2 ^L,2 ^ ^s,S,2 ^S,2
(3.16)
38
^c,L,2 = "s.L,2 - ^f,2
= "s.L,2 - ("s,L,2 \,2
' Us,S.2 ^ . 2 '
= ^s,L,2(l - C L , 2 ) -^3,3,2^3,2
(^-H)
Substituting Equations (3.12) and (3.13) into Equation (3.17)
"c,L.2 = Ut,L,2 =2"''^
(1 - C L , 2 ) - 'Jt,S,2 ^ 2 " ' ' '
S,2
^^-'^^
In the same manner,
^c,S,2 " ^s,S,2 " ^f,2
= ^'s,S,2 • (^'s,L,2 \ , 2 "" ^s,S,2 S , 2 ^
" ^"s,S,2'^ " ^S,2^ " ^s,L,2 ^L,2
= Ut,S,2 ^ 2 " ' ' '
(^ - S , 2 ^ - Ut,L,2 ^2"'''''^L,2
^''''^
Since all the terms on the right-hand side of Equations (3.18) and
(3.19) are known, U , ^ and U c; o can be calculated.
Here, U , ^
corresponds to the observed rate of fall of the interface between the
two sedimenting zones.
The concentration of the particles in the upper zone is not directly known but can be calculated using a mass balance.
The volume-
tric rate at which the small particles pass from the lower zone to the
upper zone ^'^ (U < 2 " ^c s 2^^S 2 '^t' ^''^^'^- "^t ^"^ cross sectional
area of the container.
The rate of increase in the volume of the
upper zone is (U , o - U ^^ J A . . Thus the concentration of partiC,L,^
C,o,!
C
cles in upper zone C^ , is given as
39
'^c,L.2 - "c.S,2)S,2
- u c,S,l T
(3.20)
WZ:
'S,l
Since this zone contains only uniform particles. Equation (3.1) can
be directly applied to give U
U
c,S,l
'S,l
= U.
c,S,l
t,S,l (1 -<^s.i)
(3.21)
where U^^j,, = U^,s,2 ^"d so nj^, = n^^^
Equations (3.20) and (3.21) can be solved simultaneously to calculate
U
3 , and C^ ,.
^^ c i corresponds to the observed rate of fall of
the interface separating the suspension and the clear liquid.
Sedimentation of Suspensions of Multisized Particles
Consider a suspension of m different size particles (a,b,...m,
a being the smallest); this will give rise to M(=m) zones of settling
suspensions, (1,2...M counting from top), with clear liquid above and
a sediment layer at the bottom.
All zones will exist at the beginning
of the sedimentation (Figure 3.2) and each zone will disappear in turn
during the sedimentation process as its upper boundary coincides with
that of the sediment layer.
The model developed for binary suspensions may be extended to
the sedimentation of multisized particle systems.
Three suffixes
will be used for the particle velocity 'J; the first suffix indicates
the kind of velocity (settling, slip, terminal), the second indicates
the size of particle (size a, size b, ..., size m ) , and the third
indicates the sedimenting zone in which the velocity is being
40
Settling velocities
of interfaces
Clear liquid
U c,a,l
Contains only the
smallest size oarticles
U c,b,2
U c,m-l,M-1
Zone M-1
U c, m, M
Zone M
• Contains particles of
: sizes a,b,...m-1
Contains particles of
all sizes a,b,...m
Sedimen
Figure 3.2.
Formation of zones durina sedimentation of a
multisized particle suspension.
considered (zone 1, zone 2, ..., zone M ) . The notation for the concentration C has two suffixes, the first suffix indicates the size
of the particle and the second indicates the zone in which this concentration is being considered.
Velocity of Particles in the Lowest Zone (M)
This zone contains all sizes of particles and their velocities
can be written using Equation (3.7)
^'s,i,M = ^t,i,M ^M
'"
T = a, b, ..., m
(3.22)
As there is no material crossing the sediment zone, the upward flow
rate of the liquid must equal the downward flow rate of the settling
particles.
Writing this in velocity terms
^c,a,M^a,M ^ ^c,b,M^b,M ^ • • • "^ ^c,m,M^m,M
^f,M ^M
m
= Z U . ^.C. ^,
. ^ c,i,M 1,M
(3.23)
The slip velocity of any size particle may be written as
"s.i.M^'^f.M^^Ci.M
Substituting U
i = a , b, ...,m
(3.24)
. ^ from Equation (3.24) into Equation (3.23)
m
U f . M ^ M ^ .? (Us,i,M- ^f,M'S,M
1 —a
Equation (3.25) may be rearranged to give
(2-25)
42
m
^f,M " .1^ ^s,i,MS-,M
I —d
m
n •
= Z U t,i,M ^M
i=a
M-1
(3.26)
^i,M
Substitution of U^^.^^ from Equation (3.22) and U^ ^^ from Equation
(3.26) into Equation (3.24) gives
n • M-1
U
U
c,i,M
£
1 ,M
^t,i,M ^M
m
Z U t,j,M^M
j=a
n-
M-1
C
J,m
J,M
i = a, b, ..., m
(3.27)
Since concentrations in this zone are the same as the initial concentrations, all the terms on the right-hand side of Equation (3.27)
are known and the settling velocity of each size particle can be
calculated directly.
U
,, corresponds to the rate of fall of the
interface separating the zones M and M-1.
However, it should be kept
in mind that the terminal falling velocity of a particle of size i is
calculated as if it were settling in a suspension consisting only of
particles of sizes smaller than size i.
Ut«3,i,M
where p
_ ^- g'Pp - Ps,i,M>
18 u,
Accordingly we may write
i = b, c.
(3.28)
, m
• M is the density of a suspension consisting of all particle
S , I ,n
sizes smaller than size i in zone M, and may be written as
i-1
m
J=a -^'
s,i,M
m
(1 - .^C
^ j^a ^^
^)
i = b, c ,
, m
(3.29)
^3
For the smallest size particles (i=a), the terminal falling velocity
may be written as
d g(p - p_)
^t°o,a,M "
18 y.
(3.30)
Equations (3.28) and (3.30) must be corrected f o r wall e f f e c t s .
WaT
corrected terminal v e l o c i t i e s can be obtained by using the equation
^t,i,M
^ " ^'^'^^ ^ - / ^ '^
11
— - [—T
H~Tn
]
^t«',i,M
' " ^V^
i = a ,
b, . . . , m
(3.31)
Equation (3.27) in combinations with Equations (3.28), (3.29), (3.30),
and (3.31) may be used to compute the settling velocities U • M of
C , I ,11
all different size particles in zone M.
Velocities of Particles in Zone M-1
The particles present in this zone are of sizes a, b, .... m-1.
Concentrations and settling velocities of particles of all sizes present are unknown and must be calculated.
The rate per unit area at which particles of size i cross the
interface from zone M to zone M-1
(U
M - U . .JC. ^
^ c,m,M
c,i,M' 1,M
i = a, b, ..., m-1
(3.32)
^
'
The rate per unit area at which the volume of zone M-1 is increasing
is clearly the difference in rate of fall of the interfaces forming
this zone. This is, therefore, given by
"c.m,M- Uc,m-1,M-1
(3-33)
d^
Dividing the quantity in (3.30) by that in (3.31), we get the concentration C. ,, , as
1, M- I
C
^i,M-i
= ^^c,m,M ' ^ c , i , M ) ^ - , M
Tu
~nr
^ c,m,M
,• _ ,
r m
u
^ i
^ - ^^ b, ..., m-i
(7 7A\
(3.34)
c,m-l,M-1^
For t h i s zone, equations analogous to Equations (3.27) through (3.31)
may be l i s t e d as follows
n,- M 1-1
^c,i,M-l = U t , i , M - l ^ M - l "
m-1
"j,M-l'
- .^ ' t , j , M - l ^ M - l
J~a
S,^M
i = a, b, . . . , m-1
(3.35)
' ~- '^' ^ ' • • • ' ^ - ^
(2-26)
^- 9(Po " ^s i M-P
^too,i,M-l =
s, i,M-1
18 M^ ' '
m-1
i-1
(12 C . . . T)P^ + P
L
-1=. j^,' M - r ^ f
"^p
.^^
j=a
^ j=a
m-1
C... T
J,M-1
i = b, c,. . . , M-1
^ " j ^ i S\M-1
(3.37)
d g(p - Pf)
U.„
o - ^y .
t " , a,, MM- l, = ^ - T18
U. . ,, ,
U
7-—
too,1,M-1
(3.38)
1 - 0.^75 d / D
- L 1 . d./D
-4
^
' " ^ ' ^'' • • • ' " " '
^-^-"^^
l'
The terminal v e l o c i t i e s U, . .. ^ ( j = a, b, . .. , m-1) are f i r s t computed from Equations (3.36) and (3.38).
required in Equation (3.36).
The densities p
• M •. are
These are obtained from Equation (3.37)
45
An iterative scheme is, however, required since Equation (3.37) contains the unknown concentrations C. ., , (j = a, b, ..., m - 1 ) . The
J j''-'
iteration scheme proceeds as follows:
(1)
An initial set of concentrations C. |^_i (j = a, b, ..., m-1)
is assumed and are used as starting values in Equation (3.37) to
compute p^ . j^_, (i = b, c, ..., m - 1 ) . A reasonable guess for this
set is the corresponding concentrations in the lower zone.
(2)
The terminal falling velocities U.^ . .,_-. (i = a, b, ...,m-l)
are computed using Equations (3.36) and (3.38), and are then corrected
for the wall effect using Equation (3.39).
(3)
Equations (3.34) and (3.36) are then solved sim.ultaneously
for the concentrations C. .. . and the velocities U^ . .. ,
I , rl- I
C , 1 ,r!- I
(i = a, b, ..., m - 1 ) .
(4)
The values obtained in step (3) for the concentrations
C. ^, -, (i = a, b, ..., m-1) are used in step (1) to recalculate
I 5 I I"" I
p
. f^, -, (i = b, c, ... , m-1) and the iteration cycle is repeated.
5 , I , I 1— I
In the calculation the iteration cycle is checked for completion by
the requirement that
(n+1)
C.
i,M-l
(n)
'i,M-l
< 6.
and
(n+1)
U c,m-l,M-1
(n)
U c,m-1,M-1
< 0,
hold for all values of i (i = a, b, .... m - 1 ) , where o-j and 5,^ a re
predetermined error values.
46
Tne simultaneous solution of Equations (3.34) and (3-35) can be
obtained in a direct manner by first writing Equation (3.35) for the
largest particle size in that zone.
For the present zone (zone M - 1 ) ,
the largest particle size is m-1. The resulting equation takes the
form
^c,m-1,M-1 " •^'^a,M-r S , M - r •'•' ^m-l,M-l'
'^•'^°'
which indicates that U^ ^_^ j,^_^ is a function only of all concentrations present in zone M-1. As Equation (3.32) indicates, each of
these concentrations is a function of U
T ... alone; i.e.
c,m-1,M-I
^a,M-l "^ "^a'^c,m-l,M-l^
^b,M-l " "^b'^cm-UM-l^
(3.41)
^m-l,M-l " Vl^^cm-UM-l^
When expressions for these concentrations are substituted from Equation (3.41) into Equation (3.40), there results a nonlinear equation
for U^^ ^_i ^_y
The latter may be solved for U^ ^_.^ ,,_^ using the
Newton-Raphson method or the Reguli-falsi method (Lapidus, 1962).
With U
_i M 1 known, all unknown concentrations may be obtained
directly from Equation (3.34).
Substitution of the resulting values
for the concentrations into Equation (3.35) gives the r-^naining unknown velocities and thereby the solution is com,plete.
Similar equations can be written for zones M-2, ii-O, ..., etc.;
and zone by zone computations can be carried out to calculate the
concentrations and the settlinc velocities in all zones.
47
The simultaneous solution of Equations (3.34) and (3.35) may also
be obtained in an iterative manner.
An initial value of U
, ,, . is
assumed as U^ ^__^ ^, the settling velocity of the same size particle in
the zone below the present zone.
Substitution of U
, ., -, in Eouation
c ,m- I ,:1- I
(3.34) gives the concentrations of all particle sizes in this zone.
These new values of concentrations are used to calculate U
Equation (3.35).
, ., , from
This value of U^ ^ , .. , may be used to calculate new
c,m-I ,M-1 -^
concentrations from Equation (3.34).
The iteration cycle continues until
the difference between two successive velocity values is less than a oredetermined error value.
A computer program which carries out tnese
computations for an arbitrary preassigned number of zones was written.
A sample program is shown in Appendix B.
Sedimentation of Suspension with Continuous Particle Size Distribution
Partial segregation has been observed in suspensions with continuous particle size distribution.
Unlike segregated suspensions, no
distinct settling zones are observed.
Hov/ever, the overall physical
effects may be assumed identical to those in segregated suspensions.
In general, any suspension with a broad particle size distribution can
be regarded as a mixture of several fractions, each with a narror particle size distribution.
Thus the rate of fall of the interface sepa-
rating the suspension and the clear liquid in a partially senregated
suspension may be predicted usino the theory developed -^or senregated
suspensions.
CHAPTER IV
PROCEDURE AND EXPERIMENTS
There are significant inconsistencies in the experimental observations on sedimentation of suspensions and usuallya large number of
data are required to evaluate any correlation on sedimentation.
It
is important to evaluate experimentally the proposed model and to compare it with other models as there are wide discrepancies among the
published correlations.
Although data were available in the litera-
ture , they are not sufficient and more data are essential to verify
the model. A large number of experimental data on sedimentation of
multisized particles were taken using various sizes of spherical
glass particles and different suspending media.
Details of the ap-
paratus and experimental procedure used for the sedimentation experiments is given below.
Glass Spheres
Ten sizes of glass particles were used in the sedimentation experiments.
These were spherical in shape and were obtained from
Cataphote Division of Ferro Corporation.
Each particle size was a
fraction between two consecutive sieves with at least 95% of the parti
cles having a diameter between the openings of the two sieves.
The
density of each particle size was measured using a pycnometer.
A
density of 2.43 gm/cm
was taken as an average for all sizes since
only small variations were observed in the densities of the various
48
49
sizes. The average density supplied by the manufacturer was 2.42
3
gm/cm . An arithmetic average of the sieve-openings was taken as the
diameter of each particle size.
This assumption was acceptable since
the ratio of the diameters of the largest particle to that of the
smallest particle in each fraction was 1.19.
identification,
For easy reference and
each particle size was assigned a number.
Table 4-1
lists the identification number, the sieve numbers, the size range of
the particles, the average diameter, and the density of each sizefraction.
Table 4.1 Properties of the Glass Spheres
P a r t i c l e size i n
sieve numbers
No.
Diameter
range
cm
Mean
Diameter
cm
Density
gm/cm3
1
-35 +
40
0.0500-0.0420
0.0460
2.45
2
-40 +
45
0.0420-0.0354
0.0387
2.42
3
-45 +
50
0.0354-0.0297
0.0326
2.45
4
-50 +
60
0.0297-0-0250
0.0274
2.44
5
-60 +
70
0.0250-0.0210
0.0230
2.43
-70 +
80
0.0210-0-0177
0.0194
2.47
7
-SO + 100
0-0177-0.0149
0.0163
2.4'!
8
-100 + 120
0.0149-0.0125
0.0137
2.^6
9
-120 + 140
0.0125-0.0105
0.0115
2.39
10
-170 + 200
0.0088-0.0074
0.0081
2.34
6
'
50
Coloring of Glass Particles
Segregation of particles can be observed visually if the larger
size particles are colored.
It is almost impossible to carry out ex-
perimental work on sedimentation of multisized particle suspensions
without coloring all but the smallest size particles.
Particles can
be colored by a process originally proposed and patented (U.S. Patent,
1941) for dyeing glass fibers.
The fibers are heated with various
ionic solutions and supposedly the ions are adsorbed on the glass by
a base exchange of the solution ions with the alkali and alkaline
earth metals in the glass.
It was found that blue and orange colors
were easy to produce on the glass particles.
Particles were treated
with a 3% solution of ferrous sulfate at 85°C for about 30 minutes
with occasional stirring, and were subsequently treated with either
a warm dilute solution of sodium carbonate for 10 minutes to color
them orange, or with a warm 1% solution of potassium ferrocyanide
acidified with HCl to color them Prussian blue.
Density measurements after coloration showed that the coloring
process had no significant effect upon the particle density.
Suspending Media
Ethylene glycol, diethylene glycol and 60'o aqueous glycerol were
used as suspending media.
Selection of these liquids depended upon
the following requirements:
(1)
number.
High viscosity, which helps in lowering the Reynolds
51
(2)
Transparency, which permits observation of the particles
while settling and hence monitoring the positions of the interfaces.
(3)
Chemical inertness, which is essential to prevent any chem-
ical attack on the glass particles as well as on the Plexiglas tube
in which sedimentation is taking place.
(4)
Physical inertness:
No physical properties other than den-
sity and viscosity should affect the rate of sedimentation.
Liquids
with flocculating effects or those supplying positive or negative
ions to the particles cannot be used.
(5)
Non-hygroscopicity:
Suspending media should not show a
tendency to absorb moisture from the atmosphere, which would otherwise result in significant density and viscosity changes.
Aqueous
glycerol has little tendency to absorb moisture from air but this
was prevented by keeping it in an air-tight vessel.
Viscosity mea-
surements taken before and after an experimental run showed no significant change.
The densities of the liquids were measured at 24 j^ 1°C using a
pycnometer.
The viscosities were measured in the temperature range
of 20°C to 26°C using a Brookfield Viscometer.
The resulting smooth
temperature-viscosity curves were used to obtain the viscosities o^
the liquids at 24°C.
ties at 24°C.
Table 4.2 lists the densities and the viscosi-
52
Table 4.2
Properties of Suspending Media
QMcnoo^-;.,^ m«^-,-..m
Suspending
^
^ medium
Density at
o/ior
/ 3
24 C, gm/cm^
Viscosity
at
onor
r-r.
24 C, cp
Ethylene glycol
1.108
0.184
Diethylene glycol
1.115
0.302
60% Aqueous glycerol
1.165
0.135
Experimental Setup
Sedimentation experiments were carried out in a v e r t i c a l
bottomed Plexiglas tube 3.2 cm in diameter and 76 cm long.
flat-
During an
experimental run, the tube was held in a v e r t i c a l position using a
metal stand which was kept in a controlled temperature chamber.
scales graduated to one m i l l i m e t e r were fixed to the f l a t
Paper
vertical
metal plate j u s t behind the tube so that the positions of the i n t e r faces could be e a s i l y read.
The controlled temperature chamber was
used to keep the v i s c o s i t y and density of the l i q u i d constant and to
avoid convection currents.
A l l experiments were carried out at a
constant temperature of 24 j ^ 0.5°C.
Experimental Procedure
A s l u r r y of glass p a r t i c l e s was prepared in the sedirr^entation
tube by mixing known weights of two or three sizes of p a r t i c l e s with
a known volume of the f l u i d .
Total volume of the suspension in a l l
o
experimental runs was 596 cm . A rubber cork was used to plug the
3
top opening of the tube. About 8 cm of air was deliberately allowed
to be trapped between the suspension and the cork.
When tre tube
53
was tilted upside down and then brought back to the upright position,
the trapped air helped in agitating the suspension.
The suspension
was placed in the constant temiperature chamber until its temperature
reached the desired temperature of 25°C.
Since the thermal conduc-
tivity of plexiglas is low, it took generally more than one hour for
the suspension to reach the desired temperature, but was easy to
maintain thereafter.
The tube was then taken out and the suspension
was agitated for several minutes to achieve a uniform concentration
of solids throughout the suspension.
No significant temperature
change was noted during the agitation period.
During sedimentation
the temperature of the chamber was kept between 23.5°C to 25°C.
positions of the interfaces were recorded with time.
The
At the
end of an experimental run, the solids and liquids were recovered by
filtering out the liquid through a ceramic filter using a vacuum
pump.
The particles were then washed with distilled water and were
dried in a drying oven.
Varying proportions of each particle size were used to prepare
39 different suspensions.
These included 30 suspensions of two dis-
tinct sizes of particles, 3 suspensions of three distinct sizes of
particles, and 6 suspensions with a continuous particle size distribution.
Sedimentation data were taken at five different concentra-
tions for each suspension.
Total concentrations of the solids ranged
from 12% to 45% by volume.
In all, 195 data points were collected. Each
experimental data point was reproduced twice and reproducibility was
within 3'.. The arithmetic average of the two measurements was used
in the final calculations.
These data are tabulated in Appendix A.
\
CHAPTER V
RESULTS AND DISCUSSION
A new model for the sedimentation of multisized particle suspensions was developed in Chapter 3.
In order to test the validity
of the model, it is necessary to compare results from the model with
experimental data.
For binary suspensions, the only data available
were due to Smith (1965) on sedimentation, Lockett and Al-Habbooby
(1973) on sedimentation and countercurrent operations, and due to
Mirza and Richardson (1979) on sedimentation.
The data by Smith
totalled 85 points, out of which 43 data points were used in testing
the new model.
The remaining data points were excluded since they
were only for fine particles of approximately 60 micron in diameter.
In this range of small particle size, reliable correlations for sedimentation are not available.
Lockett and Al-Habbooby collected
data on the initial sedimentation rate for binary suspensions, and
also obtained a total of 89 data points for a countercurrent solidliquid system.
Only the latter data were included since the present
model predicts only the average settling rates.
All 45 data points
collected by Mirza and Richardson on binary suspensions were used.
For ternary systems, only five data points were available from
Smith (1965).
These were excluded because of the absence of the
concentration variable.
The data on sedimentation of binary and ternary suspensions
collected by Davies (1968) were excluded because of the extremely
54
55
high settling rates encountered in his experiments.
Also, the
physical properties of the solids and the suspending medium were not
reported in his published paper.
Because the published experimental data on sedimentation of
multisized particle systems were found to be inadequate, a total of
195 new experimental data points were collected.
These included 150
data points for binary suspensions, 15 for ternary suspensions and
30 for suspensions with continuous particle size distribution.
The
published and new experimental data were compared with predictions
from the proposed model, the Lockett and Al-Habbooby model and the
Mirza and Richardson Model.
The results of the comparative analysis
are presented below.
Comparison of the Models with Experimental data for Binary Suspensions
The new experimental data on binary suspensions are listed in Tables
A.l through A.3.
Tables A.4 and A.5 contain the data acquired from
Smith (1965) and Mirza and Richardson (1979), respectively.
These
tables compare the observed values of velocities of the lower and
upper interfaces with those predicted from the present model.
The
first and second columns of each of these tables give the concentrations of the large and small size particles, respectively.
Columns
three through six list the observed and predicted interface velocities.
The last two columns show the percentage deviation of the
predicted values of velocities from the observed values.
For each
suspension the results are expressed by plotting the observed and
predicted velocities of the interfaces versus the voidage of the
56
suspension.
The new experimental results for 30 binary suspensions
are plotted in Figures 5.1 through 5.30.
Figures 5-31 through 5.33
compare the proposed model with the data acquired from Smith (1965).
Figures 5.34 through 5.36 compare the present model and MirzaRichardson model with the experimental data provided by Mirza and
Richardson (1979).
The proposed model and the two previously pub-
lished models, those of Lockett and Al-Habbooby (1973) and of Mirza
and Richardson (1979), are compared graphically in Figures 5.37
through 5.43.
It is clear from these tables and figures that the
proposed model is in good agreement with the experimental data for
both interfaces.
Figures 5-37 through 5.43 show clearly that the
Lockett and Al-Habbooby model significantly overpredicts the velocities for both interfaces.
Although these figures show that pre-
dictions from the Mirza and Richardson model are not unsatisfactory
for the upper interface, significant deviations from the experimental
data are observed for the lower interface.
In order to compare the
success of the proposed and previously published models in representing experimental results. Table 5-1 gives the average percentage
deviation of predicted values of the interface velocities from the
measured values.
The results of Table 5.1 are based on a total of
238 data points for the lower interface and an equal number for the
upper interface.
It is clear from this table that the proposed model
shows a substantial improvement over the previously published models.
The mean-error and standard deviation shown in Table 5.1 also emphasize the superiority of the proposed model.
Though not shown in the
57
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53
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
d^ = 0.0460 cm
00
d3 = 0.0194 cm
C3 = 0.0696
Suspending Medium:
Ethylene Glycol
H
CM
u
OJ
o
CM
CO
CJ
u
o
CD
nter1^ace
<v
CNJ
>—•
CO
CD
rr
o
o
y ^
1
O.AQ
0.50
0.6G
U /L
Voidage, £
w. -ru
Figure 5.1. Comparison of experimental results with model for
d^=0.0460 cm, d3=0.0194 cm and 03=0.0696-
'v
59
Experimental Results (Upper Interface)
Experimental Results (Lower Interface)
Predicted from Model
d. = 0.0460 cm
d3 = 0.0194 cm
00
CM
C3 = 0.119
CO
Suspending Medium:
Ethylene Glycol
C3
CJ
OJ
CO
O
CM
O
E
o
««
>>
+->
•r—
U
CD
o
^—
CD
>•
a
CD
u
<a
MSCD
-!->
CM
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Q.'AQ
0.50
v_v.w*J
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w.^_^—
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Figure 5.2. Comparison of experimental results with model for
d =0.0460 cm, d3=0.0194
, 03=0.119.
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
0.0460 cm
OO
CM
^S = 0.0194 cm
C. =
0.168
Suspending Medium:
Ethylene Glycol
ZT
CM
• /
sec
.20
/
/
E
u
^-^
m
>•>
•r—
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o
•^-<v •
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s_
CU
c
OvJ
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00
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c
O.UO
0.50
0.60
0.70
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Figure 5.3. Comparison o f experimental r e s u l t s w i t h model
f o r d|^=0.0460 cm. d3=0.0194 cm and C, =0.163-
^n
61
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
d^_ = 0.0460 cm
oc
d3 = 0.0194 cm
CM
C^_ = 0.217
Suspending Medium:
Ethylene Glycol
CM
O
o
U CM
CO O
1
a CO
o •~^
CU
> •
o
•
a;
u
fO
Mt-
cu rsi
OO
CD.
c:
CD
0.40
0.50
0.60
n ''n
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•1
-n
0-20
Figure 5.4. Comparison of experimental r e s u l t s w i t h model f o r
d, =0.0460 cm, dc-=0.0194 cm and C, =0.217.
62
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
= 0.0460 cm
00
':M
= 0.0137 cm
= 0.0595
Suspending Medium:
Ethylene Glycol
o
<v
O
CSJ
CO
^-^
E
u
u
o
CO
r—
*
(U
u
M-
&.
(U
•M
CM
o
y
OO
CD
o
a
CD
0.'40
0.50
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Figure 5.5. Comparison of experimental results with model for
d^_=0.0460 cm, d3=0.0137 and 03=0,0595.
63
Experimental Results (Upper I n t e r f a c e )
Experimental Results .(Lower I n t e r f a c e )
Predicted from Model
0.0460 cm
00
d . = 0.0137 cm
0.119
Suspending Medium:
Ethylene Glycol
o
a
O
CU
CO
CM
O
E
CJ
o ^-.
^
%
CU
u
<T3
scu
CM
C
—
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c
CD
o
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o.^o
0.50
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0.70
0.30
0. 30
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Figure 5.6. Comparison of experimental results with model for
d,=0.0460 cm, d^=0.0137 cm and C^=0.119.
6^
~i
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
CO
CM
0.0460 cm
d . = 0.0137 cm
0.168
Suspending Medium:
Ethylene Glycol
CM
CD
'sec
CD
CD
E
u
9k
0C11
J..)
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'ace
CU
>
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CU CM
C3
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40
0.50
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Figure 5.7. Comparison of experimental r e s u l t s with model f o r
d^=0.0460 cm, d3=0.0137 cm and C^_-0.168.
5V
65
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
d|_ = 0.0460 cm
oc
d^ = 0.0137 cm
CM
Cj_ = 0.217
Suspending Medium:
Ethylene Glycol
H
CM
o
a
U
CD
CO
' M _
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CJ
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Figure 5.8. Comparison of experimental results with model
for d,=0.0460 cm, d3=0.0137 cm and C^=0.217.
66
o
•
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
d^^ = 0.0326 cm
zr
d3 = 0.0137 cm
ft
o
C3 = 0.0797
Suspending Medium:
Ethylene Glycol
CD
C
o
CU
CO
u
o
CO
c
<v
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03
scu
4J
c
CO
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Figure 5.9- Comparison of experimental results with model for
d^=0.0326 cm, d3=0.0137 cm and 03=0.0797-
67
o
•
—
-1
Experimental Results (Upper Interface)
Experimental Results (Lower Interface)
Predicted from Model
d^ = 0.0326 cm
d3 = 0.0137 cm
C3 = 0.139
Suspending Medium:
Ethylene Glycol
CM
o
CU
to
"E
o
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ft
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Figure 5.10- Comparison of experimental results with model
for d, =0.0326 cm, d3=0.0137 cm and 03=0.139-
0 ^0
68
O
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
d^ = 0.0326 cm
7
d3 = 0.0137 cm
n
/
C^ = 0.126
Suspending Medium:
Ethylene Glycol
CM
^
u
<v
o
CO
• /
E
CJ
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CJ
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1 —
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Figure 5.11. Comparison of experimental results with model for
d|_=0.0326 cm, d3=0.0137 cm and C^=0.126
0.40
0-^0
69
o
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
d^ = 0.0326 cm
d3 = 0.0137 cm
C^ = 0.200
Suspending Medium:
Ethylene Glycol
'sec
CM
CD
E
CJ
A
+->
LOO
O
Face
CU
>•
CD
s_
CU
CO
c
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CNJ
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Figure 5.12
o ,r n
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Voidage, £
Comparison of experimental r e s u l t s with model
f o r dj^=0.0326 cm, d3=0.0137 cm and C,=0.200.
7Q>
o
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower Interface)
Predicted from Model
d^^ = 0.0326 cm
d3 = 0.0081 cm
oo
CM
C3 = 0.0624
Suspending Medium:
Ethylene Glycol
'^g
0
CU
CO
CM
0
E
CJ
A
>1
-M
•r—
CJ
0
^
t—
CU
• /
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0
CU
CJ
03
M-
s_
CU
+->
E
^—^
CM
_
—
0
00
CD
CD
CD
CD
c:
-n
u. 60
Voidage, £
Figure 5.13. Comparison of experimental results with model for
d(_=0.0326 cm, d3=0.0081 cm and 03=0.0624.
w . -tC-
^
.
/ •
71
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
d|_ = 0.0460 cm
d3 = 0.0194 cm
oc
CM
C3 = 0.0612
Suspending Medium:
Diethylene Glycol
zr
0.20
'sec
CM
E
CJ
M
LOO
CD
nte rface Vel
•*-)
0
/
I
CM
J
/
/
CO
/
c
0.40
0.60
Voidage, £
Figure 5.14. Comparison o f experimental r e s u l t s with model f o r
d|_=0.0460 cm, d3=0.0194 cm and 03=0.0612.
J . DU
'\
72
O
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lov/er I n t e r f a c e )
Predicted from f-lodel
d, = 0.0460 cm
d3 = 0.0194 cm
C3 = 0.122
Suspending Medium:
Diethylene Glycol
CNJ
CD
U
CU
CO
O
CD
I
o
o
oc
CU
o
CD
ft
:=»
CU
o
ra
t+-
scu
•M
E
CJ
c
o
o
r^j
Voidage, £
Figure 5.15. Comparison of experim.ental results with model for
d^=0.0460 cm, d3=0.0194 and 03=0.122.
\
73
O
9
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
dj_ = 0.0460 cm
ZT
d3 = 0.0194 cm
ft
O
C^ = 0.173
Suspending Medium:
Diethylene Glycol
CM
CD
0. 10
/sec
/
E
u
m
>>
•r-
U
OO
o
o
o
nte race
<U
>
ft
s-
CD
CD
/
/
CD
C^J
0.40
0.50
n ,q n
w. ru
Voidage, e
Figure 5.16. Comparison of experimental results with model for
d,=0.0460 cm, d3=0.0194 cm and 0^=0.173.
74
o
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
d|_ = 0.0460 cm
d3 = 0.0194 cm
C^ = 0.225
Suspending Medium:
Diethylene Glycol
CM
(Di
o
CU
CO
"o
o
CD
—
o
OO
o
CU
CU
o
<a
^S-
cu
CD
CD
/
rj
CD
V.^.-iCJ
l _ . - w
l _ . _ w
_
.
^
^
.
<w ^ «
Voidage, £
Figure 5.17. Comparison of experimental results with m.odel for
d^_=0.0460 cm, d3=0.0194 cm and C,=0.225.
75
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e
Predicted from Model
d^ = 0.0460 cm
d3 = 0.0137 cm
M
C3 = 0.0612
Suspending Medium:
Diethylene Glycol
'M
CJ
CD
CO
CJ
o
CU
o
CM
CD
ft
o
• /
CU
o
03
I
I
&CU
,M
CD
o
,1
J L
n "n
w'. -TL
Voidage, z
Figure 5.18. Comparison of experimental results with model for
dL=0.0460 cm, d3=0.0137 cm and 03=0.0612.
76
~i
i
o
•
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
• /
d|_ = 0.0460 cm
/
d3 = 0.0137 cm
C3 = 0.120
Suspending Medium:
Diethylene Glycol
CNJ
ft
o
O
<U
CO
'E
o
ft
o
u
u
o
CO
o
CU
CU
o
^i-
<u
CD
CO
ft
o
rr
CD
CM
CD
0.40
0, 50
0.70
0 60
Voidage, £
-(
on
Figure 5.19. Comparison of experimental results with model for
d, =0.0460 cm, d3=0.0137 cm and 03=0.12a
77
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
^L = 0.0460 cm
^ s - 0.0137 cm
C. = 0.188
Suspending Medium:
Diethylene Glycol
CM
ft
o
a
CU
CO
OCl
u
CU
>
OO
c
a
ft
CU
u
ca
•
i_
CU
•M
CD
CD
zr
CO
3/
CM
CD
r
y
o
c
n
0.60
90
Voidage, £
Figure 5.20. Comparison of experimental results with model for
d|^=0.0460 cm, d3=0.0137 cm and C,=0.188.
^.^^
0.50
78
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
d^ = 0.0460 cm
IT
d3 = 0.0137 cm
CD
Cj_ = 0.225
Suspending Medium:
Diethylene Glycol
CM
CD
a
CJ
.
CU
o
CO
E
CJ
«ft
>^
+->
o
o
r—
•r—
CO
CD
•:—1
CU
>•
CU
CJ
03
MSCU
-l-J
E
CD
CD
^
zr
CD
CM
C2
CD
^
.0
0.50
- CU
Voidage, £
Figure 5.21. Comparison of experimental results with model for
d^=0.0460 cm, d3=0.0137 cm and 0^^=0.225.
79
o
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
dj_ = 0.0326 cm
d3 = 0.0137 cm
C3 = 0.0612
Suspending Medium:
Diethylene Glycol
C\J
ft
CD
u
CU
CO
o
ft
a
"E"
CJ
-M
•r—
CJ
o
CU
00
CD
a
CU
u
<T3
S-
cu
4->
E
CO
CD
CD
ft
/
CM
CD
C
u
40
C.5L
Figure 5.22
'n
Voidage, £
Comparison of experimental results with model for
d, =0.0326 cm, d3=0.0137 cm and 03=0.0612-
80
o
^
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted w i t h Model
d^_ = 0.0326 cm
/
/
d3 = 0.0137 cm
(•>„
CD
ft
C3 = 0.122
o
Suspending Medium:
Diethylene Glycol
CD
CD
CO
CD
"E
CJ
CO
O
O
CU
CD —
•—^
ft
CU
o
u
03
cu
CD
ft
O
CM
CD
0 40
en
Voidage, £
Figure 5.23- Comparison of experimental results with model for
d^=0.0326 cm, d3=0.0137 cm and 03=0.122.
81
o
o
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lov;er I n t e r f a c e )
Predicted w i t h Model
d|_ = 0.0326 cm
d3 = 0.0137 cm
CD
C^ = 0.174
Suspending Medium:
/
Diethylene Glycol
/
/
CO
(3
u
CU
CO
m
CO
E
u
+->
CJ
o
CD
CD
CU
U
<0
<+-
s_
CU
^
-n
.~i
/
/
CM
/
dP
zn
Voidage, £
Figure 5.25- Comparison of experimental r e s u l t s with model f o r
d^=0.0326 cm, d3=0.0137 cm and C^=0.174.
82
o
o
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted w i t h Model
d^ = 0.0326 cm
d3 = 0.0137 cm
X
Csj
C3 = 0.0615
CD
Suspending Medium:
Aqueous Glycerol
^j
ft
o
CJ
CU
CO
O
cu
a
CD
D-
CD
CU
u
03
cu
CM
CO
CD
CD
CD
CD
CD
0 40
U . OL
o
un
Voidage, £
Figure 5.25- Comparison of experimental results with model for
d|_=0.0326 cm, d3=0.0137 cm and 03=0.0616.
83
Experimiental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
"1
0.0326 cm
0.0137 cm
Co =
0.123
Suspending Medium:
Aqueous Glycerol
CM
CD
u
CU
CO
O
ft
o
-M
•r-
u
o
<v
CO
CD
CU
U
03
MS-
CU
CD
CD
."-J
CD
CD
n
40
Figure 5.26
;U
n
cr
n
:"n
t;n
Voidage,
Comparison of experimental results with model for
d^=0.0326 cm. d3=0.0137 cm and 03=0.123.
84
O
<y
—
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from M.odel
d, = 0.0326 cm
d3 = 0.0137 cm
C^ = 0.174
Suspending Medium:
Aqueous Glycerol
ft
O
O
CJ
CU
CO
ft
o
u
-^
CJ
o
CO
CD
CU
CU
u
«3
cu
-M
CD
(D;
CD
3
/
;
/
/
I
CD_^
/
^J
CD
CD
0.40
0.50
en
Voidage, £
Figure 5.27. Comparison of experimental results with model for
dj_=0.0326 cm, d3=0.0137 cm and Cj_=0.174.
85
Experimental Results (Upper I n t e r f a c e )
Experimental Results (Lower I n t e r f a c e )
Predicted from Model
d^_ = 0.0326 cm
d3 = 0.0137 cm
Cj_ = 0.225
Suspending Medium:
Aqueous Glycerol
CM
ft
o
CJ
CU
Q
CO
^-^
CJ
•r—
CJ
o
^-
'CO
CD
CD
cu
>•
CU
CJ
03
M-
s_
CU
-M
CD
CD
CD
/
CD
1
aI
0.5i
w . DU
~n
;u
Voidage, £
Figure 5.28- Comparison of experimental r e s u l t s with model f o r
d, =0.0326 cm, d3=0.0137 cm and Cj_=0.225.
86
O
Experimental Results (Upper
ExperifTiental Results (Lower
Predicted from Model
Interface)
Interface)
0.0326 cm
0.0081 cm
CD
OJ
c . = 0.0615
Suspending Medium:
Aqueous Glycerol
OJ
o
CJ
CU
CO
o
CM
CD
U
G
CO
O
—'
CU
CU
CJ
sCU
CM
OO
CD
CD
ft
CD
CD
CD
0.40
0.50
0.60
0.70
Voidage, £
,^ zn
Figure 5.29- Comparison of experimental r e s u l t s with model f o r
d,=0.0326 cm, d3=0.0081 cm and 03=0.0615.
n
87
o
•
Experimental Results (Upper I n t e r f a ce)
Experimental Results (Lower I n t e r f a ce)
Predicted from Model
/
dj_ = 0.0326 cm
•I
I
I
d3 = 0.0081 cm
C3 = 0.123
Suspending Medium:
Aqueous Glycerol
OJ
ft
o
CD
CJ
<
u
CO
— 1
O
,
E
CJ
«ft
>>
4->
-r—
CJ
o
OO
CD
CU
o
r""
z=>
ft
<v
u
03
l l
Scu
+J
cc
E
CD
zr
CD
CM
C
CD
0.40
0.50
Voidage, £
Figure 5.30. Comparison of experimental r e s u l t s with model for
d^_=0.0326 cm, d3=0.0081 cm and 03=0.123.
88
OO
Experimental Results (Upper Interface)
Experimental Results (Lower Interface)
Predicted from the Proposed Model
d^ = 0.0252 cm
o
d3 = 0.0131 cm
C^_:C3 = 2:1
o
o
u
CD
CO
5
<5
o
CJ
o
CD
CD
CJ
fC
<+s-
CM
O
CD
OQ
O.
«
o
3*
o
o
o.
0.40
0.50
T
0.60
T
0.70
0.80
Voidage, £
Figure 5.31. Comparison of the proposed model with the
experimental data of Smith.
0.90
89
Experimental Results (Upper Interface)
00
•'
Experimental Results (Lower Interface)
o
Predicted from the Proposed Model
d^_ = 0.0252 cm
d3 = 0.0187 cm
o
C,_:C3 = 4:1
o
u
CD
CO
(O
CJ
u
o
<v
<v
CJ
03
4-
CM
o
sCD
00
o,
o
o.
o
o
o
0.40
0.50
T
0.60
Voidage,
T
0.70
0.80
c
Figure 5.32. Comparison of the proposed model with the
experimental data of Smith.
0.90
90
OO
CM
Experimental Results (Upper Interface)
Experimental Results (Lower Interface)
Predicted from the Proposed Model
dj_ = 0.0252 cm
d3 = 0.0187 cm
O
CJ
OJ
CO
'E"
CJ
<o
-t->
u
o
CD
CM
CD
U
03
4S-
o
CD
00
0.40
0.50
""!
0.60
T"
0.70
0.80
Voidage, £
Figure 5.33.
Comparison of the proposed model with the
experimental data of Smith.
0.90
91
LO
cn
CD
Experimental Results (Upper Interface)
Experimental Results (Lower Interface)
Predicted from the Models
d^ = 0.0463 cm
O
CO
d3 = 0.0115 cm
C3 = 0.156
LO
CM
CJ
CD
CO
o
.
o
CJ
O
CD
CD
U
03
4S-
lO
o
cu
4->
o
Present Model
o
Mirza-Richardson Model
LO
O
ilirza-Richardson Model
Present Model
o
o
0.40
0.50
0.50
0.70
0.80
0.90
Voidage, £
Figure 5.34,
Comparison of the Mirza-Richardson model and the
proposed model with the experimental data of
Mirza and Richardson.
92
in
on
O
Experimental Results (Upper Interface)
Experimental Results (Lower Interface)
Predicted from the Models
d^ = 0.0463 cm
o
on
d3 = 0.0115 cm
C,_ = 0.308
CM
CJ
CD
CO
o
CJ
O
CD
CD
U
03
<+s_
un
4
o
CD
Present Model
!'1irza-Richardson Model
o
ft
o
Mirza-Richardson Model
o
Present Model
a
o
0.40
0.50
Figure 5.35.
0.60
0.70
Voidage, £
Q.8G
0.90
Comparison of the Mirza-Richardson model and the
proposed model w i t h the experimental data of
Mirza and Richardson.
93
O
Experimental Results (Upper Interface)
Experimental Results (Lower Interface)
Predicted from the Models
d^_ = 0.0231 cm
CO
o
d3 = 0.0115 cm
Cj_ = 0.315
o
u
<v
CO
o
u
o
CD
en
OJ
CJ
03
<+S-
ci;
C50
o
Mirza-Richardson Model
-Present Model
Mirza-Richardson Model
O
«
o
— Present Model
a
o
0.40
0.50
I
i
O.SO
G.7G
G.ao
0.90
Voidage, £
Figure 5.36.
Comparison of the Mirza-Richardson model and the
proposed model with the experimental data of
Mirza and Richardson.
94
^
Experimental Results (Lower Interface)
Predicted from the Models
d, = 0.0460 cm
s_
d3 = 0.0194 cm
C3 = 0.
Suspend
« _
U
CD
CO
u
8-
U
o
CD
tf)
CU
U
ro
MSCD
-l-J
E
o
bbooby Model
8_
hardson Model
odel
8.
0.40
0.50
~[
0.60
r~
0.70
T
0.80
0.90
Voidage, c
Figure 5.37. Comparison of models with the new experimental
data for the lower interface.
95
Experimental Results (Lower Interface)
•'
o
Predicted from the Models
d|_ = 0.0460 cm
d3 = 0.0194 cm
W-.
C,_ = 0.217
Suspending Medium: Ethylene Glycol
ft
CJ
CD
CO
CJ
(O
u
o
CD
CD
CJ
CM
03
^s_
CD
tt-Habbooby Model
-Richardson Model
nt Model
—[
0.40
0.50
0.60
r"
0.70
0.80
0.90
Voidage, £
Figure 5.38.
Comparison of models with the new experimental
data for the lower interface.
96
»
-I
•
Experimental Results (Lower Interface)
Predicted from the Models
dj_ = 0.0460 cm
d3 = 0.0137 cm
Cj_ = 0.217
Suspending Medium: Ethylene Glycol
s_
CJ
CD
CO
u
(O
u
o
CD
r>
CD
CJ
O*
* ^ .
^
/-J
M-
O
SCD
8.
tt-Habbooby Model
S_
irza-Richardson Model
Present Model
s.
0.40
O.SO
•~T
0.60
r~
0.70
0.80
0.90
Voidage, £
Figure 5.39. Comparison of models with the new experimental
data for the lower interface.
97
8
Experimental Results (Lower Interface)
Predicted from the Models
d|_ = 0.326 cm
d3 = 0.0137 cm
R-
C^_ = 0.126
Suspending Medium: Ethylene Glycol
8
CJ
CU
CO
CJ
(O
>,
u
o
CU
CD
CM
CJ
03
4iCD
-M
E
s.
ockett-Habbooby Model
irza-Richardson Model
Present Model
s.
0.40
0.50
Figure 5.40.
•n
0.80
Voidage
0.70
0.80
0.90
Comparison of models with the new experimental
data for the lower interface.
98
n
«.
•
Experimental Results
Predicted from the Models
d^_ = 0.0326
d3 = 0.0137
J5_
C3 = 0.123
Suspending Medium: 60% Glycerol
8.
CJ
CD
CO
CJ
(O
u
o
CD
CD
CJ
03
4SCD
CM
S.
Lockett-Habbooby Model
s_
lirza-Richardson Model
Present Model
0.40
0.50
T
0.60
T
0.70
0.80
0.90
Voidage, £
Figure 5.41.
Comparison of models with the experimental data
for the lower interface.
99
Experimiental Results
Predicted from the Models
d|_ = 0.0460
d3 = 0.0194
C3 = 0.119
8.
Suspending Medium: Ethylene Glycol
O
8
u
CU
CO
S.
CJ
+j
CJ
o
CD
>•
CD
CJ
03
SCD
-M
E
cn
O
•'
O
s.
LOckett-Habbooby Model
Mirza-Richardson Model
Present Model
0.40
0.50
—[
0.60
T"
0.70
0.80
0.90
Voidage, £
Figure 5.42.
Comparison of models with the experimental data
for the upper interface.
100
Experimental Results
Predicted from the Models
d^ = 0.0326
d3 = 0.0137
(O
C^ = 0.200
Suspending Medium: Ethylene Glycol
in
CJ
CD
CO
F
CJ
«ft
>,
4->
•r—
U
O
r—
CD
s
•
o
>•
0)
CJ
03
M-
<-
23
o•
o
CD
-l-J
E
CM
tt-Habbooby Model
-Richardson Model
nt Model
s.
"~r~
0.40
0.50
•"1
0.60
r"
0.70
0.80
I
0.90
Voidage, £
Figure 5.43.
Comparison of models with the experimental data
f o r the upper i n t e r f a c e .
101
table, the proposed model shows a deviation of 20'. or more for just
12 data points, while such a deviation is observed for 77 data points
in the case of Mirza and Richardson model and 218 data points in the
case of Lockett and Al-Habbooby model.
Comparison of the Models with the Experimental Data for Ternary
Suspensions:
The new experimental data for ternary suspensions are
listed in Table A.6.
The table compares the observed values of velo-
cities of the lower interface, the middle interface, and the upper
interface with those predicted from the proposed model.
The first
three columns of the table give the concentrations of the large,
medium and small size particles, respectively.
Columns four through
nine list the predicted and observed interface velocities.
The last
three columns show the percentage deviation of the predicted values
from the observed values.
For each suspension the results are ex-
pressed by plotting the observed and predicted velocities of the interfaces versus the voidage.
through 5.46.
These plots are shown in Figures 5.44
In general, these figures show that the agreement of
the proposed model with the experimental data is satisfactory,
though the predicted velocities are underpredicted for some interfaces.
Using the same data points, the three models are compared in
Table 5.2.
The results in this table indicate that the present
model shows a slight improvement over the Mirza and Richardson model.
Because of the lack of sufficient experimental data (only 15 data
points are available), it is not possible to carry out a conclusive
comparative study and, obviously, more experimental data for terr.ary
102
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CO
1
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1/1
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s La£
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ro
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ro
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s:
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CD
Z2
Q-
IQ.
1—1
X
o
"
io
«5 O
CD
"O
=3
\-r.
4-1
^^
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.,
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un
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CD <J-i
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rt3
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ME
O CD
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.—.
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un
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CO
4J
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01
CJ
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3
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00
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CM
r«»
un
un
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—
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rs • —c•
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0)
cn
«3
4->
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c
OJ
u
i '^
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03 LLJ
C3.
E CD
O -E
CM
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L. M -
•!-
CD
C
i*-
0)
o s-
in
•1-
1—.
1 —
z
E
E
•vO
c
a; (O
"O
E
03
un
o
5.4
13
CO
E
Z3 CD
i
r—
1^—
CM
-3
-3
S0)
UO
•f—
4—1
y
•
uo
c
•— 1
OJ
CJ
j1
1
CM
iti
r—
i. <40) 1 .
CM
CO
•
ro
3 OJ
ro
ro
O 4-J
LO
CD
JZl
<o
-3
O
•o ~r
1) -c
w
I
-A <.—'
fT3 -a
o
103
Experimental Results (Upper Interface)
Experimental Results (Middle Interface)
cn
Experimental Results (Lower Interface)
Predicted from the Proposed Model
Suspending Medium: 60% Glycerol
d^ = 0.0460 cm
a
d^^ = 0.0194 cm
o
»
o
d^ = 0.0081 cm
C^ = 0.0518
C^ = 0.0518
in
CM
CJ
CD
CO
o
o
CM
4->
O
o
CD
CD
U
in
03
^s-
CD
o
«
o
in
o
o
o
0.40
0.50
Figure 5.44-
^1
\—
0.60
0.70
Voidage, £
0.80
Comparison of the proposed model with the
experimental data on ternary suspension.
0.90
r. 1
-«- Experimental Results (Upper Interface)
^
•
Experimental Results (Middle Interface)
•
Experimental Results (Lower Interface)
— Predicted from the Proposed Model
Suspending Medium: 60% Glycerol
d^ = 0.0460 cm
a
d^ = 0.0194 cm
d^ = 0.0081 cm
C^ = 0.104
C = 0.0518
in
CM
U
CD
CO
(J
c^
>, o
CJ
o
CD
CD
CJ
03
MS_
CD
in
o
T
0.40
0.50
0.60
0.70
0.80
Voidage, £
Figure 5-45- Comparison of the proposed model with the
experimental data on ternary suspension.
0.90
105
Experimental Results (Upper Interface)
Experimental Results (Middle Interface)
R
Experimental Results (Lower Interface)
Predicted from the Proposed Model
Suspending Medium: 60% Glycerol
d^ = 0.0460 cm
d
«
d|^ = 0.0194 cm
d^ = 0.0081 cm
C^ = 0.207
(M
.
O
C = 0.0518
u
CD
CO
F
CJ
0\
^
-l->
•r—
(O
^^
.
o
CJ
o
r^
CD
:>
CD
CJ
03
(M
^^
<+-
o
i-
CD
-M
E
a
s
•
o
o
0.40
0.50
T
T
0.60
0.70
Voidage, c
0.80
Figure 5-46. Comparison of the proposed model v.'ith the
experimental data on ternary suspension.
0.90
10'
suspensions are needed.
Nevertheless, the present model does appear
to be quite staisfactory in describing the sedimentation of ternary
particle systems.
Behavior of the Models at High Reynolds Numbers: Of particular importance is to study the validity of the model for a wide range of
Reynolds numbers. Sedimentation experiments provide data only in the
low Reynolds number range.
Data at higher values of Reynolds number
may be obtained from co- or counter-current solid-liquid vertical
flow operations.
The data collected by Lockett and Al-Habbooby
(1973) from countercurrent flow experiments covered a range of
Reynolds number of about 79 to 546. These data are 1 isted in Table
A.7.
Unlike sedimentation, continuous countercurrent operations of
binary suspensions do not give rise to two zones. However, the
physical system is similar to the lower zone of a binary sedimenting
suspension and slip velocities of each size particles are calculated
using Equations (3.12) and (3.13) developed in Chapter three.
Slip
velocities predicted from the present model are compared with the
observed values as shown in Table A.7. The first three columns of
the table list the flowrates of the liquid, the large size particles,
and the small size particles, respectively.
Columns four and five
give the concentrations of the large and small size particles.
Columns six through nine list the predicted and observed slip velocities for each size.
The last two columns show the percentage de-
viation of the predicted velocities from the observed values.
Table 5.3 shows the average percentage deviation, the m.ean-error.
107
u
CD
CO
o
LD
CO
E
O
•r—
-M
03
CD %.
- E CD
-M Q.
O
-E
+J -M
•I—
E
2 CD
SCO
r— : 3
CD CJ
s_
-a so CD
s: +EJ
13
-o
CD O
E
O
CO
-i->
E
cn
o
o
+J
o
03
• r - C\J
>
c\j
I — CD
03
N
E -f<.r) (Ji
Q.
^
CD
Q
03
Q.
X
CD
S03
"O
E
03
Z
t/)
KI
I
03
-o
O
CU
i(ZL
SCD
SO
E
ZS
CD
CD CD
SM
03-1_ J <>0
CM
cn
- E CJ
CO
"r— E
1
O
-O
Z3 > >
Q . JO
O
>^
1 —
o
o
CM
CM
o
<J^
JO
CO SO
Zi 03
3:
•r—
1
> 1—
CD c C
SQ_ • u
E
• o 03
E
03 -f-J
4->
CD
CD J^
CO CJ
O
Q
1
O
s_ ^ Q-
LO
r—
03
o
SUJ
I
E
03
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E -r-
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<y) (J-)
CD
CD CD
SM
_i
C>0
I—
03
CD
N
c
o
cr>
CD
N
00
CO
o
o
CM
I
CM
CO
CM
-o
o
o
<V 03
-E +j
-i-J 03
Q
MO r—
03
E +J
E
O
CO CD
•r—
E
S- • 1 —
03 SQ. CD
f= C L
O
X
CJ LU
ro
LO
E
O
+J
03
>
CD
Q
CM
E -r-
C/1 OO
CM
<m
CM
CM
CM
CD
CD
03
+J
E
CD
CJ
idJ
QCD
CD
r3
S_
CD
>
CTi
C30
O
CD
c n CD
SN
03 - r _ J U^
o
CO
O
CM
CD
^—
JZi
03
-D
E
03
CD
>^
-O
O
O
-M XJ
+ J - Q ..—s
O) 03 c^
^^ or r^
CJ
O
E
X3
O
E
CO
ro " O
S- - ' - ro 03 CTi
IN - E 1 ^
S- CJ CTi
1
o .—
—1 < :
CT>
r—
•r— T—
•
1
2: cr —'
o
CJ
i/";
r~^
f
V
-—\
^
Cl.
108
and the standard deviation for all three models.
The results in
Table 5.3 show that predictions from the published models do not
compare well with the experimental data.
On the other hand, the pre-
sent model represents the experimental data fairly well.
The super-
iority of the present model over the published models and its validity over a wide range of Re is thereby established.
Comparison of the Proposed Model with the Experimental Data on Suspensions with Continuous Size Distribution:
There is no reason for
the progressive settling analysis, which is apparently successful in
the case of discrete size distribution, not to be successful for the
progressive settling of continuous size distribution.
In fact, any
suspension can be regarded as being composed of a large number of
closely sized fractions.
In making the progressive settling computa-
tions with the distribution broken into various numbers of size
species. Smith (1966) found that it is sufficient to break down the
distribution such that the ratio of the species-to-species average
diameter is 1.2.
In the U.S. sieve series, the sieve interval, which
is the ratio of two successive sizes of screen openings in the series,
is V2( = 1.19).
Accordingly, two kinds of suspensions were prepared;
the first kind consisted of a mixture of two successive size-fractions,
and the second of three successive size-fractions.
Tables A.8 and
A.9 show the experimental data and further analysis for the two-species
and three-species suspensions, respectively.
The first two columns
of Table A.8 list the diameter of each size-fraction.
Columns
three and four of the table show the concentrations of the two
^4
-<^»^','
109
fractions.
Columns five and six list the predicted and observed
interface velocities.
The last column list the percentage deviation
of the predicted velocities from the observed values.
Similarly,
the first three columns of Table A.9 list the diameter of each sizefraction.
Columns four through six show the concentrations of the
three fractions.
Columns seven and eight list the predicted and ob-
served interface velocities.
The last column shows the percentage
deviation of the predicted velocities from the observed values.
Figures 5.47 through 5.49 display the plots of the observed and predicted velocities versus the voidage for the two-species suspensions.
Similarly, Figures 5.50 through 5.52 show the plots of observed and
predicted interface velocities versus the voidage for the threespecies suspensions.
It is clear from these tables and figures that
the proposed model satisfactorily represents the experimental data.
Error analysis for the two-species suspension gives an average percentage deviation of 8.23 and a mean-error of -3.24.
The average
percentage deviation and the mean-error for the three species suspensions are 16.24 and -16.24, respectively.
This indicates that the
predicted velocities are underpredicted in the latter case.
These
errors are of the same order of magnitude as those of the top interface for binary suspensions and ternary distinct sized suspensions.
Thus suspensions with continuous size distribution show the same
settling pattern as those with discrete size distribution.
Furthei--
more, the proposed model can be used to predict the sedimentation
velocities of suspensions with continuous size distribution.
no
OO
CM
•
-n
Experimental Results
Predicted from the Present Model
Particles:
Fractions 1 and 2
Suspending Medium: Diethylene Glycol
<M^
O
O
CM.
u
CD
(/)
CJ
o
+J
CJ
o
CD
CM
CD
CJ
03
MSCD
4->
E
OO
o.
«
o
3*
o
o.
0.40
0.50
T
0.80
T
0.70
0.80
0.90
Voidage, £
Figure 5.47.
Comparison of the proposed model with experimental
data on suspensions with continuous size
distribution.
Ill
O
•
Experimental Results
—
Predicted from the Present Model
Particles:
Fractions 5 and 6
Suspending Medium:
Diethylene Glycol
CO
o
in
o
CJ
CD
CO
CJ
rr
o* — » - _
-i->
•r-
u
o
CD
>
en
CD
^
ro
^s-
O,
rA
O
CD
CM
O
«
o
o
o
0.40
0.50
0.60
0.70
0.80
0.90
Voidage, £
Figure 5.48.
Comparison of the proposed model vyith experimental
data on suspensions with continuous size
distribution.
112
o
Experimental Results
o
Predicted from the Present Model
Particles:
Fractions 8 and 9
Suspending Medium: Diethylene Glycol
OJ
o
o
CD
o
o
u
CD
CO
u
'JO
o
o
CJ
o
CD
CD
O
03
M-
o
c
s_
<v
o
CM
o
o
o.
0.40
0.50
Figure 5.49.
—1
I
0.60
0.70
Voidage, £
0.80
0.90
Comparison of the proposed moael with experimental
data on suspensions with continuous size
distribution.
113
ZT
—•_
«
O
CM
•
Experimental Results
Predicted from the Present Model
Particles: Fractions 1, 2, and 3
Suspending Medium: Diethylene Glycol
O
o
o
a
CD
CO
CJ
OO
o
CJ
o
OJ
CD
CJ
03
CO
o
t °
CD
+J
3*
O
CM
O
O
o
0.40
0.50
~~1
0.60
\—
0.70
Voidage, £
0.80
0.90
Figure 5.50. Comparison of the proposed model with experimental
data on suspension with continuous size
distribution.
114
o
Experimental Results
Predicted from the Present Model
Particles:
Fractions 3. 4, and 5
Suspending Medium: Diethylene ^ilvcol
CO
o
in
o
u
CD
CO
CJ
o^ ' — j
•
O
«
>^
-M
•r—
O
O
CD
CD
U
03
MSCD
CO
o
CM
O
•
o
o
o
0.40
0.50
Figure 5.51.
0.60
0.70
Voidage, £
0.80
0.90
Comparison of the proposed model with experimental
data on suspension with continuous size
distribution.
115
Experimental Results
Predicted from the Present Model
Particles:
Fractions 5, 6, and 7
Susoending Medium: Diethylene Glycol
OO
O
u
CD
CO
a
O
4->
O
o
CD
OJ
U
03
<4SCD
+J
o,
o
o
o
0.40
0.50
O.SO
0.70
0.80
0.90
Voidage, £
Figure 5.52.
Comparison of the proposed model i.'ith experimental
data on suspension with continuous size
distribution.
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
A physical model for sedimentation of binary suspensions was first
developed by Lockett and Al-Habbooby (1973).
This model was based on
the assumption that a particle settles only according to the local voidage fraction around it, regardless of whether its neighbors are particles of the same or of other sizes and whether they are moving relative to it or not.
This model, therefore, does not take into account
particle collisions and interactions arising from the relative velocities of the particles.
When tested experimentally, the model was
found to overpredict the sedimentation velocities by between 5 and 70-o.
Mirza and Richardson (1979) extended the Lockett and Al-Habbooby model
for multisized particle suspensions.
and collisions were neglected.
Again, interparticle interactions
In order to match the predicted velo-
cities with the observed values, the predicted velocities were multiplied by a factor equal to the voidage raised to the power 0.4.
By
introducing such a correction factor, they were able to represent the
data reasonably well for binary and ternary suspensions.
However, no
0 d
physical explanation was provided for the correction factor £ ' .
Clearly, such an empirical correction cannot be aoplied over a wider
range of conditions than those used in their experimental work without
further investigation.
116
117
In the present study, a model was developed for the sedimentation
of multisized particle suspensions which takes into account interparticle interactions.
These interactions were accounted for by taking
into consideration the buoyancy effect induced by the smaller size
particles on the terminal falling velocities of large size particles.
The new model and the two previously published models, those of Lockett
and Al-Habbooby and of Mirza and Richardson, were tested against a large
number of published and newly collected experimental data on binary and
ternary suspensions.
It was found that the proposed model is the m.ost
accurate and represents the experimental data satisfactorily.
Qy suc-
cessful representation of the Lockett and Al-Habbooby data on binary
countercurrent operations, the model was proven to be valid over a wide
range of Reynolds number.
Furthermore, the model was found to apply for
suspensions with continuous size distribution.
In making computations
for continuous size distributions, it was found sufficient to break down
the distribution into closely sized fractions such that fraction-tofraction average diameter ratio is 1.19.
It is recommended that the present model be used to predict the
sedimentation velocities of suspensions with discrete or continuous
size distributions.
Also, the model can be used to predict particle
velocities in multisized co- or counter-current solid-liquid operations.
Recommendations for Further Work
1.
The model was shown to agree favorably well with the experi-
mental data for ternary suspensions and those with continuous size
distribution.
However, because of the meager data base used, more
118
experimental data and further verification of the model may be complimentary.
2.
The model should be extended to include the sedimentation of
multicomponent mixtures of particles of different densities and different sizes.
3.
Sedimentation data for fine particles of diameter less than
100 micron is required and a new correlation for the sedimentation of
equisized fine particle suspensions needs to be developed.
4.
An attempt should be made to develop the shape factor or equi-
valent diameter in the case of sedimentation of non-spherical particle
suspensions.
5.
An experimental technique for monitoring the position of the
solid-liquid interface during the sedimentation of opaque suspensions
needs to be developed.
In this connection, the method introduced by
Raffle (1976) to study the settlement and consolidation of concentrated
suspensions may be used.
By continuously monitoring the pressure vari-
ations at various depths during sedimentation, the position of the interface may be determined.
NOMENCLATURE
A
= Asymptotic value of U^/U^ at low Re (Equation 2.57)
A^
= Cross-sectional area of vessel or tube
ap
= particle acceleration
B
= Asymptotic value of U^/U^ at high Re (Equation 2.57)
C
=
r
Cp
Fractional volumetric concentration of particles
n
(TT/6)d (p_-p.p)g
= Drag coefficient =
^—i
1/2 p^(Tr/4 d^)U^
%
=
[4d(Pp-p^)g/(3, U j ) ] [ ( l - C ) / ( U C ^ / ^ ) ]
D
=
Diameter of vessel or tube
d
= diameter of spherical particle
F
=
f.
= drag force on a particle
g
= Acceleration due to gravity
K
= Constant in Equation (2.57)
n
= Exponent of £ in Equation (2.57)
R,
= Drag force per unit projected area of isolated spherical
Force on a single particle in suspension
particle
R'
= Drag force per unit projected area of spherical particle
in suspension
t
= time
U
=
S e t t l i n g v e l o c i t y of p a r t i c l e in suspension
U^
=
Velocity of displaced f l u i d
U.
=
S e t t l i n g v e l o c i t y at i n f i n i t e
U
=
Superficial v e l o c i t y of f l u i d
119
dilution
120
U^
= Terminal velocity of a single particle in finite fluid
medium
^too
" Terminal velocity of a single particle in an infinite fluid
medium
V
= Volume of a single particle
X
= distance
z
= constant in Equation (2.57)
Greek Letters
£
= voidage (fluid fraction in suspension)
<^^
= [1 - 1.21(l-£)2/^]"''
y^
=
Viscosity of f l u i d
]i
=
Viscosity of suspension
Pr
=
density of f l u i d
pr
=
Mass of f l u i d per unit volume of f l u i d - p a r t i c l e mixture
p
=
density of p a r t i c l e s
p
=
Mass of p a r t i c l e s per u n i t volume of f l u i d - p a r t i c l e mixture
r
p
= density of suspension
Dimensionless Groups
3 '
2
d p^(p -p.p)g/p.p
Ga
= Galileo number
Re
= Particle Reynolds number dU,p-,/u^
Re^
= Fluid Reynolds number dU p^p/y^
Re^
= Particle Reynolds number dU^^pVy^p
Re,
(p
=
U /U,
Modified Reynolds number Re (
- r r / o M r\- )
-^
00
exp^.
5 L / 3 ( 1 - C ) ,,
121
Subscripts
L
=
Large particle
S
=
Small particle
a-m
=
Sizes of particles (smallest to largest)
1-M
=
Settling zones (top to bottom)
BIBLIOGRAPHY
1.
Barnea, E. and J- Mizrahi, "A Generalized Approach to the Fluid
Dynamics of Particulate Systems Part 1. General Correlation f o r
F l u i d i z a t i o n and Sedimentation in Solid M u l t i p a r t i c l e Systems,"
Chem. Enq. J . . Vol. 5, pp 171-189 (1973).
2.
Becker, H. A., "The Effects of Shape and Reynolds Number on Drag
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The Can. J . Chem. Eno. . Vol. 37, pp 85-91 (1959).
3.
Bedford, A. and 0. D. H i l l , "A Mixture Theory Formulation for
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4-
Brinkman, H. C., "A Calculation of the Viscous Force Exerted by
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5.
Coe, H. S. and G. H. Clevenger, "Methods for Determining the
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6.
Davies, C. N. , " D e f i n i t i v e Equations for the Fluid Resistance of
Spheres," The Proceedings of the Physical Society, Vol. 57,
pp 259-270 (1945).
7.
Davies, R., "The Experimental Study of the D i f f e r e n t i a l S e t t l i n g
of P a r t i c l e s in Suspension at High Concentrations," Powder
Technology, Vol. 2, pp 43-51 (1968/69).
8.
F i l d e r i s , V. and R. L. Whitmore, "Experimental Determination of
the Wall Effect f o r Spheres F a l l i n g A x i a l l y in C y l i n d r i c a l
Vessel," B r i t . J . Appl. Phys., Vol. 12, pp 490-494 (1961).
9.
F i n k e l s t e i n , E., R. Leton, and J . C. E l g i n , "Mechanics of Vertical
Moving Fluidized Systems with Mixed P a r t i c l e Sizes," A.I.Ch.E.
Journal, Vol. 17, pp 867-872 (1971).
10.
Francis, A. W., "Wall Effect in Falling Ball Method for V i s c o s i t v ,
Physics, Vol. 4, pp 403-406 (1933).
11.
Garside, J . and M. R. Al-Dibouni, "Velocity-Voidage Relationship
f o r F l u i d i z a t i o n and Sedimentation in Solid-Liquid Systems,"
Ind. Enq. Chem., Process Des. Dev., Vol. 16, pp 206-214 (1977).
12.
Goldestein, S., "The Steady Flow of Viscous Fluid Past a Fixed
Spherical Obstacle at Small Reynolds Number," Proc. Roy. Soc. ,
Vol. 123 A, p 225 (1929).
122
123
13.
Gurel, S., S. G. Ward, and R. L. Whitmore, "Studies of the
Viscosity and Sedimentation of Suspensions Part 3. - The Sedimentation of Isometric and Compact Particles," Brit. J. Apol.
Phys., Vol. 6, pp 83-87 (1955).
14.
Happel, J., "Viscous Flow in Multiparticle Systems: Slow Motion
of Fluids Relative to Bed of Spherical Particles," A.I.Ch.E.
Journal, Vol. 4, pp 197-201 (1958).
15.
Happel, J. and H. Brenner, "Low Reynolds Number Hydrodynamics,"
Prentice-Hall, Englewood-Cl iffs, N.J., 1965.
16.
Hawksley, P. G. W., "The Effect of Concentration on the Settling
of Suspensions and Flow Through Porous Media," Inst, of Phvs.
Symposium, p 114 (1950).
17.
Heyv/ood, H., "Measurements of the Fineness of Powdered Materials,"
Proc. Inst. Mech. Engrs. (London), Vol. 140, pp 257-308 (1938).
18.
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19.
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20.
Jottrand, R., "An Experimental Study of the Mechanism of Fluidization," J. Appl. Chem., Vol. 2, pp S^^-S^g (1952).
21.
Jottrand, R., "Calculations of Terminal Falling Velocity of
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(1958).
22.
Kunkel, W. B., "Magnitude and Character of Errors Produced by
Shape Factors in Stokes' Law Estimates of Particle Radius,"
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23.
Lapidus, L., "Digital Computation for Chemical Engineers,"
McGraw-Hill, New York, NY, 1962.
24-
Lapidus, L. and J. C. Elgin, "Mechanics of Vertical-Moving
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25.
Lewis, E. W. and E. W. Bowerman, "Fluidization of Solid Particles
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26.
Lewis, W. K., E. R. Gilliland, and W. C. Bauer, "Characteristics
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27.
Letan, R., "On Vertical Dispersed Two-Phase Flow," Chem. Eng.
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28.
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29.
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~
30.
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31.
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^
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APPENDIX A
EXPERIMENTAL DATA AND COMPARISON WITH
PREDICTIONS FROM PROPOSED MODEL
126
127
Table A . l .
Comparison of the Proposed Model with the Experimental Data
for Binary Suspension in Ethylene Glycol
!
I n t e r f a c e V e l o c i t y , cm/sec
Concentration of P a r t i c l e s
voluTC f r a c t > o n
Laroe
Small
1
j
Lower i n t e r f a c e
Predicted
Exoerimental
i,
= 0.0460 cin
Upper i n t e r f a c e
E.xoeriniental
Predicted
dj '
a t i on
Percent Devi
^ C » 1 «* w ' \ * • '
!
LoweInterface
uppeInter-ace
-1.45
-5.72
-4.55
-4.01
- c . oS
\SE
-. . - 3.47
-1 SC
- 3. ^'
0.0194 CT
C.199
0.239
0.268
C.29a
0.33&
0.0696
0.0696
0.0696
0.0696
0.0696
0.163
0.123
0.0957
0.078C
0.0560
G. 170
0.124
0.100
0.0R05
0.0582
0.0725
0.0625
0.0547
0.0465
0.0359
G.0735
0.0663
:.0573
0.0454
o.o:':-
-4.12
-1.13
-1.3';
-3.06
-2.7C
0.139
0.179
0.219
0.259
0.295
0.119
0.119
0.119
0.119
0.119
0.170.131
0.0959
0.0706
O.C504
0.181
0.133
0.0960
0.0692
0.0494
0.0570
0.045=
C.0416
0.0336
0.0260
C.0559
0.0^93
0.414
0.0342
0.0270
-1.8'
0.97
2.01
' , 53
0.158
0.168
0.168
0.166
C.16B
0.0696
0.201
0.151
0.111
0.0804
0.0566
0.212
0."'56
0.113
0.0826
0.0581
0.0791
0.0558
0.0390
0.0271
0.0183
0.0769
0.0530
0.0373
0.0271
C.0187
-:.
0.217
0.217
0.217
0.21?
0.217
0.0696
0.144
0.106
0.0758
0.0536
0.0365
0.141
0.''03
0.0759
0.0537
0.0362
0.C681
0.0459
0.03G6
0.0204
0.0132
0.0705
0.0439
0.0311
C.0210
0.0140
:.9G
2.60
-G. ' 4
-0.26
0.69
-3.35
•1.4'
-•' .56
-2.84
-e.s:
o.no
0.151
0.191
0.232
O.no
0.151
0.191
C.232
d^ = 0.0460 cir
i-i
£ . C' •
-3.21
-1.38
-2.62
-2.64
5. 3c
- . - J.
COG
-1 .93
_ •
^Ci
d<; = 0.0137 cir
0.197
0.238
0.27S
C.319
0.359
0.0595
0.0595
0.0595
0.0595
0.0595
C.179
0.134
0.0999
0.0723
0.0517
0.180
0.132
0.101
0.0743
0.0521
0.0^50
0.0418
0.0373
0.0327
0.0270
0.0483
0.0445
C.040S
0.0348
0.0289
-0.63
^ ^C
-1.12
-2.6S
-0.84
G. 139
0.179
0.219
0.259
0.299
0.119
0.119
0.119
0.119
0.119
0.176
0.133
0.0985
0.0719
0.0513
0.173
0.129
0.0948
0.0684
0.0476
0.C313
0.02':'.5
0.0254
0.0219
0.0181
3.0327
0.0285
0.0251
0.0217
0.0182
1.5C
2.81
3.9^
5.04
7.77
0.168
0.168
C.16c
0.158
0.168
0.0696
0.110
C.151
0.191
0.232
0.203
0.153
0.113
0.0827
0.0587
0.201
0.149
0.11c
0.0322
0.0530
0.0433
0.0315
0.0225
0.0159
0.0109
0.0433
0.0306
0.0218
0.0160
D.011C
0.88
2.75
2.S7
0.59
1.13
0.00
3.03
3.12
-0.72
0.21''
C.217
0.21""
o.:i"
0.0695
0.110
O.iSi
0.191
0.145
0.107
0.0775
0.0552
0.147
0.109
0.0762
0.0520
C.0398
0.C278
0.0185
0.0123
0.0421
0.0287
C.0193
-1.40
-1.59
-5.3c
-3.3'
-1.34
-Z.59
d^ = C.0326 cn
•J .J\
JC
1 .67
6.13
-6.08
-7.37
-6.01
-o.c5
.
1 -^
c.co
1 1£
C.S^
-G.'io
.1',
01
•»
Of - CO'. 3
0.0993
0.159
0.21°
0.278
0.337
0.0797
0.0797
0.0797
0.C797
0.0797
C.'i47
G.lOO
C.C563
0.0421
0.':257
0.149
C.102
C.0686
0.0402
C.C257
0.0422
0.0372
0.0308
0.C232
0.0159
C.0131
0.0335
C.G320
0.0237
0.016"'
-1.3"
-", .93
0,0993
0.139
0.179
0.229
0.256
0.139
0.139
0.139
0.139
0.139
0.0938
0.0752
C.C552
0.0380
0.0274
0.10^
0.C743
0.0555
0.C355
0.0252
0.0274
0.C242
C.C2S4
0.0246
C.C21C
0.0162
0.0129
-2.13
;.13
1 .19
4.2c
e.9i
0.126
C.126
C.126
0.126
0.126
0.0595
0.119
G.17e
C.238
0.297
0.142
0.0950
C'.Ooi:'
C.0381
0.0226
C.146
G.."93:J.0C3E
0.0392
0.0225
0.04c&
0.0296
0.2184
0.J2C7
0.0162
0.0126
o.onc
0.00635
- 3 . 35
4.6C
O.OC
_ •"
u.j3or
0.0^9.
''.0112
3.00660
CC
" - • **
-2.91
- te .
•' '
3.45
-3.40
- 0. ^
1 •* C
- 1 . .3
-\5f
-'• 7£
-6.^9
- - , ; J
" r•- "
. u-
^ "^ 7 ^
128
Table A . l .
—r
1
(continued)
I
In : e r f a c e Ve l o c i t y , c-n/se:
1
1
t
1
Concentrat i o n o f P a r t i c l e s
volume f r a c t i o r
1
Large
S.Tiall
P-edictt'd
£ x p e n Tie;" I d 1
Predicted
0.200
0.200
0.200
0.200
0.200
C.C72S
0.120
0.159
J.199
U.C239
..0793
C.055G
0.0406
0.0238
0.ei5V
C.C695
0.0515
C.0382
0.0270
0.0179
0.335;:
G.C225
0.0155
0.010^
0.03^^92
LOwer I n t . : r f a c e
d, 0 .0326 cm
0.0993
0.0149
0.209
0.268
0.328
0.0524
0.0624
0.06J4
0.0624
0.0524
0.165
0.121
0.0311
0.0528
C.C32B
•^
r.^Zi'•i
0.122
0.0752
0.0523
0.0326
•Jcoar
'^s -
:3ev:afior
inte-'ace
Experi-'^rita 1
" , •^ *
~
0.:22b
0.0137
c.oi:^
C.GC7jt
:.ower
Inter-'oce
13.4:
7
~C
6.40
c . .'c
11 .^'
•.DV9'^
Interface
4 '7
-".33
1
•^',
G.03
-1.9/
0.0081 cr,
0.0174
C.0i67
0.0157
0.0142
0.012C
0.0155
0.0175
0.017:
0.0155
0.G130
2 . •:•?
3 . 4»
C'.5v
0.52
.-.12
- C . DC
.-.'' Q'
- -,. ; r
-
• .
T>~
129
Table A.2.
Comparison of the Proposed Model with the Experimental Data
for Binary Suspension in Diethylene Glycol
I
I n t e r f a c e Veloci t y , cm/sec
Concentration of P a r t i c l e s
volume frac; t i o n
Large
Small
t
!
Lower ! i t e r f a c e
Predicted
Upper In t e - f a c e
Experimental
Predictec
- 0.0460 cm
d^ = 0.0194 err.
i
Experimenta1
^ ^s . . ^ A n ^
re'^N-cn.
Deviation
ucwer
Interface
Upper
Interface
C.102
0.153
0.204
0.255
0.306
0.0612
G.0612
0.0612
0.0612
0.0612
0.194
0.141
0.100
0.0696
0.0469
0.196
0.142
0.0931
0.0695
0.0477
0.0579
0.0531
0.0467
0.0385
0.0294
G.0525
0.0499
G.C442
C.G35'.
0.0313
-C.3G
-0.5£
2.41
0.00
1G.2"
6.42
r.l'l
- 1.76
-6^0^
0.102
0.153
0.204
0.255
0.306
0.122
0.122
0.122
0.122
0.122
0.130
0.0922
0.0635
0.0425
C.0274
0.131
0.0901
0.0602
0.0425
0.0280
0.0374
0.0321
0.0262
0.0200
0.0143
0.C343
0.0298
C.0250
0.0203
0.0151
-0.39
2. 35
5.5C
O.OC
-2.C3
8.9C
7.65
4.59
-\33
0.173
0.173
0.173
0.173
0.173
0.0714
0.112
0.153
0.194
0.235
0.115
0.0861
0.0630
0.0451
0.0315
0.112
0.0847
0.0620
0.0447
0.0317
0.0465
0.0326
0.0226
0.0155
0.0104
0.0438
0.0314
0.0216
0.0147
0.0106
2.9S
' .57
1.53
O.G'
-0.63
6.07
3. 71
/ ^^
3^21
0.225
0.225
0.225
0.225
0.225
0.0714
0.112
0.153
0.194
0.235
0.0802
0.0586
0.0417
C.0290
0.0196
0.0761
0.0569
0.0409
0.0284
0.0188
C.0392
0.0261
0.C173
0.0113
0.00724
0.0379
0.0257
0.0173
0.0119
0.00811
= 0.0460 cn
d - = 0.0137 cm
5.4C
2.91
3.00
2.11
4.22
«r
^
--
-
T:
1 .
^
3.36
1.55
,-1
f; I-,
-5.09
-Vz.h
0.102
0.153
C.2C4
0.255
0.3C6
0.0612
0.0612
0.0612
0.0612
0.0612
0.196
0.142
0.101
0.0701
0.0473
0.230
0.151
0.0992
0.0691
0.0472
0.0300
0.0285
0.0266
0.0239
0.0204
0.0348
0.0306
0.0268
0.0252
0.0228
-14.9C
-5.84
1.S9
1.47
0.15
-13.91
-6.35
-C.91
-5.16
-10.73
0.0986
0.149
0.199
0.249
0.299
C.120
0.120
0.120
0.120
0.120
0.137
C.0977
0.0681
0.0462
0.0303
0.153
0.101
0.0657
G.C441
0.0301
0.0202
0.0184
0.0162
0.0136
0.0103
0.0211
0.0192
0.0162
0.0136
0.0113
-10.34
-3.28
2.08
4.68
0.6£
-4.13
-4.26
0.00
0.188
0.188
G.ISS
0.188
0.188
0.0707
0.110
0.150
0.189
0.229
0.106
0.0796
0.0587
0.0428
0.0303
0.103
0.0851
0.0569
0.0421
0.0300
0.0250
0.0182
0.0129
0.00901
0.00614
0.0264
0.0192
0.0127
0.00904
0.00632
2.50
-2.34
3.14
1.63
G.90
-4.50
-5.27
1.24
-0.28
-2.83
0.225
0.225
0.225
G.225
0.225
0.0714
0.112
0.153
0.194
C.235
0.0810
0.0595
0.0427
0.0299
0.0204
0.0803
0.0582
0.0414
0.0285
0.0194
0.0233
0.0161
0.0109
C.00720
0.00467
G.0248
0.0163
0.0111
0.00764
0.00503
0.81
2.19
3.09
4.99
-6.00
-1.50
-2.19
- 5 . 72
- 0.0326 cm
d . - 0.0137 cm
1.46
1.34
3.66
£. 39
1C.62
- C. 6£
0.00
Z. Ic
- . .c \
0.102
0.153
0.204
0.255
0.30C
0.0612
0.0612
0.0612
C.0612
0.0612
0.0983
0.0713
0.0506
0.0351
0.0236
G.0969
0.0688
0.0465
0.0317
0.0223
0.0290
0.0266
C.0234
0.0193
C.0148
0.0236
0.0263
:.0234
0.0189
0.0153
0.102
0.153
0.204
0.255
0.306
0.122
0.122
0.122
0.122
0.122
0.0659
0.0466
0.0320
0.0214
0.0138
0.0670
0.0435
0.0302
0.0202
0.C139
0.0187
0.0161
0.0131
0.0101
0.00720
0.0192
C.0161
0.0135
0.0105
C.00772
-1.59
0.17^
C.174
0.174
0.174
0.174
0.0714
C.112
0.153
0.194
0.235
0.0579
0.0432
0.0315
0.0225
0.0157
0.0536
0.0398
0.0288
0.0212
G.02S2
0.0163
G.0H3
0.00771
C.00519
0.0233
0.0162
n.0112
C.00813
C.005r£
7.93
£.43
U. wI3 ^
-4.57
- J . ^ I
7
•
r\:
. w
6. G'
5.55
-C.£:
5.44
f. 3 3
4.55
-I. 53
G. OG
-3.54
-"*..:.-6.7:
-b . iC
0.45
G.5t
* - . . C
-£.23'
130
Table A.3.
Comparison of the Proposed Model with the Experimental Data
for Binary Suspension in Aqueous Glycerol
1
1
Interface Veloci t y . cr,/se:
Concentration o'f' Particles
volume •fraction
Large
Small
1
Lower Interface
Predicted
Experimental
1
;
'Jpoer Interface
Predicted
C ^ ^ ^ A ^^ ^
*'M r L
TI
Deviation
|
Experimental
Lower
Interface
upper
Interface
d. = 0.0326 cm dj = 0.0137
0.102
0.154
0.205
0.256
C.307
0.0615
0.0515
0.0165
0.0C15
0.0615
0.212
0.153
0.109
0.0756
0.C510
0.0237
0.0152
0.0111
0.0759
0.0521
0.0623
0.0571
0.0502
0.0515
C.031S
0.0678
0.0632
C.0561
G.0461
0.0343
-10.41
-5.36
-1.3G
-1.5£
-2.15
0.102
0.154
C.205
0.255
0.307
0.123
0.123
0.123
0.123
0.123
0.142
0.0997
0.0687
0.0459
0.0297
0.152
0.101
0.0680
0.0459
0.0297
0.0400
0.0343
G.02S0
0.0214
G.C153
0.042S
C.0359
C.0293
0.0229
0.0168
-6.59
-1.31
0.97
G.c:
C.OC
-6.-"
.4.45
-4.58
-5.45
-£.56
0.174
0.174
0.174
0.174
0.174
0.0717
0.112
G.153
0.194
0.235
0.125
0.0938
0.0586
0.0491
0.0343
0.129
0.0962
0.0697
0.0494
0.0342
0.0500
0.0351
0.0244
0.0167
0.0112
0.0560
0.0385
0.0261
0.0180
0.0119
-2.2'
-2.50
-1.50
-0.53
G.41
-13.76
-5.3"
-6.53
-7.37
--..•;
0.226
0.226
0.226
0.226
0.226
0.0717
0.112
0.152
0.194
0.235
0.0872
0.0538
0.0455
0.0316
0.0214
0.0372
0.0538
0.0463
0.0312
0.0221
0.0422
0.0282
0.0187
0.0122
C.00781
0.0459
0.0306
0.02G0
0.0131
0.00861
G.OG
O.OC
-1.78
1.35
-3 31
-£.05
0.102
0.154
0.205
0.256
0.307
C.C614
0.0615
0.0515
0.0615
0.0515
0.214
0.155
0.0764
0.0515
0.0228
0.C219
0.0207
0.0191
0.0170
0.0253
0.0229
0.0219
0.02C4
0.0179
-C.35
1.12
7.91
5.96
4.55
-9.5^
O.no
0.216
0.153
0.102
0.0721
0.0493
0.102
C.154
0.205
0.256
0.307
C.123
0.123
0.123
0.123
0.123
0.145
0.102
C.0702
0.0470
0.0304
0.148
0.0978
0.0548
0.0427
0.0277
0.0151
0.0139
0.0124
0.0107
0.00364
0.0151
0.0132
0.0119
0.0102
0.0C845
-2.26
3.57
10.04
9.77
-o. 1 -
-5.64
-
. J . . . -
-IC.09
- c . 72
-6.73
-6.9^
-5.2i
-4* . ^ U
-5.56
-0.35
-5.25
O.OC
5.16
4.24
4.43
2.30
131
Table A.4.
Comparison of the Proposed Model with the Experimental Data
of Smith (1965) on Binary Sedimentation
1
1
— —
Interface Veloci t y , cm/sec
1
Concentra tion of P a r t i c l e s ,
__ volume f r a c t i o n
1
1
1
Large
Predicted
Small
0
Lower Interface
Experimental
'
Expe^-i mental
Percent Deviat"on
Upper
Lower
Inte'-'^ace Interface
Upper Int erface
Predicted
= 0.G252 cm d . = G.0130 cm
0.099S
0.150
0.200
0.250
0.300
0.0250
0.0374
0.0500
0.0624
0.0749
0.208
0.142
0.0941
0.0602
0.0358
0.212
0.15S
0.0819
0.0545
0.0293
0.0915
0.0765
C.0580
0.0400
0.0254
0.0771
0.0714
G.0552
C.0400
C.020C
-1.88
-15.37
14.92
1G.33
25.53
15.7:
0.0874
0.131
0.175
0.218
0.262
0.0437
0.0655
0.0874
0.109
0.131
0.200
0.133
0.0852
0.0529
0.0308
0.232
0.138
0.0334
0.0530
0.0290
0.0811
0.0630
0.0449
0.0295
0.0178
0.0740
0.0615
0.0531
0.0310
0.0191
-14.C7
^3.57
2 21
= 3.24
6.25
26.52
9.55
2. 53
-15.53
-4.73
-£.?3
0.0624
0.093C
0.125
0.156
C.0624
0.0935
0.125
0.155
0.205
0.140
0.0915
0.0580
0.215
0.186
0.101
0.0655
0.0740
0.0560
0.0397
0.0265
0.C67C
0.0559
0.0465
C.0274
--.42
-24^79
-5.21
-11.38
IG.-C. 12
-1^64
-3.15
0.0437
0.0655
0.0874
0.109
0.131
0.0874
0.131
0.175
0.218
0.262
C.197
0.130
0.0825
0.0504
0.0289
0.248
0.138
0.105
0.463
0.250
0.0650
0.0462
0.0311
0.0199
0.0118
0.0635
0.0490
0.0346
0.0186
0.0104
-20.65
-5.67
-22.36
8.94
15.62
2.23
-10.04
6.72
13.08
<^L
7.•'6
' 0.0252 dm d^ = 0.0187 cm
0.0998
0.15C
0.200
0.25C
0.500
0.0250
0.0374
0.0499
0.0624
0.0749
0.207
0.140
0.0924
0.0585
0.0355
0.246
0.191
0.0975
0.0620
0.0332
0.165
0.118
0.0792
0.0506
0.0307
0.153
0.119
0.0333
0.0521
0.0312
-15.92
-26.51
-5.26
-5.54
6.89
3.32
-0.72
-4.92
-2.97
-1.55
C.0S74
0.131
0.175
0.218
0.262
0.0437
0.0555
0.OS74
0.109
0.131
0.197
0.130
0.0823
0.0503
0.0286
0.235
0.168
0.0823
0.0455
0.0246
0.147
0.102
0.0657
0.0407
0.0235
0.137
0.105
0.0701
0.0409
0.0290
-16.75
-22.72
0.00
10.61
17.51
7.27
-3.31
-6.21
-0.59
-19.05
0.0624
0.0936
0.125
0.156
0.0624
0.0936
C.125
0.156
0.202
0.135
0.0370
0.0541
0.214
0.136
0.104
0.0555
0.139
0.0976
0.0647
0.0411
0.158
0.126
0.0720
0.0546
-5.69
-0.70
-16.61
-2.59
-11.73
-22.72
-11.28
-24.75
0.0437
0.0655
0.0S74
0.109
0.131
0.0874
0.131
0.175
0.218
0.262
0.191
0.124
0.0765
0.0454
0.0249
0.222
0.135
0.0771
0.0506
0.0286
0.125
0.0849
0.C545
0.0337
C.0195
0.131
0.0351
0.0601
0.0446
0.0195
-13.96
-3.45
-0.82
-10.37
-12.96
-4.47
-1.39
-5.32
-24.51
0.00
0.0250
0.0374
0.049?
0.0624
0.0749
0.0993
0.150
0.200
0.250
0.300
0.197
0.129
0.0817
0.0492
0.0280
0.206
0.139
0.0934
0.0563
0.0338
C.124
0.0851
0.0564
0.0353
0.0219
0.128
0.0980
0.0C07
0.0391
0.0268
-4.58
-6.£5
-12.59
-12.57
-"7.04
-13!21
-7.17
-5. 25
-18.44
132
Comparison of the Proposed Model with the Experimental Data
of Mirza and Richardson (1979) on Binary Sedimentation
Table A.5
1
r/\f%/»(a»*^fc-a • i rtrt f\^
Da»«*irlAc
volume f r a c t i o n
Large
Small
j
I nterface Veloci ty, cm/sec
i
Percent D e v i a t i o n
Lower In terface
Predicted
Experimental
UoDer Inte'-'^ace
Predicted
1
Experimental
Lower
Interface
upper
Interface
* 0.0115 cm
'^L • 0.0462 cm d.
0.201
0.246
0.289
0.335
0.380
0.0579
0.0679
0.0579
0.0579
C.0679
0.0288
0.0208
0.0145
0.0102
0.00685
0.0320
C.0227
0.0164
0.0114
0.00^38
0.00537
0.00498
0.00451
C.00387
C.00313
0.00523
0.00500
0.00434
0.00392
0.00328
-10.12
-S.35
-C.8S
-1C.2C
-7.05
2.63
-'". 35
3.53
-i'.24
-4.65
0.217
0.246
0.272
0.296
0.324
0.156
0.156
0.156
0.155
0.155
0.0130
0.0102
0.00813
0.00641
0.00499
C.0147
0.0111
0.0911
0.0630
0.0488
0.00241
0.00216
0.00152
0.00168
0.00144
0.00255
0.00223
0.00213
0.C0180
0.00156
-11.45
-£.00
-10.71
1.77
-5.35
-3.23
-9.31
- c . 7"
-:.£1
C.308
0.308
0.308
0.308
C.308
0.0396
0.0652
0.0946
0.124
0.153
C.0161
0.0131
0.0102
0.00788
0.00519
0.0173
0.0141
0.0111
0.00336
0.00516
0.00581
0.00439
0.00318
0.00229
0.00138
0.00585
0.00449
0.00323
C.00254
0.C0158
-6.73
-7.01
-7.86
-5.79
-'
T:
'- z->
-'.'.. 0 5
- 2.2 C
,'
c .*
-9.7:
':Z.8i
= 0.0327 cm d<; = 0.0115 cm
•^ "\::
0.183
0.234
0.285
0.333
0.385
0.0588
C.C688
0.0688
0.0638
0.0688
0.0163
0.0113
0.00769
0.00457
0.00326
0.0190
0.0134
C.00870
0.00590
0.00370
0.00509
0.00447
0.00365
0.00279
0.00201
0.00520
0.00433
0.00337
0.00315
0.00236
-14.37
-15.36
-11.59
-15.34
0.178
0.211
0.243
0.275
0.306
0.141
0.141
0.141
0.141
0.141
0.00992
0.00769
0.00553
0.00451
0.0C341
0.0117
0.00872
0.00664
0.00480
0.00374
0.00277
0.00243
0.00209
0.00175
0.00143
0.00296
0.00253
0.00233
0.0C193
0.00155
- 15. 20
-11.80
-10.66
-6.01
-6.82
-6.32
-7.57
-10.39
-9.56
-13.48
0, 303
0.303
0.303
0.303
0.3C3
0.0388
0.0676
0.0977
0.128
0.157
0.00850
0.00672
0.00520
0.00396
0.00301
C.0105
0.00344
0.00600
0.00460
0.00370
0.00438
0.00344
0.00242
0.00170
0.00120
0.00545
G.00375
0.00273
0.00209
0.00159
-19.05
-11.99
-13.33
-13.92
-18.74
-10.38
-8.55
-11.44
- 18. 74
-24.33
• — . - ' -
-7.39
-4.57
-11.55
-14.72
= 0.0231 cm d<; = 0.0115
0.192
0.245
0.297
0.346
0.3S6
0.0563
0.0563
0.0568
0.0568
0.0558
0.00828
0.00559
0.00383
0.00255
0.0C153
0.00939
0.00595
0.00437
0.00282
0.00179
0.00470
0.00371
0.00273
0.00193
0.00128
0.00503
0.00381
0.00311
0.0C219
0.00153
-11.31
-4.31
-12.35
-5.31
-S.73
-7.53
-2.54
-12.17
-12.03
-16.G1
0.205
o!243
C.271
0 301
o'.327
0.135
0.135
0.135
C.125
0.135
0.00414
0.00305
0.00240
0.00184
0.00144
0.00442
0.00330
0.00247
0.00195
0.00163
0.00208
0.00164
0.00135
0.00108
0.00087
0.00230
0.00188
0.00154
0.00132
0.00112
-6.36
-7.61
-2.6"
-7.55
-11.55
-9.^"
• -12.59
-12.14
-18.^8
-21.60
0.315
0.315
0.315
c'.315
0.315
C.0402
C.0690
0.0953
0.124
0.152
0.00381
0.00298
0.00236
0.00180
0.00136
0.00413
0.C0333
0.00262
0.00173
0.00134
0.00296
0.00209
0.00153
0.00109
0.00078
0.00323
0.00241
0.00131
0.00126
C.00094
-7.68
-10.44
-10.04
1.07
1 .£5
-8.2;
-13.28
1 C
1"
-15 . .^'
- 1 . ; . 2u
-16.71
133
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APPENDIX B
COMPUTER PROGRAM FOR PREDICTION OF INTERFACE VELOCITIES
IN MULTISIZED PARTICLE SUSPENSIONS
139
1-^0
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
r
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
**:^*^
LIST
PROGRAM SYMBOL
C
D
DT
EPSLON
MP
NZ
PN
REo
ROHF
ROHP
ROMS
SCS
TCOMC
uc
UT
UTI
VISF
OF
c>KlNCIPAL
VARIABLES
***••
DEFINITION
CONCENTRATION OF PARTICLES
PARTICLE DIAMETER
TUBE DIAMETER
VOIDAGE
NUMBER OF PARTICLE SIZES
NUMBER OF ZONES
INDEX N
PARTICLE REYNOLDS NUMBER
DENSUITY OF THE FLUID
DENSITY OF THE PARTICLES
DENSITY OF THE SUSPENSION
SUM OF CONCENTRATIONS OF SMALLER
SIZE PARTICLES
TOTAL PARTICLE CONCENTRATION
SETTLING VELOCITIES OF =>A.^TICLES
TERMINAL VELOCITY OF A PARTICLE IN A
FINITE FLUID MEDIUM
TERMINAL VELOCITY Or A PARTICLE IN AN
INFINITE FLUID MEDIUM
VISCOSITY OF THE FLUID
COMPUTER PROGRAM
**:4ca)c;Jc:^:ic
^^ ^
^^^^
DIMENSION
D(10),C{10»10),UTI(10),UT(10),REP(10),
ilPNdG) ,UC(10,10)tC0LD{10)tRaHS(10),CNE'/(10)
*«*
READ DATA
**•
READ NUMBER OF PARTICLE SIZES
READ(5,1G0)NP
NZ=NP
READ PARTICLE DIAMETER AND CONCENTRATION
DO 1 1=1,NP
READ(5,101)D(I)tC(ItNZ)
CONTINUE
READ PHYSICAL PRQC^ERTIES
READ(5,102)R0HP,R0HF,VISF,DT
^^"^
WRITE DATA
WRITE(6,201)
WRITE(6,202)
WRITE{6,203)ROHF
WRITE(6,204)KQHP
WRITE{6,2C5 )VISF
***
Ul
WRITE(6,206)DT
WRITE{6,207)
WRITE(6,208)
WRITE(6,211)
DO 7 I=1,NP
WRITE(6»2 09)I,0(I),C(1,NZ)
C
C
C
C
C
CONTINUE
***
COMPUTATIONS FOR THE LOWEST Z J N E
^**
CALCULATE TOTAL CONCENTRATION
TCONC=0.0
DO 2 I=ltNP
TCONC = TCONC'»-C{I,NZ)
CONTINUE
CALCULATE VOIDAGE
C
c
C
EOSL0N=1.0-TC0NC
C
C
* CALCULATE TERMINAL VELOCITIES AND THE N-INDICES *
FOR PARTICLE SIZE KSMALLEST SIZE)
c
c
c
UTI(1)=D(1)*0(1)*981.0*(ROHP-ROHF)/(13.0*VISF)
UT(1)=UTI(1 )*( ( 1 .C-0.^73*D( l)/DT)/( 1.0-D(1)/DT) )*'^(-4)
REP( 1)=R0HF*UT(1 )=)'D ( 1 )/VI SF
RE9=REP(1)**0.9
PN( !)=( 5. 10+0.27*RE9)/(1.0-»'0.1*RE9)
C
c
C
DO 4
FOR P A R T I C L E
1=2,NP
SIZES
2,3,...tNP
IS=I-1
SCS=0.0
DO 3 J = 1 , I S
SCS = SCS-»-C( J , N Z )
CONTINUE
ROHS( I ) = ( EPSLCN*ROHF+RQri?^SCS)/{EPSLC!N-»-SCS)
UTI(I)=D(I)*u(I)*931.0«{R0HP-R0HS(I))/(18.0*VlSF)
U T U ) = U T I ( I )=^( ( 1 . 0 - 0 . 4 7 5 * D ( I ) / D T ) / ( 1 . 0 - D ( 1 ) / D T ) ) ' « (
REP(I)=ROHF*UT(I)*D(I)/VISF
RE9 = REt>( I )=!=*0.9
PN(I) = ( 5 . l O + 0.27='RE9)/(1.0-»-C.l*KE9)
CONTINUE
* CALCULATE S E T T L I N G V E L O C I T I E S *
C
C
c
5
6
DO 6 1 = 1 , N P
SUM=0.0
DO 5 J = 1 , N P
SUM = SUM + UT( J)*EPSL0N*=5^( °N ( J ) - 1 . 0 ) ^C ( J , N'Z )
CONTINUE
UC( I , N Z ) = U T ( I )=^E^SLGN«=*{PN( I ) - l . 0 ) - S U M
CONTINUE
WRlTE{o,21Q)
WRITE(6,212)
142
WRITE(6,200 )NZ,UC(NP,NZ)
C
C
«** COMPUTATIONS FOR THE REMAINING ZONES
ITER=NZ-1
DO 24 K=1,ITER
NP=NP-1
NZ=NZ-1
c
C
C
ASSUME INITIAL CONCENTRATIONS AS CCRRESOONOING
CONCENTRATIONS IN THE ZONE LOWER TO THE PRESE.NT
ZONE
c
c
c
c
c
c
c
c
c
c
c
c
8
9
10
EPSLON=1.0-TCONC
CALCULATE TERMINAL VELOCITIES AND N-INDICES
12
r
c
c
c
c
c
c
DO 8 1=1,NP
C( I,NZ)=C{I ,NZ + 1)
COLD(I>=C{I,NZ)
CONTINUE
CALCULATE TOTAL CONCENTRATION
TCONC=0.0
00 10 1=1,NP
TCONC=TCONC+C(I,NZ)
CONTINUE
CALCULATE VOIDAGE
11
c
c
*^*
13
00 12 1=2,NP
IS=I-1
SCS=0.0
DO 11 J=1,IS
SCS = SCS-^C( J,NZ)
CONTINUE
ROHSC I )=( E^ SLON'J'ROHF+ROHP*SCS)/{EPSLGN-^ SCS)
UTI(I)=D(I)*D(I)*931.G*(RCHP-R0HS(I))/{18.0*VISF)
UT(I)=UTI(I)*((1.0-0.^7 5-D(I)/DT)/(1.0-0(I)/DT))**(
R E P d )=ROHF^UT(I )*D(I )/VISF
RE9 = REP(I)^^0 .9
PN(I)=(5.10+0.27«RE9)/(1.0+C.1*RE9)
CONTINUE
ASSUME VELOCITY OF THE LARGEST SIZE ^ARTICLE
TO BE THAT OF IT IN THE ZONE LOWER TO THE
PRESENT ZONE
UC(NP,NZ) =UC (N«^,NZ-»-l )
UCOLD=UC(Nt>,NZ)
CALCULATE CONCENTRATIONS ACCORDING TO
EQATION(3.34)
DO 14 1=1,NP
C ( I ,NZ) = (UC(NP-H,NZ-»-l )-UC{I ,NZ-H) )^C( I,NZ-H )/
C(UC(NP-^l,NZ-»-l )-UC(N^,NZ) )
143
C
C
14
CONTINUE
15
CALCULATE TOTAL CONCENTRATION
TCONC=0.0
DO 15 1=1,NP
TCONC = TCONC-^C( I ,NZ)
CONTINUE
c
c
c
c
c
c
c
c
CALCULATE VOIDAGE
EPSL0N=1.0-TC0NC
16
c
c
c
c
c
c
c
17
18
19
c
c
c
c
20
21
22
24
CALCULATE THE SETTLING VELOCITY OF THE
LARGEST SIZE PARTICLES
SUM=0.0
DO 16 J=1,NP
SUM=SUM>UT(J)*£PSLON*«(PN(J)-1.0)*C(J,NZ)
CONTINUE
UC(NP,NZ)=UT(NP)*EPSL0N^*IPN(NP)-1.0)-SUM
COMPARE THE NEW SETTLING VELOCITY WITH THE OLD
SETTLING VELOCITY FOR THE LARGEST SIZE PARTICLE
UCNEW=UC(NP,NZ)
ERROR=(UCNEW-UCOLD)/UCNEW
IF(A3S( ERROR) .LE.O.ODGO TO 17
UCOLD=UCNEW
GO TO 13
COMPARE THE NEW CONCENTRATIONS WITH THE OLD ONES
OERROR=0.0
DO 18 1 = 1,NP
CNEW(I)=C(I,NZ)
ERR0R=(CNEW(1)-COLD(I))/CNEW(I)
ERROR=ABS(ERROR)
IF(ERROR. GT.OERROR)OERROR = ERROR
CONTINUE
IF(0ERR0R.LT.0.01)G0 TO 20
DO 19 I=1,NP
COLD(I)=CNEW(I)
CONTINUE
GO TO 9
CALCULATE SETTLING VELOCITIES FOR ALL
PARTICLE SIZES
DO 22 1 = 1 ,NP
SUM=G.O
DO 21 J=l,r4P
SUM=SUM+Ut( J)*EPSL0N**(PN(J)-1.0)=*C(J,NZ)
CONTINUE
UC(I,NZ)=UT(I)*EPSLON-*(PN(J)-1.G)-SJM
CONTINUE
WRITE(6,200 )NZ,UC(NP,NZ)
CONTINUE
14:
C
C
c 100
lOi
102
200
201
202
203
204
205
206
207
208
209
210
211
212
FORMAT STATEMENT FOR READ AND WRITE
FORMAT( I D
FORMAT( 2F10.4 )
F10.3,F10.3,F10.2)
FORMAT(
F10.2,
12,21X,F7.3)
FOR^AT(
/,15X, X,»»** INPUT DATA
^^**)
FORMAT( ///,19
FORMAT( / / / / / , 3X,'PHYSICAL PROPERTIES: )
•FLUID DENSITY, GM/CUBIC CM =
FORMATt
F5 .3 )
FORMAT( //,5X, PARTICLE DENSITY, GM/CUBIC CM
F4 2)
?F5
F 0 R M A T ( /,5X,» FLUID VISCOSITY, POISE = •,F5.3)
TUBE DIAMETER, CM = •,F4.2)
FURMAT(
/,5X, •3Xf'PARTICLE DIAMETER AND CONCENTRATION:*
FORMAT( /f5X,»
DIAMETER
CONCENTRATION'
FORMAT( / / / / / , ,»SIZE NUMBER
FORMAT( ///,5X 2,9X,F6 4,8X,F6.4)
X,'ZONE NUMBER
FOKMAT(
INTERFACE VELOCITY')
CM
VOL.
FRACTION')
FORMAT( /,9X,I
FRACTION
1H1,10
FORMAT( 38X,
)
20X. 'CM/SEC
•
STOP
END
i;
**• INPUT DATA •••
PHYSICAL PROPERTIES:
FLUID DENSITY, GM/CUBIC CM = 1.165
PARTICLE DENSITY, GM/CUBIC CM = 2.43
FLUID VISCOSITY, POISE = 0.136
TUBE DIAMETER, CM = 3.14
PARTICLE DIAMETER AND CONCENTRATION:
SIZE NUMBER
DIAMETER
CONCENTRATION
CM
VOL. FRACTION
1
0.0081
0.0518
2
0.0194
0.0518
3
0.0460
0.1040
146
***
ZONE NUMBER
OUTPUT
***
INTERFACE
VELOCITY
CM/SEC
3
0.32009
2
0.08517
1
0.02056