University of Wollongong
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University of Wollongong Thesis Collections
1997
Velocity distribution of single phase fluid flow in
packed beds
Subagyo
University of Wollongong
Recommended Citation
Subagyo, Velocity distribution of single phase fluid flow in packed beds, Doctor of Philosophy thesis, Department of Materials
Engineering, University of Wollongong, 1997. http://ro.uow.edu.au/theses/1537
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VELOCITY DISTRIBUTION
OF SINGLE PHASE FLUID FLOW IN
PACKED BEDS
UNIVERSITY <
4GOMC
WOILONGOMC
U8RAF
_X I
A thesis submitted in fulfilment of the requirements
for the award of the degree of
Doctor of Philosophy
from
University of Wollongong
by
SUBAGYO
(Ir., U G M Yogyakarta)
Department of Materials Engineering
July 1997
Gusti Allah Mboten Sare
(God never sleeps)
n
DECLARATION
The work presented in this thesis is, to the best of m y knowledge and
belief, original except as acknowledged. I hereby declare that I have not
submitted this material, in whole or in part, for a degree at this or any
other university.
in
ACKNOWLEDGMENTS
I would like to express my sincere appreciation and gratitude to all who
contributed to this thesis with their experience, expertise and support. In
particular I a m deeply grateful to m y supervisor Professor Nick Standish
for his invaluable guidance, deep enthusiasm and encouragement
throughout the course of this project. Sincere appreciation is extended to
m y co-supervisors Dr. G.A. Brooks and Professor R J . Dippenaar for their
invaluable guidance, useful suggestions and encouragement throughout
this research work. I also thank Dr. S. Nightingale for providing a
computer facility.
I am greatly indebted to the board of management of PT Krakatau Steel,
Cilegon, Indonesia for giving opportunity and financial support, which
allowed this research to be carried out. Grateful acknowledgment is also
extended to ICMINET-ICMI, Jakarta, Indonesia for providing formative
financial support.
My sincere thanks are extended to the workshop and laboratory staff of
the Department of Materials Engineering, especially G. Hamilton and R.
Kinnell, for their assistance with various aspects of the research. Thanks
are also extended to m y student colleagues: D. Muljono, S. Street, W .
Setiadharmaji, D. Phelan, D. Rosawinarti, J. Jones, Tridjaka, Supramu, E.
Rukman, S. S o e d o m o and N. Ross for their constant encouragement and
support during the academic years. Grateful acknowledgment is also
extended to J. Giarini and A. Moerwanto for their invaluable help.
Last but not least, I would like to express my deepest gratitude to my
parents, Hardjo Suwito and Suharti, for their patience, constant support
and encouragement, which have helped m e to keep going throughout.
IV
ABSTRACT
A detailed knowledge of the flow distribution is required for further stu
of rate processes and their mechanism taking place in the packed beds. A
better understanding of the rate processes is essential for proper proces
design and process optimisation of packed bed systems to maximise the
comparative advantage and safety factor in industrial operations.
A new mathematical model of velocity distribution of single phase fluid
flow in packed beds was developed by assuming the flow characteristic is
a combination of a continuous and a discontinuous systems of fluids
between voids in the bed. In order to allow a comparison with data
measured at the downstream of the bed, the model was completed by a
new mathematical model of a developing flow profile in an empty pipe.
The model can be applied for both compressible and incompressible fluid.
The validity of the model has been checked using previous experimental
data and new measurement results. The agreement between measured
data and results predicted by the mathematical model is good. The new
model favourably compares with previous models in terms of accuracy
and its simplicity does not require new empirical constants.
It is clearly demonstrated that the fluid flow distribution in a packed bed is
influenced by the Reynolds number and the bed characteristics. However,
when the Reynolds number is higher than 500, the flow profile is mostly
determined by the bed characteristics. Moreover, it is also demonstrated
that the disagreement of previous investigators in the effect of Reynolds
number and particle diameter on fluid flow distribution in packed beds is
mostly due to limitation experimental.
Similar to the macroscopic view of fluid flow in packed beds, it can be
shown at the microscopic level that models based on discontinuous
systems are only successful for local porosity less than 0.5. In conditions
when the local porosity is higher than 0.5, especially in the vicinity of the
wall, a model based on the continuous systems approach, as used in the
present case, is more accurate.
The restriction for the flat flow profile assumption for packed bed systems
was investigated by using the present model. It is clearly shown that the
deviation of flat profile condition not only depends on the D/DP ratio but
also depends on the Reynolds number. The deviation of flat flow profile
condition was also used to investigate the possibility of generating a
distorted physical model in terms of D/DP ratio and L/D ratio.
VI
CONTENTS
DECLARATION
iii
ACKNOWLEDGMENTS
iv
ABSTRACT
v
CONTENTS
vii
LIST OF SYMBOLS
ix
1
INTRODUCTION
1
2
LITERATURE SURVEY
7
2.1
THE EXPERIMENTAL W O R K S
7
2.2
MATHEMATICAL MODELLING
33
2.2.1
The Phenomenological Approach
34
2.2.2
The Theoretical Approach
46
3
CHARACTERISATION OF A PACKED BED
76
3.1
PARTICLE SIZE
77
3.2
THE BED POROSITY
79
3.2.1
Mean Bed Porosity
80
3.2.2
Radial Distribution of the Bed Porosity
3.3
PERMEABILITY
119
4
DEVELOPMENT OF A MATHEMATICAL MODEL FOR
125
94
VELOCITY PROFILE OF FLUID FLOWING IN PACKED
BEDS
4.1
THE EQUATION OF FLOW THROUGH A SINGLE PIPE
126
4.2
THE EQUATION OF FLOW THROUGH A PACKED BED
130
4.2.1
The Equation of Continuity
135
4.2.2
The Incompressible Fluid
135
VII
4.2.3
The Compressible Fluid
4.2.4
Pressure Drop Correlation for Packed Beds
141
4.3
FLUID F L O W AT THE OUTLET O F THE BEDS
143
5
EXPERIMENTAL TECHNIQUES
157
5.1
EXPERIMENTAL APPARATUS A N D MATERIALS
157
5.2
EXPERIMENTAL P R O C E D U R E
161
6
EXPERIMENTAL
RESULTS
140
AND
MATHEMATICAL
166
M O D E L VERIFICATION
6.1
EXPERIMENTAL RESULTS
167
6.1.1
Reproducibility of Data
170
6.1.2
Measurement Results of Velocity Profile
170
6.2
MATHEMATICAL M O D E L VERIFICATION
6.2.1
Validation of the Mathematical Model
175
175
6.2.1.1 Uni-Sized Particle Packed Beds
178
6.2.1.2 Multi-Sized Particle Packed Beds
190
6.2.2
197
Comparison of the Mathematical Models of the Velocity
Profile in Packed Beds
7 DISCUSSION 205
7.1
C O M P U T E D VELOCITY PROFILES
205
7.2
SIMILARITY CRITERIA OF PACKED BED SYSTEMS
220
8
CONCLUSIONS
226
REFERENCES
228
APPENDIX
241
ALGORITHMS OF VELOCITY PROFILE CALCULATION
241
VI
LIST OF SYMBOLS
A
= area
{L~
A
= constant defined by equation (4-66)
Am
= constant defined by equation (2-41)
At
= transition region area
{L2
Aw
= wall region area
{L2
Az
= constant defined by equation (2-93)
Ao
= constant defined by equation (2-23)
Ai
= constant defined by equation (2-24)
A2
= constant defined by equation (2-25)
AE
= constant defined by equation (2-91)
a
= constant defined by equation (2-28)
a
= constant defined by equation (3-34) or (3-35)
a
= aspect ratio
at
- constant in equation (4-44)
a,
= constant in equations (6-1) and (6-3)
av
= surface are per unit volume of particle
B
= constant defined by equation (4-67)
Bp
= function defined by equation (2-71)
Bm
= constant defined by equation (2-42)
Bm
= constant defined by equation (2-42)
Be
= constant defined by equation (2-89)
Bz
= constant in equation (2-94)
B(n)
= constant in equation (2-57)
Bb(n)
= constant in equation (2-61)
Bt(n)
= constant in equation (2-63)
B w (n)
= constant in equation (2-66)
{ML"Y
{L~
{ML"Y
IX
b
= constant defined by equation (2-6)
{-}
b
= constant defined by equation (3-37)
{-}
b{
= constant in equation (4-44)
bi
= constant in equations (6-1) and (6-3)
{-}
h
= constant defined by equation (3-49)
{-}
C
= constant defined by equation (4-68)
{-}
*—m
= constant defined by equation (2-43)
{-}
C
= constant defined by equation (2-8)
{-}
Cj
= constant in equations (6-1) and (6-3)
{-}
C\
= calculation result of ith sample
H
c'
= constant defined by equation (3-24)
{-}
D
= column diameter
(L)
De
= effective channel diameter
(L)
DP
= particle diameter
{L}
Dpe
= equivalent packing diameter
{L}
D Pi
= particle diameter of ith component
{L}
Dpm
= m e a n size diameter of multi-sized particles
{L}
Dpv
= equivalent volume diameter
{L}
L-'vs
= volume-surface m e a n particles diameter
(L)
di
= constant in equations (6-1) and (6-3)
{-}
di
= data of ith sample
{-}
d
= sample m e a n
{-)
E
= constant defined by equation (2-22)
{-}
F
= force
Fh
= friction head
{L}
Fk
= friction force
{MLf 2 }
Fi
= first randomising factor defined by equation (3-
{-}
F2
41)
= second randomising factor defined by equation
{-)
(3-42)
{ML" 4 }
{MLt"2}
f
= constant defined by equation (2-82)
fk
= constant defined by equation (4-21)
fi
= constant defined by equation (2-47)
f2
= constant defined by equation (2-48)
Q
= constant defined by equation (3-18)
g = gravitational constant
gi
= constant defined by equation (3-46)
h
= constant defined by equation (3-50)
I
= inertia parameter of bed
Io
=- modified Bessel function of first kind,
Ic
= constant defined by equation (3-14)
Id
= constant defined by equation (3-13)
Jo
= Bessel function of first kind, order 0
K
= constant defined by equation (2-18)
Ko
= constant in equation (2-55)
k
= coordination number
k'
= constant defined by equation (2-4)
ki
= constant in equation (2-1)
k2
= constant in equation (2-1)
k0
= constant defined by equation (4-39)
ki
= constant defined by equation (4-40)
k2
= constant defined by equation (4-41)
4
= constant defined by equation (5-1)
fi2
= constant defined by equation (5-1)
L
= bed length
Le
= equivalent length
LM
= Prandtl's mixing length
Liu
= inlet length
1
= radial position in the bed that has porosity
{L}
equal to 0.5
k
= effective path length
lw
= loss work
M
= constant defined by equation (4-16)
m
= mass
N
= transformation function as given by equation
{L}
{ML2."2}
{-}
{M}
{MLY 3 }
(2-53)
o¥c
= digital computer number
[-:
Nj
= number of spheres with centres lying in the fh
{-}
cylindrical concentric layer
NRe
= Reynolds number
{-]
n
= number of segments
{-
nc
= constant in equation (2-27)
{-]
nd
= number of data
{-
n,
= particles number of ith component
{-
P
= dimensionless pressure defined by equation
[-
(4-61)
Ps
= pressure defect per unit length
{ML"Y2
P
= pressure
{ML"Y2
P
= particular layer of particles
Q
= flow rate
{{LV
Q.
= local flow rate
{LY1
q
= heat
R
= column radius
{MLY 2
{L
X
{Lf1
SH
= residual defined by equation (4-38)
/?
= reproducibility
{-
#"
= average error
{-
9.
= residual defined by equation (4-72)
{-
equivalent radius
{L
radial position
(L
distance from wall in particle diameters
{-
r
radius of outer edge of central core
{L
r
constant defined by equation (3-38)
{-
TH
hydraulic radius
{L
TM
constant defined by equation (2-26)
{-
rc
starting point radial position of the core region
{L
re
equivalent radial position
(L
rm
radial position of m a x i m u m superficial velocity
{L
S
entropy
S
dimensionless
Re
{MLY2T'
axial
position
defined
by
{-
equation (4-60)
s
constant in equation (2-21)
T
temperature
U
internal energy
7i
dimensionless
{{T
{ML 2 f 2
radial
velocity
defined
by
{-
equation (4-54)
uM
cross section average of superficial velocity
{Lf'
u
velocity
{Lf1
uz
local velocity
{Lf1
Ub
superficial velocity at the bulk region
{Lf1
u,
local velocity at t
{Lf1
Urn
local superficial velocity
{Lf1
XIII
u
= local superficial velocity at the edge of central
{Lf }
core
ut
= superficial velocity at the transition region
{Lf'}
uw
= superficial velocity at the wall region
{Lf'}
umb
= superficial velocity at bypass section
{Lf'}
umc
= superficial velocity at core section
{Lt"1}
Umr
= radial direction of superficial velocity
{Lf1}
umz
= axial direction of superficial velocity
{Lf1}
um0
= superficial velocity at the centre of the bed
{Lf'}
uzo(r)
= velocity profile at the top (outlet) of the bed
{Lf1}
UZLW
= fully developed flow profile in the empty pipe
{Lf1}
V
= volume
V
= dimensionless
{L }
axial
velocity
defined
by
{-}
equation (4-53)
Vj
= initial specific volume of ith component
{-}
-p
= dimensionless velocity defined by equation (4-
{-}
64)
-t
V
= specific volume
V.
= total volume of solid in the ith cylindrical
i
{L M }
{L3}
concentric layer
Vy
= volume of the solid in the ith cylindrical
{V}
concentric layer due to a sphere with centre in
the jth cylindrical concentric layer
W
= constant defined by equation (2-19)
{-}
Xb
= starting point of bulk region
{L}
Xi
= volume fraction of ith component
{-}
Xt
= starting point of transition region
{L}
x
= distance from wall in particle diameters
{-}
z
T
= axial position
{L}
= wall distance defined by equation (3-25)
{L"1}
Xl\
a
= constant defined by equation (2-15)
ag
= constant defined by equation (2-73)
ak
= constant in equation (3-27)
d
= constant in equation (4-71)
psj
= quadratic coefficient of binary synergism
Ac!}
= m a s s fraction of ilh component
e
= local porosity
eb
= average porosity at the bypass section
ec
= average porosity at the core section
ep
= constant defined by equation (2-10)
et
= porosity at the transition region
e'0
= over cross section average porosity
e L0
= average porosity at the core region
ecb
= porosity at the bulk region
siw
= average porosity at the i-region
e0w
= average porosity at the wall region
o)
= function defined by equation (2-11)
y
= constant defined by equation (2-5)
y
= surface area of material
Yij
= cubic coefficient of binary synergism
(pm
= bypass cross-sectional fraction
K
= bed permeability
K'
= constant in equation (3-55)
KS
= constant in equation (2-35)
Xi
= inertia factor defined by equation (4-33)
\2
= permeability factor defined by equation (4-34)
X3
= pressure factor defined by equation (4-35)
{ M L "iY
p.
= viscosity
{MLU't
{L2
{L
{L
{ M L-4
3
{ML"Y'
t
.
i.
XV
{ML'Y'
= effective viscosity
V
= kinematic viscosity
ve
= effective
kinematic
{LY1
viscosity
defined
by
{LY'
equation (2-88)
v<
= turbulent kinematic viscosity
{LY1
vr
= function defined by equation (2-90)
{LY1
p
= density
{MU 3
o
= surface tension
_
= shear stress tensor
{ML"'f2
= shear stress tensor
{ML"'f2
= turbulence shear stress tensor
{ML'Y2
= laminar shear stress tensor
{ML"Y2
= laminar shear stress tensor
;ML"Y
T1
r_o
{Mf2
= constant defined by equation (4-79) or (4-81)
= correction factor defined by equation (4-18)
{-
= sphericity
= dimensionless
radial
position
defined
by
equation (4-55)
= the "del" or "nabla" mathematical operator
= increase in internal energy due to chemical
j^dnij
{{ML-f
effects or changes in component or substance
i, between states 1 and 2
X*
= the s u m of square errors
{-}
x
CHAPTER ONE
INTRODUCTION
Many engineering fluid flow problems fall into one of three broad
categories, namely flow in channels, flow around submerged objects, and
the transition between flow in channels and flow around submerged
objects. Examples of fluid flow in channels are pumping oil in pipes, flow
of water in open channels and flow of fluids through a filter. Examples of
fluid flow around submerged objects are the motion of air around an
aeroplane wing, motion of fluid around particles undergoing sedimentation
and flow across tube banks in a heat exchanger. The fluid flow in packed
beds is an example of the transition fluid flow between flow in channels
and flow around submerged objects.
In many chemical and metallurgical process operations, a fluid phase
flows through a particulate-solid phase. Examples include gas-solid phase
reactors, filtration, heat transfer in regenerators and pebble heaters, mass
transfer in packed columns, chemical reactions using solid catalysts, and
gas absorption and chemical reactions in packed columns. In many
cases, the solid phase is stationary, as it is in a packed absorber column.
In some cases, the bed moves counter-current to the gas stream, as it
does in a pebble heater or in some gas-solid phase reactors. In some
1
Introduction
cases, the fluid velocity is great enough that the m o m e n t u m transferred
from the fluid to the solid particles balances the opposing gravitational
force on the particles and the bed expands into a fluid-like phase, as it
does in a fluidized bed reactor. In still other applications, the fluid phase
carries the solid phase with it, as it does in pneumatic conveying.
The fluid flow in a packed bed system has important applications as in
heat and mass transfer equipment and in chemical reactors. The constant
effort to realise the profit improvement in industrial operations has
prompted extensive studies of mathematical and physical models for this
system. In order to improve the accuracy of the evaluation of packed bed
system performance, new study efforts should be undertaken in a
microscopic view, eg. heat and mass transfer coefficient distribution over
a packing of particles, rather than a macroscopic view, eg. pressure drop
or overall mass and heat transfer coefficient.
One of the crucial factors in the study of packed bed systems is the
velocity distribution of fluid flow across the bed. A knowledge of velocity
profile in the packed bed systems is required for further studies of rate
processes and their mechanisms taking place in a packed bed. This
thorough study is required as a sound basis for process evaluation of
more complex situations where flow distribution is accompanied by heat
1
Introduction
and mass transfer with chemical reactions, as for example, in hot spot
formation in a packed bed chemical reactor.
It has been extensively studied and reported in several papers that the
velocity profile in packed beds has a significant influence on the
performance of mathematical models of the packed bed systems [Schertz
and Bischoff, 1969; Choudhary et al., 1976ab; Lerou and Froment, 1977;
Kalthoff and Vortmeyer, 1980; Vortmeyer and Winter, 1984; Vortmeyer
and Michael, 1985; Cheng and Vortmeyer, 1988; Vortmeyer and
Haidegger, 1991; McGreavy et al., 1986; Delmas and Froment, 1988;
Ziolkowski and Szustek, 1989; Kufner and Hofmann, 1990]. Figure 1-1
illustrates the influence of velocity distribution across the bed on the
performance of reactor evaluation as reflected in the temperature profile
calculation [Kufner and Hoffmann, 1990]. The results in Figure 1-1 only
account for the convective contribution arising from the variation in the
residence time over the radial cross section. Strictly, account should also
be taken of the consequential changes in the local film coefficients due t
the velocity distribution, but this only tends to exacerbate the situation.
The work undertaken in the present study investigated the mathematical
model of velocity distribution of single-phase fluid flow in packed beds.
This problem has been widely studied; however, the results obtained have
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Introduction
not yet satisfactorily considered the models performance and the methods
of solving the mathematical equations.
According to Reid and Sherwood [1966], the value of a mathematical
model of the physical phenomena depends on at least three parameters:
its accuracy, its simplicity, and the type of information necessary for its
use. Actually, it is a difficult problem to generate perfect models but the
continuous improvement of the existing models will increase their value.
Figure 1-2 shows schematically the flow chart for mathematical model
development of velocity distribution of single-phase fluid flow in packed
beds.
5
Introduction
Physical
Phenomena
1
Development of
Mathematical Model
I
Results of
Calculations
I
Compare to
Experimental Data
No
Yes
Compare to
Existing Models
No
Yes
New/Improved Model
Figure 1-2: Flow diagram of model development.
6
CHAPTER TWO
LITERATURE SURVEY
In the 1950s, mass, heat and momentum transfer in packed bed systems
has received much attention. This condition has prompted extensive
studies of mathematical and physical models for these systems. One of
the crucial factors in the study of packed bed systems is the velocity
distribution of fluid flow across the bed.
The investigations of velocity distribution of fluid flow in packed beds can
be done by two main methods: experimental investigation and
mathematical modelling. The results of these investigations are discussed
in detail in the following section.
2.1 THE EXPERIMENTAL WORK
Since 1950, numerous experimental works have been done to investigate
the velocity distribution of fluid flowing through packed bed systems.
Arthur et al. [1950] investigated the radial velocity distribution of fluid f
in packed beds by measuring the flow distribution of air through a bed of
charcoal granules (0.0009 to 0.0028 m) in a glass tube of 0.0483 m
diameter. In this investigation, the flow was separated into several parts
7
Literature Sun'ex
by the inserting of thin rings, concentric with the tube wall, at the top of
the bed.
In their experiment, the flow rate in each section was measured
simultaneously using the soap bubble technique. In this technique, the air
flow rate is measured by allowing the flow to drive a soap bubble along a
tube and observing the time taken for the bubble to sweep out a
calibrated volume.
The essential result of Arthur ef al. [1950] is that the fluid flow is not
uniform across a packed bed. Although the results obtained from this
experiment did not give point velocities but integrated flow rates over
small cross-sectional areas of the bed, they indicated that the fluid
velocity reached a maximum at a short distance from the tube wall and a
minimum at the centre of the tube. It is considered that a good qualitative
representation of non-uniform fluid flow over the cross section of a packed
bed has been shown in these results. Actual flow distribution above a
rectangular packed bed depicted by Vortmeyer and Schuster [1983] is
shown in Figure 2-1.
The authors [Arthur etal., 1950] also obtained similar results by using the
following other methods:
8
Literature Survey
Figure 2-1: Actual flow distribution above a rectangular packed
bed, NRe = 8 and Dp = 1.25 mm [Vortmeyer and
Schuster, 1983].
Literature Survey
1. Chemical estimation of the products of reaction or adsorption on
various parts of charcoal when a stream of air containing a gas which is
absorbed by, or reacts with the charcoal, is passed through the bed.
2. Qualitatively, by observing the passage of a gas-laden air stream
through a bed of granules stained with an indicator.
3. By observing the concentration of emergent gas from the charcoal
column at various parts of the cross-section by means of test papers, a
gas-laden air being used.
4. By measuring the temperature attained at the side and the middle of a
wide column.
The measured data are listed in Table 2-1. Referring to their experimental
results, Arthur et al. [1950] remarked that the main factor affecting flow
rate distribution is bed porosity. Higher flow rates of fluid at the region
near the tube wall are strong indications that this occurs because the
smooth wall increases the bed voidage near it.
Morales et al. [1951] employed a series of circular hot wire anemometers
of various diameters to measure the fluid velocity at a series of the radial
positions of the bed. Their measurements were made over a range of air
velocities (0.123 - 0.533 m/s) in a 52.5 mm diameter tube packed with
three sizes (3.175, 6.35, and 9.525 mm) of the equilateral particle
diameter and height of the cylindrical pellets. The results indicated that
io
Literature Survey
velocity distribution in packed beds is a function of air velocity and bed
height as shown in Figure 2-2.
The authors believed that two important factors bring about a velocity
distribution of the kind obtained. These are skin friction at the tube wall
and variation in the void space in the bed with radial position. These
conclusions seem reasonable since it is known that with flow in an empty
tube, a wall friction causes the velocity to decrease sharply near the wall.
In fact, for streamline flow in an empty tube, a wall friction causes the
velocity to parabolically decrease from the centre of the tube. It may
therefore be expected that when packing is introduced, the effect of wall
friction would be dampened and become negligible near the centre of the
tube. Close to the wall, however, wall friction would again become
important and is the probable cause of decreasing velocities near the
wall.
The characteristic of a maximum in the velocity profile was found to occur
for all heights of the packing. This, together with the subsequent decrease
as the centre of the bed was approached, was thought to be due to
variation in bed porosity with radial position and the resultant losses of
pressure energy. Referring to their experiment results, Morales et al.
[1951] also remarked that the velocity distribution is not independent of
the distance of the anemometer above the top of the bed.
3 0009 03201226 7
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Literature Survey
A more comprehensive investigation of velocity profiles in packed beds
was undertaken by Schwartz and Smith [1953] with the objectives of
determining under what conditions the uniform velocity assumption would
be valid, and, of correlating and explaining the magnitude of the observed
velocity profiles. Data were obtained in 50.8, 76.2 and 101.6 mm pipe,
using 3.175, 6.350, 9.525 and 12.7 mm spherical and cylindrical pellets,
corresponding to the range of D/DP from 5 to 32. The length of the
cylindrical pellets was equal to the diameter.
Their measurement velocity distribution is tabulated in Table 2-2. The
reproducibility of the data was tested without repacking of the bed; and
the maximum deviation of 2% was reported. If, however, a point velocity
was rechecked after the pipe was emptied and repacked, the variability in
packing increased the deviations. The maximum deviation of velocity
distribution data with repacking was 25%, although average deviation for
the three replications was 9%.
One of the main limitations of Schwartz and Smith [1953] study is related
to the velocity measurement points at the downstream of the bed by
assuming that the velocity distribution does not change. This distance
should be sufficient to smooth out the severe variations of velocity found
near the exit face of the bed and to eliminate non-axial components of
velocity without permitting any gross changes in velocity profile.
14
Literature Smre\
Table 2-2: Velocity distribution above the bed [Schwartz and Smith, 1953].
(Anemometer position 50.8 mm above the bed, 0.584 m bed depth).
Column Diameter = 50.8 m m .
UZ/UM
1
r/R
6.35 m m *
0.46
0.67
0.96
0.89
0.82
0.86
0.30
1.06
9.525 m m
0.67
0.46
1.02
0.94
0.82
0.92
0.32
0.30**'
1.05
0.55
1.25
1.12
1.07
1.05
1.26
1.19
1.15
1.14
0.71
1.23
1.16
1.17
1.22
1.26
1.13
1.16
1.20
0.84
1.09
1.12
1.11
1.11
1.06
1.03
0.99
0.98
0.95
0.45
0.68
0.74
0.76
0.46
0.59
0.63
0.64
Column Diameter = 76.2 m m .
UZ/UM
r/R
6.35Imm*'
12.7 mm
0.80
0.49
0.77
0.82
0.32
0.49**'
0.62
0.80
0.61
9.525 m m
0.49
0.80
0.81
0.79
0.55
0.80
0.70
0.88
0.82
0.97
0.90
0.71
1.08
1.12
1.04
1.04
1.09
1.11
0.84
1.20
1.20
1.06
1.17
1.01
1.04
0.95
0.99
0.93
0.80
0.82
0.89
0.83
: Diameter of spherical packing.
: Average velocity, m/s.
Literature Sur\>e\
Table 2-3: Velocity distribution above the bed [Schwartz and Smith, 1953].
(Anemometer position 50.8 mm above the bed, 0.584 m bed depth).
(Continued)
Column Diameter = 101.6 m m .
r/R
6.35imm*'
0.31**'
0.49
0.71
0.69
0.32
Uz/UM
9.525 m m
0.31
0.49
0.69
0.67
12.7 m m
0.31
0.49
0.62
0.70
0.55
0.83
0.71
0.98
0.83
0.89
0.81
0.71
0.99
1.00
1.09
1.05
1.06
1.07
0.84
1.38
1.31
1.25
1.20
1.27
1.24
0.95
1.19
1.10
1.14
0.97
1.08
1.03
Column Diameter = 101.6 m m .
UZ/UM
r/R
0.32
6.35imm*'
0.65**'
0.80
0.70
0.69
9.525 m m
0.65
0.80
0.64
0.64
12.7 mm
0.65
0.80
0.65
0.63
0.55
0.75
0.75
0.85
0.85
0.83
0.83
0.71
1.03
1.02
1.05
1.05
1.09
1.09
0.84
1.34
1.18
1.16
1.15
1.21
1.20
0.95
1.10
1.07
0.97
0.98
1.02
1.00
: Diameter of spherical packing.
: Average velocity, m/s.
16
Literature Suirey
Schwartz and Smith [1953] overcame the problem of measurement
distance from the packing by assuming the ratio of point velocity to
average velocity equal to 1.0 at r/R equal to 0.55. Based on this
assumption and regarding to preliminary experimental investigations, they
concluded that in their system, a distance of 50.8 mm between the bed
and the anemometer would minimise the errors and so all data were taken
at this position.
Based on their experimental data, these authors concluded the results as
follows: The maximum or peak velocity ranges up to 100% higher than the
centre velocity as the ratio of pipe diameter to pellet diameter decreases.
The divergence of the profile from the assumption of a uniform velocity is
less than 20% for ratios of pipe diameter to pellet diameter of more than
30. The maximum in the velocity profile occurred at a distance of
approximately one particle diameter from the wall, regardless of pipe and
packing size. The deviation from flat profile became more pronounced as
the ratio of D/DP decreased.
Dorweiler and Fahien [1959] used a series of circular hot wire
anemometers to determine velocity profile at the exit of packed beds. The
test column was a vertical 101.6 mm diameter pipe, packed with 6.35 mm
spherical, ceramic catalyst-support pellets. The anemometer was
17
Literature Survey
operated at 25.4 m m above the bed, after this distance w a s determined
experimentally to be an optimum height.
Their measurement result of the velocity distribution across the test bed is
shown in Figure 2-3, and this profile was reported to be independent of
total flow rate above an average superficial velocity of 0.122 m/s. Briefly,
the radial fluid velocity profiles obtained by these investigators exhibited
the similar basic trend of having a maximum near the wall as those of
previous workers. The values of the ratio of (uz)max/uM were greater than
those reported before.
In order to overcome the problem of flow changes in the open tube where
velocity profile measurement was carried out at the exit of the bed, Cairns
and Prausnitz [1959] measured velocity profile at inside of the bed. They
used electrode techniques to measure mean axial velocities over a length
of bed, using both fixed and fluidized beds with water as the fluid.
A salt tracer solution was injected into the water main stream over the
entire cross-section of the bed and the time interval necessary for
detecting a sudden change in the rate of injection at zero position was
measured at some distance downstream. Electrical conductivity cells at
various radial and axial positions in the bed detected the change in the
injection rate.
18
Literature Sune\
Column diameter = 101.6 m m
Spheres
= 6.35 m m
Superficial velocity =0.12 m/s
Figure 2-3: Velocity distribution 25.4 m m above a bed of spheres
[Dorweiler and Fahien, 1959].
19
Literature Swvey
Typical results obtained by Cairns and Prausnitz [1959] are shown in
Figure 2-4. ln each case a slight maximum was found at the centre of the
tube, but apart from this the profiles were found to be flat. It was not
possible to make measurements closer than two particle diameters from
the wall. The dotted extensions to the profiles shown are based on a mass
balance and indicate the type of behaviour that the authors expected at
the wall. The limitation of these experimental results in generating the
velocity profile near the wall by employing a material balance due to the
value of local bed porosity is assumed constant over the cross section of
the bed.
Price [1968] used a 3.8 mm diameter pitot-static tube to measure the air
velocity at the exit of packed beds. In order to overcome the problem of
flow changes in the open tube at the outlet of the bed, Price [1968]
divided the exit flow area with a honeycomb of concentric splitters with
intersecting radial vanes placed between the exit face of the bed and the
plane of measurement. The nose position of the Pitot-static tube is
approximately 1.6 mm downstream of the exit face of the honeycomb. This
is to minimise the flow blockage caused by the Pitot-static tube.
Measurements were made for all compartments of the honeycomb over
the whole cross section of the bed, as many as 1,000 readings were taken
in a single run to confirm the reproducibility of measurement.
20
Literature Su/rev
10000
Pipe diameter
=50.8 m m
Spheres diameter =3.2 m m
e'o
= 0.38
uM/e'0 = 1440 mm/s
— o
o-'
1000 -630
—o
o-
320
230
160
100 --
80
SI
3
25
-o
10--
o*'
8
-t
=8=
6.0
4.0
=0-"
7.4
2.0
0.0
(R-r)/DP
Figure 2-4: Velocity profiles for a randomly packed bed [Cairns and
Prausnitz, 1959].
21
Literature Sun-ey
The author investigated the parameters that were suspected to have
influence on velocity profile, such as Reynolds number, bed length,
packing method, sphere material and D/DP ratio. Tests also were made to
assess the reproducibility of the measurements with repacking of the bed
between tests.
A typical velocity distribution above the bed measured by Price [1968] is
shown in Figure 2-5. The results from these investigations may be
summarised as follows: The velocity profiles, normalised with respect to
uM, were independent of Reynolds number (1,470 < NRE < 4,350), beds
length (9 < L/D < 36), and spheres material over the range tested. Slight
systematic effects were observed near the walls due to packing method,
sphere properties, and vessel to sphere diameter ratio (12 < D/Dp < 48).
The maximum velocity was found to exist within one-half sphere diameter
from the walls of the containing vessel.
Newell and Standish [1973] used thermistor anemometers to measure the
velocity distribution of gas streams flowing in packed beds. Velocity
distributions for a number of air velocities (0.04, 0.08, and 0.09 m/s) wer
determined in two columns, having square and rectangular cross-section,
respectively. The square column (101.6 mm by 101.6 mm) was used to
measure velocity distributions for packing consisting of 6.35 and 16.93
IT
Literature Surrey
m m spheres, 6.35 m m Raschig Rings and 6.35 m m coke resting on a wire
gauze support. The rectangular column (533.4 mm by 152.4 mm) was
used to determine the profile for 6.35 mm spheres.
In addition, velocity distributions were determined in a slice model of a
copper blast furnace. This model was packed with 6.35 mm and 12.7 mm
coal and velocity distributions were measured at various heights above
the tuyere level [Newell, 1971].
The essential result from Newell and Standish [1973] was that the fluid
flow profile in square and rectangular packed beds is similar to that in
circular beds. Their measurements also indicated that the fluid velocity
reached a maximum at less than one particle diameter from the wall as
shown in Figure 2-6.
Szekely and Poveromo [1975] also used hot wire anemometers to
measure the velocity distribution at the exit of gas streams flowing in
packed beds. This investigation was undertaken to elucidate the
mechanism of flow maldistribution or non-uniform flow through packed
beds system. Their measurements were made, over a range of Reynolds
number (100 - 400) in 101 and 152.5 mm diameter tube packed with 1 - 6
mm diameter glass spherical particles.
23
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Literature Survey
1.5
101.6 x 101.6 m m square column
16.9 m m Spheres
1.0-
s
3
-_
3
0.5 Superficial velocity, m/s
- - o - - 0.04
-^^0.06
- -o- - 0.08
- • o - 0.09
0.0 *-»
0.0
I
0.5
1.0
1.5
l
2.0
I
I
L_J
I
2.5
i
i
i
-H-13.0
(Re-re)/Dp
Figure 2-6: Velocity distribution of fluid flow 25.4 m m above square
bed of spheres [Newell and Standish, 1973].
25
Literature Suivex
The results of parallel flow measured by Szekely and Poveromo [1975]
were also similar to previous investigations on measurements at the exit
of the beds, the smoothed profile again showed the characteristic rise in
velocity near the wall as shown in Figure 2-7. Experimental
measurements were also made of the pressure distribution at the inlet and
at the exit of the column. It was also found that for the experimental
conditions used the pressure was uniform.
In order to provide a more comprehensive data on fluid flow inside a
packed bed, Stephenson and Stewart [1986] employed the optical
measurement technique to measure the velocity and porosity distribution
inside the beds. The experiments were done in a vertical 75.5 mm
diameter fused quartz tube, randomly packed for a length of 145 mm with
cylinders cut from fused quartz rod. They used tetra-ethylene glycol, tetrahydropyran-2-methanol, and a mixture of cyclo-octane and cyclo-octene
as a fluid in order to fulfil the requirement of the range of Reynolds
number and Newtonian fluid characteristics.
The composite of superficial velocity across the test bed is shown in
Figure 2-8. The velocity profile has a peak near the wall, where the
porosity is largest and its fluctuations correspond to those of local
porosity. These results are therefore similar to the measurement at the
exit of the beds by employing flow separator of Price [1968].
26
Literature Suirey
3.00
2.50 --
2.00 -•
s
3
""Si
3
1.50-
1.00 --:
0.50 --
0.00
0.0
5.0
10.0
15.0
20.0
25.0
(R-r)/DP
Figure 2-7: Velocity profiles of fluid flow 10 m m a b o v e a circular bed
of spheres [Szekely and Poveromo, 1975].
27
Literature Surx'ev
DPe
= 7.035 m m
D/Dpe = 10.7
L/Dpe = 20.6
0
1
2
3
4
5
(R-r)/Dp
2-8: Composite superficial velocity profile inside the bed
[Stephenson and Stewart, 1986].
T
Literature Surcey
The conclusion of Stephenson and Stewart [1986] that the bed porosity
distribution determines the velocity profile is also supported by
experimental work carried out by McGreavy ef al. [1986]. These authors
measured the velocity distribution inside and at the exit of the bed by
using laser Doppler anemometry. It has the advantage that it is capable of
giving good spatial resolution so that the flow distribution can be related
to the structure of the bed. However, this method is only good to take
measurements for small values of D/DP ratio because the need to provide
a suitable optical path poses problems for fixed beds.
The characteristic of a non-single maximum or an oscillation in the
velocity profile was found to occur for all measurement points, namely in
the inside and at the exit of the beds as shown in Figure 2-9. The most
striking feature of Figure 2-9 is that the observed flow profiles at the exit
are different from those inside the bed. This is of some significance as it
can reject the assumption that velocity profile inside the bed is similar to
that at the exit, as being appropriate.
In a more recent study, Ziolkowska and Ziolkowski [1993] used thermoanemometric techniques to measure the velocity distribution at the exit of
gas flow in packed beds. The experiments were done in a vertical 94 mm
inside diameter tube, packed randomly with uniform porcelain spheres to
a height of 1050 mm and diameter of spherical particles (4.11 - 8.70 mm).
29
Literature Survey
Distance from wall, particle diameter
0.5
1
2.8
J
1.5
L.
_•
I
2
i
2.5
i_
-J
1
I
1
L
Inside the bed.
6-
Exit of the bed.
it
A
4N
3
%x
1/ \
2 Atr
___-_^_mm^_____-_
3^r*""
:__r
0- ^ T
0
\ *
^^*-4r-_.
IV
_-W
— ! — I I I I I — i — r — T — r — i — i —
0.5
1
-1
1
1.5
1
1
r—i
1
1 1
2
r-i
1
1
1 1
1
1
P
2.5
Distance from wall, particle diameter
Superficial velocity, u M (mm/s)
34 28 20
11
Figure 2-9: Comparison between corresponding velocity profiles at
the exit and inside of the packed bed (DP=16 mm, D=50
mm, Packing height = 220 mm, Measurement height =
140 m m and 222 m m ) [McGreavy etal., 1986].
30
Literature Survey
Ziolkowska and Ziolkowski [1993] measured the radial distribution of air
flowing through over a range of superficial velocity (0.4 - 1.0 m/s). A
typical velocity profile above the bed of their measurement result is shown
in Figure 2-10. Similar to previous investigations, these authors found
that, with an accuracy of ±3.2%, the shape of the local gas velocity radial
profile does not depend on flow rate while the average reproducibility of
these profiles after repacking the bed was 4.2%. The pellet diameter had
a more pronounced effect than the flow rate on the shape of velocity
profile that the smaller the pellet diameter, the flatter the profile.
From the foregoing brief summary of the results of experimental
investigations of velocity distribution in packed beds, it can be concluded
that the fluid flow is not uniform across a packed bed. Although the
number of observed flow maxima points is dependent on the
measurement technique that was employed; however, generally the
maximum value in velocity occurs near the wall of the container. This is
due to the opposing effects of wall friction and the variation in bed
porosity with radial position relative to the wall.
Generally, the dimensionless wall distance in terms of particle diameter,
(R-r)/DP, is more representative to explain the velocity distribution than
dimensionless radius, r/R. This condition agrees with the behaviour of bed
porosity in packed beds [Goodling etal., 1983; Roblee etal., 1958].
31
Literature Surve
rev
uM
0.4 m/s
0.8 m/s
1.5
2
3.
0.5
Pipe diameter
= 94 m m
Bed depth
= 1050 m m
Spheres diameter = 8.7 m m
0
l
0
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i—1_
2
(R-r)/DP
Figure 2-10: Velocity distribution at 15 m m above the beds of
spheres [Ziolkowska and Ziolkowski, 1993].
32
Literature Survey
According to this condition, it is evident that the bed porosity distribution
has significant influence on the velocity distribution in a packed bed
system. Any account of experimental investigations of velocity profile in
packed beds must therefore also include investigations of the porosity
distribution in packed beds.
Regarding to the phenomena of the developing fluid flow in a channel
[Poirier and Geiger, 1994], it is reasonable to conclude that the velocity
profile at the exit of the bed without flow separator is a developing flow
condition. This is a transition profile between a velocity profile inside the
bed and a fully developed flow profile in an empty tube, and possesses a
developing flow distribution. Therefore, the velocity distribution data that
were measured at the outlet of the bed can not directly be used to
represent the velocity profile at inside of the bed. Based on this condition,
the study of velocity distribution inside the beds by using the velocity
profile that was measured at the outlet of the bed without flow separator
needs to be corrected with developing flow phenomena.
2.2 MATHEMATICAL MODELLING
As mentioned earlier, the fluid flow problem in packed beds is a transition
between flow in channels and flow around submerged objects. According
to discontinuity of this system, an exact representation of the fluid flow
distribution in packed beds is impossible [Ziolkowska and Ziolkowski,
33
Literature Survey
1993]. The fluid mechanics equations cannot be easily applied to describe
the flow field within the whole space bounded by the tube wall. They might
be applied for the space between granules [Mickley er al., 1965], but they
could not be integrated within that space because the boundary
conditions are indefinable.
For this reason, a number of mathematical models of fluid velocity profile
in packed beds is based on experimental data, or on theoretical
considerations adjusted by relatively simplified assumptions. Although the
accuracy of phenomenological approach is good for a particular set of
data, it is difficult to apply this type of model with any confidence to oth
systems/conditions. On the other hand, although numerous theoretical
approaches have been done in this area, a consensus has not been
reached and there is a need for further investigations.
2.2.1 The Phenomenological Approach
On the basis of velocity distribution that was measured at the exit of the
beds, Schwartz and Smith [1953] tried to develop a mathematical model of
velocity profile in packed beds. The model was developed by applying two
main assumptions: uniform pressure drop over all radial positions of the
bed and the variation of bed porosity due to the wall effects over cross
section of the bed.
34
Literature Survey
In order to develop the mathematical model, Schwartz and Smith [1953]
employed the Prandtl's expression of shearing force [Bird et al., 1960] and
Leva's correlation of pressure drop [Leva, 1992]. Schwartz and Smith
[1953] assumed the driving force for velocity distribution or momentum
transfer is the difference in pressure drop between the center of the pipe
with local porosity minimum, where no momentum transfer occurs, and
that at any radial position. This pressure defect per unit length of bed, P5,
can be derived from Leva's correlation as follows:
P —
8
2
'
"DP
2(1-0
um2(1-e)
-^--u
3
"m0
3
e
£
(2-1)
LO
The total pressure drop correlation proposed by Leva is due to skin
friction between the fluid and solid particles and pipe wall and orifice
losses, as well as that due to turbulent shear in the fluid alone. Then, if
is assumed that the fraction of the total pressure drop due to turbulent
shear in the fluid is constant, the Leva's correlation can still be used to
determine the pressure defect by introducing a constant factor, k2, that
was obtained by adjusting with their experimental data to give the value of
k2 equal to 0.0096 [Schwartz and Smith, 1953].
In order to reduce the errors involved in using Leva's pressure drop
correlation, Schwartz and Smith [1953] measured the pressure drops
each time the velocity profile was determined. Considering the force
35
Literature
SUITCX
balance over the bed to develop the correlation between the pressure
defect and the shear stress gives:
k 2 Ap
uM
l-e r
u.
fl-^^
1-8
-u
LO
mO
tL
. ° yJ
= PL,
du. du.
dr dr
(2-2)
Considering equation (2-2), Schwartz and Smith [1953] failed to
distinguish between true local fluid velocity, uz for Prandti's shearing fo
on the right hand side of equation (2-2) [Bird et al., 1960] and local
superficial fluid velocity for Leva's pressure drop correlation. This
condition is acceptable only if the value of bed porosity is uniform over t
whole cross section of the bed.
Equation (2-2) defines the velocity gradient in terms of bed porosity e,
radius r, and mixing length LM (the radial distance that a small mass of
fluid travels before losing its identity). Equation (2-2) is basically a
theoretical equation other than the semi-empirical expression for Leva's
pressure drop correlation, and Prandti's shearing force [Bird et al., 1960].
The problem in using this equation is that the integration of the expressio
to obtain the velocity profile varies in complexity with radial position.
However, simplifications have been developed to overcome the above
problems. Flow through packed beds is analogised to flow through a
36
Literature Suirey
bundle of tubes, which consists of a central core, containing tubes of
constant diameter, surrounded by tubes of progressively larger diameter.
The bed porosity at central core is assumed constant and equal to e0. Also
in this region the mixing length, LM, will be assumed equal to DP/2. With
these simplifications equation (2-2) may be integrated and then
substituting dimensionless parameter um/uM, to give:
2
'u^
u. • + <
VUM ;
uM
In
(
xx
u mO
V21
\
i/
VUM J
{-
= k'
(2-3)
U mO
UM
Where:
D \K k 2 D p ApY
3 KD*J
8u M
(2-4)
ri- £ ' 0 Y £ ^
(2-5)
(
Y=
Although the value of the constant k', m a y be determined theoretically
from equation (2-4), however, the errors involved in using this model is
significant. The deviation may arise from the inappropriate assumptions in
developing the model, namely, LM = DP/2 [Schwartz and Smith, 1953], fluid
phase shear stress being a constant fraction of the pressure loss [Newell,
1971], the use of the poor analogy of mixing length by Prandtl [Bird ef al.
1960; Mickley etal., 1965], the failure to distinguish between uz and uM,for
37
Literature Suirey
Prandlt mixing length and Leva's pressure drop correlation, and the
constant properties of central core which are dependent on D/DP ratio
[Roblee etal., 1958; Benenati and Brosilow, 1962].
In order to reduce the errors involved in using this model, Schwartz and
Smith [1953] offered the experimental value of k'. The problem in using
this model is due to the lack of constant k' data of other
system/conditions. For this reason, it is appropriate to bring this model
into phenomenological model category.
With improvements in shear stress correlation and the concept of
pressure defect of the Schwartz and Smith [1953] model, Price [Price,
1968; Newell, 1971] tried to develop a mathematical model of velocity
profile in circular packed beds. By extending the Price result, Newell and
Standish [1973] employed the model for rectangular packed bed and nonferrous blast furnaces.
Price [Price, 1968; Newell, 1971] employed Boussinesq's shear force
correlation [Bird et al., 1960] to replace the Prandtl correlation that was
used in Schwartz and Smith's [1953] model. With reference to work by
Dorweiler and Fahien [1959], the correlation of eddy viscosity to fluid
velocity is performed by assuming constant value of Peclet number, then
introducing this result into Boussinesq's shear force correlation, gives:
38
Literature Surx-ey
x'=pbDpuM-^
(2-6)
Considering a cylindrical packed bed of unit length and radius, r, Price
[Price, 1968; Newell, 1971] developed a force balance on the fluid flow as
follows :
27.r.t + ..r2—= F (2-7)
dz
'
In equation (2-7), F is the force resisting motion per unit length of bed and
arises from the interactions between the solid packing and the fluid. Since
velocity varies with radius across the cylindrical section considered, the
total resistive force acting on the fluid within the cylinder is:
R
F = J27crcuM2dr (2-8)
0
In developing the force balance, Price [Price, 1968; Newell, 1971] failured
to distinguish between shear stress momentum in continuous systems, for
example, fluid flow in an empty tube, and discontinuous systems, for
example, fluid flow in packed beds. The shear stress momentum part in
equation (2-7) is an expression of continuous systems rather than
discontinuous systems, which must be corrected by local bed porosity
factor.
39
Literature SuiTev
Substituting equation (2-8) into equation (2-7), and assuming that the
value of the pressure drop is independent of the radius and then
rearranging gives:
2c
2 U
d2uv If
du-^j
+
2 U
MTT :
MM ^ v
dr'
2c (
,
ldp^
dr J pbDF v
r.
For the central core, the true velocity, uz, is related to superficial velocity
in the empty tube at the exit from the bed, um, by the expression
L
U
eU
m=77 Z
(2-10)
p
= £ uz
Rewriting equation (2-9) in terms of u m , putting 2c/(pepbDP) equal to B 2
and making the substitution gives:
AX 2
£p2d
P ,0.,-n
6 = um
y
m
(2-11)
c dz
V
;
By assuming (dum) «umd2um, which is reasonable when the value of
du m is between 0 and 1, results in
- i + --f-B 2 ()) = 0
dr
(2-12)
r dr
Equation (2-12) is a Bessel equation [Mickley et al., 1957]. By assuming
the symmetric condition of velocity profile, which is reasonable, the
solution of equation (2-12) is:
40
Literature Survey
((> =
ep dp
I„(Br)
um„ -•
m0
c dz
(2-13)
Before equation (2-13) can be solved, a further boundary condition has to
be specified. This boundary condition relates to the outer edge of the
central core where the increased velocity provides the potential for
momentum transfer into the central core of the bed. These high velocities
arise primarily from the relatively high bed porosity fraction near the wal
[Newell, 1971].
Equation (2-13) predicts a superficial velocity profile whose gradient
increases with radius, as a central core is assumed to extend to a radius
r, where the gradient of the velocity profile is no longer increasing. The
velocity at this radius is denoted by um. Substituting these boundary
ep2 dp
conditions into equation (2-13), and solving for — — -
in terms of u m , f
and um0, and then normalising with respect to um, gives:
r 2
o
m
[l„(B.)-l]= '-- _^. * [(a2-l)l0(Br) + I 0 (Br)-a 2 ]
(2-14)
.U M J
Where
a =
u.
u mO
(2-15)
41
Literature Survey
It should be noted that the use of equation (2-14) to predict the velocity
profile requires a knowledge of B, f, and a, as empirical constants. These
values must be determined by experiment with measuring of velocity
distribution. An evaluation of experimental data obtained by previous
workers [Schwartz and Smith, 1953; Dorweiler and Fahien, 1959], Price
[Newell, 1971; Newell and Standish, 1973] showed that B may be
reasonably assumed to have a value of (DP)"1. However, the values of
rand a remain to be determined by measuring the superficial velocity
profiles for the particular packing structure under consideration and over
the Reynolds number range involved. Since a number of empirical
constants in this model are required to be determined by experimental
work of velocity distribution, it is evident that the model is in the
phenomenological category.
Although the Price model was originally derived from fundamental fluid
dynamics theory, the result has a number of empirical constants. The
reason for this condition is because the urge for an analytical solution led
to oversimplification of the mathematical equations.
Newell and Standish [1973] tried to extend the Price model of fluid flow
distribution in circular packed beds to rectangular packed beds by
employing equivalent diameter concept. They assumed an equivalent
diameter term based on the bed cross section to represent a packed bed
42
Literature Survey
of non-circular geometry. Velocity profile of fluid flow in a non-circular
packed bed was developed by substituting the equivalent radius re into
equation (2-14) to give:
(
V
.UiM,
(
V
[l0(B?J-l]= ^=2. [(a2-l>0(Bre)+I0(Bfe)-a2]
I
11
V
U
(2-16)
M;
The validity of equation (2-16) w a s investigated by using square and
rectangular packed beds [Newell and Standish, 1973]. The agreement
between measured data and results predicted by this model was good for
physical model of square and rectangular packed beds. However, the
model failed validation for non-ferrous blast furnaces by using a V3-slice
model of the copper blast furnace. In the central region the measured
velocities differed widely from predicted values. This is considered to be
due to a combination of causes, which include the failure emanating from
original Price model development, as mentioned earlier, and also the
changing of the boundary conditions, for example, movement of the
burden, method of charging, and segregation in the burden.
Based on the data at the exit of the beds that was measured by Schwartz
and Smith [1953], Hennecke and Schlunder in 1973 [Tsotsas and
Schlunder, 1988; 1990] proposed an empirical correlation to predict the
velocity distribution in packed beds. The empirical correlation for circular
packed beds of spheres is as follows:
43
Literature Survey
u
uM
K + [(W + 2)/2](r/R)'
K +l
K = 1.5+0.0006
W = 1.14
_D_
DT
-2
(2-17)
(2-18)
(2-19)
The local superficial velocity at the centre of the bed can be predicted by
substituting r=0 into equation (2-17) to give:
K
_____
(2-20)
uM K+l
Similar to Hennecke and Schlunder's model, Fahien and Stankovic [1979]
proposed an empirical correlation of velocity distribution based on the
data at the exit of the beds that were measured by Schwartz and Smith
[1953]. The empirical correlation for circular packed beds of spheres is as
follows:
A0+A,rs+I+A2rs+2
um =
(2-21)
2E
Where
An
A,
(2-22)
E= —
2 + s+3 s + 2
1
0
s+2
l
M
s+1
(2-23)
Literature Sur vev
r
A
M
(2 24
'=i7T
r
A
" »
M
= ^ s+ 2
(2-25)
rM = l - 2 ^
(2-26)
2
All constants in equations (2-17) to (2-26) are empirical. These values
must be determined by experiment if the condition, such as particle size
distribution, column diameter, etc., is not similar to Schwartz and Smith's
[1953] data. Thus, using this model is not practical because of the
requirement of the experimental data of the velocity distribution.
Tien [Vortmeyer and Schuster, 1983] has derived a general analytical
expression of flow profiles in semi-infinite packed beds which are
bounded on one side by a rigid wall. He found the general analytical
expression due to the porosity function near the wall. The equation for a
tubular packed bed with R^>°° was given as:
.
=1
U
\ I r-R
Dp
f
1- n
M
.
C
r-R'e v
(2-27)
DP
p
J
where
4n.
a=
(2-28)
4-n,
45
Literature Survey
Unfortunately, the coefficient n cannot be determined theoretically. This nc
value must be determined by experimental work, and as an approximation
Vortmeyer and Schuster [1983] have developed a formula for n as a
function of NRe.The value of nc varies from 0.1 to 27.
Because of the infinite packed bed assumption in developing this model,
error is able to occur for finite packed bed calculation. Consequently,
D
calculation of fluid flow in packed beds with small value of —
by using
-Up
this model must be examined carefully.
From the foregoing brief summary of the modelling of velocity profile of
fluid flow in packed beds by using phenomenological approach it can be
concluded that this approach works better if it is calibrated with measured
data for a given system; however, the applicability may still vary
significantly for another system.
2.2.2 The Theoretical Approach
Numerous studies have been done to develop a mathematical model on
the basis of physical phenomena rules, or by extending the well-known
macroscopic model, have led to a number of mathematical models.
Because of the discontinuity of the systems a mathematical manipulation
46
Literature Survey
was required to apply the equations, so that a number of mathematical
models depended on the simplification of the system.
Szekely and co-workers [Stanek and Szekely, 1974; Szekely and
Poveromo, 1975; Poveromo et. al, 1975; Choudhary etal., 1976a; 1976b;
Szekely and Propster, 1977; Szekely and Kajiwara, 1979] tried to explain
the channelling and maldistribution of fluid flow in a packed bed reactor b
extending the Ergun [1952] pressure drop correlation into vector terms. A
good result in application of this model to the gas-solid reactor with larg
value of D/DP ratio was reported [Poveromo et ai, 1975; Szekely and
Propster, 1977; Morkel and Dippenaar, 1992],
In vector form, the Ergun [1952] equation may be written in microscopic
term [Stanek and Szekely, 1974] as:
-Vp = um(f,+f2um) (2-29)
By assuming incompressible fluid and then employing operator Vx on
equation (2-29) to eliminate the pressure term gives:
Vxum-umxV(ln(f1 + f2uj) = 0 (2-30)
The components of the velocity vector also have to satisfy the equation of
continuity [Bird etal., 1960; Poirier and Geiger, 1994] as follows:
V.um=0 (2-31)
47
Literature Survey
Upon finding the velocity field through the solution of equations (2-30) and
(2-31), the pressure distribution may be evaluated [Stanek and Szekely,
1974] from:
V2p = -um.V(f1+f2uJ (2-32)
For incompressible fluid flow through a cylindrical bed with axial symmetry
(that is, d/d§ = 0 and um$ = 0), equations (2-30) and (2-31) may be
rewritten [Stanek and Szekely, 1974; Bird et al., 1960; Jenson and
Jeffreys, 1963] as follows:
du du 3ln(f,+f7um) 3ln(f,+f,um)
" UJT
IJQ2 ,
i r - ^ r
\ I
+ u
2
m/
- — ^ —
_ u
u
\ 1
-«
2
3r
•-
~
my-
- rx
L
—=°
l3(rumr) ( 5u m z
=0
r dr
3z
/ 0
nri\
(2 33
- »
(2-34)
In general, equations (2-33) and (2-34) have to be solved numerically
[Stanek and Szekely, 1974]. For special cases, the analytical solutions do
exist when dfjdz = df2/dz = 0, when the resistance does not vary in the
direction of flow. If the bed is sufficiently long, all radial components of
velocity vanish and the flow becomes parallel. On putting umr = 0 and
Um=Umz, equation (2-33) is readily solved to obtain:
U =
» -_T;
+
1
'O
Uf;
+^
(2-35)
2J
48
Literature 5i.rv_v
The integration constant K S is determined from an overall balance on the
fluid.
R
uMR2=j2rumdr
o
y
R
= }2r
0
2f2 +
2f 2
+
r ^ r -
dr
(2-36)
;
The basis of this model is the Ergun [1952] macroscopic equation for
pressure drop of fluid flow in packed beds. Basically, the Ergun equation
was developed based on the approach of the packed bed being regarded
as a bundle of tangled tubes [Bird et al., 1960], and this approach yield
good results for bed porosity less than 0.5 [Bird et al., 1960; Cohen and
Metzner, 1981]. Another important limitation of the Ergun equation, as
cited by Gauvin and Katta [1973], is the fact that it is not appropriate for
systems containing particles of low sphericity. It follows that the uses of
this equation, or the use of relationships derived from it, become
automatically influenced by these limitations.
Comparison of Stanek and Szekely's [1974] model with experimental data
[Poveromo, et al., 1975; Szekely and Poveromo, 1975] shows a good
qualitative representation of velocity profiles in packed beds. However,
because of the limitation of the Ergun equation, the numerical values of
calculations are questionable, especially in the vicinity of wall area where
49
Literature Smrey
usually the bed porosity is greater than 0.5 [Goodling et al., 1983]. The
approach of dividing the bed into a number of incremental beds, as a
means of generating a theoretical velocity profile based on the Ergun
macroscopic approach, is also problematic because it means that the wall
effect is accounted for many times over and leads to a significant error.
In a more simple approach, Martin [1978] also developed the velocity
distribution of fluid flow in packed beds model based on the Ergun [1952]
equation. Martin [1978] divided the packed bed into a core section of
porosity, ec, and by-pass section of porosity, eb, where ec < eb. Assuming
radially constant velocities and the validity of the Ergun equation in each
of the sections the ratio umb/umc can be obtained.
Based on Benenati and Brosilow's [1962] bed porosity data, Martin [1978]
assumed that the by-pass section area is one particle diameter from the
wall of container. Since the two parts of the model packed bed offer
different resistances to the flow, a non-uniform flow distribution will resul
This can be calculated by applying the Ergun equation [Ergun, 1952] to
both parts of the packed bed:
i_P __
150^1B^
+ 1.75^^=1 (2-37)
L
ec"
Dp
ec-
Dp
and
50
Literature Survey
^=
l50(±^}^+lJ5lz3^^
2
(2-38)
D.
Dr
The superficial velocities umc and u m b are defined as the volume flow rates
divided by the empty cross-sections (1-<pm)A and cpmA. Substituting this
definition into equation (2-38) and rearranging gives:
u mb
u.
u
1 + 0.0117u mc
M
u
1 + 0.0117-u mb
M
N
' c 1-e. Y Y _
l-e
N
1-eb )
Re
l-e b
R :
V
(2-39)
For further use of the model, equation (2-39), can be solved explicitly for
the positive values of velocity ratio:
umb
(pma + B m )-l +A/((pm(l + B ro )-l) 2 -4((p m +C m A m Xl-( Pm +A m )B r
(2-40)
u
»c " 2((pm+CmAm)
where
N,
A m =0.0117—Ss_
m
(2-41)
-.
1-e.
f' Y
(2-42)
B.
vfcc;
(2-43)
1-e.
Actually, the purpose of the simplifications by Martin [1978] are to
overcome the problem of calculation when employing the model to heat
51
Literature Survey
and mass transfer calculations in packed beds and the limitation of
applicability of Ergun equation which is restricted to bed porosity less than
0.5. Because of the increased availability of computing power and
software utilising numerical methods to solve the mathematical equations,
the first reason for the simplifications is not essentially required any
longer.
By using two zone area of fluid flow in packed beds as proposed by Martin
[1978], it may overcome the original limitation of Ergun equation for bed
porosity higher than 0.5 at the vicinity of the wall. However, this
simplification reduces the microscopic view advantage, because of the
results of investigation by using this approach are more pronounced as
macroscopic view, therefore it should not be considered for systems which
are sensitive to local velocity variation.
Vortmeyer and co-workers [Kalthoff and Vortmeyer, 1980; Vortmeyer and
Schuster, 1983; Vortmeyer and Winter, 1984; Vortmeyer and Michael,
1985; Cheng and Hsu, 1986; Cheng and Vortmeyer, 1988; Vortmeyer and
Haidegger, 1991] tried to develop the mathematical model of velocity
profile in packed bed by extending the Brinkman's equation [Brinkman,
1947] into microscopic view. The model was applied in a study of mass
and heat transfer [Vortmeyer and Michael, 1985; Cheng and Vortmeyer,
52
Literature Survey
1988] and a runaway chemical reaction [Kalthoff and Vortmeyer, 1980],
which have shown good results, qualitatively.
Brinkman's [1947] equation is developed from macroscopic view of fluid
flow in a porous medium by interpolating the Stokes equation and Darcy's
law [Durlofsky and Brady, 1987]. By neglecting the potential energy
change, in microscopic view the Brinkman equation may be written
[Wilkinson, 1985] as:
Vp = li'V2um-^um (2-44)
where K is the permeability, u\ is the viscosity of the fluid, and u' is an
effective viscosity. The permeability K and viscosity ratio ji'/p are
properties of the porous material [Wilkinson, 1985].
On the macroscopic level the flow of a single fluid through permeable
materials may be describe by Darcy law [Larson and Higdon, 1986]:
^ = -^u (2-45)
dz
K
M
v
Darcy flow is an expression of the dominance of viscous force applied by
solid porous matrix on the interstitial fluid and is of limited applicability.
Post-Darcy flow is effected by inertia forces and turbulence [Kececioglu
and Jiang, 1994], for example the Ergun [1952] equation:
53
Literature Surcey
^
= -f.UM-f2UM2
(2-46)
Where
(1_8o) H
f, = 1 5 0 - — r - — —
p
t
o
D
^P
(1_eo) P
f, = 1.75-
(2-47)
^
e •
—
(2-48)
D
Substituting equation (2-45) into equation (2-46) gives:
I1 . __.
2
- U M = flUM+f2UM
K
(2-49)
Since the validity of the Brinkman equation is restricted to low flow rates,
Vortmeyer and Schuster [1983] extended it to higher flow rates by
incorporating the Ergun equation. By assuming the value of \i'/\i = 1, and
the macroscopic view result of Darcy law and Ergun equation treatment
(equation (2-49)) can be applied into microscopic view, then substituting
equation (2-49) into equation (2-44) to give:
Vp = uV2um-f.um-f2um2 (2-50)
For incompressible fluid flow through a cylindrical bed with axial symmetry
(that is, 8 /3<|> = 0 and um$=0), no radial flow direction (that is umr=0), an
no pressure gradient over the cross-section of the bed, and then equation
(2-50) may be rewritten [Vortmeyer and Schuster, 1983] as follows:
54
Literature Survey
3p
fav i9u
= -fiUm-f2um2+u.
dz
dr' + r dr
(2-51)
In order to fulfil the assumption that velocity profile has one m a x i m u m at
near the wall, Vortmeyer and Schuster [1983] proposed an exponential
relation of porosity distribution, as follows:
R-r
e = e' 1 + c'Exp 1-2
(2-52)
157
where c' has to be adjusted according to e'0.
Equation (2-51) is an elliptic partial differential equation. In order to
ensure the stability of the calculation of equation (2-51), Vortmeyer and
Schuster [1983] proposed a variational method. The solution of
mathematical equation by using variational method is based on the
optimisation of an integral [Courant and Hilbert, 1953].
Derived from equation (2-51) by employing the variational method, the
equation to calculate the velocity profiles in circular packed beds is
[Vortmeyer and Schuster, 1983]:
N = LJ.]£
[(ri+ArJ-rr'_C 1
i= l
2
mi
i 2 r
3
2
Au. A
+ 2rsM
Ar
= minimum
(2-53)
with regard to the m a s s balance equation as follows:
55
Literature Su/rev
Qicai = 2 ^ X u m i A r i r i
i
= constant (2-54)
Comparison of calculation results by using equations (2-53), (2-54), and
(2-52) with experimental data has shown that the predicted maximum
value is far higher than that measured [Vortmeyer and Schuster, 1983].
Similar result was also reported by Johnson and Kapner [1990], who
developed a model velocity profile based on Brinkman equation without
modification for the Darcy term with Ergun equation. A poor agreement
between predicted and measured results also has shown on more
advanced investigation by using this model for more complex situations
where flow distribution is accompanied by heat and mass transfer, and
chemical reaction, for example, runaway reaction [Kalthoff and Vortmeyer,
1980; Vormeyer and Winter, 1984]. This is due to a combination of
causes, which include the limitation of the Brinkman equation, the using of
macroscopic level of Ergun equation in microscopic view, and the
assumption that the value of viscosity ratio equals to 1.0.
The Brinkman equation is a superposition of Darcy's law and Stokes
equation [Saleh et al., 1993b; Durlofsky and Brady, 1987; Wilkinson,
1985]. The diffusion of momentum in the bed, via the effective viscosity u',
is given predominance to Darcy term (that was replaced by Ergun
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Literature Survey
equation for this model). The effective viscosity is a function of bed
properties [Lundgren, 1972]. Consequently, the assumption of |i'=u would
reduce the validity of Brinkman equation. Another limitation is the validity
of Brinkman equation which is restricted to highly porous media [Saleh ef
al., 1993b] and it is the fact that the bed porosity of major packed beds is
generally lower than 0.5 except in the wall region [Vortmeyer and
Schuster, 1983].
The important effect of bed porosity on the velocity profile of fluid flow in
packed beds has been recognised by numerous investigators [Schwartz
and Smith, 1953; Cairns and Prausnitz, 1959; Newell and Standish, 1973;
Stanek and Szekely, 1974; Vortmeyer and Schuster, 1983; Standish,
1984]. Cohen and Metzner [1981] followed the question by attempts to
analytically quantify the effect of bed porosity variation on radial
distribution of fluid velocity in packed bed.
These authors developed a parallel channel model of the packed bed
which they regarded as a porous medium divided into wall, transition and
bulk regions, as shown in Figure 2-11. The region extending from the wall
to xt is considered to be the wall region. The second region, which
extends from xt to xb, is defined as the transition region. The remaining
region, which extends from xb to the centre of the bed, is considered to be
the bulk region.
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Literature Survey
Considering a Newtonian fluid flowing in a packed bed and assuming that
inertial effects can be neglected, the relationship between interstitial
velocity and pressure drop becomes [Christopher and Middleman, 1965;
Cohen and Metzner, 1981]:
Ap De2
=
2 55
^ W.t < - >
Where le is the length of the channel or alternatively, the effective path
length followed by the fluid, and De is the effective channel diameter. The
factor K0 essentially accounts for the inadequacy in the choice of a proper
effective diameter, De. The usual choice of an effective diameter is based
on the concept of hydraulic radius [Bird et al, 1960], where the effective
diameter is replaced by four times the hydraulic radius.
Cohen and Metzner [1981] defined the interstitial velocity as follows:
"-it (2-56)
Substituting equation (2-56) into equation (2-55) and replacing the
effective diameter by means of the hydraulic radius, equation (2-55) may
be rewritten to give :
u =^^- (2-57)
m
L uB(n)
v
where
58
Literature Suney
l^2
B(n) = K 0 l £ l
(2-58)
Although the constant B(n) in equation (2-57) w a s claimed as a universal
constant, Cohen and Metzner [1981] suggested that constant B(n) be
determined by experimental in order to assure the accuracy of fluid flow
calculations.
In order to account for the wall effect, Cohen and Metzner [1981]
employed a different kind of hydraulic radius for each region of the bed.
For wall region, they employed the hydraulic radius definition that was
developed by Mehta and Hawley [1969], as follows:
Volume of voids
H
Volume of bed
~ Wetted surface area of spheres Wetted surface of wall
+
Volume of bed
Volume of bed
and for transition and bulk regions, they employed the hydraulic radius
definition that was developed by Bird etal. [1960], as follows:
Volume of voids
Volume of bed (2-60)
H
" Wetted surface area of spheres
Volume of bed
59
Literature Survey
bulk
region
transition
region
wall
region
bulk
region
R/2
X=0
X,
xh
Figure 2-11: A schematic representation of the tri-regional model
[Cohen and Metzner, 1981].
60
Literature Survey
For the bulk region, the equation to predict the velocity of fluid w a s
derived by substituting equation (2-60) into equation (2-57) to give [Cohen
and Metzner, 1981]:
ub =
Ap
DPV
36uL(l-e cb ) 2 B b (n)
(2-61)
The average superficial velocity, ut, in the transition region from xt to xb,
can be written as [Cohen and Metzner, 1981]:
u t =^-£u m dA
(2-62)
W h e r e A t is the cross sectional area of the transition region. Substitution
of the capillary model for um (equation (2-57)), together with the definition
of the hydraulic radius (equation (2-60)) yields:
u,
_Ap_
•D,
36uL
V
1
A, Jv
£ t dA
(2-63)
1-e.
In the wall region, the usual definition of the hydraulic radius (equation (260)) cannot be used since the presence of the wall is not considered
[Cohen and Metzner, 1981]. As the porosity tends to the value of unity
[Roblee etal., 1958 ; Benenati and Brosilow, 1962], the hydraulic radius
tends to infinity. The applicability of the capillary model (discontinuous
system) is restricted to bed porosities less than 0.5 [Bird et al., 1960]. In
61
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order to minimise the error from limitation of the capillary model, Cohen
and Metzner [1981] employed the average porosity to wall region rather
than the local porosity. The average porosity in the wall region, £0w, is
defined as:
= — iedA (2-64)
eow
A
JAxx,
The hydraulic radius for the wall region can be derived from equation (259) to give:
(
' D
X
DP
t
X
t^0w
VD P
r
H - D - r nD
A
(l-e 0w )
+ 6x
-x,
DP
'ID,
(2-65)
The average superficial velocity in the wall region can then be determined
by using equation (2-65) for hydraulic radius in equation (2-57), hence
[Cohen and Metzner, 1981]:
Ap %2 r (2-66)
^* Wl
_
-~-wx
I
\
uLB w (n)
The limitations of Cohen and Metzner's model [Cohen and Metzner, 1981]
are the presence of empirical constants (Bw(n), B,(n), and Bb(n)) and the
prediction results that are average superficial velocities at each region
rather than the local velocities. Because of the requirement of velocity
62
Literature Suivev
distribution measurement to determine the empirical constants, using this
model is not practical. In cases where heat transfer or residence time
considerations are critical, as in chemical reactors, this velocity profile
prediction also may not adequately fulfil the requirement, as for example,
analysing of hot spot formation and runaway reactions.
The three region model of Cohen and Metzner [1981] was then adapted
by Nield [1983], who assumed that near the wall only fluid exists, whose
flow may be represented by Stokes equation with the unrealistic slip
boundary condition proposed by Beaver and Joseph [1967]. For the core
region, Nield [1983] assumed that the fluid flow obeys Darcy's law.
Although the mathematical equations that were developed by Nield [1983]
may be solved analytically, however, because of the limitation of the
assumption, the results are questionable. The assumption that near the
wall only the fluid exists is unrealistic for packed beds, because the
porosity never reaches unity for any random packed bed, except for r = R,
where the velocity is zero for no-slip condition. Additionally, the Nield
equation still needs an empirical constant to be determined
experimentally.
Similar to Cohen and Metzner [1981] and Nield [1983], McGreavy et al.
[1986] also divided the flow phenomena in packed beds into regions.
Their model was based on an assumption that the flow of fluid can be
63
Literature Suivey
divided into two zones, shown as the core and annulus in Figure 2-12. In
the core the velocity is assumed to be constant, umc, and governed by the
usual equations for flow through packed beds. The annular region
extends to approximately one to two particle diameters from the wall and
because of the enhanced porosity the velocity is higher. It is also
influenced by the boundary, which will cause the fluid velocity to approach
zero at the wall.
McGreavy ef al. [1986] assumed a continuum model can be applied for
the annular region, and they derived from a momentum balance that:
Ap 1 d . . ,„ ,.
—f- + -—M
L
=0
(2-67)
r dr
The pressure drop is assumed constant over the cross section of the bed,
and the shear stress is assumed can be expressed by two components,
as follows [McGreavy etal., 1986]:
du
x = x_-\x-
dr
(2-68)
Where xd is the drag due to particles that is constant and analogous to
that arising in the core, but is based on different bed porosity.
Substituting equation (2-68) into equation (2-67) and then integrating
gives:
64
Literature Survey
Bed Centre
maximum
u m = f(r)
umm = u"mc
Figure 2-12: A two zoned bed [McGreavy etal, 1986].
65
Literature Survey
du.
i_-V dr r = C
2L
(2-69)
If the general profile in the annulus is to show a maximum at some radius
rm, then dum/dr = 0 at this point and this defines the constant C.
Integrating equation (2-69) gives the equation of u m , as follows:
um=
r2 R2 rm
pHf( " )~ 4^
(2_70)
where
p
(2-71)
2L
Which, w h e n r=R gives u m =0, as required. T h e other condition if that at
r=rc the velocity is equal to umc, and this then enables a relationship for rm
to be obtained as follows:
V
MUU,
rm
=•
m
2B.
2B n
+B p ln
+ 1+
2T,
B p (R + rc)
R
B p ln
(2-72)
^
y^JJ
In order to reduce the errors involved in using the constant value of xd,
McGreavy etal. [1986] proposed that the value of xd proportional to um, as
follows:
66
Literature Survey
x
d = agum
(2-73)
Where Og is a constant, that is determined by experiment.
The main limitations are that this mathematical model has empirical
constants and uses an inconsistent interpretation of shear stress.
Basically, the shear stress is function of fluid properties and true velocity
(interstitial velocity) difference [Bird et ai, 1960; Poirier and Geiger,
1994], so the definition of shear stress by McGreavy et al. [1986] in
equation (2-68) is unrealistic and in using it, it is possible to introduce
significant errors. This condition was proved by inconsistency in
interpretation of the value of xd [McGreavy et ai, 1986].
In order to accommodate the oscillation profile of velocity distribution
measured in packed beds, Ziolkowska and Ziolkowski [1993] developed a
model based on fluid kinetic energy dissipation that is not uniform over
the cross section of the bed. The kinetic energy dissipation is a result of
fluid friction against the interface and between fluid molecules. All of these
contributions are represented in a local effective viscosity, which is an
empirical parameter.
In order to overcome the problem of discontinuity of the packed beds
system, Ziolkowska and Ziolkowski [1993] proposed a dimensionless
distance parameter as follows:
67
Literature Survey
—
T*
R-r
DP
(2-74)
Where the parameter r* was chosen so that the reduced distances from
the tube wall, r*are multiples of DP/4 and the reduced thickness, AT*, of
each individual ring fulfils the conditions:
Ar;=r;-r;_p i = 1,2,3 n (2-75)
The model consists of fluid dynamics equations describing fluid flow
through an arbitrary annular segment of packed tube with a volume :
V.=2.iLr._.Ar. (2-76)
The shear force per unit area was assumed appropriate to modified
Newton's law by Boussinesq [Ziolkowska and Ziolkowski, 1993] as
follows:
x = x1 + x'
rz
rz
v
rz
p,
t /du^
v
Dp•(v + v\dr*
^ k jJ
(2-77)
The turbulent kinematic viscosity coefficient v1, valid for an interval of
radial position Ar* within the system, may be approximated by a constant
function. The density and the laminar viscosity v1 are assumed to be
constant because of the isothermal conditions and negligible effects of the
68
Literature Survey
pressure variations due to the flow. The local bed porosity of an arbitrary
annular bed segment of a region near the wall and extending to r* < 5 of
the tube-to-pellet diameters, was assumed as a linear function
[Ziolkowska and Ziolkowski, 1993].
The equation given by Ziolkowska and Ziolkowski [1993] for the radial
component of interstitial velocity vector of a bed of spherical particles is:
ur=±^(l-e)u (2-78)
The coordination number is positive (+1) when ur is directed toward the
tube wall and negative (-1) when ur is directed towards the symmetry axis
of the tube [Ziolkowska and Ziolkowski, 1993].
The continuity equation given by these authors to develop the
mathematical model of velocity profile across the bed may be written as
follows:
^il = 0 (2-79)
Sr*
and the equation of motion in the axial direction:
13/ r , _, lN dp
Dpdr
,{purruzz+e[x\
z])-£+PF^0
L rz z+T[
rzJ/
^
dz
(2-80)
69
Literature Survey
The axial pressure gradient is determined by using the Ergun equation
[Ergun, 1952], Ziolkowska and Ziolkowski [1993] defined the Ergun
equation as the external force, and the friction (internal) force Fk was
calculated by using Ranz [1952] correlation, given as:
Fk=f^uz2 (2-81)
k
DP
where
f==1 . 75+1 50n-ftlai
(2-82)
pDPuM
The boundary conditions of equations (2-79) and (2-80) are
- at the tube inlet, when z = 0:
0<r*<|-, u = uM, p =
Po (283)
p
r* = 0,
u = 0,
p = p0
along the tube wall, when 0 < z < L:
1 < K n,
r = r- ,
u
uz -
£
(2-84)
i = n. r*=0, uz =0
at the tube outlet, when z = L:
.
R
0<r < — ,
P = PL
Dp
i = n, r*=0, u = 0, P = Pi
:
2
1 < i < n, r. . <r* < r.\ u = -Juz +u,.
(2-85)
Literature Survey
The model solution, determining the radial distribution of the gas
interstitial velocity within a circular packed bed, has been performed by
Ziolkowska and Ziolkowski [1993] for the region near the wall and the
central region, separately. This has been done because the bed porosity
fluctuates along the tube radius up to a distance of r* < 5 from the wall and
approaches a constant value within the bed core when r* > 5 [Roblee et al,
1958; Benenati and Brosilow, 1962],
When r* > 5, the model solution for the bed core is based on the
assumption that the bed porosity within this region is uniform. The
solution of the mathematical model for this region is [Ziolkowska and
Ziolkowski, 1993]:
ApE)
p
pLf(l-e0i)
(2-86)
The model solution for the region near the wall of the bed, up to r* < 5 is
[Ziolkowska and Ziolkowski, 1993]:
1
Uz =
2
1
ve ]
ve
•+ .
.
D P e oi
D
£
J
v p 0i
ApDpA£
+ 4 _
(2-87)
pLfe0l
where
v e = -_iL( v > + v . + v n )
(2-88)
71
Literature Survey
f AeV
(2-89)
V
~4 f_^L
ri _ 7 . U M D p k £n;e
Oi'
(2-90)
I Ar*
e0i(e2+e
e=
Oi'
2(1-e]
(2-91)
Within the limits of assumptions made and of developing methods of
equations, equations (2-86) to (2-91) define the velocity profile in terms of
bed porosity, particle diameter, radial position, and local effective
kinematic viscosity, ve. The effective viscosity is a function of the radial
gradient of the bed porosity, the local bed porosity, the laminar
(molecular) and turbulent viscosities, the radial dispersivity, and the
Reynolds number.
The value of the effective viscosity, unlike the molecular and turbulent
viscosities, may be positive or negative depending upon the sign of the
bed porosity gradient, and thus the effective viscosity coefficient may also
be negative or non-negative [Ziolkowska and Ziolkowski, 1993]. The
difficulty arises from the fact that the effective viscosity is an empirical
parameter that is determined from measuring of superficial velocity
distribution at the outlet cross section of the bed.
72
Literature Survey
Based on the measured data at the downstream of the tube for the tubeto-particle diameter ratio in the range 10.8 < D/DP< 22.9, Ziolkowska and
Ziolkowski [1993] proposed an empirical correlation to predict the
effective viscosity, as follows:
ve
— = Az(Exp(r*Bz))(Cos(2.0557ir*) + 0.45) (2-92)
where
'D V
D
A z =3.419-0.148
DD
+ 0.011 ,D ;
P
6n DV
D
B z = -0.668 + 0.048
D
+ 0.004VDP ;
(2-93)
(2-94)
^p
Comparison of the model proposed by Ziolkowska and Ziolkowski [1993]
with their measured data shows the reported deviations in the range of
20- 30%. The errors probably are due to both the failures in developing
the mathematical model from the fundamentals of the physical correlation
and to turbulent fluid flow behaviour assumption. Another limitation of this
model is the empirical constant, effective viscosity. Thus, using this model
is not practical because of the requirement of the experimental data of the
velocity for the system being considered.
According to Bird etal. [1960], the frictional energy losses are included in
the Ergun pressure drop correlation [Ergun, 1952] rather than separately
73
Literature Suney
accounted by using Ranz correlation [1952], as shown in equation (2-80).
Basically, the Ergun and Ranz correlations are complementary equations
to account for energy losses of fluid flow because of friction which use
different theoretical approach in developing the correlations [Bird et ai,
1960; Poirier and Geiger, 1994]. However, the value of the parameter Fk
in equation (2-80) must be replaced by the acceleration of gravity, g, to
make this equation consistent with the basic equation of motion [Bird et
ai, 1960]. In addition, Ziolkowska and Ziolkowski [1993] also neglected
the term of -(e/(R-r*DP))xrz from the left-hand side of equation (2-80)
without any reasonable reasons. The above factors may also introduce a
significant error into the model.
The use of the turbulent fluid flow behaviour assumption is unjustified in
packed beds, as the work of Mickley et al. [1965]. Their has shown that
momentum transfer inside a packed bed is a function of the gross
properties of the bed, depending on the sidestepping of the fluid stream
as it passes between packing particles, rather than the actual turbulent
structure of the flow. The effect of turbulence is more pronounced on
flow of fluid in the voids between particles [Merwe and Gauvin, 1971a;
Matsuoka and Takatsu, 1996] rather than the flow profile over cross
section of the bed. This explanation is supported by the very much higher
74
Literature Surx-ev
deviation of effective viscosity (22%) measured with repacking, as
reported by Ziolkowska and Ziolkowski [1993],
From the above brief discussion of the available models of the velocity
distribution of flowing fluid through a packed bed reported in the literature,
it is clear that the basic requirement of a mathematical model according to
Reid and Sherwood [1966] has not been reached. This is due to the
nature of packed beds system in which the phenomena of fluid flow
through packed beds include a transition between flow in channels and
flow around submerged objects. There is, therefore, still a continuing
need for research in this area.
75
CHAPTER THREE
CHARACTERISATION OF A PACKED BED
The study of the packing behaviour of granular materials has been carried
out since many years ago over a very broad range of topics [German,
1989], whether the specific objective be the achievement of a dense
packing or the establishing and maintenance of a freely flowing condition.
In practice, there are a variety of factors, which influence the packing
behaviour as summarised by Macrae and Gray [1961] eg. particle,
container, deposition and treatment after deposition. The focus of the
following discussion is restricted to the behaviour of particle packing
related to the velocity distribution of the fluid flowing through a packed
bed.
The importance of the character of the packed beds for the velocity
distribution of fluid flowing in the bed, has been increasingly recognised
over the years by numerous investigators [Arthur et ai, 1950; Morales et
ai, 1951; Schwartz and Smith, 1953; Dorweiler and Fahien, 1959; Cairns
and Prausnitz, 1959; Newell and Standish, 1973; Szekely and Poveromo,
1975; Stephenson and Stewart, 1986; McGreavy et ai, 1986; Ziolkowska
and Ziolkowski, 1993]. According to Brown et al. [1950], the packing
76
Characterisation of a Packed Bed
variables that have significant influence on fluid flow in a packed bed are
as follows: porosity of the bed; size and shape of particles; packing
arrangement of the particles; and roughness of the particles.
3.1 PARTICLE SIZE
It is well known that the particle size, including the size distribution, is th
basic parameter of the packed beds system. For a smooth dense sphere,
the particle size can be accurately defined by a measurement of its
diameter. In most applications, however, as cited by Davies [Fayed and
Often, 1984] accurate particle size measurement is difficult, because most
practical particles are irregular in shape; and therefore, the diameter is a
function of the measurement method.
Generally, the method of particle size measurement is dependent on the
size of particle [Fayed and Otten, 1984; Levenspiel, 1984]. For a nonspherical particle, the equivalent diameter is used to represent the size of
particle. The equivalent diameter is defined by the diameter of a sphere
having the same volume as the particle [Brown et ai, 1950; Levenspiel,
1984]. Usually, for the packed bed which consists of a number of different
size particles, the size of particles is characterised by mean size or
diameter [Brown et ai, 1950]. The mean size (diameter) of multiple sized
particles, DPm, can be based on diameter, on area, or on volume
[Standish, 1979; Goodling etal, 1983];
77
Characterisation of a Packed Bed
D p (diameter) =
_£n,Dpi
(3-1)
Zni
DPm(area) =
(3-2)
D _(volume) =
(3-3)
and the harmonic m e a n diameter [Standish, 1979; Yu and Zulli, 1994] is:
D p (harmonic)
n;
n-i
^ D
(3-4)
Although the consensus has not been reached on the best definition for
the mean size diameter, considering the study of fluid flow in packed
beds, the widely used m e a n size diameter is the volume-surface m e a n
diameter D vs [Standish, 1979; Bird et ai, 1960; Poirier and Geiger, 1994].
This is reasonable because mixtures which have the s a m e value of the
volume-surface m e a n diameter have the s a m e surface area (the total
particle surface/the volume of the particles) as cited by Standish [1979].
The volume-surface m e a n diameter is defined as [Standish, 1979]:
D
_!___&:
(3-5)
In practice, the mixture data are often performed in mass fraction terms,
and therefore the volume-surface m e a n diameter may be defined as
78
Characterisation of a Packed Bed
[Fayed and Often, 1984; Poirier and Geiger, 1984; Morkel and Dippenaar,
1992]:
3.2 T H E B E D P O R O S I T Y
German [1989] defines the bed porosity or voidage as a volume fraction of
void space in a powder mass. The porosity equals one minus the
fractional density of the bed. The mathematical expression of the bed
porosity as cited by Dullien [Fayed and Otten, 1984] is:
volume of voids in packing
e=
bulk volume of packing
(3-7)
The bed porosity or voidage could be broadly categorised by two terms,
that is, mean voidage, e'0, and local voidage, e. The mean voidage is the
fractional free volume in a packed bed. The local voidage is the fractional
free volume in a point at the bed; however, because the point voidage is
not readily measurable, usually the local voidage is defined as a fractional
free volume in an element of bed volume, as in a thin strip or shell.
The nature of packed beds are random systems and cannot be exactly
duplicated [Schwartz and Smith, 1953; German, 1989], and hence, the
experiment is a key factor to describe the characteristics of random-
79
Characterisation of a Packed Bed
packed beds [Blum and Wilhelm, 1965]. But obviously, a mathematical
correlation of packed bed characteristics is often required to solve the
mathematical description of systems, which involve packed beds. A
mathematical correlation of the bed characteristics, with proper error level,
is perhaps expedient and satisfactory for several process-engineering
calculations. In general, a successful mathematical correlation of packed
bed characterisation is based on experimental data which have been
correlated by using statistical approach [Blum and Wilhelm, 1965;
German, 1989; Yu and Standish, 1993ac; Zou and Yu, 1996].
3.2.1 Mean Bed Porosity
In principle, the value of the mean bed porosity depends on size consist
(particle size distribution), handling method and container. The container
has significant influence on the mean bed porosity and is called the wall
effect [Haughey and Beveridge, 1966; Fayed and Often, 1984;
Cumberland and Crawford, 1987; German, 1989]. For example, the value
of the mean porosity of uniform sized spheres in a tube that has a
maximum value at the tube to diameter ratio of about 1.62 and gradually
tends to a constant value for the tube to diameter ratio higher than 10
[McGreavy etal., 1986] as shown in Figure 3-1.
80
-
oc
-Mr
CD
OO
CD
_
0
TO
o.
CO
CM
>
5
cc
a
03
3
1.
O
LO
CM
O
•a
03
A
CO
03
O
q
c\i
CO
a
ro
o
'sz
o
Q.
Q
o
CO
0)
JZ
a.
co
o
>
"co
o
o
Q.
c
q
0.
ro
03
E
c
o
o
LO
*-•
o
0)
M—
03
i_
03
03
E
ro
T31
CO
0)
03
13
i_
33
+J
O)
0
3
LL
£
Characterisation of a Packed Bed
Considering the random packed beds in which particles are randomly
arranged [Blum and Wilhelm, 1965], or all particles of the same size and
shape have the same probability to occupy each unit volume of the
mixture [Debbas and Rumpf, 1966], there are two reproducible states of
packing. These are random dense arrangement and random loose
arrangement [Brown et ai, 1950]. These terms characterise the
configurations, which result when a bed of particles is packed in an
apparently random manner to its densest and loosest conditions,
respectively. The packing arrangement is called dense random packing,
when the particles are poured into container then shaking for about 2
minutes to reduce the total volume. The packing arrangement is called
loose random packing, when it is tipped horizontally then slowly rotated
about its axis and returned gradually to the vertical position [Cumberland
and Crawford, 1987].
In a more detailed categorisation, Dullien [Fayed and Otten, 1984] divided
the random arrangement of packing into four categories; close random
packing; poured random packing; loose random packing; and very loose
random packing. When the bed was vibrated or vigorously shaken down,
the resulting arrangement was called close random packing. Pouring
particles into a container, corresponding to a common industrial practice
of discharging powders and bulk goods, was termed poured random
packing. Loose random packing resulted from dropping a loose mass of
82
Characterisation of a Packed Bed
particles into a container, or packing particles individually and randomly
by hand, or permitting them to roll individually into place over similarly
packed particles. The packing arrangement of the fluidised bed particles
at the minimum fluidisation is called very loose random packing.
The spherical particles is a simplest geometry of packing particles, that is
not a surprising condition if it is taken into account as a basis to develop a
method to predict the packing character [Brown et ai, 1950; Lamb and
Wilhelm, 1965; Levenspiel, 1984; Yu and Standish, 1993b; Zou and Yu,
1996]. Considering the infinite packing for which voidage (the bulk mean
voidage, e'0) is not affected by the presence of external surfaces
[Haughey and Beveridge, 1966], that has been experimentally found for a
tube to particle diameter ratio of from 10 to 15 [Lamb and Wilhelm, 1965;
McGreavy et ai, 1986; German, 1989], the mean voidage of mono-sized
spheres is only dependent on packing arrangement and independent from
particle size [Standish, 1990]. The measured data of the mean voidage of
the mono-sized spheres as a function of packing arrangement are listed in
Table 3-1.
Usually, for non-spherical particles, they are characterised in terms of an
equivalent spherical diameter [Brown et ai, 1950; Levenspiel, 1984; Yu
and Standish, 1993b; Zou and Yu, 1996], called as sphericity.
83
Characterisation of a Packed Bed
Table 3-1: Mean porosity of the packed beds of spheres.
No
Arrangement
Porosity, e 0
0.476
Reference
Brown et ai, 1950
1
Cubic
2
Face-centered cubic
3
Body-centered cubic
0.3198
German, 1989
4
Orthorhombic
0.3954
Brown etal., 1950
5
Tetragonal sphenoidal
0.3019
Brown etal., 1950
6
Rhombohedral
0.2595
Brown et ai, 1950
7
Diamond
0.6599
German, 1989
8
Close random packing
0.359 - 0.375 Fayed and Otten, 1984
9
Poured random packing
0.375 - 0.391 Fayed and Often, 1984
10
Loose random packing
0.40 - 0.41
Fayed and Otten, 1984
11
Very loose random
0.44 - 047
Fayed and Otten, 1984
packing
0.2595 - 2880 German, 1989
Blum and Wilhelm, 1965
84
Characterisation of a Packed Bed
The sphericity concept is more applicable than the concept of packing
size which was proposed by Meloy [Fayed and Otten, 1984], as cited by
Yu and Standish [1993b].
The sphericity, \|/, is defined as the surface area of a sphere having a
volume equal to that of the particle, divided by the surface area of the
particle [Brown et ai, 1950; Levenspiel, 1984], and the maximum value of
the sphericity is equal to 1.0 [Levenspiel, 1984]. On the basis of the
similarity between the packing systems of spherical and non-spherical
particles, the characteristics of the non-spherical particle can be defined
and determined [Yu and Standish, 1993b].
A graphical correlation between the particle sphericity, x\r, and the bulk
mean voidage, e'0, of mono-sized particles have been proposed by Brown
etal. [Brown et ai, 1950; Levenspiel, 1984] as shown in Figure 3-2. Zou
and Yu [1996] have proposed a mathematical formulation for estimating
the bulk mean voidage of cylinder particles and disk particles as follows:
- For the loose random packing:
ln e
( 'o)cyl!nder = ¥558ExP[5.89(l - ¥)]ln(0.40) (3-8)
ln(e'0)dlsk = Y 60Exp[o.23(l - \|/)°45]ln(0.40) (3-9)
85
Characterisation of a Packed Bed
1.0
\\\
0.8V\*M
Loose packing
/
V \ *
\ \
x
*
x
*
*
x
-- x
*
0.6
N
>_
Dense packing
/
s
0 . 4--
Nomal packing
0.2 --
1
0.0
0.0
.
.
1
0.2
1
.
1
1
0.4
L
•
•
1
0.6
•
i
i
—1—'—'—'—
0.8
1.0
Particle sphericity
Figure 3-2: Effect of particle shape on voidage for random-packed
beds of uniform-sized particles [Levenspiel, 1984].
86
Characterisation of a Packed Bed
For the dense random packing:
ln e
( 'o)cylind.. = V674Exp[8.00(l - ¥)]ln(0.36) (3-10)
ln e
( 'oL = V°63Exp[0.64(l-¥)°45|ln(0.36) (3-11)
In order to develop a mathematical correlation to predict the m e a n bulk
porosity of non-spherical particles, Zou and Yu [1996] used the cylindrical
and disk particles as an extreme condition. The mean porosity of nonspherical particles then may be predicted by using the equation as
follows:
e ' = — — (e'n)
I
+ I
+——(e'J
^cylinder
J
_|_ J
\
°/disk
(3-12)
V
'
Where
1,.= vi/-Vdisk
(3-13)
I =
(3-14)
W
V cylinder
For packed beds that consist of a mixture of particle sizes, the m e a n bulk
porosity principally depends on particle size distribution and handling
method [Standish, 1990]. A general quantitative representation of porosity
of a multi-sized packed bed system is impossible because of the nature
of this system having almost unlimited probability of particles
arrangements for a particular size distribution. It is not surprising that
numerous investigators [German, 1989] in this area tended to be
87
Characterisation of a Packed Bed
concerned either with theoretical, unreal (simplified) conditions or to be
entirely empirical to fulfill the requirement for a quantitative prediction
[Macrae and Gray, 1961]. Fortunately, for a uniform mixture of multi-sized
particles in which the number of size components is more than two, the
feature of the changing of the mean bulk porosity as a function of the
volume fraction of the components is similar to the binary system
[Standish, 1990].
Considering uniform mixtures of multi-sized particles, generally the
investigation of the mean bulk voidage is developed from the results of
investigation of two-sized mixture (binary system) of spherical particles
[Furnas, 1931; Ridgway and Tarbuck, 1968b; Standish and Borger, 1979;
Standish and Yu, 1987a'b; Yu and Standish, 1988; 1991; 1993abc]. By
introducing any arbitrary factor or function into the results of binary
spheres system then the correlation may be extended to characterise both
of multi-sizes mixture systems, namely spheres and non-spheres particles
[Yu and Standish, 1988; 1993b].
For uniform mixtures of the binary spherical particles, the mean bulk
porosity is lower than initial porosity of the former uni-sized particles
[Ridgway and Tarbuck, 1968b; Standish and Borger, 1979; Fayed and
Otten, 1984; German, 1989] as shown in Figure 3-3. The explanation of
Figure 3-3, as given by Standish [1990], is that on addition of fines to
88
Characterisation of a Packed Bed
coarse particles the voids among the coarse particles gradually fill up until
they are all filled and no more fine particles can fit in, and the voida
decreases. If more fines are still forced into the already filled space,
do that by forcing the coarse particles apart, and this increases the to
volume, therefore voidage increases.
Yu and Standish [1988] have applied a general thermodynamics concept
of solutions [Smith and Van Ness, 1975] to determine the mean bulk
porosity of the multi-sized particles bed. They introduced a definition
the initial specific volume of particles, Vi, that may be expressed
mathematically as follows:
1
V
i = 1-e'
(3-16)
Oi
Considering binary system of the bed with the fractional volume of
particles, Xi, equation (3-12) is satisfied [Yu and Standish, 1988]:
V V-X.-V2X2
'V-V-X-V
v-v.x,
+ 2$
V,
v,-i
V,
(3-17)
^V-X,-V2X2^
=2
+
v,-i
89
Characterisation of a Packed Bed
0.1 H — ' — ' — ' — I — ' — ' — ' — | — ' — ' — ' — | — ' — ' — ' — I — ' — ' — ' —
0.0
0.2
0.4
0.6
0.8
1.0
% Volume of large particles
Figure 3-3: Voidage mixtures of binary system for spheres [Yu and
Standish, 1988].
90
Characterisation of a Packed Bed
Where the coefficient Q in equation (3-17) is an unknown parameter of
the Westman equation [Yu and Standish, 1988; Yu et ai, 1993]. For
spherical particles, Q has been reported to be dependent on the size ratio
of large particle diameter (DP2) to small particle diameter (DPi) and this
dependence can be determined empirically [Yu et ai, 1993]. Yu et ai
[1993] proposed the following general correlation to predict the value
of#
f rxx V-566
D^
Lrfrx,
V Dp2 J
ID P2 J
1.355
-^Pl
< 0.824
(3-18)
> 0.824
VDP2 )
For given initial mean bed porosities of the binary systems, it is evident
that the mean voidage can be determined by employing equations (3-16)
to (3-18), simultaneously. In applying this method to predict the mean
voidage of the non-spherical systems, Yu and Standish [1993b] proposed
a concept of equivalent packing diameter. The equivalent packing
diameter, Dpe, of a non-spherical particle may be expressed as a function
of its equivalent volume diameter, DPv, and sphericity, as given by [Yu and
Standish, 1993b]:
D Pe
D Pv
3.6821 1.5040
3.17811- V + l|T=—
(3-19)
91
Characterisation of a Packed Bed
Considering multi-sized system of the bed with the fractional volume of
particles, Xi, Yu and Standish [1988] introduced some arbitrary terms into
the thermodynamics equation of solutions, that is called the binary
synergism of the mixture. For binary mixtures, Yu and Standish [1988]
approximated the specific volume equation by the equation:
V = V1X1 + V2(l-X1) + (3I2X1(l-X1) + y12X1(l-X1)(2X, -1) (3-20)
Where the coefficients p 12 and y12 are called the quadratic coefficient and
the cubic coefficient of the binary synergism, respectively [Yu and
Standish, 1988]. These coefficients are only dependent on the initial
specific volume and size ratio of binary systems, being constant for given
initial specific volumes and size ratio [Yu and Standish, 1988].
By extending equation (3-18), the specific volume of n-component
mixtures may then be represented by the following equation [Yu and
Standish, 1988]:
v = Evlxi + £ Spax.xj+ X Xy^x^-xJ (3-21)
1
l<]
Kj
Where
v... + v... - v. - v.
p..=-^
H,J
* !
0.4032
L
(3-22)
V
'
V
iii-Viii-0.44Vi+0.44 V.
v, = —
r,J
-
!
L
(3-23)
0.177408
92
Characterisation of a Packed Bed
Vy and Vyj are the specific volumes corresponding to the two points:
X; = 0.72, Xj = 0.28, and X| = 0.28, Xj = 0.72 (i < j), respectively, which can
be calculated from equations (3-17) and (3-18).
As mentioned earlier, random packing is a random (stochastic) system,
which can almost certainly not be possible to be explained in an exact
correlation. In other words, all of the representations of packing behaviour
are an approximation. This has prompted numerous types of models to
predict the mean bulk porosity for which the applicability is strictly
dependent upon the basic assumptions of the model development. Beside
the successful model that was developed on the basis of the solution
thermodynamics theory, the correlation which was developed from the
coordination number (the number of neighbouring particles forming
contacts with a given particles) also gave satisfactory results [Ouchiyama
and Tanaka, 1980; 1981; 1989; Fayed and Otten, 1984].
Actually, the components used to construct a particle mixture in
engineering practice are themselves particle mixtures and not mono-sized
particles [Yu and Standish, 1993a]. That is, the particle mixture is usually
a mixture of a number of sub-mixtures of particles and its particle size
distribution is thus a mixture of distributions. Based on intensive studies
in this area, it has been shown that the application of models which were
93
Characterisation of a Packed Bed
developed on the basis of the solution thermodynamics theory [Yu and
Standish, 1988] and the coordination number approach [Ouchiyama and
Tanaka, 1981] are satisfactory to predict the bulk mean porosity of the
multi-sized particles mixture consisting of a number of sub-mixtures of
particles [Yu and Standish, 1991; 1993ac; Ouchiyama and Tanaka, 1989],
From the above brief discussion of the bulk mean porosity of multi-sized
packed bed, it has been shown that the general assumption for model
development of the uniform particle mixture has been reached. For
conditions that the particles mixtures are not homogeneous, i.e., there is
particle size segregation, the applicability of the models still remain a
question mark. As stated by Standish [1990], there are two requirements
that must be met simultaneously for a size segregation to occur, namely,
difference in particle sizes and relative motion between particles. For
conditions for which a size segregation may be suspected to occur, then
the statement of Blum and Wilhelm [1965] that the experiment is the key
factor to describe a random packed bed, is still relevant.
3.2.2 Radial Distribution of the Bed Porosity
The wall of container used to hold a random packing material will induce a
local area of order at the region near the wall [Blum and Wilhelm, 1965;
German, 1989]. The effect is more pronounced for flat, smooth containers
[German, 1989], giving local regions of oscillating porosity in first few
94
Characterisation of a Packed Bed
particle layers near the wall and, it has been shown experimentally,
almost independent of the bulk region [Roblee et ai, 1958; Benenati and
Brosilow, 1965; Thadani and Peebles, 1966; Kondelik et ai, 1968; Scott
and Kovacs, 1973; Goodling et ai, 1983]. A knowledge of this local
variation is important since the microscopic evaluation of fluid flowing
through a packed bed can only be obtained from a knowledge of the local
bed structure and not from the use of bulk properties.
The earliest reported comprehensive experimental investigation of radial
bed porosity distribution was carried out by Roblee et al. [1958]. They
designed experiments to study the influence of the confining wall on bed
voidage in a cylindrical column with randomly packed uniform particle
beds of spheres, cylinders, Raschig rings, and Berl saddles. They
investigated the radial voidage distributions by using the following
method. A packing material was poured into a cardboard cylinder, which
was then filled slowly with hot wax, and then allowed to solidify. After the
wax had solidified, the bed was sawed into circular slabs, which were in
turn sawed into concentric rings. Analysis for bed porosity was made by
first removing the wax from the packing material by dissolving the wax in
boiling benzene, then distilling the benzene to recover the wax. The void
fractions was then determined by calculating the mass of wax recovered
and its density.
95
Characterisation of a Packed Bed
By employing the similar method and filling materials to Roblee et ai
[1958], the porosity distribution of random close packing of uni-sized
spheres was investigated by Scott [1962]. He used about 4,000 steel balls
with 3.175 mm diameter that were poured into a 45 mm diameter and 150
mm long cylinder column. The study was continued [Scott and Kovacs,
1973] to investigate the porosity distribution of an equal number of two
sizes (3.172 mm and 3.567 mm diameters) of steel balls in 45 mm and
125 mm of cylinder columns.
Benenati and Brosilow [1962] investigated the radial bed voidage
variation of spherical particle beds in cylindrical, concave, and convex
columns. They used epoxy resin as filling material, which was introduced
into the bed from the bottom and allowed to flow upwards through the bed.
The function of this method is to avoid the air from being trapped inside
the filling agent, so the more accurate result of bed porosity may be
achieved. After curing the resin, the bed was machined and the layer of
approximately one sixth particle diameter removed each time. Porosity
was determined for each layer by means of the simple material balance
based on the mean density. The packing used consisted of uniform sized
lead spheres and measurements were made for D/DP ratios varying from
2.6 to infinity.
96
Characterisation of a Packed Bed
A non-destructive method on the basis of different absorption of the Xrays or y-rays in the sphere material and the matrix material was
employed by Thadani and Peebles [1966] to determine the variation of the
local bed porosity over a cross section of spherical particles in cylinder
column. The cylinder vessel was charged with 9.525 mm diameter red
Plexiglas spheres which was then filled slowly with epoxy resin mixed with
araldite catalyst, and then allowed to cure. After curing the resin, the bed
was sawed into slices two particles diameter thick of circular slabs.
Analysis for bed porosity was made by scanning on the micro-photometer
scanning unit.
The similar non destructive technique was applied by Mueller [1992] to
study the radial porosity distribution of randomly packed beds of uniform
sized spherical particles in a cylindrical container. The local bed porosity
of Lucite Plexiglas spheres was analysed by using X-ray radiography. He
investigated the radial porosity distribution of 12.751 mm diameter
Plexiglas spheres that were packed into four sizes of cylindrical
containers. The diameters of cylindrical columns were 25.75 mm, 50.50
mm, 76.00 mm and 101.88 mm (corresponding to D/Dp ratio of 2.02, 3.96,
5.96 and 7.99) and with the height of each of the different cylindrical
columns is approximately 100.00 mm (corresponding to H/Dp ratio of
7.84). The study was continued [Mueller, 1993] to investigate the angular
porosity variation in randomly packed beds of uniformly sized spheres in
97
Characterisation of a Packed Bed
cylindrical containers. The materials, equipment and analysing methods
for the investigation of the angular distribution bed porosity were similar to
his investigation for radial distribution.
The minimum local bed porosity in the near wall region of a cylindrical
column packed with equilateral 7x7 mm cylinders was investigated by
Kondelik et ai [1968]. They used a technique which consisted of pouring
cylindrical particles into a container and then filling all the interstices with
a solution of poly (methyl methacrylate) in methyl methacrylate (Dentacryl,
Dental, Prague). Upon curing the resin, the bed was cut into cylindrical
layers 2-3 mm thick and 10 DP long. The removal of poly (methyl
methacrylate) was carried out by using acetone, the quantity of bed
particles in each fraction was determined by directly weighing.
A non-destructive method on the basis of the fluorescence of a slightly
impure organic liquid and on the refractive index matching of the packed
bed components was employed by Buchlin et ai [1977] to determine the
local voidage of uniformly sized spherical particles in a rectangular
vessel. The vessel was charged with glass spheres which was then filled
with a liquid having same refractory index as the glass spheres to allow
a light beam to cross the bed without scattering. With ethyl salicylate
liquid the light excites a fluorescent re-emission. The interstitial volumes
are therefore selected by taking advantage of this property and then the
98
Characterisation of a Packed Bed
local voidage distribution can be m a d e by manually marking of the photo
that was observed using a camera.
Goodling et ai [1983] investigated the radial bed voidage variation of
uniform and non-uniform spherical particle beds in a cylindrical column.
They used polystyrene spheres as particles and an epoxy resin was used
to fill the void matrix. The packing material was poured into a plastic pipe
of 50.8 mm inside nominal diameter, fixing a small-mesh wire screen over
the top to prevent flotation of the particles in the denser liquid and then
filled with liquid epoxy together with hardener, from the top. After curing
the resin, the bed was cut from the outer periphery over the entire length
of the sample. The local bed porosity was determined for each layer by
means of the simple materials balance based on the mean density. The
measurements were made for D/DP ratios varying from 7 to 17 for uniform
size of spheres and from 7 to 13.5 for multi-sized of spheres.
In a more recent study Stephenson and Stewart [1986] studied the bed
porosity distribution over the cross section of a cylindrical column packed
with cylindrical particles, by employing the optical measurement
technique. Analysis for bed porosity was made by manually marking of the
photo that was observed using a television camera. The markings were
transferred to punched cards via digitising, and then reduced to standard
coordinates by data reduction program.
99
Characterisation of a Packed Bed
Based on the above cited reports of experimental measurements of bed
porosity distribution in packed beds it has been shown that the local
voidage has non-constant or oscillation pattern over the bed cross
sections, especially in the region close to the walls [Roblee et al, 1958;
Benenati and Brosilow, 1962; Scott, 1962; Thadani and Peebles, 1966;
Kondelik et ai, 1968; Scott and Kovacs, 1973; Goodling et al, 1983;
Stephenson and Stewart, 1986]. The oscillation pattern is dependent on
the shape and size distribution of particles [Roblee et al, 1958; Scott,
1962; Scott and Kovacs, 1973; Goodling et ai, 1983] but almost
independent of the shape of the container wall [Benenati and Brosilow,
1962]. For spherical particles, the local bed porosity with ratio of the
particle diameter to container diameter greater than 6.0 is independent of
angular positions [Mueller, 1993].
In the case of uniform size spherical particles, it would be expected the
measured porosity would have a limiting value of unity at the wall,
reaching a minimum at one particle radius from the wall, and a maximum
at one particle diameter. The porosity continues cycling until four to five
particle diameters from the wall before the constant value is reached
[Roblee etal., 1958; Benenati and Brosilow, 1962; Scott, 1962; Thadani
and Peebles, 1966; Goodling et ai, 1983] as shown in Figure 3-4.
100
a
o
d
c
•2
a
tj
2
a
CO
q
oo
a.
a
-«->
_.
re
<N
_.
ON
£
a
vo
•xO
os
,—1
o
_3
c_
s/s
u. n
I-I
J-t
00
M-J
o
3
tr.
cn
q
Q
SO __T,
r-H
a)
"c.
a
T3
a.
c
C
c.
M—<
a
C-
d)
a.
OS
a,
CQ
o
_.
a
-C
<U 00
CQ
.3
OJ
a.
<u
_^
re
75
•t
E
r-l
T3
O
.r, a
E-
CN
Os
a
o
<u
3
0)
o
c
CQ
q to
*5
Q.
CO
i_
o
c
o
H—
3
-Q
'v.
(fl
T3
>
CO
o
-o
a
ro
TJ
ro
OC
•sf
o
cs
•
CO
Q
J
53
a>
in
ires
Characterisation of a Packed Bed
The radial variations of the voidage are due to the confining effect of the
wall of the bed. In a randomly packed bed, the layer of spheres nearest to
the wall tends to be highly ordered, in which most of the spheres make a
point contact with the wall of the container with the result of the unity
value of voidage [Goodling et ai, 1983]. The next layer builds up on the
surface of the first, in a less ordered fashion. The subsequent layers are
less and less ordered, until a fully randomised arrangement is attained in
regions far removed from the wall. In condition the particles are
surrounded by a container wall of small ratio of D/Dp, the opposite wall
also affects the particle arrangement. It explains why the measured data
of oscillation pattern [Benenati and Brosilow, 1962; Goodling et ai, 1983]
for the wall distance greater than one particle diameter is dependent on
the ratio of D/Dp.
The similar results also were reported for uniform size of cylindrical
particles [Roblee et al, 1958; Kondelik et ai, 1966; Stephenson and
Stewart, 1986]. The bed porosity has a value of unity at the wall, and then
reaching a minimum at 0.5 - 0.7 particle diameter from the wall [Kondelik
et ai, 1966], and a maximum at about one particle diameter. The porosity
continues cycling until four to five particle diameters from the wall before
the constant value is reached [Roblee et ai, 1958] as shown in Figure 3-5.
102
Characterisation of a Packed Bed
1.25
Stephenson and Stewart, 1986
Kondelik et al., 1966
Roblee etal., 1958
1.00 -
0.25 -
0.00
+
0.0
1.0
J
I
I
I
2.0
I
I
I
I
3.0
L.
' ' I
4.0
5.0
6.0
Distance from wall in particle diameters
Figure 3-5: Radial variation of bed porosity for cylindrical particles.
103
Characterisation of a Packed Bed
For highly irregular shapes such as Berl saddles and Raschig rings,
results indicate that the bed porosity decreases regularly from unity at the
wall to the constant porosity at about one particle radius from the
container wall [Roblee et ai, 1966], as shown in Figure 3-6. The constant
bed porosity almost over all of the cross section of the bed could be
expected because of the irregularity in the shape of materials, which does
not allow any appreciable orientation which might result in a definite
pattern.
For multi-sized spherical particles in a cylindrical column, the
measurements indicate that the bed porosity oscillation over the cross
section of the bed is function of the number of sphere sizes that were
mixed as shown in Figure 3-7. Based on Scott and Kovacs [1973] and
Goodling et al. [1983] data, obtained on equal number and equal volume
of multi-sized spherical particles, it may be concluded that for mixtures of
two sizes, regular oscillations are detected only up to 2 or 3 diameters
from the wall and for three sizes the effect of the wall is observed only
within a distance of one particle diameter. The behaviour of multi-sized
spherical particles bed approached the behaviour of the highly irregular
shape particle together with the increasing of the number of particle sizes.
104
Characterisation of a Packed Bed
2.0
Raschig rings
1.5
Berl saddles
1.0
CO
0.5-•
-I
0.0
0.0
I
I
I
I
I
I
-H2.0
I
I
L.
+
4.0
_i
•
•
i
i
i
i
i
6.0
i
i
i_
8.0
Distance from wall in particle diameters
Figure 3-6: Radial variation of bed porosity for Raschig rings and
Berl saddles in cylindrical columns [Roblee etal., 1958].
105
Characterisation of a Packed Bed
1.0
Binary-mixture
0.8
0.6
Oo
0.4
o ^ A / *_
0.2 4
0.0
0.0
1.0
2.0
3.0
4.0
5.0
0.0
1.0
2.0
(R-r)/Dp
Dpi
=
Dp2
Dvs
D
=
=
=
6.35
7.94
7.06
52.6
3.0
4.0
5.0
(R-r)/Dp
-
Dpi
Dp2
Dvs
D
mm
mm
mm
mm
1.0
4.76
7.94
5.95
52.6
=
=
=
mm
mm
mm
mm
1.0
Ternary-mixture
Quaternary-mixture
0.8
0.8
0.6
0.6Qo
0.4
0.4
0
°>Do r°r
Oo
0.2
0.2
0.0 4
0.0
0.0
1.0
2.0
3.0
4.0
5.0
0.0
1.0
Dp2
Dp3
Dvs
D
=
=
=
=
=
4.76
6.35
7.94
6.08
52.6
mm
mm
mm
mm
mm
Ci-p <
o
2.0
(R-r)/Dp
Dpi
°
9>
0
3.0
4.0
5.0
(R-r)/Dp
Dpi
Dp2
Dp3
Dp4
Dvs
D
=
=
=
=
=
=
3.18
4.36
6.35
7.94
4.86
52.6
mm
mm
mm
mm
mm
mm
Figure 3-7: Radial variation of bed porosity for multi-sized spherical
particles in a cylindrical column [Goodling etal., 1983].
106
Characterisation of a Packed Bed
From the foregoing brief summary of the experimental work on radial bed
porosity distribution it can be concluded that the behaviour of the bed
porosity distribution near the wall is deterministic. This behaviour is due to
the wall effect, in that the presence of the wall allows to order the
arrangement of packing particles into a certain condition [Blum and
Wilhelm, 1965; German, 1989; Mueller, 1992]. The first layer of particles
in contact with the wall tends to be well ordered with most of the particles
touching it. Subsequent layers are less and less ordered as one moves
away from the container wall. Particles in layers far removed from the wall
display a randomised configuration or called as stochastic system. Thus,
any mathematical description of bed porosity distribution over the cross
section of the bed must be based on a combination of deterministic
approach for the vicinity of wall area, and stochastic approach, for the
centre of the bed area, to achieve a satisfactory explanation of this
system.
In order to develop a method of correlating the local porosity behaviour,
especially in the vicinity of the wall area, an extensive study has been
carried out by many investigators [Haughey and Beveridge, 1966;
Kondelik etal., 1968; Ridgway and Tarbuck, 1968a; Pillai, 1977; Gotoh, et
ai, 1978; Martin, 1978; Cohen and Metzner, 1981; Vortmeyer and
Schuster, 1983; Govindarao et ai, 1986; 1988; 1990; Kubie, 1988;
Johnson and Kapner, 1990; Kufner and Hofmann, 1990; Mueller, 1991;
107
Characterisation of a Packed Bed
1992; 1993]. There are two main categories in mathematical correlations,
namely, phenomenological approach and theoretical approach.
Numerous empirical correlations have been proposed to predict the local
bed porosity of the packed bed systems. Some representative models are
summarised below. Each of these equations contains empirical constants,
which were fitted based upon a particular set of experimental data, and
therefore, the validity of these kinds of correlation is restricted to the
original source of experimental data.
The simplest empirical correlation of radial bed porosity distribution is an
exponential function that was proposed by Vortmeyer and Schuster
[1983]. They employed the exponential function to fit their measured data
of a packed bed consisting of glass spheres with small deviation from the
spherical structure, which in this case only shows one oscillation in order
to approach the average porosity [Vortmeyer and Schuster, 1983]. The
porosity distribution was expressed by the following exponential function:
— = 1 + c'Exp' i - _ * - ' x
.
(2-52)
DP
where the constant c' can be determined as
c'=
l
£
'°
2.71828e',o
(3-24)
108
Characterisation of a Packed Bed
In order to accommodate the oscillation porosity profile at near the wall
region of a packed bed, Martin [1978] employed the combination of
exponential function and cosinus function to fit the Benenati and Brosilow
[1962] measured data of spherical particles bed. Martin [1978] divided the
cross section of the bed into two zones and introduced a new wall
distance term, Z', that was defined by the following equation:
Z'=2
R-r
(3-25)
-1
Dt
and for - 1 < Z < 0 :
e = e . +(l-e . )Z'2
min
V
(3-26)
min /
for 0 < Z :
e = e'o+(emin -e'„)Exp (
Z'^Cos
4
r J
f
\
7.
.
k
(3-27)
J
where the value of ock is equal to J— for the ratio of D/D p equal to °o
[Martin, 1978].
A similar treatment for fitting the mathematical correlation of porosity
distribution has been proposed by Cohen and Metzner [1981]. They
developed a correlation of experimental data for spherical particles that
109
Characterisation of a Packed Bed
were measured by Roblee et ai [1958] by dividing the cross section of the
bed into three regions, and the correlation for each region is as follows:
R-r
For — — < 0.25:
DD
1-e
R-r
= 4.5
1-e'
D,
7 R-r
2^
(3-28)
v DP ,
R-r
For 0.25 < — — <8.0:
DD
6-6,
For 8.0 <
A
f
R-r A (
R-r
= 0.3463Exp -0.4273•-2.2011 n
Cos 2.4509D,
Dp /
)
(3-29)
R-r
DD
(3-30)
6 = 6'
A single equation, by combining the exponential function and cosinus for
fitting the experimental data of porosity distribution was proposed by
Johnson and Kapner [1990] for spherical particles and Kufner and
Hofmann [1990] for cylindrical particles. The measurement data of
Benenati and Brosilow [1962] were fitted by the following equation by
Johnson and Kapner [1990]:
/
R-r
e = 0.38 + 0.62Exp -1.70
DD
V
-\U}\
0.434 N
Cos 6.67
R-r
Dp
(3-31)
_
no
Characterisation of a Packed Bed
For cylindrical particles, Kufner and Hofmann [1990] extended the
exponential correlation of Vortmeyer and Schuster [1983] by introducing a
cosinus function term to fit their measured data, as follows:
A
f
R - r ^ ( R-01
'
1--V,
1+
Exp 1v2.71828e'oy
Cos 27,
V
DPe ,
I
DPe )
As mentioned earlier, the D/Dp ratio has significant influence on the
oscillation patterns of local voidage, especially for the wall distance
greater than one particle diameter [Benenati and Brosilow, 1962;
Goodling et ai, 1983]. Therefore, the applicability of empirical correlations
become strictly restricted to systems that have the same value of the
aspect ratio as the original data for fitting the correlation [Mueller, 1991;
1992; Govindarao et ai, 1986; 1988; 1990]. Mueller [1991; 1992]
overcame this problem by incorporating his correlation with an aspect
ratio parameter. Based upon the experimental data from Roblee et al.
[1958], Benenati and Brosilow [1962], Goodling et ai [1983] and Mueller
[1992], he has described the following correlation for predicting the radial
variation of voidage:
e = eb+(l-eb)J0(flr)Exp(-^r), for D/Dp> 2.02 (3-33)
Where
3.15
a = 7.45 - — — ,
for 2.02 < D/D P < 13.0
(3-34)
111
Characterisation of a Packed Bed
a = 7.45 - — — , for D/D P > 13.0
(3-35)
jU/L) p
0.725
Y=
/D
'
for0
^r/DP
(3-37)
0.220
8b=0365+
D/D7
(3 38)
"
Although the accuracy of the empirical correlation is good for a particular
set of data, it is difficult to apply this type of correlation with any
confidence to other conditions. Thus, using this type of correlation is not
practical because of the requirement of experimental data on same
condition. It was that reason which motivated numerous investigators to
study the question theoretically, in order to develop a mathematical
correlation of radial distribution of voidage based on the knowledge of
statistics and geometry.
A critical study of local bed porosity variation of spheres was carried out
by Haughey and Beveridge [1966], who developed the semi-theoretical
correlation of voidage distribution in a packed bed based on the
distribution of number of points of contact made by a reference sphere
with adjacent spheres (the coordination number). The coordination
number is determined by sphere center position and particle arrangement.
112
Characterisation of a Packed Bed
The coordination number is not an intensive property of a packed bed, but
is determined by method of packing, geometry of packing, and geometry
of the container [Haughey and Beveridge, 1966; German, 1989].
According to the local bed porosity prediction, Haughey and Beveridge
[1966] proposed a mathematical method only for spherical particles
bounded by a spherical wall container. Considering that the most
common type of packed bed system is bounded by cylindrical or
rectangular column [Perry and Green, 1984], this model is not useful
because its applicability is restricted to spherical wall container.
In a more recent study, Ridgway and Tarbuck [1968a] developed a
mathematical correlation of voidage variation over a cross section of
randomly packed beds of spheres in a cylindrical column. They used an
analytical derivation of bed porosity variation over the cross section of a
regular close-packed hexagonal array of spheres for general voidage
profile correlation of spherical particles. This generalisation is carried out
by introducing two empirical randomising parameters into the equation
that was derived from regular close-packed hexagonal packing.
In an array of close-packed spheres aligned on a flat wall, the bed
porosity profile is an oscillatory function as shown in Figure 3-8. If the bed
is cut by planes parallel to the wall, it can be divided into two types of
113
Characterisation of a Packed Bed
region, namely, A, B, C, etc. are occupied by one layer of spheres only,
whereas a, b, c, etc. are occupied by two interpenetrating layers.
By employing the analytical geometry technique, the local bed porosity for
any distance x of a regular close-packed hexagonal arrangement of
spheres may be expressed by the following general equation [Ridgway
and Tarbuck, 1968a]:
(
^
V
K X-..3PDP Dp-x + J-pDp
.3 J
=1-X-
^
'
-
(3-39)
2
4
-P
Where p represents a particular layer, with p = 0 for the first layer, p = 1
for the second layer and so on. Accordingly, as the value of porosity is
between zero and unity, the bracketed terms must be positive or zero; if
one is negative, it is taken to be zero. In addition, similar results were also
proposed by Pillai [1977], Gotoh et ai [1978] and Kubie [1988] to
determine the wall effect on the bulk density variation.
In order to extend the applicability of equation (3-38) for a random packed
bed, Ridgway and Tarbuck [1968a] introduced two correction factors that
were called as randomising factors. The first randomising factor, F^ is to
allow for the voidage increase within a layer over that for a close-packed
array. The second randomising factor, F2 is to allow for the closer
approach of a given layer to the wall compared with a close-packed array.
114
Characterisation of a Packed Bed
0.00
1.00
2.00
3.00
4.00
5.00
Distance from wall in sphere diameters
Figure 3-8: Bed porosity variation of an array of close-packed unisized spheres aligned on a flat wall [Ridgway and
Tarbuck, 1968a].
115
Characterisation of a Packed Bed
By introducing the randomising factors into equation (3-39), the Ridgway
and Tarbuck [1968a] relation for the random packed bed becomes:
Fill
p
l
(2
X_
'
PDp
V3
D p - x + ^-pDpF2
1-1 —
(3-40)
^
Ridgway and Tarbuck [1968a] used the measurement data of Benenati
and Brosilow [1962] to fit the following correlation of randomising factors.
9 F\
F = -^(0.62 + 0.18e-0J6p) (3-41)
F2=0.991p (3-42)
The validation of the correlation, performed by making comparison of the
predicted results with the measurement data of Benenati and Brosilow
[1962] was reported by Ridgway and Tarbuck [1968a]. The root mean
square deviation between experimental results and prediction was 0.02
and the standard deviation of the experimental results was 0.01. It may be
thus concluded that the performance of the correlation is quite good;
however, it requires two empirical factors to be known from measured
data of porosity distribution.
A multi-channel model for estimating the local voidage profile in a
randomly packed bed of uniformly sized spheres in cylindrical column has
11.5
Characterisation of a Packed Bed
been proposed by Govindarao and Froment [1986]. The bed w a s divided
into a number of concentric cylindrical layers q of equal thickness. They
provided procedures for predicting the voidage variation up to distances
of five particle diameters from the wall. However, this model still requires
an empirical constant, but since they incorporated the aspect ratio
(a = D/Dp)into the model, therefore the applicability can be extended.
In order to simplify the mathematical manipulations, Govindarao and
Froment [1986] chose the thickness of cylindrical concentric layers, such
that m = DP/2Ar, as a suitable integer, and for realistic aspect ratios, the
effect of the curvature of a cylindrical concentric layer was neglected.
Consider the ith cylindrical concentric layer, the volume of spheres whose
center is in jth cylindrical concentric layer v^ may be calculated by
[Govindarao and Froment, 1986]:
. iAr
v , = — Jv(r)dr
A I
(3-43)
(i-l)Ar
If Nj is the number of spheres whose center is in jth cylindrical concentric
layer, the total volume of solids in the ith cylindrical concentric layer is
given by [Govindarao and Froment, 1986]
V1=^NJviJ (3-44)
JI
117
Characterisation of a Packed Bed
where ji = 1 + m and j2 = i +rafor i < 2m,'^= 1 - m and j2 = i +rafor i > 2 771.
The voidage in the ith cylindrical concentric layer is then [Govindarao and
Froment 1986]
(3-45)
6i = 1 -
TcAr'Lgj
where
g- = 2am-2i + l
(3-46)
By solving equation (3-43) for different spherical slices and caps and then
substituting the results into equation (3-45) and defining x\, as the numbe
fraction of spheres with centers in the jth cylindrical concentric layer, gi
[Govindarao and Froment 1986]:
_h_
e ; =_.-•
Si
(
1 "\
i+m-I
n : j m —— + 3Xnjb,
4; j=m+l
V
i < 2m
(3-47)
and
\
6; = 1 —
i+m-l
( n i - ™ + n i + J m - - +3Xi-.b
Si
i > 2m
(3-48)
j=m+l
where
b a =/n 2 -i 2 +j(2i-j)-, N T Ar
h= —!—
3L
(3-49)
(3-50)
118
Characterisation of a Packed Bed
The value of the number fraction {ni = N j / N T ) is an empirical parameter
[Govindarao and Froment, 1986]. Based upon the experimental data from
several investigators Govindarao and Froment [1986] have shown that
n, =n2...= nm = n„,+2 = nm+3 = ..= n3n(_1 =0 (3-51)
and Govindarao and Ramrao [1988] have described the following
correlations for predicting the two number fractions in terms of the aspect
ratio:
3 08
"
(3-52)
"m+l
a
2.60
niffl —
(3-53)
a
Based upon the comparison of the local bed porosity, predicted by using
the procedure proposed by Govindarao and Froment [1986], to
measurement data has shown that the applicability of the procedure is
restricted up to distances of about two particle diameters from the wall
[Govindarao and Froment, 1986]. Additionally, this procedure also gives
an uncertain result for infinite value of the aspect ratio.
3.3 PERMEABILITY
Standish [1979] defined the bed permeability as the ability of a given
packing to allow a fluid to flow through it under given conditions. The
permeability is a function of four variables, namely particle size and
shape, bed voidage and geometric factor [Standish, 1979]. For packed
119
Characterisation of a Packed Bed
beds of multi-sized particles, the permeability is also affected by the
spread and size range of a particle size distribution [Yu and Zulli, 1994].
The permeability of the bed K is defined by Darcy's law [Bird et ai, 1960;
Fayed and Otten, 1984; Wilkinson, 1985; Kececioglu and Jiang, 1994]
uM=--(Vp + pg) (3-54)
M-
A large number of efforts has been expended on determining K for various
packed bed systems [Bo et ai, 1965; Standish and Leyshon, 1981;
Standish and Collins, 1983; Leitzelement et ai, 1985; MacDonald et ai,
1991; Yu and Zulli, 1994; Kececioglu and Jiang, 1994]. The following
semi-empirical expression has been found to accurately represent many
experimental data [Bird et ai, 1960; Kececioglu and Jiang, 1994; Poirier
and Geiger, 1994]. It is
where av is the surface area per unit volume of particles, and K' is an
experimentally determined constant and it has been found to equal 4.17
[Ergun, 1952].
The quantity av for spherical particles is defined by the following equation
[Bird etal., 1960; Poirier and Geiger, 1994]:
Characterisation of a Packed Bed
a
v
6
=D—P
(3-56)
Substituting equation (3-56) into equation (3-55) and then inserting of the
value of the constant K" equal to 4.17 into the result, then gives
K=
2.3
D/E
I_x7-_F
(3 57)
"
The constant K" equal to 4.17 is not universally selected, s o m e believe the
value to be as high as 5.0 [Kececioglu and Jiang, 1994; Poirier and
Geiger, 1994].
The applicability of equations (3-56) and (3-57) is restricted to spherical
particles. As stated earlier, for non-spherical particles, equations (3-56)
and (3-57) may be used by introducing the sphericity concepts [Poirier
and Geiger, 1994].
According to Forchheimer's generalisation for pressure drop correlation of
fluid flowing in a packed bed [Kececioglu and Jiang, 1994], the energy
losses of fluid flow is due to inertia energy losses and kinetic energy
losses as shown in Ergun equation [Ergun, 1952]. The kinetic energy
losses and the inertia energy losses are expressed by bed permeability
and inertia parameter, respectively [Fayed and Otten, 1984]. According to
121
Characterisation of a Packed Bed
Ergun equation [Ergun, 1952], the inertia parameter appears to be
independent of the fluid properties and may be written as follows [Fayed
and Otten, 1984]:
Dp£3
I=
mTi)
0-58)
Comparing the permeabilities given by equations (3-57) and (3-58) with
equation (2-37), the Ergun equation, it is obvious that these equations
give the characteristics of the bed for viscous and inertial flow in the
Ergun equation, while the other terms, namely, p, \x and uM characterise
the flowing fluid.
An important assumption in the above definitions of permeability is that
the porosity is uniformly distributed throughout the bed. This condition is
equivalent to the geometric factor of the bed being equal to unity
[Standish, 1979]. However, as also pointed out by Standish [1979], this
condition is rather unusual and rarely met with in practice, where the
geometric factor is hardly ever unity due to the use of packings having a
size distribution and/or non spherical shape, causing particle segregation
of one kind or another. This definitely complicates prediction of
permeability distribution, at least by any model that seeks to achieve this
without any prescribed information.
[22
Characterisation of a Packed Bed
Recently, it w a s demonstrated [Yu and Zulli, 1994] that a good prediction
of radial permeability distribution in a blast furnace is possible if a
measured radial particle size distribution is given. Considering the
complexity of the system involved, namely coke and sinter of different size
distributions and absolute sizes (sinter: 5-20 mm and coke: 25-70 mm),
the reported good result may be regarded as an important achievement
that will undoubtedly be improved and extended in time to beds with
different geometric factors.
It is of interest to observe that in this regard, Yu and Zulli [1994] noted the
need for understanding the microstructure of packing of particles, giving
as an example a binary size system in a mixed state and in a segregated
state. The large difference in permeability in this example as given in the
paper [Yu and Zulli, 1994], is a direct result of a different geometric
arrangement of the same packings in the bed, ie. a different geometric
factor, viz \|/=1.0 f°r tne uniformly mixed bed and \j/<1.0 for the
segregated bed used.
It is also of interest to observe that Yu and Zulli [1994] were motivated in
their research "Because the radial permeability distribution is directly
related to the radial gas distribution in a blast furnace. It provides a more
quantitative and useful information for the process control then the radial
particle size distribution". Noting the above stated radial permeability-gas
123
Characterisation of a Packed Bed
distribution connection, it is suggested that a possibility of employing a
mathematical model of velocity distribution of a fluid in packed beds, as,
for example, the model proposed in the present work, be investigated
further.
124
CHAPTER FOUR
DEVELOPMENT OF A MATHEMATICAL
MODEL FOR VELOCITY PROFILE OF FLUID
FLOWING IN PACKED BEDS
Generally, there have been two main theoretical approaches for studying
flow conditions in packed bed systems. In the first approach the packed
bed is regarded as a bundle of tangled tubes; the theory is then
developed by applying the previous results for single straight tubes to the
collection of crooked tubes. In the second approach the packed beds is
visualised as a collection of submerged objects, and the point of view is
extended from conditions of submerged particles. For macroscopic level,
the first approach has been successful for bed porosities less than 0.5
[Bird etal., 1960].
In order to develop a general mathematical equation which may be used
to evaluate the velocity distribution of single phase fluid flow in the packed
bed, the fluid flow phenomena is treated by using the above first
approach. Considering the limitation of this approach that the applicability
is restricted to the bed voidage less than 0.5 [Bird et ai, 1960; Cohen and
Metzner, 1981; Foscolo etal., 1983] and the variation of the voidage over
the cross section of the bed as stated in the previous chapter, it has
125
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
seemed desirable to introduce s o m e modification to improve the
performance of the model. The improvement can be carried out by
considering the fluid flow phenomena in a packed bed as a continuous
system for the local voidage greater than 0.5, and as a discontinuous
system for the local voidage less than 0.5.
4.1 THE EQUATION OF FLOW THROUGH A SINGLE PIPE
The energy relationships of a fluid flowing through a pipe may be obtained
by an energy balance. Energy is carried with the flowing fluid and also is
transferred from the fluid to the surrounding, or vice versa [Brown et ai,
1950]. The energy carried with the fluid includes the internal energy, U,
and the energy carried by the fluid because of its condition of flow or
position, namely potential energy, kinetic energy and pressure energy.
The energy transferred between a fluid or system in flow and its
surrounding is the heat, q, [Brown etal., 1950; Foust et ai, 1960].
An energy balance around a flow system, such as between points 1 and
2 in Figure 4-1 and the surroundings, assuming steady state condition
(no accumulation of material or energy) at any point in the system is given
by equation (4-1) [Bird etal., 1960]:
Energy
Accumulation
Rate of
Rate of
Rate of
> = <
Energy
Input
>• — <
Energy
(4-1)
Output
126
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
Point 2
Fluid Outlet
-i
— z2
Zi
Point 2-
Fluid Inlet
Figure 4-1: Flow diagram of the fluid flow in a pipe.
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
Assuming the direction of fluid flow is only in the z-direction, then equation
(4-1) gives:
AU + -muz2+A(mgz)+A(pV)=q (4-2)
The increase in internal energy is the s u m of the increases due to all
changes considered as taking place in the material in flow, including heat
effects, compression effects, surface effects, and chemical effects [Brown
etal., 1950].
AU = j*TdS + |2p(-dV)+|2odt+|2KAdmA + f KBdmB +etc. (4-3)
The pressure energy term, A(pV) is complete differential:
A(pV) = Ji2pdV + jVdp (4-4)
Combining equations (4-2), (4-3) and (4-4), including surface and
chemical effects in the etc. term, gives:
J] TdS + A -muz2 +A(mgz) + J Vdp + etc.= q (4-5)
The increase of the internal energy due to heat effects, TdS, as cited by
•M
Brown et al. [1950], is equal to the sum of the heat absorbed from the
surroundings and all other energy dissipated into heat effects within the
128
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
system due to irreversibility, such as overcoming friction occurring in the
process,
Ji2TdS = q + (lw) (4-6)
where the lost work (Iw) is energy that could have done work but was
dissipated in irreversibility within the flowing material.
Combining equations (4-5) and (4-6) gives,
r2 fl }
J Vdp + A - m u z 2 +A(mgz) + etc.= -(lw)
(4-7)
Equation (4-7) is a general equation of fluid flow in a pipe and is
unrestricted in application to material flowing or transferred from state 1 to
state 2, except for unsteady state condition and the presence of shaft
work.
Assuming the fluid flowing through a pipe is free of chemical change,
surface effects, etc., equation (4-7) may be written, for a unit mass of
material as:
f - d p + ^ - A u z 2 + A Z = -F h
g
(4-8)
2g
Where:
F
h
= ^
(4-9)
mg
129
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
The energy per unit mass lost as frictional conversion into heat, Fh, m a y
be calculated by the following equation [Brown et ai, 1950; Bird et ai,
1960; Foustefa/., 1960].
Fh=^^ (4-10)
2gd p
Substituting equation (3-10) into equation (4-9) gives:
r2 V Au
2
fXu
2
—dp + —^- + Az+
Jl
g
2g
k
/ =0
4-11
2gd p
Equation (4-11) is applicable for straight pipes with constant diameter and
with no heat or work transferred. Brown et a/.[1950] introduced the
concept of equivalent length, Le, for describing fluid flow through nonregular piping sections such as bends or sections with changing cross
sectional area. The value of Le is the length of pipe itself plus an
equivalent length allowance for non-regular piping sections, the friction
energy losses equal to straight line pipe of length Le.
4.2 THE EQUATION OF FLOW THROUGH A PACKED BED
Considering fluid flowing through a packed bed with voidage less than
0.5, it can be assumed that the flow phenomena is similar to fluid flow
inside a bundle of tangled tubes [Bird et ai, 1960; Cohen and Metzner,
1981] with radius re. By using a definition of hydraulic radius, rH, [Bird et
130
Development of a Mathematical Model for Velocity Profde of Fluid Flowing in Packed Beds
ai, 1960; Foust, etal., 1960; Poirier and Geiger, 1994] then the quant
may be expressed in terms of the hydraulic radius, as follows:
re = 2rH
_d^
2
(4-12)
The hydraulic radius is defined by Bird etal. [1960] as follows:
cross section available for flow
H
wetted perimeter
(4-13)
volume available for flow
total wetted surface
By neglecting the wall effect, the hydraulic radius for a bed composed of
spherical particles can be shown [Bird et ai, 1960] to be:
EDD
r_
H =
"6(l-e)
(4-14)
Mehta and Hawley [1969] have attempted to account for the wall effect by
modifying the expression for the hydraulic radius. Their modification takes
the form:
ED,
rHu = ... \..
6(1-e)M
(4-15)
where
4D
M
= 1
+
f l &
(4
'16)
131
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
Although originally the approach of Mehta and Hawley [1969] w a s
proposed to apply over the entire cross-section of the column, Cohen and
Metzner [1981] suggested to apply this approach only at the wall region,
because the effect of the wall should be confined to the wall region
without affecting the nature of the hydraulic radius in the bulk of porous
bed.
W h e n the bed contains non-spherical particles and the material is
screened, and then the particle diameter, DPs, may be taken as the
arithmetic mean of the openings of the two screens. By introducing the
sphericity concept into equation (4-14), and then the hydraulic radius may
be calculated by the following equation [Poirier and Geiger, 1994]:
r =^^ (4-17)
H
6(1-e)
{
'
Considering the fluid flow in packed beds, the value of equivalent length,
Le, can be defined by the following equation:
Le=r;Az (4-18)
Interstitial velocities can be calculated using the following equation
[Cohen and Metzner, 1981] which relates the interstitial velocity with bed
voidage:
132
Development of a Mathematical Model for Velocity Profde of Fluid Flowing in Packed Beds
uM
uz=-fL
(4-19)
By assuming the value of the correction factor, l%, is constant over the
cross section of the bed with local voidage less than 0.5 and replacing L
in equation (4-11) with Le, then substituting equation (4-12) into equation
(4-11) gives:
nV
Au7
'1 + f_W^= 0
J —gd p + —_-^+
Az
2g
(
^ 8grH /
(4-20)
The friction factor fk in equation (4-20) is calculated by using the Blake's
correlation [Ergun, 1952], as follows:
4 = 1.75+150^ (4-21)
Where
NRe="^L^ (4-22)
H
Because the applicability of equations (4-20) to (4-22) is restricted to bed
voidage less than 0.5, and considering the voidage higher than 0.5
usually occurs only at the wall region of the random packed beds,
equation (4-14) or equation (4-17) is taken into account for calculating the
hydraulic radius.
133
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
As discussed earlier, the local bed porosity has a limiting value of one at
the wall for the particles which have possibility to make point contact with
the wall of the container and then continuously decrease to reach a
minimum value at a distance about a half particle diameter from the wall.
In order to overcome the limitation of equations (4-20) to (4-22), which is
restricted to the bed voidage less than 0.5, then for the wall region the
mathematical equation for the velocity profile calculation may be
formulated by assuming the flow characteristics as a continuous system.
Consider the fluid flowing through a packed bed at the wall region, no-slip
condition is assumed at the wall or the velocity is equal to zero at r = R
and .is a radial position in the bed which has voidage equal to 0.5.
Employing the momentum balance [Bird et ai, 1960; Poirier and Geiger,
1994] in the wall region with the local voidage higher than 0.5, assuming
the flow of fluid only driven by the difference of momentum and that the
fluid is Newtonian, the following equation can be generated:
3u,
3r
re
z
3r
=0
(4-23)
The boundary conditions are;
1. at r = £, 6 = 0,5 uz = u, (4-24)
2. at r = R, e = 1.0 uz = 0 (4-25)
134
Development of a Mathematical Model for Velocity Profde of Fluid Flowing in Packed Beds
4.2.1 The Equation of Continuity
For a volume element (Figure 4-2), the material balance of the fluid flow is
as follows [Bird etal., 1960]:
Rate
of
Input
Rate
r-<
Rate
of
r =
<
of
Output
(4-26)
Accumulation
Which, with assumed steady state condition and no chemical reaction,
gives;
nre2(puz), )-[nre2{puz\
' <)
(4-27)
|z / \ U + AZ
and for Ar,Az -> 0, from equation (4-27) gives:
a(puz)
=0
3z
(4-28)
Or
dU,
az
d
P
z
rx
+ UM,-^ =
az
0
(4-29)
4.2.2 The Incompressible Fluid
a
Considering that for the incompressible fluids, the value of —P is equal to
zero then equation (4-29) may be rearranged to give:
9UM
3z
=0
(4-30)
135
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Bed.
FLUID O U T L E T
_.Ar „
Z=L
r
R
**—•
Z=Z+AZ
Z=Z
z=o
FLUID INLET
Figure 4-2: Diagram of the fluid flow in a packed bed.
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
For an incompressible fluid the value of V is constant and equal to P
then by substituting equation (4-30) into equation (4-20) gives:
1 9p f&u2
~^T+ _
+1 = 0
Pg dz 8grH
4-31)
Substituting equations (4-14), (4-19), (4-21) and (4-22) into equation (431) and then the result solved for positive value of uz is;
Uz
= -^+A-2-4^
(4_32)
2A,,
Where
(1-e) pt
...=1.3125- £ -f*Dp
x2
(1-E) 2
0.
— 1
1 11Z.0
r,
£
^
^
Dp 2
dp
=-^-+pg
dz
(4-33)
(4-34)
(4-35)
Considering the incompressible fluid, because the interstitial velocity is
independent of the axial position, equation (4-23) with the two boundary
conditions, equations (4-24) and (4-25), may be solved by using a
polynomial approximation [Burnett, 1987] and assuming that in the region
of concern porosity is a linear function of r, as follows:
(r-R)
8= 1
"2T7-R)' for£<r<R
(4-36)
137
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
For the problem at hand, the following form of a complete linear
polynomial is chosen:
uz = k0 + k.r + k2r2 (4-37)
Where the coefficients k0, h and k3 are constants to be determined which
all boundary condition must be satisfied exactly. By using the collocatio
method [Burnett, 1987] to optimise the best value of k0, k: and k3, a
quantity called the residual, SH, is required. The residual is derived by
inserting equation (4-37) into equation (4-23) and gives:
9.= ^2i~R'kx +[2(21 -R)k2-k, }-3k2r2
2
= 0
(4-38)
By using the boundary conditions, two of the constants can be
determined. Then the rest of the constants may be determined by forcing
the residual to be exactly zero at a point, n in the domain. The followin
approximate solution of the constants for equation (4-37) is obtained by
employing this procedure.
/.0=-Rf>.+_-2R) (4-39)
5.+R
*,= ,, ^ 2u ,
3(.-R)
(4-40)
*a = U ' - * ' ( ' T R )
(4-41)
2
2
. -R
2
138
Development of a Mathematical Model for Velocity Profde of Fluid Flowing in Packed Beds
The value of the correction factor, \%, may be evaluated by using the
equations derived from a material balance of the system and assuming
steady state condition, using the following equations:
7.R2puM
m=
P1-
(4-42)
4
and
R
m = 27ipJ r£uzdr (4-43)
o
By minimising the deviations between the macroscopic result (equation
(4-42)) and the microscopic result (equation (4-43)) of the mass rate, the
value of the correction factor, £, can be determined implicitly.
Equations (4-32) to (4-35), (4-37) and equations (4-39) to (4-43) are a
complete set of the equations to predict the velocity profile of singlephase incompressible fluid flow in packed beds. Use of these equations
requires only the knowledge of physical properties of the fluid, bed
characteristics and macroscopic data. Although these equations are
derived for uni-sized spherical particles, the applicability can be extended
for multi-sized spherical particles by employing the volume-surface mean
diameter for particle size or for non-spherical particles by employing the
sphericity concept.
139
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
4.2.3 T h e Compressible Fluid
Because the density or the specific volume of a compressible fluid is not a
r V
constant and is a function of the pressure, the term — d p in equation (4J
g
19) can not be directly solved. The solution of this term requires a
knowledge of the correlation between specific volume, V, and pressure.
This difficulty is complicated by the fact that the fluid velocity, uz, besides
being a function of bed radius, also depends upon the axial position in the
bed. These conditions lead to an increase in the complexity of the solution
of equations (4-20) and (4-23) to get the velocity profile of compressible
fluid flow in a packed bed.
On the other hand, based upon the measured data that were obtained at
both the inside of the bed [Stephenson and Stewart, 1986] and the outlet
of the bed [Price, 1968; Newell and Standish, 1973; Szekely and
Poveromo, 1975; Ziolkowska and Ziolkowski, 1993], for compressible and
incompressible fluid, it is shown that the velocity is more a function of the
bed character rather than that of the superficial velocity or the flow rate of
the fluid. This condition makes the assumption that a constant value of the
specific volume of compressible fluid over a small value of axial distance
of the bed becomes reasonable. This assumption is considered to simplify
the algebra considerably for the solution of the equations (4-20) and (423).
140
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
The bed with large value of L/D is divided into finite number, N, of short
beds and the assumption of small pressure drop would be valid for every
short bed. The approximation is expected to approach the exact condition
as N^oo. For conditions of a small value of the pressure drop or the
value of the length increment (Az->0), equations (4-32) to (4-35), (4-37)
and equations (4-39) to (4-43) can be employed to calculate the velocity
distribution of compressible fluid flow in packed beds.
4.2.4 Pressure Drop Correlations for Packed Beds
A very large number of relations have been proposed for estimating the
pressure losses of a fluid flowing through a packed bed as discussed by
Brown, et ai [1950]; Bird et al. [1960]; Foust et al. [1960]; Perry and
Green [1984]; Agarwal and O'Neill [1988]; Leva [1992]; Poirier and Geiger
[1994] and Kececioglu and Jiang [1994], but only a notable few will be
described here. The principal reasons for not discussing the others were
their poor validity, limited applicability and complexity of the correlations.
However, because of the pressure drop correlation being an additional
equation for predicting the velocity distribution, the other correlations can
also be applied with regard to the availability of data and the validity of
correlations, if required.
The most widely used mathematical correlation for single phase fluid flow
in packed beds is that advanced by Ergun [1952], who proposed the
I4l
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
prediction of pressure drop due to friction losses based on mechanical
energy balance by using Blake equation of friction factor (equation (421)). Although the theoretical explanation of Ergun correlation has been
made by Bird et ai [1960], actually this correlation is an extension of the
empirical correlation that was proposed by Forchheimer in 1901
[Kececioglu and Jiang, 1994]. Generalising Forchheimer's equation gives;
-^^afuM+bfuM2 (4-44)
dz
where a{ and bf are empirical constants.
In 1952, Ergun [1952] examined this general expression for gas flow
through crushed porous solids, based on its dependence upon the flow
rate, properties of fluid, porosity and character of the bed particles. He
obtained the following equation:
__-.15Q.k£^__f.M.751-E,'°PU"i (4-45)
e'0J D p 2
L
e'0!
Dp
When a bed contains a mixture of different-size particles and if the
material is screened to determine the diameter of each size faction,
Poirier and Geiger (1994) suggested to use the volume-surface mean
diameter, Dvs and then the sphericity, y, is introduced as a factor into
equation (4-45) to give:
142
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
1
2
^ = 150%^i^
+ 1.75 "^^ ,
2
L
e'0> Dv/
E V DVSV
(4-46)
As mentioned earlier, the applicability of the Ergun correlation is restricted
to the bed porosity of less than 0.5 [Bird et ai, 1960], and it is not
appropriate for systems containing particles of low sphericity [Gauvin and
Katta, 1973; MacDonald ef ai, 1979].
4.3 FLUID FLOW AT THE OUTLET OF THE BEDS
Although a considerable progress has been made in the development of
experimental techniques for investigating velocity distribution inside a
packed bed, they were limited to special situations. The optical technique
as employed by Stephenson and Stewart [1986] is restricted to
incompressible fluids. The use of a laser Doppler anemometry to measure
the velocity profile [McGreavy et ai, 1986] is only good for small values of
D/Dp ratio. The use of velocity sensing probes inside the bed could disturb
the packing arrangement and because the fluid velocity between the
packing particles is not uniform, as stated by Mickley et al. [1965], many
measurements of local velocity are necessary to give a true indication of
the mean axial velocity. The measurement of velocity profile by noting the
time taken for a step change in the electrical conductivity of the fluid to
travel between two fixed points in the bed [Cairns and Prausnitz, 1959] is
143
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
not satisfactory if radial mixing occurs in regions where velocity gradients
exist.
The measurement in the down stream of the bed is not directly
representative of the flow profile inside the bed because of the changing
of flow profiles. Based upon the result of previous investigators [Schwartz
and Smith, 1953; Newell and Standish, 1973; Cohen and Metzner, 1981;
Nield, 1983; Vortmeyer and Schuster, 1983; McGreavy et ai, 1986;
Tsotsas and Schluder, 1987, 1990; Stanek and Szekely, 1974; Szekely
and Poveromo, 1975; Ziolkowska and Ziolkowski, 1993], the difference in
the velocity profiles at the inside and the exit of the packed beds is clearly
shown. Use of the flow divider [Arthur et al. 1949; Price 1968; Newell
1971] is faced by problems in equalising the pressure losses through the
different passages. A schematic diagram of the fluid flow phenomena in
packed beds is shown in Figure 4-3. Clearly, the model of the velocity
distribution of the fluid flowing in packed beds needs to be corrected to
allow comparison with the data that were obtained at the outlet of beds.
A velocity profile at the outlet of the bed is a transition profile between the
inside bed profile and the fully developed flow of the fluid in an empty
pipe, which may be assumed as a developed flow profile in a pipe.
Numerous studies have been conducted for investigating the developed
flow profile [Langhaar, 1942; Foust et ai, 1960; Christiansen and
144
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
FULLY DEVELOPED FLOW
T T T TTTTTTTTT T T
EXIT OF BED FLOW
PACKED
BED
INSIDE BED FLOW
TTfTTTTTTTTTTTTTTT
FLUID INLET
Figure 4-3: Flow of a single-phase fluid in a packed bed.
145
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
Lemmon, 1965; Vrentas etal., 1966; Vrentas and Duda,1967; Atkinson et
ai, 1969; Chen, 1973; White, 1986].
In order to develop the velocity distribution model at the exit of the b
is assumed that there is no-slip at the wall, steady state conditions ap
the fluid is Newtonian with constant density and viscosity, no pressure
gradient for r-direction, and any angular motion is negligible (axisymme
flow). The following equations can be derived from momentum balance
and material balance [Bird et ai, 1960]:
f
duz
U
.'"37
f dur
u.1 dr
du z ^
+UZ
"CJ7
U,
1 d
9p
^3z
<___
+ pg + H
dr
Vf dr v
-„
2
d I a
a
u
(m +
=
n
dz
L3r7a7 ^ "a?
2
a2„
u
+• dz2
>
(4-47)
A
(4-48)
dUr
M+
rd~r Tz
=0
(4-49)
The boundary conditions are;
1. at z=0
u z = uzo(r) and
2. at z=oo
uz = uZL(r) and ur =0
(4-51)
3. at z>0
uz(R,z) = 0 and ur(0,z) = 0
(4-52)
ur = 0
(4-50)
146
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
The value of uzo(r) is the velocity distribution at the top of the bed, that
can be calculated from the velocity distribution inside the bed model
(Equations (4-32) to (4-35), (4-37) and equations (4-39) to (4-43)) times
the local bed voidage. The value of uZL(r) is the fully developed flow
distribution. The equations of motion for developed flow have been solved
numerically and analytically for many simplified cases by many
investigators, as discussed by Atkinson et ai [1969], Vrentas and Duda
[1967], Vrentas et ai [1966], Christiansen and Lemmon [1965], and
Langhaar [1942]. However, all of these solutions use the boundary
condition that the velocity at the entrance, uzo, is independent of the
radius, which is different for flow conditions at the exit of the bed. There is
a need to solve the equation of motion, equations (4-47) to (4-49) by
using different boundary conditions (equations (4-50) to (4-52)).
dimensionless
In order to simplify the problem at hand, the following
variables are introduced into equations (4-47) to (4-52);
u
z
(4-53)
<p = -^-
U
r
(4-54)
(4-55)
z
(4-56)
147
n= —
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed
v=U
ap
R
2
M
P
(4-57)
az + p g
Then, the partial differential equations and the boundary conditions
become:
dry
dv
ac
as
2
1a
f dfU
_*^
N R e ^ a c i C ac
dfU du
2 (a i a
K— + V—--.—ac C ac ^
ac
1
^ tr \
a2^
+ ds2
a2^
' as
(4-58)
(4-59)
as N R e
dP
r.
(4-60)
and the boundary conditions are;
1. at 5 = 0
V =%
and
«= 0
(4-61)
2. at 5 = oo
f = #
and
« = 0
(4-62)
3. at s > 0
A
tf(l,s) = 0 and «(0,s) = 0
(4-63)
complete procedure for numerical solution by employing the
dimensionless stream function and the vorticity vector of equations
(4-58) to (4-60) has been developed by Vrentas et ai [1966]. However,
the applicability of this procedure is restricted to a flat velocity profil
5 = 0, therefore it could not be applied to predict the velocity profile in
downstream of the packed bed.
148
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
After the fluid is far downstream from the outlet of the bed, one expects,
intuitively, that the flow profile will not undergo any further change of s
(fully developed flow profile condition). Hence, introducing the following
dimensionless parameter into equations (4-58) to (4-63) seems
reasonable for simplifying the problem:
<p = =- (4-64
%-Vw.
Because the parameter if is independent on C it is reasonable to assume
that the radial component of the equation of motion (equation (4-59)) is
negligible. Hence, the flow equation may be reduced to give the following
ordinary differential equation for if:
d
^-(A+E^)^-C = 0 (4-65)
ds
' ds
where
A =
B =
N If
ili*Lf__.
N^V*0
(4_66)
(4.67)
N 7>
C = , *" x
(4-68)
and the boundary conditions become;
1. at s = 0 if = \ (4-69)
2. at s = °o i/ = o (4-70)
149
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
By inspection, it is evident from equations (4-65) to (4-70) that the if is a
function of S and will be damped out exponentially with s. Equation (465) with boundary conditions of equations (4-69) and (4-70) is a boundary
value type equation [Burden and Faires, 1993] and may be solved by
using the approximate methods of the weighted residuals [Burnett, 1987],
According to the method of weighted residuals, the solution of equation
(4-65) is approximated by functions. These functions are chosen so that
all boundary conditions are satisfied exactly, although the solution of
equation (4-65) itself is only approximate [Chow, 1979]. For the problem
at hand, the solution of equation (4-65) is approximated by the following
exponential equation;
it = €*s (4-71)
which automatically satisfies all the boundary conditions (equations (4-69)
and (4-70)). The parameterd is a constant to be determined. By using the
collocation method [Burnett, 1987] to optimise the best value of d, which
is carried out by choosing a point s in the domain and then forcing the
residual to be zero. The residual, Si, is derived by inserting equation (471) into equation (4-65), and by choosing s = 1 to give:
ft = aV°+(de-(kXA + Ae-")-C ,^-jry,
=0
Solution of equation (4-72) for d, leads to a complete description of the
velocity field for the system under consideration. However, there arises a
150
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
mathematical difficulty in the derivation of the desired solution for
equation (4-72). This equation is a non-linear equation and hence a trialand-error technique is required to solve the problem. Besides, a time
consuming calculation is needed, and the convergence also cannot be
guaranteed [Burden and Faires,1993].
In order to overcome the above problem, the parameter d was
determined from experimental data of the entry length as given by
Atkinson ef ai [1969] and Bowlus and Brighton [1968]. The values of the
Reynolds number for the entry length data range from 1.0 to 3.88 x 105.
The correlation of d may be represented by the following equation:
-for NRe<2100
d = 2.41NRe-°'5 (4-73)
-for NRe>2100
d = 0.05 (4-74)
In order to perform a complete solution of the fluid velocity profile in the
entry region requires the knowledge of the fully developed velocity profile,
the pressure losses in an empty pipe and the entry length. Fortunately,
these had much attention from a theoretical and an experimental
viewpoint for exploring the physical phenomena; therefore, the availability
of the mathematical correlations is adequate. By using this equation, the
151
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
data that were measured at the exit of the bed may be used to evaluate
the model of the velocity profile inside the bed.
Consider a steady flow of a Newtonian fluid of constant density in a very
long tube of radius R, so the end effects are negligible. For the value of
the Reynolds number less than 2100, the equation for velocity profile is as
follows [Bird et ai, 1960]:
^=2(l-C2) (4-75)
For Reynolds number greater than 2100 the velocity profile may be
predicted by the following empirical correlation [Bird et ai, 1960]:
*.- 1.25(1 -,P (4-76)
The pressure gradient in the developing flow region is higher than in the
fully developed flow region due to the increasing friction and kinetic
energy losses [Langhaar, 1942; White, 1986]. Therefore, the pressure
drop in equation (4-68) may not be calculated by using the pressure drop
correlation that was formulated based upon measurement of fully
developed flow as proposed in Brown et al. [1950], Bird et al. [1960],
Foust etal. [1960], and Perry and Green [1984]. Considering the smooth
tube, the pressure drop in equation (4-68) can be calculated by the
following empirical correlation [Christiansen and Lemmon, 1965; White,
1986]:
152
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packe
2(-Ap) . L-M,
-pu..
— T ^ f k - MD^ + G-
(4-77)
[Schmidt and Zeldin, 1969]
(4-78)
'M
where:
- For laminar flow:
64
fk = —
,
IN'r Re
05 = 1.0 -1.41, [Christiansen and L e m m o n , 1965]
(4-79)
For turbulent flow:
0.3164
f
k=
0.25 .
[Bird et ai,1960]
(4-80)
•^Re
05 = 1.314, [Schmidt and Zeldin, 1969] (4-81)
The entry length, which is defined as the distance along the axis of the
flow where the centre-line velocity reaches 99% of its fully developed
value [Chen, 1973], depends on the inlet profile and on the Reynolds
number [Foust et ai, 1960; Berman and Santos, 1969; Brady, 1984],
Numerous studies have been conducted for correlating the entry length to
its variables [Langhaar, 1942; Foust et ai, 1960; Christiansen and
Lemmon, 1965; Vrentas et ai, 1966; Bowlus and Brighton, 1968; Atkinson
et ai, 1969; Chen, 1973]. Based upon the survey of their results, it can be
shown that the following equation is satisfactory to predict the entry length
for uniform inlet profile:
153
Development of a Mathematical Model for Velocity Profile of Fluid Flowing in Packed Beds
- For 226 > NRe [Atkinson et ai, 1969]:
^L = 1.18 + 0.112NRe
(4-82)
D
- For 104 < NRe < 107 [Bowlus and Brighton, 1968]
(4-83)
It is instructive to compare the calculation results of the present
correlation (equations (4-64), (4-71), (4-73) and (4-74)) with previous
models and experimental data. In order to facilitate comparisons, the
computed velocity profiles and the predicted entry length, together with
data from several previous solutions and experimental studies, are plotted
in Figures 4-4 and 4-5, respectively. Although the available experimental
data are inadequate for a very rigorous test of the present mathematical
model of velocity distribution in the entry region of an empty pipe, the
reasonable agreement of computed with experimental velocity profile and
entry length data indicates that a close approximation to reality for the
prescribed conditions has been achieved. Therefore, it may be concluded
that the present model of velocity profile under developing flow condition
can be used to fulfill the need of correction factor for study of flow profile
inside a packed bed which measurement is carried out at the downstream
of the bed.
154
Development
of a Mathematical
Model for Velocity Profile of Fluid Flowing in Packed
Beds
1.5 -
1.5 --
l . O . l Q Q D Q Q Q - I Q O i
if
.1
8 8 Q 6 A A A
I.0--
o „
o 8
|
if
f
s = o.oo
0.5
S = 0.49
0.5 -
0.0 I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' '
0.0 - 1 ' ' ' i ' ' ' ' i ' ' ' ' i ' ' ' ' i ' ' '
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
C
0.6
o.;
c
1.5
" 2 8 _ _ G £ A A
° o
1.0-
A
o
if
S =0.71
0.5 -
O Predicted
A Data
I ' ' ' ' I ' ' ' ' I ' ' ' 'ft
0.0
0
0.2
0.4
0.6
0.8
1
c
1.5 -•
8
g fi A
A
A
A
A
° O A
1.0
O
O
if
0.5
S = 0.92
1
0.0 --------
0
0.2
' ' l ' ' ' ' l ' ' ' ' I ' ' ' '6
0.4
0.6
0.8
1
c
Figure 4-4: Comparison of the developing flow profile, at NRe = 47,
calculated by equation (4-71) with Berman and Santos
[1969] measurements.
155
-J
_
_
(J
a
a.
00
o
XO xo
Ox Ox
r.
„
5)
-C
u
is
_
c
c
T3
tL
C
C3
Z
_J
=:
_
Ir
cd
c.
F
CC -J
Ox
xO
Ox
c
o
c
1
3
<f
c_
O
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
CO
c
g
"+-I
ss.
<D
i_
!_
o
u
_c
c
cCD
+-I
cn
cu
o
c
o
(fl
"SZ
CO
o
o
a
E
o
o
LO
I
a>
.. O
O
o
Q
!_.
3
D)
CHAPTER FIVE
EXPERIMENTAL TECHNIQUE
In order to provide the data for validating the model of velocity profile of
single-phase fluid flowing in a packed bed, experimental work was
conducted to measure the velocity distribution. The measurements were
carried out in a vertical cylindrical tube, randomly packed with glass
spherical particles. The experimental equipment was designed and built to
allow the best possible observation of fluid flow profile in the random
packed beds of the spherical particles in which the particles are uni-sized
and multi-sized.
5.1 EXPERIMENTAL APPARATUS AND MATERIALS
A schematic diagram of experimental apparatus is shown in Figure 5-1.
The packed column that was used for the measurement of the velocity
distribution of the fluid flow in packed beds was made of clear perspex to
allow easy viewing of the packing material. The column dimensions were
3.17 mm wall thickness, 144.14 mm inside radius and 1000 mm length.
The cylindrical column was provided with an inlet pipe made from brass
with inside diameter of 4.28 mm, as a connector with a "AS 1335-996
157
Experimental Techniques
C O M W E L D " rubber hose with inside diameter 5.01 m m . The position of
the fluid inlet was in the center at the bottom of the column.
In the experiment, compressed air as the fluid was supplied from a bottle
(G size) of industrial grade compressed air. Air was chosen because of its
ready availability, its well known rheological and thermodynamic properties
[Bird etal., 1960; Reid et ai, 1977; Perry and Green, 1984], its Newtonian
fluid behaviour and its low cost. The inlet line of the fluid was completed
by a "CIGWELD COMET 500" gas regulator with two pressure gauges (0 27.58x106 kgm'V2 and 0 - 9.65x105 kgm"1s"2 of the gas bottle and outlet
pressures, respectively) and a "KEY INSTRUMENTS" air flow meter type
FR 4500 rotameter (2.4-16.8 m3/hr). The accuracy of this type flow meter,
as given by the manufacturer (KEY INSTRUMENTS, PA) is + 3 %. Air was
delivered from the bottle, at a pressure 6.89x104 - 41.37x104 kgnrfV2 and
a temperature of 26°C, to the inlet pipe via a "AS 1335-996 COMWELD"
rubber hose with inside diameter 5.01 mm.
Glass spherical particles were used to randomly pack the cylindrical
column to provide a packed bed system. Two different beds were used in
the column; namely the test bed and the inlet flow straightening bed, with
steel wire mesh with openings of 1.0 mm that separated them.
158
Data Processing Unit
350 m m
Test Bed
1000 m m I
Wire Mesh
210 mm
Straightening Bed
1 1
Wire Mesh
Rotameter
\^J Pressure Gauge
AIR
Figure 5-1: A schematic diagram of the experimental apparatus.
Experimental Techniques
Three sizes (15.69 + 0.19, 23.71 ± 0.59 and 34.33 ± 0.41 m m ) of spherical
diameter of "PHANTOMS" marbles and a 6.04 ± 0.05 mm of spherical
glass particles were used as the test bed and a mixture of sizes (4.0 - 7.0
mm) of spherical glass particles was used as the flow straightening bed.
The densities of spherical particles which used as the test bed were 2670,
2550, 2622 and 2453 kg/m3 for particles diameter of 6.04 ± 0.05, 15.69 ±
0.19, 23.71 ± 0.59 and 34.33 + 0.41 mm, respectively.
The height of the flow straightening bed was 210 mm in order to achieve
the H/D ratio greater than 1.0. The ratio of H/D greater than 1.0 was also
used for the test beds. This value is a minimum value to ensure a parallel
flow of the fluid inside the bed as stated by Poveromo and Szekely [1975].
The measurement of the air flow velocity was carried out by using glass
encapsulated thermistors with diameter of 1.4 mm, which is sufficiently
smaller than the bed particles to minimise flow disturbance. The accuracy
of this type of flow meter, as given by Bolton [1996], is ±1.0 %. These are
temperature dependent resistors that enable temperature variation, due to
the cooling effect on the thermistor, by the media flowing round it, to be
recorded as changes in the voltage across the thermistor [Benard, 1988].
A number of electronic data cables was used as connectors between the
thermistors and a "COMPAQ ProLinea 4/25S", IBM-compatible personal
160
Experimental Techniques
computer, 8 M B R A M and 540 M B H D . Glass tubes with an outside
diameter of 3.08 mm were used as sheaths for the connecting wires
between the thermistors and electronic data cables. Computer Board
"PPIOAI8" was used as an analog-digital interface to convert the voltage
changes from thermistors to digital numbers, and also to generate the heat
for the thermistors in order to keep the thermistors temperature higher
than ambient temperature.
5.2 EXPERIMENTAL PROCEDURE
In order to provide a correction factor for the rotameter reading, the total
equivalent length, Le, of line between the pressure indicator and the
rotameter was determined by using the mechanical energy balance
(equation (4-11)) and measured data of the inlet pressure and the gas flow
rate. The value of the total equivalent length is needed to determine the
actual pressure in the rotameter and then by employing the material
balance correlation, the gas flow rate at inside and at down stream of the
bed can be determined properly.
Before any velocity measurements were made the thermistors were
calibrated by using the flow of air in an empty pipe. The column
dimensions were 144.14 mm inside diameter and 2000 mm length, and a
mixture of sizes (4.0 - 7.0 mm) of spherical glass particles with 210 mm
height was used as the flow distributor. The thermistors were calibrated by
161
Experimental Techniques
placing them at 1750 m m above the flow distributor. Similar to the hot-wire
and hot-film anemometer, resistance-temperature relationship of a
thermistor is not linear [Bolton, 1996]. Hence, the similar type equation to
the hot-wire and hot-film anemometer for air flow measurement, as given
by Wasan and Baid [1971], was chosen to make a correlation of the
velocity and the digital computer data. Considering that the fluid properties
of air for calibrating the thermistor are similar to the air for experiment, th
Wasan and Baid [1971] equation may be simplified to give:
u0M = wi\oVe + T^ (5-1)
where the value of 6-\ and 4 is determined by using the experimental
relationship of the digital computer numbers, cMz and velocities.
The mean bed voidage was determined by weighing the column before
and after packing was added. From density measurements of the packing
and a knowledge of the total volume occupied by the packing plus voids,
the average volume fraction of voids was calculated. A good agreement
between measurement of the mean bed voidage was obtained by using
this technique, compared with the volume displacement technique as
reported by Newell [1971]. Additionally this technique is non destructive for
the packing arrangement.
162
Experimental Techniques
W h e n the air flowing in the packed bed had reached a steady state
condition, as indicated by constant position of the rotameter float, the
velocity measurement was started with a sampling rate 1 s"1. For each
position of thermistors, data logging was carried out with sampling time 60
seconds or 60 readings were taken in a single run.
The following parameters were investigated: fluid flow rate, D/DP ratio, and
particle size distribution. Test programs were also made to investigate the
changes of fluid distribution at the outlet of the bed. In addition, tests were
made to assess the reproducibility of the measurements with repacking of
the bed between tests for the same value of the mean bed voidage.
The thermistors position of 300 mm above the bed as a basis of
measurement was chosen to avoid the errors from the high turbulence
intensities of flow [Mickley et ai, 1965] and the axial component of flow
[Schwartz and Smith, 1953] at the exit of the bed. At a distance of 300 mm
above the bed, any gross changes in the flow distribution would be
expected. Although the flow distribution was changing from the inside of
the bed, validation of the mathematical model of fluid flow distribution at
inside the bed still can be done because the model includes developing
flow profile model, as discussed in the previous chapter. The
measurements for the thermistors positions of 250, 350, 400 and 450 mm
163
Experimental Techniques
above the bed also were carried out in order to provide data for validating
the developing flow profile mathematical model above the bed.
The study of the effects of the fluid flow rate and the D/Dp ratio on the
velocity profile was carried out by variation of air flow rate and particle
diameters. The air flow rate varied from 4.02 to 19.62 m3/hr where the unisized particle diameters were 15.69 ±0.19 mm. The measurements of the
velocity profile for the uni-sized particles diameters of 23.71 ± 0.59 and
34.33 ± 0.41 mm also were performed to provide the validating data for
different value of the D/DP ratio. Binary sizes (23.71 ± 0.59 and 34.33 ±
0.41 mm, and 15.69 + 0.19 and 34.33 + 0.41 mm) and ternary sizes (6.04
± 0.05, 23.71 ± 0.59 and 34.33 ± 0.41 mm) mixtures of particles were also
used to investigate the effect of the mixture of packing particles on the
velocity profile of the fluid flowing in packed beds.
Ideal gas behaviour assumption for air was applied for the data analysis.
This is reasonable because of the low pressure and temperature of the
experimental conditions as stated by Reid and Sherwood [1966].
According to Bird et ai [1960], Newtonian fluid behaviour also can be
applied for the air under these conditions.
Since packed beds are random systems and cannot be exactly duplicated
[Schwartz and Smith, 1953; German, 1989], hence, the reproducibility is a
164
Experimental Techniques
key factor to verify the validity of the experiment. The reproducibility of the
measurements were calculated according to the following equation
[Davies and Goldsmith, 1977]:
R = - ^ — - — x 100%
l
d
n.
(5-2)
As stated earlier, the objective of the experiment is to provide the data for
validating the model of velocity profile of single phase fluid flowing in a
packed bed. The error probability of a good model is minimum [Reid et al,
1977]. In order to develop a good model there is a need to achieve
minimisation of errors by adjusting the form of the equation and the values
of constants that are included in the equation. In considering how to
guarantee the stability of error minimisation steps, Mickley et al. [1957]
suggested "the sum of square errors". The calculation of the sum of
square errors is made by using the following equation:
X/?2=Ife-^)2 (5-3)
i
i=l
While the average deviation of calculated results compared with
measurements is determined as follows [Davies and Goldsmith, 1977]:
R =
H
f
d,-c*
i=i
-xl00%
(5-4)
nd-l
165
CHAPTER SIX
EXPERIMENTAL RESULTS AND
MATHEMATICAL MODEL VERIFICATION
In order to minimise the possibility of generation of unwieldy mathematical
equations, usually some assumptions are required in the development of a
mathematical model [Franks, 1972]. Of course, introducing any
assumptions has a consequence of carrying through an error or a
deviation from real situation, simultaneously. Therefore, verification of the
solution obtained from the mathematical model, by making a comparison
with measured data, is needed to check the validity and the consistency of
the mathematical model.
The requirement for experimental data involving an incompressible fluid for
validating the mathematical model has been fulfilled by the measured data
of Stephenson and Stewart [1986]. To provide the measured velocity
profile data of compressible fluid for validating the mathematical model, as
described at previous chapter, an experiment has been carried out by
measuring the velocity profile of the air at the downstream of a packed
bed.
166
Experimental Results and Mathematical Model Verification
6.1 E X P E R I M E N T A L R E S U L T S
Based on the data of the inlet pressure and the rotameter reading
relations, the total equivalent length, Le of the line between the pressure
indicator and the rotameter was determined. It was carried out by using
the mechanical energy balance (equation (4-11)), for which the average
value of Le was 5.152 m. Hence, the actual pressure of the fluid inside the
rotameter can be determined by using the Le value and then by employing
the material balance correlation, the gas flow rate at inside and at down
stream of the bed can be determined properly.
The thermistors were calibrated by placing them in a vertical column with
L/D ratio of 12.14. From the measured average velocity, obtained from the
rotameter reading and the cross-section of the pipe, the calibration of the
thermistors was made using equation (4-71) to determine the point
velocities of the thermistor's position. The calibration data of the
thermistors were fitted as a straight line by means of equation (5-1) and
the result demonstrates a good agreement, as illustrated in Figure 6-1.
For each position of the thermistor, digital computer number {cAQ is taken
as the average of 60 readings in a single run. This is to reduce the drift
errors of thermistor and high turbulence of fluid. Figure 6-2 represents an
example of the data read by a thermistor.
167
Experimental Results and Mathematical Model Verification
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
CfVsz
Figure 6-1: Typical calibration curve for thermistors.
168
Experimental Results and Mathematical Model Verification
2.8
2.7 -
2.6
:
cVc 2.5 -
2.4 -
2.3 -
0
10
20
30
40
50
60
Time, s
Figure 6-2: Typical reading data of the thermistor.
Experimental Results and Mathematical Model Verification
6.1.1 Reproducibility of Data
Five tests were made using three identical packed beds (34.33 ± 0.41 mm
diameter spherical particles, 390 mm bed length, 0.437 average voidage
and 0.1246 m/s superficial velocity). Measurements were carried out for
two of the packed beds twice and a measurement for the last packed bed,
once.
As shown in Figure 6-3, a good reproducibility of velocity measurement
was obtained. The absolute deviation from the mean of the five
replications was 12%, although maximum values as high as 28% were
observed. The reproducibility, which is on average 12%, is considered
very good for this type of experiment due to the nature of packed beds,
which are random systems and cannot be exactly duplicated [Schwartz
and Smith, 1953; Blum and Wilhelm, 1965; Fayed and Otten, 1984].
6.1.2 Measurement Results of Velocity Profile
Experimental work was conducted to investigate the effects of the
Reynolds number, the ratio D/DP, the particle size distribution and the
change of fluid distribution at the downstream of the packed bed. The
objective of this work was to provide the data to validate the mathematical
model of the velocity profile of a compressible fluid flowing in a packed
bed. For the case of an incompressible fluid, the model will be validated by
the commonly available data in the literature.
170
r-
-O
O
T3
"O
ii
T3
r-.
---
c,
ed
C-M
OH
OI
MI
II
II
d
c
-a
—M
-
X)
s
-a
_
•s
_
r-M
o
O <KX
O
-J
K
<
i
no. 3
ooa o
no. 2
8
c
c_
oo-
-
a:
<© o <<
s
cc
+-*
CO
c
03 O
o
r
•_3
0)
-C
o
>
<D
<o o
Q
i
ooo
<w
p_
o
.3
"O
o
i-
Q.
Mr
1?
SO
Tf
oo
C
oo o <
r- co
h-
d depth =
o
3
OJ
Br
CO
CD
O Gd<K>
J_
0
o
1)
QQm
<&0
o<]
TT
o
oo
O
«2> <
- r —
Ox
"cc
o
"EL
0303 O
— CO
<N co ^. Tt
Tt ON
' Tt
O o CO — CO
II II
0)
_.
r-
d
_*
rd
d
ro
d
o
d
3
LL.
Experimental Results and Mathematical Model Verification
Figure 6-4 presents measurements of point velocities, uz, which is
normalised with respect to average velocity, uM, as function of wall
distance for mono-sized spherical particles. The velocity profile was
measured at 300 mm above the bed with superficial velocity 0.1246 m/s.
The bed particles in Figure 6-4 have diameters 15.69 mm, 23.71 mm and
34.33 mm which correspond to D/DP ratios 4.2, 6.1 and 9.2.
Figure 6-5 exhibits the velocity profile measured at 300 mm above the bed
with superficial velocity 0.1246 m/s for multi-sized spherical particles.
Figures 6-5a and 6-5b show the results for an equal volume binarymixture, whereas Figure 6-5c represents the result for an equal volume
ternary-mixture. In Figure 6-5a, the bed particles consist of diameters
34.33 mm and 15.69 mm. These give the following values: volume-surface
mean diameter, Dvs, of 21.42 mm, DP1/DP2 ratio of 2.2 and D/Dvs ratio of
6.7. The bed particles in Figure 6-5b comprise the diameters of 34.33 mm
and 23.71 mm, which correspond to Dvs of 28.15 mm, DPi/DP2 ratio of 1.4
and D/Dvs ratio of 5.1, while in Figure 6-5c, the bed particles comprise
diameters of 34.33 mm, 23.71 mm and 6.04 mm. These correspond to Dvs
of 12.43 mm, DP1:DP2:DP3 equal to 1.0:3.9:5.7 and D/Dvs ratio of 11.6.
The data in Figures 6-4 and 6-5 will be used to compare experiment with
prediction in the next section.
172
Experimental Results and Mathematical
Model Verification
.
0
•
1.5
if 1
0
0
•
0
-:o
o o
p
0
o
0
o
0
o
0
oo
0.5 -•
•
-1—1—'—
j
D
= 144.14 m m
Dp
= 15.69 m m
Bed depth = 322.5 m m
1
1
' — l — '
1.0
0.0
' — ' — ' — l — ' —
u M =0.1246 m/s
e'o = 0.43
L
-
1
1
3.0
2.0
1
s
1
4.0
(R-r)/Dp
•
•
1.5
-
o
0
o
O °
if i -
~: o
-
o
o
o
o
o
o
o
0.5 --••
-
o
"
—'—'—1—L-
D
, , .— 1 — L -
0.5
0.0
= 144.14 m m
= 23.71 m m
DP
Bed depth = 350 m m
1
•
1
1.0
— I — • —
-I
1
1
1
1
u M = 0.1246 m/s
e'0 = 0.425
L _
2.0
1.5
i—,—;—,—,—,—,—-]
3.0
2.5
(R-r)/Dp
2.0
o
1.5 -
o
if
o
o
1.0
o o
o
0.5 -•
o
o
°°o °
D
Dp
Bed depth
= 144.14 m m
= 34.33 m m
= 390 m m
u M = 0.1246 m/s
e'o = 0.437
0.0
0
a5
1
(R-r)/Dp
15
2
Figure 6-4: Velocity profile data at 300 mm above a bed of monosized spherical particles of different diameter, DP.
173
Experimental Results and Mathematical Model Verification
z.u -
: a) Binary-mixture
1.5- :
0
if
0
o o
1.0-
o
0
0
0.5-
1
0.00.0
0
o
0
° o
0
D
= 144.14 m m
= 21.42 m m
Dvs
Bed depth = 405 m m
' 1 ' ' ' ' 1 '' ' ' 1 ' '
0.5
1.0
1.5
u M = 0.1246 m/s
e'0 = 0.383
'1 I ' ' ' ' l — ' —
2.0
2.5
• — ' — • — f — ' — ' — • -
3.0
(R-r)/Dp
2.0
l
; b) Binary-mixture
l
1.5
o
-T'l
O
l
l
0
if
o
o
o
1.0 -
oo
0.5 ,
0.0
,
,
0.0
,
,
o
,
,
0.5
o
o
D
=144.14 m m
Dvs
=28.15 m m
Bed depth = 350 m m
o
u M =0.1246 m/s
e'0 =0.428
_.—,—,—,—,—|—
_—,—,—,—,—,—,—,—,—,—__
1.0
1.5
2.0
2.5
L _
(R-r)/Dp
2.0
c) Ternary-mixture
O
1.5
O
if
0.5
°
OO
1.0
°
0
0
O
0
O
o
OO
oo
o
D
=144.14 m m
Dvs
= 12.42 m m
Bed depth = 370 m m
u M =0.1246 m/s
e'0 = 0.306
0.0
0.0
1.0
2.0
3.0
4.0
5.0
(R-r)/Dp
Figure 6-5: Velocity profile data at 300 mm above a bed of
multi-sized spherical particles (see text for
details of a, b and c).
174
Experimental Results and Mathematical Model Verification
The measured data of the effects of the air flow rate and the distance
above the bed on the velocity distribution at the downstream of the bed
are presented in Figures 6-6 and 6-7, respectively. These effects will be
discussed later together with other pertinent effects.
6.2 MATHEMATICAL MODEL VERIFICATION
The purpose of this verification is to examine the performance of the
present mathematical model of the velocity distribution in a packed bed.
The verification begins with a comparison of the predicted results with
measured data, at the inside and at the downstream of the incompressible
and compressible fluid flowing in a packed bed, and then continuing with
the discussion of advantages and disadvantages of the present mode!
compared with other previous models. The criteria for a good
mathematical model as given by Reid and Sherwood [1966] is used in
verification of the mathematical model.
6.2.1 Validation of the Mathematical Model
The tests of the present mathematical model of velocity profile were
carried out by employing the measurement data taken from the literature
and from the present experimental work. The purpose of this procedure is
to test the present model, especially for the consistency of the model for
different experimental methods and materials.
175
Experimental Results and Mathematical Model Verification
2.0
o--C=0.5
1.5-
#1.0-•o
o•o.
•o-
144.14 m m
15.69 m m
322.5 m m
0.43
Column diameter
Spheres diameter
Bed depth
Average bed voidage
0.5-
0.0
50
100
+
150
+ -1
200
I
I
I
1
1-
250
300
350
Air flow rate, L/min.
Figure 6-6: The effect of air flow rate upon the velocity profile at 300
m m above the bed.
176
Experimental Results and Mathematical Model Verification
2.0
o- - • £=0.5
1.5-
#1.0--
0.5
-o.
Column diameter
Spheres diameter
Bed depth
Average bed voidage
Average velocity
144.14 m m
15.69 m m
322.5 m m
0.43
0.1246 m/s
0.0
5
7
S
ure 6-7: T h e effect of thermistors positions o n the velocity profile
a b o v e the bed.
177
Experimental Results and Mathematical Model Verification
For beds packed with uniform and non-uniform size of spherical particles,
the model can be validated by the present experimental data. For beds
packed with uniform size of cylindrical particles, the model can be
validated by comparison with the measured data of Stephenson and
Stewart [1986].
6.2.1.1 Uni-Sized Particle Packed Beds
For the incompressible fluid flow in beds of uniform size cylindrical
particles, the model has been validated by using the data of Stephenson
and Stewart [1986]. Their velocity and porosity distribution data were
obtained by using optical measurements for particle Reynolds numbers
from 5 to 280 in the beds with D/Dp = 10.7 and velocity was measured
inside the beds. Table 6-1 shows the important parameters for
Stephenson and Stewart [1986] data, which were used for calculation by
the present model.
Because of the lack of mathematical correlation of porosity distribution for
cylindrical particles bed, the natural cubic spline method [Burden and
Faires, 1993] was used to fit a correlation for the measured data (Figure
3-5) of Stephenson and Stewart [1986]. Although the use of the natural
cubic spline method is cumbersome, this method is more accurate than
polynomial regression when the data contain regions of sharply different
178
Experimental Results and Mathematical Model Verification
behaviour [Tao, 1987a]. The value of local voidage m a y be approximated
by the following equations:
for r*j < r* < r*i+1:
e = -.i+fei(r*-r*i)-r-ci(r*-r*)2+c?i(r;t:-r*i)2 (6-1)
where
Table 6-2 contains a tabulation of the constants (a-,, h, cu d, and r*,) for the
natural cubic spline correlation of porosity distribution of equation (6-1).
Figure 6-8 shows the comparison of the measured velocity distribution
data inside the bed and those predicted by the present mathematical
model. The agreement between the experimental data with those
predicted by the model is good, with average deviation of 10% (30%
maximum) and the sum of square errors of 1.44 for the total number of the
data points of 120.
The deviation between the predicted and the measured data may be due
to the assumption of fitting the measured data (Figure 2-8) by Stephenson
and Stewart [1986], in which they assumed that the velocity profile,
normalised with respect to uM, is independent of the Reynolds number.
179
Experimental Results and Mathematical Model Verification
Table 6-1: T h e experimental conditions of Stephenson and Stewart
[1986] data.
Parameters
Value
Cylindrical column
- Diameter, m m
75.5
- Bed height, m m
145.0
Cylindrical particle
- Diameter, m m
7.011
- Length, m m
7.085
- Volume-surface mean diameter, m m
7.035
Mean bed voidage
0.354
Fluids:
- Tetra ethylene glycol
- density, kg/m3
1,125
- viscosity, g/cm s
0.474
- superficial velocity, cm/s
3.1 and 12.4
- Tetra hydropyran-2-methanol
- density, kg/m3
- viscosity, g/cm s
- superficial velocity, cm/s
1,027
0.114
5.9 and 11.8
- Mixture of cyclo octane and cyclo octene
- density, kg/m3
- viscosity, g/cm s
- superficial velocity, cm/s
834.3
0.0242
6.0 and 12.0
180
Experimental Results and Mathematical Model Verification
Table 6-2: The value of the constants of equation (6-1).
i
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
-
i
0.00
0.13
0.27
0.54
0.80
1.07
1.34
1.60
1.87
2.14
2.41
2.68
2.94
3.21
3.48
3.74
4.01
4.28
4.55
4.82
4.95
5.08
5.22
5.35
a\
bi
1.00
0.35
0.33
0.33
0.35
0.34
0.34
0.37
0.36
0.35
0.34
0.37
0.37
0.33
0.33
0.36
0.36
0.32
0.38
0.37
0.36
0.34
0.34
0.35
-4.89
-0.21
-0.06
0.02
-0.10
-0.06
0.05
-0.10
-0.10
-0.10
0.06
-0.06
-0.21
-0.05
0.06
-0.06
-0.21
0.17
-0.11
-0.12
-0.21
-0.06
0.03
0.00
Ci
0.00
0.59
0.25
0.19
0.23
0.21
0.21
0.24
0.22
0.22
0.20
0.24
0.24
0.20
0.20
0.23
0.23
0.18
0.23
0.30
0.50
0.37
0.54
0.00
d\
1.47
-0.85
-0.08
0.05
-0.02
-0.01
0.04
-0.02
-0.00
-0.02
0.04
0.00
-0.05
0.00
0.03
0.01
-0.07
0.07
0.08
0.51
-0.33
0.42
-1.35
0.00
Experimental Results and Mathematical
Model Verification
2.5
2.5
N R e = 20
NRe = 5
2.0-
2.0
31.5 -:
1.5 -:
1.0 -;
1.0
0.5 -:
Measured
Predicted
0.5
i '' ' i
0.0
0
3
1 2
4
5
4
5
I
0.0
i
I
1
(R-r)/Dp
0
1 2
3
0
I
L_
I '' ' I
fe-i)ibp
3
1 2
4
5
(R-r)/Dp
(R-r)/Dp
2.5
2.5
N R e = 280
N R P = 145
2.0 -
2.0-;
21.5
3
1.0
0.5
L
i ' ' ' i ' I ' '' I ' '' I
0.0
0
1
2 3 4 5
(R-r)/Dp
0.0
i ' '' i
0
1
2
3
4
5
(R-r)/Dp
Figure 6-8: Comparison of velocity profile calculated by the
present model with the Stephenson and Stewart [1986]
measurements.
182
Experimental Results and Mathematical Model Verification
This assumption is not exactly true as demonstrated by measured data of
Morales et ai [1951], Schwartz and Smith [1953], Newell and Standish
[1973], Szekely and Poveromo [1975], and McGreavy etal. [1986].
For the compressible fluid, the model of the flow profile at inside and the
developed flow profile at the downstream of the bed has been validated
by using measurement data of air flow in a bed of spherical particles. The
flow profiles were taken from measurement at the downstream of the bed.
In accounting for the distribution of voidage over the cross section of the
bed, the empirical correlation proposed by Mueller in 1992 (equations (333) to (3-37)), was used. This correlation was chosen because it is simple
and valid over a wide range of D/DP ratio as given in the papers by
Mueller [1990; 1992]. However, considering that the value of the bulk
porosity, eb, for a random packed bed is highly dependent on the method
of charging [Blum and Wilhelm, 1965; Cumberland and Crawford, 1987].
Therefore, It seems reasonable to use measurement value of the average
voidage rather than a value predicted by means of equation (3-38) for eb
[Mueller, 1991].
Figure 6-9 is a plot of the developed flow profile downstream of the bed
versus the position from the top of the bed. The solid curves represent the
calculated values by using the present model, whilst the symbols
183
Experimental Results and Mathematical Model Verification
represent the experimental values. It can be shown that there is
reasonable agreement between the experimental data and the predicted
values.
Figure 6-10 shows the effect of the flow rate variation on the velocity
profile of the fluid at 300 mm above the bed. The agreement between
measurements (symbols) and predictions (solid curves) is again quite
reasonable. It is seen, furthermore, that the fluid flow distribution,
normalised with respect to average velocity, is not independent of the flow
rate over the range tested.
Figure 6-11 shows the effect of the bed particle diameter variation on the
velocity profile of the fluid at 300 mm above the bed. The agreement
between measurements (symbols) and predictions (solid curves) is again
quite good.
It is seen, furthermore, that the oscillation pattern of flow profile has a
similar tendency for all bed particle diameters used. The velocity has a
zero value at the column wall, reaching the first peak value at about 0.2
particle diameter from the wall, and has a minimum at 0.5 particle
diameter. The velocity continues cycling with interval distance of the
peaks about 1.0 particle diameter.
184
Experimental Results and Mathematical Model Verification
2.0
O
1.5C = 0.0
O
O
O
#1.0 -£ = 0.5
0.5
Column diameter
Spheres diameter
Bed depth
Average bed voidage
Average velocity
0.0
-J
I
l_
0
=
=
=
=
=
-J
144.14 m m
15.69 m m
322.5 m m
0.43
0.1246 m/s
•
•
8
Figure 6-9: Typical developed flow profile at the downstream of the
mono-sized particle bed.
185
Experimental Results and Mathematical Model Verification
3.0
C o l u m n diameter
Spheres diameter
B e d depth
Average bed voidage
= 144.14 m m
= 15.69 m m
= 322.5 m m
= 0.43
2.0 --
o
if
o
^o
V
1.0 --
\
£ = 0.5
I
0.0
10
I
•
I
90
•
•
t
I
170
•
•
I
I
250
I
•
+
330
•
J
—
\
-
410
490
Airflow rate, L/min.
Figure 6-10:
Comparison between measurements and predictions
for the effect of air flow rate upon the velocity profile at
300 m m above the bed.
186
Experimental Results and Mathematical Model Verification
It also can be seen that there is a rising tendency of the oscillation
patterns. This is because the flow conditions at 300 mm above the bed
represents the developing flow profile as discussed in section 2.1.
A complete comparison of the measured velocity distribution results and
those predicted by the present mathematical model is presented in Figure
6-12. The agreement between the experimental data with those predicted
by the model is good, with average and maximum deviation of 17% and
39%, respectively, and the sum of square errors of 1.750 for the total
number of the data points of 64.
Although the maximum deviation is 39%, it is still reasonable because the
difference between the maximum value of the calculation error and the
maximum reproducibility (28%) remains below the average error of
calculations. The deviation between the experimental data and the
predicted values may be due to a number of factors, particularly problems
caused by the use of Mueller's correlation [Mueller, 1992], which was
fitted from other packed bed systems to account for the radial voidage
profile. As stated by Blum and Wilhelm [1965], packed beds of spheres
are a random system, which almost certainly can not be duplicated, even
by repacking of the same particles.
187
Experimental Results and Mathematical
Model Verification
2.0
o
1.5
-
Predicted
Measured
O
•
O
0.5 -\
O \^^/^~b
/ °\70
if 1.0 -
0
0
rV
' 0
D = 144.14 m m
\ol
Dp = 15.69 m m
1
0.0
0.0
u M = 0.1246 m/s
'' ' ' ' 1
' '' ''' ' ' ' l
1.0
2.0
3.0
4.0
(R-r)/Dp
2.0
O
Predicted
Measured
1.5
if 1.0 -144.14 m m
23.71 m m
0.1246 m/s
0.5 --
_J
0.0
0.0
I
I
0.5
L_
1.5
1.0
2.5
2.0
3.0
(R-r)/Dp
_. - -
- Predicted
O Measured
/ O \
1.5 -
0
/
<p 1.0-
Ofc
0.5 -
0.0 -
0.0
,
—
,
—
,
—
•
D = I44.l4mm
D p = 34.33 m m
u M = 0.l246m/s
_
— I
1
1
1
1
1
l.O
0.5
L _
— ' — ' — i — ' — ' — ' — ' — i — '
1.5
2.0
(R-r)/Dp
Figure 6-11: Comparison between measurements and predictions
for the effect of D/DP ratio on the velocity profile at 300
m m above the bed.
188
Experimental Results and Mathematical Model Verification
£..\J -
0
•
o
/
o/
o
1.5 -
&Jf" X A +
AA
•
^ > +
TJ
&
/
A
AA
1 1.0-
o
__
o
Q.
As
0.5 -
/ °
0.0- _L_,
0.0
1 , , 1 _.
1
0.5
•
—'—1—L—
1
1.0
#measured
Diagonal
X Variation of s
O Dp = 23.71 m m
—
1
—
1
—
1
—
1
—
1
—
1.5
1
—
1
—
2.0
+ Variation of flow rate
A Dp = 15.69 m m
O Dp = 34.33 m m
Figure 6-12: Comparison of predicted velocity distribution and
measurements at downstream of a bed of mono-sized
spherical particles.
189
Experimental Results and Mathematical Model Verification
6.2.1.2 Multi-Sized Particle Packed B e d s
In order to test the validity of the mathematical model for beds of multisized particles, the predicted results were compared with experimental
data for beds of equal-volume of binary and ternary mixtures of spherical
particles. T h e experimental measurements of Goodling et ai [1983] were
used to account for the variation of local bed porosity over a cross section
of the multi-sized particle bed.
Because of the lack of mathematical correlation of porosity distribution for
multi-sized particle beds, the natural cubic spline method [Tao, 1987a;
Burden and Faires, 1993] w a s used to fit a correlation for measured data
(Figure 3-7) of Goodling et al. [1983]. However, because of the difference
of particles sizes and the relative motion between particles when pouring
into the cylindrical column to form a packed bed system, occurrence of a
size segregation is possible [Williams, 1976; Standish, 1990; Rhodes,
1990; Yu and Zulli, 1994]. Additionally, the value of local bed voidage is
also highly dependent on the method of charging [Blum and Wilhelm,
1965; Fayed and Otten, 1984; Cumberland and Crawford, 1987]. Hence,
the data of the radial voidage distribution should be chosen from the bed
with similar characteristics to that being considered. Table 6-2 represents
the comparison of bed characteristics between present work and Goodling
et al. [1983] measurement condition that is used for accounting of the
radial voidage distribution.
190
Experimental Results and Mathematical Model Verification
Table 6-2: Comparison of the bed characteristics between present
work and Goodling etal. [1983] measurement.
No.
Parameters
1
Binary-mixture A
equal volume
- Dpi, m m
15.69
4.76
- Dp2, m m
34.33
7.94
- DPI:DP2
1.0:2.2
1.0:1.7
- Dvs, m m
21.46
5.95
6.7
8.8
0.383
0.392
equal volume
equal volume
- Dpi, m m
23.71
6.35
- Dp2, m m
34.33
7.94
- DPI:DP2
1.0:1.4
1.0:1.3
- Dvs, m m
28.15
7.06
5.1
7.4
0.428
0.426
equal volume
equal volume
- Dpi, m m
6.04
4.76
- Dp2, m m
23.71
6.35
- DP3, m m
34.33
7.94
1.0:3.9:5.7
1.0:1.3:1.7
12.43
6.08
11.6
8.6
0.306
0.415
- D/DvS
-e'o
Binary-mixture B
- mixture basis
- D/Dv S
-e'o
3
Goodling etal., 1983
equal volume
- mixture basis
2
Present work
Ternary-mixture
- mixture basis
- Dpi:Dp2:Dp3
- Dvs, m m
- D/Dvs
-e'o
191
Experimental Results and Mathematical Model Verification
As discussed in Chapter three, for studying fluid flow in a packed bed, the
volume-surface mean diameter is commonly used to represent the mean
diameter of multi-sized particles. Therefore, equation (2-74) can be
modified into the following form:
The values of the constants (a,, b\, c„ d\ and r*j) for binary-size mixtures
and ternary-size mixtures are given in Tables 6-3and 6-4, respectively.
A comparison of the measured velocity distribution results and those
predicted by the present mathematical model for binary-mixtures and
ternary-mixtures is presented in Figure 6-13. The agreement between the
experimental data and those predicted by the model is reasonably good.
For the total number of the data points of 43, the deviation between
experimental data and the calculated values is on average 21%, the
maximum deviation is 39%, and the sum of squares error is 1.564.
The deviation between the calculated results and observed values, as
shown in Figure 6-13, may be due to a combination of causes, which
include the method of charging, segregation of particles and the
distribution of particles. Moreover, because as noted earlier, a packed
bed is a random system which almost certainly can not be duplicated.
192
Experimental Results and Mathematical Model Verification
Table 6-3: The value of constants of equation (6-1) for beds of
binary-mixture of particles.
1 Binary-mixture A
i
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
r*i
0.000
0.162
0.233
0.314
0.422
0.485
0.601
0.673
0.745
0.862
0.933
1.041
1.086
1.158
1.274
1.346
1.418
1.508
1.579
1.660
1.759
1.831
1.947
2.019
2.091
2.181
2.253
2.333
2.432
2.531
2.621
2.692
2.764
2.845
2.944
3.033
3.114
3.204
3.294
3.662
3.967
4.038
4.487
a\
1.000
0.717
0.547
0.413
0.333
0.283
0.316
0.340
0.443
0.500
0.440
0.367
0.329
0.329
0.350
0.367
0.367
0.383
0.350
0.316
0.300
0.337
0.350
0.397
0.383
0.359
0.337
0.340
0.318
0.350
0.350
0.383
0.395
0.425
0.413
0.383
0.375
0.325
0.337
0.383
0.325
0.383
0.325
bi
C\
-1.824
-2.463
-1.746
-0.812
-0.835
0.224
0.289
1.376
0.396
-0.901
-0.763
-0.886
-0.065
0.120
0.185
-0.062
0.112
-0.518
-0.479
-0.217
0.472
0.043
0.604
-0.264
-0.334
-0.360
-0.021
-0.281
0.271
-0.059
0.404
0.101
0.304
-0.197
-0.399
-0.160
-0.621
0.087
0.047
-0.253
0.783
-0.232
0.000
1.334
1.316
0.617
0.690
0.500
0.517
0.876
0.714
0.951
0.810
0.752
1.120
0.474
0.600
0.953
0.699
0.810
0.815
0.533
0.596
0.615
0.536
1.062
0.745
0.735
0.762
0.642
0.486
0.635
0.697
0.944
0.862
0.796
0.706
0.760
0.747
0.657
0.248
0.139
0.342
0.344
ffj
2.753
-0.086
-2.884
0.226
-1.008
0.049
1.669
-0.754
0.678
-0.657
-0.178
2.728
-2.998
0.362
1.637
-1.179
0.412
0.025
-1.166
0.213
0.086
-0.225
2.442
-1.471
-0.039
0.124
-0.493
-0.527
0.504
0.229
1.146
-0.381
-0.272
-0.302
0.201
-0.056
-0.334
-1.519
-0.098
0.221
0.011
-0.256
Experimental Results and Mathematical Model Verification
Table 6-3: The value of constants of equation (6-1) for beds of
binary-mixture of particles (Continued).
2 Binary-mixture B
i
r*i
fli
b\
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
0.000
0.078
0.166
0.244
0.400
0.448
0.526
0.614
0.731
0.809
0.897
0.975
1.053
1.180
1.297
1.384
1.462
1.540
1.628
1.706
1.833
1.950
2.028
2.115
2.193
2.271
2.359
2.437
2.515
2.846
3.412
3.656
4.387
4.874
1.000
0.925
0.733
0.540
0.392
0.283
0.233
0.250
0.315
0.427
0.525
0.583
0.502
0.483
0.367
0.300
0.315
0.323
0.384
0.447
0.450
0.483
0.438
0.407
0.375
0.367
0.315
0.325
0.392
0.427
0.342
0.367
0.367
0.367
-1.025
-2.371
-2.567
-1.055
-2.269
-0.697
0.160
0.506
1.386
1.035
0.654
-1.132
-0.241
-1.066
-0.815
0.143
0.050
0.637
0.745
-0.063
0.207
-0.648
-0.429
-0.473
-0.165
-0.654
0.078
0.808
0.014
-0.226
0.068
-0.090
-0.036
C\
0.000
2.420
1.399
0.729
0.574
0.885
0.394
0.379
0.514
0.900
1.010
1.429
0.751
0.655
0.584
0.582
0.717
0.589
0.819
0.780
0.502
0.909
0.896
0.800
0.810
0.770
0.555
0.816
0.350
0.127
0.148
0.128
0.112
d\
10.346
-3.880
-2.863
-0.332
2.124
-2.096
-0.058
0.384
1.650
0.416
1.792
-2.897
-0.252
-0.202
-0.011
0.579
-0.548
0.874
-0.166
-0.731
1.159
-0.056
-0.363
0.039
-0.169
-0.818
1.116
-1.992
-0.224
0.012
-0.028
-0.007
-0.077
Experimental Results and Mathematical Model Verification
Table 6-4: The value of constants of equation (6-1) for beds of
ternary-mixture of particles.
i
r*i
a\
bi
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
0.000
0.070
0.190
0.350
0.460
0.530
0.650
0.760
0.880
0.990
1.130
1.220
1.370
1.450
2.000
2.500
3.500
4.700
5.000
1.000
0.933
0.767
0.500
0.360
0.283
0.277
0.383
0.475
0.467
0.433
0.400
0.392
0.400
0.400
0.367
0.380
0.380
0.380
-1.009
-1.591
-1.777
-1.341
-1.146
-0.104
0.918
0.692
-0.149
-0.328
-0.422
-0.136
0.059
-0.104
-0.118
-0.060
-0.073
-0.023
C\
0.000
2.224
0.733
0.560
0.731
0.503
0.338
0.557
0.748
0.583
0.665
0.506
0.647
0.229
0.108
0.093
0.034
0.113
d\
10.589
-4.140
-0.361
0.519
-1.084
-0.460
0.664
0.531
-0.499
0.194
-0.586
0.313
-1.742
-0.073
-0.010
-0.019
0.022
-0.125
195
Experimental Results and Mathematical Model Verification
3.0
Binary-mixture:
Dvs
=21.46 m m
D
=144.14 m m
UM
=0.1246 m/s
Dpi:DP2 = 1.0:2.2
— Predicted
O Measured
2.0
if
1.0
0.0
0.0
• M -
0.5
1.5
1.0
2.0
2.5
3.0
(R-r)/Dp
3.0
Binary-mixture:
Dvs
=28.15 m m
D
= 144.14 m m
uM
=0.1246 m/s
Dpi:DP2 = 1.0:1.45
O Measured
— Predicted
2.0-
V
O
1.0-
0.0
0.0
0.5
1.0
1.5
2.0
2.5
(R-r)/Dp
3.0
— Predicted
O Measured
2.0 -
Ternary-mixture:
Dvs
= 12.43 m m
= 144.14 m m
D
uM
=0.1246 m/s
DPi:Dp2:Dp3 =1.0:3.9:5.7
O
O
-CTQ
O
0.0 I ' ' ' ' l ' ' ' ' l ' ' ' ' I ' ' 'i i'' I' ''i ' '' ''' i'• I ' ' ' ' I '
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
(R-r)/Dp
Figure 6-13: Comparison of velocity distribution predicted by the
model with those measured at downstream of beds of
multi-sized spherical particles.
[96
Experimental Results and Mathematical Model Verification
Using porosity profile data of Goodling et ai [1983], measured under
different conditions to the present velocity profile measurements for this
calculation m a y also have introduced significant errors.
It should also be noted that unlike for mono-sized particles for which the
model predicts a smooth result (eg. Figure 6-11) for the case of binary
and ternary particle mixtures the model predicts a non-smooth line as in
Figure 6-13.
6.2.2 Comparison of the Mathematical Models of the Velocity Profile
in Packed beds
A velocity profile model of a single-phase fluid flow in packed beds w a s
developed in the present work by assuming that the flow characteristic is
a combination of a continuous and discontinuous system of fluid between
voids in the bed. Employing the proposed model of a single phase fluid
flow in the packed beds only requires the physical properties of the fluid
and those of the packed bed to be known, and does not require new
experimental variables.
The validity of the model has been checked by using previous and new
experimental data, and a reasonable agreement w a s obtained. S o m e
deviation of the model prediction from measured data is expected
because of the nature of packed beds, which are random systems and
197
Experimental Results and Mathematical Model Verification
cannot be exactly duplicated. However, based on the accuracy, simplicity,
and the requirement of empirical data, it is believed that the proposed
model provides a useful velocity distribution model.
It is instructive to compare the value of the present mathematical model
with previous mathematical models. As mentioned in the preceding
chapter, although the models that are based on phenomenological
approach are more accurate for a particular set of data, it is difficult to
apply this type of models with any confidence to other systems and
conditions. For these reasons, it would seem appropriate to make
comparison of the present model only with the previous models that were
based on a theoretical approach.
Of the many proposed mathematical models that have been developed
based on a theoretical approach, only Stanek and Szekely's model
[Stanek and Szekely, 1974; Szekely and Poveromo, 1975; Poveromo et
ai, 1975] and Vortmeyer and Schuster's model [Vortmeyer and Schuster,
1983] will be used to make comparison here. The principal reasons for not
using the other models for comparison were poor agreement between
their experimental and their predicted values, and the requirement of
empirical constants, which reduce the model's reliability, generality and
applicability.
198
Experimental Results and Mathematical Model Verification
All of these three models (Stanek and Szekely [1974], Vortmeyer and
Schuster [1983] and the present model) have almost similar condition of
model's simplicity and not require any new empirical constants. Hence, it
would seem reasonable to make comparison only of the accuracy of the
model in predicting velocity profiles. Because Stanek and Szekely's model
[1974] and Vortmeyer and Schuster's model [1983] only provide a
prediction for velocity profile inside a packed bed, the measured data of
Stephenson and Stewart [1986] were used as a basis in this model's
comparison.
The basis of Stanek and Szekely's [1974] model is the Ergun [1952]
macroscopic equation for pressure drop of fluids flowing in packed beds,
where the fluid pathways are regarded as bundles of tangled tubes
(discontinuous system) which are then treated in relation to individual
straight tubes. Originally, this model used the experimental measurements
of Benenati and Brosilow [1962], Figure 3-4, to account for the porosity
oscillation at the grid point adjacent to the wall [Szekely and Poveromo,
1975]. However, the experimental data of voidage profile measured by
Stephenson and Stewart [1986], Figure 3-5, are quite different from those
of Benenati and Brosilow [1962]. Hence, equation (6-1) with constant
values tabulated in Table 6-6 is used to predict the radial voidage
distribution.
199
Experimental Results and Mathematical Model Verification
The point superficial velocities, normalised with respect to average
superficial velocity predicted by Stanek and Szekely's model [1974],
compared with the measurements data of Stephenson and Stewart are
shown in Figure 6-14. Deviation between prediction and measurement is,
on average, 5 5 % , the maximum deviation is 3 1 9 % , and the s u m of
squares error is 59.9 for 120 data points.
The principal reason for this large deviation is probably the effect of the
bed voidage in the vicinity of the wall which has a value higher than 0.5.
This is a maximum value for which the Ergun [1952] equation is
appropriate [Bird et ai, 1960; Cohen and Metzner, 1983]. Another error
may also c o m e from the limitation of Ergun [1952] equation, as stated by
Gauvin and Katta [1973], which is not appropriate for systems containing
particles of low sphericity, whereas cylindrical particles were used by
Stephenson and Stewart [1986] as bed particles for their experiment.
Vortmeyer and Schuster [1983] have extended Brinkman's equation
[Brinkman, 1947] to develop a mathematical model of velocity distribution
in packed beds. The Brinkman equation [1947] is a macroscopic equation
for pressure drop of fluid flow in packed beds, in which the pressure drop
is obtained by summing the resistances of all the individual submerged
particles [Foscolo et ai, 1983], and it is an interpolation between Stokes
equation and Darcy's law [Durlofsky and Brady, 1987].
200
Experimental Results and Mathematical Model Verification
6.0
6.0—|
NRe = 5
4.0-
4.0-
Measured
Predicted
I
E
3
2.0-
2.0•• '
0.0
-1—1—I
H '-H-
0
L
3
1 2
N R e = 20
I.
II — I' !__-.' ' I
4
0.0
5
I ' ' ' I- J — I — l — l
0
1
(R-r)/Dp
I
I
l
I
1
i
(
i.
fc-rvbp
6.0
6.0
N R e = 75
N R e = 37
4.0
4.0-
E
3
2.0 4
E
3
2.0i.- *• •
0.0
0
t.
... J
.
_ ..
i ' ' 'i '' 'i ''' i ' ' ' i
0.0
12 3 4 5
0
-1 \ 1^—1 L_
i ' ' ' i
2
3
4
5
(R-r)/Dp
(R-rVDp
6.0 i
6.0 f
N R e = 145
4.0
£
3
2.0-
0.0
• i
1 2
3
4
' ' ' i *
5
0
(R-r)/Dp
1
2
3
4
(R-rVDp
Figure 6-14: Comparison of velocity profile predicted by Stanek and
Szekely's model [1974] with Stephenson and Stewart
[1986] measurement data inside the beds.
201
Experimental Results and Mathematical Model Verification
This model has better validity than Stanek and Szekely's [1974] model for
experimental data measured by Stephenson and Stewart [1986] as shown
in Figure 6-15. The deviation of predicted value from measurements is
16% on average and 94% maximum, with the sum of square errors being
59.9 for 120 data points. The deviation may be due to the exponential
profile assumption of the radial porosity profile (equation (2-52)) of
Vortmeyer and Schuster's [1983] model. Additional error is probably the
assumption in fitting the measured data by Stephenson and Stewart
[1986], in which they assumed that the uz/uM is independent of the
Reynolds numbers. This assumption is not exactly true as discussed
earlier in section 6.2.1.1.
Figure 6-16 demonstrates the validity comparison of the Stanek and
Szekely's model [1974] and Vortmeyer and Schuster's model [1983] with
the present model, which were tested by using the Stephenson and
Stewart [1986] data. It is clearly seen that the present model has better
validity than other models examined. This result may be expected to be
useful in furthering studies concerned with velocity distribution in packed
bed systems.
202
Experimental Results and Mathematical Model Verification
3.0
3.0
NRe = 5
Measured
Predicted
2.0
1.0
0.0
- -
_1
0
I
I
|_l
I
1
L__l
I i_
2.0-
,
m ,
4
1.0
•• -
0.0
I '' ' I
3
1 2
N R e = 20
5
.
•
>•
i '' ' i
0
3
1 2
4
5
(R-r)/Dp
(R-rVDp
3.0
3.0
N R e =-37
-
N R e = 75
-
2.0
E
3
--V
1.0-
*•
1
: ....
_.
__-..'
0
1 2
3
1.0 - ^ :• -
i
i
4
5
.
.
,
_
.
.
.
_.
I _•- -'
-H-1-1 1 ' ''1 ' ' '" i11 '
0.0
E
3
0.0 - ' ' ' 1 ' ' ' i
0
1 2
' ' ' i ' ' ' i ' ' i i '1
3
4
5
(R-rVDp
(R-r)/Dp
3.0
N Re = 280
2.0
E
3
T
-.
r -
1.0-
0.0
0
1 2
(R-rVDp
3
4
5
I
I
' |
I II I I
' | I
' I
| ' ' ' | ' ' ' |
0 12 3 4 5
(R-r)/Dp
Figure 6-15: Comparison of velocity profile predicted by Vortmeyer
and Schuster's model [1983] with Stephenson and
Stewart [1986] measurement data inside the beds.
203
Experimental Results and Mathematical Model Verification
v50U 1
300 1
250 1
200
150 I
100 I
P
i
I
50-
r\ _
U
AE, %
• Stanek and Szekely, 1974
ME, %
SSE
I Vortmeyer and Schuster, 1983
B Present work
Figure 6-16: Comparison of the velocity distribution models (AE =
average error, M E = maximum error, and SSE = the sum
of squares error).
204
CHAPTER SEVEN
DISCUSSION
A mathematical model of the velocity profile of a single-phase fluid flow in
packed beds and developing flow profile in the downstream of the bed has
been proposed. The model was tested against the experimental data
obtained from previous investigations and present measurements, and was
found to give an excellent fit. Verification of the model carried out by
comparing it with the previous models, showed an improvement in terms of
accuracy and simplicity, and the model does not require new empirical
constants. Based on these results, an attempt has been made to obtain a
more complete, comprehensive understanding of the fluid flow phenomena in
packed bed systems.
7.1 COMPUTED FLOW PROFILES
Figure 7-1 exhibits a velocity profile computed by the present model for a
cylindrical packed bed of spherical particles. It can be seen that the predicted
flow profile inside of the bed is a non-single peak, or oscillating profile. The
profile is in good agreement with the experimental data of McGreavy et al.
[1986] in Figure 2-9 and is qualitatively similar to the actual flow distributio
Figure 2-1. Moreover, the first maximum value is at about 0.2 particle
diameters, and this is consistent with the data of McGreavy et al. [1986] and
205
Discussion
Ziolkowska and Ziolkowski [1993], and is also consistent with the predicted
value of Vortmeyer and Schuster [1983].
Since the particles are mono-sized, then according to equation (3-57) the bed
permeability variation is represented by the local porosity. As would be
expected, the distribution of the bed permeability determines the velocity
profile, as shown by Figure 7-2. However, it also can be seen that, for
porosity above 0.5, the bed permeability has practically no effect on velocity
compares with its effect for porosity below 0.5. This is not surprising, because
equation (3-57) has been developed based upon an assumption that the
voids in the bed are not connected which each other, or as a discontinuous
system. This is almost true for small voidage beds in which the fluid in a void
is not interacting with the fluid inside the neighbouring voids. Hence, for the
beds with high voidage in which the interaction of the fluid between
neighbouring voids is possible to occur, the validity of the discontinuous
system assumption is no longer tenable. This confirms the findings in
macroscopic study of Foscolo et al. [1983]. In their study, for fluidised beds
system in which the bed voidage is higher than 0.5, the continuous approach
is more appropriate than the discontinuous approach. The kink in the velocity
curve at about K = 0.4, which is related to porosity about 0.5, represents the
transition between continuous and discontinuous behaviour of fluid between
voids.
206
rr.
(ft
0)
o
r
CO
a
CQ
O
'£CD
-C
a
to
•o
0)
N
'w
6T3O
a
9
JO
c
CO
o
0)
TJ
CO
E
c
<D
a
^i,
>
O
u
o
CD
o
o
>
TJ
CO
MM
3
a
E0
u
CO
o
n
I..
N
r-
Wn^n
CU
i_
3
O.
LL
II
0)
cc
Z
•a
c
ca
o
CM
II
CJ.
a
Q
Discussion
-T
K,
2.5
mm
Figure 7-2: The effect of the bed permeability upon the fluid velocity
in a packed bed (D/DP = 12.0 and NRe = 1000).
208
Discussion
Although the effect of Reynolds number on the radial velocity profile in
packed beds has been studied by many investigators, a consensus has
not been reached. Therefore, it has seemed reasonable to investigate the
effect of NRe upon the velocity distribution by using the present model. The
results are presented in Figures 7-3 and 7-4.
As shown in Figure 7-3, the results clearly demonstrate that the
dimensionless flow profile is dependent on the Reynolds number if its
value is less than 500. This confirms the findings of Vortmeyer and
Schuster [1983], though their calculated distributions differ from that
shown in Figure 7-1. However, as shown in Figure 7-4, the interaction
between the flow profile with D/DP ratio and the Reynolds number is more
complicated than discussed by Vortmeyer and Schuster [1983]. This may
be due to the exponential porosity profile of Vortmeyer and Schuster
[1983] in which, as discussed in the preceding chapter, the porosity profile
is not exponential but follows an oscillation pattern. From Figures 7-3 and
7-4, it is also clearly shown that the flow profile is independent of the
Reynolds number greater than 500. The independency of the flow profile
on the Reynolds number, concluded by Price [1968], is discounted here
since this author measured the flow profiles only between NRe= 1470 and
4350.
209
Discussion
3.5
-3
2.5
2
•©-e-eee-o
1.5
A A AM A
1
^-•-•<->-^
0.5
*-*
_i
10
100
i
I I I I
0
10000
1000
NRe
Distance from wall, particle diameter :
—A—6.00
--•©•--1.38
--O--0.96
0.54
— a — 0.18
Figure 7-3: The effect of Reynolds number on the flow velocity for
D/Dp = 12.0.
210
Discussion
3.5
- e — D/Dp = 3
-a— D/Dp = 6
ZJ
1.
_:
"A
1.0
_i
i
-A
I I I M
100
10
1000
N Re
Figure 7-4: The effect of Reynolds number on the flow velocity at 1.10
particle diameters from the wall.
2ll
Discussion
Computer simulations of fluid flow in a bed of mono-sized of spherical
particles were made using the present model to investigate the effect of
the LVD ratio on the flow profile. Both compressible and incompressible
fluids were used as a fluid for this investigation. As shown in Figure 7-5,
for a bed of mono-sized spherical particles with uniform axial properties of
the bed, the flow profiles is independent of the L/D ratio. This result agrees
with the finding of Price [1968], even though he measured the flow profile
at the downstream of the bed.
As discussed by Bird et al. [1960], for pressures up to critical pressure the
fluid viscosity is almost independent of pressure. Also uniform properties in
axial direction can be applied for mono-sized packed bed system. For an
incompressible fluid, if the parallel flow condition is achieved, it is clear tha
the Reynolds number of the flowing fluid is independent of the axial
direction in the bed, therefore, the flow distribution is independent of the
L/D ratio. To explain why the flow profile for a compressible fluid is also
independent of the L/D ratio, it is assumed that there is no significant
variation of bed properties in the axial direction and the fluid viscosity is
also independent of pressure. It is well known that the density of a
compressible fluid is a function of pressure and by assuming that the ideal
gas behaviour can be applied to correlate the density-pressure relation,
the increasing of the fluid density is linear with the increasing of the
pressure. On the other hand, based upon mass balance equation, the
212
Discussion
10
20
30
40
50
60
L/D
Reynolds number:
--•<>-• 10
---O-- 100
---A-- 1000
Figure 7-5: The effect of the L/D ratio upon the flow velocity at 1.0
particle diameters from the wall of a bed of mono-sized
spherical particles.
213
Discussion
average superficial velocity is linearly decrease with increasing pressure.
Therefore, the Reynolds number is independent of the pressure and as a
result the flow profile is also independent of L/D ratio for a compressible
fluid.
The effect of temperature on the fluid flow distribution also has been
studied by using the present mathematical model. Figure 7-6 exhibits this
temperature effect for various Reynolds numbers of a bed of mono-sized
spherical particles with D/DP ratio of 12.0. It is obvious from Figure 7-6 that
the flow profile is independent of temperature for a given Reynolds
number.
The dependency of the flow profile on temperature concluded by
Vortmeyer and Schuster [1983] is discounted here since these authors
investigated the effect of temperature by maintaining a constant average
superficial velocity, rather than a constant Reynolds number. Therefore,
the variation of the flow profile obtained in their investigation, is actually a
effect of the Reynolds number variation rather than of the temperature
variation.
Usually, in the study of mass and heat transfer in packed bed systems, it is
assumed that the effect of the column diameter can be neglected. It can
be seen that the average superficial velocity, particle diameter, bed
214
Discussion
E
3
Temperature, K
Reynolds number:
_g_ 1 _*_ 10 -&- ioo -e-1000
Figure 7-6: The effect of the temperature on the flow profile at 1.10
particle diameters from the wall of a bed of mono-sized
spherical particles.
215
Discussion
voidage and physical properties of the fluid, are usually used to predict the
mass and heat transfer coefficients [Perry and Green, 1984]. It may be
appropriate for packed bed systems with very big value of the D/DP ratio,
but probably is poor for systems having small D/DP ratio. For packed bed
systems with a small value of D/DP ratio, the velocity profile is far from the
flat profile condition, therefore, average superficial velocity can not
properly represent the local condition of fluid dynamics.
However, it is also a fact that the flat profile velocity assumption has been
very helpful, and much reducing the calculation time for analysing and
designing of the packed bed systems [Himmelblau and Bischoff, 1968].
For these reasons, it has seemed desirable to investigate the values of
D/Dp in which a flat profile assumption is applicable.
The present model is further used to investigate the effect of the D/DP ratio
on the deviation of local superficial velocity from average superficial
velocity (Figure 7-7). An inspection of Figure 7-7 reveals qualitatively the
deviation from flat profile flow condition as a function of the D/DP ratio for
mono-sized particles bed. At a Reynolds number below 500 the percent
deviation from the flat profile is found to decrease with corresponding
increase in the D/DP ratio and the Reynolds number. Moreover, for a
Reynolds number above 500, the deviation from the flat profile condition is
independent of the Reynolds number. It agrees with Figures 7-3 and 7-4 in
216
Discussion
100
Reynolds number
—B—1
—•—10
80-
60c
g
_
0)
a
40-
20-
0
0
- 1 —I20
-I40
_l
60
80
J^
L.
100
D/Dp
Figure 7-7: The effect of D/D P ratio on the deviation from the flat
profile condition.
217
Discussion
which there are no more significant changes in the dimensionless flow
profiles for Reynolds numbers above 500.
Considering that the recommended safety or over-design factor for packed
columns is about 15% [Peter and Timmerhaus, 1968], then for D/DP ratios
above 75, it is reasonable to apply the flat profile flow assumption. The
statement of Cairns and Prausnitz [1959] that the flat profile assumption
can be applied for systems with D/DP ratio above 10 is discounted here
since the deviation is far higher than 15%, and even for a Reynolds
number above 500 it is about 25%. However, when the condition of the
packed bed system is sensitive to the flow profile, for example hot spot
formation and nuclear reaction, the flat profile assumption should be
avoided.
Figure 7-8 illustrates the effect of particle diameter on flow profile inside
packed beds. It is clearly shown that the variation of flow profile is more
evident as an effect of the D/DP ratio rather than of particle diameter. The
significant effect of particle diameter on flow profile, reported by
Ziolkowska and Ziolkowski [1993], is argued here as being incorrect. This
is because these authors used the same column diameter to investigate
the effect of particle diameter, so their data is more representative of the
effect of the D/DP ratio rather than of DP. The significant effect of D/DP ratio
rather than of DP itself on the radial porosity in packed beds, as has been
218
Discussion
2.5
-©••• (R-r)/Dp=0.2;D/Dp=10
- D - (R-r)/Dp=1.0;D/Dp=10
- © — (R-r)/Dp=0.2;D=160 m m
2.4 -
-_$_ — (R-r)/Dp=1.0;D=160 m m
2.3
2.2 -
5
.3
-£2.1
2.0 --
1.9 -
1.8 -
-a
1.7 -I
3.0
'
H
5.0
7.0
9.0
11.0
13.0
15.0
17.0
Dp, m m
Figure 7-8: A typical effect of particle diameter on flow profile inside a
bed of mono-sized spherical particles (uM = 0.1 m/s).
219
Discussion
reported by Govindarao and Froment [1986] and Mueller [1992], is an
additional reason for the above conclusion.
The present model of fluid flow distribution, both inside and downstream of
packed beds, was developed by assuming the flow characteristic is a
combination of continuous and discontinuous systems. A good
performance of the model in predicting the flow phenomena in packed
beds also was clearly demonstrated. Therefore, it is suggested that this
mathematical model can then be directly incorporated into mass and
energy balance equations for the further mathematical simulation of
packed bed systems. Moreover, based upon investigation by the present
model, it is clearly shown that the Reynolds number and the bed
characteristics influence the flow distribution. However, when the Reynolds
number is higher that 500, the flow distribution is almost independent of
the Reynolds number.
7.2 SIMILARITY CRITERIA OF PACKED BED SYSTEMS
For economic and safety reasons, usually a physical modelling of systems
is required [Johnstone and Thring, 1957; Fleming, 1958; Peter and
Timmerhaus. 1968; Szucs, 1980; Euzen et ai, 1993]. Moreover, as stated
by Geldart [Rhodes, 1990], there is too little understanding of background
theory of packed bed characteristics, as an essential factor for fluid flow
distribution, in which the occurrence of a size segregation is possible.
220
Discussion
Therefore, experimental work becomes a key factor in studying the
system.
In order to achieve a greater confidence in studying of physical
phenomena by using a physical model, a knowledge of similarity criteria is
essential [Schuring, 1977; Szucs, 1980; Zlokarnik, 1991]. Therefore, it
seems reasonable to investigate the similarity criteria of a packed bed
system based on the fluid flow distribution point of view. In this
investigation, the two similar conditions of phenomena (systems) are
defined such that their corresponding characteristics (features,
parameters) are connected by bi-unique (one-to-one) mappings
(representations), as stated by Szucs [1980],
Studying flow phenomena in a packed bed system is usually based upon
dynamic and geometric similarity. If the flow distribution factor can be
neglected, then the Reynolds number, P"M p , and the Froude number,
—P-Y , are acceptable as similarity criteria [Hoftyzer, 1957]. if the fluid is
U
M
D2
liquid, the Bond number,
pg p
, also should be considered [Hoftyzer,
o
1957]. However, as also reported by Hoftyzer [1957], the performance of a
model based on the above similarity to predict the prototype (actual size)
221
Discussion
behaviour is only about 2 5 % . Therefore, it is a continuing need for
investigation of similarity criteria on the basis of the fluid flow distribution
For fluid flow in packed beds, the characteristic of the bed can be
represented by the bed permeability. The bed permeability as noted in
Section 3.3, is influenced by the particle size and the particle size
distribution (the spread and the size range), as discussed by Yu and Zulli
[1994]. Therefore, the similarity of particle size and the particle size
distribution should be satisfied between the model and the prototype.
Considering that for the physical model with geometrical size smaller than
prototype, it is often impossible in practice to satisfy the requirement of
similar D/DP ratio together with the particle size distribution (the spread
and the size range). This is because, in engineering practice, the
components used to construct a particle mixture are themselves particles
mixtures [Yu and Standish, 1993a]. Accordingly, packed bed models are
easier to study by using actual (prototype) particle bed rather than
generate smaller particles with similar size distribution.
As discussed earlier, the fluid flow distribution is strongly dependent on
bed character and the Reynolds number. The Reynolds number has been
recognised by Hoftyzer [1957]; therefore, it seems reasonable to explore
110
Discussion
the bed character for similarity criteria and its possibility in developing of a
distorted model.
The similarity requirement of the D/DP ratio is still more difficult to satisfy
even for mono-sized particle beds since the pressure drop becomes very
high for small values of particle diameter. For these reasons, it has
seemed desirable to investigate the possibility of distortion for the similarity
criteria of the D/DP ratio.
To facilitate the investigation of the distorted model for D/DP ratio in
physical modelling, it is assumed that the deviation from flat profile
condition can represent the variation of flow profile. Hence, the minimum
D/DP ratio of the model is defined by the value of the D/DP ratio in which
the difference of the deviation from the flat flow profile condition with
prototype is equal to 10%. Figure 7-9 illustrates the effect of D/DP ratio and
Reynolds number on the minimum D/DP ratio of the model.
If a parallel flow condition and uniform condition in axial direction of the
bed is achieved, the flow distribution is independent of LVD ratio. However,
a L/D ratio higher than 1.0 is required to achieve a parallel flow distribution
[Szekely and Poveromo, 1975]. Therefore, in order to minimise the end
effects (at inlet and outlet), it seems reasonable to maintain the L/D ratio of
the model higher than 3.0.
223
Discussion
70
Reynolds number:
60
••--•IO
--—100
•-.--•500
-1000
50-
40-o
A
Ow
Q
Q
30-
20-
10
_i
50
i
i
100
150
200
(D/Dp)prototype
Figure 7-9: Minimum D/D P ratio of a physical model as a function of
D/Dp ratio of prototype and Reynolds number.
224
Discussion
Although, the use of a physical model without any distortion of similarity
criteria is more preferable, it is often difficult to satisfy all requirements
unless it is by using the model with conditions exactly the s a m e as the
prototype conditions. Therefore, the results of this investigation are
expected to be useful in the development of physical models of packed
bed systems.
225
CHAPTER EI-3HT
CONCLUSIONS
A mathematical model of the flow distribution at inside and at downstream of
packed beds for single-phase fluid has been developed. The model is based
on the analysis of a packed bed viewed as a combination of continuous and
discontinuous systems of fluid between voids in the bed. The model is
applicable to both compressible and incompressible fluids. There is a
reasonable agreement between the prediction of the model and the
measured values. The new model compares favorably with previous models
in terms of accuracy and simplicity, and does not require new empirical
constants.
The disagreement of the previous investigators of the effect of the Reynolds
number and the particle diameter on fluid flow distribution in packed beds is
clearly explained by using the present mathematical model. It is demonstrated
that the Reynolds number has a significant effect on flow distribution only fo
NRe less than 500, and when the Reynolds number is higher than 500, the
flow profile is determined by bed characteristics. Moreover, that model result
226
Conclusions
has also demonstrated that it is the D/D P ratio that has a significant effect on
the flow profile rather than the particle diameter, as hitherto believed.
The results have also shown that models based on discontinuous systems
are only successful for local porosity of less than 0.5. When the local porosity
is higher than 0.5, especially at the vicinity of the wall, a model based on the
continuous systems approach, as used in the present case, is more accurate.
The results of the present model regarding the flat flow profile assumption for
packed bed systems have clearly shown that the deviation from the flat profile
condition does not only depend on the D/DP ratio, but is also depends on the
Reynolds number. This conclusion was also used to suggest some rules of
how a distorted physical model may be generated in term of D/DP ratio and
L/D ratio.
Finally, one practical conclusion that suggests itself is that the present
mathematical model of fluid flow distribution, together with energy and mass
balances equations and other rate processes equations, may be expected to
be useful in process design and process optimization of packed bed systems,
in general.
227
REFERENCES
Agarwal, P.K., 1988, Chem. Engng. Sci, 43, 2501-2510.
Agarwal, P.K., Mitchell, W.J. and Nauze, R.D.L., 1988, Chem. Engng.
Sci, 43, 2511-2521.
Agarwal, P.K. and O'Neill, B.K., 1988, Chem. Engng. Sci, 43, 2487-2499
Arthur, J.R., Linnett, J.W., Raynor, E.J. and Sington, E.P.C., 1950, Tr
Faraday Soc, 46, 270-281.
Atkinson, B., Brocklebank, M.P., Card, C.C.H. and Smith, J.M., 1969,
AlChEJ., 15, 548-553.
Beavers, G.S. and Joseph, D.D., 1967, J. Fluid Mech., 30, 197-207.
Benard, C.J., 1988, Handbook of Fluid Flow Metering, The Trade and
Technical Press Ltd., Surrey.
Benedict, R.P., 1977, Fundamentals of Temperature, Pressure, and Flow
Measurements, 2nd Ed., John Wiley and Sons, New York.
Benenati, R.F. and Brosilow, C.B., 1962, AlChEJ., 8, 359-361.
Berman, N.S. and Santos, V.A., 1969, AlChEJ., 15, 323-327.
Bey, O. and Eigenberger, G., 1997, Chem. Engng. Sci, 52, 365-1376.
Bird, R.B., 1965, Proceeding of the A.I.Ch.E.-l.Chem.E. Symposium
Series, No. 4, London, pp. 4.3-4.13.
Bird, R.B., Stewart, W.E. and Lightfoot, E.N., 1960, Transport
Phenomena, John Wiley and Sons, Inc. New York.
228
Bibliography
Blevins, R.D., 1984, Applied Fluid Dynamics Handbook, Van Nostrand
Reinhold Co., Melbourne.
Blum, E.H. and Wilhelm, R.H., 1965, Proceeding of the A.I.Ch.E.l.Chem.E. Symposium Series, No. 4, London, pp. 4.21-4.27.
Bo, M.K., Freshwater, D.C. and Scarlett, B., 1965, Trans. Instn. Chem.
Engr., 43, T228-T232.
Bolton, W., 1996, Instrumentation and Measurement, 2nd ed., Newnes,
Oxford.
Brady, J.F., 1984, Phys. Fluids, 27, 1061-1067.
Brinkman, H.C., 1947, Appl. Sci. Res. Sect. A1, 27-34.
Brown, G.G., Foust, A.S., Katz, D.L., Schneidewind, R., White, R.R.,
Wood, W.P., Brown, G.M., Brownell, L.E., Martin, J.J., Williams,
G.B., Banchero, J.T. and York, J.L., 1950, Unit Operations, John
Wiley and Sons, Inc., New York.
Buchlin, J.M., Riethmuller, M. and Ginoux, J.J., 1977, Chem. Engng. Sci
32, 1116-1119.
Burden, R.L., and Faires, J.D., 1993, Numerical Analysis, 5th. Ed., PWS
Publishing Co., Boston.
Burnett, D.S., 1987, Finite Element Analysis, from Concepts to
Applications, Addison-Wesley Publishing Co., Massachusetts.
Bowlus, D.A., and Brighton, J.A., 1968, J. Basic Engng., 90, 431-433.
Cairns, E.J. and Prausnizt, J.M., 1959, Ind. Engng. Chem., 51, 14411444.
229
Bibliography
Chen, R.Y., 1973, J. Fluids Engng., 95, 153-158.
Cheng, P. and Hsu, C.T., 1986, Int. J. Heat Mass Transfer, 29, 18431853.
Cheng, P. and Vortmeyer, D., 1988, Chem. Engng. Sci, 43, 2523-2532.
Choudhary, M., Szekely, J. and Weller, S.W., 1976a, AlChEJ., 22, 10211027.
Choudhary, M., Szekely, J. and Weller, S.W., 1976b, AlChEJ., 22, 10271032.
Chow, C.Y., 1979, An Introduction to Computational Fluid Mechanics,
John Wiley and Sons, New York.
Christiansen, E.B. and Lemmon, H.E., 1965, AlChEJ., 11, 995-999.
Christopher, R.H. and Middleman, S., 1965, Ind. Engng. Chem. Fund., 4
422-426.
Cohen, Y. and Metzner, A.B., 1981, AlChE J., 27, 705-715.
Courant, R. and Hilbert., D., 1953, Methods of Mathematical Physics,
Vol. 1, Interscience Publishers, Inc., New York.
Cumberland, D.J. and Crawford, R.J., 1987, The Packing of Particles,
Elsevier, Amsterdam.
Davies, O.L. and Goldsmith, P.L., 1977, Statistical Methods in Researc
and Production, 4th ed., Longman Group Limited, London.
Debbas, S. and Rumpf, H., 1966, Chem. Engng. Sci, 21, 583-607.
Delmas, H. and Froment, G.F., 1988, Chem. Engng. Sci, 43, 2281-2287.
230
Bibliography
Dorweiler, V.P. and Fahein, R.W., 1959, AlChEJ., 5, 139-144.
Durlofsky, L. and Brady, J.F., 1987, Phys. Fluids, 30, 3329-3341.
Durlofsky, L. and Brady, J.F., 1984, Phys. Fluids, 27, 3329-3341.
Ergun, S., 1952, Chem. Engng. Prog., 48, 89-94.
Ergun, S. and Orning, A.A., 1949, Ind. Engng. Chem., 41,1179-1184.
Euzen, J.P., Trambouze, P. and Wauquier, J.P., 1993, Scale-Up
Methodology for Chemical Processes, Gulf Publishing Co., Paris.
Fahien, R.W. and Stankovic, I.L., 1979, Chem. Engng. Sci, 34, 350-1354
Fayed, M.E. and Otten, L., 1984, Handbook of Powder Science and
Technology, Van Nostrand Reinhold Co., New York.
Fleming, R., 1958, Scale-Up in Practice, Chapman and Hall, LTD, London
Foscolo, P.U., Gibilaro, L.G. and Waldram, S.P., 1983, Chem. Engng.
Sci, 38, 1251-1260.
Foust, A.S., Wenzel, L.A., Clump, C.W., Maus, L. and Andersen, L.B.,
1960, Principles of Unit Operations, John Wiley and Sons, Inc., New
York.
Franks, R.G., 1972, Modeling and Simulation in Chemical Engineering,
John Wiley and Sons, Inc., New York.
Furnas, CC, 1931, Ind. Engng. Chem., 23, 1052-1058.
Gauvin, W.H. and Katta, S., 1973, AlChEJ., 19, 775-783
German, R.M., 1989, Particle Packing Characteristics, Metal Powder
Industries Federation, New Jersey.
231
Bibliography
German, R.M., 1981, Powder Technoi, 30, 81-86.
Givler, R.C. and Altobelli, S.A., 1994, J. Fluid Mech., 258, 355-370.
Goodling, J.S., Vachon, R.I., Stelpflug, W.S., Ying, S.J. and Khader,
1983, Powder Technoi, 35, 23-29.
Gotoh, K., Jodrey, W.S. and Tory, E.M., 1978, Powder Technoi, 20, 257260.
Govindarao, V.H. and Froment, G.F., 1986, Chem. Engng. Sci, 41, 533539.
Govindarao, V.H. and Ramrao, K.V.S., 1988, Chem. Engng. Sci, 43,
2544-2545.
Govindarao, V.H., Subbanna, M., Rao, A.V.S. and Ramrao, K.V.S., 1990,
Chem. Engng. Sci, 45, 362-364.
Haughey, D.P. and Beveridge, G. S. G., 1966, Chem. Engng. Sci, 21,
905-916.
Himmelblau, D.M. and Bischoff, K.B.,1968, Process Analysis and
Simulation Deterministic Systems, John Wiley and Sons, Inc.,
New York.
Hoftyzer, P.J., 1957, Proceeding of the Joint Symposium the Scaling-U
Chemical Plant and Processes, Instn. Chem. Engrs., London, pp.
S73-S77.
Jenson, V.G. and Jeffreys, G.V., 1963, Mathematical Methods in Chemic
Engineering, Academic Press, London.
232
Bibliography
Johnson, G.W. and Kapner, R.S., 1990, Chem. Engng. Sci, 45, 329-339.
Johnstone, R.E. and Thring, M.W., 1957, Proceeding of the Joint
Symposium the Scaling-Up of Chemical Plant and Processes, Instn.
Chem. Engrs., London, pp. S7-S8.
Kalthoff, 0. and Vortmeyer, D., 1980, Chem. Engng. Sci., 35,1637-164
Kececioglu, I. and Jiang, Y., 1994, J. of Fluids Engng., 116, 165-170
Keller, H.B., 1968, Numerical Methods for Two-Point Boundary Value
Problems, Blaisdell Publishing Co., Waltham.
Kondelik, P., Horak, J. and Tesarova, J., 1968, Ind. Engng. Chem. Pr
Des. Dev., 7, 250-252.
Kubie, J., 1988, Chem. Engng. Sci, 43, 1403-1405.
Kufner, R. and Hofmann, H, 1990, Chem. Engng. Sci, 45, 2141-2146.
Lamb, D.E. and Wilhelm, R.H., 1963, Ind. Engng. Chem. Fund., 2, 173182
Langhaar, H.L., 1942, J. Appl. Mech., 9, A55-58.
Larson, R.E. and Higdon, J.J.L., 1986, J. Fluid Mech., 166, 449-472.
Leitzelement, M., Lo, CS. and Dodds, J., 1985, Powder Technoi, 41,
159-164.
Lerou, JJ. and Froment, G.F., 1977, Chem. Engng. Sci, 32, 853-861.
Leva, M., 1992, Chem. Engng. Prog., 88, 65-72.
Levenspiel, O., 1984, Engineering Flow and Heat Exchange, Plenum
Press, New York.
233
Bibliography
Lundgren, T.S., 1972, J. Fluid Mech., 51, 273-299.
MacDonald, M.J., Chu, C.F., Guilloit, P.P. and Ng, K.M., 1991, AlChE
37,1583-1588.
MacDonald, I.F., El-Sayed, M.S., Mow, K. and Dullien, F.A.L., 1979,
Engng. Chem. Fund. 18, 199-208.
Macrae, J.C. and Gray, W.A., Brit. J. Appl. Phys., 12, 164-172.
Martin, H., 1978, Chem. Engng. Sci, 33, 913-919.
Masuoka, T. and Takatsu, Y., 1996, Int. J. Heat Mass Transfer, 39, 28
2809.
McGreavy, C. and Cresswell, D.L., 1968, Proceeding of the Fourth
European Symposium of Chemical Reaction Engineering, 9-11
Sept., Brussels, pp. 59-71.
McGreavy, C, Foumeny, E.A. and Javed, K.H., 1986, Chem. Engng. Sci,
41, 787-797.
Mehta, D. and Hawley, M.C, 1969, Ind. Engng. Chem. Proc. Des. Dei/.,
280-282.
Merwe, D.F.V.D. and Gauvin, W.H., 1971a, AlChEJ., 17, 402-409.
Merwe, D.F.V.D. and Gauvin, W.H., 1971b, AlChEJ., 17, 519-528.
Mickley, H.S., Smith, K.A. and Korchak, E.I., 1965, Chem. Engng. Sci
237-246.
Mickley, H.S.,Sherwood, T.K. and Reed, C.E., 1957, Applied Mathemati
in Chemical Engineering, 2nd Ed., McGraw-Hill Book Co., New York.
234
Bibliography
Miconnet, M., Guigon, P. and Large, J.F., 1982, Int. Chem. Engng., 22
133-141.
Morales, M., Spinn, C.W. and Smith, J.M., 1951, Ind. Engng. Chem., 43
225-232.
Morkel, W.S.T., and Dippenaar, R.J., 1992, 1(f PTD Conference
Proceedings, pp. 77-89.
Mueller, G.E., 1993, Powder Technoi, 77, 313-319.
Mueller, G.E., 1992, Powder Technoi, 72, 269-275.
Mueller, G.E., 1991, Chem. Engng. Sci, 46, 706-708.
Murphy, G., 1950, Similitude in Engineering, The Ronald Press Co.,
New York.
Murphy, D.E. and Sparks, R.E., 1968, l&EC Fund., 7, 642-645.
Newell, R.,1971, Velocity Distribution in Packed Beds of Rectangular
Geometry, M.Sc. Thesis, University of New South Wales, Australia.
Newell, R., and Standish, N., 1973, Met. Trans., 4B, 1851-1857.
Nield, D.A., 1983, AlChEJ., 29, 688-689.
Nield, D.A., Juqueira, S.L.M. and Lage, J.L., 1996, J. Fluid Mech., 3
201-214.
Ouchiyama, N. and Tanaka, T., 1989, Ind. Engng. Chem. Res., 20, 66-71
Ouchiyama, N. and Tanaka, T, 1981, Ind. Engng. Chem. Fund., 20, 6671.
Bibliography
Ouchiyama, N. and Tanaka, T., 1980, Ind. Engng. Chem. Fund., 19, 33
340.
Ouchiyama, N. and Tanaka, T., 1975, Ind. Engng. Chem. Proc. Des. De
14,286-289.
Ouchiyama, N. and Tanaka, T., 1974, Ind. Engng. Chem. Proc. Des. De
13, 383-389.
Perry, A.E., 1982, Hot-wire Anemometry, Clarendon Press, Oxford.
Perry, R.H. and Green, D., 1984, Perry's Chemical Engineers' Handboo
6th ed., McGraw-Hill Book Co., New York.
Peter, M.S. and Timmerhaus, K.D., 1968, Plant Design and Economics f
Chemical Engineers, 2nd Ed., McGraw-Hill Book Co., New York.
Pillai. K.K., 1977, Chem. Engng. Sci, 32, 59-61.
Poirier, D.R. and Geiger, G.H., 1994, Transport Phenomena in Materia
Processing, TMS, Pennsylvania.
Poveromo, J.J., Szekely, J. and Propster, M., 1975, Proceedings of B
Furnace Aerodynamics Symposium, Wollongong, pp. 1-8, 171-172.
Prausnitz, J.M. and Wilhelm, R.H., 1957, Ind. Engng. Chem., 49, 978Price, J., 1968,. Mech. Chem. Eng. Trans. Aust, MC4, 7-14.
Propster, M. and Szekely, 1977, Powder Technoi, 17, 123-138.
Puncochar, M. and Drahos, J., 1993, Chem. Engng. Sci, 48., 2173-2175
Ranz, W.E., 1952, Chem. Eng. Prog., 48, 247-253.
Bibliography
Reid, R.C. and Sherwood, T.K., 1966, The Properties of Gases and
Liquids, Their Estimation and Correlation, 2nd ed., McGraw-Hill
Book Co., New York.
Reid, R.C, Prausnizt, J.M. and Sherwood, T.K., 1977, The Properties o
Gases and Liquids, 3rd ed., Mc Graw-Hill Book Co., New York.
Rhodes, M., 1990, Principles of Powder Technology, John Wiley and
Sons, Chichester.
Ridgway, K., and Tarbuck, K.J., 1968a, Chem. Engng. Sci, 23,1147-1155.
Ridgway, K., and Tarbuck, K.J., 1968b, Chem. Proc. Engng., 49, 103-105
Ridgway, K., and Tarbuck, K.J., 1967, Brit. Chem. Engng., 12, 384-388.
Roblee, L.H.S., Baird, R.M. and Tierney, J.W.,1958, AlChEJ, 4, 460-464
Saleh, S., Thovert, J.F. and Adler, P.M., 1993a, Chem. Engng. Sci, 48,
2839-2858.
Saleh, S., Thovert, J.F. and Adler, P.M., 1993b, AlChEJ., 39, 1765-177
Saunders, O.A. and Ford, H., 1940, J. Iron Steellnst, CXLI, 291p-329p.
Schertz, W.W. and Bischoff, K.B., 1969, AlChEJ., 15, 597-604.
Schuring, D.J., 1977, Scale Models in Engineering Fundamentals and
Applications, Pergamon Press, Oxford.
Schmidt, F.W. and Zeldin, B., 1969, AlChEJ., 15, 612-614.
Schwartz, CE. and Smith, J.M., 1953, Ind. Engng. Chem., 45, 1209-1218.
Scott, G.D., 1962, Nature, 194, 4831-4832.
237
Bibliography
Scott, G.D. and Kovacs, G.J., 1973, J. Phys. D: Appl. Phys., 6, 10071010.
Slattery, J.C, 1969, AlChEJ., 15, 866-872.
Slattery, J.C, 1968, AlChEJ., 14, 50-56.
Smith, J.M. and Van Ness, H.C., 1975, Introduction to Chemical
Engineering Thermodynamics, 3rd Ed., McGraw-Hill, New York.
Standish, N., 1990, Bulk Density of Coal, A Booklet of NERDDP Project,
University of Wollongong, Australia.
Standish, N., 1984, Chem. Engng. Sci, 39, 1530.
Standish, N., 1979, Principles in Burdening and Bell-Less Charging,
Nimaroo Publisers, Wollongong.
Standish, N. and Borger, D.E., 1979, Powder Technoi, 22, 121-125.
Standish, N. and Collins, D.N., 1983, Powder Technoi, 36, 55-60.
Standish, N. and Leyshon, P.J., 1981, Powder Technoi, 30, 119-121.
Standish, N. and Yu, A.B., 1987a,Powcter Technoi, 49, 249-253.
Standish, N. and Yu, A.B., 1987b,Powder Technoi, 53, 69-72.
Stanek, V. and Szekely, J., 1974, AlChEJ., 20, 974-980.
Stephenson, G., 1973, Mathematical Methods for Science Students, 2nd
Ed., Longman Scientific and Technical, London.
Stephenson, J.L. and Stewart, W.E., 1986, Chem. Engng. Sci, 41, 21612170.
Szekely, J. and Kajiwara, Y., 1979, Met. Trans., 10B, 447-453.
23X
Bibliography
Szekely, J. and Poveromo, J.J., 1975, AlChEJ., 21, 769-775.
Szekely, J. and Propster, M., 1977, Ironmaking & Steelmaking, 1, 15Szucs, E.. 1980, Similitude and Modeling, Elsevier Sci. Publishing
Amsterdam.
Tao, B.Y., 1988a, Chem. Engng., 95, 107-110.
Tao, B.Y., 1988b, Chem. Engng., 95, 85-92.
Tao, B.Y., 1987a, Chem. Engng., 94, 109-113.
Tao, B.Y., 1987b, Chem. Engng., 94, 145-148.
Thadani, M.C. and Peebles, F.N., 1966, Ind. Engng. Chem. Proc. Des.
Dev., 5, 265-268.
Tsotsas, E. and Schlunder, E.U., 1990, Chem. Engng. Sci, 45, 819-83
Tsotsas, E. and Schlunder, E.U., 1988, Chem. Engng. Sci, 43, 12001203.
Vortmeyer, D. and Haidegger, E., 1991, Chem. Engng. Sci, 46, 26512660.
Vortmeyer, D. and Michael, K., 1985, Chem. Engng. Sci, 40, 2135-213
Vortmeyer, D. and Schuster, J., 1983,C..em. Engng. Sci, 38, 1691-16
Vortmeyer, D. and Winter, R.P.,1984, Chem. Engng. Sci, 39, 1430-143
Vrentas, J.S. and Duda, J.L., 1967, AlChEJ., 13, 97-101.
Vrentas, J.S., Duda, J.L and Bargeron, K.G., 1966, AlChE J., 12, 837
844.
Wasan, D.T. and Baid, K.M., 1971, AlChE J., 17, 729-731.
239
Bibliography
White, F.M., 1986, Fluid Mechanics, 2nd Ed., McGraw-Hill Book Company,
New York.
Wilkinson, D., 1985, Phys. Fluids, 28, 1015-1022.
Williams, J.C, 1976, Powder Technoi, 15, 245-251.
Wit, A. D., 1995, Phys. Fluids, 7, 2553-2562.
Yu, A.B. and Standish, N., 1993a, Powder Technoi, 76, 113-124.
Yu, A.B. and Standish, N., 1993b, Powder Technoi, 74, 205-213.
Yu, A.B. and Standish, N., 1993°, Ind. Eng. Chem. Res, 32, 2179-2182.
Yu, A.B. and Standish, N., 1991, Ind. Eng. Chem. Res, 30, 1372-1385.
Yu, A.B. and Standish, N., 1988, Powder Technoi, 55, 171-186.
Yu, A.B. and Standish, N., 1987, Powder Technoi, 49, 249-253.
Yu, A.B. and Zulli, 1994, Powder Handling Process., 6, 171-177.
Yu, A.B., Standish, N. and McLean, A., 1993, J. Am. Ceram. Soc.,76,
2813-2816.
Ziolkowska, I. And Ziolkowski, D., 1993, Chem. Engng. Sci, 48, 32833292.
Ziolkowski, D. and Szustek, S., 1989, Chem. Engng. Sci, 44,1195-1204.
Zlokarnik, M., 1991, Dimensional Analysis and Scale-up in Chemical
Engineering, Springer-Verlag, Berlin.
Zou, R.P. and Yu, A.B., 1996, Powder Technoi, 88, 71-79.
240
APPENDIX
ALGORITHMS OF VELOCITY PROFILE CALCULATION
241
INPUT
bed properties
fluid properties
superficial velocity
COMPUTE
pressure drop
GUESS
COMPUTE
radial porosity distribution
O
COMPUTE
radial velocity profile
CHECK
total mass-flow rate
UPDATE
Figure A-1: Flow diagram of the velocity profile calculation by the
present model.
242
INPUT
bed properties
fluid properties
superficial velocity
GUESS
uz at r = 0
COMPUTE
radial porosity distribution
O
COMPUTE
radial velocity profile
CHECK
total mass-flow rate
UPDATE
uz at r = 0
Figure A-2: Flow diagram of the velocity profile calculation by
Vortmeyer and Schuster [1983] model.
243
INPUT
bed properties
fluid properties
superficial velocity
COMPUTE
GUESS
pressure drop
Ks
COMPUTE
radial porosity distribution
o
COMPUTE
radial velocity profile
CHECK
total mass-flow rate
UPDATE
No
OUTPUT
uz=/(r)
ure A-3:Flow diagram of the velocity profile calculation by Stanek
and Szekely [1974] model.