AN ABSTRACT OF THE THESIS OF
Pajongwit Pongsivapai for the degree of Master of Science in Chemical
Engineering presented on January 14, 1994.
Title: Residence Time Distribution of Solids in a Multi-Compartment
Fluidized Bed System.
Abstract approved:_
Redacted for Privacy
VV
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In a conventional well-mixed fluidized bed the solids are almost
completely mixed and the residence time for individual particles may vary
between zero and infinity. For materials especially sensitive to the processing
time, a wide distribution of residence time is extremely undesirable. A multicompartment fluidized bed was proposed to minimize this problem. The twocompartment fluidized bed system was designed and experimentally
investigated with positive results. Residence time distributions were
measured with glass bead particles of 379 p.m mean diameter size. Tracer
particles were colored glass beads which had exactly the same properties as the
particles used for bed material. Two theoretical models were proposed to
predict the flow behavior of solids through the two-compartment fluidized
bed vessel. The results show the solids flow pattern can be described by the
axial dispersion plug flow model and the tanks-in-series model. The
parameters of both models can be determined by minimizing the sum of the
squared differences between experimental data and model predictions curves.
The influence of fluidization velocity, diameter of the orifice connecting the
two adjacent compartments, and the height of the overflow exit orifice were
investigated. The two-compartment fluidized bed can improve the residence
time distribution of solids from a single fluidized bed. The residence time
distribution of solids was also improved with decreasing fluidizing gas
velocity. Orifice diameter and height of the overflow exit orifice did not
influence significantly on particles flow in the two-compartment fluidized
bed system. However, a solids mass flow through the partition orifice is
predicted reasonably well, within moderate range of variance, by equation
developed by de Jong (1965) [12].
Residence Time Distribution of Solids
in
a Multi-Compartment Fluidized Bed System
by
Pajongwit Pongsivapai
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirement for the
degree of
Master of Science
Completed January 14, 1994
Commencement June 1994
APPROVED:
Z)
Redacted for Privacy
Professor op enemiczngineering in cnarge of major
_
in
Redacted for Privacy
Head of Dtefiartnient of Chemical Engineering
Redacted for Privacy
Dean of th. .... . .0..6-
0
Date thesis is presented
January 14,1994
Typed by Pajongwit Pongsivapai for Pajongwit Pongsivapai
TABLE OF CONTENTS
CHAPTER 1 : INTRODUCTION
1.1 Objectives
1
CHAPTER 2 : LITERATURE REVIEW
8
CHAPTER 3 : THEORETICAL APPROACHES
Tracer techniques
3.2 Residence time distribution (RTD) analysis
3.3 Model of flow of solids in a non-ideal system
3.3.1 Axial dispersion plug flow model
3.3.2 Tanks-in-series model
3.4 Model of the fluidized solids flow through the orifice
3.1
CHAPTER 4 : EXPERIMENTAL EQUIPMENT AND PROCEDURES
4.1
Experimental apparatus
4.1.1 Fluidized bed
4.1.2 Fluidized bed material
4.1.3 Air supply system
4.1.4 Pressure measuring system
4.1.5 Feed system
4.1.6 Tracers
4.2
4.3
4.4
Experimental mechanism
Experimental procedures
Experimental plan
CHAPTER 5 : RESULTS AND DISCUSSION
6
15
15
16
21
21
26
29
34
34
34
35
36
36
37
37
37
39
41
43
5.1
5.2
5.3
5.4
5.5
5.6
The measurement of minimum fluidizing velocity
Tracer response curves
Axial dispersion plug flow modeling
Tanks-in-series modeling
The effect of the fluidized bed comparmentation
The effect of the orifice size in the partition
5.7 The effect of the height of exit orifice
43
44
53
63
73
76
78
5.8 The effect of fluidizing gas velocity
5.9 The modeling of solid flow through the orifice
81
5.10 Tracer recovery
CHAPTER 6 : CONCLUSIONS AND RECOMMENDATIONS
Conclusions
6.2 Recommendations for future work
6.1
BIBLIOGRAPHY
84
85
86
86
87
88
APPENDICES
Appendix 1 : Dispersion program code
Appendix 2 : The paired t-analysis
Appendix 3 : Calculation of minimum fluidizing velocity
Appendix 4 : Percentiles of t-distributions
Appendix 5 : Gamma function table
Appendix 6 : Apo and W
91
98
103
104
105
106
LIST OF FIGURES
Page
Figure
1-1
Flow regime in fluidized bed vessel with increasing
number of compartments
3
1-2
Solids flow from dense bed to lean bed
4
3-1
E curve for solids flowing through a vessel, RTD curve
17
3-2
F curve, the cumulative distribution curve
17
3-3
E curves in closed-closed vessels for various extents
of dispersion
26
3-4
E curves for various extents of number of tanks in series
29
3-5
The illustration of solid flow through the orifice
30
4-1
Schematic of apparatus for the two-compartment
fluidized bed system
38
4-2
Concentration of tracer versus absorbance calibration curve
41
5-1
Pressure drop across a fluidized bed versus gas
fluidizing velocity of glass beads (379 gm)
43
5-2
C curve of experiment #1
45
5-3
C curve of experiment #2
45
5-4
C curve of experiment #3
46
5-5
C curve of experiment #4
46
5-6
C curve of experiment #5
47
5-7
C curve of experiment #6
47
5-8
C curve of experiment #7
48
5-9
C curve of experiment #8
48
5-10
C curve of experiment #9
49
Page
Figure
5-11
C curve of experiment #10
49
5-12
C curve of experiment #11
50
5-13
C curve of experiment #12
50
5-14
C curve of experiment #13
51
5-15
C curve of experiment #14
51
5-16
C curve of experiment #15
52
5-17
C curve of experiment #16
52
5-18
C curve of experiment #17
53
5-19
Optimization approach on experiment #9
54
5-20
Dispersion modeling for experiment #1
55
5-21
Dispersion modeling for experiment #2
55
5-22
Dispersion modeling for experiment #3
56
5-23
Dispersion modeling for experiment #4
56
5-24
Dispersion modeling for experiment #5
57
5-25
Dispersion modeling for experiment #6
57
5-26
Dispersion modeling for experiment #7
58
5-27
Dispersion modeling for experiment #8
58
5-28
Dispersion modeling for experiment #9
59
5-29
Dispersion modeling for experiment #10
59
5-30
Dispersion modeling for experiment #11
60
5-31
Dispersion modeling for experiment #12
60
5-32
Dispersion modeling for experiment #13
61
5-33
Dispersion modeling for experiment #14
61
5-34
Dispersion modeling for experiment #15
62
Page
Figure
5-35
Dispersion modeling for experiment #16
62
5-36
Dispersion modeling for experiment #17
63
5-37
Tanks-in-series modeling for experiment #1
64
5-38
Tanks-in-series modeling for experiment #2
64
5-39
Tanks-in-series modeling for experiment #3
65
5-40
Tanks-in-series modeling for experiment #4
65
5-41
Tanks-in-series modeling for experiment #5
66
5-42
Tanks-in-series modeling for experiment #6
66
5-43
Tanks-in-series modeling for experiment #7
67
5-44
Tanks-in-series modeling for experiment #8
67
5-45
Tanks-in-series modeling for experiment #9
68
5-46
Tanks-in-series modeling for experiment #10
68
5-47
Tanks-in-series modeling for experiment #11
69
5-48
Tanks-in-series modeling for experiment #12
69
5-49
Tanks-in-series modeling for experiment #13
70
5-50
Tanks-in-series modeling for experiment #14
70
5-51
Tanks-in-series modeling for experiment #15
71
5-52
Tanks-in-series modeling for experiment #16
71
5-53
Tanks-in-series modeling for experiment #17
72
5-54
Comparison of single- and two-compartment RTD curves
73
5-55
Effect of compartmentation by axial dispersion
plug flow model
74
5-56
Effect of compartmentation by tanks-in-series model
75
Figure
Page
5-57
Effect of orifice size by tanks-in-series model
76
5-58
Effect of orifice size by axial dispersion plug flow model
77
5-59
The E curves conducted with different height of
exit orifice
78
5-60
Effect of exit orifice height by axial dispersion plug flow model
79
5-61
Effect of exit orifice height by tanks-in-series model
80
5-62
Effect of fluidizing gas velocity by tanks-in-series model
81
5-63
Effect of fluidizing gas velocity by axial dispersion
plug flow model
82
5-64
Measured versus predicted mass flow rate of solids
through the orifice
84
5-65
The percent recovery of tracer
85
A2-1
t-distribution curve
99
A6-1 Pressure drop across the orifice for 16 experiments
106
A6-2 Weight of solid hold-up after finishing 16 experiments
107
LIST OF TABLES
Em
Table
2-1
Tracers used in a liquid-carrier system
2-2
Tracers used in a gas-carrier system
11
2-3
Tracers used in a two-phase system
13
2-4
Tracers used in a solid system
14
3-1
Dispersion models, Bischoff and Levenspiel
4-1
The glass beads material size distribution
36
4-2
The physical characteristics of bed material
36
4-3
Experimental plan
42
A2-1
Effect of exit orifice height when d.= 5.1 mm
100
A2-2
Effect of exit orifice height when d.= 3.5 mm
100
A2-3
Effect of orifice size when h = 10 cm
100
A2-4
Effect of orifice size when h =
cm
100
A2-5
Effect of exit orifice height when d.=
5.1
mm
101
A2-6
Effect of exit orifice height when d.=
3.5
mm
101
A2-7
Effect of orifice size when h = 10 cm
101
A2-8
Effect of orifice size when h = 20 cm
101
20
8
(1962) [14]
22
NOMENCLATURES
A
Surface area of orifice, m2
Ab
Crossectional area of bed, m2
C
Tracer concentration, kg of tracer/kg of sample
CD
Orifice discharge coefficient, s.(m/kg)"
CD
Modified orifice discharge coefficient, s.(m/kg)"
dp
Particle diameter, m
do
Orifice diameter, m
Tube diameter, m
D
Dispersion coefficient, m2/s
Di,
Axial dispersion coefficient, dispersion plug flow model, m2/s
D'L
Axial dispersion coefficient, axial dispersion plug flow model, m2/s
Db.
Mean value of DL(R), m2/s
D'Ln,
Axial dispersion coefficient, uniform dispersion model, m2/s
DL(R) Axial dispersion coefficient at radial position R,
general dispersion in cylindrical co-ordinates, m2/s
DR
Radial dispersion coefficient, dispersion plug flow model, m2/s
DR,
Mean value of DR(R), m2/s
D 'IN,
Radial dispersion coefficient, uniform dispersion model, m2/ s
DR(R) Radial dispersion coefficient at radial position R,
general dispersion in cylindrical co-ordinates, m2/s
E(t)
Exit age distribution function, s"'
E(0)
Exit age distribution function, dimensionless
F,
Flow rate of solids, kg/s
g
Gravitational acceleration, m/s2
h
Height of exit orifice (measure from distributer plate), m
hd
Height of dense bed, m
hi
Height of lean bed, m
L
Distance between measurement points for dispersion experiment
M
Amount of tracer introduced into the vessel, kg
N
Number of tanks in series
Apb
Pressure drop cross the bed of solids, Pa
Apo
Pressure drop cross the orifice, Pa
R
Radial position, m
r,
Rate of chemical reaction, mole/time-volume
S
Source term
t
Time, s
Mean residense of solids in vessel, s
Average interstitial velocity of fluid in axial direction, m/s
Fluidizing gas velocity, m/s
u-und Excess gas velocity, m/s
U-U
Relative excess gas velocity, m/s
mf
uL
Dispersion number
2D
V
Volume of tank, m3
W
Average mass of soids in the vessel, kg
x
Axial position measured from start of test section, m
Greek symbols
0
.t
e
Porosity or fraction voids in a fluidized bed, m3 gas/m3 bed
A
Solids density, kg/m3
Os
Flow rate of solids through the orifice, kg/s
t
,
reduced time, dimensionless
Subscripts
b
Bed
d
Dense bed
f
At bubbling fluidizing condition
/
Lean bed
mf
At minimum fluidizing condition
RESIDENCE TIME DISTRIBUTION OF SOLIDS
IN
A MULTI-COMPARTMENT FLUIDIZED BED SYSTEM
CHAPTER 1
INTRODUCTION
Fluidization is a subject which engineers and scientists have been
studying since its impact on industrial application of a catalytic cracking
process in 1942. Kunii and Levenspiel (1991) [27] define the term gas
fluidization as "the operation by which solid particles are transformed into a
fluidlike state through suspension in a gas". This mechanism of gas-solid
contacting has some unusual characteristics which depicts several advantages
of fluidization.
Fluidized bed reactors have been widely acclaimed in the process
industry for their highly advantageous characteristics of good mixing and
contacting, high rates of heat and mass transfer, mechanical robustness, and
capability of continuous operation. However, fluidized bed also has
undesirable characteristics. The rapid mixing of solids in the bed gives
nonuniform residence times of solids in continuous gas-solid flow vessels
which is a consequence of individual solid particles having different lengths
of stay in the bed. For continuous treatment of solids this leads to a
nonuniform product and lower performance especially at high conversion
levels. For example, in particle drying processes, the wide distribution of
residence times yields a variation in moisture concentration in dried-particle
products. The effect may be uneven and too high final moisture content.
2
This is a disadvantage especially when the time to achieve the desired level of
moisture content is longer than the mean residence time. Solids residing too
long in a dryer may deteriorate if the material of solids is sensitive to heat.
Furthermore, in catalytic reactions, the movement of porous catalyst partides
which continually capture and release reactant gas molecules, contributes to
the backmixing of gaseous reactant, thereby reducing yield and performance,
Kunii and Levenspiel (1991) [27].
The fundamental idea of this research is the improvement of residence
time distribution of solid particles flowing through a fluidized bed. From the
viewpoint of the residence time distribution (RTD) of solids the flow of solids
through a single-compartment fluidized bed vessel can be considered as flow
through a single ideal mixed vessel, Kunii and Levenspiel (1991) [27]. The
best RTD of solids that we can expect in a single-compartment fluidized bed is
the RTD obtained in an ideal mixed vessel. Very often, due to the short
circuiting of solid flow, between inlet and outlet, much less favorable RTD's
are obtained. However, our goal is to obtain RTD which are much more
favorable than RTD for ideal mixing. To create conditions where all particles
may have the same residence time, a basic idea of plug flow is considered.
Plug flow is defined as the flow where all solids have the same residence time
in the vessel. There is no diffusion relative to the solid bulk flow and no
backmixing of solids of different ages. For most gas-solid reactions where
high conversion of solids is expected the idea conditions of plug flow of solids
is desirable.
One way to approach the plug flow of solids is to connect several
fluidized beds in series. Then the RTD is narrowed and moved towards that
3
(a)
(b)
(c)
(d)
Figure 1-1 Flow regime in fluidized bed vessel with increasing
number of compartments
4
have been reported in literature (Guanglin et a/.(1985)[18] and Kato et
ai.(1985)[24]) design to overcome the unfavorable backmixing. This idea is
illustrated in Figure 1-1. Figure 1-la shows a possibility of bypassing the flow
of solids which can be prevented by partitioning a single fluidized bed vessel
as shown in Figure 1-1b, c and d. This approach can reduce a gross
backmixing and improve the flow pattern close to plug flow.
In this study the design of a two-compartment fluidized bed vessel is
proposed to improve the RTD of solids in the fluidized bed vessel. The
schematic of the two-compartment fluidized bed vessel is given in Figure 1-2.
Solids in
Solids out
.A)
4
Gas
Alif i ..."'
Gas
Figure 1-2 Solids flow from dense bed to lean bed.
5
Two fluidized beds are separated from each other by a partition. The
driving force that establishes the flow of solids through the system is
generated from the pressure difference across the orifice surface between two
compartments. From Figure 1-2, bed 1 (left bed) and bed 2 (right bed) will be
refered to as a dense and lean bed respectively. The fresh solids are fed from
the top of the dense bed and discharged through the orifice to the lean bed
then overflow out of the vessel at the exit orifice.
RTD measurements have become an essential analytical tool in the
study of solid flow through fluidized bed vessel. To study the flow pattern in
a vessel it is necessary to have a good method of measuring the residence
time distribution. Tracer technique which has been used by chemical
engineers for almost 50 years is also employed in this work. It is a powerful
tool for determining the parameters of any particular flow model
representing the system. Monitoring and interpreting the actual flow pattern
of individual particles is usually impractical. Therefore, the approach taken is
to postulate a flow model which reasonably approximates real flow, and then
use this flow model for predictive purposes. If a flow model closely reflects a
real situation, its predicted response curves will closely match the tracer-
response curve of the real system. This is only one of the requirements in
selecting a satisfactory model.
The extent of the deviation from ideal flow model will depend mainly
on the geometric characteristics of the bed, the mean residence time of
particles, the fluidizing gas velocity, and the size and density of the particles.
Individual particles of the same size have different lengths of stay within the
fluidized bed. All these effects must be taken into account if we wish to
6
control and predict the behavior of solids that pass through the system.
Besides the main future of the design of the two-compartment fluidized bed
vessel, this research is concerned about the influence of fluidizing gas
velocity, orifice diameter in partition and mean residence time of solids.
1.1 Objectives
The study presented in this thesis focuses on the development of the
design of the multi-compartment fluidized bed reactor and the parameters
influencing its. We are primarily interested in RTD curves of flowing solids
inside the multi-compartment fluidized bed vessel. The RTD curves are
considered to be a measure of a potential performance of the fluidized bed
multiphase (gas-solid) chemical reactor. The specific objectives of the study
are:
1. to improve RTD of solids in a single-compartment fluidized bed
vessel,
2. to investigate the effects of the orifice size connecting adjacent
compartments,
3. to investigate the effects of the bed height, or mean residence time of
solids,
4. to investigate the effects of the fluidizing gas velocity on the RTD of
solids in two-compartment fluidized bed,
5. to develop models to predict the residence time of solids in the muticompartment fluidized bed vessel,
6. to develop an equation expressing the flow rate of solids passing
through the orifice.
7
There are 5 chapters in this thesis. Chapter 2 summarizes a literature
review on the relevant work about tracer techniques. Chapter 3 describes the
theoretical approaches of tracer techniques and the RTD analysis. A detailed
description of experimental apparatus, experimental mechanism,
experimental procedures and experimental plan are discussed in chapter 4.
The results from experiments are discussed in chapter 5. Two models
describing the flow pattern are also presented in this chapter. Finally, chapter
6 contains the conclusions of this study and recommendations for future
work.
8
CHAPTER 2
LITERATURE REVIEW
The idea of applying the distribution of residence time in the analysis
of chemical reactor performance was apparently first systematically attempted
by Mac Mullin, et al .(1935) [33]. It was the case of a number of identical
completely mixed tanks in series. However, this concept was not to be used
until 1952, when Danckwerts (1952) [6] explained how residence-time
distribution functions can be defined and measured for real system under the
assumption that the flow through out the system was steady. The use of the
distribution-functions is illustrated by incorporating them in the calculation
of the performance of reactors and blenders, Danckwerts (1952) [6].
A literature survey on tracers used in different studies is summarized
in Table 2-1, 2-2, 2-3 and 2-4.
Method
of tracer
detection
Conduc-
tivity
salts
Carrier
fluid
Tracer
H2O
NaNO3
Instruments
Thermal or electrical
conductivity cells,
recorder or
potentiometer
Thermal or electrical
NaC1 or
conductivity cells,
KC1 or
Cl- + NaCl recorder or
potentiometer
or HC1
Type of
reactors
References
packed
bed
[40]
fluidized
bed
tubes
tanks
packed
bed
fluidized
[40]
bed
Table 2-1 Tracers used in a liquid-carrier system
[2]
[35]
[40]
[40]
9
Method
of tracer
detection
Conduc-
tivity
salts
References
Carrier
fluid
Tracer
Instruments
Type of
reactors
Ethyl
Acetate
NaC1
tubes
[40]
Na2S203
KC1
Thermal or electrical
conductivity cells,
recorder or
potentiometer
Thermal or electrical
conductivity cells,
recorder or
potentiometer
colorimeter
tanks
[40]
tubes
[40]
+ H2O
black dye
Color and H2O +
40% of
light
sensitive sugar
solution
dyes
1120
Color and H2O
light
sensitive
packed
bed
packed
red dye
bed
KMnO4
tubes
spectrophotometer
fluorescein phototube mlliamper fluidized
bed
recorder
packed
Alizarin
phototube
bed
Saphirol
[40]
fluidized
[31]
blue dye
photoelectric
colorimeter
colorimeter
SES
[40]
[40]
[40]
[31]
bed
dyes
sulphite
spectrophotometer
iodate,
or photograph
tubes
[40]
starch
Color and H2O
sensitive
dyes
Table 2-1 (continued)
light
sensitive
solution
[40]
colorimeter
tubes
10
Method
carrier
fluid
of tracer
detection
Color and 1120
sensitive
dyes
Radioactive
tracers
Titration
and
others
Tracer
Instruments
type of
reactors
sodium per
sulphate,
potassium
iodide,
sodium
thiosulphate,
starch
fi-naphthol
spectrophotometer
tubes
[40]
packed
bed
fluidized
bed
tubes
[40]
or
photograph
spectrophotometer
petrole-
Ba'
u rn
sb124
F120
radioactive Geiger counter
F120
NaCl or
Geiger counter
titration with AgNO3
HC1
Benzoric
acid
H2O
Teepol
Xylene
cyanol FF
solution
benzene ethyl
n-butyrate
Table 2-1 (continued)
References
titration with NaOH
absorbtometer
refractive index
measurements
[40]
[40]
[40]
tanks
fluidized
bed
packed
bed
fluidized
bed
packed
bed
[40]
packed
bed
[40]
[40]
[40]
[40]
[40]
11
Carrier
fluid
Air
Type of
Referenreactors
ces
CO2
thermal conductivity cells packed bed
[40]
packed bed
anemometer
[40]
fluidized bed
[40]
gas chromatograph
tubes
gas analyzer
[40]
packed bed
[40]
thermal conductivity cells packed bed
He
[40]
fluidized bed
[40]
fluidized bed
(40]
spectrometer
thermal conductivity cells gas-liquid
[40]
142
contractor
multistage
(113]
gas chromatograph
fluidized bed
tubes
modified Orsat analyzer
[40]
N2
fluidized bed
[40]
gas analyzer
packed bed
Ammo- gas analyzer
[30]
fluidized bed
[30]
ni a
fluidized bed
[40]
0-scintillation detector
Kr85
packed bed
Hg vapor U.V. absortion
[40]
Tracer
SO2
butane
sulfur
hexafluoride
N2
CO2
H2
KC1
Instruments
measurement
autometer
flame ionization detector
not reported
thermal conductivity
measurement
gas chromatograph
gas chromatograph
potentiometer titration
Table 2-2 Tracers used in a gas-carrier system
packed bed
tubes
fluidized bed
[40]
packed bed
fluidized bed
tubes
tubes
packed bed
fluidized bed
[40]
[40]
[26]
[40]
[16]
[16]
[1]
[1]
12
Carrier
fluid
Tracer
N2
KMnO4
Instruments
spectrophotometer
FeC13
spectrophotometer
C2 H4
katharometer
SF6
katharometer
H2
N2
analytical cell (ionizes the
gases passing through cell)
, electrical conductivity cell
SF6
katharometer
He
1,3
U.V. sensitive photobutadiene electric tube,
spectrophotometer
A
gas chromatograph
gas chromatograph
H2
NH3
gas chromatograph
N2
gas chromatograph
ammonia KC1 or
potentimetric titration
synthesis NaC1
1,3 butadi- 1 butyne
He
ene
A
Octadeane 04
CO2
H2
N2
A
oil gas
H2
Sc46, C'33
Celt 3 0
Table 2-2 (continued)
U.V. sensitive
photoelectric tube
spectrophotometer
gas analyzer
gas chromatograph
gas chromatograph
katharometer
scintillation counter
Type of
reactors
packed bed
packed bed
tube
tube
packed bed
References
[1]
[1]
[13]
[13]
[40]
tube
tube
[13]
tube
tube
tube
tube
packed bed
fluidized bed
[16]
tube
tube
tube
tube
tube
tube
tube
regenerator
[40]
[40]
[16]
[16]
[16]_.
[1]
[1]
[40]
[40]
[40]
[16]
[16]
[13]
[40]
13
Carrier fluid
benzene - H2O
-H2O
Tracer
Instruments
HC1 for H2O ,
electric conductivity
oil red dye for photoelectric cells
benzene
sargent osillometer
He for air,
MgSO4 or KC1 (measures the dielectric
strength titration for
or HCl for
H2O phase
kerosene - H2O NaNO3 for
H2O, oil blue
Air - H2O
HC1 for H2O
isobutanol -
NaNO3 for
H2O
H2O, oil blue
dye for
Refer-
reactors
packed
bed
ences
packed
bed
[40]
solvent
[40]
[40]
H2O phase
electric conductivity
photoelectric cells
extrac-
tion
column
dye for
kerosene
He for Air,
Type of
electric conductivity,
thermal conductivity
cells, millivolt recorder
electric conductivity,
special electric
conductivity cells
isobutanol
Table 2-3 Tracers used in a two-phase system
packed
bed
[40]
packed
bed
[40]
14
Carrier fluid
Tracer
Instruments
Type of reactors
Air
colored Si02
not reported
gas-solid
[18]
multistage
fluidized bed
fluidized bed
[25]
polycarbonate not reported
References
resin
Air - H2O
colored glass
not reported
spheres
Kerosene - H2O oleophilic dye not reported
1120
Benson-Lehner
y- rays
'Oscar' film
darkened
glass particles analyzer
Table 2-4 Tracer used in a solid system
gas-liquid-solid
multistage
fluidized bed
gas-liquid-liquid
multistage
fluidized bed
liquid-solid
fluidized bed
[24]
[24]
[37]
15
CHAPTER 3
THEORETICAL APPROACHES
3.1 Tracer technique
The residence time distribution (RTD) for a flowing fluid can be
determined easily and directly by a widely used method of inquiry, the so-
called "stimulus-response" technique. With this technique, the RTD is
determined experimentally by injecting an inert chemical, particle, molecule
or atom, called a tracer, into the inlet stream at some time t = 0 and then
measuring the corresponding response, the tracer concentration, at the
effluent stream of the reactor as a function of time. Then, a suitable flow
model can then be selected by matching the experimental residence time
curve with that obtained from the mathematical model. To successfully
measure a RTD the tracers must have certain properties. Therefore, the type
of tracer and techniques used in experimental work are of interest. The basic
requirements for a satisfactory tracer experiment can be outlined as follows:
1. The tracer compound should be miscible and have similar physical
properties as the main fluid/solid stream when its mixing behavior
is under investigation.
2. The tracer compound should be accurately detectable in small
concentrations so that the introduction of tracer does not effect the
flow pattern of the main stream.
3. Generally, the tracer should be nonreactive, so the analysis of the
RTD curve can be kept simple.
4. During an experiment in a multiphase system the tracer should not
be transferred from one phase to another phase.
16
5. The tracer itself, its detection device and other auxiliary equipment
should be relatively inexpensive.
3.2 Residence time distribution (RTD) analysis
To predict the performance of a system such as a chemical reactor, we
must know the pattern of fluid/solids motion through the reactor. This
pattern can be determined by finding how long the individual molecule of
flowing fluid/solids stay in the system; however, the complexity of the flow
pattern make this method impractical and often impossible. A better
approach to overcome this problem about the flow behavior in a process is to
find the age distribution of the elements of the fluid/solids in the exit stream
or the residence time distribution (RTD). The RTD describes the distribution
of lengths of time spent by elements of the fluid inside the vessel before
reaching the exit. This is termed as an exit age distribution function (E curve).
The integral form of RTD of solids at the vessel exit can be expressed as a
cumulative distribution function (F curve), Danckwerts (1952) [6].
An exit age distribution function describes the fraction of solids, E(t)dt,
in the exit stream which has resided a time between t and (t + dt) in the
vessel, Figure 3-1. It is the most used of the distribution functions connected
with reactor analysis because it characterizes the lengths of time various
solids spends within the reactor. The other expression has been defined as
the fraction of solids which has spent a time t or less than t , Figure 3-2.
17
Fraction of solids leaving the
/vessel spends time between t
and t+dt in the vessel.
E(t)
Total area under
E curve =1
t+dt
t
Figure 3-1 E curve for solids flowing through a vessel, RTD curve.
1.0
The fraction (F(t) ) of
solids spends time t or
less in the vessel.
F(t)
0.0
t
t
Figure 3-2 F curve, the cumulative distribution curve.
18
Thus, the relation between E and F function is
F(t) = 5 E(t) dt
(3-1)
o
Besides the E and F functions, measured at the exit stream, it is possible
to define, on the other hand, a distribution of residence times for which the
solids currently in the vessel. It is termed as the internal age distribution
function, I(t). It is a function such that I(t)dt is the fraction of solids inside the
vessel that has been inside the vessel for a period of time between t and t+dt .
The function E(t) is viewed outside the vessel and I(t) is viewed inside the
vessel. It follows from the definitions of E(t), I(t) and F(t) that
5 E(t) dt = 1
(3-2)
o
1 I(t) dt = 1
(3-3)
o
F(t) = 1 at t ....
and
(3-4)
As stated previously in the tracer technique section, the RTD curve can
be established by the stimulus-response technique. The stimulus can be of
different forms but the most commonly used methods are pulse input and
step input. If the form of stimulus is a pulse input, then the observed
response at the vessel exit is called C curve, the tracer concentration versus
time curve, which will be modified to E curve. The exit response of a step
input yields the F curve.
The selection of these two tracer-injection methods is dependent upon
the applications. In this study the pulse technique is applied because it is
19
probably the most convenient for conducting this experiment. To satisfied
requirements of the pules input method the injection must take place over a
short period. Therefore, one of the basic assumptions in conducting this
experiment is a negligible amount of dispersion between the point of
injection and the entrance to the system. An amount of tracer M is injected
into the solids entering the system. This experiment analyzes the injection of
a tracer pulse for a single-input and single-output system in which only the
flow of solids carries the tracer material across system boundaries, and where
the tracer is not deminished from its flowing phase.
The other basic assumption on this study is to restrict a steady-state
flow with one entering stream and one leaving stream of a single fluid of
constant density, Danckwerts(1952) [6]. We focus only on the unchanging size
and density of particle throughout the continuous two-compartment
fluidized bed system. There is no reaction taking place inside the vessel.
From material balance equations, Levenspiel (1989) [28]:
M
Area under the C curve = F
w
(3-5)
S
t
=T
(3-6)
where M is the amount of tracer introduced into the solids entering the
vessel, W is the mass of solids in the vessel, and F, is mass flow rate of solids.
Principally, the area under the C curve and the mean residence time can be
calculated by equation(3-5) and (3-6) but the reactor characteristics may then be
the reason for making the theoretical values deviate from experimental
values. Therefore, we apply the mathematical tools which are widely used in
20
tracer work so we can calculate the numerical values from the experimental
data.
00
Area under the C curve = IC dt
(3-7)
0
and
00
itC dt
i.
o____.
(3-8)
fC dt
0
Since the distribution curve is known at a number of discrete time values
ti
then
Area under the C curve = I Cl At
i
(3-9)
all i
and
I, ti cieti
all i
t
(3-10)
I, CAti
all i
It is more convenient to normalize the C curve which will lead to the
other type of RTD curve, as stated previously as E curve, in such a way that
the area under the curve is equal to one, equation(3-1). Since the flow rate of
solids throughout the system is assumed to be steady, we can define the RTD
function, E(t), from the tracer concentration C(t) as:
C(t)
(3-11)
E(t) =
IC dt
0
21
The integral in the denominator is the area under the C curve, equation(3-7).
Frequently, E(t) function is normalized respect to the dimensionless
time scale,9 =
t
t
.
Therefore, a dimensionless function E(8) can be defined as
E(8) = i E(t)
(3-12)
3.3 Model of flow of solids in a non-ideal system
There are several types of models that have found useful applications,
but two are most common, at least when found to be adequate to represent
the physical situation of the real system. The first is usually mentioned the
"axial dispersion" or "axial dispersion plug flow" model. The second
visualize various flow regions connected in series by perfectly mixed tanks,
and is called tanks-in-series model.
3.3.1 Axial dispersion plug flow model
The fundamental ideas of the axial dispersion plug flow model are
derived on the basis of homogeneous quality of flow conditions throughout
the vessel. In this study the quality of flow conditions of solids are
interrupted in the orifice connecting the two compartments. Therefore, the
dispersion model can not exactly explain the flow characteristics of solids in
the two-compartment fluidized bed vessel. The influences of an orifice on
the quality of solids flow may be eventually incorporated in flow
characteristics if enough orifices are present in the vessel. This suggests that
with increase in the number of compartments the dispersion model will do a
better justice to the experimental data. The interruptive effect of orifices will
Name of model
Simplifying assumptions or
restrictions in addition to
those for the model above
General dispersion :
includes chemical Constant density
reaction
and sourse term
Bulk flow in axial
Genaral dispersion
direction only.
in cylindrical
co- ordinates
Radial symmetry
Uniform dispersion
Dispersion coefficients
independent of position
hence constant
Dispersion plug flow Fluid flowing at mean
velocity, hence plug
flow
Parameters
of model
D, i4
Defining differential equation
dC
_
Tt + u DC = V . (D. VC) + S + rc
dC
dC
a
D (R) D (R) dC
R
' L ' at + ii (R) Tx = ax DL(R) Tx
ii (R)
D' Rm, D'L.,,
+
dC
_
at + u
dC
(R) ax
1d
dC
R dR RD R(R) -dR + S + 7. , (T3-2)
d2C
No variation in properties
in the radial direction
D'L' 14
Table 3-1 Dispersion models, Bischoff and Levenspiel (1962) [4].
a
+S+/.c
ii (R)
DR, DL, ll
D DR
= D Tx 7 + --'"L"
R dR
d2C
X
ax = 131.--ap- +
at + u.. X
.
PA d
_
X
2C
R
dC
dR
(T3-3)
43C
aR R A
+S+rc.
Axial dispersion
plug flow
(T3-1)
5 T. + u Tx = DLTx-2- +S+l-c
at
(T3-4)
(T3-5)
23
be evenly spread throughout the vessel and eventually accounted as
homogeneous property of the flow quality.
The most widely accepted approach for representing the mixing in
flowing systems has been based on models that use diffusion equations with
modified diffusion coefficients. These are termed dispersion models, and the
coefficients are termed dispersion coefficients. Table 3-1 lists the
mathematical equations corresponding to the various dispersion models
from the most general and unwieldy to the most restricted but most easily
handled. Equation(T3-4) and (T3-5) are the most commonly used because of
their simplicity. Naturally, the axial dispersion model, as just stated, are
useful mainly to represent flow in tubes and packed beds, however, this
model can also be successfully applied to describe a variety of other types of
vessels, Bischoff (1990) [15]. An attempt will be made in this thesis to make
use of the axial dispersion plug flow model to predict the RTD of solids in the
two-compartment fluidized bed vessel.
The phenomenon of longitudinal or axial mixing is assumed to be
described by the equation(T3-5). If there is no reaction in the system, the
aC
_
dC
crC
-ai, + u Ti = rY,.-7 + S + rc
(T3-5)
reaction rate (re) term is equal to zero. Typically, there is no source term
within the system boundary, the tracer injection point is upstream from the
section over which the equations are used; thus the source term (S) is equal to
zero. The differential equation for the axial dispersion plug flow model is
then reduced to:
24
X
gC
_
dC
Ti. =Ddx2 -u ax
(3-13)
where t is the time from the commencement of the displacement, x is the
distance from the point of introduction of displacement, C = C(x,t) is the solid
tracer concentration, D is the axial or longitudinal dispersion coefficient
which uniquely characterizes the process, it is the characteristic velocity
(very often interstitial velocity) of the carrier phase, and L is the characteristic
length of the system.
Equation(3-13) can be put in dimensionless form by introducing
z=
(u t+ x)
L
t
tu
and 0 = = r the basic differential equation representing this
t
axial dispersion plug flow model becomes.
dC _( D )d2C .dC
de
itL)W- az
where the dimensionless group
1_3'
(3-14)
is the parameter which reflects the
uL
extent of axial dispersion. To simplify subsequent formulas, U
14
L
2D
is
introduced into equation(3-14) and termed the dispersion number. Therefore,
the axial dispersion plug flow differential equation can be written as :
X
do
1
gC X
21/ di
dz
(3-15)
25
The two-compartment fluidized bed system in this study is considered
as "closed-dosed vessel". We assume that solids enter and leave the system
in plug flow manner. The entrance and exit point are designed in such a way
that once the solids enter the vessel they do not diffuse back out of the system
and once they leave the vessel they do not move upstream back to the system.
Based on the aforementioned boundary conditions, called "closed-closed
vessel", Yagi, et al .(1953) [41] presented the solution of equation(3-15) for
initial condition given in the form of "S (Dirac's) function" and x=L as:
E(8) = 2 I
pn (Usin(pn) + pn cos(pn))
(112+ 211 +,412)
pn2 )
+
x exp [ U -((1.122U
]
(346)
with pn positive roots of
cot(pn)
LI
pn
11-1
9
2
=
(3-17)
U is the parameter which measures the extent of axial dispersion. Thus
it
LI=
L
2D
--) e°
: negligible dispersion (D=0), hence plug flow
: 0 # 1, then E(0) = 0
: 0 = 1, then E(0) = ...
U=
ic
L
2D
>0
: large dispersion(D =oo), hence mixed flow
: E(0) = exp (-0)
Figure 3-3 shows the E(8) versus 8 curves of the model for various
extents of back-mixing as predicted by the axial dispersion plug flow model.
26
1.5
i
4
I
1
i
i
1.1=0.00
I
1.2
I
U = 0.20
,
II
!;
,
t
0.9 1.
t
,
1
1
0
.
,
1
1
11=0.50
%
I
1
. f, ;
"
1
t
E(0)
I
st
.
0.6
I
8
I
1
=1.00
1
1
U = 5.00
I
. .
,
11
i
%
kAt,
,
11=0.69
I
1
,.
t.1,,
U = 20.00
d1
%
%
%
%
i:
0.3
%
i
;
II
1
1
i
i
i
'%
%
1ili :
I
la
%
i
I
I
:
%
I/
i
.
.
\,%,
0.0
'
0.0
20
10
.. .
30
40
5.0
0
Figure 3-3 E curves in closed-closed vessels for various extents of dispersion.
3.3.2 Tanks-in-series model
The tanks-in-series model is another one-parameter model frequently
used to simulate behavior of non-ideal flow. In this case the volume of the
real system is represented by a series of identical ideal stirred tanks, and the
only parameter of this model is the number of tanks (N) in series. The total
volume of these equal-sized tanks is the same as that of the real system.
V real = N VCSTR
(3-18)
27
It happens that the tanks-in-series model gives tracer response curves
that are somewhat similar in shape to those found from the axial dispersion
plug flow model. Thus, this model can be applied whenever the axial
dispersion model is applied. For not too large a deviation from plug flow,
both models give identical results, Levenspiel (1989)[28]. The simplicity is the
advantage of the tanks-in-series model. Starting from first principles one can
easily develop the function representing the response of the system to Dirac's
function type stimulus, Levenspiel (1972) [29].
Ea?)
N(NO)(N_i),N-1
ex p (-NO)
;N =1, 2, 3,
...
(3-19)
The model suffers from one shortcoming, it allows only integer values
of N
.
The usefulness of the model can be improved if N is allowed to
assume non-integer values. The idea of using non-integer values of N in
tanks-in-series model is reported by Hill (1977) [19]. The most obvious
advantage of using non-integer values of N is to match an experimental RTD
curve with the theoretical one that lies between those of adjacent N values of
the tanks-in-series model. To resolve this problem the gamma function is
introduced . The basic relations are:
rag + 1). Nr(N)
(3-20)
and
RN). (N-1)!
then, the equation(3-19) becomes
(3-21)
28
E(0) -
N(N 0)14-1
1-(N)
exp ( -N0)
;N
0
(3-22)
The values of gamma function with non-integer values of N can be
found in tables, see appendix 5. Figure 3-4 shows the E(0) versus 0 curves
correspond to tank-in-series model, equation(3-22).
N -4 1
: single well-mixed tank, hence ideal mixed flow
N -4 oo
: ideal plug flow
: 0 * 1, then E(0) = 0
: 0 = 1, then E(0) = co
29
2.0
I
i
i
1
I
I
iP
I
II
i
I!
,
f
'II
1.6
N=
------
I
'.I
I
)
il
il
!I
il
9
1
N=2
N=5
ii
N = 10
1.2
i
1
1
I
!I
I I,
MI
,
.
t
N = 50
s.
ft
,
0.8
-------
.
I
I..
,
a
,
0.4
i
4,
1
i
I
I
1
tN.
e
0.0
0.0
,
4.
t
I
; I
; )
i
N = 1.5
1
t.
.'. r\
,
---------- N = 20
t
.1
E(0)
.%
i
-...
%
....
%.
...
10
20
3.0
40
5.0
0
Figure 3-4 E curves for various extents of number of tanks in series.
3.4 Model of the fluidized solids flow through the orifice
Several investigators, J. A. H. de Jong (1965) [11], Davidson (1985) [8]
and Korbee et al. [20], studied the discharge of fluidized solids through the
orifice. They all agreed that the discharge flow rate of solids depends on:
1. the pressure drop across the orifice,
2. the density of the fluidized solids,
3. the orifice diameter,
4. the particle size and shape.
30
Gas
Gas
The circle area is enlarged below
r
.
:
a,. .11 64, ..
.4.
., i. ,-i ..,.4
., --...,. :. -. 4.
...
Partition
.
Lean bed
,
.1%,
i A % I'
I
11
I?
.
1
I
I
1.1
,,,
'
'4 t Dense bed i.,- "
i
..,..
,..
a. 4, Y-
'
'....
.
-,
4
s :l':1,.%3- , .......
4 -1
'.1.
* .:
....... 4`; T
v
I .r;.t4,*
P.:;:.' :. ..:, d aPd ,:,. .
.../ .,,e.,'
e 'v:4 :
" ... ,'. or, ', 4,, ,... .... ' '
y: .; ' : e :-, .,
I
ft
-.
.. .:.,...- )., ...,. ..
-..-'
I
,.
I
I
"
A
4
%
, 44
..
Pt
g
04,
4
i
,7
'I
I...
s
Solid flow
direction
/. -.4
I.
1
9
1.
..p.:
... i
,,
.
4
4,....
.4
.
A, : :4
..-..
I.
*
1.4
44% .
r 'i ,1.4
di
.0 %.
...
`
I
.
e
I
%%4 :?.-
.8
!I. Dense bed
I .4
c.:4*
,
I
4.
t
It
Lean bed
%.
a,
t ,..r
b
t
Partition
..
Figure 3-5 The illustration of solid flew through the orifice.
The driving force for this flow is the pressure drop across the orifice
(Apo) between the dense and lean bed, Figure 3-5. On the basis of the
mechanical energy balance, a semi-empirical equation of solid mass flow has
31
been derived. In order to describe a solid mass flow through the orifice by a
simple model it is assumed that particles in the vicinity of the orifice are in
the state of minimum fluidization. The flow of the solids through the orifice
can be viewed as a flow of liquid with a density of the fluidized bed at
minimum fluidization. By assuming the uniform velocity through the crosssectional area of the orifice, the mechanical energy balance may be written in
the form:
0s2
CD2 A:
(3-23)
where
psnl = density of the fluidized solids at minimum fluidization, kg/m3
= ps (1- Cm?
Ao
1/
= ivc102) = area of the orifice, m2
J. A. H. de Jong (1969) [11] concludes from his experimental data that
values for discharge coefficients (CD) in equation(3-23) range from 0.45 to 0.51.
In the two-compartment fluidized bed situation, the solids flowing through
the orifice from the dense bed to the lean bed are assumed to have the same
composition as the dense bed itself, Korbee, et al [10]. Therefore, equation(323) can be applied to derive a relation for the solids discharge rate:
Co A02 2ps (1- Ed) Apo
(3-24)
Inside the orifice there is an area where particles can not pass or where
the concentration of particles is low, Figure 3-5. Therefore, in order to correct
the solids mass flow rate, Davidson (1985) [8] suggests the use of an effective
32
orifice diameter given by (do - kd,). Equation(3-24) can then be expressed as,
de Jong (1965) [12] :
= C,132 A.,2 2ps (1.. Ed) Apo
(3-25)
(1-k L
do )2
(3-26)
with
A'. = A
ci
and de Jong (1969) [11] reports that
k = 2.9 for sand
= 1.4 for spherical particle
leading to
2/
Os = CD Ao(1 -kd; k2p, (1- Ed)" (Apo)"
(3-27)
Then, all experiments give the constant orifice discharge coefficient
(C'D ) value of 0.53 ± 0.01, de Jong (1969) [11]. In this work to determine the
solid flow rate(Os) from equation(3-27), the pressure drop across the orifice
(AO and the voidage of dense bed (Ed) are considered. The pressure drop
across the orifice is measured directly by using the u-tube manometer while
performing an experiment. The voidage of dense bed can then be calculated
by using the following equation.
E
d
= 1-
APa
hd(ps-Pg) g
(3-28)
33
where
Apd
dense bed.
is the pressure drop across the dense bed and hd is the height of the
34
CHAPTER 4
EXPERIMENTAL EQUIPMENT AND PROCEDURES
4.1 Experimental apparatus
4.1.1 Fluidized bed
Experimental measurements of solid flow behavior were conducted in
a Plexiglass cylinder with a 100.0 mm internal diameter and 6.3 mm wall
thickness. The length of the cylinder measured from above the distributor
plate to the top is 590.0 mm and below the distributor plate to the bottom
where the air entrance holes were located is 190.0 mm. The fluidized bed
cylinder was divided into two compartments by a 6.3 mm-thick Plexiglass
partition; both the upper and the lower part of the vessel were partitioned.
To facilitate the flow of the bed material through the system an orifice is
provided at the bottom of the partition near the distributor plate. The size of
the orifice was one of the investigated system parameters. The center of the
orifice was located vertically 10.0 mm above the distributor plate and near the
center of the partition.
The fluidizing air was introduced at bottom of the fluidized bed. The
air flow rate, i.e. the fluidizing velocity, was controlled independently for
each compartment by a control valve and measured by a calibrated rotameter.
A perforated "sandwich" design plate was used as a gas distributor. The
distributor was a 3.6 mm-thick Plexiglass plate covered with a metal screen to
prevent solids from raining through the orifices when the gas flow is stopped.
The size and number of openings depend upon the diameter and the density
35
of particles and the height of the bed. The plate has 86 of 1.1 mm-diameter
holes positioned in a triangular pitch.
4.1.2 Fluidized bed material
Glass beads were used as a fluidized bed material. The minimum
fluidizing velocity (um? was determined by measuring the pressure drop
across the bed as a function of a fluidizing gas velocity. Data are shown in
Figure 5-1. The experimental value for minimum fluidizing velocity was
compared to the calculated unif from theoretical equation in appendix 3.
Since the solids had a distribution of sizes, it was necessary to calculate
the mean diameter for a mixture of particles of different sizes. The mean
surface particle diameter was determined from the sieve size analysis. The
following formula is applied to calculate the surface average diameter for
mixture particles, Kuni and Levenspiel (1991) [27].
(IP
1
=
(4-1)
x1
all i
dpi
where
x.
= the mass fraction of particles of size interval i
dpi
= the average diameter of particles in interval i
1
The glass beads particles were seived and the size distribution of the
batch of glass beads particles is shown in table 4-1.
36
Sieve Aperture (p.m)
Weight Percent (%)
125-250
8
297-417
32
417-500
60
Table 4-1 The glass beads material size distribution.
By using equation(4-1) the mean surface partide diameter of glass beads
is calculated to be 379 gm. The density of glass beads was determined from
volumetric and weight measurements in water. Table 4-2 summarizes the
physical characteristics of the glass beads.
Bed
Materials
Glass beads
d range
(I
P
P.
u mf (exp)
u mf (theo)
(gm)
(gm)
(kg/m3)
(cm /s)
(cm/s)
125-500
379
2415.9
16.1
15.0
P
Table 4-2 The physical characteristics of bed materials.
4.1.3 Air supply system
Air was used as fluidization gas. It was supplied from the compressor
available in the chemical engineering building.
4.1.4 Pressure measuring system
The pressure measuring system consists of pressure taps and U-tube
manometers. The pressure taps used to determine the pressure difference
are positioned at both side of the vessel along the vertical axis. Pressure taps
were installed right above and below the distributor plate and along the wall
37
of the fluidizing column, figure 4-1. Fine screen was added at each tap to keep
the solids from plugging the lines.
4.1.5 Feed system
The bed material is introduced into the fluidized bed by a gravity feed
through the conical hopper and 300.0 mm long glass standpipe which are
located 400.0 mm above the dense side of the two-compartment fluidized
bed.
4.1.6 Tracers
The basic requirements for a satisfactory tracer, are described in chapter
3. The tracer particles were prepared from glass beads and food color. These
particles are identical to the particles used in the fluidized bed. The particles
are coated with a solution of the Durkee-French Foods Company green food
coloring and propanol. After soaking particles long enough to retain color
solution on their surface the mixture is poured into a tray, spread evenly, and
allowed to dry at room temperature.
4.2 Experimental Mechanism
The experimental equipment is installed as shown in Figure 4-1. The
fluidizing air is supplied from the building pipeline and flows through a
pressure regulator which allows a maximum pressure of 24.13 kPa to reach
the rest of the equipment. The air is divided into 2 lines and passed through
38
Ink
1. Two-compartment
fluidized bed vessel
2. Partition
3. Partide hopper
4. Rotameter
5. Pressure probe
6. U-tube manometer
7. Pressure guage
8. Sampling cup
9. Exit orifice
10. Pressure regulator valve
Figure 4-1 Schematic of apparatus for the two-compartment
fluidized bed system.
39
two rotameters. Two independent lines are connected to each side of the
fluidized bed. The particles are fed continuously from the hopper located
above the system. Small orifice placed at the bottom of the partition, allows
the particles to flow from one compartment to the other, Figure 1-1b. The
pressure drop across the orifice is the driving force for this flow. The partides
leave the system via an adjustable overflow exit orifice, Figure 1-2.
4.3 Experimental Procedures
To begin the procedure for each experiment the preliminary
experimental set-up procedures are required.
1. Determine the minimum fluidizing velocity (IQ by using the
conventional fluidized bed vessel. The pressure drop across the bed
(Ap b) is measured at fluidizing gas velocities uo being increased
(aeration) from 0.0 cm/s to 28.0 cm/s and then at U. being decreased
(deaeration) back to 0.0 cm/s. The results are shown in chapter 5,
Figure 5-1.
2.
Assemble and inspect the two-compartment fluidized bed vessel as
shown in Figure 4-1.
3.
Set the pressure regulator to 24.13 kPa.
4.
Adjust the control valves of both rotameters to allow the fluidizing
gas to fluidize the partides in each compartment.
5.
Select the orifice size in the partition wall by changing the partition,
and
6.
Adjust the height of the exit orifice.
After all preliminary set-up procedures are complete, we proceed as follows:
40
7.
Fill the hopper with particles.
8. Allow the flow of partides reaches the steady state,
9. At time equal to zero (t = 0), instantaneously introduce the tracer (10
grams of colored partides) into the system through the top of the
vessel together with the solids entering the system from the hopper.
At the same time, start to collect the all particles at the exit orifice at
prescribed intervals of time.
10. Measure the pressure drop across the dense (Apd) and lean (Apt) bed
by the u-tube manometers at both sides of the vessel.
11. When almost all of tracer particles are out of the vessel (typically
after 60 minutes of run), turn off the fluidizing gas on both sides of
the bed.
12. Measure the height of dense (hd) and lean (hi) bed, and weigh the
amount of solids remaining in the vessel.
13. Measure the weight of all collected sample and analyze the content
of each sample for tracer particles.
To find the amount of tracer in each sample, use the absorption ability
of samples. Each sample is mixed with 250 cm3 of distilled water, stired and
left overnight to insure that the color on the surface of the tracer particles is
dissolved. Absorbance of the colored solution is then measured by using the
" spectrophotometer " (Beckman, model UD-62). This instrument operates
from 200 to 900 nm. It uses stable beam technology to take reading in either
absorbance or transmittance. Using a visible lamp, the wavelenght of light is
set to 420 nm. To learn how much of tracer was in each sample cup, the
absorbance reading is converted to the concentration units by the calibration
curve given in Figure 4-2.
41
7.0
0'
1
6.0
y = 3.960x - 0.000 r2 =1.000
,
x
o
1
5.0
a
4.0
W
w
uets
3.0
1...
.,..
,....
o
....
0
o
5
<4
2.0
1.0
0.0
0
02
04
06
08
1
12
14
1.6
Absorbance ( at wave length = 420 nm)
Figure 4-2 Concentration of tracer versus absorbance calibration curve.
4.4 Experimental plan
To satisfy all aforementioned objectives of this study, the experimental
plan is created and listed as follows.
1. To investigate the improvement of the RTD in the twocompartment fluidized bed, the RTD experiment in the single
compartment fluidized bed was conducted for comparison with
two-compartment fluidized bed. Both experiments were set-up
42
with identical 20.8 cm/s fluidizing gas velocity (u0) and 100.0 mm
exit orifice height (h).
2. To investigate the effect of the fluidizing gas velocity (u0), two sizes
of the partition orifice (do = 3.5 and 5.1 mm), and two overflow exit
orifice heights (h = 100.0 mm and 200.0 mm) are experimented.
3. To investigate the effect of the size of partition orifice (d.) , four
levels of the gas velocity (uo = 16.2, 20.8, 25.6 and 30.5 cm/s) in both
compartment, and two overflow exit orifice heights (h = 100.0 mm
and 200.0 mm) are experimented.
4. To investigate the overflow exit orifice height (h), four levels of the
gas velocity (uo = 16.2, 20.8, 25.6 and 30.5 cm/s), and two sizes of
partition orifice (do = 3.5 and 5.1 mm) are experimented.
Table 4-3 summarize the experimental plan. Each run is coded as an
experimental number.
Gas velocity
h = 10.0 cm
h = 20.0 cm
u 0 (cm/s)
d0 = 5.1 mm
d0 = 3.5 mm
d0 = 5.1 mm
d0 = 3.5 mm
16.2
Exp. # 1
Exp. # 2
Exp. # 3
Exp. # 4
20.8
Exp. # 5
Exp. # 6
Exp. # 7
Exp. # 8
25.6
Exp. # 9
Exp. # 10
Exp. # 11
Exp. # 12
30.5
Exp. # 13
Exp. # 14
Exp. # 15
Exp. # 16
Table 4-3 Experimental plan.
Experiment #17 was conducted in a single compartment fluidized bed
vessel with 20.8 cm/s of fluidizing gas velocity and 10.0 cm of the height of
exit orifice.
43
CHAPTER 5
RESULTS AND DISCUSSION
5.1 The measurement of minimum fluidizing velocity
The minimum fluidizing velocity (um? for glass beads can be
determined from the pressure drop-versus-velocity diagram. Figure 5-1 is the
pressure drop-versus-velocity diagram of glass beads with the particle size
379 gm.
1.4
1.2
0
-. - -,
-7-'Ci
1.0
i
0.8 -
I
-0-
0.6
-a-
Areation
Deareation
0.4
0.2
.
0.0
114Inf
50
10.0
15.0'
= 16.1 cm/s
20.0
25.0
30.0
U. (cm/s)
Figure 5-1 Pressure drop across a fluidized bed versus
fluidizing gas velocity of glass beads (379 gm)
44
By increasing and decreasing the fluidizing gas velocity (u0), it has been
observed that, for the relatively low flow rates in a fixed bed, the pressure
drop is approximately proportional to gas velocity and then remains
practically unchanged when the bed is fluidized. It is obviously seen from the
diagram that the effect of inter-particle force does not make a big difference
between the plot of increasing and decreasing in gas velocity. Therefore, we
can neglect the effect of inter-particle force for this particles. The minimum
fluidizing velocity Wm? is taken as the intersection of the pressure drop-
versus-u 0 line for the fixed bed with the horizontal line. From the
aforementioned procedures the minimum fluidizing velocity for glass beads
can simply be determined to be 16.1 cm/s.
5.2 The tracer response curve
A total of 17 RTD experiments were carried out. Investigated
parameters were the fluidizing gas velocity (u), the size of partition orifice
(do) and the height of overflow exit orifice (h), i.e. mean residence time. One
of 17 experiments was conducted in a single compartment fluidized bed
vessel. There are 4 experiments which were performed with the fluidizing
gas velocity equal to 16.2 cm/s, close to the unit. No significant quantities of
solids were elutriated. The original RTD curves are obtained directly from
the tracer concentration, C curve. Figures 5-2 to 5-18 show the C curves
obtained for different experimental conditions.
45
0.007
a
0.006
as
0.005
0.004
= 16.2 cm/s
h
=10.0 cm
do = 5.1 mm
F's = 2.37 g/s
Apo = 0.4 an of H2O
Uo
A
A
A
A
A
0.003
A
0.002
.6A
AAA
0.001
as
DD
0.000
0.0
10.0
20.0
AA
30.0
40.0
50.0
t (min)
Figure 5-2 C curve of experiment #1
0.007
A
A
A
A
0.006
0.005
0
to
uo =16.2 cm/s
A
=10.0 cm
= 3.5 mm
Fs = 2.32 g/s
h
A
0.004
do
A
0.003
bo
CI
Apo = 1.4 cm of H20
A
0.002
0.001 A
0.000
0.0
10.0
20.0
30.0
t (min)
Figure 5-3 C curve of experiment #2
40.0
50.0
46
0.0
0.0
uo = 16.2
)3
,
do
Ps =
A
0.0 )2
=20.0 cm
= 5.1 mm
h
,n?6,
2.54
Apo = 0.3
ds
-
cm/s
g/s
cm of H2O
A
A
0.0 )1
AAA
/-
AA
AAAA,
A
0.0 )0
A
0.0
AAAAAA AA
.
.
20.0
10.0
30.0
40.0
50.0
60.0
70 .0
t (min)
Figure 5-4 C curve of experiment #3
0.004
0.003 -
A
A
:A
0.002 -
A
A
A
A
A
A
uo
h
= 16.2 cm/s
=20.0 cm
do = 3.5 mm
Fs = 2.37 g/s
4),
Apo = 1.3 cm of H20
A
0.001 -s
?N
0.000
0.0
10.0
20.0
30.0
40.0
50.0
t (min)
Figure 5-5 C curve of experiment #4
60.0
70.0
47
0.007
0.006
uo = 20.8 cm/s
0.005
=10.0 cm
h
0.004
do =5.1 mm
A
Fs = 2.26 g/s
AA
0.003
Apo = 0.3 cm of H20
A
6.
00
0.002
A
A
0.001
AA
*6AAA,n,
A
tAAAAAAAL
0.000
30.0
20.0
10.0
0.0
40.0
50.0
t (min)
Figure 5-6 C curve of experiment #5
0.007
0.006
Uo
A
0.004
0.003
Apo = 1.5 CM Of
0
H2O
A
A
A
A
0.002
,.....
.1..
tl
_
_
= 20.8 cm/s
h =10.0 cm
do = 3.5 mm
Fs = 2.38 g/s
0.005
AA
0.001
'4AA
i
0.000
LAAAAAAAAA
i
0.0
10.0
20.0
30.0
t (min)
Figure 5-7 C curve experiment #6
40.0
50.0
48
0.004
,e4A6a
.
A
0.003
4,
A
A
A
A
= 20.8
h
=20.0 cm
= 5.1 nun
do
Fs = 2.24
A
A
0.002
Apo
A
A
A
tiA
A
0.001
cm/s
zio
g/s
___
= 0.5 cm of H2O
A
AAA.°
n
0.000 4
AAAA'AAA
-AAAAL AA
.
10
.
t
t
20.0
10.0
30.0
40.0
50.0
60.0
70 0
t (min)
Figure 5-8 C curve of experiment #7
0.004
AA
Fs = 1.92
bn
0.002
_
Uo = 20.8 cm/s
=20.0 cm
h
do = 3.5 mm
A
A
A
0.003
Apo
g/s
= to cm of H20
0.001
A
AA
AAA
AAAAAAA
0.000
.
.
/0
10.0
20.0
30.0
40.0
50.0
t (min)
Figure 5-9 C curve of experiment #8
60.0
70 .0
49
0.007
0.006
.
AAp
0.005
A
= 25.6 cm/s
=10.0 cm
h
do = 5.1 mm
Fs = 2.31 g/s
Apo= 0.4 cm of H2O
uo
A
A
aA
0.004 .
0.003
A
A
0.002
A
A
A
A
°°°
0.001
66,Ap
0.000
.
0.0
i
-
30.0
20.0
10.0
40.0
50.0
t (min)
Figure 5-10 C curve of experiment #9
0.007
0.006
A
0.005
A
.
5
= 3.5 mm
Fs = 2.33 g/s
Apo = 1.5 cm of H2O
do
A
A
0.003
0.002
h
A
0.004
A
A
=25.6 cm/s
=10.0 cm
uo
°°
A
A
a
A
46,
0.001
0.000
AAA
.
Y
0.0
10.0
AAA
A
20.0
A666,66A6,6,
ise\mA
.
30.0
40.0
t (min)
Figure 5-11 C curve of experiment #10
50.0
50
0.005
aE
(1)
AA
0.004
A
A
u)
'15
00
li0 = 25.6
LA
A
0.003
$.4
A
V
ett
I::
cm/s
= 20.0 cm
do = 5.1 mm
Fs = 1.70 g/s
Apo = 0.3 cm of H2O
h
_
A
0.002
'15
b0
AAA
z.----%
0.001
A
A
AAAA
A
A
(..)
0.000
0.0
AAA
LAAA AAAA
_
-
-
A
AAA,
.AAAA
20.0
10.0
30.0
40.0
50.0
60.0
70.0
t (min)
Figure 5-12 C curve of experiment #11
0.004
A%
A
0.003
=25.6 cm/s
=20.0 cm
h
do = 3.5 mm
Fs = 2.05 g/s
Apo = 1.1 cm of H2O
6/1
U0
A
6A
A
41
0.002
AA
0.001
"AAAAAAAA,
--A-AAAAAA-AA AAA
0.000
0.0
10.0
20.0
30.0
40.0
50.0
t (min)
Figure 5-13 C curve of experiment #12
60.0
70.0
51
0.007
A
0.006
far
A
0.005
bo
A
A
0.004
AA
A
0.003
0
bO
C)
uo = 30.5 cm/s
=10.0 cm
h
do = 5.1 mm
Fs = 2.33 g/s
Apo = 0.4 cm of H2O
6.°
A
0.002
A
A
0.001
0.000
TA6° 46.6ATA6A/A61\6616'
0.0
30.0
20.0
10.0
40.0
50.0
t (min)
Figure 5-14 C curve of experiment #13
0.007
ai
0.006
A
(3
uo
°
44
bel
= 30.5 cm/s
=10.0 cm
h
do = 3.5 mm
Fs = 2.32 g/s
0.005
0.004
A
c.9
br
0.003
Apo
_
= 1.4 cm Of H2O
A
bo
0.002
tl
0.001
A
°°
°°
AMA6'6466A6'66AAXP
0.000
0.0
10.0
20.0
30.0
40.0
t (min)
Figure 5-15 C curve of experiment #14
50.0
52
0.005
w
a
4,
0.004
A LA
E
A
cs
V3
,.....
o
top
A
A
0.003
038
A
0.002
h
=20.0 cm
= 5.1 mm
_
Ps = 2.23 g/s
A
A
A
i..4
--- 30.5
do
_
cm/s
Uo
Apo = 0.3
cm of H2O
66,
4
sr-t
0
66,
bo
AAA
0.001
.A
A
AAA
AAAAAA
t
0.000
AhhA"AAAAA
20.0
10.0
0.0
60.0
50.0
40.0
30.0
70.0
t (min)
Figure 5-16 C curve of experiment #15
0.004
uo = 30.5 cm/s
0.003
, ,
0.002
do
Fs = 1.96 g/s
A,
A
A
=20.0 cm
= 3.5 mm
h
AAA
Apo
= 1.9 cm of H2O
YA
4n,
0.001
AAAA
AAA
,
LA A
I AAAA AALAiiii AA
0.000 -l
).0
.
1
10.0
20.0
30.0
40.0
50.0
t (min)
Figure 5-17 C curve of experiment #16
60.0
70
53
0.009
0.008 a
0.007
u0 =20.8 cm/s
A
=10.0 cm
Single fluidized bed
h
A
0.006
A
A
0.005
A
00
0.004
A
0.003
66,6
A6:68
0.002
0.001
666,6,66.664
0.000
0.0
10.0
20.0
30.0
.40.0
50.0
t (min)
Figure 5-18 C curve of experiment #17
5.3 Axial dispersion plug flow modeling
To evaluate the model parameter U which represents the best fit to the
experimental data, numerical method is applied to resolve equation(3-16) and
(3-17). A FORTRAN program is developed, using half-interval approach, to
calculate a series of pn . The details of this numerical program are described in
appendix 1. The least-square method is used to determine model parameters
from experimental data. However, to obtain the best-fit an optimization
procedure must be applied. The sum of square differences are calculated for
several discreate values of dispersion number (U). Several values of the sum
of square differences versus the dispersion number (U), experiment #9, are
shown in Figure 5-19. The quadratic equation while represents this set of data
is obtained by using curve-fit function in Cricket Graph III Software (version
1.1).
54
0.148
0.147
0.146
0.145
0.144
0.143
0.142 0.76
0.80
0.88
0.84
Ti
0.92
0.96
= 0.86
Dispersion number, U = D/2uL
Figure 5-19 Optimization approach on experiment #9.
By setting the first derivative of this equation to zero, the dispersion
number (U) which yields the best-fit between the measured and the simulated
RTD curve is determined. Once the best value for dispersion number is
found, the simulated RTD curve can be calculated. Figures 5-20 to 5-36
illustrate the results for the best-fit dispersion number (U) in all 17
experiments.
55
1.0
0.8
uo =16.2
h =10.0 cm
do = 5.1 mm
.
0.6
cm/s
11 =0.87
E(0)
0.4
0.2
.
0.0
0.0
y
'ilk
.
ionloW
05 10 15 20 25 30 35
4.0
45
5.0
8
Figure
5-20
Dispersion modeling for experiment #1
1.0
0.8uo = 16.2
cm/s
h = 10.0 cm
do = 3.5 mm
0.6-
U = 0.89
E(0)
0.4-
0.2
'A
0.0
.
0.0
AA
.
A
05 10 15 20 25 30 35
A
4.0
45
8
Figure
5-21
Dispersion modeling for experiment #2
5.0
56
1.0
0.8
uo = 16.2
cm/s
h =20.0 cm
0.6
do = 5.1 mm
U=0.71
E(8)
0.4
0.2
eAAAA
A AAAAAA
0.0
.
0.0
w
Aaa
Akik
.
AA /A
w
05 10 15 20 25 30
3.5
4.0
45
5.0
e
Figure 5-22 Dispersion modeling for experiment #3
1.0
0.8
uo = 16.2 cm/s
h =20.0 cm
0.6
do = 3.5 mm
u =0.84
E(0)
0.4
0.2
0.0
0.0
05 10 15 20 25
3.0
3.5
40 45 5.0
0
Figure 5-23 Dispersion modeling for experiment #4
57
1.0
0.8
uo = 20.8 cm/s
h =10.0 cm
0.6
do = 5.1 mm
E(6)
11 = 0.53
0.4
0.2
.
0.0
0.0
05
10 15
A
20 2.5
AA,..AAA
A.
m
30 35
4.0
45 5.0
8
Figure 5-24 Dispersion modeling for experiment #5
1.0
0.8 -
uo = 20.8 cm/s
h =10.0 cm
do = 3.5 mm
U = 0.75
E(0)
0.0
05
10 15 20 25 30 35 40 45
0
Figure 5-25 Dispersion modeling for experiment #6
5.0
58
1.0
0.8
AA
o
i
U0 = 20.8
h =20.0 cm
0.6
E(8)
A
0.4
cm/s
do = 5.1 min
U = 0.71
p
A
-
0.2
AA,
A
1hiAaa
'
20 25 30 35 40 45
AA,
ahALAaA
0.0
A
L
0.0
05
10 15
5.0
9
Figure 5-26 Dispersion modeling for experiment #7
1.0
0.8 -
uo = 20.8 cm/s
h =20.0 cm
0.6
do = 3.5 mm
E(0)
U = 0.76
0.4
0.2
Atha
AhAia
0.0
/ALA,
Av
0.0
05 10 15 20 25
3.0
35 40 45
9
Figure 5-27 Dispersion modeling for experiment #8
5.0
59
1.0
0.8
uo = 25.6 cm/s
h =10.0 cm
0.6
do = 5.1 mm
E(0)
11 = 0.86
0.4
0.2
A
0.0
05 10 15 20
0.0
. Av
. ,
i
30 35 40 45
2.5
5.0
9
Figure 5-28 Dispersion modeling for experiment #9
1.0
0.8
64
A
A
uo = 25.6
cm/s
h =10.0 cm
0.6
do = 3.5 mm
U = 0.75
E(0)
0.4
0.2
A
0.0
0.0
.,
05 10 15 20 25 30 35
--4.0 4.5
e
Figure 5-29 Dispersion modeling for experiment #10
5.0
60
1.0
0.8
uo = 25.6 cm/s
h =20.0 cm
0.6
do = 5.1 mm
U = 0.69
E(0)
0.4
0.2
0.0
0.0
10 15 20 25 30 35 40 45
05
5.0
Figure 5-30 Dispersion modeling for experiment #11
1.0
0.8
IJAhk
AA
UO = 25.6 cm/s
h =20.0 cm
._
0.6
,A
E(8)
do = 3.5 mm
U = 0.73
A,
0.4
va,
..
A
A
0.2
-AAA
1
AA,A
AA
AAAAAA
0.0
ialakAiAL4M/A
!
0.0
0.5
10 15 20 25
3.0
35 40 45
0
Figure 5-31 Dispersion modeling for experiment #12
5.0
61
1.0
0.8
A
uo = 30.5
cm/s
h =10.0 cm
0.6
do = 51 mm
E(8)
1.1 = 0.78
0.4
A
6
0.2
A,
AvA.A.
0.0
0.0
05
10 15 20 25 30 35
--
-A
4.0
.\1611416
45 5.0
e
Figure 5-32 Dispersion modeling for experiment #13
1.2
A
1.0
Uo = 30.5
0.8
cm/s
h =10.0 cm
do = 3.5 mm
E(0)
Li = 0.01
0.6
0.4
0.2
.
0.0 A .
0.0 0.5
.
.
10 15 20 25 30
3.5
40 45
9
Figure 5-33 Dispersion modeling for experiment #14
5.0
62
1.0
0.8
uo = 30.5 cm/s
h =20.0 cm
0.6
do = 5.1 mm
lI = 0.63
E( 0)
0.4
0.2
" AAAAAAA,
AIAAA
0.0
05
0.0
10
30
25
20
15
35
4.0
4.5
5.0
0
Figure 5-34 Dispersion modeling for experiment #15
1.0
0.8
0.6
o
uo = 30.5 cm/s
a,.A
h =20.0 cm
3.5 mm
U =0.36
A
dO =
A
E(0)
IA
0.4
Ill,
0.2
Ilkl
*A
IL
ImA1141,A
i hith
0.0
1
0.0
.
1
nr
05 10 15 20 25 30 35 40 45
I
.
0
Figure 5-35 Dispersion modeling for experiment #16
5.0
63
1.0
0.8
uo = 20.8
.
=10.0 cm
Single fluidized bed
h
0.6
u = 0.00001
.
E(9)
0.4
cm/s
AA
.
i
0.2 ...
a
AA
0.0
:-.'
0.0
1
05
10 15
.
20 25 30
3.5
40 45
5.0
Figure 5-36 Dispersion modeling for experiment #17
5.4 Tanks-in-series modeling
The number of tanks (N) which is a parameter in equation (3-22) is
calculated on computer spread sheet by Microsoft Excel (version 3.0). Using
the sum of the squared differences technique to compare the experimental
points with the calculated points on the RTD curve. The identical
optimization approach in the dispersion modeling is also applied in order to
determine the value of the best-fit N value. The N values obtained base on
this optimization approach were round off to a single decimal number.
These outcomes are shown in Figures 5-37 to 5-53.
64
1.0
0.8
A A
Uo = 16.2
0.6
cm/s
h =10.0 cm
A
do = 5.1 ram
N =2.0
E(8)
0.4
:
0.2
A
0.0
.
.
.
A
A
&AAP
05 10 15 20 25 30 35 40 45
0.0
5.0
9
Figure 5-37 Tanks-in-series modeling for experiment #1
1.0
A
A
0.8
A
A
uo =16.2 cm/s
h =10.0 cm
do = 3.5 mm
0.6
E(0)
N =2.1
A
0.4
A
A
0.2
A
A*AA
0.0
AA_AAA
05 10 15 20 25 30 35 40 45
5.0
e
Figure 5-38 Tanks-in-series modeling for experiment #2
65
1.0
0.8
uo = 16.2 cm/s
h =20.0 cm
0.6
do = 5.1 mm
N = 1.9
E(9)
0.4
0.2
0.0
05 10 15 20
0.0
2.5
30 3.5
40 45
5.0
Figure 5-39 Tanks-in-series modeling for experiment #3
1.0
0.8
IAA.
uo = 16.2
0.6
cm/s
h =20.0 cm
A
dO =
3.5 min
N . 2.0
E(6)
0.4
r
0.2
4A,
'A
AA
AAAA4A 'AAAA/AAAAAA
- AAA
0.0
.
0.0
05 10 15 20 25 30
3.5
APA
40 45
5.0
0
Figure 5-40 Tanks-in-series modeling for experiment #4
66
1.0
0.8
uo = 20.8
0.6
h =10.0 cm
A
E(0)
cm/s
do = 5.1 mm
N =1.90
AAA
°
0.4
A
0.2
L
A
AA
4p
A,A6AL,
AAAAA AA
0.0
05 10
0.0
30 35 40 45
20 2.5
15
A
5.0
9
Figure 5-41 Tanks-in-series modeling for experiment #5
1.0
0.8
uo = 20.8 cm/s
h = 10.0 cm
0.6
do = 3.5 mm
N = 2.0
E(0)
0.4
0.2
0.0.4
0.0
.
.
05 10
15
.
.
20 2.5 3.0
35 40
4.5
5.0
9
Figure 5-42 Tanks-in-series modeling for experiment #6
67
1.0
0.8
AAIA
Aw
=20.8 cm/s
h =20.0 cm
do = 5.1 mm
N =1.9
UP
A A
A
0.6
A,
E(0)
p
A
A
0.4
At
0.2
AAA
,AA
I/AAA/AAA
AAAPAAA i
0.0
.
.0
05 10 15 20
2.5
3.0
35 40 45
5.0
0
Figure 5-43 Tanks-in-series modeling for experiment #7
1.0
0.8
A AA
uo =20.8 cm/s
A
h =20.0 cm
0.6
.
3.5 min
N =1.9
dp
E(0)
0.4
AAA
0.2
PA
Aik
A,
AkAAA AkiAlAi
hApA AA
0.0
AA f
).0
f
05 10 15
1
20 25
i
3_0
3.5
4n ac ci
0
Figure 5-44 Tanks-in-series modeling for experiment #8
68
1.0
0.8
A
uo = 25.6
cm/s
h =10.0 cm
0.6
do = 5.1 mm
N . 1.9
E(0)
0.4
0.2
A
A
0.0
A
A
0.0
05 10
15
25 30 35
2.0
4.0
45
5.0
Figure 5-45 Tanks-in-series modeling for experiment #9
1.0
0.8
a
uo = 25.6
A
cm/s
h =10.0 cm
0.6
do . 3.5 mm
N = 1.9
E(9)
0.4
0.2
AA
A
0.0
1
0.0
05
1.0
1
15 20
AA
.
2.5
AA
A
30 35 40 45
5.0
9
Figure 5-46 Tanks-in-series modeling for experiment #10
69
1.0
0.8
o
uo = 25.6
cm/s
h =20.0 cm
0.6
do = 5.1 nun
N . 1.8
E(9)
0.4
A
AA
AA
A
A
1AAA
0.2
A
L IIAeAA LAir
/"Li AWAA
0.0
1
0.0
05 10
15 2.0
25 30 35 40 45
5.0
9
Figure 5-47 Tanks-in-series modeling for experiment #11
1.0
0.8
AA.
AAA
U0 = 25.6
0.6
cm/s
h = 20.0 cm
A
do = 3.5 mm
A
N = 1.8
E(0)
0.4
A
IA.
0.2
AA
r
1,A
AAiA1
A/AAAA11
-AA
AAALAA
0.0
I
0.0
I
A
05 10 15 20 25 30 35 40 45
5.0
Figure 5-48 Tanks-in-series modeling for experiment #12
70
1.0
A
0.8
A
uo = 30.5 cm/s
h =10.0 cm
0.6
do = 5.1 mm
N = 1.7
E(0)
0.4
0.2
A
0.0
.
0.0
1
.
05 10
15
20
2.5
A
30 35 40 45
5.0
0
Figure 5-49 Tanks-in-series modeling for experiment #13
12
1.0
uo =30.5 cmis
0.8
we)
h =10.0 cm
do = 3.5 mm
N = 1.1
0.6
0.4
A
A
0.2
AVA
0.0
05 10
15
,
AA
20 2.5
30 35 40 45
A
5.0
e
Figure 5-50 Tanks-in-series modeling for experiment #14
71
1.0
0.8
uo = 30.5 cm/s
h =20.0 cm
0.6
do = 5.1 mm
N =1.8
E(0)
0.4
A
0.2
0.0
05 10
0.0
15
2.0
25 3.0 35 4.0 45
5.0
8
Figure 5-51 Tanks-in-series modeling for experiment #15
1.0
0.8
uo = 30.5 cm/s
h =20.0 cm
0.6
do = 3.5 mm
N = 1.6
E(0)
0.4
0.2
/AA
-A A
A. A
A'Aii,#,AA, A
0.0
.
0.0
1
I
05 10
15 2.0
25 30 35 40 45
5.0
8
Figure 5-52 Tanks-in-series modeling for experiment #16
72
1.2
1.0
A
0.8
cm/s
h =10.0 cm
Uo
A
A
E(8)
Single fluidized bed
N = 1.1
o
o
0.6
= 20.8
A
A
'
0.4
A
A
A
A
0.2
.
A
A
AAA
0.0 .Z1
0.0
.
I
1
05
1.0
.
15
A A A 4.4
-A
A
20 25 30 35 40 45
1
5.0
0
Figure 5-53 Tanks-in-series modeling for experiment #17
73
5.5 The effect of fluidized bed compartmentation
The tracer experiment in a single compartment fluidized bed vessel is
conducted (experiement #17). The obtained RTD curve is compared to the
RTD curve obtained from the two-compartment fluidized bed vessel under
exactly the same experimental conditions (experiment #6). Figure 5-54 shows
that the RTD curve of solids improves substantially to the direction of plug
flow when fluidized bed vessel is partitioned into two compartments.
1.2
a
1.0
0.8
o
o
oath
A
E(8)
0.6
A
A
A
0
0
0
A
Single-compartment
fluidized bed
,,,'.1.
.0
5,
.
0.0
0.0
0
A
00a
0.2
Two-compartment
fluidized bed
A
00 A
0 A
oo AA
o
0 A
0.4
A
05
10
15
20
25
30
A
t
35
Yv..
40
45
5.0
0
Figure 5-54 Comparison of single- and two-compartment RTD curves.
74
To decribe the improvement of RTD curves in Figure 5-54, the axial
dispersion plug flow and the tanks-in-series models are used to simulate the
curves. Figure 5-55 shows the best-fit curves and model parameter values on
experiment #6 and #17 by the axial dispersion plug flow model. The
dispersion number (II) is improved from 0.00001 (single compartment) to 0.75
(two compartments). By tanks-in-series model, Figure 5-56 also illustrates the
same trend of improvement from the number of tanks (N) 1.1 to 2.0.
1.0
I
0.8
Exp. #6
A
%'s
t
'
I
I
4.
.
U = 0.75
r4
0
44
0
0.6
Exp. #17
u = 0.00001
.
E(0)
_
\A.6,
..t,
0.4
*.
,
0.2
.
0.0
i
0.0
.
-,,, sr .... ire
ti72,A'VAFA7A7AT
I
I
05
10
15
20
25
3.0
3.5
V' te TA1
40
45
5.0
9
Figure 5-55 Effect of compartmentation by axial dispersion
plug flow model.
75
1.0
0
0.8
As
,&
;
4
ti
E(
,
0.4
0.2
;
Exp. #6
N = 2.0
o
A
0.6
"
A
o.
..
A
Exp. #17
_
_
N = 1.1
.A.
4
fA
tA4p,
S
A<Ie
A i-AvAt,,,,,,..
Af
0.0
0.0
05 10
15 2.0
25 3.0
I 7 -4° °
35 40
4.5
5.0
8
Figure 5-56 Effect of compartmentation by tanks-in-series model.
76
5.6 The effect of the orifice size in the partition
An increase in gas velocity leads to a decrease in model parameter
values. The model parameters-versus-relative excess gas velocity diagrams
are shown in Figures 5-57 and 5-58. They demonstrate the effect of orifice size
for the same overflow height. Both axial dispersion plug flow and tanks-inseries models give the same tendency of the results. In each figure the slope
of both lines are negative, however, the slope for the small orifice diameter
(do = 3.5mm) is steeper.
2.4
2.2
2.0
....
6- -----1.8
-
------ - -------- --- --6--
1.6 -
1.4 -
0
---&-1.2
h= 200.0 mm, do = 5.1 mm, do/dp = 14
h = 200.0 mm, do = 3.5 mm, do/dp = 9
1.0
0.0
0.2
0.4
0.6
0.8
1.0
(U-Umf)/ Umf
Figure 5-57 Effect of orifice size by tanks-in-series model.
77
1.2
1.0
0.8
O
--2 ........
A
0.6
0.4
0.2 _
.
0.0
0
0
---16.--
h = 200.0 mm, do = 5.1 mm, do/ dp = 14 _
h = 200.0 mm, do = 3.5 mm, do/dp = 9
i
0.0
0.2
0.4
1
1
0.6
0.8
1.0
(U-Umf)/Umf
Figure 5-58 Effect of orifice size by axial dispersion plug flow model.
Therefore, model parameters are more sensitive to variations of the
fluidizing gas velocity for the 3.5 mm diameter orifice size. However, the
influence of orifice size can not be supported by a statistical analysis of data.
The paired t-analysis (appendix 3) is applied to test the effect of orifice sizes. It
is demonstrated in appendix 3 that there is no difference between the orifice
size (d) 3.5 mm and 5.1 mm. Thus, the variation of orifice size in this study
can not show the influences on model parameters.
78
5.7 The effect of the height of exit orifice
The primary effect of the height of the exit orifice can be seen from E(t)-
versus-t curve in Figure 5-59. Naturally, the mean residence time of solids is
0.12
Exp # 10
Fs = 2.33 gis
h =100.0 nun
0.10 1:P
0.08 -
w
Exp # 12
Fs = 2.05 gis
Mean residence time
= 8.85 min (h =100.0 mm)
-
h = 200.0 nun
Mean residence time
= 16.18 min (h = 200.0 mm)
oA
1:1646AA
0.03 OA
.A
EI:13
bA66666
A
0.00 ,0
0.0
466A&AA
ChICPCP
.
10.0
20.0
1666606
PEISITMEITLED
40.0
30.0
50.0
time (min)
Figure 5-59 The E curves conducted with different height of exit orifice.
longer for the bed with longer solids hold-up and roughly the same solids
flow rate; equation(3-6).
However, the statistical results show no difference between the exit
orifice heights 100.00 mm and 200.00 mm (appendix 3). The paired t-analysis
is employed to test the hypothesis that the bed with longer mean residence
time (longer exit orifice height, h = 200.0 mm) and the bed with shorter mean
79
residence time (h = 100.0 mm) give identical model parameter values. The
results of the statistical analysis show no difference in model parameters due
to different bed heights, different mean residence times (appendix 3). Figures
5-60 and 5-61 support this results and illustrate no significant influences of
the exit orifice height for orifice size (do) 5.1 mm and 3.5 mm respectively.
1.2
1.0 -....
.....
o
0.8
-..
-....
e
0.6
-...
0.4
-...
.. -.,
.
0.2
.
0.0
--IN--
h = 100.0 mm, do = 3.5 mm
0--
h = 200.0 mm, do = 3.5 mm
A
1
0.0
-%..
0.2
.
1
0.4
-
1
0.6
.
t
08
1.0
(14-Umf)/limf
Figure 5-60 Effect of exit orifice height by axial dispersion
plug flow model.
80
2.4
2.2
2.0
0
1.8
1.6
1.4 -
A - -- h= 100.0 mm, do = 5.1 mm
1.2
.0 h = 200.0 mm, do = 5.1 mm
1.0
0.0
0.2
0.4
0.6
08
1.0
(14-Unst)/ Unit
Figure 5-61 Effect of exit orifice height by tanks-in-series model.
81
5.8 The effect of fluidizing gas velocity
Our visual qualitative observations in all experiments support the
statement that the quality of fluidization depends on fluidization velocity.
2.4
2.2
2.0
.... ...
ft. ..........
.. ..11....... . .....
..ACA
1.8 -
..
....,
... ..
...
....4..., . .
.....
...
...,
A
1.6
h = 100.0 nun , do = 5.1 mm
1.4
h = 200.0 mm, do = 5.1 mm
0-- h= 100.0 mm , do = 3.5 mm
1.2
-0- 1.0
h = 200.0 nun , do = 3.5 mm
1
0.0
0.2
.
1
0.4
.
1
0.6
.
08
1.0
(U-Umf)/ Umf
Figure 5-62 Effect of fluidizing gas velocity by tanks-in-series model.
The particle mixing in the bed is very sensitive on the excess gas velocity.
The mixing quality of the beds are obviously varied by changing the level of
fluidizing gas velocity (u0). Therefore, one of the major parameters
influencing the solids mixing rate in the bed is the fluidizing gas velocity (u0).
82
When operating at gas velocities close to the minimum fluidization
velocity (um ?, it was found that the solids could not be readily fluidized. Often
the dense side of the bed would defluidize. However, if the fluidization
velocity is increased beyond um! ,the solids would fluidize smoothly.
1.2
1.0
0.8
CY ---------
----- ---.........
0.6
...................
h = 100.0 mm , do = 5.1 mm
0.4
h = 200.0 mm , do = 5.1 mm
0 h= 100.0 mm , do = 3.5 mm
0.2
h = 200.0 mm , do = 3.5 mm
0.0
1
0.0
0.2
0.4
0.6
08
1.0
(U-Umf)/ Umf
Figure 5-63 Effect of fluidizing gas velocity by axial dispersion
plug flow model.
The influences of fluidizing gas velocity on the model parameters, both
the number of tanks in series and the dispersion number, are shown in
Figures 5-60 and 5-61 respectively. An increasing gas velocity results in a
83
decreasing values of the model parameters; the dispersion number(U) and
the number of tanks (N) in series. On the other hand, a decrease in gas
velocity leads to higher values for model parameters, thus, improving the
flow pattern of solids.
84
5.9 The modeling of solid flow through the orifice
The measured mass flow rate of solid through the orifice is plotted
against the predicted value in Figure 5-64. The predicted value is calculated
from equation(3-27) in chapter 3. The predictions are shown to be in very
satisfactory agreement with the experimental results. Raw data of pressure
drop across the orifice are plotted in Figure A6-1 (appendix 6).
4.2
3.8
3.4
0
3.0
O/e
ie
,
1.8
1.4
1.0
,,
1.0
,
,
,,
,-
14
,,
,
cy,o
o
0
,-'o
,
18
/e
,
/
/
/
0
2.6
2.2
/
//
/
//
/e
/
,/
/
//
,,
,
,,
o
11:6411".
o o
22
26
30
34
38
Measured mass flow rate (g/s)
Figure 5-64 Measured versus predicted mass flow rate of solids
through the orifice.
4.2
85
5.10 Tracer recovery
The material balances on the amount of tracers are calculated for 16
experiments. In each experiment the amount of tracers are recovered from
sample in each sample cup and the bed material remained in the vessel after
finishing that experiment. The percent of recovered tracers are plotted in
Figure 5-65. The average of tracer recovery is approximate around 92%,
therefore, the tracer technique used in this study is trustworthy.
130
E3 Recovery from sample
120 -
Recovery from remained bed
80
70
60
50
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
Experiment #
Figure 5-65 The percent recovery of tracer.
86
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions
Both the axial dispersion plug flow and the tanks-in-series models
were proposed to describe the residence time distribution of solids flowing
through the two-compartment fluidized bed vessel. Experiments for
investigating the influences of the fluidizing gas velocity, the size of the
partition orifice, and the height of the overflow exit orifice were carried out to
test the models and evaluate the dispersion number (U) and the number of
tanks (N). The results of this study can be summarized as follows.
1. The axial dispersion plug flow and the tanks-in-series models satisfactorily
describe the solids flow in the two-compartment fluidized bed vessel.
2. The dispersion number (U) and the number of tanks in series (N) decrease
with the increase in gas velocity. The lower the excess gas velocity (uo-umf),
the more favorable RTD can be obtained.
3. Within the range of experiments performed in this study, the size of the
orifice diameter in the partition and the height of the overflow exit orifice
do not effect the model parameters,U andN.
4. The solid mass flow rate through the orifice between two adjacent
compartments can be predicted reasonable well by equation(3-27).
87
6.2 Recommendations for future work
This study is an initial investigation of multi-compartment fluidized
bed reactor. Future work will necessarily incorporate longer number of
compartments. We believe that the increase in number of compartments can
further improve the residence time distribution of solids. Recommendations
for future experimentation are given as follows:
1. To investigate the influence of the orifice size and the exit orifice
height. Longer variations of orifice diameter and different exit
orifice location are recommend.
2. The location of the orifice in the partition does not necessarily need
to be close to the distributor plate. The effect of the position of the
orifice may also be of interest.
3. The design of an orifice which would allow solids to move only one
direction (one way valve) from one compartment to the other is of
extreme importance.
88
BIBLIOGRAPHY
1. Argo, W. B. and Cova, D. R. , Ind. Eng. Chem., Process Design and
Development, Vol. 4, pp. 352 (1965).
2. Allen, C. M. and Taylor, E. A. , Trans. Amer. Soc. Mech. Engrs, Vol. 45, pp.
285 (1923).
3. Brenner, H. , "The diffusion model of longitudinal mixing in beds of
finite length (numerical value)", Chemical Engineering Science, Vol. 17
pp. 229-243 (1962).
4. Bischoff, K. B. and Levenspiel, 0. , "Fluid dispersion-generalization and
comparison of mathematical models_I (Generalization of Models)",
Chemical Engineering Science, Vol. 17, pp. 245-255 (1962).
5. Butt, J. B. , Reaction Kinetics and Reactor Design, Prentice-Hall, New Jersey
(1980).
6. Danckwerts, P. V. , "Continuous Flow Systems Distribution of Residence
Times", Chemical Engineering Science, 1(2):1-13 (1952).
7. Danckwerts, P. V. , Jenkins, J. W. and Place, G. , "The distribution of
Residence-Times in an industrial fluidized reactor", Chemical Engineering
Science, (3):26-35 (1954).
8. Davidson, J. F. , Clift, R. and Harrison, D. , Fluidization Second Edition ,
Academic Press, London (1985).
9. Denbigh, K. G. , Turner, J. C. R. , Chemical Reactor Theory Third Edition,
Cambridge University Press, London (1984).
10. Dudukovic, M. P. , Mills, P. L. , Chemical and Catalytic Reactor Modeling,
American Chemical Society, Washington D.C. (1984).
11. de Jong, J. A. H., "Vertical Air-Controlled Particle Flow from a Bungker
Through Circular Orifices", Powder Technol. , 3:297-286 (1969/70).
12. de Jong, J. A. H.,"Aerated Solids Flow Through a Vertical Standpipe Below
a Pneumatically Discharged Bunker", Powder Technol. , 12:197-200 (1975).
13. Evan, E. V. and Kenney, C. N. , Proc. Roy. Soc. (London), Vol. A284, pp. 540
(1965).
14. Fogler, H. S., Elements of Chemical Reaction Engineering Second Edition ,
Prentice-Hall, New Jersey (1992).
89
15. Froment, G.F. , Bischoff, K. B. , Chemical Reactor Analysis and Design
second Edition, John Wiley & Sons, New York (1990).
16. Giddings, J. C. and Seager, S. L. , Ind. Eng. Chem. Fundamentals, Vol. 1, pp.
277 (1962).
17. Gilliland, E. R. , Mason, E. A. , Ind. Eng. Chem., 41:1191 (1949).
18. Guanglin, S. , Lin, N. , Xiaoyu, C. and Gantang, C. , "Flow and
Performance Study in a Gas-Lift Multistage Fluidized Bed",
Fluidization'85 Science and Technology, Science Press, Beijing (1985).
19. Hill, C. G.,Jr. , An Introduction to Chemical Engineering Kinetics &
Reactor Design , John Wiley & Son, New York (1977).
20. Korbee, R. , Schouten, J. C. and Van den Bleek, C.M.,"Modelling
Interconnected Fluidized Bed Systems", AIChE Symposium Series
No. 281, Vol. 87.
22. Klinkenberg, A. , "Residence Time Distributions and Axial Spreading in
Flow Systems (With their Application in Chemical Engineering and Other
Fields)", Trans. Instn Chem. Engrs., Vol. 43, pp. T141-T156 (1965).
23. Kreyszig, E. , Advanced Engineering Mathematics seventh edition, John
Wiley & sons, New York(1993).
24. Kato, Y. , Morooka, S. and Nishiwaki, A. ,"Behavior of Dispersed- and
Continuous-Phase in Multi-stage Fluidized Beds for Gas-Liquid-Solid and
Gas-Liquid-Liquid systems" ,Fluidization'85 Science and Technology,
Science Press, Beijing, pp.217-227 (1985).
25. Kececioglu, I. , Wen, C. Y. and Keairns, D. L. , "Fate of Solids Fed
Pneumatically through a Jet into a Fluidized Bed", AIChE Journal, Vol. 30,
No. 1, pp. 99-110 (1984).
26. Krambeck, F. J. , Avidan, A. A. , Lee, C. K. and Lo, M. N. , "Predicting
Fluid-Bed Reactor Efficientcy Using Absorbing Gas Tracers", AIChE
Journal, Vol. 33, No. 10, 1727 (1987).
27. Kunii, D. , Levenspiel, 0. , Fluidization Engineering Second Edition,
Butterworth-Heinemann, Stoneham (1991).
28. Levenspiel, 0. , The Chemical Reactor Omnibook, OSU Book Stores,
Corvallis (1989).
29. Levenspiel, 0. , Chemical Reaction Engineering Second Edition, John
Wiley & Sons, New York (1972).
90
30. Muchi, J. , Mamuro, T. and Sasaki, K. , Kagakukogaku, Chemical
Engineering (Japan), Vol. 25, pp. 747 (1961).
31. Murphree, E. V. , Alexis, V., Jr. and Mayer, F. X. , Ind. Eng. Chem., Process
Design and Development, Vol. 3, pp. 381 (1964).
32. Molodtsof, Y. , Ould-Dris, A. and Bognar, G.,"Particle Flow Through
Orifices and Solids Discharge Rate from Hoppers", Fluidization VII ,
363-370.
33. MacMullin, R. B. and Weber, M. , Chem. Met. Eng., 42:254 (1935).
34. Naor, P. and Shinnar, R. , "Representation and Evaluation of Residence
Time Distributions", I &EC Fundamentals, 278-286.
35. Parent, J. R. and Roy, P. H. , Symposium on New Data in Fluid Mixing,
AIChE 57th Meeting, Boston, Dec. 6-10 (1964).
36. Resnick, W. , Heled, Y. , Klein, A. and Palm, E. , "Effect of differential
Pressure on Flow of Granular Solids Through Orfices", I & EC
Fundamentals , 5(8):392-396 (1966).
37. Richardso, J. F. and Carlos, C. R. , "Partide Speed Distribution in a
Fluidized system", Chemical Engineering Science, Vol. 22, pp. 705-706
(1967).
38. Smith, J. M. , Chemical Engineering Kinetics Third Edition, McGraw Hill, New York (1981).
39. Shah, Y. T. , Gas-Liquid-Solid Reactor Design, McGraw-Hill, New York
(1979).
40. Wen, C. Y. , Fan, L. T., Models for Flow Systems and Chemical Reactors ,
Marcel Decker, New York (1975).
41. Yagi, S. and Miyauchi, T. , "On the Residence Time Curves of the
Continuous Reactors", Kagakukogaku, Chemical Engineering (Japan),
Vol. 17, No. 10, pp. 382-386 (1953).
42. Zhang, J. Y. and Rudolph, V., 'Packed Solids Downflow in Standpipes",
Fluidization VII , 371-379.
43. Mendenhall, W. and Sincich, T. ,Statistics for the Engineering and
Computer Sciences Second Edition, Dellen, San Francisco (1988).
APPENDICES
91
APPENDIX 1
DISPERSION PROGRAM CODE
Variable Listing
Program Symbol
Definition
E
E(8) values which are calcylated from equation(11)
ETHETA
E(0) values read from input file for each experiment
IMAX
Maximum number of iterations
mu
p
N
Number of roots, p, from equation(12)
PI
it
XL
Left boundary of current interval
XLO
Original left boundary of search interval
XN
Midpoint of current interval
XR
Right boundary of current interval
XRO
Original right boundary of search interval
THETA
0 values read from input file for each experiment
TOL
Tolerance criterion
U
tiL
=D
, Dispersion number (U)
FORTRAN Code for the dispersion program
PROGRAM DISPERSION
*
*
The purpose of this program is to determine the dispersion number
*
(U) which is the parameter in the axial dispersion plug flow model,
92
*
equation(11). The result of this analysis is the parameter U that
*
is the sum of the squared differences between theoretical E(0) values by
*
equation(11) and experimental E(0) values.
*
*
Main program
*
REAL XL, XR, XN(1000), TOL, TERM(1000), THETA(100), U
REAL mu(1000), E(100), PI, SUM(100), ETHETA(100)
INTEGER IMAX, M, N
CALL INPUT (N, U, THETA, ETHETA, PI)
CALL HALF (N, U, mu, PI)
CALL RTD (mu, N, E, U, THETA, ETHETA)
CALL LEAST (THETA, ETHETA, E, SUM)
END
*
*
The data input subroutine which read the value of 0 and E(0) from
*
each experimental data. Both values are located in EXPERIMENT file.
*
SUBROUTINE INPUT (N, U, THETA, ETHETA, PI)
REAL U, THETA(100), ETHETA(100), PI
INTEGER N
PI = 22.0/7.0
N = 30
OPEN (UNIT=111, FILE.'EXPERIMENT.', STATUS='OLD')
DO 10 I = 1,100
READ (111,20) THETA(I), ETHETA(I)
10
CONTINUE
93
20
FORMAT (F6.4,1X,F6.4)
PRINT *,'
ENTER THE VALUE OF "U" ?'
READ *, U
END
The half-interval method is applied as a root-solving technique. The
basic idea is initialized with an interval containing a root and reduces
that interval by half.
SUBROUTINE HALF (N, U, mu, PI)
REAL XL, XR, XN(1000), TOL, mu(1000), U, PI
INTEGER I, IMAX, M, N
TOL =1E -9
IMAX =1000
XL0 = PI/2.
XRO = PI
XL = XLO
XR = XRO
INDEX =1
DO 30 M=1,N
IF ((INDEX .EQ. 1) .AND. (XL .GT. U)) THEN
XRO = XLO
XR = XRO
XLO = XRO-PI/2.
XL = XL0
INDEX = 2
94
ENDIF
DO 40 I=1,IMAX
XN(M)=(XL+XR)/2.0
FL =FUN (XL, U)
FR=FUN (XR, U)
FN=FUN (XN(M), U)
IF ((FN *FL) .LT. 0.0) THEN
XR=XN(M)
FR=FN
ELSE
XL=XN(M)
FL=FN
ENDIF
IF (ABS(XR -XL) .LT. TOL) GOTO 50
40
CONTINUE
50
XL0 = XL0 + PI
XR0 = xizo + PI
XL = XLO
XR = XRO
mu(M)=XN(M)
OPEN(1,FILE='OUTPUT1',STATUS=1UNKNOWN')
WRITE(1,60) M, mu(M), INDEX
30
CONTINUE
60
FORMAT (9X,I3,',1,4X,F15.4,',',1X,I2,',$)
END
*
*
The theoretical E(0) values corresponding to 8 values which read from
95
input file are evaluated by using equation(11).
*
SUBROUTINE RTD (mu, N, E, U, THETA, ETHETA)
REAL TERM(1000), THETA(100), E(100), mu(1000)
REAL ETHETA(100), U
INTEGER I, M, N, Q
DO 90 Q=1,100
E(Q) = 0.0
DO 70 M=1,N
TERM(M)=2.0
1 *(mu(M)*(U*SIN(mu(M))+mu(M)*COS(mu(M))))
1 /(U **2.+ 2.*U + mu(M)**2.)
1 *EXP(U-((U**2.+mu(M)**2.)/(2.*U))*THETA(Q))
E(Q) = E(Q) + TERM(M)
70
CONTINUE
DO 80 N=1,100
IF (THETA(N) .EQ. 0.0) THEN
ETHETA(N) = 0.0
E(N) = 0.0
ENDIF
80
CONTINUE
OPEN(1,FILE=tOUTPUT',STATUS='UNKNOWN')
WRITE(1,100) THETA(Q), ETHETA(Q), E(Q)
90
100
CONTINUE
FORMAT (8X,F9.5,',',7X,E12.7,',',7X,E12.7,1;)
END
96
*
The experimental data set points is compared to the theoretical one
*
which calculated from the model. The sum of the squared
*
differences of all data points is to be minimized.
*
SUBROUTINE LEAST (THETA, ETHETA, E, SUM)
REAL THETA(100), E(100), ETHETA(100)
REAL DIFF(100), SUM(100)
INTEGER N, I
DO 110 N=1,100
IF (THETA(N) .EQ. 0.0) THEN
ETHETA(N) = 0.0
E(N) = 0.0
ENDIF
110
CONTINUE
SUM(0) = 0.0
DO 120 I=1,100
IF (E(I) .LT. 0.0) THEN
E(I) = 0.0
ENDIF
DIFF(I) = (ETHETA(I)-E(I))*42.
SUM(I) = SUM(I-1) + DIFF(I)
120
CONTINUE
PRINT 130,SUM(I-1)
130
FORMAT ('The sum of the squared differences = ',F12.5)
END
*
*
This is the function to determine the vulues of "p", equation(12)
97
REAL FUNCTION FUN( X, U )
REAL X, U
IF (X .EQ. 0) THEN
X =1E -06
ENDIF
FUN = 1./TAN(X) - (X/U-U/X)/2.
RETURN
END
98
APPENDIX 2
THE PAIRED tANALYSIS
To perform a paired t-analysis, Mendenhall (1988) [43], we simply
calculate the treament (group) difference separately for each pair. The set of
differences constitutes a random sample from a single population of such
differences. When there is no difference in the original groups, the
population of differences will be centered at zero. Therefore the analysis
estimates the mean difference with particular attention paid to the possibility
that the mean might be zero. In the situation we are concerning, the potency
of the population of differences can also be less or greater than the mean
difference. So a two-tailed test of a paired t-analysis is applied.
Let introduce X and X2 as the data from population 1 and 2
1
respectively, and n is the number of pairs. The population of difference
(denoted by D) can be calculated from:
Di = X11 X2 1
i = 1,
n
The null and alternative hypotheses can be presented as follows:
Ho
:
µD =0
H
:
AD* 0
The null hypothesis (Ho) ) states that the mean of the population 1(/./.1) is
equal to the mean of the population 2 (µ2). Therefore, the mean of
population of differences (lip ) is equal to zero. On the other hand, the
alternative hypothesis (Ha) states that the mean of the population 1(iti) is not
99
equal to the mean of the population 2 (222). Thus, the mean of population of
differences (up ) is not equal to zero.
For a small-sample with two-tailed test, a test statistic can be decribed by:
b
t-statistic = SD/(n
,
05)
(b is the average of the population of differences =
n
with v = n -1
, pc, is the
hypothesized mean difference which is equal to zero in this case, and SD is
the standard deviation of the population of differences.
In this analysis the critical value (a) which is commonly accepted to be
0.05, indicate the level of significance of t-analysis (95%). The region of
rejection can be shown as:
f(t)
a/2 = 0.025
t
0.0
Rejection re:;11
Figure A2-1 t-distribution curve
t 0.975
t
ection region
100
The following is a calculation of t-statisic values. The hypothesized mean
difference (pp ) is equal to zero and the degrees of freedom are equal to 3.
Data from axial dispersion plug flow model
h=10 cm
h=20 cm
difference
h=10 cm
h=20 cm
difference
0.87
0.71
0.16
0.89
0.84
0.05
0.53
0.71
-0.18
0.75
0.76
-0.01
0.86
0.69
0.17
0.75
0.73
0.02
0.78
0.63
0.15
0.10
0.36
-0.26
D
0.0750
D
-0.0500
SD
0.1702
SD
0.1421
t-statistic
0.8813
t-statistic
-0.7036
Table A2-1 Effect of exit orifice
height when do = 5.1 mm
5
d 0=5.1mm d =3.mm
difference
Table A2-2 Effect of exit orifice
height when do = 3.5 mm
1
d0 =5.mm
d 0=3.5 rrun difference
0.87
0.89
-0.02
0.71
0.84
-0.13
0.53
0.75
-0.22
0.71
0.76
-0.05
0.86
0.75
0.11
0.69
0.73
-0.04
0.78
0.10
0.68
0.63
0.36
0.27
D
0.1375
D
0.0125
SD
0.3863
SD
0.1763
t-statistic
0.7119
t-statistic
0.1418
Table A2-3 Effect of orifice size
when h = 10 cm
Table A2-4 Effect of orifice size
when h = 20 cm
101
Data from tanks-in-series model
h=10 cm
h=20 cm
0.1
2.1
2.0
0.1
1.9
0.0
2.0
1.9
0.1
1.9
1.8
0.1
1.9
1.8
0.1
1.7
1.8
-0.1
1.1
1.6
-0.5
h=10 cm
h=20 cm
2.0
1.9
1.9
difference
difference
D
0.0250
D
-0.0500
SD
0.0957
SD
0.3000
t-statistic
0.5222
t-statistic
-0.3333
Table A2-5 Effect of exit orifice
height when do = 5.1 mm
d0=5.1mm d 0=3.5mm
difference
Table A2-6 Effect of exit orifice
height when do = 3.5 mm
d0=5.1min
d 0=3.5mm
difference
2.0
2.1
-0.1
1.9
2.0
-0.1
1.9
2.0
-0.1
1.9
1.9
0.0
1.9
1.9
0.0
1.8
1.8
0.0
1.7
1.1
0.6
1.8
1.6
0.2
D
0.1000
D
0.0250
SD
0.33665
SD
0.1258
t-statistic
0.5941
t-statistic
0.39736
Table A2-7 Effect of orifice size
when h = 10 cm
Table A2-8 Effect of orifice size
when h = 20 cm
Because the degrees of freedom are equal to 3 and the critical value (a)
is equal to 0.05, the tabulated values (appendix 5) for the regions of rejection
102
are t-statistic < -3.182 andt-statistic > 3.182. Therefore, all above tests show that
there are no sufficient reasons to belive that the variations of orifice size (3.5
and 5.1 mm) and the height of exit orifice (100.0 mm and 200.0 mm) are
influence to the dispersion number (U) and the number of tanks (N).
103
APPENDIX 3
Calculation of Minimum Fluidizing Velocity
The minimum fluidizing velocity (u..? can be determined by using the
equations from Kunii and Levenspiel (1991) [27] which are recommended by
Chitester et al..
dup
p
mi
d3 p
g
[ (28.7)2 + 0.0494( v
g(p9
It
where
d
P
= mean diameter of particle, m
Pg
= density of fluidizing gas, kg/m3
It
= viscosity of fluidizing gas, kg/m.s
p,
= density of solids partide, kg/m3
g
= acceleration of gravity, 9.8
m/s'
-p)g
'
Hun
- 28.7
104
APPENDIX 4
Percentiles of t-distributions
t (A; v)} = A
Entry is t (A; v) where P
t (A; v)
A
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
.60
.70
.80
.85
.90
.95
.975
0.325
0.289
0.277
0.271
0.267
0.727
0.617
0.584
0.569
0.559
1.376
3.078
1.886
1.638
0.920
1.533
1.476
6.314
2.920
2.353
2.132
2.015
12.706
0.978
1.963
1.386
1.250
1.190
1.156
0.265
0.263
0.262
0.261
0.260
0.553
0.549
0.546
0.543
0.542
0.906
0.896
0.889
0.883
0.879
1.134
1.119
1.108
1.100
1.093
1.440
1.415
1.397
1.383
1.372
1.943
1.895
1.860
1.833
1.812
2.447
2.365
2.306
2.262
2.228
0.260
0.259
0.259
0.258
0.258
0.540
0.539
0.537
0.537
0.536
0.876
0.873
0.870
0.868
0.866
1.088
1.083
1.079
1.076
1.074
1.363
1.356
1.350
1.345
1.796
1.782
2.201
0.258
0.257
0.257
0.257
0.257
0.535
0.534
0.534
0.533
0.533
0.1165
1.071
0.863
0.862
1.069
1.067
1.066
1.064
1.337
11.333
0.257
0.256
0.532
0.532
0.532
0.531
0.531
0.859
0.858
0.858
0.857
0.856
1.063
0.531
0.531
0.256
0.256
0.256
0.256
0.256
0.256
0.256
0.256
40
60
0.255
120
0.254
0.253
40
0.25.4
1.061
0.941
1341
1.771
1.761
1.753
4.303
3.182
2.776
2.571
2.179
2.160
2.145
2.131
1.746
1.740
1.734
1.729
1.725
2.120
2.110
1.721
1.060
1.059
1.058
1.323
1.321
1.319
1.318
1.316
2.080
2.074
2.069
2.064
2.060
1.058
1.057
1.056
1.055
1.055
1.315
1.314
1.313
1.311
1.310
1.706
1.703
0.530
0.530
0.530
0.856
0.855
0.855
0.854
0.854
1.699
1.697
2.056
2.052
2.048
2.045
2.042
0.529
0.527
0.526
0.524
0.851
0.848
0.845
0.842
1.050
1.045
1.303
1.296
1.289
1.282
1.684
2.021
1.671
2.000
1.980
1.960
0.861
0.860
1.061
1.041
1.036
1.330
1.328
1.325
1.717
1.714
1.711
1.708
1.701
1.658
1.645
2.101
2.093
2.086
105
APPENDIX $
Gamma function table
x
T(x)
x
1(x)
1.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1.00000
.99433
.98884
.98355
.97844
.97350
.96874
.96415
.95973
.95546
1.50
1.51
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
.88623
.88659
.88704
.88757
.88818
.88887
.88964
.89049
.89142
.89243
1.10
1.13
1.14
1.16
1.16
1.17
1.18
1.19
.95135
.94740
.94359
.93993
.93642
.93304
.92980
.92670
.92373
.92089
1.60
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
.89352
.89468
.89592
.89724
.89864
.90012
.90167
.90330
.90500
.90678
1.20
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
.91817
.91558
.91311
.91075
.90852
.90640
.90440
.90250
.90072
.89904
1.70
1.71
1.72
1.73
1.74
1.75
1.76
1.77
1.78
1.79
.90864
.91057
.91258
.91467
.91683
.91906
.92137
.92376
.92623
.92877
1.11
1.12
1.30
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
.89747
.89600
.89464
.89338
.89222
.89115
.89018
.88931
.88854
.88785
1.80
1.81
1.82
1.83
1.84
1.85
1.86
1.87
1.88
1.89
.93138
.93408
.93685
.93969
.94261
.94561
.94869
.95184
.95507
.95838
1.40
1.45
1.46
1.47
1.48
1.49
.88726
.88676
.88636
.88604
.88581
.88566
.88560
.88563
.88575
.88595
1.90
1.91
1.92
1.93
1.94
1.95
1.96
1.97
1.98
1.99
.96177
.96523
.96877
.97240
.97610
.97988
.98374
.98768
.99171
.99581
1.50
.88623
2.00
1.00000
1.41
1.42
1.43
1.44
106
APPENDIX 6
Apo and W
Pressure drop across the orifice (Apo) ) and the amount of solid hold-up
(W) measured after turning of the fluidization gas for all 16 experiments are
shown in the curves below.
2.5
h = 100.0 mm
2.0 -
h = 200.0 mm
O
0
1.5 -
Q
0
0
1.0:
o
0.5
GI
0.0
I
0
0
CI
1
o
I
I
o
I
I
2345
o
I
I
i
I
I
I
El
iii
o
I
I
6 7 8 9 10 11 12 13 14 15 16 17
Exp #
Figure A6-1 Pressure drop across the orifice for 16 experiments.
107
2400.0
o
2000.0-
0
0
0
In
0
Q
0
1600.0-
A
1200.0-
4
A A
800.0-
400.0
0
A
A
1
2
A
h= 100.0 mm
0
h = 200.0 mm
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17
Exp #
Figure A6-2 Weight of solid hold-up after finishing 16 experiments.