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Experimental Results and Models for SolidLiquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids.

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Dev. Chem. Eng. Mineral Process. 12(3/4), pp. 403-426, 2004.
Experimental Results and Models for
Solid/Liquid Fluidized Beds Involving
Newtonian and Non-Newtonian Liquids
M. Aghajad3,H. Muller-Steinhagen2*and
M. Jamialahmadi3
I
Dept of Chemical and Process Engineering, University of Surrey,
Guildford, Surrey, UK
Institute for Thermodynamics and Thermal Engineering,
University of Stuttgart, Stuttgart, Germany
Petroleum University of Technology, Ahwaz, Iran
Fluidization technology relies almost solely on fluid/particle interaction wherein the
liquid phase may exhibit Newtonian or non-Newtonian behavior. The steady motion
of particles and the velocity-voidage relationship are the most important design
parameters f o r fluidization, providing the basis for the prediction of heat and mass
transfer coeflcients and information on hydrodynamic conditions. A summary of the
literature on particle settling velocity, minimum fluidization velocity and velocityvoidage relationship is supplemented by new experimental results, which extend the
range of investigated solid and liquid phase physical properties. Correlations for
particle settling velocity and velocity-voidage relationship are developed and ver$ed
against experimental data.
Introduction
Solidlliquid fluidized beds are used throughout the process industries for catalytic
cracking, hydrometallurgical operations, crystallization and sedimentation. In
processes where severe fouling of the heat transfer surfaces is expected, installation of
a fluidized bed heat exchanger where the solid phase is cylindrical stainless steel or
tantalum particles is also recommended. Over the years, fluidized beds have been the
subject of intensive research, covering both fundamental and applied aspects. As a
result, considerable progress has been made towards understanding the detailed
hydrodynamics as well as in the development of sound methodologies for the design
of fluidized bed systems. Most of these research efforts have been focused on low
viscosity liquids with Newtonian behavior, in which the solid phase consists of
spherical glass particles. Previous investigations on the hydrodynamic behavior of
these systems have been documented and discussed, most recently by Jamialahmadi
and Muller-Steinhagen [ 11.
* Author for correspondence (hms@itw.uni-stuttgart.de).
403
M.Aghajani, H. Muller-Steinhagen and M.Jamialahmadi
In recent years, solidliquid fluidized beds are also finding increasing application
in the treatment of aqueous wastes, heavy oil cracking, polymerization, biological
oxidation and fermentation. Here, the liquid phase is viscous with non-Newtonian
behavior. Only limited information is available on the hydrodynamic and thermal
performance of fluidized beds with non-Newtonian liquids. Fluidized beds involving
high viscosity liquids and non-spherical heavy particles have been the subject of even
less investigations.
It is generally agreed that the terminal settling velocity of the particles, the
minimum fluidization velocity and its corresponding bed voidage and the velocityvoidage relationship are the most important parameters of fluidization technology.
Considerable empirical progress has been made on the understanding of these
parameters and in establishing the velocity-voidage relationship in fluidized beds
involving low viscosity Newtonian liquids. Correlations have been proposed, which
may be found in several papers, e.g. Davidson et al. [2], and in a book recently
published by Chhabra [3]. For non-Newtonian liquids, no such information is
available to design engineers. For t h s reason, more detailed experimental
investigation and analyses of fluidized beds are necessary. The aim of the present
investigation was, therefore, to measure the hydrodynamic parameters over wide
ranges of particle size, density and shape using liquids with Newtonian and nonNewtonian behavior. The predictions of various published correlations are compared
with these experimental data, and new correlations and methodologies are presented
for prediction of the terminal settling velocity of the particles, the minimum
fluidization velocity and its corresponding voidage, and the velocity-voidage
relationship in beds which are fluidized with Newtonian and non-Newtonian liquids.
Experimental Equipment and Procedure
Particle settling velocity apparatus
Experiments to determine terminal falling velocities of different spherical and
cylindrical particles are performed in vertical columns of different diameter ranging
fiom 2.5 to 10 cm, and variable height (see Figure 1). All tests were performed in a
random mode according to the following procedure.
Initially, the columns were cleaned and the test liquid introduced. The system was
then left for about 12 hours to allow trapped air bubbles to escape and homogenous
conditions to be established throughout the system. The test particles were soaked in
the test liquid for about 6 hours before they were introduced below the surface of the
liquid, as close to the center of the column as possible. The terminal settling velocity
of each particle was measured by timing its descent over a pre-marked distance using
an electronic timer and a video camera. Each terminal settling velocity represents an
average of ten repeat measurements. In the case of distilled water which has a low
viscosity, the particle Reynolds number of heavy particles was above 1000, and hence
sinking occurred very quickly and measurements were difficult and subject to error.
To minimize the error, in this case each value of terminal falling velocity represents
an average of at least twenty repeat measurements. Some measurements were
repeated later to check the reproducibility of the experiments, which proved to be
excellent.
404
SolidILiquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
Figure 1. Schematic diagram of experimentalfluidized bed system.
Fluidized bed system
A schematic diagram of the fluidized bed system is shown in Figure 1. The setup
consists of fluidized bed column, pump, supply tank, liquid flow meter, control
valves, AP cell and U-tube manometer with mercury under water as the manometric
liquid. The column consists of calming, test and expansion sections. A stainless steel
screen fitted above the calming section of the bed supports the solid particles. The
liquid passing through the column and the return lines is discharged into the supply
tank. The liquid temperature between the inlet and outlet of the test section was
measured for determination of the liquid viscosity and density. A personal computer
was used for data acquisition. Various types of spherical and cylindrical particles
were used as the solid phase in this investigation. The physical properties of the
particles are listed in Table 1.
The minimum liquid fluidization velocity was measured from the intersection of
pressure drop-velocity plots in fixed and fluidized bed regimes. For measurement of
liquid velocity at very low volumetric flow rates, liquid from the column was passed
through a graduated cylindrical column before returning to the supply tank. No
significant difference was found when the minimum liquid fluidization velocity was
determined with increasing or decreasing liquid velocity. Bed voidage is defined as:
405
M. Aghajani, H. Muller-Steinhagen and M. famialahmadi
For a given mass of particles, therefore, only the bed height has to be measured.
To start the experiment, about 0.3 kg of particles are fed into the column and the
minimum height of the bed is recorded as the static bed height. Liquid is admitted into
the system and the average bed expansion height is recorded up to the limit when the
particles are at the point of being carried away from the system. Expanded bed height
data were obtained using both increasing and decreasing liquid flow rates to
investigate the possibility of a hysteresis effect. Static bed voidages were also
measured separately for each particle type, in a column of the same diameter but of
smaller height, by measuring the volume of water required to fill the void space. The
results of these measurements are also included in Table 1.
-
Table 1. Physical properties of the solid particles.
Cylindrical
Spherical
/Aluminum
Aluminum
Brass
1I
Stainless Steel
Tantalum
Glass
Glass
Glass
Lead
Lead
Carbon Steel
Carbon Steel
Stainless Steel
2x3 mm
3x3mm
:;:
I
I
PP
[kg 1 m31
2.62
3.43
4
3
3.7
2600
2600
8500
7900
7900
17600
2700
2700
2700
11350
11350
7800
7800
8100
Y
ESB
[-I
[-I
0.86
0.87
0.87
0.87
0.87
0.87
0.40
0.4 1
0.41
0.4 1
0.40
0.41
6
9
9
9
4
16
1
1
1
1
1
1
1
1
0.39
0.39
0.40
0.39
0.40
0.40
0.39
0.40
3.14
7.1
12.6
6.6
12.6
12.6
7.1
10.7
A*
[m21
-- -
I
I
Test liquids
In order to cover a wide range of particle Reynolds numbers, a series of aqueous
solutions of sugar and carboxymethylcellulose (CMC) were used as Newtonian and
non-Newtonian liquids. The concentration of sugar and CMC was varied from 0 to
60 wt% and 0 to 1 wt% respectively. Rheology of these solutions was determined on
a Carrimed 50 controlled stress rheometer equipped with a cone and plate assembly.
The temperature was controlled by a Peltier element situated in the plate. As
expected, the sugar solutions exhibited a constant shear viscosity whereas the CMC
solutions displayed varying levels of pseudoplastic behavior. An examination of the
steady shear stress-shear rate data suggested that the two-parameter power law fluid
406
SolidILiquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
model provides an adequate representation of their pseudoplastic behavior. For steady
shear, the power law is written as:
r = kZ;"
(2)
where the best values of k and n were estimated using a nonlinear regression
approach. The resulting values, with the density of each solution, are given in Table 2.
It has been assumed that the average shear rate over the entire particle surface is
d d P .With this definition, the apparent viscosity is given by:
\"PI
Table 2. Physical properties of Newtonian and non-Newtonian solutions.
CMC solutions (25°C)
Sugar solutions (25°C)
wt%
20
40
60
PI
Power Law Model : r = k y"
[ kg/m3 ]
P
[ Pa. s ]
wt%
1070
1150
1300
0.001714
0.005359
0.044410
0.2
0.4
0.6
0.8
1
PI
[kg/m3]
998.3
998.3
998.3
998.3
998.3
k
[Pa.s"]
0.0697
0.2084
0.3413
0.5756
2.538
n
[-]
0.7468
0.6953
0.6882
0.6729
0.5519
i
Results and Discussion
Particle terminal velocity corrected for wall effect
Particle settling velocity is essential for the prediction of bed voidage, and heat and
mass transfer coefficients. The substantial number of investigations on this topic
reflects the importance of particle settling velocity in solid-liquid systems (e.g. Lali et
al. [4]; Chhabra [5]). The previous findings, particularly with respect to the influence
of particle shape, density and the behavior of the liquid phase on the terminal
velocity, are often of a contradictory nature. To clarify these uncertainties, extensive
new experimental results are presented in t h s work for regions where the published
data are insufficient.
It is well known that the walls of the column exert an additional retardation effect
on a settling solid particle. The extent of this wall effect is usually quantified by
introducing a wall factor defined as:
407
M. Aghajani, H. Miiller-Steinhagen and M. Jamialahmadi
Terminal settling velocity in the presence of wall effect =-u,
= Terminal settling velocity in the abscence of wall effect u,
(4)
It is obvious that f has a positive value less than 1. Figure 2 display representative
results illustrating the relationshp between the measured terminal velocity and the
settling column diameter for 3x3 mm aluminum particles in solutions with various
physical properties. In all cases, the variation is almost linear. Therefore, it can easily
be extrapolated to Dh+ m to estimate the corresponding terminal fall velocity of the
particles in the absence of any wall effect.
0.35
1I
c 1
-
0.3
1
Aluminum particle 1
d, = 3.43 mm
v)
2
0.25
.-d
............
?*..
U
-=>
0
0.2
.-
(0
0
€
0
0.15
c)
0
U
.-
5
0.1
................
............
................
................
L
0.05
0
10
20
30
40
50
l/Dh [l/ml
Figure 2. Variation of measured settling velocity of an aluminum particle with
column diameter in various solutions.
The extracted wall effect results are plotted as a f i c t i o n of d,,/Dh in Figure 3. The
results conform (with an absolute mean average error of less than 6%) to the
following equation proposed by Richardson and Zaki [6]:
);(
;
Log,, - =-
408
SolidILiquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
1-00
- Richardson and Zaki [6]
0.60
1
0.00
0.05
1
1
1
1
1
1
1
1
1
0.10
1
0.15
1
1
1
1
0.20
Diameter ratio, dJDh
Figure 3. Variation ofthe wall factor with diameter ratio.
Correlation of particle free fall velocity data
The forces acting on a particle moving with its terminal velocity through a fluid are in
dynamic equilibrium Namely, the effective weight (the gravitational force minus
buoyancy force) is equal to the drag force. Thus for the free terminal velocity yields:
Equation (6) may be written in terms of fiee fall particle Reynolds number:
Equation (6) shows that the terminal velocity of a particle is inversely proportional to
the drag coefficient, Co. Theoretically, the drag coefficient can be obtained from the
solution of the equation of momentum for the system. in the absence of the inertial
terms it yields:
c, =-
24
Re p m
409
M.Aghajani, H. Miiller-Steinhagen and M. Jamialahmadi
As the particle Reynolds number increases, the inertial terms become increasingly
significant in the momentum equation and no analfical solutions are possible under
these conditions. Therefore, almost all drag coefficients reported for higher Reynolds
number have been obtained from experiments. These results are generally presented
in graphical form as a complex function of the flow conditions. Most of this work has
been reviewed and critically evaluated by several investigators (e.g. Clift et al. [7];
Khan and Richardson [8]). Clift et al. [7] and Lydersen [9] fitted these curves to a
series of straight lines for calculating the value of CD for a given value of Re,
embracing the complete standard drag curve. Description of these equations and
conditions for which their application has been recommended are summarized in
Table 3. For a given value of Reynolds number, calculation of CD using one of these
correlations is a straightforward matter. Unfortunately, the form of these correlations
is not convenient for the calculation of the free settling velocity for a given solidliquid system as the unknown velocity appears in both Repmand CD. This difficulty is
overcome by writing Equation (7) in terms of the Arclumedes number, which is not a
finction of the terminal velocity:
3
gP, (Pp - PI ) d ;
A r = - C D R e 2Pm =
4
P2
(9)
Equation (9) shows that the particle free fall Reynolds number is a function of
Archimedes number only, and can be better presented in the form:
Re, = F (Ar)
(10)
Several attempts have been made to establish this functionality between the
Archimedes number and the particle Reynolds number. Most of the work is
documented and critically evaluated by Khan and Richardson [8]. These authors
proposed the following correlation based on a large body of experimental data
extracted from the literature:
Hartman et al. [15] proposed the following explicit relation for the prediction of the
free fall velocity of a spherical particle in an infinite medium, which avoids the
iterative solution of Equation (7) and the equations for CD:
.&I,
Repa = P(C)+log,o
m
where
P(C)= ((0.0017795C - 0.0573)C + 1.0315)C - 1.26222
R(C)= 0.99947+ 0.01853sin(1.848C -3.14)
and C = Log,,Ar
410
(12)
(13)
(14)
Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
Table 3. Recommended correlations for drag coeflcients.
Re, < 0.01
C
, =
2(I+ 0. I3 I5Ref 82-o
0.01 <Re,520
Re,
C, = "(I
+0.1935Re;
,OS)
20 5 Re, 5 260
Re,
log c, = 1.6435- 1.12420+0.155802
IogC, = -2.4571+2.55580-0.92950~
260 < Re,
+0.1O49w3
< 1500
1500<Re,< I.2x104
IOgCD = -1.91 81 + 0.6370- 0.0636W2
1 . 2 ~ 1<Re$
0~ 4.4~10~
log c, = -4.339
4 . 4 ~ 1 <Reps
0 ~ 3.38~10'
+ I ,5809W - 0. I 546 O 2
CD = 29.78- 5.30
3 . 3 8 1~0' <Re, 5 4x 10'
CD = 0.10 - 0.49
4x10' <Re,< lo6
C, =0.19--
8x104
lo6< Re,
Re,
Chhabra [ 131 R ~ ,= a Arb
For non-Newtonian liquid;
n = Rate index of
power law model
b = -0'954
0.16
Table 4 shows the percentage of absolute error, from a comparison between
measured and calculated particle settling velocities, for the most frequently
recommended correlations and models for Newtonian and non-Newtonian solutions.
The best agreement for Newtonian solutions is obtained with the method suggested by
Hartman et al. [ 141, followed by the standard equation of Lydersen [9].
41 I
M. Aghajani, H. Miiller-Steinhagen and M. Jamialahmadi
Table 4. Absolute relative error (96) of correlations for the prediction of urn.
Khan and Richardson [8]
Few attempts have been made to establish the functional dependence of
Archimedes number on particle Reynolds number for non-Newtonian solutions.
Chhabra [13]presented a model for non-Newtonian fluids in terms of flow behavior
index, which is also listed in Table 3. The best agreement for non-Newtonian fluids is
obtained with the correlation of Khan and Richardson [8], as shown in Table 4.
However, the variation between the predictions of the various correlations is quite
considerable. Improved values of the constants used in Equation (1 1) are determined
by non-linear regression analysis using all available data:
Re,,
= 0.334Ar0.654
(15)
For non-Newtonian solutions, the apparent viscosity (pa) must be used in both
Rep, and Ar. Therefore, Equation (15) is implicit with respect to the fiee particle
terminal settling velocity (u,) and must be solved in parallel with Equation (3).
Equation (15 ) predicts the fiee falling velocity of particles in non-Newtonian
solutions with an absolute mean average error of less than 10%.
Minimum fruidization vefocity
The incipient or minimum fluidization point represents the transition between the
fixed and fluidized states. It is readily recognized that the minimum fluidization
velocity is one of the main design variables in such applications. Several investigators
have studied this situation and developed correlations that can be used for Newtonian
fluids. Most of these correlations have been compiled and critically evaluated by
Couderc [ 151 who concluded that the minimum fluidization velocity for Newtonian
liquids can be predicted with an average accuracy of about 15 to 20%, and larger
errors may be encountered for non-spherical particles. In contrast, very little
information is available on fluidized bed systems involving non-Newtonian fluids.
The limited amount of work which is available in this area has been documented by
Chhabra [3].
412
Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
In this present study, the minimum fluidization velocity (ud) was obtained by
plotting pressure drop gradient versus liquid phase Reynolds number for fixed and
fluidized regimes. The transition point is designated as the minimum fluidization
velocity, and the corresponding bed voidage is denoted by E&. Thus for Rep-Red, the
pressure drop across the bed remains constant as illustrated in Figure 4 for typical
experiments with cylindrical particles. All other results obtained in this study conform
to this behavior.
I00
-E
.
a"
25
90
80
70
Y
a
E
60
20
10
0
0
200
400
600
800
1000
Particle Reynolds number, Re,
Figure 4. Typical variation ofpressure drop with particle Reynolds number for
cylindrical particles.
Most attempts to develop models for the estimation of minimum fluidization
velocity are based on the fact that at the point of incipient fluidization, the pressure
drop for a fixed bed is equal to the apparent buoyancy weight of solid particles. The
most widely used correlation for fixed beds is that of Ergun [ 161:
M.Aghajani, H. Muller-Steinhagenand M.Jarnialahrnadi
At incipient fluidization, this pressure drop is equal to:
AP
-=(I - Em/ 1( P , - P I )g
L
Combining Equations (1 6 ) and (1 7) yields:
1.75
-Re$+
3
150( 1- zm/ )
YEmf
v/
2 3
Em/
Remf- Ar = 0
Equation (1 8) is a dimensionless equation for the minimum fluidization velocity. The
main problem with the solution of this equation is that ~dis unknown. Based on
experimental observations, Wen and Yu [ 171 suggested that:
1
= 14
3
Y
Em/
-
and
Y
'
2
= 11
3
Em/
(19)
With these approximations the solution of Equation (18) yields:
Re,/ = /(33.7)*
+ 0.0408Ar - 33.7
(20)
Comparison of all experimental data for Newtonian fluids with Reynolds numbers
calculated according to Equation (20) shows an absolute mean average error of about
7%. This is a good indication that Equation (20) is suitable for the prediction of the
minimum fluidization velocity of solidliquid fluidized beds with Newtonian
behavior. Chhabra [3]compiled all the available correlations for the prediction of u d
for systems with a non-Newtonian nature. Table 5 shows a comparison between
measured and calculated minimum fluidization velocities for these correlations.
Evidently, none of these correlations seems to predict the experimental data for nonNewtonian solutions satisfactorily. The present investigation shows that Equation (20)
predicts the experimental data for non-Newtonian solutions with an absolute mean
average error of lo%, if the apparent viscosity is used in the Red and Ar numbers.
Table 5. Performance of correlationsfor predicting minimum fluidization velocity.
Kumar and Upadhyay [2 11
Kawase and Ulbrecht [22]
414
Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
Velocitpvoidage relationship
Theoretical and empirical correlations available for the prediction of design
parameters, such as heat and mass transfer coefficients, are strong functions of the bed
voidage. Therefore, accurate knowledge of t h s relationship is crucial for the reliable
estimation of transfer coefficients. Considerable progress has been made in
establishing the velocity-voidage relationship in fluidized bed systems, and several
correlations have been proposed for its prediction. Jamialahmadi and MiillerSteinhagen [25] compiled the published correlations and conditions for which their
application has been recommended. Most of these correlations are empirical and
apply only over a restricted range of Reynolds number, for specific particles, or for
Newtonian fluids. Furthermore, the prediction of bed voidage requires the use of
iterative solutions for most of these correlations.
Development of a new bed-voidage correlation
When a liquid flows upwards through a bed of particles at low velocity (u < ud), a
fixed bed exists where the solid particles rest on top of each other and on the bottom
of the column. In this regime, the height and, therefore, the voidage of the bed
remains constant at fixed bed voidage
while the pressure drop increases with
liquid velocity. If the velocity of the fluid is sufficiently high (u > ud), the solid
particles will be freely supported in the liquid to create a fluidized bed.
In fluidized beds, the height of the bed increases while the pressure drop balances
the buoyant weight of the fluidized particles and remains constant. Thus it can be
written:
) = cons tan t
W ( U , E - gSE
(21)
Differentiating Equation (2 1) gives:
aAP
dAP = -du
au
aAP
+
a(E
-
d(& - E S B ) = 0
Pressure drop is a function of bed voidage and liquid velocity and can be expressed
as:
The velocity dependency may be justified on theoretical grounds with the
exponent x talung values of 1 and 2 for laminar conditions where Stokes' Law is
applicable and for turbulent conditions where Newton's law is valid, respectively. On
the other hand, the voidage dependency is chosen quite arbitrarily. The partial
derivatives in Equation (22) can be evaluated from Equation (23). Differentiating
Equation (23) partially with respect to velocity and keeping the bed voidage constant
produces:
415
M. Aghajani, H. Miiller-Steinhagenand M. Jamialahrnadi
Similarly:
Substituting the values given by Equations (24) and (25) into Equation (22), and after
some algebraic manipulation and rearrangement, results in:
Equation (26) is linear and subject to the condition that, at u = u,, the bed voidage is
unity. Its solution in the range of umf S u Iu, yields:
where the exponent z is known as the fluidization index, and is equal to:
7
Y
= -X
Rearranging Equation (27) gives the following explicit expression for the bed
voidage:
It is well known that the hydrodynamic behavior of fluidized beds is different and
independent from that of the corresponding fixed bed. However, this is only correct
when fluidization is fully developed and the superficial liquid velocity is considerably
larger than the minimum fluidization velocity. For low superficial velocity close to
the minimum fluidization velocity (u,f), the hydrodynamic behavior of fluidized beds
is close to that of a static bed. The results of this investigation show that there is a
smooth transition between fixed and fluidized beds. This is confirmed by several
other investigators, e.g. Foscolo et al. [26] who stated that all the empirical evidence
indicates a smooth transition from fixed to fluidized bed behavior. Therefore, in the
range of low superficial velocities close to the minimum fluidization velocity, the
fluidized bed voidage in Equation (29) approaches the constant static bed voidage.
Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
Prediction of static bed voidage (ssB)
The static bed voidage (ESB ) in Equation (29) can be calculated from the correlation
of Fan and Thinakaran [27]:
€SB
=
0'15'
+ 0.365
;
Dh
2 2.033
dP
Equation (30) generally under-predicts the static bed voidage. Considering that in
solidtliquid fluidization the diameter of particles is relatively large (e.g. d, >2 mm),
the interaction forces between particles, and between particles and walls, are not taken
into account in the following equations for predicting static bed voidage. At the
expense of a slight loss in accuracy this is a reasonable assumption. Improved values
of the constants used in Equation (30) are determined for spherical and cylindrical
particles by regression using all available data. For spherical particles, the static bed
voidage is obtained from:
and for cylindrical particles:
0.15
Equations (31) and (32) predict the static bed voidage for spherical and cylindrical
particles with an absolute mean average error of less than 3%.
Prediction of thefluidization index (zj
When the particles are fluidized with non-Newtonian CMC solutions, the behavior of
the bed was similar to that observed for Newtonian solutions. Figure 5 shows typical
plots of Equation (27) for various particles fluidized with Newtonian and nonNewtonian solutions. The results obtained with the remaining solidliquid
combinations also conform to this behavior, and no noticeable differences were
observed. For each experimental run, the value of z has been determined by
performing regression analysis on velocitylvoidage data. Examination of the
calculated values suggests that:
z=p($,
Re,,)
(33)
417
M. Aghajani, H.Miiller-Steinhagen and M. Jamialahmadi
1
0.9
0.8
0.7
0.6
-A
5
0.5
0.4
\
1”
0.3
0.2
0.1
0.2
0.1
0.3
0.4
(E-ESB)/(~-ESB)
0.5
0.6 0.7 0.8 0.9 1
1-1
Figure 5. Typical plot of Equation (27) for Newtonian and non-Newtonian solutions.
The effect of the wall is considered when the terminal velocity is corrected for
wall effect using Equation ( 5 ) from Richardson and Zaki [ 6 ] . Hence, Equation (33)
reduces to:
z = F(Re,)
(34)
The calculated fluidization index (z) for various particle sizes, types and shapes, and
fluids with Newtonian and non-Newtonian nature, plus all the z values which are
recalculated from the reported experimental data by Richardson and Zaki [ 6 ] , are
plotted as a function of particle terminal Reynolds number (Re,, ) in Figure 6. The
general shape of the curve is similar to that observed for the variation of the
Richardson and Zaki exponent, n, with Re,,. At low and high particle terminal
Reynolds numbers (Re,, < 0.2 and Re,, > 500), the fluidization index (z) is almost
independent of particle Reynolds number. Between these boundaries, the value of z
decreases graduaIly as Re,, increases. This functionality can be well represented by
the following equation:
0.65(2 + 0.5Rei:’)
Z=
418
(1 + 0.5 Re’:;
)
(35)
Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
Equation (35) is independent of the nature of the fluids and it can equally well be used
for Newtonian and non-Newtonian solutions. The nature of the solutions is taken into
consideration by using the apparent viscosity in the calculation of parameters such as
the Archimedes number and the particle Reynolds number.
1.6
1.4
4
0
- I .2
L
N
J
P)
1
‘El
.-
‘C
0
0.8
cp
0.6
E
0.4
: Newtonian solution 0: non - Newtonian solution
0.2
PI = 998-1 300 Kdm’
p = 0.001- 0.05 Pa. s
Solid physical properties
p, = 998 Kg/m’
pp= 2600 - 17600 Kg/m’
p = 0.01 - 0.55 pa.
d,= 1 - 5 mm
0
0.0001
0.001
0.01
0.1
1
10
Particle Reynolds number, Rep,
100
1000
10000
[-I
Figure 6. Variation ofjluidization index with particle terminal Reynolds number for
Newtonian and non-Newtonian solutions.
Comparison with experimental data
The predictions from Equations (29) and (35) for 3x3 mm brass particles and fluids
with Newtonian and non-Newtonian behavior are shown in Figures 7a and 7b,
respectively. The calculated trends are in excellent agreement with the experimental
results of all investigators. Furthermore, for low liquid velocity, Equation (29)
approaches the static bed voidage (ESB ) as a limiting condition. The applicability of
the presented model for Newtonian and non-Newtonian (shear-thinning power law)
fluids is demonstrated in Figures 8 and 9, where the experimental data of different
investigators and also the data measured in this investigation are compared with those
predicted from Equation (29). Table 6 shows the content of the database used for this
comparison.
419
M.Aghajani, H. Muller-Steinhagen and M.Jamialahmadi
1
0.9
--
0.8
6
0.7
I
0
M
I
e0
>
2
0.6
0.5
G
0.3
0
0.1
0.2
0.3
0.5
0.4
0.6
0.7
0.8
Superficial liquid velocity, us Ids]
Figure 7a. Comparison of measured and predicted bed voidages for
Newtonian solutions.
1
A
9/
/
0.9
--
0.8
w
6 0.7
4 = 3.43 mm
I
pp = 8500 Kg/m'
M
.-
U
0
r
z
0.6
.
Solutions Experiment Predicted
l.Owt%CMC
...._.........
0.8wt% CMC
A
-..0.6wPhCMC
A
0.4 wt%CMC
0
0.5
(j
n?wP/nrMr
-
------
0.3
0
0.1
0.2
0.3
0.4
0.5
0.6
Superficial liquid velocity, us Im/s]
Figure 76. Comparison of measured and predicted bed voidages for
non-Newtonian solutions.
420
Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
0.4
0.5
0.6
0.7
E ( experimental
0.8
0.9
1
)
Figure 8. Comparison of measured bed voidages with values calculated
from Equation (29).
0.4
0.5
0.6
0.17
0.8
0.9
1
E ( experimental )
Figure 9. Comparison of measured bed voidages with values calculated
from Equation (29)for Newtonian and non-Newtonian solutions.
421
M. Aghajani, H. Muller-Steinhagen and M. Jamialahmadi
Table 6. Summaly of database usedfor comparison with the present model.
Liquid
Pure water
Pure water
Richardson and
"aki
L
1
30
1
0.5,1,0.3,6.4
1
Pure water
2745,1060,7740
Pure water
Pure water
Loeffler and Ruth
Pure water
~
Pure water
Pure water
Garside and Al-
Pure water
Pure water
Pure water
Jamialahmadi et
Pure water
Present work
Present work
464
I
422
same as Table 1
1 same
as Table 1
same as Table 1
1
sameas Table I
Pure water and
Sugar solutions
same as Table2
CMC solutions
same as Table 2
Ten of the most frequently recommended correlations from the literature have
been compared with the experimental data. The results of thls comparison are
summarized in Table 7 in terms of average relative error and the standard deviation of
prediction. The model developed in the present investigation clearly out-performs all
other correlations. This table also indicates whether correlations tend to under-predict
"- " or over-predict
the measurements. Correlations with "- - " or "+ + '' have a
high tendency to under-predict or over-predict the measurements, and for correlations
with Y " no clear tendency was found.
I'+"
422
Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
Table 7. Average relative error of values predicted by published models as compared
to experimental data.
Newtonian liquids
Average Standard
relative deviation
rrror(%)
(%)
Prediction
-
Non -Newtonian liquids
Average
relative
'rror (%,
itandam
reviatioi 3redictio
(%)
3
e
19.98
5.61
f
15.12
1 I .87
9.51
4.75
f
16.81
25.79
8.22
--
6.52
1.50
11.16
8.74
-
All liquids
Average Standard
relative deviation
?rror(%)
(%)
17.56
5.08
13.45
13.08
5.46
15.14
12.96
20.61
9.57
f
18.21
14.52
12.79
8.36
--
8.61
10.45
10.89
5.62
13.48
12.44
-
50.93
55.74
++
34.83
36.93
12.66
9.57
f
10.36
8.21
9.59
7.47
f
9.90
18.09
15.09
f
6.95
4.44
f
f
f
t
1
6.64
- -423
M. Aghajani, H. Muller-Steinhagen and M. Jamialahmadi
Bed voidage at minimum fluidization velocity
Bed voidage at minimum fluidization velocity is one of the important factors to be
considered in the design of fluidized bed systems. The main problem with the
prediction of minimum fluidization velocity is that Equation (1 8) is highly sensitive to
the value of ~pllrandthe shape factor, w. For spheres, Wen and Yu [33] selected a
value for ~pllrof 0.42, and Bamea and Mednick [43] used a value of 0.415.
Theoretically it should be possible to determine the bed voidage at minimum
fluidization velocity by using one of the velocity-voidage relationships for fluidized
bed systems. Equation (29) is general and includes the effect of various operational
and geometrical parameters, and the nature of the fluids, on the velocity-voidage
relationship. Bed voidage at minimum fluidization velocity predicted from Equation
(29) for various solidliquid combinations is:
The minimum fluidization velocity ( u d ) can be calculated from Equation (20),
and E~~ from Equations (31) and (32) for spherical and cylindrical particles,
respectively. It is worthwhile to note that Equation (18) can be used as a cross-check
because the predicted upllr and E& should conform to this equality. The prediction of
Equation (36) is verified against experimental data for various particle sizes, types
and shapes, and for fluids with different natures. The absolute mean average errors of
6% illustrate the excellent applicability of this model for the prediction of bed voidage
at minimum fluidization velocity.
Conclusions
An experimental and theoretical investigation of particle settling velocity and
velocity-voidage relationship in solid/liquid fluidized bed systems has been
undertaken using distilled water and sugar solutions as Newtonian fluids, and CMC
solution as non-Newtonian fluids, and a variety of solid particles as solid phase. New
and simple correlations for the prediction of particle settling velocity and bed voidage
for both Newtonian and non-Newtonian fluids are presented. The present
experimental data as well as a database containing a large number of published bed
voidages over a wide range of operational parameters, and liquid and solid phase
physical properties, are compared with the predictions from various correlations in the
literature. The best prediction is obtained using the correlations recommended in the
present investigation.
Nomenclature
A
A,
Ar
424
cross-sectional area of the
column, m2
projected area of particle, m2
Archimedes number,
(= gdP3 ( P p -Pi) PI / PI3
CD
d,
d,,
Dh
drag coefficient
particle diameter, m
equivalent particle, m
hydraulic diameter of fluidized
bed, m
Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids
wall factor
acceleration due to gravity,
Ids2
k
viscosity coefficient in power
law model, Pa. S"
k
constant
L
bed height, m
M
mass of particles, kg
n
Richardson and Zaki exponent
n
rate index in power law model
Remf minimum fluidization,
(= PI UmfdP 1PI)
particle Reynolds number,
(= PI us dP 1 PI)
particle terminal Reynolds
number in an infinite fluid,
(= PI u,dP / PI)
superficial liquid velocity, d s
minimum fluidization velocity,
m/S
particle terminal velocity
corrected for wall effect, m/s
particle terminal velocity in an
infinite fluid, m/s
v
z
volume, m3
fluidization index
Greek symbols
q~
shape factor
E
bed voidage
y
shear rate, s-'
p
dynamic viscosity, kg/m. s
pa
apparent viscosity, kg/m. s
p
density, kg/m3
T
shear stress, Pa
Subscripts and superscripts
a
apparent
D
drag
h
hydrodynamics
1
liquid
mf
minimum fluidization
p
particle
RZ Richardson and Zaki
s
superficial velocity
SB static bed
t
terminal velocity
T
total
00
infinity
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Received: 24 April 2003; Accepted after revision: 11 September 2003.
426
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