7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Investigation of the shiftingparameter as a function of density in the fluidization
of a packed bed.
C. Rautenbach* M.C. Melaaen* and B.M. Halvorsen*
Institute for Process, Energy and Environmental Technology, Telemark University College, Norway
christo.rautenbach@hit.no, morten.melaaen@hit.no and britt.halvorsen@hit.no
Keywords: Fluidization, poweraddition, regime transition, drag models
Abstract
Accurate predictions of pressure drops in fluidized beds are of great importance in the industry. Up to date no sat
isfactory correlation exists to predict the pressure drop in a fluidized bed as the bed is traversing from one regime to
another.
In the present study experiments have been performed in an experimental fluidized bed reactor. The experimental
tower has been equipped with a set of nine pressure sensors located at different positions along the height of the tower.
The tower has a diameter of 7.2 cm and a height of 1.5 m.
A procedure providing a correlation for data in the transition region between asymptotic solutions or limiting
correlations have been described by Churchill & Usagi (1947). This correlation can generally be expressed as
y{x} = y{x} + y {x}, where yo{x} and yoo{x} represents the asymptotic solutions for large and small val
ues of the independent variable x and s is the shifting parameter. ( Ii.iiigiig the value of s shifts the correlation given
by y{x} closer to or away from the asymptotic solutions. This procedure has been proven to give good correlations in
a wide range of applications.
A series of different powders have been used to investigate the influence of a particular parameters on the shifting
parameter, s. Up to date no expression has been stated for this shifting parameter to govern the transition from fixed
to fluidized bed. Two powders have been used in the present study and they are Zirconium Oxide (ZrO) and spherical
glass particles. The powders have the same size distributions but very different densities. The effect of different
densities on the shifting parameter was investigated. Several different drag models were used to serve as a control for
investigating the shifting parameter. The results are given in the form of pressure drop data versus superficial velocity
data. Experimental data are presented with the drag model correlations and the investigated values of the shifting
parameter, s. Some of the drag models that were used were the Syamlal O' Brien drag model (Syamlal, Rogers &
O'Brien (1993)) and the extended HillKochLadd drag correlation (Benyahia, Syamlal & O'Brien (2006)). The
results are evaluated and discussed.
Nomenclature c critical point
f fluid
Roman symbols mf point of minimum fluidization
dp mean surfacevolume mean diameter (m) p particle
g gravitational constant (ms 2) o limiting condition for small values
L bed height (m) of the independent variable
p pressure (Nm 2) oo limiting condition for large values
q superficial velocity (ms1) of the independent variable
Re Reynolds number ()
y canonical dependent variable (Nm 3) Superscipts
s shifting parameter
Greek symbols
6 porosity ()
p density (kg m 3)
Subscripts
Introduction
Fluidized bed reactors are widely employed in the chem
ical, petrochemical, metallurgical and pharmaceutical
industries (Stein, Ding, Seville & Parker (2000)). Bet
ter understanding of the complex multiphase fluid and
solid movement are essential for optimal reactor design.
The powered addition technique serves as a method to
correlated data in the transition region between two lim
iting conditions. This technique has got the possibility
of a wide range of applications as described in the work
by Churchill & Usagi (1947). Applying the technique
to a fluidized bed traversing from a fixed to fluidize bed
proved very useful. In general the shifting parameter, s,
can be defined as follows
yS {q}= y {q} + y{q}, (1)
where yo q} and yf {q} represents the asymptotic solu
tions for large and small values of the superficial veloc
ity, q, and s is the shifting parameter. In fluidized beds
the lower bound, yo{q}, is described by drag models.
Over the years numerous drag models have been pro
posed. In general two types of experimental data can be
used to create a fluidsolid drag model (Syamlal, Rogers
& O'Brien (1993)). The first type is with packedbed
pressure drop data expressed in the form of a correlation
and the second is provided in the form of correlations
for the terminal velocity in a fluidized or settling bed,
expressed as a function of porosity and Reynolds num
ber (Syamlal, Rogers & O'Brien (1993)). A well known
example of a drag model based on the packed bed pres
sure drop data is the Ergun equation (Kunii & Levenspiel
(1991)) and an example of a drag model using the ter
minal velocity correlation is the Syamlal O'Brien drag
model (Syamlal, Rogers & O'Brien (1993)). It will be
discussed later in the present study how these to basic
formulations of drag models may influence the pressure
drop predicted by these models in the fixed bed regime
of a fluidizedbed reactor.
At minimum fluidization the total weight of the
packed bed is supported by the upward force created by
the gas moving upward through the porous structure. As
the superficial velocity is increased above minimum flu
idization velocity, the pressure drop remains practically
the same (Kunii & Levenspiel (1991)). In some cases
the pressure drop does not remain constant in the fully
fluidized regime but actually increases. The explanation
for the slight increase of pressure drop with an increase
of superficial velocity may be attributed to wall effects
which occurred due to the physical dimensions of the
experimental tower used. More specifically, some slug
ging can occur and due to the formation of slugs addi
tional potential energy is required to move the slug ver
tically. The result is an approximate linear increase in
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
pressure drop across the fluidized bed (Chen, Gibilaro
& Foscolo (1997)). In the present study the pressure
drop in the fluidized regime will be assumed constant.
This constant pressure drop will be assumed as the up
per bound, y, {q}, in our poweredaddition correlation
given in equation (1). At minimum fluidization velocity
the pressuredrop is given by
Ap= (1 )(pp pf)Lg,
with pp the particle's density, pf the fluid density and L
the bed height.
In previous studies it has been found that the shift
ing parameter s was relatively independent of the par
ticle sizes that were investigated (Rautenbach, Melaaen
& Halvorsen (2010)). The three particle size distribu
tions that were used were 100200 pm, 400600 pm and
7501000 pm. It was found that a shifting parameter
value greater than about 12 but smaller than about 20
produced an acceptable correlation in the transition re
gion between the fixed to fluidized regime.
Experimental setup
The experiments that were carried out in the present
study were performed at the TUC (Telemark University
College) in Porssgrun Norway. A 1.5 m long experimen
tal fluidized bed reactor were used. The pressure drop
data were acquired using a set of nine pressure probes
located at different height along the bed. This setup is
presented in Figure 1. A porous plate distributor was
used in this study to produce an uniform entry profile to
the bed.
In the present study the influence of the particle den
sity on this shifting parameter, s, was investigated. The
two powders used were Zirconium Oxide (ZrO) and
glass particles. Both were spherical particles with a size
distribution of 400600 pm. The particle size distribu
tions are given in Figure 2. The void fraction of the beds
as well as the minimum fluidization velocities and the
particle densities are presented in Table 1. The void
fractions refer to the void fractions after the bed has
been fluidized. This void fraction will be used in cal
culating the predictions of the drag models in the fixed
bed regime. Two mixtures of the ZrOpowder and glass
powder were used to create a powder mixture with a dif
ferent effective density than the two original powders.
One mixtures consisted of one third ZrOpowder and
two thirds glass powder. The other mixture was half
half ZrOpowder and glasspowder. The mean particle
diameters presented in Table 1 are the surfacevolume
mean diameter and can be expressed as
dp mean i xidi
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
p
0
A
100 200 300 400 500 600 700 800 900 1000
Particle size (pm)
Figure 2: Particle size distributions for the ZrOpowder
(.) and the glass powder ().
x 104
25
Figure 1: Experimental fluidization reactor equipped
with nine pressure probes and fed through a
porous plate distributer.
where xa is the mass fraction of the particles with a di
ameter di. An estimation of the minimum fluidization
velocity, Uf, was determined by equating equation (2)
with the pressure drop prediction of the Ergun equation
(Kunii & Levenspiel (1991)). The intersection between
these two predictions gives a fairly good correlation for
Uf. For the ZrOpowder the surfacevolume mean di
ameter were used and for the glass particles the large
particle size (600 pm) were used as this gives the best
correlation to experimental data using this particular in
tersection method. For both the ZrOpowder and the
glass powder only the first term of the Ergun equation
were used as this produced a good estimation of the min
imum fluidization velocity. This approximation with the
Ergun equation is usually only used with low superficial
velocities or very small particles (Rep,mf < 20) (Ku
nii & Levenspiel (1991)). This one term approximation
of the Ergun equation did not give good enough corre
lations for the two mixture powders' minimum fluidiza
tion velocities and thus the total Ergun equation were
used. The results are given in Table 1. For the mix
ture consisting of one third ZrOpowder a porosity of
0.38 were used and not the measured porosity of 0.42.
This value is just the average value of the porosities of
the original powders. This value of 0.38 was assumed
to compensate for the suspected incapability of the drag
models to compensate for the drag effect associated with
segregation. This topic will be discussed later in the
present work. In both of the mixtures' calculations of
the minimum fluidization velocities an effective particle
++x .ZrO
+ Glass
33 3% ZrO and
66 67% Glass
x 50% ZrO and 50% Glass
01 02 03 04 05
q [m/s]
Figure 3: Pressuregradient against superficial velocity
data for all the powders investigated.
size of 600 pm were assumed as this produced an ade
quate result.
Results
The pressuredrop data retrieved for the powders investi
gated are given in Figure 3. It is very clear to see that the
higher density Zirconium Oxide (ZrO) produces a much
larger pressure drop across the bed and also fluidizes at
a higher value for the superficial velocity, q.
In the present study several drag model have been
used to provide a pressure drop correlation in the fixed
bed regime. Some of these models differ in the way that
they have been derived but most of them make use of
some sort of empirical basis. The models used in the
present study was the well know Ergun equation (Kunii
& Levenspiel (1991)), the Syamlal O' Brien drag model
(Syamlal, Rogers & O'Brien (1993)) and the extended
Table 1: Summary of particle and bed properties.
Variable Value Units
Zirconium Oxide
e 0.39
pp 3800.0 kg/m3
Um, 0.3 m/s
dp mean 503.29 pm
Glass
e 0.37
pp 2500.0 kg/m3
Umf 0.21 m/s
dp mean 482.93 pn
33.3% Zirconium Oxide
and 66.7% glass
e 0.42
pp 2933.33 kg/m3
U,7f 0.24 m/s
dp mean 489.53 pmr
50% Zirconium Oxide
and 50% glass
e 0.38
pp 3150.0 kg/m3
Umf 0.25 m/s
dp mean 492.9 pm
HillKochLadd drag correlation (Benyahia, Syamlal &
O'Brien (2006)). In all of the Figures the limiting condi
tion for large values of q are represented by the dashdot
line (equation (2) is represented by ()). These differ
ent drag models are represented in Figure 4 and 5 along
side the pressure drop data acquired by using the ZrO
powder and glasspowder. For conciseness the models
predictions are shown using the largest and smallest par
ticle size of the distributions used. The mean value of
the particle distributions were not used because it did not
give good correlations in any of the cases investigated in
the present study.
It is clear that the different models have varying ac
curacy with different values of the effective particle di
ameter. Previous research has found that in a fluidized
bed, consisting of a particle size distribution, it is the
smaller particle sizes that have the largest contribution
I1.i\.iI.iIlin.i & Halvorsen (2009)). This followed from
data that were collected by using different mixtures of
particles. In the work by LT.i.1l.iilii & Halvorsen (2009)
it was found that only after about 40% of the mixture
consisted of the larger particles did the minimum flu
idization velocity differ considerably from the value for
Umf found with just the smaller particles. Even after
40% of the bed consisted of large particles the mini
mum fluidization velocity was closer to the smaller par
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
x 10
Data
The Ergun equation
0 Modified HillKochLadd drag correlation
Syamlal & O Bnen drag model
3 Powered addition correlation
25
E 2 r ^
01 6o
2f
1 0 13
005 01 015 02 025 03 035 04 045 05
q [m/s]
(a)
x 104
35
Data
S The Ergun equation
3 0 Modified HillKochLadd drag correlation
S Syamlal & O Bnen drag model U
Powered addition correlation
25 3 0
15
05
1 0
05
0 01 02 03 04 05
q [m/s]
(b)
Figure 4: Investigation of effective particle diameter on
the different dragmodel predictions with the
ZrOpowder. Drag models with an effec
tive particle diameter of (a) 400 pm and (b)
600 pm respectively.
tides minimum fluidization velocity than to the value
for the larger particles iT.i\.iI.nlii.,i & Halvorsen (2009)).
Given the data as represented in Figure 4 it seems that
the Syamlal O' Brien drag model (Syamlal, Rogers &
O'Brien (1993)) produces the best correlation at a par
ticle diameter close to the smallest particle diameter in
the range. This is in agreement with previous research
IT.\.i .lllii.a & Halvorsen (2009)) and from Figure 2 it
is clear that there were some particles with a diameter
even smaller than 400 pm. This fact makes the good
correlation found with the Syamlal O' Brien drag model
(Syamlal, Rogers & O'Brien (1993)) even more feasible
as 400 pm is not necessarily the smallest particle size in
the distribution.
In all the cases investigated the Ergun equation (Kunii
& Levenspiel (1991)) and the extended HillKochLadd
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
x10 ____
+ Data
 *Equation (1)
25 The Ergun equation
O Modified HillKochLadd drag correlation
Syamlal & O Bnen drag model
Powered addition correlation
Y 1 5     '" '
os
1 0
05
0
005 01 015 02
q [m/s]
x 10
45
4
35
3
j25
' 2
15
025 03 035
Data
t The Ergun equation
O Modified HillKochLadd drag correlation
* Syamlal & O Bnen drag model
X O
O0
    ... .........
8 V
S* *
005 01 015 02 025 03
q [m/s]
035 04
x10
25
+ Data
S Equation (1)
X The Ergun equation
2 O Modified HllKochLadd drag correlation
S Syamlal & 0 Bnen drag model
Powered addition correlation 0
1 5        
1
0 005 01 015 02 025
q [m/s]
03 035
Figure 5: Investigation of effective particle diameter on
the different dragmodel predictions with just
the glass powder. Drag models with an effec
tive particle diameter of (a) 400 pm and (b)
600 pm respectively.
drag correlation (Benyahia, Syamlal & O'Brien (2006))
corresponded to the larger particle size in the distribu
tion. This result agrees with previous findings by de Wet,
Halvorsen & du Plessis (2009). The results for the glass
particles and mixtures are similar to that given in Figure
4 for the ZrO and are given in Figure 5 to 7.
Results using powered addition. If we assume that
the fully fluidized regime can be given by equation (2)
then the following equation is produce using the pow
ered addition technique
1
A = (Drag model' + equation (2)) s (4)
L
were any adequate drag model can be used. The negative
powers of s is because the data is a decreasing power of
q. The powered additioned results are depicted in Figure
x10
2 x
Data
1 8 The Ergun equation
0 Modified HillKochLadd drag correlation  
1 6 Syamlal & O Bnen drag model .
1 4 0
12 O 0
"o *
S1 0 *0
< 08 0 .*
06 a **
04
02 *
0 005 01 015 02 025 03 035 04
q [m/s]
(b)
Figure 6: A mixture powder consisting of one third
ZrOpowder and two thirds glass powder.
Drag model comparisons to data assuming a
particle size of (a) 400 pm and (b) 600 pm.
4, 5 and 8. First the correlation is given when the Syam
lal O' Brien drag model (Syamlal, Rogers & O'Brien
(1993)) is used and a particle diameter equal to 400 pm
(Figure 4 and 5 (a)). Secondly the powered addition cor
relations are given using the Ergun equation and a par
ticle diameter of 600 pm (Figure 4 and 5 (b)). In all
the cases a value of 15 was used as the shifting param
eter although values with in a range from 12 to about
20 would have serviced. These results seem to indicate
that the shifting parameter s is insensitive to the density
of the particles. In an attempt to confirm this suspicion
the same correlations were made but now with the pow
der mixtures. In Figure 6 and 7 the correlation with the
data are given with the large and smallest particle sizes,
namely 400 pm and 600 pm respectively. In both cases
poor agreement were found between the drag models
and the experimental data. In the case of the mixture
x o10
Data
S The Ergun equation
0 Modified HillKochLadd drag correlation a
S Syamlal & 0 Bnen drag model o
2 ... ..... ... .. _4 .._
S 4
005 01 015 02 025 03 035
q [m/s]
(a)
x io
Data
# The Ergun equation
S 0 Modified HllKochLadd drag correlation
Syamlal & 0 Bnen drag model S
# 0
2 0
2 ..... . . ....... *.
So
15 0
04
05 S
0
005 01 015 02 025
q [m/s]
03 035 04
(b)
Figure 7: A mixture powder consisting of fifty percent
ZrOpowder and fifty percent glass powder.
Drag model comparisons to data assuming a
particle size of (a) 400 pm and (b) 600 pm.
consisting of one third ZrO particles a relatively high
void fraction value is obtained as presented in Table 1.
This is considered a high value because the powders that
this mixture is made up of has void fractions below 0.4.
The main suspicion for this high value is due to segrega
tion. The heaver ZrO particles can move to the bottom
of the tower while the lighter glass particles can move to
the top of the bed. As all of the drag models investigated
has got some sort of assumption of only one particle size
it is not a bad assumption that segregation could be the
cause of this discrepancy. Although most of the inves
tigated models should cover all porosities they still did
not take into account the effect of particle size distribu
tions and segregation. When a void fraction of about
0.38 was used (average void fraction of the base pow
ders) for the powder consisting of one third ZrO, very
good correlations were found with the data as presented
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
in Figure 8. Much the same result as depicted in Figures
4 and 5. Thus the drag models predicted a more accurate
drag with a lower void fraction supporting the suspicion
that the drag models do not take into account the influ
ence of segregation. The possible reason why the one
powder mixture formed some segregated effects and the
other powder mixture (50/50 mixture of glass and ZrO)
did not is explainable in light of initial fluidization. If
the superficial gas velocity, q, is slowly decreased af
ter fluidization the heavy particles have time to settle to
the bottom while the lighter particles are force upwards.
Precaution was taken to avoid this, for example to close
the gas inlet quickly after fluidization, but this approach
is not guaranteed to always work equal effectively.
The aim of the present study is to investigate the shift
ing parameter, s. To be able to do so a fairly accurate
prediction of the pressure drop in the fixed bed regime
is required. Thus a porosity of 0.38 will be assumed for
the mixture powder having a volume that consist of one
third ZrO. In Figure 8 the correlation using the powered
addition technique is given with the mixture powders'
pressure drop data. In both cases the Syamlal O' Brien
drag model (Syamlal, Rogers & O'Brien (1993)) were
used with a particle diameter of 400 pm. In both cases
a value of 15 were used for the shifting parameter, s. A
possible explanation for the discrepancy of the Syam
lal O' Brien drag model (Syamlal, Rogers & O'Brien
(1993)) in the fixed bed regime of Figure 8 (b) be can
be because the smallest particle size in the distribution
range were used in the drag correlations and this particle
size is just an assumption. The characteristics of a par
ticle bed seems to be mainly determined by the smaller
particles LT.,\ ..ii.lia. & Halvorsen (2009)) but of course
the larger particles will also still have an effect on the
over all pressure drop. It is also possible that the mix
ture's particle size distribution did not consist out of an
approximate bellshaped curve but that there were more
of the bigger particles than in the other powders that
were investigated. A sieve analysis has to be performed
to confirm or disconfirm this ]i ,i llici,
Discussion
Effective particle size is of great importance when work
ing with a particle size distribution. In a lot of practical
applications fluidized beds consist of such powders. It is
clear from the results produced in the present study and
from results in previous work IT.,a .,.lili., & Halvorsen
(2009)) that the smaller particle sizes in the distribu
tion plays a bigger role in the estimation of the drag.
More research is needed to find a better way of estimat
ing the representative particle diameter for particle size
distributions. I seems clear from the present study and
from previous research done by de Wet, Halvorsen & du
SData
# The Ergun equation
3 0 Modified HllKochLadd drag correlation
SSyamlal & 0 Bnen drag model *
Powered addition correlation
25
2 6
25    ^
S *
15
o5
005 01 015 02 025 03
q [m/s]
(a)
x 10
35
+ Data
The Ergun equation
3 O Modified HllKochLadd drag correlation
Syamlal & O Bnen drag model
Powered addition correlation
25
035 04
Sn.. ... . .. .
015
05
01 015 02 025 03 035
q [m/s]
(b)
Figure 8: Powered addition correlation using the Syam
lal O' Brien drag model (Syamlal, Rogers &
O'Brien (1993)) and a particle size of 400 pm
compared to (a) the mixture powder consist
ing of one third ZrO and two third glass and
(b) the mixture powder consisting of fifty per
cent ZrO and fifty percent glass.
Plessis (2009) that a normal average representative value
for the particle diameter is not appropriate. Using a par
ticle size equal to 400 pm together with the Syamlal O'
Brien drag model (Syamlal, Rogers & O'Brien (1993))
did however produce acceptable results.
In a case where data in the transition region is know a
different approach can be taken to calculate the shifting
parameter. At the point where the two asymptotes in
tersect a critical value is obtained (de Wet, Halvorsen &
du Plessis 121 a i,,. In the present study this point forms
an estimation for the minimum fluidization velocity with
the lower bound being a drag model and the upper bound
equation (2). This can also be expressed as
y,{q} = yo{q}. (5)
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
At this point equation (1) simply becomes
ys= + y s =2ys 2. (6)
with y, the functional value at the critical point (de Wet,
Halvorsen & du Plessis (2009)). Solving for s produces
the following equation
In 2 In 2
y6 1 y(7)
In y, In yo In y, In y (7)
Thus if the functional value y, is known a suitable value
for the shifting parameter, s, can be calculated.
The problem with this procedure is that in the indus
try one usually wants to determine this transition regime
not prescribe it. As the physical meaning of the shift
ing parameter is not known further research is needed to
describe the value for s more precisely.
A possible explanation for the high pressures pre
dicted by the Ergun equation could be found in the man
ner in which it was derived. It was derived on a fixed bed
model and then later adapted empirically using fixed bed
pressure drop data.
The inaccuracies of the HillKochLadd drag correla
tion (Benyahia, Syamlal & O'Brien (2006)) could prob
able be based on the empirical way in which it was de
rived. Singling out the exact cause for the over estima
tion of the drag is not a trivial task.
It must be noted though that equation (2) did produce
a reasonable result with a particulated bed void fraction
equal to 0.42 in the case of the mixture consisting of a
third ZrO. This actually leads to the same suspicion that
the problem is mainly with the drag models in the case
of a segregated bed.
Conclusions
In fitting an appropriate curve the shifting parameter is
relatively insensitive as found by previous research (de
Wet, Halvorsen & du Plessis 1,21 i"n. A range of val
ues produce an appropriate result. This range can be
anywhere between 12 and 20 but 15 was chosen in the
present study. The shifting parameter, s, seems to be in
sensitive to changes in density. With all four powders
investigated a value of 15 serviced. Further research is
needed to determine the physical meaning of s, but it
seems that a value of 15 is a good estimation for the
shifting parameter in most practical application.
Thus an effective correlation is produced to give an
adequate prediction of pressure drop data for a fluidized
beds traversing from fixed to fluidized regime
References
Benyahia S., Syamlal M. and O'Brien T.J., Extension
of the HillKochLadd drag correlation over all ranges
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
of Reynolds number and solids volume fraction, Powder
Technology, Vol. 162, pp. 166174, 2006
Chen Z., Gibilaro L.G. and Foscolo PU., Fluid pressure
loss in slugging fluidized beds, Chemical Engineering
Science, Vol. 52, No. 1, pp. 5562, 1997
Churchill S.W. and Usagi R., A Standardized procedure
for the production of correlations in the form of a com
mon empirical equation, Ind. Eng. Chem. Fundamen.,
Vol. 13(1), pp. 3944, 1947
de Wet P.D., Halvorsen B.M. and du Plessis J.P., Pow
ered addition applied to the fluidization of a packed
bed, Computational methods in Multiphase flow V Pa
per 431441, WITPress (UK), 2009
T.i\.iI.niii.i C. and Halvorsen B.M., Experimental and
computational study of particle minimum fluidization
velocity and bed expansion in a bubbling fluidized bed.,
SIMS 50 conference, Fredericia, Denmark, October
2009
Kunii D. and Levenspiel O., Fluidization Engineering,
Second edition, ButterworthHeinemann (USA), 1991
Rautenbach C., Melaaen M.C. and Halvorsen B.M., In
vestigation of the shiftingparameter as a function of par
ticle size distribution in a fluidized bed traversing from a
fixed to fluidized bed., Fluidization XIII: New Paradigm
in Fluidization Engineering conference, Hotel Hyundai,
Gyeongju, Korea, May 2010
Stein M., Ding Y.L., Seville J.PK. and Parker D.J.,
Solids motion in bubbling gas fluidized beds, Chemical
Engineering Science, Vol. 55, pp. 52915300, 2000
Syamlal M., Rogers W. and O'Brien T.J, MFIX Doc
umentation theory guide, Technical note, U.S. Depart
ment of Energy, Office of Fossil Energy, Morgantown
Energy Technology Center, Morgantown, West Virginia,
December 1993
