7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The effect of the wall on the fixedfluidized state transition of binarysolid liquid beds
Renzo Di Felice and Carla Scapinello
University degli Studi di Genova, Dipartimento di Ingegneria Chimica e di Processo "G.B. Bonino"
via Opera Pia, 15, Genova, 16145, Italy
renzo.difelice@unige.it
Keywords: liquid fluidized beds, binarysolid systems, minimum fluidization velocity, Janssen approach
Abstract
Minimum fluidization velocity can be defined as the "smallest" fluid velocity at which all the particles in the bed are fully
supported by the upflowing fluid, with the pressure drop coincident with the soli defective weight. The phenomenon is more
complex when binarysolid mixtures are fluidized, specifically when the solid possessing the highest minimum fluidization
velocity is placed on top of the other solid layer. For a start, once the minimum fluidization velocity of the bottom particles is
reached nothing happens, as the bottom bed is kept in the fixed condition by the weight of the material above, which is still not
completely supported by the fluid. As the water velocity increases and the overall pressure drop reaches the value sufficient to
support the overall solid weight, the bed still remains fixed and stays so until a certain critical velocity, u, is reached. The
overall pressure drop, may, in this case, be much larger than the solid effective weight.
The approach followed here is that named as "differential slices", which simply consists in writing a force balance for a
differential horizontal slice of material layer. The simplifications suggested by Janssen (1895) and nowadays still very popular
have also been used, in particular the assumption of constant ratio throughout the solid layer of the solid vertical pressure to the
solid horizontal pressure, quantified by the Janssen constant K. Model predictions and experimental measurements have been
found to be in excellent agreement.
Introduction
density (kg m3)
The concept of minimum fluidization velocity is probably
the first one a researcher comes across when starting to
engage in fluidization studies. It can be defined as the
"smallest" fluid velocity at which all the particles in the bed
are fully supported by the upflowing fluid: for smaller fluid
velocities the bed will be in the fixed state (with some of its
weight supported by the column wall and by the distributor
plate) and for larger fluid velocities the bed will be
completely fluidized, either in a homogeneous or
heterogeneous manner (Gibilaro, 2001).
Nomenclature
d particle diameter (m)
D column diameter (m)
g gravitational constant (ms2)
K Janssen constant ()
H layer height (m)
P pressure (Nm2)
u velocity (ms1)
z vertical coordinate (m)
Greek letters
a voidage ()
v kinematic viscosity (m2s1)
IL friction coefficient ()
Subscripts
E
i
s
t
W
Ergun
interface
solid
total
weight
For beds made up of monocomponent solids the behaviour
is quite well known: a gradual increase of the fluid velocity
through the bed leads to an increase of the pressure drop,
quantified for example by the Ergun (1952) equation,
2 u = l 150 (1e+1.75u (1)
H d er d
up to the point where the buoyant weight of the particles
K =F(1 )(p)g
is balanced and the solid becomes fully supported by the
fluid. Further velocity increases lead simply to bed
expansion, or to bubble appearance, but has no effect on the
overall pressure drop. Equation (2) represents therefore the
maximum pressure drop observable for a bed of
monocomponent solids. Many experimental observations
on mono component solid beds have been made which
agree with the description summarized above.
When the solid phase can no longer be considered
monocomponent, such as for the case of binarysolid
mixtures, exhibiting a variation of size or density or both,
the first impulse is to treat the system as pseudo
monocomponent by defining some average size and
density (Gauthier et al., 1999, Leu & Wu, 2000). It is easy to
understand that such an approach will only be valid when
the difference in solid physical characteristics are not too
pronounced; as a matter of fact previous works have
demonstrated that when the difference in diameter or
density are considerable, the whole idea of a single
minimum fluidization velocity is no longer valid (Chen &
Keairns, 1975). It has been shown, when considering for
example a binarysolid mixture fluidized by a gas, that the
transition from the fixed to the fluidized state is
characterized by at least two velocities: the initial
fluidization velocity, and the complete fluidization velocity,
(Formisani et al., 2001). The first value corresponds to the
first solid being fluidized, and is determined experimentally
from the first observed deviation of the measured pressure
drop from the expected fixed bed curve; whereas the second
value is reached when all the material in the column is fully
fluidized.
The works of Vaid & Sen Gupta (1978) and Asif & Ibrahim
(C'" ') do address the problem of minimum fluidization
velocity of binarysolid liquid beds, but they do so by
investigating experimentally slow defluidization of initially
fluidized suspensions rather than considering the case of the
fluid flux being gradually increased. In a recent publication
(Di Felice & Scapinello, 2009) we showed qualitatively that
the transition from the fixed to the fluidized state in
binarysolid mixtures depends on the way the solid mixture
has been charged into the column and very different results
were reported for extreme cases of the initial mixing
condition: complete segregated and perfect mixing. The
completely segregated case implies that one of the solid
components can be initially in either the bottom or the top
region of the bed giving in all three different situations. Of
these three cases the one where the solid possessing the
highest fluidizing velocity is placed on top on the solid
possessing the smallest fluidizing velocity was observed to
be the more interesting and will be investigated in depth in
the present work.
Experimental
The experiments reported here were carried out in
cylindrical, constant cross section, transparent columns, of
three different internal diameters, namely 35, 50 and 90
mm. The fluidized solids were uniform diameter spheres,
whose diameter and density are summarized in Table 1.The
fluidizing liquid was tap water at ambient temperature. Its
density and viscosity were assumed constant at 1000 kg/m3
and 0.001 Pa s, respectively. A finetuning valve was used
to regulate water flow rates, which were measured with a
series of rotameters. The fluidizing column was equipped
with a probe positioned just above the supporting distributor
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
plate (a simple mesh followed by a 50 mm layer of lead
shot, which remained in the unfluidized state for the whole
range of water flow rates investigated). The probe was
connected to a piezoelectric pressure transducer from which
the instantaneous value of the pressure was obtained, both
from a visual display and in digital form through an A/D
signal converter connected to a PC. The overall piezometric
pressure drop in the bed was obtained by subtracting from
the actual measure a reference value, corresponding to the
reading when no water was flowing through the column.
The experimental procedure simply consisted in charging
the column with a defined quantity of particles, then
initiating the water flow in the rig, starting from the lowest
possible flow rate and increasing it progressively in fixed
steps. For each water flow rate the pressure drop was
recorded along with the status of bed material in the
column. Visual observation were complemented with still
photographs of the column.
Table 1 Solid physical characteristics
Solid
Glass spheres
Glass spheres
Glass spheres
Lead spheres
Density (kg/m')
2500
2500
2500
10800
Size (mm)
1.7
3.0
5.0
1.7
Experimental observation
Before working with the solid mixtures, a check on the
monocomponent bed minimum fluidization velocities was
carried out in order to verify the correct functioning of the
basic equipment and a typical result is presented in Figure 1.
This conforms exactly to expected patterns: for low liquid
flow rates the beds were in the fixed condition with the
pressure drop varying with flow rate in accord with the
Ergun equation; thereafter the bed attained the fluidized
state when the overall pressure drop exactly balanced the
particles effective weight. As expected, the measured
pressure drop is never larger than the particle effective
weight.
0 00 0 01 0 02
u (m/s)
Figure 1: Pressure drop function of increasing water
velocity for a bed of 1.7 mm glass particles.
i
a0
S* Measured
Ergun equation
Effective weight
000 0 02 004 006
u (m/s)
Figure 2: Pressure drop function of increasing water
velocity for a layered binarysolid bed. Bottom layer 1.7
mm lead particles; top layer 1.7 mm glass particles.
Another trivial case is represented when the two solid layers
are charged in the column with the one possessing the
lowest minimum fluidization velocity on top. For the lowest
water flow rates both solid layers are in the fixed conditions
and the overall pressure drop is the sum of the two Ergun
correlation predictions for the two sections. For larger water
velocities only the top layer is fluidized, the bottom layer
remaining in the fixed condition. The overall pressure drop
in this case is thus the sum of the Ergun correlation value for
the bottom section plus the effective weight of the top
particles. For the even larger water velocities both materials
are fluidized in a completely segregated manner and the
total pressure drop thus amounts to the sum of the two
particle component effective weights. Figure 2 summarises
this behaviour. Needless to say in this case also the overall
pressure drop never exceeded the total solid effective
weight.
0 00 0 02 0 04
u (m/s)
0 06
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
minimum fluidization velocity of the bottom particles is
reached nothing happens, as the bed is kept in the fixed
condition by the weight of the material above, which is still
not completely supported by the fluid. More surprisingly, as
the water velocity increases and the overall pressure drop
reaches the value sufficient to support the overall solid
weight, the bed still remains fixed and stays so until a
certain critical velocity, uc, is reached. Beyond this point the
glass layer manages to expand, lifting the lead particles,
which then are able to move through the glass layer to settle
at the bottom of the column transforming the system to
that considered previously with the glass on top in the
fluidized state. The specific case of a mixture made up of
equal volumes of glass and lead particles is depicted in
Figure 3. From this Figure the very large difference between
particle effective weight (just over 6 kPa) and maximum
measured pressure drop (greater than 12 kPa) is strikingly
evident.
It is evident that some other effect must intervene in order to
balance this difference in forces and this can only originate
from the interaction between the solid particles and wall as
static friction. In the next section a quantification of this
effect in attempted and put in a quantitative footing.
Model description
The system under consideration is schematically depicted
in Figure 4. It is made up of two distinct horizontal layers
of particles, the solid possessing the lowest minimum
fluidization velocity placed at the bottom. The height of
the bottom layer is H, and the overall bed height is H,. The
range of fluid velocities of interest is that which yields
overall pressure drops larger than the overall particle
effective weight, up to the critical velocity uc.
Figure 4:
investigated.
Figure 3: Pressure drop function of increasing water
velocity for a layered binarysolid bed. Bottom layer 1.7
mm glass particles; top layer 1.7 mm lead particles
The phenomenon is completely different when the solid
possessing the highest minimum fluidization velocity is
placed on top of the other solid layer. For a start, once the
Schematic representation of the system
The approach followed here is that named as "differential
slices" (Nedermann, 1992), which simply consists in
writing a force balance for a differential horizontal slice of
material layer introducing, when some physical
characteristics do not have a constant value, some average
over the layer. The simplifications suggested by Janssen
(1895) and nowadays still very popular have also been used,
in particular the assumption of constant ratio throughout the
solid layer of the solid vertical pressure to the solid
horizontal pressure, quantified by the Janssen constant K.
As in each slice the pressure drop and the particle weight do
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
not balance, solidsolid must intervene and play an
important role.
With this simplification the force balance over a solid layer
dPS 4
dz D K)P
dz D
H HE
In Eq (3) Ps represent the solid pressure originated through
solidsolid contact in the vertical direction, the second term
is the solidwall friction effect, the third is the particle
effective weight (Eq. 3) and the last the fluidsolid
interaction force brought about by the fluidsolid relative
motion and quantified for example by the Ergun
relationship, Eq. (2). This right hand side of the equation is
therefore a function, for a given system, of the fluid
superficial velocity only. The differential force balance
reported in Eq. (3) is applicable to both solid layers, the only
difference arising from the fact that the right hand side will
assume for our specific case a negative value for the bottom
layer and a positive value for the top layer.
Integration of Eq. (3) is fairly straightforward and can be
done either analytically or, without any practical
inconvenience, numerically. We are interested in applying
this force balance at the critical velocity u, when the
solidwall friction just balance the difference between
overall pressure drop and particle effective weight. In this
situation boundary condition will be simply given as
P = 0
at z = 0
for the bottom layer, and
P = 0
at z = Ht
for the top one.
In order to have a result with a physical significance, the
solid pressures in the two layers must assume the same
value at the interface, z H,. Recollecting that the only
parameter which can be varied in Eq. (3) is the fluid
velocity, the specific value of fluid velocity which result in
equal solid pressure at the interface is our solution and will
yield the sought critical velocity uc.
Model validation
For a given system all the numerical parameters appearing
in Eq. (3) are easily measured and known with the
exception of the product (pK). In the original Janssen
approach (Sperl, 2006) / represent the solidwall friction
factor and K the ratio between horizontal and vertical solid
pressure. In that work Janssen found that for the system he
had experimentally investigated (corn in a wooden square
column) a value of 0.21 pK produced an excellent fitting of
the measured solid pressure at the bottom of the silo. In a
recent work (Di Felice and Scapinello, 2010) it was found
rather surprisingly that the summentioned product assumed
always a value of around 0.21 for all the systems
investigated, regardless of their physical characteristics,
and therefore such value has been assumed to be valid in
the present numerical calculations.
'2
E 004
002
0 02
0
N
20 u (m/s)
0.050
0.053
 0.055
15
10
05
00
0 1000 2000 30
s (Pa)
Figure 5: Calculated solid pressure in the bottom and the
top solid layers function of water velocity.
Figure 5 reports calculated solid pressures in the two solid
layers for the system of Figure 3 at three different water
superficial velocities. As clearly seen in the Figure only a
water velocity of 0.053 m/s yields an identical value for the
solid pressures at the interface, and therefore that value is
the sought predicted critical velocity, in excellent
agreement with the experimental value.
0 00 4
00
02 04 06 08 10
1H,/Ht
Figure 6: Experimental and calculated
velocity. System: 1.7 mm lead particles,
particles. Total bed height: 0.2 m.
water critical
1.7 mm glass
Further model testing have also been carried out. Figure 6
depicts experimental and calculated critical velocity for
systems where the total bed height was kept constant but the
relative amount of the two material was varied. Water
critical velocity ranges, as expected, from the two material
minimum fluidization velocities, although not linearly
dependent on the solid fraction. The behaviour reported in
Figure 6 should not lead to incorrect conclusions: fluid
critical velocity is not simply a function of the ratio of
material charged in the column but also on the overall
quantity. This illustrated in Figure 7 where experimental
and calculated u, are depicted for binarysolid systems
where the fraction of one solid respect to the other is kept
constant but the total solid amount is varied.
* Measured
Model
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
006 
004
E
002
00
00
on
004
E
" 0 02
000
0
0
02 04 06 08
Ht
Figure 7: Experimental and calculated water critical
velocity. System: 5 mm/1.7 mm glass particles.
Inspection of Eq. (3) clearly indicate that the wallsolid
friction effect increases as the column diameter is
decreased. It is therefore to be expected an influence of the
column diameter on fluid critical velocity and this is shown
in Figure 8 where the same system, employed in the 30 and
in the 50 mm i.d. columns, exhibits different experimental
critical velocities. At the same time, when the column
diameter becomes large enough no effect should be
observed and the fluid critical velocity should simply
coincide with the velocity at which the pressure drop
balances the total particle effective. This behaviour has been
observed when the 90 mm i.d. column was used and is
reported in Figure 9.
Ht=0.2
A 
004
S ,
3il
o00 I
00 02
/A
^**0
D (mm)
30 50
A <
04 06
08 10
1HI/Ht
Figure 8: Experimental and calculated water critical
velocity for two different column diameters. System: 5 mm
glass particles, 1.7 mm glass particles.
02 04 06 08
1H,/Ht
Figure 9: Experimental and calculated water critical
velocity function of mixture overall composition. System: 5
mm glass particles, 1.7 mm glass particles.
Finally, it is worth mentioning that the proposed is easily
extendible to fixed bed made up of more than two layers.
The basic equation is still the same used before, which can
be integrated with the condition of equal solid pressure at
each layer interface. Excellent agreement (0.042 vs 0.041
m/s) was found for the critical velocity of the system tested,
a three layer bed (0.1 m individual layer height) of 1.7, 3
and 5 mm glass particles in the 50 mm diameter column.
Discussion
For all the cases investigated the agreement between
experimental and calculated fluid critical velocity has been
excellent. The correct estimation of u, allows, as a
consequence, the estimation of the maximum pressure
drop before the bed becomes fluidized, by simply applying
the Ergun equation to each bed layers at that velocity.
The critical velocity will always assume a value in
between the two solid minimum fluidization velocities.
The overall behavior of a typical binarysolid system (5
mm glass particle, 1 mm glass particle fluidized with
water) is summarized in Figure 10 and will be explained at
length as is representative of any system.
For any given system the critical velocity is function of
only two parameters: the overall solid bed height scaled by
the column diameter, H/D, and the relative amount of solid
charged in the column, HH,. In Figure 10 two boundaries
for the critical velocity are indicated with a dotted line.
The bottom boundary represent the case for which the
effect of the solidwall friction is negligible and the critical
velocity corresponds simply the value obtained by
equating Ergun equation to the overall solid particle
effective weight. This is the case for H/D very small (large
column diameter compared to the overall bed height) or,
alternatively, of very small solid wall friction, pK = 0. In
the specific case reported the calculated bottom boundary
is a curve but, if we were in viscous flow regime, it would
be a straight line joining the two minimum fluidization
velocities. The top boundary is nothing else than the
higher minimum fluidization, as in any case at that point
* Measured
Model
S H/Ht=0.2
H,=0.2
A
rz
A Experimental
Model
the top layer can offer no resistance through the solid
wall contact.
00 02 04 06 08 10
1H,/Ht
Figure 10: Calculated water critical velocity for various
overall bed height to column diameter ratio. System: 5 mm
glass particles, 1 mm glass particles.
Figure 10 indicates that in order to have some tangible
effect of the solidwall friction the bed aspect ratio (H/D)
must be larger than one. Conversely at bed aspect ratio
larger than 30, critical velocity always assumes values
coincident with the upper boundary, regardless of the bed
composition.
Conclusions
The approach based on the Janssen approximation regarding
solid pressure distribution in a stationary granular material
has proved to be very effective in quantitatively describing
the transition from fixed to the fluidized state of layered
binarysolid beds. The critical velocity, and consequently
the pressure drop overshot, can be predicted quite accurately
in the most general manner without the use of any
adjustable parameters.
Acknowledgements
We are indebted to Proff. Diego Barletta and Massimo
Poletto of the University of Salerno for very constructive
discussions on the present subject.
References
Asif, M. & Ibrahim, A.A., Minimum fluidization velocity
and defluidization bahaviour of binarysolid
liquidfluidized beds. Powder Technology Vol. 126,
241254 .'1 '2).
Chen J. L.P., & Keairns, D.L., Particle segregation in a
fluidized bed. The Canadian Journal of Chemical
Engineering Vol. 53, 395402 (1975).
Di Felice, R. & Scapinello, C., On the interaction between
a fixed bed of solid material and the confining column
wall: the Janssen approach Granular Matter, Vol. 12, 4955
(2010)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Ergun, S., Fluid flow through packed columns. Chemical
Engineering Progress 48, 11791184 (1952).
Formisani, B., De Cristofaro, G. & Girimonte, R., A
fundamental approach to the phenomenology of
fluidization of size segregating binary mixtures of solids.
Chemical Engineering Science Vol. 56, 111 (2001).
Gauthier, D., Zerguerras, S. & Flamant, G, Influence of
the particle size distribution of powders on the velocities of
minimum and complete fluidization. Chemical
Engineering Journal Vol. 74, 181196 (1999).
Gibilaro, L.G., FluidizationDynamics. Butterworth 
Heinemann, Oxford (2001).
Leu, L.P & Wu, C.N. Prediction of pressure fluctuation
and minimum fluidization velocity of binary mixtures of
Geldart group B particles in bubbling fluidized beds. The
Canadian Journal of Chemical Engineering Vol. 78,
578585 (2 '"" i ).
Nedderman, R.M., Statics and kinematics of granular
materials, Cambridge University Press, Cambridge (1992)
Sperl, M. Experiments on corn pressure in silo cells 
translation and comment of Janssen's paper from 1895.
Granular Matter Vol, 8, 5965 (2006)
Vaid, R.P & Sen Gupta, P., Minimum Fluidization
velocities in beds of mixed solids. The Canadian Journal of
Chemical Engineering Vol. 56, 292296 (1978).
