
Full Citation 
Material Information 

Title: 
16.3.3  Bubbly flow through fixed beds contactors. Experiments and modelling in the dense regime Granular Media 

Series Title: 
7th International Conference on Multiphase Flow  ICMF 2010 Proceedings 

Physical Description: 
Conference Papers 

Creator: 
Sechet, P. Cartellier, A. Bordas, L.M. Boyer, C. 

Publisher: 
International Conference on Multiphase Flow (ICMF) 

Publication Date: 
June 4, 2010 
Subjects 

Subject: 
packed bed bubbles size distribution relative velocity pressure drop modeling 
Notes 

Abstract: 
Multiphase reactors operated in fixed bed configuration are widely used in petrochemical industry but their hydrodynamic is still not well
understood. In a previous paper (Bordas et al, 2006), a new onedimensional model able to predict the pressure drop and the mean void
fraction for bubbly flows in packed beds was proposed. The equations required closure laws accounting for the liquidsolid and the
gasliquid interactions. Compared to previous models, these laws had been completely revisited, in order to better account for the flow
dynamics at the pore scale. In particular, it had been demonstrated that (i) in dilute conditions, the bubble size remains of the order of the
pore (ii) and that the mean bubble dynamics is somewhat similar to that of a slug, with a relative velocity at mesoscale linearly increasing
with the liquid superficial velocity. Besides, that relative velocity monotonically increases with the gas flow rate ratio, a behaviour that is
tentatively attributed to the formation of preferential paths for the gas phase. (iii) Based on the motion of a bubble train in capillary tubes,
the twophase flow pressure drop fls scaled by its singlephase flow counterpart flsf at the same superficial liquid velocity is predicted to
linearly increase with the void fraction, with a prefactor l evolving with the Capillary number: Y= fls/ flsf
=1+l(Ca)a. Considering the
capillary pressure contribution to the pressure drop, one expects l~Ca1/3 Bretherton 1961. These proposals were only partly validated and
required further confirmation. Thus, new experiments have been performed at LEGI and at IFP in order to cover an enlarged range of flow
conditions while gathering all the necessary information to test the model. Attention has been paid to the measuring techniques, both to
understand the exact meaning of the measured quantities and to control the uncertainty. So far, the results confirm the postulated dynamic.
The model presented assumes that all the bubbles contributing to the void fraction are mobile while in some circumstances, bubbles can be
trapped within the bed. Fixed bubbles can also contribute to the pressure drop, and the closure proposed does not take such contributions
into account, This question was addressed by way of a thorough data analysis. Data gathered on the void fraction and pressure loss shows
that the proposed closures laws for the gas dynamic and twophase flow pressure drop are still relevant for moderate pressure conditions
(up to 10 bars) and gas volumetric ratio b ranging from 20% to 70%. In the range of operating conditions considered (capillary number, gas
ratio b), bubblesbubbles interactions probably limit such blockage. Those interactions would explain also why the size distribution remains
controlled by the pore characteristic dimension, through a process of continuous coalescence/breakup. Using these closures, the model
sensitivity was studied and its ability to predict the void fraction and pressure loss addressed. 

General Note: 
The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: BioFluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and NanoScale Multiphase Flows; Microgravity in TwoPhase Flow; Multiphase Flows with Heat and Mass Transfer; NonNewtonian Multiphase Flows; ParticleLaden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows 
Record Information 

Bibliographic ID: 
UF00102023 

Volume ID: 
VID00399 

Source Institution: 
University of Florida 

Holding Location: 
University of Florida 

Rights Management: 
All rights reserved by the source institution and holding location. 

Resource Identifier: 
1633SechetICMF2010.pdf 

Full Text 
Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Bubbly flow through fixed beds contractors. Experiments and modelling in the dense regime
Sechet Philippe'), Cartellier Alain) Bordas Marie Laure') and Boyer Christophe(2)
(')LEGI, INPG/CNRS/UJF, 1025 rue de la Piscine, 38400, Saint Martin d'Heres, France,
(2)Institut Francais du Petrole Lyon BP3, 69390, Vernaison, France
email: philippe.sechet(hmg.inpg.fr, alain.cartellier(hmg.inpg.fr, christophe.bover(@ifp.fr
Keywords: packed bed, bubbles size distribution, relative velocity, pressure drop, modelling
Abstract
Multiphase reactors operated in fixed bed configuration are widely used in petrochemical industry but their hydrodynamic is still not well
understood. In a previous paper (Bordas et al, 2006), a new onedimensional model able to predict the pressure drop and the mean void
fraction for bubbly flows in packed beds was proposed. The equations required closure laws accounting for the liquidsolid and the
gasliquid interactions. Compared to previous models, these laws had been completely revisited, in order to better account for the flow
dynamics at the pore scale. In particular, it had been demonstrated that (i) in dilute conditions, the bubble size remains of the order of the
pore (ii) and that the mean bubble dynamics is somewhat similar to that of a slug, with a relative velocity at mesoscale linearly increasing
with the liquid superficial velocity. Besides, that relative velocity monotonically increases with the gas flow rate ratio, a behaviour that is
tentatively attributed to the formation of preferential paths for the gas phase. (iii) Based on the motion of a bubble train in capillary tubes,
the twophase flow pressure drop fis scaled by its singlephase flow counterpart fi1sl at the same superficial liquid velocity is predicted to
linearly increase with the void fraction, with a prefactor X evolving with the Capillary number: '= fis fis, =l+X(Ca)a. Considering the
capillary pressure contribution to the pressure drop, one expects XCa1 3 [Bretherton 1961]. These proposals were only partly validated and
required further confirmation, Thus, new experiments have been performed at LEGI and at IFP in order to cover an enlarged range of flow
conditions while gathering all the necessary information to test the model. Attention has been paid to the measuring techniques, both to
understand the exact meaning of the measured quantities and to control the uncertainty. So far, the results confirm the postulated dynamic.
The model presented assumes that all the bubbles contributing to the void fraction are mobile while in some circumstances, bubbles can be
trapped within the bed. Fixed bubbles can also contribute to the pressure drop, and the closure proposed does not take such contributions
into account, This question was addressed by way of a thorough data analysis. Data gathered on the void fraction and pressure loss shows
that the proposed closures laws for the gas dynamic and twophase flow pressure drop are still relevant for moderate pressure conditions
(up to 10 bars) and gas volumetric ratio 3 ranging from 20% to 70%. In the range of operating conditions considered (capillary number, gas
ratio 3), bubblesbubbles interactions probably limit such blockage. Those interactions would explain also why the size distribution remains
controlled by the pore characteristic dimension, through a process of continuous coalescence/breakup. Using these closures, the model
sensitivity was studied and its ability to predict the void fraction and pressure loss addressed.
Introduction
Dispersed twophase flows in confined geometry such as
packed beds are widely used in the chemical and the
biochemical industry. Indeed, this type of reactor is often
preferred to mobile beds because of its high conversion
efficiency for rather low investments and maintenance costs.
Yet, reliable tools are lacking for dimensioning and
optimizing such systems. Many correlations are proposed in
the literature aiming at predicting the pressure loss and the
gas/liquid retention in packed beds, but they provide poorly
reliable results when extrapolated to operating conditions
outside the range for which they were established. Such
drawbacks demonstrate that some hidden parameters
accounting for the complexity of the phenomena occurring
at the pore scale are missing. Another practical issue
concerns the occurence of improper functioning in these
systems, which arise from maldistributions those origin is
still unclear.
To address these questions, an accurate modelling of the
hydrodynamics of bubbly flows in packed beds is needed.
Attempts have been made in that direction using
mechanistic approaches. In particular, Attou et al. (1999)
were the first to propose a model inspired from an Eulerian
twofluid formulation. Yet, their model suffered from
serious drawbacks, and we therefore proposed a new one
dimensional twofluid model that better accounts for
phenomena arising at the pore scale (Bordas et al. 2006).
The first comparisons with experiments proved encouraging.
Yet, they were achieved over a limited range of flow
conditions. The purpose of the present contribution is to test
further the validity of our proposal using new data gathered
in two different experimental facilities.
Summary of the 1D model
The model itself was presented in Bordas et al (2006). It
was grounded on local instantaneous Eulerian twofluids
balance equations averaged at a mesoscale scale. That
mesoscale was selected so as to be intermediate between a
pore characteristic size and the outer dimensions of the
fixed bed. Accordingly, all the relevant variables were
averaged at that mesoscale (this averaging is denoted by
<.> in the sequel). The main features of this model are
recalled here. Assuming quasi onedimensional (along the z
direction), and stationary motions at mesoscale for both
phases, the continuity balances are automatically satisfied,
Paper No
and the momentum balances for each phase writes:
dp _Mz
dpPLg+f S< 1>
dz 1a
< > < >=pLg+ (2)
V, 1a
In the above equations, z is directed along the main motion
so that dp/dz and the liquidsolid force density fLS are
always positive quantities. Moreover, g should be
understood as g cos(z,g) where (z,g) denotes the angle
between the main motion direction and gravity. The average
momentum exchange term )> corresponds to
the liquidgas force density fLG. F* denotes the average
resisting force acting on a test bubble of volume Vp. The
remaining term represents an additional
contribution of the unconditional continuous phase stress
to the generalized Archimedian force acting on the bubble.
Indeed, the operator Ri is of the order (a/L)2, where a is the
inclusion dimension and L the characteristic length scale of
the unconditional continuous phase flow. In packed beds, a
and L are of the same order since the bubble size happens to
be comparable to that of the pore (at least, when the
characteristic pore dimension is smaller than the capillary
length see Bordas et al. (2006), Jo et al. 2009). Therefore,
the contribution cannot be neglected and
the set of equations (1) and (2) must be kept as such. The
drastic simplifications of the momentum equations that arise
when a<< L (for example for bubbly flows in large tubes or
columns) are not feasible here. The set of momentum
balances (1) and (2) would allow the determination of the
pressure loss and of the mean void fraction <(x> in a fixed
bed if the closures for the four terms fLs, fLG, and
are available. As shown in Bordas et al.
(2006), the later three closures cannot, for the time being, be
made explicit in terms of the considered mesoscale
variables. To bypass this limitation, the equation (2) which
amounts for the gas phase dynamics, has been replaced by a
kinematical relationship relating the average void fraction
with the volumetric gas ratio P and with the apparent
relative velocity . Namely:
P8 (a) _(a) E (3)
1/3 1(a) Vs
Here, E is the porosity, P is defined as VsG/(VsG+VsL), VSL
(respectively VSG) is the liquid (respectively the gas)
surperficial velocity (the superficial velocities are estimated
as the volume flow rate divided by the entire cross section
of the column). The apparent relative velocity
involved in equation (3) is defined as the difference between
the dispersed phase axial velocity averaged at mesoscale
() and the continuous phase axial velocity averaged at
mesoscale (). In view of experimental evidences,
was equated to the mean liquid velocity in a pore, i.e. VsL/s
(Bordas et al. 2006). On another hand, is not readily
accessible and the apparent relative velocity is the new
unknown.
An additional simplification of eq.(1) can be done. Indeed,
the average momentum exchange term )> is
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
expected to be of order (Achard and Cartellier,
2000). Therefore, and at least for small to moderate void
fractions, it can be assumed that its contribution to equation
(1) can be neglected compared with fLS. In these conditions,
the model combining the equations (1) and (3) should be
completed by the closures for the two quantities E/VSL
and fLs. The closures proposed in Bordas et al (2006) for
these two quantities, which were based on the physical
mechanisms occurring at the pore level, are recalled
hereafter.
Concerning the apparent relative velocity, one expects that
bubbles with a size comparable to that of the pore should
behave similarly as slugs i.e. strongly confined gas
inclusions, that is their relative velocity should be
proportional to the continuous phase mean velocity.
However, in single capillary ducts, the resistance to the slug
motion is controlled by the liquid flow around the bubble,
so that the ratio /VSL is a Capillary (Ca = IL (VsL/E) /
o) dependent quantity. In fixed beds, the liquid can bypass
the bubble using nearby connected pores, and that feature
leads to a ratio /VsL that is insensitive to Ca as shown
in Bordas et al. (2006) by the analysis of various data sets
from the literature with the help of the kinematic relation (3).
That analysis also shown that this ratio monotonously
increases with the gas flow rate fraction P. For P below 0.6,
the increase was nearly linear, and follows the trend:
e(U, )/V,= f(f) with f(fI)=6.7f1.9
where the coefficients have been identified from four data
sets corresponding to various flow conditions (Bordas et al.
2006). This peculiar behaviour may be interpreted as an
evolution of the bubble size with the gas content.
Alternatively, and more probably, it could be the trace of a
channeling effect. Whatever its origin, the combination of
the kinematic law along with the above closure for
allows to predict the mean void fraction for known
phasic superficial velocities VSL and VSG.
Concerning liquidsolid force density fLS, and using an
analogy between the bubble motion in a packed bed and
bubble trains in capillaries, it was expected that the
twophase flow pressure drop fLs scaled by its singlephase
flow counterpart fLslat the same superficial liquid velocity
should linearly increase with the void fraction, namely:
S= fLS /fLSl = 1+ (a)
Here fLSl is given by the Ergun law:
fLSl, L/(/u L d 2d,) = [(1 E)2 / 3][A + B. Re,]
with A=180 and B=1.8 for a rhombohedric arrangement
according to MacDonald et al (1979), and with a pore liquid
Reynolds number defined as Rep=PL.VsL.dbeads/[IL (10)].
Moreover, X is expected to evolve with the Capillary
number, according to a scaling XCa'/3 (Bretherton (1961).
The analysis of available data has demonstrated the validity
of the expected linear variation of V with the averaged void
fraction . In addition, the coefficient X was found to
Paper No
decrease with the Capillary number according to a power
law, with an exponent close to the expected value of 1/3
(Bordas et al. 2006).
Although encouraging, the underlying hypotheses need to
be ascertained and the range of validity of the proposed
closures deserves to be investigated. Among the main issues,
let us recall that the proposed model is based on the
assumption of a fixed mean bubble size which uniquely
controlled by the packed bed geometry when the pore scale
is much smaller than the capillary length scale. Such a
behaviour has been demonstrated in dilute conditions (up to
0.02 in void fraction, Bordas et al. 2006) and recently
confirmed by Jo and Revankar (2' "' 'i on a 2D experimental
setup. However, the question remains open for dense
regimes. In particular, the increase of the apparent relative
velocity with P may be attributed either to a change in the
bubble size or to an evolution of the bubble dynamics
because of the formation of preferential paths. In the same
perspective, some effects of the absolute pressure level on
the flow dynamics have been reported in the literature. It is
not clear whether bubbles expanding in a strong pressure
gradient continuously break to keep their equilibrium size or
whether their dynamics is actually modified. Concerning the
limitations of the proposed closures, the proposed model
predicts a void fraction less than the gas flow rate fraction
for P above 10 to 20 %. In experiments, an opposite trend is
systematically observed at lower gas flow rates that may be
due to measurement uncertainties or that may correspond to
a different gas dynamics, such as for example trapped
bubbles. Indeed, the proposed model assumes that all the
bubbles contributing to the void fraction were mobile (the
void faction involved in the kinematical relationship (3)
corresponds to mobile bubbles only). Yet, it is known that
bubbles can be trapped within the bed, and such bubbles
may alter the void fraction gas flow rate fraction
relationship. They can also bring a contribution to the
pressure drop, which is not accounted for by the closure
proposed in equation (5).
To address these questions, new experiments have been
performed in order to cover an enlarged range of flow
conditions and flow parameters while gathering all the
necessary information to test the model. In that scope, a
great attention has been paid to the measuring techniques,
both to understand the precise meaning of the measured
quantities, and to control the uncertainty.
Experimental Facility
To gather new sets of experimental results, experiments
have been performed in parallel at LEGI and at IFP.
LEGI experiment. The experiments at LEGI (denoted
ExpLEGI in the sequel) have been performed in a 5 m high
vertical cylindrical column with a 50 mm inner diameter.
Airwater cocurrent, upflow conditions were considered.
The column was made of plexiglass, except its lower part
which was made of steel because of pressure constraints.
The measurement section was 2.395 m long. It was
completely filled with glass beads (diameter 2 mm) and this
packed bed was tightly maintained between two fixed grids.
The geometrical arrangement was close to be rhombohedric
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
and the porosity equal to e=0.34.
The gas was injected at the bottom of the column through a
porous material, that allowed the formation of small bubbles,
with a typical diameter about one millimeter. In order to
ensure almost fully developed conditions from the very
beginning of the fixed bed, the bubbles produced by this
porous material were forced to breakup in a short (20 cm
long, representing about 100 bead layers) packed bed socket
inserted in the steel tube and located 15cm upstream the
main bed entrance. Since the bead diameter and the
arrangement were the same in the socket and in the test
section, one expects that the bubbles have already reached
their equilibrium size (as stated in Bordas et al. 2006) before
entering the measuring section.
*e4r
/I
Figure 1 : Experimental facilities (a)LEGI (b) IFP
The measured quantities consisted of the pressure drop
along the test section and of the mean void fraction. The
absolute pressure level was recorded at the level of the gas
injector. The mean void fraction was accessible by way of
the gas volume fraction measurement. The later was
achieved using two quickclosing ball valves located at each
extremities of the measuring section, that is about 2 m and
4.5 m above the gas injector respectively. The simultaneous
closure of the valves was ensured by a mechanical link
driven by a pneumatic command. The closure time was
sufficiently short (less than 0.1 s) for the flow conditions
considered. The void fraction measurement requires the
determination of the volume of liquid trapped inside the test
section. Because of the column design, the liquid volume
between the two valves could not be directly measured.
Instead, the measuring section was drained using the lower
pressure port. Sufficient time was allowed for the draining
process to be completed. The volume comprised between
that location and the upper valve was emptied and the
corresponding liquid volume was measured by weighting.
Knowing the available free volume within this test section,
the mean void fraction could be estimated. The fact that all
the measuring section is not drained is the main factor of
uncertainties : indeed the gas trapped below the draining
device can move in the measuring section. An upper and
lower estimation of this uncertainty was computed assuming
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
that all the gas moved in the measuring section or remained
trapped outside the measuring section. Another source of
uncertainties is the residual liquid which remains trapped
within the test section when the column is drained. This
source of uncertainty was estimated using results found in
OrtizArroyo et al (21 ') : from the results found in this
publication, the static liquid holdup is estimated to be about
5% and the corresponding uncertainty on the void fraction
was computed accordingly.
IFP experiment. The experiments at IFP were performed for
cocurrent downflow conditions. The experimental facility
was composed of a 400mm I.D. and 2.250 m long vertical
column. The packed bed was installed over a height of 1.55
m. Experiments were performed either with heptane and
nitrogen with 2.5 mm beads (refer to as ExpIFP1 in the
sequel, and with water and air with 2 mm beads (refer to as
ExpIFP2 in the sequel. The porosity was measured
considering the weight of the beads filling the column. It
was found equal to 0.348 for ExpIFP1 and 0.35 for
ExpIFP2. The pressure drop between different sections
was measured with differential pressure sensors (range
0300 mb and 3003000mb, relative uncertainty less than
2.5%) installed along the test section. The absolute pressure
level was also recorded at the column inlet. The average
void fraction was measured by yray tomography at a
location 80 cm downstream the injector. The yray
tomography, which was developed at IFP, provides a
timeaveraged value of the gas phase fraction in a cross
section. The measurement principle is described in Boyer et
al ( '2) and a analysis of the uncertainties can be found in
Boyer and Koudil (1999). The main source of error arise
from statistical errors due to random y photons emission,
from the dynamic bias associated with the gas fraction
fluctuations and from reconstruction errors. The intrinsic
performance of the system with regards to these potential
error sources was studied in Boyer et al. (2 i"" ) and Boyer
et al. (2001). As shown in Boyer et al (2 I"" ), the maximum
absolute error on the gas fraction measurement is 3%.
o~so
8 0,50
> 0,40
0,30
0,20
0,10
0,00
0.00
A Exp LEGI Water/Air 2mm Glass Beads Upflow
SExpIFP2 Bordas et al (2002) Water/Air 2mm Glass Beads
Downflow
SExpIFP1 Boyer et al (2002), Hepta/N2, 2,5 mm beads,
Downflow
SI
0,10 0,20 0,30 0,40 0,50 0,60
mean gasflowrate ratio ,BGlEGIibcalOEIFP,
Figure 2 : Evolution of the void fraction a with p
For each operating conditions, the reference volumetric gas
ratio P used in figure 1 was taken as its mean value between
the bed inlet and outlet. For sake of clarity, the horizontal
extend of the uncertainties on p are not drawn on figure 2.
Lets stress that the variation on p between the inlet and
outlet are such that in some configurations, p can be
multiplied by a factor 3. The uncertainty on a was computed
according to the error sources identified in section
LegiExperiment.. In both experiments, the flows are
stationary and fully developed. Therefore, neglecting flow
inhomogeneities that may arise close to external walls, the
gas surface or volume fractions provide indeed a measure of
the void fraction.
Whatever the experiment or the flow conditions considered,
the void fraction a is always significantly less than the gas
flow rate fraction p provided that the later is not too small
(say for p above about 0.15). This trend is the same as the
one already noticed by Bordas et al (2006), one observes a<
p. In addition, both data series that correspond to upflow
and downflow behave very similarly, indicating that gravity
is not a key parameter governing the bubble dynamics.
Results
Relative velocity
Void fraction
Figure 2 shows the evolution of the void fraction a with p
for the three sets of experiments. For the IFP experiments,
the void fraction in the ordinate represents the mean gas
surface fraction in a cross section located 800 mm
dowstream the bed entrance. Accordingly, the P parameter
has been computed in the same crosssection by accounting
for the absolute pressure level. The uncertainties on a have
been reported in figure 1. For the LEGI experiments, the
void fraction represents the gas volume fraction (relative to
the total pore volume) between the two quickclosing
valves. Clearly, the volumetric gas ratio P evolves along the
column height because of the change in the absolute
pressure. Since the flow conditions are diabetic, one can
compute P(z) for known phasic mass flow rates and from
the measurements of the pressure drop and the absolute
pressure at the gas injection.
The kinematical relationship (3) was then used to derive the
quantity Ur/Vpore where Vpore is the interstitial velocity
defined as VSL /F. The results are shown on figure 2 along
literature data already presented in Bordas et al. (2006).
As stated in Bordas et al (2006), only data's in the range
0.2
eq.(3). Below P=0.2, the uncertainty on the void fraction
can be significant depending on the sensitivity of the
measurement techniques. Moreover, as stated in Bordas et al
(2006) and in the beginning of this paper, in this range, the
proportion of blocked bubbles can be significant and add a
supplementary source of error on the estimation of the void
fraction associated to mobile bubbles. However, despite
these uncertainties, in the range 0.2
kinematical relationship is assumed to be valid, in figure 3
the same trends as in Bordas et al, (2006) are observed and
Ur /Vpore happens to a be a unique function of the gas flow
rate fraction:
Paper No
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
25
with the prefactor K independent on the superficial
velocities.
vC
p mean (LEGI)/ local (IFP)
Figure 3 : Evolution of Ur/Vpore with P
This linear behaviour accounts for downflows as well as for
upflows (see ColiSerano data, along those taken in LEGI
and IFP on figure 3). Therefore, at the first order, U, /Vpore
is not only a unique linear function of P, but the K
coefficient seems not to depend on the flow direction. That
gives weight to the assumption of a "sluglike" bubble
dynamic. The linear fit presented in Bordas et al. (2006)
gave K=6.7 and CO=1.9. This fit was performed on all data
including data point below P=0.2. As the bubble dynamic in
a capillary can be expressed as Ububble/Vpore = C or
Ur/Vpore=C1.Vpore, we get C=K.3 and so Co=l.
Assuming that Co=l, the "best" fit corresponds to K=5 Two
others values for K or Co are also proposed on the figure:
given the uncertainties on the measurements, they
correspond to the probable domain of variation of K in the
range 0.2<3 <0.7. The model sensitivity to these values of
K will be analyzed further when it will be compared to
Attou's model. Given the experimental results, the C
coefficient depends on 3 (C=K.3) with value ranging from 1
to 3.5(K=5) in the range 0.2
correspond particularly well on the typical value of the C
coefficient for an isolated slug in capillaries. Figure 3
presents results gained from a postprocesssing of data
given in Jo and Revankar (2i I r' '. In this paper, the size and
dynamic of bubbles moving in a 2D experimental setup was
studied. Bubbles velocities were directly measured uisng a
high speed camera. At a given liquid superficial velocity, it
is found that the ratio Ububble/VS1 ranges from 1.2 to 2.5
with increasing 3 ranging from 25% to 50%. At a given
superficial gasvelocity, results are less monotonous (see
figure 4), but the ratio Ububble/Vsl globally increases with P,
for p ranging 30 to 60%: this behaviour is in accordance
with the present results.
0,5
0
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
Figure 4 : evolution of bubble size with P in a 2D in
packed bed made of cylinder (Jo and Revankar, 2009)
Above that value, the bubble velocity behaviour experiences
a drastic change but the tendency is opposite to the one
observed on figure 3. In our case, this may be attributed to
the fact that above 6070%, we are no longer in bubbly
configuration.
The bubble velocity increase with the volumetric gas ratio P
seems then to be a dynamic specific to packed bed. The
physical processes underlying this behaviour are however
difficult to ascertain. The proposed closure deals with the
mean behaviour of the bubble cloud and is based on average
global parameters. Locally, the bubble dynamic is strongly
influenced by the local fluid velocity which can experience
strong fluctuations (fluid redistribution in adjacent
interconnected pore, preferential path...): those fluctuations
are strongly connected to the dispersed phase content.
Liquidsolid interaction and model closure
In this section, the objective is to test further the
propositions made in Bordas et al. (2006) concerning the
liquidsolid interaction term. From the total pressure loss
and void fraction measurement, the twophase pressure loss
fls* is expressed as:
upflow f ,*=dpldz_ (1a)p g
downflow f,* =dpldzlmeasd +(a)p, g
In the above equation, fls holds for the quantity
fls (see equation (1)). The ratio V between the
two phase pressure drop fis* and its counterpart in single
phase flow fis,1 at the same superficial liquid velocity was
plotted as a function of a for various pore Reynolds
numbers Rep. The results are presented in figure 5a and
figure 5b.
Paper No
Sf(f)= K/l+Co
Vpore
 FixedVsl
o Fixed Vsg
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
CollSerano, dp=2mm upflow, ETG/N2
CollSerano, dp=2mm, upflow, Water/N2
IFP Data, dp=2 mm, downflow, heptan/N2
ExpIFP2, Downflow, Water/AIR
ExpIFP1, Downflow Heptane/N2
ExpLEGI UpflowWater/Air
0 7 Ca"1/3
+ o oa
Rep = 70 Ca=0,00089
* Rep110Ca=0,0014
a Rep 150Ca=0,00194
* Rep=190 Ca=0,00242
x Rep=250 Ca=0,00319
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40
a mean
Figure 5a : evolution of with a Upflow (ExpLEGI)
0,0001
0,001
o ExpFP2 Rep = 123 Ca=0,00158
4
A ExpFP2 Rep = 153 Ca=0,00198
,5 o ExpFP2 Rep = 184Ca=0,00238
# ExpFP1 Rep=290 Ca=0,0048
3
SExpIFP1 Rep=373 Ca=0,0064
2,5 ExpFP1 Rep = 476 Ca=0,0080
o
0,000 0,050 0,100 0,150 0,200 0,250 0,300 0,350 0,400
a
Figure 5b : evolution of wwith a Downflow. (ExpIFP1
and ExpIFP2
At first order, the linear evolution of V for a <0,40
suggested in Bordas et al (2006) is confirmed whatever
the flow direction:
Figure 6 : evolution of X with Ca.
On this figure, are plotted data from literature and the
present data. For these data, at a given liquid flow rate,
several gas operating conditions were tested. For ExpLEGI
data, the value of omean was used in the calculation of X. At a
fixed capillary number, the discrepancy on X value is not so
high which seems to valid the X dependency on Ca alone at
the first order and the closure law structure presented above
for fis. Let's notice also that as X = (w1) /c, its value is very
sensitive to the accuracy of the void fraction measurement.
This explains the discrepancy of data when all the data are
taken as a whole. However, the general trend corresponds to
a global variation with Ca1/3
Strictly speaking the pore velocity should be corrected by
the space available for the fluid inside the pore after
accounting for bubbles: Ca=(VsL.dp)/(e.o.(1a))
Thus equation (9) should write
y =1+ A(Ca, a).a
/ = 1+ 2(Rep).a
This result gives weight to the physic postulated at the pore
scale and is consistent with the assumption that the two
phase pressure drop is mainly controlled by the liquidsolid
friction term fis and that the gasliquid interaction term
has a weak contribution. This behaviour is also
encountered for twophase pressure drop in duct (Riviere an
Cartellier, 1999). In the sequel the term fis* will be
assimilated to fis.
The effect of the Reynolds number on V (or X) is also
clearly apparent. But if we keep the analogy between bubbly
flow in fixed bed and twophase flow in capillaries made in
Bordas et al (2006), this number may not be the correct
scaling parameter and, following the analysis of Betherton
(1961), the capillary number should be involved too. Figure
6 shows the evolution of X with the capillary number
Ca=(VsL.dp)/(e.o) where dp is the beads diameter, e the
bed porosity a the surface tension and VsL/E the pore
velocity.
and the model is not completely linear with the void fraction.
Figure 7 shows the evolution of X with the corrected
Capillary number
SColliSerano, dp=2mm, upflow, ETG/N2
CollSerano, dp=2mm, upflow, Water2
IFP Data dp2 mmdownflow, heptanN2
Sene3
SExpIFP2 (ater/AirDownflow)
ExpIFP1 2002 (Heptan/NDownflow)
+ExpLEGI Upflow WaterAir
o^ ^ ,
0.0001
0,001
Corrected Ca
Figure 7 : evolution of X with the corrected Ca
Given the uncertainties on the data and sensitivity of X with
a, the trends already observed are not changed and if there
are any nonlinearity due to the void fraction, its effect
seems weak and not easily noticeable from global
measurements.
Paper No
ExpLEGI Water Air Upflow dp=2mm
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Model testing and discussion
along the data of Yang et al (1993).
Model consistency with the bubble dynamic in packed bed: the
problem of the static void fraction
Although void fraction measurements were not performed
with similar techniques at IFP and at LEGI, both methods
give access to the total voidage in the packed bed. The total
void fraction can be decomposed as a dynamic void fraction
and a possible static void fraction due to bubbles blocked in
the granular media. The model presented above considers
only the dynamic void fraction. The relative importance of
this static void fraction is fundamental to assess conclusions
presented here and to estimate the limits of validity of the
model presented above.
Liquid saturation measurements performed by Larachi et al.
(1991) showed a different behavior depending on the liquid
viscosity. For high viscous liquid, the liquid saturation goes
toward 1 (i.e. the void fraction goes toward zero) when the
gas flowrate goes toward zero. However, for low viscous
liquid, the void fraction goes toward a non zero value when
the gas flow rate decreases toward zero. This different
behaviour is probably due to blocked bubbles. The shear
rate is higher for viscous liquid, which favours the bubble
detachment. To characterize this phenomenon, the capillary
number seems to be the appropriate dimensionless number.
Measurement performed in LEGI on the index matching
column presented in Bordas et al. (2006) showed that, below
a given critical Reynolds number, a fraction of the bubble
population was blocked in the porous media. This critical
Reynolds number depends on the fluid used (Cyclooctene,
Cargille code 5095). For experiments performed with
cyclooctene (which has a low viscosity) and nitrogen, in the
dilute regime, it was shown that the critical Reynolds
number was Rep=170 (Bordas (2" '2). This critical Reynolds
number corresponds to a capillary number Ca=2.76e03. In
the case of the Cargille liquid, whose viscosity is ten times
greater than the cyclooctene viscosity, no blockage was
observed (Bordas ,21 ,2)). The operating conditions lied in
the range of Rep=8 to 64 which corresponds to Capillary
numbers Ca=0.03 to 0.074. These values are above the
critical Capillary number defined from the Cyclooctene
experiments.
Concerning the experiments presented in this paper (non
dilute regime), it was not possible to perform the estimation
of the static void fraction in the case of the data acquired in
LEGI (the measurement technique didn't allow a visual
direct access in the column core). But some tests (for the
waterair case performed by Bordas) were carried out at IFP
with the tomographic technique for ExpIFP1 and ExpIFP2.
The methodology was the same as the methodology
followed in Yang et al (1993): the reactor was feeded with
gas and liquid. After a given time, the gas feeding was
stopped and the residual void fraction measured. This
fraction can be then compared to the total void fraction in
normal operating conditions.
Given the operating conditions, the evolution of the static
void fraction with the capillary number is shown on figure 6
0,16
0,14 Yang (1993) WaterAir Upflow dp=2 2 rrm
S0 12 oYang (1993) Desel FuelN2 Upflow dp=28 mm
cc 0,12 
0,1 A Bordas (2002) waterair downflow dp=2 mm
S0,086
S0,04 A
(0.044 J
0,02 **
0 0,001 0,002 0,003 0,004 0,005
Ca
Figure 8 : Static gas retention EXPIFP2 and Yang
experiments.
These later authors used several techniques to measure the
liquid saturation : the liquid static retention was measured
using gammaray adsorption. A careful analysis of their data
led to the computation of the static void fraction presented
on figure 8. This figure shows that the static void fraction
decreases drastically with the capillary number. In
downflow, even if this decrease is very steep, the amount of
blocked bubbles can be quite significant. Unfortunately, the
operating conditions in Yang et al (1993) (namely the gas
flow rate) are not completely known to fully compare Yang
data and ExpIFP2 data. Figure 9 present the ratio between
the static void fraction and the total void fraction versus the
volumetric gas ratio 3 using ExpIFP2 data.
o Bordas (2002) Vsl=3 cm/s Rep=92
* Bordas (2002) Vsl=4 cm/s Rep=122
o Bordas (2002) Vsl=5 cm/s Rep=155
A Bordas (2002) Vsl= 6 cm/s Rep=186
o
So
o
0
A o
A
A
0 0,2 0,4 0,6 0,8
Figure 9 Residual void fraction. ExpIFP2
Figure 9 Residual void fraction. ExpIFP2
Results are consistent with figure 8. At a given 3, the
amount of residual gas decreases with the superficial
velocity (and thus with the capillary number). This confirms
results obtained by visualization on the index matching
experiments presented in Bordas et al (2006). In ExpIFP2,
the capillary number lies between 0.001 and 0.0024 which
is below the critical capillary number Ca=2.76e03. That
could explain the relatively high ratio "static void
fraction/total void fraction". For ExpLEGI1, the value of
the static void fraction is not given as this phenomena could
not be quantified. Provided the experimental conditions, the
Capillary number lies between, 0.003 and 0.008 which
would correspond to no or a weak gas static retention.
The static void fraction decrease with 3, for a given
superficial velocity, is also consistent with the observations
presented in Bordas et al ( '2) for the dilute regime and
Paper No
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Larachi et al (1991) observations: the gas flow rate increase
leads to more bubbles interactions which eventually leads to
detachment. Those interactions can be direct or indirect, as
the bubble motion induces pressure and velocity fluctuations
in the adjacent pores. Lets point out that the methodology
used by Yan et al (1993) (and applied for IFP experiments)
leads certainly to an overestimation of the static void
fraction presented on figure 8 and 9: measurements are
performed when the gas flow is stopped which reduce the
effect of bubbles/bubbles interactions.
To conclude, the following remarks can be made:
 whatever the measurement technique, it will be difficult to
give a quantitative estimation of the static void fraction in
our experiments: although this fraction could be estimated
with the tomographic technique, it depends at least on two
parameters : the volumetric gasflow rate (which account for
bubble interactions) and the capillary number(which
account for the force balance on the bubble). A specific
study, beyond the scope of this paper, should be undertaken
to fully understand the mechanism and the coupling leading
to bubbles blockage. Qualitatively, it can be said that the
bias on the data will be less important for the highest
superficial velocity, high viscous fluid and above all, for
highest volumetric gas ratio (typically > 20%, value
corresponding to the lower limit of the model validity range
presented in Bordas et al, (2006). For these values, bubbles
interactions limit the static void fraction.
 In the worst case (waterair), the maximum ratio between
static void fraction and the total void fraction in this P
range is around 50%, which seems very high. But this value
is probably greatly overestimated because of the
measurement methodology, which doesn't account for
bubbles interactions when the gas flow.
 Results presented in section Results are based on the total
void fraction measured by both techniques (quick closing
valve and tomographic technic). The results (for 3>20%)
seems consistent with the presented model which doesn't
include effect due to blocked bubbles which is coherent with
the discussion above. For smaller p (3<20%), some
deviation from the model (for example the a behaviour with
p) could be partly explained by a different dynamic, where
the lack of interaction between bubbles could promote gas
blockage within the porous matrix.
As a conclusion, the physics invoked in Bordas et al (2006)
to express the closure law for the void fraction and the
twophase pressure drop is thus not invalidated by the
present experimental data and that physics seems to hold for
non dilute regime.
Model sensitivity and model comparison with Attou's model
In the section above, the structure of the closure law infered
in Bordas et al (2006) have been validated on experimental
data. The model is compared here to another mechanistic
model namely Attou's model, which was also based on a
twofluid formulation at the pore scale.
Void fraction prediction
1,00
0,90
0,80
E
S0,70
S0,60
0,50
0,40
o 0,30
0,20
0,10
0,00
a=p
SExpFP2, Water/Air, 2 mm beads, Downflow
LEGI's model prediction K=5
 LEGI's model prediction K=6
LEGI"s model prediction K=4
SAttou's Model(1999)
/ *< *
_ 
++
0 0,1 0,2 0,3 0,4 0,5
gas flowrate ratio p atz=800 mm
1,00
0,90
0,80
S0,70
0,60
0,50
0,40
S0,30
0,20
0,10
0,00
0
Boyeret al (2002), Heptan/N2, 2,5 mm beads,
Downflow
LEGI's model prediction K=5
 LEGI's model prediction K=6
LEGI"s model prediction K=4
Attou's model(1999)
p. I
0,1 0,2 0,3 0,4 0,5
gas flowrate ratio p at z=800 mm
0.6 0.7
0.6 0.7
Figure 10: a prediction. Comparison with
experimental data and Attou's model. (a) ExpIFP2 (b)
ExpIFP1
Data from ExpIFP1 and ExpIFP2 are plotted on Figure
10a and 10b along prediction given by the model. The aim
is to check the model sensitivity to the parameter K. CO was
fixed to 1 for the reasons explained in the preceding
paragraph. The model was run with the beads diameter and
physical properties of the fluid corresponding to ExpIFPI
and ExpIFP2. As parameter K was fit with experimental
values including ExpIFP1 and ExpIFP2 data, the values
K=5 and CO=1 (which correspond to the "best" fit on figure
9) gives obviously correct predictions. The two others
curves (K=6 and K=4) are representative of the maximum
error and incertitude on data, when K is fitted from the plot
Ur/Vpore versus P. The figure shows that the predictions are
sensitive to the parameter K. The bandwidth around the
"mean" prediction (K=5) is such that ca~a(K5) +10% .
However, despite the model sensitivity to K, the predictions
given for K=6 and K=4 follows the same tendency observed
on the experimental data. Furthermore, the bandwidth on the
void fraction lies in the range of incertitude of the measured
value of a Finally, although the incertitude on the parameter
K, the model is robust and its relevancy not invalidated in
the frame of engineering applications. However, the correct
derivation of K requires very accurate data in equation (3),
that is to say very accurate measurement on a. The model is
also compared to Attou's model on the same figure. As we
can see, the model fails to predict the correct void fraction
Paper No
Paper No
behaviour and underestimates the experimental value of
both experiments. This difference is attributed to the bubble
dynamic implemented in Attou's model which is not correct:
the drag force which drive the bubble relative velocity is
computed from a drag coefficient modified to take into
account hindering effects. Furthermore the characteristic
bubble diameter is obtained from consideration about the
inclusion breakup by turbulence. Figure 11 shows the
bubble diameter evolution with P for each operating
conditions for both experiments. As for those operating
conditions, the computed 3 and bubbles diameter doesn't
evolve much between the inlet and outlet, results are
presented in term of mean value between the inlet and outlet.
The diameter is normalized by 6 the characteristic pore size
defined in Bordas et al 2006. The predicted ratio db/6 is far
greater than 1 for most operating conditions which is not
consistent with the hypothesis of hindering effects taken in
the model. As well as the experimental results presented in
Bordas et al 2006.
4 Attou's Model Exp IFP1
VSL=0,04 m/S
3,5 o Attou's Model ExpIFP1 data
VSL=0,06 m/S
S 3 eAttou's Model ExpIFP1 data
VSL=0 08 m/S
2,5 A Attou's Model ExpIFP1 data
2 Vsl=0,1 m/s
2 D
1,5 *
0 1 A A 
o
0,5 
0
0 0,1 0,2 0,3 0,4 0,5 0,6
Mean
* 0
0 0,1 0,2 0,3 0,4 0,5 0,6
0 mean
* Attou's Model, ExpIFP2 data
VSL=0,03 m/s
SAttou's Model, ExpIFP2 data
VSL=0,04 m/S
SAttou's Model, ExpIFP2 data
VSL=0,05 m/s
SAttou's Model, ExpIFP2 data
VSL=0,06 m/s
0,7
(b)
Figure 11: Normalized bubble diameter as predicted by
Attou's Model (a) ExpIFP1 (b) ExpIFP2
T and pressure prediction
As in the case of EXPIFP1 experiments, pressure profiles
along the column were also available, the model was run
and the computed pressure profiles were compared to the
experimental profiles. Furthermore, compared to the initial
proposal made in Bordas et al (2006), the correction of Ca
with the void fraction a introduce a non linearity in the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
closure law for Y. As p evolves also with the pressure, the
model consistency with quasi linear pressure profiles was
checked back
The governing equations are recalled below:
 Kinematical relationship (model equation 1 : EqM1)
fl(z) a(z)
1 f(z) 1 a(z) (z)[K (z)+C
with K=5 and CO=1 (best fit on figure 8).
ratio between the twophase pressure drop and the
single flow pressure drop
/ = fls / flsl = 1 + 2(Ca, (z)).a(z) (EqM2)
with = 0.7Ca(z) 13 (best fit on figure 12 )
V,.d
and Ca = V 
(co(1 a(z))
The single flow pressure drop is given by the Ergun law
fLsl= /LSL / d ) = [(1 e)2 / ] [A + B. Re]
 relationship between the two phase liquidsolid friction
and total pressure gradient
f, = dp/dz +(1a)pg
(EqM3)
The effect of pressure on the gas density is taken into
account using the perfect gas law. The inlet data were the
liquid flowrate at the column injection, the mass gas
flowrate at the inlet (or the volumetric gas flowrate given at
the normal condition of pressure and To) and the relative
pressure at the injection. The corresponding volumetric gas
ratio in the measured section at 800 mm from the injection
will be recalled on the figure accounting for the results.
Knowing the value of P(z) in a given section, equation
EqM1 allows the computation of the corresponding void
fraction x(z). p(z) is computed knowing the mass gas
flowrate at the inlet, the liquid flowrate and the gas density
p(z) which depends on the pressure p(z). We get:
Q Pinlet /(z Q (z) (EqM4)
Q,(z) = .Qn inlet and Y(z) 
P(z) Q+QG(z)
c(z) being known equation EqM2 with its closure allows
the computation of fls. Then (EqM3) is integrated with space
step dz using a simple Euler scheme to compute P(z+dz).
First, the model was compared with Attou's model in term
of Y (figure 12).
Paper No
2
.g1 LEGI's model prediction Vsl=0,04 m/s Vsg=[0,00380,01280,0332 ms]
LEGs model Prediction Vsl =0,06 m/s Vsg=[0,00890,02950,0498]
A LEGI's model prediction Vsl0,08 m/s Vsg[00060,02670,04710,068
0,5 * 's model prediction Vsl ,1 m/s Vsg=[000520,02290,04430,0655
0,086 ms]
0 1 be /(1ta)^ 2
0 0,1 0,2 0,3 0,4
Figure 12 : I prediction
Although the nonlinearity introduced by the evolution of P
with the column location and the dependency of the
capillary number with a, the linearity Y with a and its
dependency with the Reynolds number still holds.
Prediction of Attou's model are also presented on the figure.
Because of its structure, this model predict that Y evolves as
l/(la)2. To distinguish between this behaviour and a linear
behaviour can be difficult on experimental data because of
measurements uncertainties. However, the Attou's model
doesn't allow to recover the dependency of the twophase
pressure drop with the Reynolds number as observed on the
experimental data.
Results on the pressures profile, for the present model, are
given on figure 13(a),(b),(c),(d) for each operating
conditions.
EXPIFP1 (2003) Vsl=3 98 cmis Qg=40Nm31h, 72Nm31h and 145 Nm31h
... .....
...6, 45 ......9...
* L3BG1IFP
L3 9B 4 oFP
944 IF'p
EXPIFP1 (2003) Vsl=6 19 cmis 72 Nm3h, 145 Nm31n and 217 Nm31h
o p=13 4 pinle=12,55
intlete =32,24
*4196lFP
 L6 19G1 9calcule
SLs 199 IFP
Le I 10
(c)
EXPIFP1 (2003) Vsl=10 17 cmis 80 Nm31h,142 Nm31h, 217Nm3ih, 290 Nm31h and 361 Nm31h
(d)
Figure 13 : Pressure profiles predictions
The results present interesting feature. The pressure profiles
are reasonably well predict in the case of high P value
whereas the difference between the predicted and measured
profiles increases with decreasing p.
Below that value, because of the uncertainties on the
measured value of a, it could be argued that a different
bubble dynamic is involved. In Bordas et al (2006) however
it was shown that in the dilute regime, the ratio Ur/Vpore
was of the order one for bubbles whose size was around the
pore size 6. The kinematic law should be still valid to
low gas fraction (Ububble/Vpore >1 when 3>0 which is
compatible with the motion of a bubble in confined
capillary) : the model assumes that the momentum exchange
and mean bubbles dynamic is controlled by the bubbles
belonging to that size class.
In EXPIFP experiments, the injection system is such (two
phase jet) that maybe the equilibrium size distribution is not
reached which can explain that the model can't account
perfectly for the measured data. However we expect that
this effect is more characteristic of high gas fraction that is
to say relatively high 3 at the injection.
For low gas fraction, another phenomena which is probably
important is the notion of static/dynamic void fraction
evoked in the first paragraph. As said in this part, this static
retention, although overestimated because of the
measurement procedure, seems to be predominant for low
p. It is difficult to estimate the effect of this static gas
fraction on the pressure drop. One argumentation, given by
Brenkrid (ref) would consist in considering a correction to
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
FP1 (2003) Vsl=7,96 cmls 72 Nm31h, 145 Nm31h, 217 Nm31h and 290 Nm31h
SP=7.33 pnle7,im
SL7G :acul6
L604u IF
L79 asalc cule
L7GGB,1 acule
L7eGGB,01 FP
Paper No
the porosity to compute f ,,
lcsorrected rs i/a ,, (14)
fis. = fA o/(I^t) (1)
However this correction does not take into account the
change in the effective solid specific surface due to fluid
redistribution in adjacent pore (because of the blocked
bubbles). The physics in equation is (14) then too simple
and probably doesn't account well for the pressure drop due
to the blocked bubbles.
To summarize, even if the discrepancies at low gas fraction,
correspond to phenomena not taken into account in the
model, the maximum discrepancy between the computed
pressure and measured pressure at the column bottom is
20% (VsL=10,17 cm/s pinlet=4,82%).
Conclusions
In Bordas et al (2006), a new onedimensional model able to
predict the pressure drop and the mean void fraction for bubbly
flows in packed beds was proposed. The equations required closure
laws accounting for the liquidsolid and the gasliquid interactions.
Those closures and some feature of the model were deduced from
experiments in the dilute regime or from the analysis of literature
data. Therefore, the model needed to be validated on a larger set of
experiments. In this paper, experiments performed in LEGI
and IFP, in the dense regime, were thus presented.
The relationship between the void fraction and the
volumetric gas ratio P, as well as the evolution of the
apparent relative velocity, confirms a behaviour postulated
from the analysis of previous literature data : the overall
dynamic is mainly controlled by large bubbles who behave
like "slug" in capillaries. Yet, and contrary to bubbles
confined in a single duct, the relative velocity at mesoscale
happens to be weakly dependent on the local flow
organization, because the liquid can freely bypass the gas
inclusion through neighboring channels: such a flow
organization almost eliminates any dependency of the
relative velocity with the capillary number. However, the
analysis of experiments indicates that the relative velocity at
mesoscale monotonically increases with the gas flow rate
fraction (between 1 up to 4 times the liquid superficial
velocity), which seems to be a dynamic specific to packed
beds. This phenomena is tentatively attributed to the
formation of preferential path for the gas, but the physical
processes underlying this behaviour are difficult to
ascertain.: the proposed closure deals with the mean
behaviour of the bubble cloud and is based on average
global parameters. Locally, the bubble dynamic is strongly
influenced by the local fluid velocity which can experience
strong fluctuations (fluid redistribution in adjacent
interconnected pore, preferential path...). Those fluctuations
are in return strongly connected to the dispersed phase
content
Concerning the pressure drop in the liquid phase, the later is
mainly attributed to capillary excess pressure due to the
presence of bubbles those size scales as 6 Consequently, the
ratio of twophase pressure drop to the onephase pressure
drop at the same superficial velocity is expected to linearly
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
increase with the void fraction. In addition, the
proportionality coefficient should evolve with the capillary
number. These anticipations are presently validated on both
experiments in LEGI and IFP.
When tested on available pressure profile, the model
reproduce relatively well the experimental data, provided
that the volumetric gas ratio is sufficiently high, which
corresponds to the range of validity of the closure laws when
the data were analysed. The discrepancy at law P is attributed
to the static gas fraction whose effect on the pressure drop is
not taken into account in the model. It is postulated that,
because of direct or indirect bubbles/bubbles interactions,
this static fraction, in our case (Ca number greater than the
critical capillary number), is negligible at higher P.
Acknowledgements
The authors thanks the Institut Francais du Petrole for
having funded this work
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