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Powder Technology 14
Discrete characterization of cohesion in gas–solid flows
Kunal Jain, Deliang Shi, J.J. McCarthy*
Department of Chemical and Petroleum Engineering, University of Pittsburgh, 1249 Benedum Hall, Pittsburgh, Pennsylvania 15261, United States
Received 31 December 2002; received in revised form 2 August 2004; accepted 3 August 2004
Available online 23 September 2004
Abstract
Despite the fact that a sizable portion of gas–solid flows are cohesive in nature, the mechanics of cohesive flowing gas–particle systems is
still poorly understood and manipulation/control of the flow variables is still largely done on a trial-and-error basis. While recent advances
have been made in our understanding of liquid-induced cohesion at the macroscopic level, in general, it is still not possible to directly connect
this macroscopic understanding of cohesion with a microscopic picture of the particle properties and interaction forces. In fact, conventional
theories make no attempt to distinguish between these modes of cohesion, despite clear qualitative differences (lubrication forces in wet
systems and electrostatic repulsion are two good examples). In this work, we introduce a discrete characterization tool for gas–solid flow of
wet (cohesive) granular material—the Granular Capillary Number (Cag). The utility of this tool, which is a ratio of the capillary force to the
drag force, is computationally tested over a range of cohesive strengths in two prototypical applications of gas–solid flows. It is shown that
rescaling our results in this way yields a collapse of data for varying surface tensions and fluidization velocities, and that a clear transition
from free-flowing to cohesive behavior occurs at a distinct value of Cag.
D 2004 Elsevier B.V. All rights reserved.
Keywords: Particle dynamics; Cohesion; Fluidization; Mixing
1. Introduction
Gas–solid transport is crucial in a variety of industrially
important applications. In particular, fluidization and the
transport of solid particles either by gravity or by pneumatic
means are used in fluid catalytic cracking, fluid hydro-
forming and solid fuel processes such as coal gasification
and liquefaction [1]. In many instances, the cohesive nature
of a powder sample is a prime factor contributing to
difficulties in powder flowability causing, for example,
channeling and defluidization in combustion/feeder systems.
Despite recent advances [2,3], an understanding of the flow
and characterization of cohesive gas–solid flows remains
poor.
The most widely used classification system for particles
in gas–solid flows was introduced by Geldart [4] based on
the density difference between the particles and the gas
0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.powtec.2004.08.001
* Corresponding author. Tel.: +1 4126247362; fax: +1 4126249639.
E-mail address: [email protected] (J.J. McCarthy).
(qp�qg) and the average particle diameter dp—where
particles are generally considered cohesive (C), aeratable
(A), sand-like (B), and spoutable (D) as particle diameter
increases. However, a first principles basis for the group
boundaries and/or how they change as a function of
particle interaction forces is elusive. In particular, the
group A/C demarcation is by no means absolute and in
addition to the particle size and density also shows a
dependence on a multiplicity of other factors such as the
width of the particle size distribution, the particle shape and
the surface texture and composition [5]. Recently, computa-
tional studies of gas-fluidized beds [2,3,6,7] based on the
combined approach of Particle Dynamics (PD) modeling
and computational fluid dynamics have been used to
investigate the impact of liquid-induced cohesion in these
systems.
In this paper, we first review several discrete character-
ization tools [8] for cohesive particle systems then
introduce a novel characterization tool specifically for
gas–solid flows. Our goal is to use this tool, combined
6 (2004) 160–167
Table 1
Simulation parameters
Parameter Soda-lime glass
Number of particles 1020
Density 2700 kg/m3
Poisson ratio 0.33
Young’s modulus 68.95 GPa
Particle radius 4.7�10�4 m
Friction coefficient 0.3
Parameter Air
Fluid density 1.23 kg/m3
Viscosity 1.80�10�5 kg/m s
K. Jain et al. / Powder Technology 146 (2004) 160–167 161
with computations of gas–solid flows, to examine the
transition from noncohesive to cohesive behavior in gas–
solid flows. Specifically, we vary gas velocities and
bridging liquid surface tension in order to explore a range
of the possible fluidization parameter space. The paper is
organized as follows. First, we describe the particle
dynamics method, including the implementation of the
various interparticle forces necessary to model liquid-
induced cohesion, and briefly explain the numerical
schemes used in our CFD model as well as how the
discrete/continuum coupling is handled. Next, character-
ization criterion based on the physical picture of liquid-
induced particle-level cohesion is reviewed for static and
sheared beds and is developed for gas–solid flows. We test
the characterization tools in the following section by
measuring both the minimum fluidization velocity as well
as the mixing rate in fluidized systems. Finally, we close by
examining the limitations and potential extensions of the
current work.
2. Discrete/continuum modeling
2.1. Particle dynamics
Particle dynamics, a discrete method of simulation, has
emerged as one of the most important tools in probing
granular flows [2,6–11]. The method is extremely general in
that Newton’s second law of motion is used to determine the
trajectories of individual particles and the time evolution of
these trajectories then determines the global flow of the
granular material. The equations that describe the particle
motion, therefore, are:
Linear motion:
mp
dvp
dt¼ � mpg þ Fn þ Ft ð1Þ
Angular motion:
Ipdxp
dt¼ Ft � R ð2Þ
where Fn and Ft are the interparticle forces—normal and
tangential, respectively—acting on the particle and are
functions of contact, drag, pressure and capillary interac-
tions. The interparticle forces for cohesionless systems are
typically determined from contact mechanics considerations,
so that in their simplest form they include normal (often,
Hertzian) [12] repulsion and some approximation of
tangential friction [13]. In the present work, normal
interactions are modeled as elastoplastic contacts after the
work of Thornton [14], while a single-parameter history-
dependent friction is used in the tangential direction [15]—
details can be found in Ref. [8]. Physical parameters of both
the particles and fluids used in the simulations can be found
in Table 1.
2.1.1. Capillary forces
Moisture is a common cause of cohesion in particle
flows. Several models based on the solution of the Young–
Laplace equation are available in the literature [6,7,16] for
the case when the degree of saturation is low enough that
discrete bridges are present at the points of solid contact
(pendular regime, see Fig. 1). The capillary force, Fc, due to
both the surface tension of the bridge fluid as well as the
pressure difference arising from neck curvature may be
expressed as
Fc ¼ 2pRcsinbsin b þ hð Þ þ pR2DPsin2b ð3Þ
where b is the half filling angle, h is the contact angle, c is
the fluid’s surface tension and DP is the pressure difference
across the air–liquid interface.
Mikami et al. [7] provide an empirical fit to the
numerical solution of the Laplace–Young equations
expressed as
Fc ¼ exp Ahþ BÞ þ C:�
A ¼ � 1:1V�0:53 ð4Þ
B ¼ � 0:34lnV� 0:96Þh2 � 0:019lnVþ 0:48�
C ¼ 0:0042lnVþ 0:0078 ð5Þ
where Fc is the normalized capillary force (Fc/2pRc); V is
the bridge volume made dimensionless by the particle
radius (R); 2h is the separation distance between the
particle made dimensionless by the particle radius (R); and
A, B and C are constants. In our simulations, the moisture
content is assumed to be sufficiently low that bridges only
form upon contact of the solid surfaces. These bridges
Fig. 1. Schematic of a symmetric liquid bridge.
K. Jain et al. / Powder Technology 146 (2004) 160–167162
remain in place, however, after solid contact has ceased,
until the particles reach a critical separation (rupture)
distance (hc) given by:
hc ¼ 0:62h þ 0:99ð ÞV 0:34: ð6Þ
2.1.2. Viscous forces
Dynamic formation/breakage of liquid bridges results in
a viscous force resisting motion, derived from lubrication
theory. It is essential that any liquid-induced cohesion
simulation include these effects as they may become large
relative to the capillary force as the particle velocity
increases [17]. In the limit of rigid spheres, Adams and
Perchard [18] derive the viscous force in the normal
direction (Fvn) to be
Fvn ¼ 6plRvnR
2hð7Þ
where l is the bridge fluid’s viscosity, and vn is the relative
normal velocity of the spheres. In the tangential direction
(Fv t), Lian et al. [6] suggest the use of the solution due to
Goldman et al. [19] for the viscous force between a sphere
and a planar surface
Fvt ¼8
15ln
R
2hþ 0:9588
�6plRvt
�ð8Þ
where vt is the relative tangential velocity of the spheres.
2.1.3. Drag force
Drag between the fluidizing medium (gas) and the
particle(s) couples the discrete simulation to the (contin-
uum) fluid flow and represents the primary mode of
interphase momentum transfer. The drag force not only
depends on the local fluid flow field but also on the
presence of the neighboring particles [20].
In this work, we use the drag force (Fd) suggested by Di
Felice [21]. In the formulation
Fd ¼1
2CdqgpR
2ju� vpj u� vp�e�vþ1
�ð9Þ
where u is the local gas velocity, vp is the particle
velocity, and Cd and v are functions of the particle
Reynolds Number Re ¼ 2Rqgej u�vpð Þjlg
� �
Cd ¼ 0:63þ 4:8
Re0:5
� �2ð10Þ
v ¼ 3:7� 0:65exp � 1:5� log10Reð Þ2
2
#:
"ð11Þ
2.2. Fluid dynamics
Anderson and Jackson [22] formulated continuum
equations representing mass and momentum balances from
the point Navier–Stokes and continuity equations using the
concept of local mean variables. The point variables are
averaged over regions that are large compared to the particle
diameter but small with respect to the characteristic
dimension of the complete system.
Continuity equation:
B eqg
� �Bt
þ 5d eqgu�¼ 0
�ð12Þ
Momentum equation:
B eqgu� �Bt
þ 5d eqguu� �
¼ � e5p�X
eFdð Þ
� 5d esg þ eqgg ð13Þwhere
P(eFd) is a summation of all particles’ drag force in
the control volume.
The flow domain is divided into cells of width three
times the particle diameter and discretized using a finite
volume method. An upwind scheme for convective, a
central differencing scheme for diffusive, and a fully
implicit scheme for temporal terms is used. The equations
are then solved on a staggered grid using the SIMPLE
algorithm suggested by Patankar [23]. Periodic boundary
conditions are employed on the left and right sides of the
fluidized bed and a Dirichlet boundary condition is used at
the bottom with a uniform gas inlet velocity. At the top,
Neumann boundary conditions (zero gradient) are applied
assuming the flow to be fully developed. The pressure is
fixed to a reference value at the bottom.
3. Characterization tools
The relevant variables that need to be considered in
studying gas–solid flows include (R, qs, qg, g, c, Dv, g, d,lg), where d is a characteristic length of any shearing
regions and Dv is alternatively the magnitude of the relative
velocity of the particle with respect to the fluid velocity
(Dv=u�vp) or the relative velocity between neighboring
particles (Dv=vpi�vpj). It should be noted that, in this study,
Fig. 2. Pressure drop versus time for different gas velocities. It is clear that
fluctuations in the calculated pressure are dramatically smaller prior to the
onset of fluidization than in fluidized systems (see inset).
Fig. 3. Standard deviation of the pressure drop versus gas velocity for dry
and wet systems. A step-change in the standard deviation of the calculated
pressure drop is seen for both wet and dry systems. This change coincides
with the onset of fluidization and is used as a fluidization criterion in this
work.
K. Jain et al. / Powder Technology 146 (2004) 160–167 163
the viscosity of the liquid bridge fluid, l, is maintained
constant as the effects of dynamic viscous forces in wet
media has been aptly explored elsewhere [17,24].
By a Buckingham Pi analysis, five dimensionless groups
are determined:
/1 ¼dR;/2 ¼
qs
qg
;/3 ¼c
qsgR2;/4 ¼
Dv2
gR;/5 ¼
clgDv
ð14Þ
The trivial dimensionless groups arising from the
density, length-scale, and relative velocity ratios do not
directly factor into studying cohesion and are ignored.
The remaining three may be thought of as combinations
of forces acting in the system: the cohesive force, the
force due to particle collisions, the weight of a particle,
and the drag force. Previous work [8] detailed the
significance of the third and fourth (using the relative
interparticle velocity) group of variables for character-
ization of wet granular systems, so they are only briefly
reviewed below.
The third group (/3), the Granular Bond Number (Bog)
[8], represents the ratio of the maximum capillary force and
the weight of a particle. This group has been shown to be
dominant in characterizing the effects of cohesion in static or
near-static systems [8]. The fourth group (/4) can be
combined with Bog to yield the Collision Number (Co)
[8,25], which represents the ratio of maximum cohesive
force and the collisional force due to Bagnold [26]. This
number has been shown to be dominant in highly sheared or
collisional granular flows where BogNCoN1.
It is the fifth group (/5) and its derivatives (i.e., a
combination of groups 4 and 5) which are examined in this
context for the first time and are of primary importance here.
This group may be easily interpreted as a ratio of the
magnitude of the drag force of the particles (Eq. (9)) to the
maximum capillary force
Fcmax¼ 2pRc: ð15Þ
This yields the Granular Capillary Number (Cag) as
Cag ¼Fc
Fd
¼ 4c
CdqgRju� vpj u� vp�e�vþ1:
� ð16Þ
4. Results and discussion
Computer simulations are run using soda lime glass
particles and air with properties as given in Table 1.
Dimensions of the bed are 30 mm (width)�510 mm
(height)�1 mm (thickness). All simulations have the same
initial condition in order to get values for comparison to
differing degrees of cohesion. The initial condition is
created by loading the particles in a randomly perturbed,
rectangular lattice. Cag is varied by changing the surface
tension of the liquid bridge fluid (as might be done with
surfactants, experimentally), while maintaining its viscosity
as a constant.
4.1. Fluidization
As a first test of the utility of the Granular Capillary
Number, we examine the onset of fluidization—the
Fig. 4. Increased fluidization velocity versus Granular Capillary Number.
Plotting the values of the minimum fluidization velocity versus Cag shows a
clear transition point where this value increases exponentially.
K. Jain et al. / Powder Technology 146 (2004) 160–167164
minimum fluidization velocity—in cohesive gas–solid
systems. The minimum fluidization velocity is typically
defined as the velocity at which the bed pressure drop
goes through a maximum value. A critical component of
this definition is that, while the pressure drop is ultimately
determined solely by the weight of the fluidized particles,
the value of the pressure drop can exceed this limit prior
to fluidization. In the small fluidization systems examined
here, a simpler, but equivalent, definition of the minimum
fluidization velocity is used. The approach used for
Fig. 5. Mixing progress for noncohesive/cohesive materia
determining the minimum fluidization velocity is similar
in spirit to that followed by Kafui et al. [27] which is
based on monitoring the state of the particle connectivity
network. Fig. 2 shows a plot of pressure drop versus time
for different gas velocities. For ubumf, the pressure drop
essentially remains constant and for uzumf, the pressure
drop varies with time. The amplitude and standard
deviation of the pressure disturbance also increase sharply
as the fluidization velocity becomes greater than minimum
fluidization velocity (Fig. 3). Hence, the minimum fluid-
ization velocity can be defined as the velocity at which
the fluctuations (standard deviation) of the pressure drop
go through a step change. This technique provides
reproducible results and avoids difficulties in averaging
for small systems.
We find that, using this definition of the minimum
fluidization velocity, an increase in the Cag (surface
tension) increases the velocity necessary to achieve a
fluidized system relative to that of the completely dry
(noncohesive) case. Fig. 4 shows a plot of the percentage
increase in the minimum fluidization velocity as a function
of the Cag. For values of surface tension where the Cagb1,
changes in the fluidization velocity from that of a
completely dry granular material are essentially unmeasu-
rable; however, for larger surface tensions, where the values
of CagN1, the fluidization velocities increases markedly
requiring as much as a 30% increase in umf at the highest
Cag examined.
4.2. Mixing
The mixing in gas–solid systems is often extremely rapid
compared to mixing in surface-dominated flows [9]. In this
ls. Particle clusters exist in the cohesive materials.
Fig. 7. Mixing rates at different surface tensions. The slope of the intensity
of segregation curves (or mixing rate) decreases markedly with increasing
liquid bridge surface tension.
K. Jain et al. / Powder Technology 146 (2004) 160–167 165
section, we examine changes of the mixing rate of
mechanically identical particles with changes in gas velocity
as well as liquid bridge surface tension.
In all cases, the system is initially completely segregated
with the right-half of the bed consisting of one type (color)
of particle and the left-half another. Typical snapshots of a
dry and wet simulation can be seen in Fig. 5. We measure
the intensity of segregation, essentially the standard devia-
tion of the local concentration, where we define the local
concentration to consist of each particle’s 10 nearest
neighbors. If the intensity of segregation is plotted as a
function of time, the value—initially at 0.5 for completely
segregated—will decrease as the system proceeds toward a
completely mixed state. Fig. 6 shows the evolution of the
intensity of segregation at several different gas velocities,
while Fig. 7 shows similar results for a fixed gas velocity
and varying liquid bridge surface tension. By fitting these
data to an exponential function, a mixing rate constant can
be extracted.
Examining both Figs. 6 and 7, it is clear that, for steady
mixing progress, higher velocities and/or lower surface
tensions result in larger (faster) mixing rates. Our definition
of the Granular Capillary Number (Cag) then suggests that
the importance of cohesion to mixing is determined by an
interplay between the capillary force and the fluid drag.
Plotting the resultant mixing rate constants as a function of
Cag, in Fig. 8, shows that this assertion is valid. That is,
mixing rates are high for small Cag and drop dramatically as
Cag increases. It is interesting to note, however, that under
certain conditions the noncohesive case (where Cag=0)
mixing rates may be lower than for small, but finite values
Fig. 6. Mixing rates at different gas velocities. As might be expected, for
steady mixing progress, the slope of the intensity of segregation curves
(which may be interpreted as the mixing rate) decreases with increasing
fluidization velocity.
of Cag. While this is most likely due to the fact that Cag=0
may be satisfied at essentially any fluidization velocity, a
detailed explanation of this observation is beyond the scope
of the current work.
5. Outlook
Here we examine the transition from free-flowing to
cohesive behavior in gas–solid flows. We have shown in
Fig. 8. Mixing rate versus Granular Capillary Number. Plotting the mixing
rate as a function of Cag causes the data from the previous trials to collapse
on one curve.
K. Jain et al. / Powder Technology 146 (2004) 160–167166
previous work [8] that discrete characterization tools are
extremely useful for studying the practical applications of
both static and sheared granular materials. Again, with the
introduction of the Cag, we note that changes in the
fluidization onset are aptly captured by these types of tools.
While this simple approach works surprisingly well for
predicting the transition point for the minimum fluidization
velocity, at present it does not address the nature of the
change—although there is clearly an exponential change in
its value. Moreover, the mixing behavior is surprisingly
sensitive to these changes as well and, in fact, the transition
seems dramatically sharper for this unit operation. Ulti-
mately, this simple characterization tool may serve as a
useful a priori test of the fluidization character to be
expected in a wet gas–solid system.
6. Nomenclature
Symbols
A, B, C constants in the empirical model of the capillary
rce
foBog Granular Bond Number
Cd drag coefficient
Cag Granular Capillary Number
F interparticle force, N
Fc capillary force, N
Fcmaxmaximum capillary force, N
Fd drag force, N
Fv viscous force, N
Fc normalized capillary force
g gravity acceleration, m/s2
h half separation distance between particles, m
h dimensionless half separation distance between
rticles
pahc dimensionless critical rupture distance
Ip particle moment of inertia, kg m2/s
mp particle mass, kg
DP pressure difference across the air–liquid interface, Pa
r1 bridge meridional radius of curvature, m
r2 bridge neck radius, m
R particle radius, m
Re Reynolds number
u gas velocity, m/s
Dv relative velocity of the spheres, m/s
vp particle velocity, m/s
V dimensionless bridge volume
Greek letters
b half filling angle
c surface tension, kg/s2
d characteristic length, m
e void fraction
h contact angle
l liquid viscosity, kg/m s
lg gas viscosity, kg/m s
qg gas density, kg/m3
qs particle density, kg/m3
sg gas viscous stress tensor, Pa
/1, /2, /3, /4, /5 dimensionless groups
xp particle angular velocity, 1/s
v a function of Reynolds number
Subscripts
n normal component
t tangential component
Acknowledgments
The authors would like to acknowledge the support of the
Petroleum Research Fund as administered by the American
Chemical Society, the Department of Energy (DOE-NETL),
and the National Science Foundation (CTS-0105688).
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