(157a) The Influence of Binary Drag Laws on MP-PIC Simulations of Segregation

            Gas-fluidized
beds with particles of different sizes and/or densities are common in industry,
and known to exhibit segregation under some operating conditions. Numerous studies have been carried out to investigate the segregation
behavior in fluidized beds. The degree of segregation depends on the
differences in density and size of the particles, as well as the gas velocity
(Chiba et al., 1979). For systems
composed of equal density particles, small particles tend to concentrate near
the surface of the bed while large particles fall to the bottom (Rowe and Nienow, 1976; Wu and Baeyens,
1998; Goldschmidt et al., 2003).
Segregation by size increases with increasing bed height, decreasing size of
fines, increasing mean size, and as the gas velocity approaches the minimum
fluidization velocity of the smaller particle (Geldart
et al., 1981). For systems composed
of different density particles, denser particles tend to fall to the bottom of
the bed. For these systems, a large degree of segregation is present at low gas
velocities and this degree of segregation decreases as velocity increases (Rowe
and Nienow, 1976). A phenomenon
which is not well understood is layer inversion, which refers to systems in
which a given species may behave as either flotsam or jetsam, depending on the
operating condition (Rasul et al.,
1999).

            Mathematical
models have been used to study segregation in gas-solid fluidized beds. Both Eulerian (van Wachem et al., 2001a; Huilin
et al., 2003; Cooper and Coronella, 2005) and Lagrangian
models (Hoomans et
al., 2000; Limtrakul et al., 2003; Feng et al., 2003; and Bokkers et al., 2004; Dahl and Hrenya, 2005) have been able to predict segregation with a certain
degree of success and have provided insight on the contributing mechanisms. Van
Wachem et al.
(2001a) used binary models for drag and solid phase stress in an Eulerian framework and was able to
model layer inversion. The inversion phenomena was explained through the
dominating mechanisms of the system, namely at low gas velocities the system is
dominated by gravity and drag force and at high velocities by pressure drop and
gradients in the granular temperature.

            A
key component of both Eulerian and Lagrangian models is the drag force, which couples the
fluid and solid phases. Numerous experimental studies have been carried out to measure
drag force in different monodisperse systems and
several correlations have been proposed. These correlations have been developed
from packed-bed measurements (Ergun, 1952 and
Macdonald et al., 1979), settling
experiments (Richardson and Zaki, 1954, Syamlal and O'Brien, 1994), fluidized-bed experiments (Wen and Yu, 1966), and Lattice-Boltzmann
simulations (Koch and Hill, 2001). Recently, Pirog
(1998) developed a drag force correlation for polydiperse
systems based on a voidage-velocity correlation
obtained from settling and creaming experiments in liquid-solid systems. Van der Hoef et al. (2005) developed a drag force correlation for binary mixtures
based on Lattice-Boltzmann simulations of low gas
flow past arrays of random spheres. Van Wachem et al.
(2001a) and Dahl and Hrenya (2005) used Pirog's correlation in conjunction with the Ergun (1952) and the Wen and Yu (1966)
drag models to describe drag force in binary systems.

            The use of different drag force
models significantly impacts the simulation results, modifying the bed
expansion and solid concentration in the bed (van Wachem,
et al., 2001b). Monodisperse
models have been traditionally employed in polydisperse
systems, adapting them by replacing the particle diameter by a species
diameter, the slip velocity of the monodisperse
system by that of a single species in a polydisperse
system, and assuming that the individual species drag force is equal to the
drag force of a monodisperse system at the same
volume fraction (van der Hoef
et al., 2005). These assumptions have
no physical basis but have been used due to the lack of adequate drag models
for polydisperse systems. The focus of the present work
is to determine the impact of various drag laws on fluidized beds composed of
binary mixtures, with special emphasis on the prediction of segregation
patterns. As described below, the simulations performed as part of this work
indicate the critical role of the drag force in predicting segregation.
Furthermore, a companion experimental work (Joseph et al., 2005) demonstrates the relative abilities of the existing
drag laws, as applied to binary systems.       In
order to isolate the effect of the drag force and hence minimize the
contribution of the collisional (particle) stress, the
simulation study was performed at low velocities, in which the drag force is
one of the dominating forces, as noted by van Wachem
et al. 2001a). A new drag force model is proposed which combines the approach
taken by Gidaspow (1994) and the recent work of van der Hoef et al. (2005).  The former is
a ?stitching? together of monodisperse drag laws for
the packed and fluidized regions, whereas the latter provides a correction to monodisperse drag laws for binary systems in order to
account for the presence of particles with different diameters. A comparison between
drag laws with and without the binary correction was completed.

            To
assess the relative merits of the various drag laws, simulations were performed
using a novel alternative for modeling fluid-solid systems, namely the
Multi-Phase Particle-in-Cell (MP-PIC) method (Andrews and O'Rourke, 1996). This
implementation is a combination of the Eulerian and Lagrangian models for the solid phase and possesses
advantages of both approaches; particles are grouped in parcels that are
individually tracked, but the particle-phase stress is calculated from an Eulerian description, which thereby eliminates the need to
resolve individual collisions between particles. Since particles are tracked in
groups and individual collisions are not resolved, the modeling of many-particle
systems is performed in a more computationally efficient manner than a strict Lagrangian treatment. Arena-flow^TM
is being used as a framework.

            To
assess the impact of the various drag force treatments on the segregation
behavior, simulations were performed for a defluidizing
bed in a 10 cm diameter column. The initial condition was a bed 40 cm high with
100 and 200 micron particles, both at a solid volume fraction of 0.15 and uniformly
mixed. The density of both components was 2600 kg/m³. The initial gas
velocity was set to 0.25 m/s, and was decreased in steps of 0.025 m/s in
periods of 0.2 s. The gas velocity was then kept constant for 2 s, and the time
average properties were calculated over the second half of this period. Both
drag laws predicted the minimum fluidization velocity around 0.05 m/s. For the
drag law without the binary correction, the simulation results show total segregation
(i.e., complete separation of species) for all gas velocities. The particles
segregated, with the big particles on top of the small particles, at the initial
velocity and remained unmixed. When using the drag law with the binary correction,
however, a relatively homogeneous mixture is obtained for the higher velocities
and segregation is observed as the gas velocity is decreased from 0.1 m/s. The
final state, at a zero gas velocity, presents partial mixing; i. e., a layer of small particles is
present at the top of the bed, a layer of large particles is present at the
bottom, and the middle portion of the bed has both components present. For gas
velocities above 0.05 m/s, the pressure drop equals the total weight of the bed
divided by its area. As the gas velocity decreases, a defluidized
layer of coarse particles forms in the bottom of the bed. This defluidized layer is no longer supported by the fluid. For
a gas velocity of 0.025 m/s a smaller pressure drop than that of the weight
divided by its area was calculated by the drag law with the binary correction. More
particles had fallen into the defluidized layer in the
simulation without the binary correction causing an even smaller pressure drop.
In summary, the MP-PIC simulation results indicate that the form of the drag
law plays a crucial role in the qualitative and quantitative nature of
segregation predictions. A test of the correctness of the drag laws employed,
and a validation of the simulation results discussed here, are determined via a
direct comparison between simulations and experiments, as is detailed in the
companion contribution by Joseph et al.
(submitted).

            To verify that
the results obtained with the MP-PIC approach were not influenced by the semi-empirical,
solid-phase stress model (Snider, 2001) in Arena-flow^TM,
two-dimensional, discrete-particle simulations were carried out following the
development by Dahl and Hrenya (2005).  In this treatment, individual particle
collisions are resolved and hence no solid-phase stress model is required
(unlike the MP-PIC approach). For this strictly Lagrangian
formulation, a higher degree of segregation was observed when using the model
without the binary correction; this behavior qualitatively mimics that obtained
in the MP-PIC simulations.