XXIV ICTAM, 21-26 August 2016, Montreal, Canada
THE ROLE OF THE PARTICLE STRESS IN THE SIMULATION OF FLUIDIZED BEDS
Mohammad Reza Haghgoo∗1 , Donald J. Bergstrom1 , and Raymond J. Spiteri2
1
Mechanical Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada
2
Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada
Summary A two-fluid model based on the kinetic theory of granular flow, for the rapid-flow regime, and the Coulomb friction law, for the
slow-plastic regime, is applied to predict the hydrodynamics of dense gas-particle flow in a large-scale three-dimensional fluidized bed. Two
different models for the particle stress tensor, in the elastic-inertial regime, are examined to assess their ability to predict the dynamics of
the bed. The numerical results are validated against published experimental data and demonstrate the significance of the stress tensor in the
elastic-inertial regime. Different flow regimes are identified, in the context of fluidized beds, through the dimensionless Inertial number and
the main characteristics of each regime are discussed to better understand the complex dynamics of fluidized beds.
INTRODUCTION AND NUMERICAL MODELING
Fluidization of a dense system of particles by a fluid is widely employed in industry for processes involving drying,
separation, and mixing of particles and chemical reactions. In turbulent fluidized beds, where the interstitial fluid plays a
significant role in the dynamics of the system, the major challenge is to model the complicated interaction of particles and
the turbulence of the fluid. In such systems, the granular particles rarely reach a volume fraction where the friction between
particles starts to play a role, i.e., εmin
. Consequently, the particle stress tensor can be described by the kinetic theory of
p
granular flows. However in bubbling fluidized beds, where the turbulence of the fluid, if any, is damped by the inertia of the
particles, not only does the particle volume fraction reach to εmin
, but it is also very likely to succeed the packed bed limit,
p
i.e., εmax
.
Therefore
in
bubbling
beds,
in
addition
to
the
rapid
flow
regime, a quasi-static slow regime is likely to happen.
p
Obviously, there are some regions in the bed where the granular flow undergoes a transient regime that is neither rapid nor
slow in nature. Accordingly, in modeling of bubbling beds, the main challenge is to have a comprehensive model that is able
to distinguish the different behaviors of the granular flow under each regime and appropriately describe the stresses.
The two-fluid model along with the constitutive relations from [1] are used to simulate an isothermal gas-particle system.
There is no general consensus in modeling the stresses in dense regions where there is sustained contact between particles. In
one approach, referred to as the Schaeffer model in the literature, Syamlal et al. [1] proposed that frictional stresses need to
be considered only for regions where the solid volume fraction is higher than the maximum packing limit, εmax
. On the other
p
hand, Srivastava and Sundaresan [3] proposed another model, referred to as the Princeton model, that adds the frictional stress
to the kinetic-collisional stress tensor in the intermediate regime. Note that both the Princeton and Schaeffer models use the
same granular rheology for the rapid and slow regimes.
The height of the simulated bed is 160 cm with a diameter of 30.6 cm. The air is injected at the very bottom of the bed at a
velocity of 63 cm/s. All simulations were performed in parallel using 60 cores. The first 10 seconds of the fluidization process
is simulated. The numerical code Multiphase Flow with Interphase eXchanges (MFIX) is used to perform the simulations [1].
The results are validated against published experimental data by Laverman et al. [2]
RESULTS AND DISCUSSION
Figures 1a and 1b represent the azimuthally and time-averaged axial velocity profiles for particles at three different heights.
At the lowest section, there is an overall downward flow of the particles near the center and a small upward flow in the nearwall region. At the highest section, i.e., for a height of 31 cm, the particle flow is opposite. The results from using the
Princeton model are closer to the experimental data than the results from using the Schaeffer model.
To understand the differences between the results predicted by each model, it is helpful to discuss the behaviour of the
granular materials under different circumstances. At a large applied stress, particles deform at the contact point(s) and are
compressed into the interparticle vacant space. In this quasi-static regime, the particle pressure, p, and shear stresses, τ are
independent of the strain rate, γ̇. In contrast, at high shear strain rates for the rapid flow regime, the momentum transport
is due to the instantaneous binary collisions as well as the inertia of the interparticle motions. Therefore in this regime, the
rheology of the particles is described by the coefficient of restitution and granular temperature and p, τ ∼ γ̇ 2 . Between these
two asymptotic regimes is the so called intermediate or elastic-inertial regime [4], where p, τ ∼ γ̇ n , with 0.5 ≤ n ≤ 1. In the
rapid-flow regime, the collisions are assumed to be infinitely rigid. However, it is obvious that once a particle is in multiple
contact with surrounding particles, the elastic forces do play a role in the momentum transport. Therefore, to understand and
quantify the dynamics of dense gas-particle flows, it is helpful to define a dimensionless parameter whose value can help
∗ Corresponding
author. Email: reza.haghgoo@usask.ca
Azimuthally and time−avergaed axial particle velocity (cm/s)
Azimuthally and time−avergaed axial particle velocity (cm/s)
60
Exp. h = 11 cm
Princeton h = 11 cm
Exp. h = 21 cm
Princeton h = 21 cm
Exp. h = 31 cm
Princeton h = 31 cm
50
40
30
20
10
0
−10
−20
0
5
10
15
r (cm)
(a) Princeton model
60
Exp. h = 11 cm
Schaeffer h = 11 cm
Exp. h = 21 cm
Schaeffer h = 21 cm
Exp. h = 31 cm
Schaeffer h = 31 cm
50
40
30
20
10
0
−10
−20
0
5
10
15
r (cm)
(b) Schaeffer model
Figure 1: Averaged axial particle velocity profiles at different heights
identify the regime of the flow. Following da Cruz et al. [5], the Inertial number, I = |γ̇|dp /(p/ρp )0.5 is defined to be the
ratio of the inertial timescale to the macroscopic deformation timescale. For the rapid or inertial regime, I has large values on
the order of unity, whereas for the quasi-static regime, I has small values in the order of 10−3 ; values in between belong to
the elastic-inertial regime. Figure 2 shows instantaneous values of the particle volume fraction, the Inertial number, and the
particle viscosity in a single plane. Both models predict that the flows inside the bubbles and free-board have the largest value
of I corresponding to the rapid-flow regime and, lowest value of the particle viscosity. For the regions around and between the
bubbles, i.e., mostly near the center and lower part of the bed, I has moderate values, mapping the flow to the elastic-inertial
regime. In these regions, the values of viscosity are also moderate. The regions in dark blue indicate slow flow corresponding
to the highest values of viscosity. The slow-flow regime predicted by the Princeton model is mostly located near the wall
region, whereas the Schaeffer model predicts one region above the highest bubble and another one at the height of 40 cm.
Note that the extent of the regions in which the flow is elastic-inertia is considerable. However, the Schaeffer model treats
these regions as a rapid flow, whereas the the Princeton model tries to improve their treatment by adding a frictional stress to
the kinetic stresses. Although the Princeton model predicts a more realistic behaviour, there remains a need to introduce the
elastic properties of the particles into the constitutive equations of the elastic-inertial regime.
(a) Princeton model
(b) Schaeffer model
Figure 2: Particle volume fraction, Inertial number, and particle viscosity at t = 5 s
CONCLUSIONS
This study demonstrates the significance of the particle stress in prediction of dense gas-particle flows. It shows that the
Princeton model yields more realistic results compared to the Schaeffer model because it differentiates between the rapid and
intermediate regimes. Further study is required to investigate the effects of elastic collisions in the intermediate regime.
References
[1]
[2]
[3]
[4]
[5]
Syamlal M., et al.: MFIX documentation: theory guide. Technical Note DOE/METC-94/1004, Morgantown Energy Technology Centre, 1993.
Laverman J., et al.: Experimental study on the influence of bed material on the scaling of solid circulation patterns. Powder Technol., 224, 2012.
Srivastava A., Sundaresan S.: Analysis of a frictional-kinetic model for gas-particle flow. Powder Technol.,129:72-85, 2003.
Campbell, C. S.: Granular shear flows at the elastic limit. J. Fluid Mech., 465:261-291, 2002.
F. da Cruz, et al.: Rheophysics of dense granular materials: discrete simulation of plane shear flows. Phys. Rev. E, 72:021309, 2005.