(54b) Constitutive Model for the Fluid-Particle Drag Coefficient in Filtered Two-Fluid Models for Gas-Particle Flows

AIChE
2012 Annual Meeting

Special
Session to Celebrate Tom O'brien's Career Long
accomplishments

Constitutive
model for the fluid-particle drag coefficient in filtered two-fluid models for
gas-particle flows

Chris Milioli^*,
Fernando Milioli^*, William Holloway, Sankaran Sundaresan

Department of
Chemical and Biological Engineering, Princeton University, Princeton, NJ

Abstract

It is well known that gas-particle flows manifest
fluctuations over a wide range of length and time scales.  Filtered two-fluid model formulations,
where the coarse flow structures are resolved and the effects of fine
structures are modeled, are beginning to appear in the literature [1-5].  These filtered models require
constitutive relations to capture the effect of sub-filter scale
inhomogeneities on the effective drag force and the particle and fluid phase stresses.
 Such closures have been derived
from the results obtained from highly resolved simulations [1-5].  Both Parmentier et al. [5] and Igci and
Sundaresan [3,4] model the correction to the fluid-particle drag coefficient as
a function of the filtered particle volume fraction and filter size.  Such models average over all possible
sub-filter scale meso-scale structures.

It is reasonable to expect that there should be additional
variables characterizing the sub-filter scale meso-scale structure, which are
not recognized in these models.  In
the present study we ask if one can formulate a more elaborate model for the filtered
drag force containing additional variables characterizing the sub-filter scale
meso-scale structures, thereby allowing us to differentiate between the filtered
drag coefficients corresponding to appreciably
different sub-filter scale structures.  To this end, we have examined the
drag coefficient values obtained by filtering the results of highly resolved kinetic
theory based simulations.  We have classified
the filtered drag coefficient as a function of solid fraction and various
additional filtered quantities. We found no systematic
dependence of the filtered drag coefficient on the variances of the solid
fraction, gas velocity, or the solid velocity.  However, it is quite sensitive to
changes in the filtered slip velocity; as the filtered slip velocity increases,
the filtered drag coefficient decreases and approaches an asymptotic dependence
on particle volume fraction at high slip velocities. This trend is opposite of
what one would obtain when inertial effects become important in flow past
individual particles.  We
rationalize it as a consequence of the formation of larger meso-scale
structures at larger slip velocities. We also propose a model for the filtered
fluid-particle drag coefficient in terms of the filtered solid fraction, filter
size and filtered slip velocity.   

References

1.              
K. Agrawal, P. N. Loezos, M. Syamlal, and S.
Sundaresan, J. Fluid Mech., 445, 151 (2001).  

2.              
D. Z. Zhang and W. B. VanderHeyden, Int.
J.  Multiphase Flow, 28, 805 (2002).

3.              
Y. Igci and S. Sundaresan, Ind. Eng. Chem. Res., 50,
13190–13201 (2011).

4.              
Y. Igci and S. Sundaresan, AIChE J., 57, 2691-2707 (2011).

5.              
J.F. Parmentier, O. Simonin, and O. Delsart, AIChE J.,
2011, DOI: 10.1002/aic.12647.

^*Department of Mechanical Engineering, University of S?o Paulo, S?o
Carlos-SP, Brazil