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263063 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*

**Extended Abstract:**File Not Uploaded

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