(29b) Closures For Filtered Two-Fluid Models Of Gas-Particle Flows

It is well known that gas-particle flows exhibit large fluctuations in velocities and local suspension density. In riser flows, these fluctuations are associated with the random motion of the individual particles (typically characterized through the granular temperature) and with the chaotic motion of particle clusters, which are repeatedly formed and broken apart. These clusters occur over a wide range of length scales and their dynamics span a broad range of time scales.

The origin of these clusters is well understood, and two-fluid model equations are able to capture their existence in a robust manner; however, to resolve the clusters at all length scales, extremely fine spatial grids are necessary [1]. Due to computing limitations, the grid size used in simulating industrial scale gas-particle flows is invariably much larger than the length scales of the finer particle clusters. Such a coarse-grid simulation will clearly not resolve the structures which exist on sub-grid length scales. The need to account for the consequences of these unresolved structures through suitable sub-grid models is now well established [1?3].

The goal of our study is to construct filtered two-fluid models that average over small scale structures. In these filtered equations, the consequences of the flow structures occurring on a scale smaller than a chosen filter width appear through correlations for which closure relations should be derived or postulated. If constructed properly, the filtered equations should produce a solution with the same macroscopic features as the finely resolved kinetic theory model results; however, as the filtered equations place less stringent requirements on the grid resolution than the original two-fluid model equations, they would be easier to solve.

In the present study, we have shown that filtering of the statistical data obtained from highly resolved 2-D and 3-D simulations ? employing a kinetic theory based two-fluid model for uniformly sized particles [1?4] implemented into the MFIX platform [5] ? in a periodic domain is a fruitful way to develop closures for coarse-grid simulation of fluidized gas-particle flows. We have completed several highly resolved 2-D and 3-D simulations in various domain sizes with different spatial resolutions to extract filtered quantities for various filter sizes. Using the computational ?data? generated through such simulations, we have extracted filtered drag coefficient and particle-phase stresses as functions of the local particle volume fraction and the size of the spatial averaging window (i.e. filter size).We found that both the filtered drag coefficient decreased systematically with increasing filter width both in 2-D and 3-D simulations, whereas the particle-phase stresses increased with increasing filter width. We have also found that these filtered statistics become less dependent on spatial resolution and essentially independent of the domain size for both 2-D and 3-D simulations provided that the filter size is appreciably smaller than the domain size and is much larger than the grid size.

In addition, we have studied the effects of various model parameters on these filtered quantities to expose the robustness of the region-averaged statistics against small changes in these parameters. We have found that the dimensionless parameters of the kinetic theory model have a weaker effect on the dimensionless filtered drag coefficient than a dimensionless group involving the filter size; so, the filtered closures can be modeled to a good accuracy simply as functions of the dimensionless filter size and particle volume fraction.

As a metric of the spatial inhomogeneity observed in the kinetic theory model simulations (at various times in the course of the simulation under statistical steady state conditions), we examined the particle volume fraction fluctuations. Here, we define two types of inhomogeneities: the external fluctuation in particle volume fraction <φ'²>_ext, where ?external? refers to inhomogeneities on a scale larger than the filter size, and the internal fluctuation in particle volume fraction <φ'²>_int, where ?internal? refers to the spatial inhomogeneities occurring on a scale smaller than the filter size. We have seen that <φ'²>_ext decreases as the filter size increases as one would indeed expect, whereas <φ'²>_int increases as the filter size increases and eventually reaches an asymptotic value. It seems reasonable to hypothesize that a filter (region) size where <φ'²>_ext and <φ'²>_int are comparable sets the lower threshold for a meaningful filter size. One can also argue that the filter size at which <φ'²>_int begins to reach its plateau is a physically meaningful estimate (as most of the inhomogeneities occur on a scale smaller than this, as evidenced by the approach to the asymptotic value). According to these qualitative arguments, a rational choice of filter size may lie in the range of 8 ? 16 dimensionless units for a 2-D system and between 2 - 4 dimensionless units for a 3-D system. [The filter size is made dimensionless using v_t²/g and as a characteristic length, where v_t refers to the terminal settling velocity of a single particle.]

We have examined the stability of a uniformly fluidized gas-particle suspension through a 1-D linear stability analysis (LSA) of the filtered model equations. As expected the filtered model predicts that the state of uniform fluidization will give way to nonuniform time-dependent solution through a Hopf bifurcation. The wavenumber corresponding to the Hopf bifurcation point, k_HB, is a function of the filter size; in particular, the k_HB decreases as the filter size increases. We have also carried out test MFIX simulations of the filtered model equations with fairly high spatial resolutions and ascertained that as one increases the filter size the inhomogeneous structures become coarser and coarser. These findings are reassuring, as they show that filtering is indeed erasing the fine structure and only presenting coarser structures.

The details of these results will be described in the presentation.

References:

[1] Agrawal, K., Loezos, P. N., Syamlal, M. & Sundaresan, S., 2001 The role of meso-scale structures in rapid gas-solid flows. J. Fluid Mech. 445, 151 ? 185.

[2] Zhang, D. Z. & VanderHeyden, W. B. 2002 The effects of mesoscale structures on the macroscopic momentum equations for two-phase flows. Int. J. Multiphase Flow. 28, 805 ? 822.

[3] Andrews, A.T. IV, Loezos, P. N. & Sundaresan, S. 2005 Coarse-grid Simulation of Gas-Particle Flows in Vertical Risers. Ind. Eng. Chem. Res., 44, 6022 ? 6037 (2005).

[4] Gidaspow, D., 1994 Multiphase Flow and Fluidization, Academic Press, CA. 31-58, 197 ? 238.

[5] Syamlal, M., Rogers, W. & O'Brien, T. J., 1993 MFIX Documentation, U.S. Department of Energy, Federal Energy Technology Center, Morgantown, WV.