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461920 Modeling the Effect of Inhomogeneities on the Fluid-Particle Drag Force

^{1}and Wen & Yu

^{2}, are derived under the assumption that the distribution of particles is homogeneous. Prior studies focusing on particle-scale dynamics, such as those of Kriebitzsch et al

^{3}and Zhou et al

^{4}, have found that inhomogeneities in the particle distribution have a significant effect on the computed fluid-particle drag force. Therefore, the goal of the current study is to quantify the effect of inhomogeneities on the drag in the form of a new constitutive model.

In this work, the dynamics of fluid-particle systems are simulated using a fully-resolved lattice Boltzmann method (LBM), as was previously done by Derksen & Sundaresan^{5}. In order to ascertain the effect of particle structures at different length scales, the fluid-particle drag results are computed over a range of filter, or averaging, sizes. Inspired by the work of Rubinstein et al^{6}, we find that, in the low Reynolds number (Re) regime, the drag force within a fluidized bed with a particular extent of inhomogeneities can be quantified relative to the drag in the high and low Stokes number (St) limit beds, with identical extents of inhomogeneities, using only the local St and particle volume fraction. With this insight into the relative fluidized bed drag behavior, we are able to simplify our analysis of the effect of inhomogeneities on the drag force by focusing solely on the high and low St limit beds, rather than having to separately analyze the effect of the particle configurations on each of the different types of fluidized beds.

By analyzing the drag results over a variety of particle configurations, we find that in the low Re regime, where the effects of fluctuations in the particle velocity on the drag force are negligible, changes in the drag as compared to the random, homogeneous case are due to structures that are unresolved at the scale of the filter. The extent of these sub-filter scale inhomogeneities can be quantified using the squared fluctuations in the particle volume fraction. In both the high and low St limits, there is a clear reduction in the drag due to increases in the extent of sub-filter scale inhomogeneities that are present at the length scale of just a few particle diameters. Thus, we have formulated models for the high and low St drag forces, respectively, as functions of the inhomogeneities in the particle configuration.

Since the extent of sub-filter scale inhomogeneities cannot be directly computed in a larger-scale fluidized bed simulation, we have employed a test filter approach to estimate this sub-filter quantity. This test filter approach is based on a dynamic sub-grid scale modeling technique that was developed by Germano et al^{7} to estimate sub-filter scale stress in single-phase turbulence, and adapted by Parmentier et al^{8} and Ozel et al^{9} to coarse-grained drag in multiphase flows. In this technique, the extent of inhomogeneities is computed at the base filter size by demanding scale similarity between the base and test filters. In this manner, we have developed a method for incorporating the effects of inhomogeneities into the constitutive relation for the drag force, which can be used in coarser simulations based on both two-fluid model and Euler-Lagrange approaches, such as the Computational Fluid Dynamics-Discrete Element Method simulations^{10,11}. It is hoped that this improved drag model will lead to more accurate and grid-size independent predictions of fluidized bed behavior; however, these improvements remain to be established.

1. Beetstra, R., van der Hoef, M. A., & Kuipers, J. A. M. (2007). Drag Force of Intermediate Reynolds Number Flow Past Mono- and Bidisperse Arrays of Spheres. *AIChE J.* **53**(2), 489–501.

2. Wen, C. Y., & Yu, Y. H. (1966). Mechanics of fluidization, *Chem. Eng. Prog.** **Symp. Ser*. **62**, 100.

3. Kriebitzsch, S. H. L., van der Hoef, M. A., & Kuipers, J. A. M. (2013). Drag Force in Discrete Particle Models-Continuum Scale or Single Particle Scale? *AIChE J*. **59**(1), 316-324.

4. Zhou, G., Xiong, Q., Wang, L., Wang, X., Ren, X., & Ge, W. (2014). Structure-dependent drag in gas-solid flows studied with direct numerical simulation. *Chem. Engng. Sci*. **116**, 9-22.

5. Derksen, J. J., & Sundaresan, S. (2007). Direct numerical simulations of dense suspensions: wave instabilities in liquid-fluidized beds. *J. Fluid Mech.* **587**, 303–336.

6. Rubinstein, G. J., Derksen, J. J., & Sundaresan, S. (2016). Lattice Boltzmann simulations of low-Reynolds number flow past fluidized spheres: Effect of Stokes number on drag force. *JFM*. **788**, 576-601

7. Germano, M., Piomelli, U., Moin, P., & Cabot, W. H. (1991). A dynamic subgrid-scale eddy viscosity model. *Phys. Fluids A*. **3**, 1760.

8. Parmentier, J.-F., Simonin, O., & Delsart, O. (2012). A functional subgrid drift velocity model for filtered drag prediction in dense fluidized bed. *AIChE J*. **58**(4), 1084-1098.

9. Ozel, A., Fede, P., & Simonin, O. (2013). Development of filtered Euler-Euler two-phase model for circulating fluidised bed: High resolution simulation, formulation and a priori analyses. *Int. J. Multiph. Flow*. **55**, 43-63.

10. Pepoit, P. & Desjardins, O. (2012). Numerical analysis of the dynamics of two- and three-dimensional fluidized bed reactors using an Euler–Lagrange approach. *Powder Technol*. **220**, 104–121.

11. Radl, S. & Sundaresan, S. (2014). A drag model for filtered Euler–Lagrange simulations of clustered gas–particle suspensions. *Chem. Engng. Sci*. **117**, 416-425.

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