Modeling the effect of inhomogeneities on the fluid-particle drag force

In a fluidized bed, the fluid-particle drag force balances the force of gravity, and therefore provides the main mechanism for fluidization. Thus, the ability to accurately simulate large-scale fluidized systems depends on the implementation of a quantitatively-accurate constitutive drag model. Despite the fact that inhomogeneities, such as bubble-like voids and particle clusters, are often observed in fluidized beds, commonly-employed drag relations, such as those of Beetstra et al1 and Wen & Yu2, 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 al3 and Zhou et al4, 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 & Sundaresan5. 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 al6, 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 al7 to estimate sub-filter scale stress in single-phase turbulence, and adapted by Parmentier et al8 and Ozel et al9 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 simulations10,11. 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.