In a fluidized bed, the fluid-particle drag force provides the main mechanism for fluidization. Constitutive models that describe the drag force between the fluid and particles are necessary for simulating the complex dynamics of large-scale fluidized systems. While prior constitutive models have been defined over a wide range of particle volume fraction and Reynolds number (Re), these drag models have significant limitations in their applicability. The current work seeks to address the inability of prior drag models to account for particle translation and rotation, as well as for inhomogeneities in the particle distribution.
In this study, the effect of the Stokes number (St), which quantifies the resistance of particles to changes in their translation and rotation, on the fluid-particle drag force is elucidated. The dynamics of fluid-particle systems are simulated using a fully-resolved lattice Boltzmann method (LBM), as was done previously by Derksen and Sundaresan1. Through these LBM simulations in which the particles are free to move around, a new drag model that is dependent on St is proposed. This drag model, which is valid in the low Re regime, bridges the transition from the low St regime (sedimentation drag models, like that of Wen and Yu2) to the high St regime (fixed bed drag models, like that of Beetstra et al3).
In addition, the current work looks to analyze the effect of spatial inhomogeneities in the particle distribution on the interactions between fluid and particles. The distribution of particles is typically assumed to be homogeneous at the length scale at which the constitutive drag models are applied to simulations of larger-scale fluidized systems. However, as suggested by the work of Parmentier et al4, such an assumption is often invalid. This study seeks to obtain a relationship for the fluid-particle drag force that is, unlike prior drag models, dependent on non-local effects. The significance of non-locality is expressed in terms of spatial gradients of particle volume fraction and fluid-particle slip velocity. These different non-local contributions are studied with LBM simulations of fluidized systems, using data obtained from an Eulerian grid, where the filter, or averaging, size is smaller than the domain size. From the current analysis, the drag force has been found to increase monotonically with the Laplacian of the fluid-particle slip velocity. Furthermore, the dependence of the drag force on gradients in the particle volume fraction has been determined.
1. J.J. Derksen, S. Sundaresan. (2007). Direct numerical simulations of dense suspensions: wave instabilities in liquid-fluidized beds. J. Fluid Mech., 587, 303–336.
2. C.Y. Wen, Y.H. Yu. (1966). Mechanics of fluidization. Chem. Eng. Prog. Symp. Ser., 62, 100.
3. R. Beetstra, M.A. van der Hoef, J.A.M. Kuipers. (2007). Drag Force of Intermediate Reynolds Number Flow Past Mono- and Bidisperse Arrays of Spheres. AIChE J., 53(2), 489–501.
4. J.-F. Parmentier, O. Simonin, O. Delsart. (2012). A functional subgrid drift velocity model for filtered drag prediction in dense fluidized bed. AIChE J., 58(4), 1084-1098.