Most of the computational approaches for simulating particulate flows in industrial devices such as fluidized beds and circulating fluidized beds treat the suspension as a mixture of two or more inter-penetrating fluids. The multi-fluid model equations, which are obtained by averaging the point equations over a scale of many individual particles, rely on constitutive models, among which the model for fluid-particle drag is particularly important, to describe the details of fluid-particle interactions. Constitutive models for fluid-particle drag in a suspension of identical particles (monodisperse) have received a great deal of attention in the literature [e.g. Hill et al. (2001a, 2001b)]. However, when the suspension contains particles of different sizes or densities, the constitutive models for the fluid-particle drag are not very well established. Many recent studies on fluid-particle drag in binary particle suspensions [e.g. van der Hoef et al. (2005), Beetstra et al. (2007)] consider the fluid-particle drag force for the special case where there is no mean relative motion between the two solid phases, i.e., the drag for a fixed bed. However, the inter-penetrating continua models for flowing binary gas-solid suspensions require more general drag force models, where different types of particles can have different local averaged velocities relative to the interstitial gas. The goal of our study is to construct such general drag force models from direct numerical simulations.

This study is a continuation of the work presented in the AIChE Annual Meeting last year [Yin and Sundaresan (2007)]. It was proposed that in a bidisperse gas-solid suspension, the drag forces acting on the two particle species could be expressed as

f_D1 = - β₁₁ΔU₁ - β₁₂ΔU₂

f_D2 = - β₂₁ΔU₁ - β₂₂ΔU₂

where f_Di is the total fluid-particle drag per unit volume of suspension acting on particles of species i, and ΔU_i is the average velocity of particles of species i relative to the interstitial gas. The friction coefficient β_ij, in general, is a function of particle volume fractions φ₁ and φ₂, particle sizes d₁ and d₂, particle velocities ΔU₁ and ΔU₂, particle velocity fluctuations <u₁²> and <u₂²>, and a length scale λ at which the lubrication force between particles begins to saturate:

β_ij = β_ij (φ₁, φ₂, d₁, d₂, ΔU₁, ΔU₂, <u₁²>, <u₂²>, λ)

In the previous study [Yin and Sundaresan (2007)], we examined the limit of small Reynolds numbers Re_i = ρΔU_id_i / μ (i = 1, 2) and equal particle sizes, d₁ = d₂. Due to the linearity of the Stokes flow, the drag forces are linear functions of ΔU_i and are not influenced by the velocity fluctuation <u_i²>. As a result, the friction coefficient β_ij only depends on particle volume fractions and λ / d.

In this work, we conducted simulations to characterize (a) the drag forces in low-Re bidisperse suspensions where d₁ ≠ d₂, and (b) the drag forces in bidisperse suspensions with moderate Reynolds numbers 1 < Re_i < 10, with d₁ = d₂, and in the absence of velocity fluctuations (<u_i²> = 0). We characterized the drag forces in binary suspensions using a lattice-Boltzmann method developed by Ladd (1994a, 1994b). This method solves the Navier-Stokes equations for the continuous phase on a rectangular lattice; the forces on the particles are obtained by integrating the hydrodynamic stresses over particle surfaces. In our simulations, the two particle species were randomly distributed and mixed in cubic periodic domains. We employed Monte-Carlo relaxation steps to ensure that the spatial configuration of particles satisfies the hard-sphere distribution. Each particle phase has a characteristic velocity ΔU_i. The particles were then equilibrated with the fluid by allowing the fluid to relax under a bulk pressure gradient to obtain a net zero flow rate through particle assemblies. The total particle volume fraction φ in our study ranges from 0.1 to 0.4. For simulations with particles of different sizes, the size ratio d₁ : d₂ was chosen to be 1 : 1.5 and 1 : 4. Our periodic suspensions contain large number of particles varying from several hundred (φ = 0.1) to several thousand (φ = 0.4). Moreover, we ensemble averaged results from 10-30 different configurations to ensure that the drag forces are statistically accurate.

For low-Re suspensions containing particles of different sizes (d₁ ≠ d₂), we show that there is only one free parameter in the friction coefficient matrix β_ij that needs to be fitted as a function of particle volume fractions and diameters, and λ. The fitting function for this free parameter, however, requires accurate determination of the drag forces in bidisperse fixed beds. Therefore, we conducted lattice-Boltzmann simulations of pressure-driven flows through fixed bidisperse assemblies of particles (ΔU₁ = ΔU₂), and proposed a drag formula for fixed suspensions upon which fitting functions for the drag in flowing suspensions (ΔU₁ ≠ ΔU₂) can be constructed.

For bidisperse suspensions with finite fluid inertia, our simulations assume static and random configuration of particles. This method provides a computationally inexpensive means to interrogate the effects of inertia on the drag force when there exists two different particle species in relative motion that are intimately mixed. To ensure the computational accuracy of this method, the drag force obtained from these frozen-particle realizations was compared against those found through more expensive, dynamic simulations where particle positions are updated after each time step. The drag forces obtained from the simulations indicate that in the Reynolds number range of interest 1 < Re_i < 10 the dependence of the friction coefficient β_ij on the relative velocities ΔU_i is weak. Therefore, the inertial effect in this Reynolds number regime is small, and can be included as a small correction in the Stokes drag formula developed in the previous study [Yin and Sundaresan (2007)].

References:

1. R. J. Hill, D. L. Koch, A. J. C. Ladd, The first effect of fluid inertia on flows in ordered and random arrays of spheres, J. Fluid Mech., 448, 213-241, 2001.

2. R. J. Hill, D. L. Koch, A. J. C. Ladd, Moderate-Reynolds-number flows in ordered and random arrays of spheres, J. Fluid Mech., 448, 243-278, 2001.

3. M. A. van der Hoef, R. Beetstra, J. A. M. Kuipers, Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse array of spheres: results for the permeability and drag force, J. Fluid Mech., 528, 233-254, 2005.

4. R. Beetstra, M. A. van der Hoef, J. A. M. Kuipers, Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres, AICHE J., 53, 489-501, 2007.

5. A. J. C. Ladd, Numerical simulations of particulate suspensions via a discretized Boltzmann-equation. 1. Theoretical foundation, J. Fluid Mech., 271, 285-309, 1994.

6. A. J. C. Ladd, Numerical simulations of particulate suspensions via a discretized Boltzmann-equation. 2. Numerical results, J. Fluid Mech., 271, 311-339, 1994.

7. X. Yin, S. Sundaresan, Fluid-particle drag in binary Stokes gas-solid suspensions, presented at the AIChE Annual Meeting (Salt Lake City, Utah) in November 2007. Manuscript "A drag law for bidisperse gas-solid suspensions containing equally-sized spheres" was accepted by Ind. Eng. Chem. Res. in April 2008.