(100c) Fluid-Particle Drag In Low-Reynolds-Number Flows Of Binary Gas-Solid Suspensions

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 various constitutive models, among which the model for fluid-particle drag is particularly important, to describe the detail of fluid-particle interactions. Constitutive models for fluid-particle drag in a suspension of identical particles have received a great deal of attention in the literature [e.g. Hill et al. (2001)]. However, when the suspension contains particles of different sizes or densities, the constitutive models for fluid-particle drag are not 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 packed 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.

In this study, it is assumed that the particles are heavy enough (high Stokes number) such that their motions are not affected by the hydrodynamic stresses in the fluid, yet small enough such that the gas flow can regarded as an inertia-less one (Reynolds number ≈ 0). With these approximations, the net fluid-particle drag only depends on the current particle configuration and velocities. In addition, the fluctuations in particle velocities do not affect the average drag exerted on each particle phase. These assumptions are realistic for many gas-particle systems; they are also attractive in the sense that they allow for efficient computations of the drag coefficients for the binary mixture. Our initial focus has been set on the simple case of equally sized particles.

We characterized the fluid-particle drag in binary suspensions using a lattice-Boltzmann method developed by Ladd (1994). This method solves the Navier-Stokes equation in 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 types of particles were randomly distributed and mixed in cubic periodic domains, the size of which is much larger than the size of individual particles. We employed Monte-Carlo relaxation steps to ensure that the spatial configuration of particles satisfies the hard-sphere distribution. The particle volume fraction φ in our study ranges from 0.1 to 0.4. Our periodic suspensions contain large number of particles varying from 728 (φ = 0.1) to 2914 (φ = 0.4). Moreover, we ensemble averaged results from 10-15 different con-figurations to ensure that our drag is statistically accurate.

We found that the fluid-particle drag in such suspensions is a linear function of fluid-particle relative velocities. The proportionality constants can be arranged into a matrix form. Compared to the mobility matrix formulation by Revay and Higdon (1992) for binary low-Reynolds-number and low-Stokes-number suspensions, where the particle velocities are expressed as functions of force, our formulation is inverted in that the forces are expressed as functions of velocities, which is the appropriate choice for binary low-Reynolds-number and high-Stokes-number suspensions.

Interestingly, the off-diagonal components in our inverse-mobility matrices are not much smaller than the diagonal components. Thus, the particle-particle momentum exchange occurs not only through collisions and contacts, as many existing models for multi-component gas-solid mixtures have assumed [e.g. Gera et al. (2004), Fan et al. (2004)], but also through hydrodynamic interactions. We found that volume-averaged off-diagonal elements are strictly symmetric, which satisfies the principle of action-reaction between the two particle phases. The magnitude of the off-diagonals increases with increasing particle concentration, and is proportional to the velocity difference between the two particle phases. These characteristics suggest that the off-diagonal elements are originated from lubrication interactions. In the end, we propose a new correlation for the fluid-particle drag in binary gas-solid suspensions in the form

F₁ = - β₁₁ΔU₁ - β₁₂ΔU₂

F₂ = - β₂₁ΔU₁ - β₂₂ΔU₂

where F_i is the fluid-particle drag acting on a particle belonging to phase i, and ΔU_i is the average velocity of particles of phase i relative to the interstitial gas. β_ij are given by

β₁₂ = φ₂g(φ) β

β₂₁ = φ₁g(φ) β

β₁₁ = β - β₁₂

β₂₂ = β - β₂₁

where g(φ) is a weak function of φ varying between 0.23 and 0.28 when φ is increased from 0.1 to 0.4, and β is the drag coefficient in a binary packed bed. In our simulations, φ₁₂ / φ₁₁ is in the range of 0.1-0.5 depending on the volume fraction ratio of the two species. Thus, the off-diagonals indeed provide a sizable contribution to the drag.

References:

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