Abstract
A linear stability analysis is performed for the homogeneous state of a monodisperse gas-fluidized bed of spherical particles undergoing hydrodynamic interactions and solid-body collisions at small particle Reynolds number and finite Stokes number. A prerequisite for the stability analysis is the determination of the particle velocity variance which controls the particle-phase pressure. In the absence of an imposed shear, this velocity variance arises solely due to the hydrodynamic interactions among the particles. Since the uniform state of these suspensions is unstable over a wide range of values of particle volume fraction φ and Stokes number St, full dynamic simulations cannot be used in general to characterize the properties of the homogeneous state. Instead, we use an asymptotic analysis for large Stokes numbers together with numerical simulations of the hydrodynamic interactions among particles with specified velocities to determine the hydrodynamic sources and sinks of particle-phase energy. In this limit, the velocity distribution to leading order is Maxwellian and therefore standard kinetic theories for granular/hard-sphere molecular systems can be used to predict the particle-phase pressure and rheology of the bed once the velocity variance of the particles is determined. The analysis is then extended to moderately large Stokes numbers for which the anisotropy of the velocity distribution is considerable by using a kinetic theory which combines the theoretical analysis of Koch (1990) for dilute suspensions (φ [Lt ] 1) with numerical simulation results for non-dilute suspensions at large Stokes numbers. A linear stability analysis of the resulting equations of motion provides the first a priori predictions of the marginal stability limits for the homogeneous state of a gas-fluidized bed. Dynamical simulations following the detailed motions of the particles in small periodic unit cells confirm the theoretical predictions for the particle velocity variance. Simulations using larger unit cells exhibit an inhomogeneous structure consistent with the predicted instability of the homogeneous gas–solid suspension.
Highlights
One of the simplest and yet most stringent challenges in the modelling of gas– particulate flows is to predict the conditions under which a homogeneous fluidized bed will be unstable to volume fraction variations. Jackson (1963) showed that equations of motion which include particle-phase inertia and a drag coefficient that is a function of volume fraction lead to a prediction that the bed will always be unstable
Batchelor (1988) suggested that the particle pressure be related to the hydrodynamic particle self-diffusivity in a low Reynolds number liquid–solid suspension and used measurements and plausible reasoning concerning this quantity to suggest a model for the particle pressure
As the Stokes number is increased the Brinkman screening of the hydrodynamic interactions among the particles leads to a velocity variance that decreases in proportion to S t−2/3
Summary
One of the simplest and yet most stringent challenges in the modelling of gas– particulate flows is to predict the conditions under which a homogeneous fluidized bed will be unstable to volume fraction variations. Jackson (1963) showed that equations of motion which include particle-phase inertia and a drag coefficient that is a function of volume fraction lead to a prediction that the bed will always be unstable. Φ is the particle volume fraction, S t = mUt/(6πμa2) is the Stokes number, m and a are the mass and radius of the particles, Ut = mg/(6πμa) is the terminal velocity of an isolated particle, g is the acceleration due to gravity, and μ is the gas viscosity Using this source to determine the particle temperature and pressure, Koch considered the limits of stability for a dilute sedimenting gas–solid suspension. Koch (1990) determined the anisotropic source in this regime for dilute suspensions where the collisional distribution of fluctuation energy into vertical and horizontal velocity moments is negligible. We first consider relatively dilute suspensions with a small system size for which we expect the suspension to be stable The simulations for this case allow us to validate the theory for velocity variance over a wide range of Stokes numbers. Our simulations show clear evidence for the instabilities arising in these dense suspensions
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