Fully developed travelling wave solutions and bubble formation in fluidized beds

Abstract

It is well known that most gas fluidized beds of particles bubble, while most liquid fluidized beds do not. It was shown by Anderson, Sundaresan & Jackson (1995), through direct numerical integration of the volume-averaged equations of motion for the fluid and particles, that this distinction is indeed accounted for by these equations, coupled with simple, physically credible closure relations for the stresses and interphase drag. The aim of the present study is to investigate how the model equations afford this distinction and deduce an approximate criterion for separating bubbling and nonbubbling systems. To this end, we have computed, making use of numerical continuation techniques as well as bifurcation theory, the one-and two-dimensional travelling wave solutions of the volume-averaged equations for a wide range of parameter values, and examined the evolution of these travelling wave solutions through direct numerical integration. It is demonstrated that whether bubbles form or not is dictated by the value of Ω l (ρ s $ t \Ag)" /# , where ρ s is the density of particles, t is the terminal settling velocity of an isolated particle, g is acceleration due to gravity and A is a measure of the particle phase viscosity. When Ω is large ( " 30), bubbles develop easily. It is then suggested that a natural scale for A is ρ s t d p , so that Ω# is simply a Froude number.

Coarse-Grid Simulation of Reacting and Non-Reacting Gas-Particle Flows

“…The observation that the fluctuating state could be sustained if the simulations began with a nonuniform initial condition reveals coexistence of a homogeneous solution and a high-amplitude nonuniform solution. [Such coexistence of solutions for these equations have been demonstrated previously, see Glasser et al (1996Glasser et al ( , 1997.] We found that adding stochastic, lateral fluctuations in particle mass flux at the inlet could eliminate the homogeneous state; thus, the homogeneous solution appears to have a small attractor basin.…”

“…Thus, a homogeneously fluidized bed gives way to nonuniform structures in coarse-grid simulations much more slowly (provided it is not stabilized by numerical viscosity), than in highly resolved simulations. As a result, appreciable amplification of the initial disturbances (which, in turn, can give way to lateral nonuniformities through secondary gravitational overturning instability, e.g., Batchelor, 1993;Glasser et al, 1996Glasser et al, , 1997Glasser et al, , 1998 does not occur in the coarse-grid simulations with free-slip boundaries within the time available in the riser flow and the disturbance gets washed out of the riser. Our observation that a highly nonuniform initial state (coupled with uniform inlet conditions) could not sustain the fluctuating state suggests that the basin of attraction for the homogeneous state is fairly large and/or that a fluctuating statistical steady state does not exist for the microscopic equations discretized on a coarse grid.…”

Fluidization of fine and ultrafine particles using nitrogen and neon as fluidizing gases

“…11 for a detailed discussion). In a numerical analysis, Glasser, Kevrekidis and Sundaresan 21 have examined the evolution of traveling wave solutions through direct numerical integration of the volume-averaged equations. In this analysis, the particle-phase pressure and viscosity were assumed to be monotonically increasing functions of the particle volume fraction.…”

“…It was clearly demonstrated that the distinction between bubbling and nonbubbling systems was linked with high-amplitude solutions as seen in liquid-fluidized beds, in which small bubbles that do not develop into large visible bubbles have been observed. 9,11 Glasser et al 21 carried out a limited parametric study and concluded that the parameter…”

Development, Verification, and Validation of Multiphase Models for Polydisperse Flows

“…These studies revealed that the state of homogeneous fluidization would first give rise to one-dimensional waves with no horizontal structures. These one-dimensional waves undergo subsequent bifurcations leading to the formation of bubble-like voids in dense fluidized beds, and particle clusters in dilute gas-solid systems (Anderson et al 1995;Glasser et al 1996Glasser et al , 1997. In the present study, we demonstrate that when anisotropic friction coefficient is taken into consideration the state of uniform fluidization is predicted to be unstable over a much wider parameter space and that in some regions of the parameter space the dominant mode has both vertical and lateral structures.…”