Secondary instability of a gas-fluidized bed

Abstract

It is known that stationary fluid with density that varies sinusoidally with small amplitude and wavenumber κ in the vertical direction is unstable to disturbances that are sinusoidal in a horizontal direction with wavenumber α. Small values of α/κ are the most unstable in the sense that a neutral disturbance exists at sufficiently small α/κ however small the Rayleigh number may be. The non-uniformity of density in the undisturbed state may be regarded as being a consequence of non-uniformity of concentration of extremely small solid particles in fluid. This paper is concerned with the corresponding instability of such a non-uniform dispersion when the particle size is not so small that the fall speed relative to the fluid is negligible. In the undisturbed state, which is an outcome of the well-known primary instability of a uniform fluidized bed with particle volume fraction ϕ₀, the sinusoidal distribution of concentration propagates vertically, and in the steady state relative to this kinematic wave particles fall with speed V (= |ϕdU/dϕ|_ϕ0), where U(ϕ) is the mean speed of fall of particles, relative to zero-volume-flux axes, in a uniform dispersion with volume fraction ϕ. This particle convection with speed V transports particle volume and momentum and tends to even out variations of a disturbance in the vertical direction and thereby to suppress a disturbance, especially one with small α/κ. Analysis of the behaviour of a disturbance is based on the equation of motion of the mixture of particles and fluid and an assumption that the disturbance velocities of the particles and the fluid are equal (as is suggested by the relatively small relaxation time of particles). The method of solution used in the associated pure-fluid problem is also applicable here, and values of the Rayleigh number as a function of α/κ for a neutral disturbance and a given value of the new non-dimensional parameter involving V are found. Particle convection with only modest values of V stabilizes all disturbances for which α/κ < 1 and increases significantly the Rayleigh number for a neutral disturbance when α/κ > 1. It appears that under practical conditions disturbances with α/κ above unity are unstable, although ignorance of the values of parameters characterizing a fluidized bed hinders quantitative conclusions.