SECONDARY INSTABILITY OF A GAS-FLUIDIZED BED

It is known that stationary fluid with density that varies sinusoidall y with small amplitude and wavenumber kappa in the vertical direction is unstable to disturbances that are sinusoidal in a horizontal direct ion with wavenumber alpha. Small values of alpha/kappa are the most un stable in the sense that a neutral disturbance exists at sufficiently small alpha/kappa however small the Rayleigh number may be. The non-un iformity of density in the undisturbed state may be regarded as being a consequence of non-uniformity of concentration of extremely small so lid particles in fluid. This paper is concerned with the corresponding instability of such a non-uniform dispersion when the particle size i s 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 prim ary instability of a uniform fluidized bed with particle volume fracti on phi0, the sinusoidal distribution of concentration propagates verti cally, and in the steady state relative to this kinematic wave particl es fall with speed V (= \phi d U/dphi\phi0), where U(phi) is the mean speed of fall of particles, relative to zero-volume-flux axes, in a un iform dispersion with volume fraction phi. This particle convection wi th speed V transports particle volume and momentum and tends to even o ut variations of a disturbance in the vertical direction and thereby t o suppress a disturbance, especially one with small alpha/kappa. Analy sis of the behaviour of a disturbance is based on the equation of moti on of the mixture of particles and fluid and an assumption that the di sturbance velocities of the particles and the fluid are equal (as is s uggested by the relatively small relaxation time of particles). The me thod of solution used in the associated pure-fluid problem is also app licable here, and values of the Rayleigh number as a function of alpha /kappa for a neutral disturbance and a given value of the new non-dime nsional parameter involving V are found. Particle convection with only modest values of V stabilizes all disturbances for which alpha/kappa < 1 and increases significantly the Rayleigh number for a neutral dist urbance when alpha/kappa > 1. It appears that under practical conditio ns disturbances with alpha/kappa above unity are unstable, although ig norance of the values of parameters characterizing a fluidized bed hin ders quantitative conclusions.