Journal of Fluid Mechanics

A new theory of the instability of a uniform fluidized bed

G. K.  Batchelor a1
a1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver, Street, Cambridge CB3 9EW, UK

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The form of the momentum equation for one-dimensional (vertical) unsteady mean motion of solid particles in a fluidized bed or a sedimenting dispersion is established from physical arguments. In the case of a fluidized bed that is slightly non-uniform this equation contains two dependent variables, the local mean particle velocity V and the local concentration φ, and several statistical parameters of the particle motion in a uniform bed. All these parameters are functions of φ with clear physical meanings, and the important ones are measurable. It is a novel feature of the equation that it contains two explicit contributions to the bulk modulus of elasticity of the particle configuration, one arising from the transfer of particle momentum by velocity fluctuations and one arising from the effective repulsive force exerted between particles in random motion. This latter contribution, which proves to be the more important of the two, is related to the gradient diffusivity of the particles, a key quantity in the new theory.

The equation of mean motion of the particles and the equation of particle conservation are sufficient to determine the behaviour of a small disturbance with sinusoidal variation of V and φ in the vertical direction. Particle inertia forces in such a propagating wavy disturbance may promote amplitude growth, whereas particle diffusion tends to suppress it, and instability occurs when the particle Froude number exceeds a critical value. Rough estimates of the relevant parameters allow the criterion for instability to be put in approximate numerical from for both gas-fluidized beds (for which the flow Reynolds number at marginal stability is small) and liquid-fluidized beds of solid spherical particles (for which the Reynolds number is well above unity), although more information about the particle diffusivity in particular is needed. The predictions of the theory appear to be in qualitative accord with the available observational data on instability of gas- and liquid-fluidized beds.

(Published Online April 21 2006)
(Received August 25 1987)
(Revised February 10 1988)