Meso-scale structures that take the form of clusters and streamers are commonly observed in dilute gas–particle flows, such as those encountered in risers. Continuum equations for gas–particle flows, coupled with constitutive equations for particle-phase stress deduced from kinetic theory of granular materials, can capture the formation of such meso-scale structures. These structures arise as a result of an inertial instability associated with the relative motion between the gas and particle phases, and an instability due to damping of the fluctuating motion of particles by the interstitial fluid and inelastic collisions between particles. It is demonstrated that the meso-scale structures are too small, and hence too expensive, to be resolved completely in simulation of gas–particle flows in large process vessels. At the same time, failure to resolve completely the meso-scale structures in a simulation leads to grossly inaccurate estimates of inter-phase drag, production/dissipation of pseudo-thermal energy associated with particle fluctuations, the effective particle-phase pressure and the effective viscosities. It is established that coarse-grid simulation of gas–particle flows must include sub-grid models, to account for the effects of the unresolved meso-scale structures. An approach to developing a plausible sub-grid model is proposed.

The role of meso-scale structures in rapid gas–solid flows

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 theoryThe 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

A new theory of the instability of a uniform fluidized bed

Experiments on fluidization with water of spherical particles falling against gravity in columns of rectangular cross-section are described. All of them are dominated by inertial effects associated with wakes. Two local mechanisms are involved: drafting and kissing and tumbling into stable cross-stream arrays. Drafting, kissing and tumbling are rearrangement mechanisms in which one sphere is captured in the wake of the other. The kissing spheres are aligned with the stream. The streamwise alignment is massively unstable and the kissing spheres tumble into more stable cross-stream pairs of doublets which can aggregate into larger relatively stable horizontal arrays. Cross-stream arrays in beds of spheres constrained to move in two dimensions are remarkable. These arrays may even coalesce into aggregations of close-packed spheres separated by regions of clear water. A somewhat weaker form of cooperative motion of cross-stream arrays of rising spheres is found in beds of square cross-section where the spheres may move freely in three dimensions. Horizontal arrays rise where drafting spheres fall because of greater drag. Aggregation of spheres seems to be associated with relatively stable cooperative motions of horizontal arrays of spheres rising in their own wakes.

Nonlinear mechanics of fluidization of beds of spherical particles

Hydrodynamics of liquid fluidisation

Vigorous homogeneous fluidization of 12-nm silica particles was easily achieved by coupling aeration with vibration. Vibration (with frequency in the range of 30 to 200 Hz, and vibrational acceleration in the range of 0 to 5 g) was found to be necessary to achieve smooth fluidization. The minimum fluidization velocity, defined as the lowest gas velocity at which the pressure drop across the bed reaches a plateau, was approximately 0.3‐0.4 cm/s, and essentially independent of the vibrational acceleration. However, the bed expanded almost immediately after the air was turned on, reaching bed expansions of three times the initial bed height or higher. Thus the bed appeared to exhibit a fluidlike behavior at velocities much lower than the minimum fluidization velocity. Fluidization of nanoparticles was achieved as a result of the formation of stable, relatively large, and very porous agglomerates. Practically no bubbles or elutriation of particles was observed. A fractal analysis combined with a modified Richardson‐Zaki approach is proposed for prediction of agglomerate size and voidage. © 2004 American Institute of Chemical Engineers AIChE J, 50: 1776 –1785, 2004

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Kozo Misima

Papers

Fluidization of Solid Particles