Numerical flow simulation utilising a full multiphase model is impractical for a suspension possessing wide distributions in the particle size or density. Various approximations are usually made to simplify the computational task. In the simplest approach, the suspension is represented by a homogeneous single-phase system and the influence of the particles is taken into account in the values of the physical properties. This study concentrates on the derivation and closing of the model equations. The validity of the mixture model is also carefully analysed. Starting from the continuity and momentum equations written for each phase in a multiphase system, the field equations for the mixture are derived. The mixture equations largely resemble those for a single-phase flow but are represented in terms of the mixture density and velocity. The volume fraction for each dispersed phase is solved from a phase continuity equation. Various approaches applied in closing the mixture model equations are reviewed. An algebraic equation is derived for the velocity of a dispersed phase relative to the continuous phase. Simplifications made in calculating the relative velocity restrict the applicability of the mixture model to cases in which the particles reach the terminal velocity in a short time period compared to the characteristic time scale of the flow of the mixture. (75 refs.)

On the mixture model for multiphase flow

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

Gas-solid fluidized beds are widely applied in many chemical processes involving physical and/or chemical transformations, and for this reason they are the subject of intense research in chemical engineering science. Over the years, researchers have developed a large number of numerical models of gas-fluidized beds that describe gas-solid flow at different levels of detail. In this review, we discriminate these models on the basis of whether a Lagrangian or a Eulerian approach is used for the gas and/or particulate flow and subsequently classify them into five main categories, three of which we discuss in more detail. Specifically, these are resolved discrete particle models (also called direct numerical simulations), unresolved discrete particle models (also called discrete element models), and two-fluid models. For each of the levels of description, we give the general equations of motion and indicate how they can be solved numerically by finite-difference techniques, followed by some illustrative examples of a fluidized bed simulation. Finally, we address some of the challenges ahead in the multiscale modeling of gas-fluidized beds

Numerical Simulation of Dense Gas-Solid Fluidized Beds: A Multiscale Modeling Strategy

Eulerian two-phase flow theory applied to fluidization

An Euler-Lagrange strategy for simulating particle-laden flows

This paper reports on flow regimes of dense-phase vertical pneumatic transport of solids, referred to in the literature as circulating fluidized beds, computed using a generalization of the Navier-Stokes equations for two fluids. In the less dense regime corresponding to volume fractions of solids of about 1%, the flow consists of centrally upward moving solids and downward moving clusters. The computations agree with observations made by high-speed motion pictures and with measurements of radial solids concentrations and velocities. In the dense regime, corresponding to volume fractions of about 10%, a core-annulus type of regime is obtained, with solids descending down at the wall.

Yuan P. Tsuo

Papers

Computation of flow patterns in circulating fluidized beds