On the mixture model for multiphase flow

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. The multiphase nature of the flow cannot, however, be avoided when the concentration gradients are large and the dispersed phases alter the hydrodynamic behaviour of the mixture or when the distributions of the particles are studied. In many practical applications of multiphase flow, the mixture model is a sufficiently accurate approximation, with only a moderate increase in the computational effort compared to a single-phase simulation. 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. However, an additional term in the mixture momentum equation arises from the slip of the dispersed phases relative to the continuous phase. 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. The terms for the viscous and turbulent stresses in the mixture momentum equation are usually combined to a generalised stress. The mixture model applications reported in the literature are briefly summarised. The areas of application include gravity settling, rotational flows and turbulent flows. The multiphase models in three commercial codes, PHOENICS, FLUENT and CFX 4, are reviewed. The mixture model approach, in a simplified form, is implemented only in PHOENICS.