Computational model for multiphase flow:

Advance in computational fluid mechanics have provided the basis for further insight into the dynamics of multiphase flow. Currently there are two approaches for the numerical calculation of multiphase flow: The Eulerian-Lagrangian approach and the Euler-Euler approach. In the Euler-Euler approach, the different phases are treated mathematically as

interpenetrating continua. Since the volume of a phase cannot be occupied by the other phase the concept of phasic volume fraction is introduced. These volume fractions are assumed to be continuous function of space and time and their sum is equal to one. Conservation equations for each phase are derived to obtain a set of equations which have similar structure for all phase. The equations are closed by providing constitutive relations that are obtained from empirical information or in the case of granular flow by application of kinetic theory. For volume averaged information on any hydrodynamic property the Euler-Euler approach is suitable for its simplicity.

3.1.1. Choosing an appropriate Eulerian model:

There are three different Euler-Euler multiphase models are available. The volume of fluid (VOF) model: The VOF model is a surface-tracking technique applied to a fixed Eulerian mesh. It is designed for two or more immiscible fluids are of interest. In the VOF model, a single set of momentum equation is shared by the fluids, and the volume fraction of each of the fluids in each computational cell is tracked throughout the domain. The mixture model: The mixture model is designed for two or more phases (fluid or particulate). As in the Eulerian model, the phases are treated as interpenetrating continua. The mixture model solves for the mixture momentum equations as prescribes relatives velocity to describe the dispersed phases. The Eulerian model: The Eulerian model is the most complex of the multiphase model. It solves a set of n momentum and continuity equations for each phase. Through the pressure and interphase exchange coefficients coupling are achieved. The manner in which this coupling is handled depends upon the type of phases involved; granular (fluid solid) flows are handled differently than non-regular (fluid-fluid) flows. For granular flows, the properties are obtained from the application of kinetic theory. Momentum exchange between the phases is also depends upon the type of mixture being modeled.

An appropriate multiphase model for the multiphase system can be determined from the flow regime. For slug, and stratified/free surface flows VOF model are used. For slurry flow, hydro transport, bubbly, droplet, and particle-laden flows in which the phase mix and/or dispersed phase volume fractions exceed 10% either mixture model or Eulerian model are used. For

general, complex multiphase flows that involve multiple flow regimes, select the aspect of flow that is of most interest and choosing of model that is of most appropriate. There are some parameters that help to identify the appropriate multiphase model: the particulate loading, β, and Stoke number, St.

Particulate loading effect:

It is defined as the mass density ratio of the dispersed phase (d) to that of carrier phase (c). It has a major impact on phase interactions. Depending on the particulate loading, the degree of interaction of phases can be divided into the following three categories. For very low loading, the coupling between the phase is one-way (i.e. the fluid carrier influence the particle via drag and turbulence, but particles have no influence on the fluid carrier). The discrete phase, mixture and Eulerian models can handle this type of problem correctly. For intermediate loading, the coupling is two way (i.e. the fluid carrier influences the particulate phase via drag and turbulence, but the particles in turn influence the carrier fluid via reduction in mean momentum and turbulence). All three models are applicable in this case, but some other factors are needed to take into account to decide which model is more appropriate. For high loading, there is two-way coupling plus particle pressure and viscous stresses due to the particles (four-way coupling). Only the Eulerian model will handle this type of problem correctly.

Significance of Strokes number:

The Stokes number can be defined as the relation between the particle response times the system response time. For the system with intermediate particulate loading, the value of Strokes number can help to select the appropriate model.

where and ts is the based on the characteristics (Ls) and characteristic velocity (Vs) of the system under investigation: ts= Ls / Vs . For St << 1.0 or = 1.0, the particle will follow

closely and any of the three model is applicable. For St > 1.0, the particle will move independently of the flow, either the discrete phase or Eulerian model is applicable.

In the present work, an Eulerian granular multiphase model is adopted where gas, liquid, and solid phase are all treated as continua, interpenetrating and interacting with each other everywhere in the computational domain. With the Eulerian multiphase model, the number of secondary phase is limited only by memory requirement and convergence behavior. Any number of secondary phases can be modeled provided that sufficient memory is available. Eulerian multiphase model does not distinguish between fluid-fluid and fluid-solid (granular) multiphase flows. A granular phase is simple one that involves at least one phase that has been designated as a granular phase. The pressure field is assumed to be shared by all the three phases, in proportion to their volume fraction. Solid-phase shear and bulk viscosities are obtained by applying kinetic theory of granular flows.

Limitation of the Eulerian multiphase model: The Reynolds Stress turbulence model is not available on a per phase basis, inviscid flow is not allowed, melting and solidification are not allowed, and Particle tracking (using the Lagrangian dispersed phase model) interacts only with the primary phase. Streamwise periodic flow with specified mass flow rate cannot be modeled when the Eulerian model is used, and when tracking particles in parallel, the DPM model cannot be used with the Eulerian multiphase model if the shared memory option is enabled.