In the standard formulation of the Lagrangian multiphase model, described in Discrete Phase, the assumption is that the volume fraction of the discrete phase is sufficiently low: it is not taken into account when assembling the continuous phase equations. The general form of the mass and momentum conservation equations in Ansys Fluent is given in Equation 14–495 and Equation 14–496 (and also defined in Continuity and Momentum Equations):

(14–495)

(14–496)

To overcome this limitation of the Lagrangian multiphase model, the volume fraction of the particulate phase is accounted for by extending Equation 14–495 and Equation 14–496 to the following set of equations (see also Conservation Equations, written for phase ):

(14–497)

(14–498)

Here, Equation 14–497 is the mass conservation equation for an individual phase and Equation 14–498 is the corresponding momentum conservation equation. The momentum exchange terms (denoted by ) are split into an explicit part, , and an implicit part, in which represents the particle averaged velocity of the considered discrete phase, and represents its particle averaged interphase momentum exchange coefficient [531]. Currently, these momentum exchange terms are considered only in the primary phase equations.

In the resulting set of equations (one continuity and one momentum conservation equation per phase), those corresponding to a discrete phase are not solved. The solution, such as volume fraction or velocity field, is taken from the Lagrangian tracking solution.

In versions prior to Ansys Fluent 13, one drawback of the dense discrete phase model was that it did not prevent the actual concentration of particles from becoming unphysically high. Hence, the model had only a limited applicability to flows close to the packing regime, such as fluidized bed reactors. To overcome this, Ansys Fluent now applies a special treatment to the particle momentum equation as soon as the particle volume fraction exceeds a certain user specified limit. Thus, the unlimited accumulation of particles is prevented. In turn, this will allow you to simulate suspensions and flows like bubbling fluidized bed reactors, operating at the packing limit conditions, allowing for polydispersed particle systems. However, no Discrete Element Method (DEM) type collision treatment is applied, which would otherwise allow the efficient simulation of systems with a large number of particles.

In the context of the phase-coupled SIMPLE algorithm (Solution Method in Ansys Fluent) and the coupled algorithm for pressure-velocity coupling (Selecting the Pressure-Velocity Coupling Method) in the User’s Guide), a higher degree of implicitness is achieved in the treatment of the drag coupling terms. All drag related terms appear as coefficients on the left hand side of the linear equation system.

For multiphase flows with energy transfer, the enthalpy equation for the phase has the following form:

(14–499)

The interphase heat exchange terms (denoted by), like the momentum equation, are split into an explicit part, , and an implicit part, . The latter term is assumed to be a function of the temperature difference between the phases and the interfacial area, .

(14–500)

is the particle averaged temperature of the considered discrete phase, and represents the particle-averaged interphase heat exchange coefficient. More information on all available heat exchange coefficients in DDPM context can be found in Inert Heating or Cooling (Law 1/Law 6). These interphase energy exchange terms are considered only in the primary phase enthalpy equation.

All other terms in Equation 14–499 correspond to heat transfer phenomena as described in Equation 14–197.

 

14.5.23.2. Granular Temperature

The solids stress acting on particles in a dense flow situation is modeled via an additional force in the particle force balance Equation 12–1.

(14–501)

The term models the additional force acting on a particle, resulting from interparticle interaction. It is computed from the stress tensor given by the Kinetic Theory of Granular Flows as

(14–502)

Following the same theory, the granular temperature is used in the model for the granular stress. For a more detailed description on the Kinetic Theory of Granular Flows refer to Solids Pressure, Maximum Packing Limit in Binary Mixtures, and Solids Shear Stresses.

The granular temperature can be estimated using any of the available options described in Granular Temperature. By default, the algebraic formulation is used. The conservation equation for the granular temperature (kinetic energy of the fluctuating particle motion) is solved with the averaged particle velocity field. Therefore, a sufficient statistical representation of the particle phase is needed to ensure the stable behavior of the granular temperature equation. For details, refer to Conservation Equations – Granular Temperature.

The main advantage over the Eulerian model is that, there is no need to define classes to handle particle size distributions. This is done in a natural way in the Lagrangian formulation [531].