It can be seen in Equations 23.5-13 and 23.5-14 that momentum exchange between the phases is based on the value of the fluid-fluid exchange coefficient and, for granular flows, the fluid-solid and solid-solid exchange coefficients .
Fluid-Fluid Exchange Coefficient
For fluid-fluid flows, each secondary phase is assumed to form droplets or bubbles. This has an impact on how each of the fluids is assigned to a particular phase. For example, in flows where there are unequal amounts of two fluids, the predominant fluid should be modeled as the primary fluid, since the sparser fluid is more likely to form droplets or bubbles. The exchange coefficient for these types of bubbly, liquid-liquid or gas-liquid mixtures can be written in the following general form:
where , the drag function, is defined differently for the different exchange-coefficient models (as described below) and , the "particulate relaxation time'', is defined as
where is the diameter of the bubbles or droplets of phase .
Nearly all definitions of include a drag coefficient ( ) that is based on the relative Reynolds number ( ). It is this drag function that differs among the exchange-coefficient models. For all these situations, should tend to zero whenever the primary phase is not present within the domain. To enforce this, the drag function is always multiplied by the volume fraction of the primary phase , as is reflected in Equation 23.5-15.
and Re is the relative Reynolds number. The relative Reynolds number for the primary phase and secondary phase is obtained from
The relative Reynolds number for secondary phases and is obtained from
where is the mixture viscosity of the phases and .
The Schiller and Naumann model is the default method, and it is acceptable for general use for all fluid-fluid pairs of phases.
The Morsi and Alexander model is the most complete, adjusting the function definition frequently over a large range of Reynolds numbers, but calculations with this model may be less stable than with the other models.
The symmetric model is recommended for flows in which the secondary (dispersed) phase in one region of the domain becomes the primary (continuous) phase in another. Thus for a single dispersed phase, and . For example, if air is injected into the bottom of a container filled halfway with water, the air is the dispersed phase in the bottom half of the container; in the top half of the container, the air is the continuous phase. This model can also be used for the interaction between secondary phases.
You can specify different exchange coefficients for each pair of phases. It is also possible to use user-defined functions to define exchange coefficients for each pair of phases. If the exchange coefficient is equal to zero (i.e., if no exchange coefficient is specified), the flow fields for the fluids will be computed independently, with the only "interaction'' being their complementary volume fractions within each computational cell.
Fluid-Solid Exchange Coefficient
The fluid-solid exchange coefficient can be written in the following general form:
where is defined differently for the different exchange-coefficient models (as described below), and , the "particulate relaxation time'', is defined as
where is the diameter of particles of phase .
All definitions of include a drag function ( ) that is based on the relative Reynolds number (Re ). It is this drag function that differs among the exchange-coefficient models.
where the drag function has a form derived by Dalla Valle [ 74]
This model is based on measurements of the terminal velocities of particles in fluidized or settling beds, with correlations that are a function of the volume fraction and relative Reynolds number [ 306]:
where the subscript is for the fluid phase, is for the solid phase, and is the diameter of the solid phase particles.
The fluid-solid exchange coefficient has the form
where is the terminal velocity correlation for the solid phase [ 115]:
for , and
and Re is defined by Equation 23.5-32.
This model is appropriate for dilute systems.
When , the fluid-solid exchange coefficient is of the following form:
This model is recommended for dense fluidized beds.
Solid-Solid Exchange Coefficient
The solid-solid exchange coefficient has the following form [ 362]:
|=||the coefficient of restitution|
|=||the coefficient of friction between the and|
|=||the diameter of the particles of solid|
|=||the radial distribution coefficient|
Note that the coefficient of restitution is described in Section 23.5.5 and the radial distribution coefficient is described in Section 23.5.5.