
It can be seen in Equations 23.513 and 23.514 that momentum exchange between the phases is based on the value of the fluidfluid exchange coefficient and, for granular flows, the fluidsolid and solidsolid exchange coefficients .
FluidFluid Exchange Coefficient
For fluidfluid 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, liquidliquid or gasliquid mixtures can be written in the following general form:
where , the drag function, is defined differently for the different exchangecoefficient models (as described below) and , the "particulate relaxation time'', is defined as
(23.516) 
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 exchangecoefficient 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 23.515.
(23.517) 
where
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 fluidfluid pairs of phases.
(23.521) 
where
and Re is defined by Equation 23.519 or 23.520. The 's are defined as follows:
(23.523) 
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.
(23.524) 
where
and
(23.526) 
where
and Re is defined by Equation 23.519 or 23.520. Note that if there is only one dispersed phase, then in Equation 23.525.
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 userdefined 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.
FluidSolid Exchange Coefficient
The fluidsolid exchange coefficient can be written in the following general form:
(23.528) 
where is defined differently for the different exchangecoefficient models (as described below), and , the "particulate relaxation time'', is defined as
(23.529) 
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 exchangecoefficient 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 fluidsolid exchange coefficient has the form
(23.533) 
where is the terminal velocity correlation for the solid phase [ 115]:
(23.534) 
with
(23.535) 
and
(23.536) 
for , and
(23.537) 
for .
This model is appropriate when the solids shear stresses are defined according to Syamlal et al. [ 364] (Equation 23.564).
(23.538) 
where
and Re is defined by Equation 23.532.
This model is appropriate for dilute systems.
When , the fluidsolid exchange coefficient is of the following form:
(23.540) 
where
When ,
(23.542) 
This model is recommended for dense fluidized beds.
SolidSolid Exchange Coefficient
The solidsolid exchange coefficient has the following form [ 362]:
where  
=  the coefficient of restitution  
=  the coefficient of friction between the and  
solidphase particles  
=  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.