It can be seen in Equation 14–204 and Equation 14–205 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
.
Note: Note that in Ansys Fluent, all the available interphase exchange coefficient models are empirically based. At the present, there is no general formulation in the literature, and attention is required in some situations, such as:
air / liquid / solids configurations
polydispersed flow
porous media
compressible flows
temperature variation
dense situations
near-wall flows
The interphase exchange coefficients for such cases may need modification, which can be dealt with using user-defined functions.
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:
 | (14–212) |
where 
is the interfacial area (see Interfacial Area Concentration), 
, the drag function, is defined differently for the different
exchange-coefficient models (as described below) and 
, the "particulate relaxation time", is defined as
 | (14–213) |
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.
For each pair of phases in fluid-fluid flows you may use one of the available drag function models in Ansys Fluent or a user-defined function to specify the interphase exchange coefficient. If the exchange coefficient is equal to zero (that is, 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. You can specify different exchange coefficients for each pair of phases.
The following drag models are available in Ansys Fluent.
For the model of Schiller and Naumann [577],
 | (14–214) |
where
 | (14–215) |
and Re is the relative Reynolds number. The relative Reynolds number for the primary phase
and secondary phase 
is obtained from
 | (14–216) |
The relative Reynolds number for secondary phases 
and 
is obtained from
 | (14–217) |
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.
For the Morsi and Alexander model [461] :
 | (14–218) |
where
 | (14–219) |
and Re is defined by Equation 14–216 or Equation 14–217. The constants, 
, are defined as follows:
 | (14–220) |
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.
For the symmetric model, the density and the viscosity are calculated from volume averaged properties:
 | (14–221) |
 | (14–222) |
and the diameter is defined as
 | (14–223) |
in turn
 | (14–224) |
 | (14–225) |
 | (14–226) |
 | (14–227) |
and 
is defined by Equation 14–215.
Note that if there is only one dispersed phase, then 
in Equation 14–223.
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. 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.
The Grace et al. model is well suited to gas-liquid flows in which the bubbles can have a range of shapes.
For the model of Grace, et al. [111] :
 | (14–228) |
where
 | (14–229) |
 | (14–230) |
In Equation 14–230, 
is the volume fraction of the continuous phase; 
is the volume fraction correction exponent; and 
, 
, and 
are defined as:
 | (14–231) |
 | (14–232) |
 | (14–233) |
where
where 
is the Morton number given by:
is given by the piecewise function:
where 
is the Eötvös number:
and 
.
Sparsely distributed fluid particles
In flows with sparsely distributed fluid particles,
in
Equation 14–230
is zero.
Densely distributed fluid particles
For high bubble volume fractions, a non-zero value for
should be used depending on the bubble size.
See Specifying the Drag Function in the Fluent User's Guide for more information.
For the model of Tomiyama, et al. [643]
 | (14–234) |
where
 | (14–235) |
 | (14–236) |
In Equation 14–236,
Like the Grace et al. model, the Tomiyama et al. model is well suited to gas-liquid flows in which the bubbles can have a range of shapes.
For boiling flows only, you can use the model of Ishii [270]. For the
Ishii model, the drag coefficient 
is determined by choosing the minimum of the viscous regime 
and the distorted regime 
, defined as follows:
 | (14–237) |
where 
and 
are given by the following formulas:
 | (14–238) |
 | (14–239) |
where 
is the relative Reynolds number, 
is the surface tension, and 
is gravity.
The bubble diameter 
is determined as in Bubble and Droplet Diameters.
The Ansys Fluent implementation of the Ishii-Zuber drag coefficient automatically accounts for different particle distribution regimes.
Sparsely Distributed Fluid Particles
For very small Reynolds numbers, fluid particles (bubbles or droplets) tend to behave as solid spherical particles. In this case, the drag coefficient can be accurately approximated by the Schiller-Naumann correlation (Equation 14–214.
As the particle Reynolds number increases, the effect of surface tension becomes more important in what is known as the inertial or distorted particle regime. The particles at first become distorted into ellipsoids, and then take the shape of a spherical cap.
Distorted Regime
In the distorted particle regime, the drag coefficient is independent of Reynolds number, but highly dependent on particle shape. To calculate the drag coefficient, the Ishii-Zuber model uses the Eotvos number, which measures the ratio between gravitational and surface tension forces:
(14–240)
where
is the density difference between the phases, 
is the gravitational acceleration, and 
is the surface tension coefficient between each phase pair.For the ellipsoidal fluid particles, the drag coefficient is calculated by:
(14–241)
Spherical Cap Regime
In the spherical cap regime, the drag coefficient is approximated by:
(14–242)
Densely Distributed Fluid Particles
The correlations Equation 14–241 and Equation 14–242 are valid for flows in which fluid particles move in a sparsely distributed fashion. However, these correlations are not applicable for high void fraction flows. As the volume fraction of the gas increases, bubbles begin to accumulate and move as a swarm of particles, thus modifying the effective drag. The Ishii-Zuber drag model automatically accounts for dense particle effects by using a swarm factor correction.
Viscous Regime
In the viscous regime, where fluid particles can be considered spherical, the Schiller Naumann correlation Equation 14–214 is modified using the mixture Reynolds number based on a mixture viscosity as follows:
(14–243)
Distorted Regime
In the distorted particle regime, the single particle drag coefficient is multiplied by the Ishii-Zuber swarm factor correction:
(14–244)
Spherical Cap Regime
(14–245)
Ansys Fluent automatically calculates the drag coefficient as:
 | (14–246) |
The universal drag laws [319] are suitable for the calculation of the drag coefficients in a variety of gas-liquid flow regimes. The drag laws can apply to non-spherical droplets/bubbles with the constraint of a pool flow regime, that is, the hydraulic diameter of the flow domain which is far larger than the averaged size of the particles.
The exchange coefficient for bubbly and droplet flows can be written in the general form
 | (14–247) |
Where 
represents the primary phase and 
the dispersed phase and 
is the interfacial area (see Interfacial Area Concentration). The dispersed phase relaxation time
is defined as
 | (14–248) |
The drag function 
is defined as
 | (14–249) |
The relative Reynolds number for the primary phase 
and the secondary phase 
is obtained based on the
relative velocity of the two phases.
 | (14–250) |
Where 
is the effective viscosity
of the primary phase accounting for the effects of family of particles
in the continuum.
The Rayleigh-Taylor instability wavelength is
 | (14–251) |
Where 
is the surface tension, 
the gravity, and 
the absolute value of the density difference between phases 
and 
.
The drag coefficient is defined differently for bubbly and droplet flows.
 | (14–252) |
 | (14–253) |
 | (14–254) |
In the viscous regime, the following condition is satisfied:
(14–255)
The drag coefficient,
, is defined as 
(14–256)
In the distorted bubble regime, the following condition is satisfied:
(14–257)
The drag coefficient is calculated as
(14–258)
In the strongly deformed, capped bubble regime, the following condition is satisfied:
(14–259)
The drag coefficient can be written as
(14–260)
The effective viscosity for the bubble-liquid mixture is
 | (14–261) |
The fluid-solid exchange coefficient 
can be written in the following general form:
 | (14–266) |
where 
is the volume fraction of the solid phase, and 
is the density of the solid phase. 
is defined differently for the different exchange-coefficient models (as
described below), and 
, the "particulate relaxation time", is defined as
 | (14–267) |
where 
is the diameter of particles of phase 
, and 
is the dynamic viscosity of fluid phase.
Here and below, the subscripts 
and 
denote the 
fluid phase and the 
solid phase, respectively.
All definitions of 
include a drag function (
) that is based on the relative Reynolds number (
). It is this drag function that differs among the exchange-coefficient
models.
Syamlal-O’Brien model
For the Syamlal-O’Brien model [639]:
(14–268)
where
is the volume fraction of fluid phase, and the drag function has a form
derived by Dalla Valle [131]: 
(14–269)
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 [557]:
(14–270)
where
and 
are the velocities of solid and liquid phases, respectively, 
is the density of the fluid phase, and 
is the diameter of the 
solid phase particles.The fluid-solid exchange coefficient has the form
(14–271)
where
is the terminal velocity correlation for the solid phase [197]: 
(14–272)
with
(14–273)
and
(14–274)
for
, and 
(14–275)
for
.This model is appropriate when the solids shear stresses are defined according to Syamlal et al. [640] (Equation 14–364).
Parametrized Syamlal-O’Brien model
The parametrized Syamlal-O’Brien model is an enhancement of the Syamlal-O’Brien model in which the values of 0.8 and 2.65 in Equation 14–274 and Equation 14–275 are replaced by parameters that are adjusted based on the fluid flow properties and the expected minimum fluidization velocity [638]. This overcomes the tendency of the original Syamlal-O’Brien model to under/over-predict bed expansion in fluid bed reactors, for example.
The parameters are derived from the velocity correlation between single and multiple particle systems at terminal settling or minimum fluidization conditions. For a multiple particle system, the relative Reynolds number at the minimum fluidization condition is expressed as:
(14–276)
where
is the Reynolds number at the terminal settling condition for a single
particle and can be expressed as: 
(14–277)
The Archimedes number,
, can be written as a function of the drag coefficient, 
, and the Reynolds number, 
: 
(14–278)
where
can be obtained from the terminal velocity correlation, 
, and the Reynolds number: 
(14–279)
is found from
Equation 14–272
with
from
Equation 14–273
and B rewritten as
follows:
(14–280)
where
Once the particle diameter and the expected minimum fluidization velocity are given, the coefficients
and 
can be found by iteratively solving Equation 14–276
– Equation 14–280.This model implementation is restricted to use in gas-solid flows in which the gas phase is the primary phase and is incompressible. Furthermore, the model is appropriate only for Geldart Group B particles.
Wen and Yu [700] model
For the model of Wen and Yu [700], the fluid-solid exchange coefficient is of the following form:
(14–281)
where
(14–282)
and
is defined by Equation 14–270.This model is appropriate for dilute systems.
Gidaspow model
The Gidaspow model [205] is a combination of the Wen and Yu model [700] and the Ergun equation [164].
When
, the fluid-solid exchange coefficient 
is of the following form: 
(14–283)
where
(14–284)
When
, 
(14–285)
This model is recommended for dense fluidized beds.
Huilin-Gidaspow model
The Huilin-Gidaspow model [263] is also a combination of the Wen and Yu model [700] and the Ergun equation [164]. The smooth switch is provided by the function when the solid volume fraction is less than 0.2:
(14–286)
where the stitching function is of the form
(14–287)
Gibilaro model
The Gibilaro model [202] is of the form
(14–288)
with the Reynolds number as
(14–289)
EMMS model
The energy-minimization multi-scale (EMMS) drag model is a heterogeneous approach derived from the mesoscale-structure-based methods. In the EMMS method, the mesoscale structures are broken down into a cluster phase and a dilute phase ([690], [391]). While homogeneous drag laws (such as Wen and Yu model [700] and Gidaspow model [205]) have the tendency to over-predict the solids flux, the cluster-based EMMS drag correctly evaluates the solid flux and axially S-shaped profiles of voidage. The EMMS drag model proposed by Lu et al. [391] is suitable for modeling two-phase granular flow in fluidized beds.
The fluid-solid exchange coefficient is of the following form:
(14–290)
where
where
is defined by Index
is expressed as: 
(14–291)
where the coefficients
, 
, and 
are functions of gas phase volume fraction. Formulas for calculating
, 
, and 
over different ranges of 
are shown in Table 14.3:
Fitting Formulas for Index
[391].Table 14.3: Fitting Formulas for Index
Range of
Coefficients
0.4≤ <0.46 
0.46≤ <0.545 
0.545≤ <0.99 
0.99≤ <0.9997 
0.9997≤ <1 
Filtered two-fluid model
The filtered two-fluid model can be used for cases with coarse meshes as described in The Filtered Two-Fluid Model .
The solid-solid exchange coefficient 
has
the following form [637]
:
 | (14–292) |
|
where | |
|
| |
|
| |
|
| |
|
|
Note that the coefficient of restitution is described in Solids Pressure and the radial distribution coefficient is described in Radial Distribution Function.
When using the Eulerian or Mixture multiphase models, Ansys Fluent can
include a user-specified drag modification term in the calculation
of interphase momentum exchange. This is introduced by replacing 
in Equation 14–204
with 
where
 | (14–293) |
and 
is defined as in Equation 14–212. You can
specify the modification factor, 
, as a constant, a user-defined function, defined by the Brucato et al.
correlation, or the near-wall drag modification.
The Brucato et al. correlation, [84] , is appropriate for dilute gas-liquid flows [333] and solid-liquid flows [451] where the drag coefficient is increased by the liquid phase turbulence. For the Brucato et al. correlation, the drag modification factor is expressed as
 | (14–294) |
where
 | (14–295) |
Here, 
, 
is the bubble diameter and 
is the Kolmogrov length scale given by:
 | (14–296) |
where 
is the specific molecular viscosity of the liquid phase and 
is the average liquid phase turbulent eddy dissipation.
The Ansys Fluent near-wall drag enhancement is effective for accurate predictions of the slip velocity in multiphase near-wall flows. The drag modification is well suited, for example, for air-water systems where a layer of equal-sized bubbles slides along the wall. In this scenario, the bubbles are in the flow region with the highest average strain ratio and highest turbulence, which can cause large bubbles to burst into smaller ones. The smaller bubbles can slide down the wall for a long period of time without breaking, only showing some deviation from an ideal spherical shape. When simulating such flows without the drag modification, the local accumulation of bubbles may lead to numerical problems.
The near-wall drag enhancement maintains a smooth transition from wall to bulk fluid, thus avoiding volume fraction stagnation and velocities overshoots. This improves the numerical robustness and the accuracy of the predicted near-wall velocity and void fraction values. It also decreases the instabilities and non-physical high velocity peaks close to the wall.
The enhancement formulation is proprietary to Ansys and therefore not published.