For spherical particles, the coefficients 
may be derived analytically. The area of a single
particle projected in the flow direction, 
,
and the volume of a single particle 
are
given by:
 | (5–37) |
where 
is the mean diameter. The number
of particles per unit volume, 
, is given by:
 | (5–38) |
The drag exerted by a single particle on the continuous phase is:
 | (5–39) |
Hence, the total drag per unit volume on the continuous phase is:
 | (5–40) |
Comparing with the Momentum Equations for phase 
, where
the drag force per unit volume is:
 | (5–41) |
you get:
 | (5–42) |
which can be written as:
 | (5–43) |
This is the form implemented in CFX.
The following section describes drag correlations specific to dispersed multiphase flow.
At low particle Reynolds numbers (the viscous regime), the drag coefficient for flow past spherical particles may be computed analytically. The result is Stokes’ law:
 | (5–44) |
For particle Reynolds numbers, such as Equation 5–9, that are sufficiently large for inertial effects to dominate viscous effects (the inertial or Newton’s regime), the drag coefficient becomes independent of Reynolds number:
 | (5–45) |
In the transitional region between the viscous and inertial
regimes, 
for spherical particles, both
viscous and inertial effects are important. Hence, the drag coefficient
is a complex function of Reynolds number, which must be determined
from experiment.
This has been done in detail for spherical particles. Several empirical correlations are available. The one available in CFX is due to Schiller and Naumann (1933) [6]. It can be written as follows:
 | (5–48) |
Note that this has the same functional form as the Schiller
Naumann correlation, with a modified particle Reynolds number, and
a power law correction, both functions of the continuous phase volume
fraction 
.
You may also change the Volume Fraction Correction Exponent from its default value of -1.65, if you want.
 | (5–49) |
This uses the Wen Yu correlation for low solid volume fractions 
, and switches
to Ergun’s law for flow in a porous medium for larger solid
volume fractions.
Note that this is discontinuous at the cross-over volume fraction.
In order to avoid subsequent numerical difficulties, CFX modifies
the original Gidaspow model by linearly interpolating between the
Wen Yu and Ergun correlations over the range 
.
You may also change the Volume Fraction Correction Exponent of the Wen Yu part of the correlation from its default value of -1.65, if you want.
At sufficiently small particle Reynolds numbers (the viscous regime), fluid particles behave in the same manner as solid spherical particles. Hence, the drag coefficient is well approximated by the Schiller-Naumann correlation described above.
At larger particle Reynolds numbers, the inertial or distorted particle regime, surface tension effects become important. The fluid particles become, at first, approximately ellipsoidal in shape, and finally, spherical cap shaped.
In the spherical cap regime, the drag coefficient is well approximated by:
 | (5–50) |
Several correlations are available for the distorted particle regime. CFX uses the Ishii Zuber [19] and Grace [35] correlations.
In the distorted particle regime, the drag coefficient is approximately constant, independent of Reynolds number, but dependent on particle shape through the dimensionless group known as the Eotvos number, which measures the ratio between gravitational and surface tension forces:
 | (5–51) |
Here, 
is the density
difference between the phases, 
is the gravitational acceleration,
and 
is the surface tension coefficient.
The Ishii-Zuber correlation gives:
 | (5–52) |
In this case, CFX automatically takes into account the spherical particle and spherical cap limits by setting:
 | (5–53) |
The Ishii Zuber Model also automatically takes into account dense fluid particle effects. For details, see Densely Distributed Fluid Particles.
The Grace drag model is formulated for flow past a single bubble. Here the drag coefficient in the distorted particle regime is given by:
 | (5–54) |
where the terminal velocity 
is
given by:
 | (5–55) |
where:
 | (5–56) |
and:
 | (5–57) |
 | (5–58) |
is the molecular viscosity
of water at 25°C and 1 bar.
In this case, CFX automatically takes into account the spherical particle and spherical cap limits by setting:
 | (5–59) |
The Ishii Zuber [19] drag laws automatically take into account dense particle effects. This is done in different ways for different flow regimes.
In the viscous regime, where fluid particles may be assumed to be approximately spherical, the Schiller Naumann correlation is modified using a mixture Reynolds number based on a mixture viscosity.
 | (5–60) |
Here, 
is
the user-defined Maximum Packing value. This is defaulted to unity
for a dispersed fluid phase.
In the distorted particle regime, the Ishii Zuber modification takes the form of a multiplying factor to the single particle drag coefficient.
 | (5–62) |
The Ishii Zuber correlation, as implemented in CFX, automatically selects flow regime as follows:
The Grace [35] drag model is formulated for flow past a single bubble. For details, see Sparsely Distributed Fluid Particles (Drops and Bubbles).
For high bubble volume fractions, it may be modified using a simple power law correction:
 | (5–64) |
Here, 
is the single bubble Grace drag coefficient. Advice
on setting the exponent value for the power law correction is available
in Densely Distributed Fluid Particles: Grace Drag Model in the CFX-Solver Modeling Guide.