For granular flows in the compressible regime (that is, where the solids volume fraction is
less than its maximum allowed value), a solids pressure is calculated independently and used for
the pressure gradient term, 
, in the granular-phase momentum equation. Because a Maxwellian velocity
distribution is used for the particles, a granular temperature is introduced into the model, and
appears in the expression for the solids pressure and viscosities. The solids pressure is
composed of a kinetic term and a second term due to particle collisions:
 | (14–345) |
where 
is the coefficient of restitution for particle collisions, 
is the radial distribution function, and 
is the granular temperature. Ansys Fluent uses a default value of 0.9 for
, but the value can be adjusted to suit the particle type. The granular
temperature 
is proportional to the kinetic energy of the fluctuating particle motion, and
will be described later in this section. The function 
(described below in more detail) is a distribution function that governs the
transition from the "compressible" condition with 
, where the spacing between the solid particles can continue to decrease, to the
"incompressible" condition with 
, where no further decrease in the spacing can occur. A value of 0.63 is the
default for 
, but you can modify it during the problem setup.
Other formulations that are also available in Ansys Fluent are [640]
 | (14–346) |
and [404]
 | (14–347) |
where 
is defined in [404].
When more than one solids phase are calculated, the above expression does not take into account the effect of other phases. A derivation of the expressions from the Boltzman equations for a granular mixture are beyond the scope of this manual, however there is a need to provide a better formulation so that some properties may feel the presence of other phases. A known problem is that N solid phases with identical properties should be consistent when the same phases are described by a single solids phase. Equations derived empirically may not satisfy this property and need to be changed accordingly without deviating significantly from the original form. From [204], a general solids pressure formulation in the presence of other phases could be of the form
 | (14–348) |
where 
is the average diameter, 
, 
are the number of particles, 
and 
are the masses of the particles in phases 
and 
, and 
is a function of the masses of the particles and their granular temperatures.
For now, we have to simplify this expression so that it depends only on the granular temperature
of phase 
 | (14–349) |
Since all models need to be cast in the general form, it follows that
 | (14–350) |
where 
is the collisional part of the pressure between phases 
and 
. In Equation 14–349, 
.
The above expression reverts to the one solids phase expression
when 
and 
but also has the
property of feeling the presence of other phases.
The radial distribution function, 
, is a correction factor that modifies the probability of collisions between
grains when the solid granular phase becomes dense. This function may also be interpreted as the
nondimensional distance between spheres:
 | (14–351) |
where 
is the distance between grains. From Equation 14–351
it can be observed that for a dilute solid phase 
, and therefore 
. In the limit when the solid phase compacts, 
and 
. The radial distribution function is closely connected to the factor
of Chapman and Cowling’s [99] theory of
nonuniform gases. 
is equal to 1 for a rare gas, and increases and tends to infinity when the
molecules are so close together that motion is not possible.
In the literature there is no unique formulation for the radial distribution function. Ansys Fluent has a number of options:
For one solids phase, use [204] :
(14–352)
This is an empirical function and does not extend easily to
phases. For two identical phases with the property that 
, the above function is not consistent for the calculation of the partial
pressures 
and 
, 
. In order to correct this problem, Ansys Fluent uses the following
consistent formulation: 
(14–353)
where
is the packing limit,
is the total number of solid
phases,
is the diameter of the phases, and 
(14–354)
and
are solid phases only.The following expression is also available [266]:
(14–355)
Also available [404], slightly modified for
solids phases, is the following: 
(14–356)
The following equation [640] is available:
(14–357)
When the number of solid phases is greater than 1, Equation 14–353 , Equation 14–355 and Equation 14–356 are extended to
 | (14–358) |
It is interesting to note that Equation 14–355 and Equation 14–356 compare well with [14] experimental data, while Equation 14–357 reverts to the [93] derivation.