16.5.5 Solids Pressure

For granular flows in the compressible regime (i.e., 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, $\nabla p_s$, 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:

where $e_{ss}$ is the coefficient of restitution for particle collisions, $g_{0,ss}$ is the radial distribution function, and $\Theta_{s}$ is the granular temperature. ANSYS FLUENT uses a default value of 0.9 for $e_{ss}$, but the value can be adjusted to suit the particle type. The granular temperature $\Theta_{s}$ is proportional to the kinetic energy of the fluctuating particle motion, and will be described later in this section. The function $g_{0,ss}$ (described below in more detail) is a distribution function that governs the transition from the "compressible'' condition with $\alpha < \alpha_{s,{\rm max}}$, where the spacing between the solid particles can continue to decrease, to the "incompressible'' condition with $\alpha = \alpha_{s,{\rm max}}$, where no further decrease in the spacing can occur. A value of 0.63 is the default for $\alpha_{s,{\rm max}}$, but you can modify it during the problem setup.

Other formulations that are also available in ANSYS FLUENT are  [ 343]

and  [ 213]

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 [ 109], a general solids pressure formulation in the presence of other phases could be of the form

where $d_{pq}=\frac{d_p+d_q}{2}$ is the average diameter, $n_p$, $n_q$ are the number of particles, $m_p$ and $m_q$ are the masses of the particles in phases $p$ and $q$, and $f$ 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 $q$

Since all models need to be cast in the general form, it follows that

where $p_{c,qp}$ is the collisional part of the pressure between phases $q$ and $p$.

The above expression reverts to the one solids phase expression when $N=1$ and $q=p$ but also has the property of feeling the presence of other phases.

Radial Distribution Function

The radial distribution function, $g_0$, 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:

where $s$ is the distance between grains. From Equation  16.5-69 it can be observed that for a dilute solid phase $s \rightarrow \infty$, and therefore $g_0 \rightarrow 1$. In the limit when the solid phase compacts, $s \rightarrow 0$ and $g_0 \rightarrow \infty$. The radial distribution function is closely connected to the factor $\chi$ of Chapman and Cowling's [ 49] theory of nonuniform gases. $\chi$ 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 [ 254]:

  
  

  
  

  This is an empirical function and does not extend easily to $n$ phases. For two identical phases with the property that $\alpha_q = \alpha_1+\alpha_2$, the above function is not consistent for the calculation of the partial pressures $p_1$ and $p_2$, $p_q = p_1 + p_2$. In order to correct this problem, ANSYS FLUENT uses the following consistent formulation:

  
  

  
  

  where

  
  

  
  

  and $k$ are solid phases only.

• The following expression is also available [ 137]:

  
  

  
  

• Also available [ 213], slightly modified for $n$ solids phases, is the following:

  
  

  
  

• The following equation [ 343] is available:

  
  

  
  

When the number of solid phases is greater than 1, Equation  16.5-71, Equation  16.5-73 and Equation  16.5-74 are extended to

It is interesting to note that Equation  16.5-73 and Equation  16.5-74 compare well with [ 3] experimental data, while Equation  16.5-75 reverts to the [ 46] derivation.