S-Gamma

The S-Gamma model is a method of moments that solves for the zeroth and the second moments of the number density $n (d_{p})$ . One transport equation is solved for each moment. Particles change size due to breakup and coalescence as they interact with each other.

The S-Gamma model (see Lo and Rao [505] and Lo and Zhang [507]) assumes that the particle size distribution conforms to a pre-defined log-normal shape function. The log-normal particle size distribution (probability density function) is represented by the mean particle diameter and its variance:

Figure 1. EQUATION_DISPLAY

P (d_{p}) = \frac{1}{d_{p} σ \sqrt{2 π}} \exp (\frac{{(\ln d_{p} - μ)}^{2}}{2 σ^{2}})

(2171)

where:

$μ$ is the mean logarithm of the particle diameter
$σ$ is the standard deviation of the particle diameter

The S-Gamma model defines a volumetric conserved quantity $S_{γ}$ that is related to the moment $M_{γ}$ of the particle size distribution:

Figure 2. EQUATION_DISPLAY

S_{γ} = n M_{γ} = n \int_{0}^{\infty} d_{p}^{γ} P (d_{p}) d (d_{p})

(2172)

where:

$γ = 0, 1, 2, 3$ is the order of the moment
$n$ is the number of particles per unit volume.
$d_{p}$ is the particle diameter
$P (d)$ is the probability density function of particle diameter [505], [507].

Simcenter STAR-CCM+ solves a transport equation for each of the three moments of $P (d)$ :

Zeroth moment: the particle number density $n$
Second moment: related to the interfacial area density
Third moment: related to the dispersed phase volume fraction

The zeroth moment is given by:

Figure 3. EQUATION_DISPLAY

S_{0} = \int n (d_{p}) d (d_{p})

(2173)

The second moment reads:

Figure 4. EQUATION_DISPLAY

S_{2} = \int d_{p}^{2} n (d_{p}) d (d_{p})

(2174)

The choice of moments is dictated by physical reasons. The zeroth moment

S_{0}

corresponds to the total number of particles. Assuming that the particles are spherical, the interfacial area can be calculated as:

Figure 5. EQUATION_DISPLAY

A = π \int d_{p}^{2} n (d_{p}) d (d_{p}) = π S_{2}

(2175)

The volume fraction of the dispersed phase can be calculated as:

Figure 6. EQUATION_DISPLAY

α = \frac{π}{6} \int d_{p}^{3} n (d_{p}) d (d_{p}) = \frac{π}{6} S_{3}

(2176)

As the third moment is based on the dispersed phase volume fraction, it is not calculated by the S-Gamma method, but it is imported from the Segregated EMP Flow solver. The Segregated EMP Flow solver solves for the volume fraction of the dispersed phase.