Transport Equations

The S-Gamma model is applicable in the dispersed regions of a particular phase for which size-distribution is being modeled. To enable the S-Gamma modeling for a situation where a phase can be locally dispersed or continuous, the dispersed phase fraction of a phase k is defined as $k=\frac{{\alpha }_d}{\alpha }$ .

In case of a Eulerian continuous-dispersed topology, the phase is dispersed everywhere, therefore $\alpha ={\alpha }_d$ , which yields $k=1$ . In case of multiple flow regime topology, a phase can be locally continuous or dispersed and $0\le k\le 1$ . The dispersed phase fraction k is then calculated based on the blending weight function as described in Phase Interaction Topology.

From the ${\alpha }_d=k\alpha$ assumption, the governing equations of S-Gamma model are written as below.

Transport Equation for the Zeroth Moment (Particle Number Density)

The instantaneous transport equation for $S_0$ is given by [505], [507]:

$$\frac{\partial {{\alpha }_{d}S}_0}{\partial t}+\nabla \cdot \left({{\alpha }_{d}S}_0{\mathbf{\text{v}}}_d\right)=s_{br}+s_{cl}+{\dot{N}}_0-s_{diss}$$

(2185)

where:

• ${\mathbf{\text{v}}}_d$ is the dispersed phase velocity that is calculated by the multiphase velocity solver
• $s_{br}$ is a source term that represents the effect of breakup
• $s_{cl}$ is a source term that represents the effect of coalescence
• $s_{diss}$ is a source term that represents the dissolution (the negative contribution). This source term is introduced to the $S_0$ transport equation to ensure a finite particle disappearance rate. When the surface growth interphase mass transfer rate $\dot{M}>0$ , particles grow and their number density is not affected by the mass transfer. However, for $\dot{M}<0$ the diameter of some particles decreases until it finally disappears as a result of the dissolution. The dissolution rate $s_{diss}$ is then modelled as:

$$s_{diss}=(\frac{d_{32}}{d_{30}}-1)\cdot \max (0,\frac{\dot{M}}{\rho })\cdot \frac{S_0}{{\alpha }_d}$$

(2186)

Reynolds averaging of Eqn. (2185) yields:

$$\frac{\partial {\overline{\alpha }}_d{\overline{S}}_0}{\partial t}+\nabla \cdot \left(\langle {\alpha }_d{S}_0{\mathbf{\text{v}}}_d\rangle \right)=\langle s_{br}\rangle +\langle s_{cl}\rangle +\langle \stackrel{\cdot }{N_0}\rangle -\langle s_{diss}\rangle$$

(2187)

For any scalar $\varphi$ (for example, temperature, concentration, kinetic energy of turbulence), the Reynolds-averaged flux is modeled as:

$$\langle {\alpha }_d\rho \varphi \mathbf{v}\rangle =\overline{{\alpha }_d}\rho \langle \mathbf{v}\rangle \overline{\varphi }+\langle {\alpha }_d\rho {\varphi }^{\prime }{\mathbf{v}}^{\prime }\rangle \approx \overline{{\alpha }_d}\rho \langle \mathbf{v}\rangle \langle \varphi \rangle -\overline{{\alpha }_d}\rho D_T\nabla \overline{\varphi }$$

(2188)

where $D_T$ is the coefficient of turbulent diffusion, calculated from the turbulent viscosity, density, and the turbulent Prandtl number as:

$$D_T=\frac{{\nu }_T}{\rho \mathrm{Pr}}$$

(2189)

Rewriting $S_0$ as ${\alpha }_d\rho \frac{S_0}{\alpha \rho }$ , the Reynolds-averaged transport equation for $S_0$ is obtained as:

$$\frac{\partial {\overline{\alpha }}_d{\overline{S}}_0}{\partial t}+\nabla \cdot \left({\overline{\alpha }}_d{\overline{S}}_0\langle {\mathbf{\text{v}}}_d\rangle -\rho {\overline{\alpha }}_d{D}_T\nabla \frac{{\overline{S}}_0}{\rho RA}\right)=\langle s_{br}\rangle +\langle s_{cl}\rangle +\langle \stackrel{\cdot }{N_0}\rangle -\langle s_{diss}\rangle$$

Variable density does not play a role in the conservation of the number density of particles.

(2190)

Transport Equation for the Second Moment

For $S_2$ , Wei and Morel [569] give an equation for the interfacial area $a_d$ of the dispersed phase in the form:

$$\frac{\partial a_d}{\partial t}+\nabla \cdot \left(a_d{\mathbf{\text{v}}}_d\right)=s_{a,br}+s_{a,cl}+s_{a,m}-\frac{2a_d}{3\rho }(\frac{\partial \rho }{\partial t}+{\mathbf{\text{v}}}_d\nabla \rho )$$

(2191)

$s_{a,br}$ and $s_{a,cl}$ represent the source of area due to breakup and coalescence respectively, and $s_{a,m}$ represents the source of area due to mass transfer between phases. The final term on the right-hand side represents the effect of variable density on interfacial area. By multiplying through by ${\rho }^{2/3}$ , performing some algebraic manipulations, and finally dividing by $\pi$ to convert to $S_2$ , it can be shown that the transport equation for $S_2$ is:

$$\frac{\partial {{\rho }^{2/3}{\alpha }_{d}S}_2}{\partial t}+\nabla \cdot \left({{\rho }^{2/3}{\alpha }_{d}S}_2{\mathbf{\text{v}}}_d\right)=s_{br}+s_{cl}+s_m+s_{nuc}$$

(2192)

$s_{nuc}$ is a source term that represents the nucleation rate. This source is introduced into the $S_2$ transport equation to represent the growth of the total particles area as a result of nucleation (a positive contribution only).

The nucleation rate $s_{nuc}$ is defined as:

$$s_{nuc}=\stackrel{\cdot }{{{\rho }^{\frac{2}{3}}N}_0}{d}_{nuc}^2$$

(2193)

where $d_{nuc}$ is the diameter of the nuclei.

The Reynolds-averaged version of the transport equation is then defined as:

$$\frac{\partial {\rho }^{2/3}{\overline{\alpha }}_d{\overline{S}}_2}{\partial t}+\nabla \cdot \left({\rho }^{2/3}{{\overline{\alpha }}_d\overline{S}}_2\langle {\mathbf{\text{v}}}_d\rangle -\rho {\overline{\alpha }}_d{D}_T\nabla \frac{{\overline{S}}_2}{{\rho }^{2/3}}\right)=\langle s_{br}\rangle +\langle s_{cl}\rangle +\langle s_m\rangle +\langle s_{nuc}\rangle$$

(2194)

The source term for S-Gamma due to the mass transfer reads:

$${\dot{S}}_2=4G\sum n_i{d}_i=4G{S}_0{d}_{10}$$

(2195)

But $\dot{M}=\rho \pi S_{2}G$ , and therefore the interphase mass flux $G$ is given as:

$$G=\frac{\dot{M}}{\rho \pi S_2}$$

(2196)

Combining Eqn. (2195) and Eqn. (2196) yields:

$${\dot{S}}_2=\frac{4\dot{M}}{\pi \rho }{\left(\frac{\pi S_0}{6{\alpha }_d}\right)}^{1/3}$$

(2197)