## 2 Mathematical Modeling
### Computational Fluid Dynamics
#### 2.4.1 Continuous Phase  
Large Eddy Simulation  
The LES adopts a filtering approach to distinguish between large and small eddies, with the filter function which is represented as [35] follows:  
\[\overline{\phi}(x)=\int_{D}\overline{\phi}(x^{\prime})\ G(x,x^{\prime})dx^{\prime} \tag{6}\] \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Geometrical Parameter** & **Notation** & **Lower Bound** & **Upper Bound** \\ \hline Cyclone diameter & * \(D_{c}/D\) & 1.000 & 1.300 \\ Total cyclone length & \(H_{t}/D\) & 4.000 & 8.000 \\ Vortex finder diameter & \(D_{x}/D\) & 0.300 & 0.700 \\ Length of conical segment & \(H_{c}/D\) & 1.000 & 4.000 \\ Cone tip diameter & \(B_{c}/D\) & 0.010 & 0.040 \\ Insertion length of vortex finder & \(S/D\) & 0.345 & 0.690 \\ \hline \hline \end{tabular}  
* \(D\) = 0.29 m corresponds to the diameter of the standard cyclone.  
\end{table} 
Table 1: Details of the upper and lower bounds used to optimize cyclone. \begin{table}
\begin{tabular}{l c} \hline \hline
**Solver** & **Gamultiobj** \\ \hline Population type & Double vector \\ Population size & 90 (defined as 15\(\times\) number of variables) \\ Number of variables & 6 \\ Selection operation & Tournament (size equal to 2 (default)) \\ Crowding distance fraction & 0.35 \\ Crossover fraction & 0.8 (default) \\ Crossover operation & Intermediate (with a ratio equal to 0.2 (default)) \\ Number of generations (iterations) & 1200 (defined as 200\(\times\) number of variables) \\ User function evaluation & In parallel \\ \hline \hline \end{tabular}  
\end{table} 
Table 2: Genetic operators and parameters. The integration over the flow domain (\(D\)) through the computational grid is represented as  
\[\overline{\phi}(x)=\frac{1}{V}\int_{v}\phi\big{(}x^{\prime}\big{)}\ dx^{\prime},x ^{\prime}\in \tag{7}\]  
where \(V\) represents the volume of the computational cell. The filter function is given as follows:  
\[G\big{(}x,x^{\prime}\big{)}=\left\{ \begin{array}{ll}\frac{1}{V},&x^{\prime} \in v\\ 0,&x^{\prime}\ otherwise\end{array} \right. \tag{8}\]  
The application of filtering on Navier-Stokes equations yields the following:  
\[\frac{\partial\overline{u_{i}}}{\partial x_{i}}=0 \tag{9}\]  
\[\frac{\partial\overline{u_{i}}}{\partial t}+\overline{u_{j}}\frac{\partial \overline{u_{j}}}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\overline{P}}{ \partial x_{i}}+v\frac{\partial^{2}\overline{u_{i}}}{\partial x_{j}\partial x _{j}}-\frac{\partial T_{ij}}{\partial x_{j}} \tag{10}\]  
The over-bar is used to represent the filtering quantity. The subgrid-scale Reynolds stress, \(T_{ij}=\overline{u_{i}}\ \overline{u_{j}}-\overline{u_{i}u_{j}}\) is modeled using the Boussinesq hypothesis as follows:  
\[T_{ij}-\frac{1}{3}\tau_{ik}\delta_{ij}=-2u_{sgs}\overline{\overline{s}}_{ij} \tag{11}\]  
where \(\mu_{sgs}\) represents the sub-grid scale viscosity. The strain rate in the resolved flow is given as follows:  
\[\overline{S}_{ij}=\frac{1}{2}\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+ \frac{\partial u_{j}}{\partial x_{i}}\Bigg{)} \tag{12}\]  
Modeling the Particulate Phase  
For a discrete particle traveling in a continuous fluid medium, the net force acting on the particles is calculated using [PERSON]'s second law [36]:  
\[\frac{du_{pi}}{dt}=F_{D}\big{(}\ u_{i}-\ u_{pi}\big{)}+\frac{\Big{(}\ \rho_{p}-\rho\Big{)}}{\rho_{p}}\ g_{i}+\ F_{i} \tag{13}\]  
Here, the density of the discrete phase is given as \(\ \rho_{p}\), \(F_{D}\big{(}\ u_{i}-\ u_{pi}\big{)}\) represents the drag force on each particle per unit mass (here, \(i\) represents the flow direction), \(\ u_{pi}\) is the particle's velocity, \(\ g_{i}\) is the acceleration due to gravity, \(\ F_{i}\) represents the additional forces (e.g., the Saffman lift force, Brownian force, etc.).