## Nomenclature \begin{tabular}{l l} \(a\) & height of the cyclone inlet \\ \(A\) & inlet area of the cyclone \\ \(b\) & width of the cyclone inlet \\ \(B_{c}\) & cone tip diameter of the cyclone \\ \(C_{D}\) & drag force coefficient \\ \(C_{s}\) & Smagorinsky constant \\ \(d_{p}\) & particle diameter \\ \(d_{50}\) & cut-off size \\ \(D\) & diameter of the standard cyclone model/characteristic diameter \\ \(D_{c}\) & (optimized) cyclone diameter \\ \(D_{L,\beta}\) & molecular diffusive transport term \\ \(D_{T,\beta}\) & turbulent diffusive transport term \\ \(D_{x}\) & vortex finder diameter \\ \(f\) & frequency of oscillation of the PVC \\ \(F_{i}\) & additional forces per unit mass \\ \(F_{D}\) & drag force \\ \(g\) & gravitational accelerations \\ \(H_{c}\) & length of the cylindrical segment of the cyclone \\ \(k\) & von [PERSON] constant \\ \(l\) & characteristic length \\ \(L_{s}\) & length scale of subgrid-scales \\ \(L_{\varphi}\), \(S\) & insertion length of the vortex finder diameter inside the cyclone \\ \(m_{p}\) & particle mass \\ \(p\) & instantaneous gas static pressure \\ \(P\) & mean static pressure of the gas \\ \(p_{0}\) & pressure at the symmetric axis \\ \(p^{\prime}\) & fluctuating gas pressure \\ \(\overline{p}\) & resolved static pressure \\ \end{tabular} \begin{tabular}{l l} \(\Delta P\) & differential pressure measurement \\ \(P_{ij}\) & stress generation term \\ \(Q\) & the volume flow rate of air \\ \(R\) & the radius of the cyclone body \\ \(Re\) & Reynolds number \\ \(S\) & vortex finder insertion length \\ \(S_{w}\) & geometric swirl number \\ \(t\) & time \\ \(t_{\eta}\) & the time scale for smaller eddies \\ \(T_{L}\) & Lagrangian integral time scale \\ \(u\) & characteristic velocity \\ \(u_{i}\) & velocity vector in the i-coordinate direction/instantaneous velocity \\ \(\overline{u}_{i}\) & the mean velocity component \\ \(u^{\prime}_{i}\) & the fluctuating velocity component \\ \(u_{pi}\) & particle velocity \\ \(U\) & mean gas velocity \\ \(V\) & the volume of a cell \\ \(v_{\theta}\) & tangential velocity \\ \(x_{i}\) & position in the i-direction \\ \(f\) & friction factor \\ \(\rho\) & gas density \\ \(Q\) & volume flow rate \\ \(v_{\theta w}\) & velocity at the wall \\ \(v_{\theta cs}\) & tangential velocity of the rotating fluid in the inner core radius \\ \(A_{R}\) & the total inside surface area \\ \(v_{c}\) & average axial velocity of the gas exiting via the vortex finder \\ \(v_{in}\) & gas velocity at the inlet \\ \(\rho_{p}\) & density of the particulate phase \\ \(\mu\) & fluid viscosity \\ \(H_{t}\) & cyclone length \\ \(S\) & Insertion depth of the outlet tube \\ \(\mu_{ggs}\) & sub-grid scale viscosity \\ \(u_{pi}\) & velocity of the particle \\ \(g_{i}\) & acceleration due to gravity \\ \(F_{i}\) & additional forces (e.g., the Saffman lift force, Brownian force, etc.) \\ \(Re_{p}\) & Reynolds number of the particle \\ \(T_{rns}\) & residence time \\ \end{tabular} 
If \(R_{in}\) is the radial position of the center of the inlet, then for a slot inlet, the following holds:  
\[R_{in}=R-b/2\] (A1)  
If \(D\) is the cyclone diameter, then its radius is as follows:  
\[R=D/2\] (A2)  
If \(D_{x}\) is the vortex finder diameter, then its radius is as follows:  
\[R_{x}=D_{x}/2\] (A3)  
If \(B_{d}\) is the vortex finder diameter, then its radius is as follows:  
\[R_{d}=B_{d}/2\] (A4)  
If \(Q\) is the volume flow rate, then the mean axial velocity in the vortex finder is as follows:  
\[v_{x}=Q/\pi R_{x}{}^{2}\] (A5)  
For the friction factor \(f\), the following relation can be used for a hydraulically smooth cyclone surface according to [PERSON] and [PERSON] (2008):  
\[f=0.05(1+3\sqrt{C_{0}}).\] (A6)  
where \(C_{0}\) is the ratio of the mass of incoming solids to the mass of incoming gas in the stream feeding the cyclone. According to the MM model, the pressure loss across a cyclone occurs primarily due to wall friction and irreversible losses within the vortex finder.