# CFD Investigations of Cyclone Separators with Different Cone Heights and Shapes
## 2 Materials and Methods
### Solid-Phase Transport  
To define the trajectory of the solid phase, the particle force equilibrium equation is configured within the Lagrange frame of reference. The force balance equation can be written as follows:  
\[\frac{du_{pi}}{dt}=F_{D}\big{(}u_{i}-u_{pi}\big{)}+\frac{\big{(}\rho_{p}-\rho \big{)}}{\rho_{p}}g_{i}+F_{i} \tag{8}\]  
where: the subscript \(i\) denotes the direction of the flow parameter, \(u_{pi}\)--the particle velocity; \(F_{i}\)--additional force (e.g., Brownian force, Saffman lift, etc.) expressed as force per unit mass, the term \(F_{D}\big{(}u_{i}-u_{pi}\big{)}\). --drag force per particle mass unit; \(\rho_{p}\)--the particle density; \(g_{i}\)--the grav. acceleration. For spherical particles, the drag force (Stokes) can be defined as follows:  
\[F_{D}=\frac{18\mu}{\rho_{p}d_{p}^{2}}\frac{C_{D}Re}{24} \tag{9}\]  
where: \(d_{p}\)--the diameter of the particle; \(Re\)--the Reynolds number; \(\mu\)--the molecular viscosity of the fluid phase. \[Re=\frac{\rho d_{p}\big{|}u_{pi}-u_{i}\big{|}}{\mu} \tag{10}\]  
In Equation (8), the second term on the right-hand side viz. \(C_{D}*Re/24\) approaches unity for Stokes flow. The first term bears a unit of reciprocal of time and is a representation of the particle response time, given as:  
\[\tau_{p}=\frac{\rho_{p}d_{p}^{2}}{18\mu} \tag{11}\]  
The drag coefficient for smooth particles is given as (\(a_{1}\), \(a_{2}\), and \(a_{3}\)--constants) [44]:  
\[C_{D}=a_{1}+\frac{a_{2}}{Re_{p}}+\frac{a_{3}}{Re_{p}^{2}} \tag{12}\]A single particle velocity is determined by integrating Equation (8) in discrete time steps, and its position in the non-stationary flow is determined from the equation:  
\[\frac{d_{xi}}{d_{t}}=u_{pi} \tag{13}\]  
Then the Equation (8) can be expressed in the following form:  
\[\frac{du_{pi}}{dt}=\frac{1}{\tau_{p}}\big{(}u_{i}-u_{pi}\big{)}+a_{i} \tag{14}\]  
where: the term \(a_{i}\) contains accelerations due to all force terms; \(\tau_{p}\)--the particle relaxation time. Fluent includes several options for tracing (or discretizing) the trajectory of solid particles. In most of the research on flow in cyclone separators carried out so far, higher-order schemes, especially trapezoidal ones, have been recommended [45]. In trapezoidal diagrams, the particle's progress in each successive discrete time step is determined. In Equation (12), the right part is averaged as:  
\[\frac{u_{pi}^{n+1}-u_{pi}^{n}}{\Delta t}=\frac{1}{\tau_{p}}\Big{(}u_{i}^{*}-u_ {pi}^{*}\Big{)}+{a_{i}}^{n} \tag{15}\]  
where: \(u_{pi}^{*}\), \(u_{i}^{*}\)--mean values of variables \(u_{i}\), \(u_{pi}\)--whose values can be expressed as appropriate: \(u_{pi}^{*}=\Big{(}u_{pi}^{n}+u_{pi}^{n+1}\Big{)}/2\), \(u_{i}^{*}=\Big{(}u_{i}^{n}+u_{i}^{n+1}\Big{)}/2\), and \(u_{i}^{n+1}=u_{i}^{n}+\Delta tu_{pi}^{n}\
abla u_{i}^{n}\). The particle velocity at the new location \(n+1\) is expressed as:  
\[u_{pi}^{n+1}=\frac{u_{pi}^{n}\Big{(}1-\frac{\Delta t}{2\tau_{p}}\Big{)}+\Big{(} u_{i}^{n}+\frac{1}{2}\Delta tu_{pi}^{n}\
abla u_{i}^{n}\Big{)}\frac{\Delta t }{\tau_{p}}+\Delta ta}{1+\frac{\Delta t}{2\tau_{p}}} \tag{16}\]  
Since the solid particles used here are in the range of 0.4-8.0 microns and present in the dilute phase, it was necessary to take into account the turbulent dispersion effect caused by instantaneous velocity (defined as the sum of average velocity and variable velocity). For this reason, a stochastic tracking model (random walk) has been used to account for the effect of instantaneous velocity fluctuations on the trajectories of particles. The dispersion of turbulent particles is defined by integrating the trajectory equations for each solid particle in the particle path. The discrete random walk (DRW) model--the stochastic model--has been used to account for the effect of the instantaneous velocity of the carrier fluid on the solid particles.