# Detection of Gas-Solid Two-Phase Flow Based on CFD and Capacitance Method
## 3 Numerical Simulation and Analysis  
Discrete Phase Model (DPM) is used to calculate the particle motion in the annular space. DPM is based on a Euler-Lagrangian framework. In this model, the fluid is a continuous medium, and the particle phase is discrete. The particles are categorised into two groups. One is characterised by instantaneous collision movement driven by momentum and the other by suspension movement controlled by the fluid drag force. Thereby a particle motion decomposition model is established [12]. In this model, the movement of the fluid is described by \(N\)-\(S\) equations of a continuous medium. Particle motion is obtained by calculating the trajectories of every particle. The trajectory of the particle can be calculated by the following formula,  
\[\frac{du_{p}}{dt}=F_{D}\big{(}u-u_{p}\big{)}+\frac{g_{x}(\rho_{p}-\rho)}{\rho_ {p}}+F_{x} \tag{1}\]  
where \(u_{p}\) is the velocity of particle, \(u\) is the velocity of fluid, \(\rho_{p}\) is density of the particle, \(\rho\) is the density of the fluid, and \(g_{x}\) is the gravity of the particle. The first item on the right side of the equal sign is the drag of the unit mass particle, the second item is the resultant force of gravity and buoyant force, and the third item is the resultant force of other additional forces. And \(F_{D}\) is calculated by using the following formula,  
\[F_{D}=\frac{3\mu C_{D}Re_{p}}{4\rho_{p}d_{p}^{2}} \tag{2}\]  
The drag force item in the formula is obtained by the correlation proposed by [PERSON] et al. [20],  
\[C_{D}=a_{1}+\frac{a_{2}}{Re_{p}}+\frac{a_{3}}{Re_{p}^{2}} \tag{3}\]  
where \(a_{1}\), \(a_{2}\), \(a_{3}\) are constants, \(Re_{p}\) is Reynolds number of particle. Figure 3: (**a**) Meshing in annular space; (**b**) Cross section of meshing in annular space. For the coupling between the discrete phase and the continuous phase, there is mainly the particle cloud model and the stochastic particle model proposed by [PERSON] et al. [21]. The second model is used by most researchers, which is also used in this paper. In addition to DPM, the Two-Fluid model is commonly used for multiphase flows. The Two-Fluid model is based on the Euler framework. Each phase will be seen as a continuous medium. Each of the two phases flow with different characters and can penetrate into the other, and the forces are coupled. A set of equations contain momentum equations and a continuity equation [22]. The continuity equation for the fluid phase is,  
\[\frac{\partial}{\partial t}\big{(}\alpha_{q}\rho_{q}\big{)}+\
abla\cdot\big{(} \alpha_{q}\rho_{q}\overset{\rightharpoonup}{v_{q}}\big{)}=\sum_{p=1}^{n}\dot{m }_{pq} \tag{4}\]  
where \(\alpha_{q}\) is the volume fraction of phase \(q\), \(\rho_{q}\) is the physical density of phase \(q\), \(\overset{\rightharpoonup}{v_{q}}\) is the speed of phase \(q\), and \(m_{pq}\) is the mass transfer flux from phase \(p\) to phase \(q\). Moreover, the momentum conservation equation is,  
\[\frac{\partial}{\partial t}(\alpha_{q}\rho_{q}\overset{\rightharpoonup}{v_{q} })+\
abla\cdot(\alpha_{q}\rho_{q}\overset{\rightharpoonup}{v_{q}}\overset{ \rightharpoonup}{v_{q}})=-\alpha_{q}\
abla p+\
abla\cdot\overset{\rightharpoonup} {v_{q}}+\sum_{p=1}^{n}(\overset{\rightharpoonup}{R}_{pq}+\dot{m}_{pq} \overset{\rightharpoonup}{v_{pq}})+\alpha_{q}\rho_{q}(\overset{\rightharpoonup }{F}_{q}+\overset{\rightharpoonup}{F}_{liftq}+\overset{\rightharpoonup}{F}_{ vm,q}) \tag{5}\]  
where \(\overset{\rightharpoonup}{v_{q}}\) is pressure strain tensor of phase \(q\), \(\overset{\rightharpoonup}{F}_{q}\) is the external volume force, \(\overset{\rightharpoonup}{F}_{lift,q}\) is the lift force, \(\overset{\rightharpoonup}{F}_{Vm,q}\) is the virtual mass force, \(\overset{\rightharpoonup}{R}_{pq}\) is the interaction force between phases, \(p\) is pressure shared with all phases, and \(\overset{\rightharpoonup}{v_{pq}}\) is the speed difference between the two phases. Standard \(k\)-\(\varepsilon\) model is used throughout the simulation. The wall boundary conditions in this paper are non-slip conditions.