# CFD Simulation of Solid Suspension for a Liquid-Solid Industrial Stirred Reactor
## 2 Materials and Methods  
In this paper, the Eulerian multiphase model was chosen to describe the flow behavior of each phase of the two phases, liquid and solid. This model is the most complex, and it  
Figure 1: Schematic representation of the stirred reactor. solves a set of 2 momentum equations for each phase. Coupling is achieved through the pressure and interphase exchange coefficients. The way this coupling is handled depends on the type of phases involved, and for granular flows, the properties are obtained from the application of kinetic theory. The derivation of the conservation equations for mass and momentum for each of the two phases is performed by phase-weighted averaging the local instantaneous balances for each of the phases, and then no additional turbulent dispersion term is introduced into the continuity equation. The pressure field is assumed to be shared by the two phases, in proportion to their volume fraction. The motion of each phase is governed by respective mass and momentum conservation equations. The continuity equation for each phase is:  
\[\frac{\partial\big{(}\alpha_{q}\rho_{q}\big{)}}{\partial t}+\
abla\cdot\Big{(} \alpha_{q}\rho_{q}\overset{\rightarrow}{v}_{q}\Big{)}=0, \tag{1}\]  
where \(\rho\) is the density, \(\alpha\) is the volume fraction and \(v\) is the velocity vector for each phase. The conservation of momentum for the liquid and solid phase is given by the following equations:  
\[ \begin{array}{l}\frac{\partial\big{(}\alpha_{q}\rho_{q}\overset{ \rightarrow}{v}_{q}\big{)}}{\partial t}+\
abla\cdot\Big{(}\alpha_{q}\rho_{q} \overset{\rightarrow}{v}_{q}\overset{\rightarrow}{v}_{q}\Big{)}=-\alpha_{q} \
abla p+\
abla\cdot\overline{\tau}_{q}+\alpha_{q}\rho_{q}\overset{ \rightarrow}{\delta}+\\ +\sum\limits_{q=1}^{n}\Big{(}K_{pq}\big{(}\overset{\rightarrow}{v}_{p}- \overset{\rightarrow}{v}_{q}\big{)}+\dot{m}_{pq}\overset{\rightarrow}{v}_{ pq}-\dot{m}_{qp}\overset{\rightarrow}{v}_{qp}\Big{)}+\left(\overset{ \rightarrow}{F}_{q}+\overset{\rightarrow}{F}_{lifl}+\overset{\rightarrow}{F }_{wl,q}+\overset{\rightarrow}{F}_{vm,q}+\overset{\rightarrow}{F}_{ul,q} \right)\\ \\ \frac{\partial\big{(}\alpha_{q}\rho_{q}\overset{\rightarrow}{v}_{s}\big{)}}{ \partial t}+\
abla\cdot\Big{(}\alpha_{q}\rho_{q}\overset{\rightarrow}{v}_{s} \overset{\rightarrow}{v}_{s}\overset{\rightarrow}{v}_{s}\Big{)}=-\alpha_{s} \
abla p-\
abla p_{s}+\
abla\cdot\overline{\tau}_{s}+\alpha_{s}\rho_{s} \overset{\rightarrow}{\delta}+\\ +\sum\limits_{l=1}^{N}\Big{(}K_{ls}\Big{(}\overset{\rightarrow}{v}_{l}- \overset{\rightarrow}{v}_{s}\Big{)}+\dot{m}_{ls}\overset{\rightarrow}{v}_{ ls}-\dot{m}_{sl}\overset{\rightarrow}{v}_{sl}\Big{)}+\left(\overset{\rightarrow}{F}_{s}+ \overset{\rightarrow}{F}_{lifl}+\overset{\rightarrow}{F}_{vm,s}+\overset{ \rightarrow}{F}_{td,s}\right)\end{array}  \tag{2}\]  
\[\overline{\tau}_{q}=\alpha_{q}\mu_{q}\Big{(}\
abla\overset{\rightarrow}{v}_{q} +\
abla\overset{\rightarrow}{v}_{q}^{T}\Big{)}+\alpha_{q}\Big{(}\lambda_{q}- \frac{2}{3}\mu_{q}\Big{)}\
abla\cdot\overset{\rightarrow}{v}_{q}\overline{t} \tag{4}\]  
where \(p\) is the pressure shared by all phases, \(p_{s}\) is the solid pressure, \(t_{q}\) is the \(q\)-th phase stress-strain tensor, \(g\) is the gravitational acceleration, \(\mu\) and \(\lambda\) are the shear and bulk viscosity of phase \(q\), \(K_{pq}\) is the interphase momentum exchange coefficient, \(K_{ls}\) is the momentum exchange coefficient between fluid or solid, \(F_{q}\) is an external body force, \(F_{lifl}\),\(g\) is a lift force, \(F_{wl,q}\) is a wall lubrication force, \(F_{vm,q}\) is a virtual mass force, \(F_{td,q}\) is a turbulent dispersion force and \(v_{pq}\) is the interphase velocity. Only the contribution of the drag force was taken into consideration because previous studies [2; 10; 12] showed that the other forces have no major effect on both the liquid-gas and liquid-solid hydrodynamics in stirred reactors. Almost all definitions of the drag function (\(f\)) include a drag coefficient (\(C_{D}\)) that is based on the relative Reynolds number (\(Re\)).