# Eulerian-Eularian Model for Agglomeration Behavior of Combusted Iron Particles
## 2 Model Description
### Modeling of Bed Hydrodynamics  
The equations of continuity and momentum are solved for both gas and solid phases with the Eulerian-Eulerian modeling approach. Equations (1) and (2) depict the continuity equations for gas and solid, illustrating mass flow from and into the controlled volume, as well as the net mass transfer in and out of the control volume. Equations (3) and (4) depict momentum transfer for gas and solid phases, encompassing all factors associated with momentum transfer into and out of a regulated volume.  
\[\frac{\partial\big{(}\epsilon_{g}\rho_{g}\big{)}}{\partial t}+\
abla\cdot \big{(}\epsilon_{g}\rho_{g}u_{g}\big{)}=S_{gs} \tag{1}\]  
\[\frac{\partial\big{(}\epsilon_{s}\rho_{s}\big{)}}{\partial t}+\
abla\cdot \big{(}\epsilon_{s}\rho_{s}u_{s}\big{)}=S_{sg} \tag{2}\]  
\[\frac{\partial\big{(}\epsilon_{g}\rho_{g}u_{g}\big{)}}{\partial t}+\
abla\cdot \big{(}\epsilon_{g}\rho_{g}u_{g}\big{)}=-\epsilon_{g}\
abla P+\
abla\cdot \overline{\overline{\tau_{g}}}+S_{mom}\big{(}u_{g}-u_{s}\big{)}+\epsilon_{g} \rho_{g}g+S_{gs} \tag{3}\]  
\[\frac{\partial\big{(}\epsilon_{g}\rho_{g}u_{g}\big{)}}{\partial t}+\
abla \cdot\big{(}\epsilon_{s}\rho_{s}u_{s}\big{)}=-\epsilon_{s}\
abla P+\
abla \cdot\overline{\overline{\tau_{s}}}-S_{mom}\big{(}u_{g}-u_{s}\big{)}+\epsilon _{s}\rho_{s}g+S_{sg} \tag{4}\]  
Interphase Momentum Transfer  
Drag force calculation is conducted using two approaches in this study. The calculation is based on Equations (5) and (8), utilizing the solid volume percentage in each cell of the domain. The flow resistance is calculated using the empirical pressure drop correlation proposed by [PERSON] [21] when the solid volume fraction exceeds 0.2 (\(\epsilon>0.2\), dense regions). The Gidaspow model [21] is utilized to determine the pressure drop for regions with a solid volume fraction below 0.2 (\(\epsilon<0.2\), dilute regions). The coefficients of permeability and inertial are denoted by \(\eta\) and \(\gamma\), computed in Equations (6) and (7), respectively. They are utilized to determine the \(S_{mom}\) for the dense phase in Equation (5). The gas phase flow resistance factor is added to the solid phase momentum equation with a sign that is opposite. Equations (9) and (10) determine the drag coefficient of the gas phase and the Reynolds number of the gas flow.  
\[S_{mom}=-\bigg{(}\frac{\mu_{g}}{\eta}|v_{\infty}|+\gamma\frac{1}{2}\rho_{g}v^{2 }\bigg{)}\hskip 28.452756 pt(\text{for }\varepsilon_{s}>0.2) \tag{5}\]  
\[\eta=\frac{\psi^{2}d_{\sigma_{0}}^{2}}{150}\frac{\left(1-\varepsilon_{s} \right)^{3}}{\varepsilon_{s}^{2}} \tag{6}\]  
\[\gamma=\frac{3.5}{\psi d_{\sigma_{0}}}\frac{\varepsilon_{s}}{\left(1- \varepsilon_{s}\right)^{3}} \tag{7}\]  
\[S_{mom}=\frac{3}{4}C_{d}\frac{\varepsilon_{g}\rho_{g}|u_{g}-u_{s}|\varepsilon_ {g}}{d_{s}}\varepsilon_{g}^{-2.65}\hskip 28.452756 pt(\text{for}\varepsilon_{s} \leq 0.2) \tag{8}\]  
\[C_{d}=\left\{\begin{array}{cc}\frac{24\left(1+0.15R\rho_{0}\kappa 687\right)}{Re},&Re\leq 1000\\ 0.44,&Re\geq 1000\end{array}\right. \tag{9}\]  
\[Re=\frac{\rho_{g}|u_{g}-u_{s}|\varepsilon_{g}d_{s}}{\mu_{g}} \tag{10}\]