# Eulerian-Eularian Model for Agglomeration Behavior of Combusted Iron Particles
## 2 Model Description
### Cohesive Force Model  
The particle agglomeration model requires a model of the chosen interparticle cohesive force and adding it to the momentum equation in addition to the other main forces. The contact force can be an electrostatic force, a liquid bridge force, etc. In this model, micron-sized iron powder is combusted for different temperatures. Therefore, solid bridge force is recognized as the major contributor to the particle agglomeration in the DRI method in a fluidized bed. When two or more particles are bonded together at a temperature lower than that particle's melting point, the solid bridge force emerges among those agglomerated particles. The DRI process is carried out at elevated temperatures of more than 500 \({}^{\circ}\)C. These temperatures cause the outer layer of the iron oxide particles to become adhesive and malleable. Hence, the environment favorable for the particle sintering is created in the DRI method when in higher temperatures. According to [PERSON] et al., surface diffusion is driven by the atoms of the particles moving to the junction of the solid bridge [23]. [PERSON] et al. explain the surface tension mechanism in detail based on the balance between the buoyancy force of the bubbles and the breaking force of the solid bridge [24]. Equation (11) represents the solid bridge force between two particles.  
\[F_{\text{sb}}=\pi\sigma x^{2}, \tag{11}\]  
where tensile strength and the neck radius are represented by \(\sigma\) and \(x\), respectively. The neck radius is modeled by using [PERSON]'s surface diffusion model, which is explained in Equation (12) [25].  
\[x=\bigg{(}\frac{56\gamma\delta^{4}}{KT}D_{s}a^{3}t\bigg{)}^{1/7}, \tag{12}\]  
where the variables \(\gamma\), \(\delta\), \(K\), \(T\), \(a\), and \(t\) stand for surface tension, lattice constant, Boltzmann constant, temperature, curvature radius, and contact time, respectively. The surface diffusion coefficient (\(D_{s}\)) in Equation (13) is determined by the frequency factor (\(D_{o,s}\)) and activation energy (\(E_{s}\)).  
\[D_{s}=D_{o,s}\text{exp}\bigg{(}\frac{-E_{s}}{RT}\bigg{)} \tag{13}\]Moreover, a general equation for solid bridge force modeling can be derived using Equation (14) [12; 26].  
\[F_{\text{sb}}=\text{C}\pi\sigma\bigg{(}\frac{56\gamma\delta^{4}}{KT}\,D_{s}r_{p} ^{3}t\bigg{)}^{2/7}, \tag{14}\]  
where \(r_{p}\) is the minimum radius value of two particles in contact.  
In this model, solid bridge force is modeled in three different impact models by varying the C value in Equation (14). The model for value C = 0.417 is called the three-micro contact-point model. Similarly, C = 1 represents the smooth surface model, and C = 1.251 refers to the nine-microcontact-point model. This classification for the model for different intensities of sintering force is similar to the CFD-DEM study conducted by [PERSON] et al. [26].  
Another key parameter in Equation (14) is the contact time, \(t\). Initially, the contact time is zero. Then, if the consecutive cells have similar velocities within the (+/\(-\)5%) range, solid mass in those cells is said to be in contact, and the contact time is added to the stored contact time of the previous time step. If the velocities fall out of the boundary, the contact time remains the same.