# Two-Parameter Probabilistic Model and Experimental Research on Micron Particle Deposition
## Particle Deposition Model
### Two-Parameter Particle Probability Deposition Model
#### 2.3.1 Two-Parameter Determination of Particle Deposition  
As shown in Figure 1, \(\theta_{i}\) represents the angle between the particle velocity and the normal direction of the wall surface. The particle velocity \(v_{P}\) is decomposed into \(v_{in}\) and \(v_{it}\). \(v_{in}\) is the normal incident velocity perpendicular to the wall surface, and \(v_{it}\) is the tangential incident velocity parallel to the wall surface. Through the energy conservation situation in the collision deformation process of particulate matter, the two dimensions of normal deposition and tangential deposition are used to obtain the determination criteria for the deposition desorbed. Among them, the degree of collision deformation and the mutual transformation of the kinetic energy and elastic potential energy of the particulate matter during the collision are mainly affected by the normal collision speed, so it needs to be characterized by the critical speed of facies deposition. The friction and sliding between the particulate matter and the sedimentary wall are mainly affected by the tangential collision speed, so it is necessary to use the tangential critical speed. Measured from the angle of collision, the two together determine whether particulate matter is deposited. When the particulate matter meets tdesorbed conditions during the collision process, the particulate matter will desorb to the sedimentary wall at the speed of rebound \(v_{r}\); otherwise, it will stick to the sedimentary wall to form a particulate matter sedimentary layer. Figure 2a shows the elastic deformation of the particulate matter during the collision, Figure 2b shows the critical elastic deformation, and Figure 2c shows the elastic-plastic deformation. The degree of deformation is determined by the center sinking scale \(h\) of the particulate matter, the radius of the deformed contact surface \(r\), and the contact load \(F\). In the event of a critical elastic deformation collision, the corresponding deformation contact radius and contact load and the initial degree of center sinking are characterized by \(r_{el}\), \(F_{el}\), and \(h_{el}\) respectively. In the collision process of the normal dimension, according to the deformation process of particle collision, the normal energy conservation equation can be used to describe the mutual conversion process of the kinetic energy-elastic potential energy-plastic potential energy of the particle. Taking into account the two possibilities of the elastic deformation or elastic-plastic deformation of particulate matter during the collision, it can be used to describe the introduction of the critical elastic deformation characterization parameters \(r_{el}\), \(F_{el}\), and \(h_{el}\) described earlier. The energy conservation equation for the collision deformation process of particulate matter is as follows [36]:  
\[\left\{ \begin{matrix}Q_{in}+Q_{a}(F)=Q_{el}+Q_{e}(F)+Q_{\omega o}(F)\\ Q_{in}=\frac{1}{2}mv_{in}{}^{2}\\ Q_{a}(F)=\zeta\pi\Big{(}r_{el}a^{2}+\frac{F-F_{el}}{\pi y}\Big{)}\\ Q_{el}=\frac{\pi^{5}d_{p}{}^{2}y^{5}}{4800\varepsilon^{4}}\\ Q_{e}(F)=\frac{1}{2}h_{el}(F-F_{el})\\ Q_{\omega o}(F)=\frac{(F-F_{el})^{2}}{4\pi d_{p}y}\end{matrix} \right. \tag{11}\]  
where \(Q_{in}\) is the normal kinetic energy of the particulate matter; \(Q_{a}\) is the surface adhesion energy of the particulate matter deposition wall facing the particulate matter; \(Q_{el}\) is the critical elastic energy; \(Q_{e}\) is the elastic energy stored when the particulate matter undergoes plastic deformation; \(Q_{eo}\) is the energy lost by the plastic deformation of the particulate  
Figure 1: Particle–wall collision escape process. (**a**) Particulate deposition; (**b**) particle rebound desorption. Figure 2: Particle–wall collision deformation process. (**a**) Elastic deformation of particles; (**b**) elastic–plastic deformation of particles occurs; (**c**) plastic deformation of particles.