# 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
matter; \(\zeta\) is the surface free energy of the particulate matter; \(E^{*}\) is the equivalent Young's modulus; and \(y\) is the critical elastic stress, which can be expressed as particulate matter yield stress \(Y\) as \(y=3.2Y\). The relationship between the equivalent [PERSON]'s modulus \(E^{*}\) and the [PERSON]'s ratio \(\
u_{P}\) of the particulate matter, the elastic modulus \(E_{P}\), and the Poisson's ratio \(\
u_{1}\) and the elastic modulus \(E_{1}\) of the dust accumulation wall is as follows [37]:  
\[\frac{1}{E^{*}}=\frac{1-\
u_{P}{}^{2}}{E_{P}}+\frac{1-\
u_{1}{}^{2}}{E_{1}} \tag{12}\]  
When only the elastic deformation of the particulate matter occurs, the sinking scale of the center of the particulate matter satisfies \(h<h_{el}\), and the particulate matter conforms to the Hertz elastic contact theory. The pressure in the contact surface is elliptically distributed, and the maximum stress \(p_{\text{max}}\) appears in the center of the contact surface corresponding to the center of the particulate matter. At this time, the contact load \(F\) of the particulate matter on the contact area, the deformation (elasticity) contact radius \(r\), and the sinking scale of the center of the particulate matter \(h\) are as follows [38]:  
\[F=\frac{2}{3}\pi r^{2}p_{\text{max}} \tag{13}\]  
\[r=\left(\frac{R^{*}F}{\frac{4}{4}E^{*}}\right)^{\frac{1}{3}} \tag{14}\]  
\[h=\frac{r^{2}}{R^{*}}=\frac{F^{2/3}}{\left(R^{*}\right)^{1/3}\left(\frac{4}{ 3}E^{*}\right)^{2/3}} \tag{15}\]  
where \(R^{*}\) is the effective contact radius of curvature, and the relationship between the radius of curvature of the particulate matter \(d_{P}\) and the radius of curvature of the deposited wall surface \(R_{1}\) of the particulate matter can be expressed as [37]:  
\[\frac{1}{R^{*}}=\frac{1}{d_{P}}+\frac{1}{R_{1}} \tag{16}\]  
When the particulate matter is transformed from elastic deformation to the critical elastic state of plastic deformation, the critical sinking scale \(h_{el}\), the critical elastic contact radius \(r_{el}\), and the critical contact load \(F_{el}\) of the particle center can be expressed as follows [38]:  
\[F_{el}=\left(\frac{2}{3}\pi\right)^{3}\frac{R^{*2}}{\left(\frac{4}{3}E^{*} \right)^{2}}y^{3} \tag{17}\]  
\[r_{el}=\frac{\pi R^{*}}{2E^{*}}y \tag{18}\]  
\[h_{el}=\left(\frac{2}{3}\pi\right)^{2}\frac{R^{*}}{\left(\frac{4}{3}E^{*} \right)^{2}}y^{2} \tag{19}\]  
When the particle is in elastic-plastic deformation, the sinking scale of the particle center satisfies \(h>h_{el}\), and the contact radius of deformation (elastic-plastic) satisfies \(r>r_{el}\). The deformation of particles accords with [PERSON]'s elastic-plastic cavity theory. With the increase in the subsidence scale of the center point, plastic deformation occurs in the middle part of the contact region between the particles and wall surface, the plastic deformation region expands outward with the increase in the subsidence scale of the center point, and elastic deformation occurs in the surrounding annular region. The stress in the plastic deformation region is uniformly distributed, and the magnitude is the critical elastic load \(y\), while the stress in the surrounding annular elastic deformation region is generally considered to conform to the Hertz contact theory. The corresponding particle contact radius \(r\) and contact load \(F\) can be expressed as [38]:  
Figure 2:0  
Figure 2:1  
where \(r_{P}\) is the radius of the contact region of the particle for plastic deformation, and the radius of the critical elastic region \(r_{el}\) is equivalent to the converted radius of the annular elastic deformation region around the plastic deformation contact region. When the plastic deformation stage of the particles is over, it enters the elastic unloading stage of the contact load, which is still in line with the Hertz elastic contact theory. At this time, the particles overcome the adhesion of the deposited wall surface and work until desorbed.