# 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
Let \(k_{t}\approx 8G^{*}a\), where \(G^{*}\) is the effective shear modulus, which can be expressed as:  
Figure 2:9  
The contact radius \(r\) when no slippage occurs can be expressed as:  
\[\frac{1}{E^{*}}=\frac{1-\
u_{P}{}^{2}}{E_{P}}+\frac{1-\
u_{1}{}^{2}}{E_{1}} \tag{12}\]0  
By introducing the equivalent coefficient of the particulate contact radius \(\beta=a/r\), the tangential critical collision angle \(\theta_{cr}\) of the particulate matter can be described:  
\[\frac{1}{E^{*}}=\frac{1-\
u_{P}{}^{2}}{E_{P}}+\frac{1-\
u_{1}{}^{2}}{E_{1}} \tag{12}\]1  
where \(f\) is the coefficient of friction of the wall surface. Based on the above analysis, after the collision between the particulate matter and the particulate matter deposition wall, there will only be two kinds of particulate matter-particulate matter deposition wall relationship (i.e., deposition or desorbed). In order to facilitate subsequent analysis, the state coefficient PP is introduced to describe the relationship between the particulate matter and the particulate matter deposition wall at this time. If the particulate matter collides with the deposited wall of the particulate matter and is attached to the wall, the particulate matter is deposited on the surface (i.e., \(v_{r}=0\)), and \(\Upsilon_{P}\) takes 0, indicating that the relationship between the particulate matter and the deposited wall of the particulate matter is deposited at this time; if the particulate matter collides with the deposited wall of the particulate matter, it is not deposited by the wall. If the surface is attached, the particulate matter is desorbed to the surface (i.e., \(v_{r}\
eq 0\)). At this time \(\Upsilon_{P}\) takes 1, which means that the relationship between the particulate matter and the deposited wall of the particulate matter is desorbed. The established two-parameter determination criteria for desorbed particulate matter deposition based on the critical velocity \(v_{intl}\) of normal-phase deposition and the tangential critical collision angle \(\theta_{cr}\) can be described as:  
\[\Upsilon_{P}=\left\{ \begin{array}{cc}0&,\theta_{i}\leq\theta_{cr}\wedge v _{in}\leq v_{intl}\\ 1&,\theta_{i}>\theta_{cr}\vee(\theta_{i}\leq\theta_{cr}\wedge v_{in}>v_{intl} )\end{array} \right. \tag{32}\]If the particle collision angle \(\theta_{i}\) is less than or equal to the tangential critical collision angle \(\theta_{cr}\) (\(\theta_{i}\leq\theta_{cr}\)) and the normal incident velocity \(v_{in}\) of the particle is less than or equal to the normal critical deposition velocity \(v_{inl}\) (\(v_{in}\leq v_{inl}\)), the particle will show a wall deposition state (\(\Upsilon_{P}=0\));  
If the collision angle \(\theta_{i}\) of the particulate matter is greater than or equal to the tangential critical collision angle \(\theta_{cr}\) (\(\theta_{i}>\theta_{cr}\)), the particulate matter will show a wall desorbed state (\(\Upsilon_{P}=1\)). If the normal incidence velocity \(v_{in}\) is greater than the normal critical deposition velocity \(v_{inl}\) (\(v_{in}>v_{inl}\)), the desorption of particles will rebound from the wall surface and return to the continuous-phase fluid. Otherwise, the desorption of particles will be parallel to the wall surface;  
When the collision angle \(\theta_{i}\) is less than or equal to the tangential critical collision angle \(\theta_{cr}\) (\(\theta_{i}\leq\theta_{cr}\)) and the normal incidence velocity \(v_{in}\) is greater than the normal critical deposition velocity \(v_{inl}\) (\(v_{in}>v_{inl}\)), the particles still exhibit a wall desorption state (\(\Upsilon_{P}=1\)), and the desorption behavior of particles is that they rebound from the wall and return to the continuous-phase fluid.