# Investigation of Roller Press Surface and Stud Based on FEM Simulation
## 2 Mathematical Model of Roller Press
### Mathematical Model of Roller Press  
Figure 3 shows the working diagram of the roller press. The material is loaded from the gap between the two rollers through the funnel and is in contact with the roller press on the horizontal plane \(A_{1}A_{2}\). The gap width here is \(b\), and the relative density of the material is \(\delta_{0}\). At this point, the material starts to be driven by the surface friction of the roller press. The material is accelerated in the vertical direction, when the material reaches plane \(B_{1}B_{2}\) places, gap width is \(h_{0}\). Assume that the material speed here is equal to the speed of the rolls surface. The material starts to be compressed by the roller press until it reaches \(C_{1}C_{2}\) circumferential speed at point 2 passes \(C_{1}C_{2}\) plane, where the gap is \(s\), and the relative density \(\delta_{s}\) reaches the maximum value. Here, the compressed material layer is rolled at \(C_{1}C_{2}\) circumferential speed at point 2 passes \(C_{1}C_{2}\) plane, and then bounce in the horizontal direction. When the compressed material layer passes through the \(D_{1}D_{2}\) plane, the relative density becomes \(\delta_{r}\), and the flakes leave the roller press and become products. Figure 3: Working diagram of roller press. Figure 2: Schematic diagram of material curve measuring device. The whole process can be divided into three zones [2] according to the dynamic characteristics of materials in the roller press. (a) Acceleration zone: \(A_{1}A_{2}\)-\(B_{1}B_{2}\); (b) Compression zone: \(B_{1}B_{2}\)-\(C_{1}C_{2}\); (c) Rebound zone: \(C_{1}C_{2}\)-\(D_{1}D_{2}\); In order to facilitate theoretical analysis, the following assumptions are made:  
(1) The relative density of materials in the acceleration zone is constant \(\delta_{0}\);  
(2) There is no relative sliding between the material and the surface of the roller press in the compression zone, and the material deformation only occurs in the horizontal direction;  
(3) The end of the compression zone is at \(C_{1}C_{2}\). The gap width here is the smallest;  
(4) The \(p\)-\(\delta\) characteristic of the material in the middle of the roller gap (\(Z=0\)) is the same as that of the uniaxial compression. The surface pressure of the roller press in the \(Z\) direction can be described by the following equation. \[p(\alpha,\lambda)=p_{m}(\alpha)\cdot\big{(}1-(2\cdot|\lambda|)^{m}\big{)} \lambda=\frac{Z}{L} \tag{3}\]  
where:  
\(p_{m}(\alpha)\)--The surface pressure of the roller press corresponding to the nip angle (\(\alpha\)) of the roller press;  
\(Z\)--The distance from a point on the surface of the roller press to the middle of the roller press;  
\(m\)--Pressure axial distribution coefficient, \(m\geq 1\). (1) Acceleration zone  
The mass flow rate in any section from plane \(A_{1}A_{2}\) to plane \(B_{1}B_{2}\) in the acceleration zone is constant. In the acceleration zone, the mass flow rate in any section from the plane to the plane is constant, and the relative density of the material is also constant, set to \(\delta_{0}\). The pressure on the surface of the roller press is mainly caused by the gravity of the material. compared with the  
(3)  
The mass of any horizontal layer in the rebound area should be equal. The relative density \(\delta\) corresponding to any central angle \(\alpha\) in this region is:  
\[\delta_{\alpha}=\delta_{s}\cdot s/\left[s+D(1-\cos\alpha)\right] \tag{8}\]  
The relative density \(\delta_{r}\) of the material at the end of the rebound is:  
\[\delta_{r}=k_{e}\cdot\delta_{s} \tag{9}\]  
In the formula, \(k_{e}\) is the rebound rate of the material. According to hypothesis 4 and uniaxial rebound equation, the horizontal specific pressure in the middle region of the roller press is:  
\[p_{m}(\alpha)=p_{c}\left[\ln(\frac{1-\delta_{0}}{1-\delta_{\alpha}})\right]^{ 1/n}\cdot\left(\frac{\delta_{\alpha}-\delta_{r}}{\delta_{s}-\delta_{r}}\right)^ {k}0\geq\alpha\geq-r \tag{10}\]  
In the formula, \(r=\cos^{-1}\left[1-\frac{(\delta_{s}-\delta_{r})\cdot\mathrm{s}}{D}\right]\).