## 3 Results and Discussion
### Mathematical Description of Pressure Self-Regulation Mechanism for Press Equipment
Let us determine the forces acting on any point of the pressure self-regulation mechanism as a function of the variation of the gap between the gap nozzle and the screw coil depending on the operation of the press equipment. Proceeding from Equation (6), the following equations are obtained:  
\[F_{E}=k\cdot\delta_{0},\ F_{1}=k\cdot\delta_{1} \tag{7}\]  
Figure 12: Scheme of forces of the pressure self-regulation mechanism. 1—pressing screw; 2—serrated nozzle; \(\delta_{i}\)—distance between gap nozzle and screw coil, m; \(L_{i}\)—distance of displacement of the pressure screw in the longitudinal axis, m; \(l_{i}\)—distance at absolute spring deformation, m, \(F_{P}\)—pressure force, N; \(F_{S}\)—shear force, N; \(F_{E}\)—elastic force, N; \(P\)—pressing pressure, Pa. It follows that if we write down the following expressions, then,  
\[\frac{F_{E}}{F_{1}}=\frac{k}{k}=\frac{\delta_{0}}{\delta_{1}} \tag{8}\]  
Taking into account the invariance of the spring stiffness, i.e.,  
\[\frac{F_{E}}{F_{1}}=\frac{\delta_{0}}{\delta_{1}} \tag{9}\]  
Consequently, let us determine the force on which the pressure self-regulation mechanism acts as a function of the variation of the gap between the gape nozzle and the screw coil:  
\[\Delta S=\pi\Big{(}\Delta R^{2}-\Delta r^{2}\Big{)},\mathrm{m}^{2} \tag{3}\]0  
Using such equal relationships, the forces acting on any point of the pressure self-regulation mechanism can be determined, even through the distance \(L_{i}\) of displacement of the pressure screw along the longitudinal axis at the absolute spring deformation. Let us describe the change in the density and volume of the product in the channel of the pressing screw due to the change in the action of displacement forces applied to the spring of the pressure control mechanism. If, based on the Equation (5), we write down the displacement forces in two cases associated with the movement of the press screw,  
\[ \begin{array}{l}F_{1}=\rho_{1}\cdot g\cdot V_{1}\\ F_{2}=\rho_{2}\cdot g\cdot V_{2}\end{array} \biggr{\}},\mathrm{N} \tag{11}\]  
If this is the case, it can be seen from the schematic in Figure 2 that the total displacement force is equal to the sum of these forces:  
\[F_{3}=F_{1}+F_{2}\rho_{1}\cdot V_{1}=\rho_{2}\cdot V_{2}\]  
On this basis, let us find the volumetric change, \(V_{2}\) :  
\[V_{2}=\frac{\rho_{1}\cdot V_{1}}{\rho_{2}}, \tag{12}\]  
Given that the total volume is equal to the sum of the first volume and the second volume, we obtain the following equations:  
\[V_{3}=V_{1}+\frac{\rho_{1}\cdot V_{1}}{\rho_{2}}\Rightarrow V_{3}=V_{1}\biggl{(} 1+\frac{\rho_{1}}{\rho_{2}}\biggr{)} \tag{13}\]  
It follows that the volume change in the screw ground coil would be as follows:  
\[V_{3}=V_{1}\cdot\frac{\rho_{1}\cdot\rho_{2}}{\rho_{2}} \tag{14}\]  
Hence, we determine the volumetric change \(V_{1}\):  
\[V_{1}=\frac{\rho_{1}\cdot V_{3}}{(\rho_{1}+\rho_{2})}, \tag{15}\]  
Using Equations (1)-(4) respectively, it is possible to determine the density of the product in the channel of the pressing screw due to the change in the action of the displacement forces applied to the spring of the pressure regulating mechanism. We can write the value of the forces in the pressing operation by using the following conditions:  
\[F_{P}\geq F_{S},\ F_{S}=F_{E} \tag{16}\]  
For a better pressing process, the pressure force, \(F_{P}\), must be greater than or equal to the shear force, \(F_{S}\). If this condition is not met, the pressure required for the pressing process will not be enough. The shear force, \(F_{S}\), and the counter-elastic force, \(F_{E}\), are then equal to each other. Because the force with which the spring is pressed, the spring force is counteracted again with the same force. Consequently, we must select a spring that can maintain the pressure force required for the optimum pressing pressure. In case the pressure force exceeds the optimum pressure limit required for pressing, it should be released by shear force.