# A Study of Magnetic Mill Productivity
## 6 Indicators of Force Interaction of Millstones in a Rotating Magnetic Field  
In order to find out the factors affecting the force interaction of millstones with each other, with the WS and with the walls of the WC, an analysis of the magnetic field inside the WC was conducted, and a mathematical model was created to calculate the quantitative indicators of this interaction. It has been established that the forces of magnetic origin acting on a separate millstone depend on its position in the WC, the intensity of the magnetic field, the material from which the millstone is made, the level of the filling of the WC with millstones, etc. Let us attach a fixed polar coordinate system to an arbitrary cross-section of the working chamber that allows setting the coordinates of the position of the center of mill mass using the length of the radius vector \(r_{m}\) and the angle of its inclination \(\gamma_{m}\) to the axis \(+r\) (Figure 2). The position of the millstone will set the angle of inclination of its longitudinal axis \(\alpha_{m}\) to the direction of the radius vector. Hereinafter, cylindrical millstones were considered, the dimensions of which were designated as \(d_{m}\) and \(l_{m}\). Figure 2: The coordinate system of the model and a schematic of force application. A number of mathematical experiments were conducted to establish the functional interrelationships between forces, moments and the aforementioned indicators. The inductor with PM was selected; the volume of the WC was 300 cm\({}^{3}\). The main dimensions of the mill are as follows: the calculated length is 80 mm; the diameter of the WC is 120 mm. The pole tips of the inductor are made of electrotechnical steel M600-50A; permanent magnets of the N38 brand are made of rare earth powder material based on neodymium, iron and boron (residual induction 1.2 T, coercive force 800 kA/m), and the millstone is made of carbon steel A283/C. The average turnover value of the magnetic induction in the WC is 0.052 T. The sizes of the millstones used in the experiment were \(d_{m}=1.5\) mm and \(l_{m}=12\) mm. The shape of the cross-section and the level of discretization of the computational domain are shown in Figure 3.  
The dependencies of the force applied to the center of millstone mass and the torque acting on a separate millstone relative to the coordinates determining its position inside the working chamber (\(r_{m}\), \(\gamma_{m}\)), the angle of rotation of the millstone \(\alpha_{m}\) and as the angle of rotation of axis \(\gamma_{i}\)\(+d\) of the magnetic system of the inductor were determined. To reduce the number of scalar arguments, the relative location of the angular position of the center of the millstone and the axis \(+d\) was determined by the difference  
\[\gamma=\gamma_{i}-\gamma_{m}. \tag{5}\]  
We shall hereinafter refer to the angle \(\gamma\) as the angular displacement between the inductor and a millstone or, in short, the angular displacement. It is obvious that in real conditions each millstone is surrounded by others, which, in turn, affects the distribution of the magnetic field, the magnitude and direction of the force \(\overline{F}_{m}\). To a certain extent, we have taken this influence into account by placing'surroundings' of 8 similar millstones located around the millstone under study. In our opinion, this will increase the adequacy of the obtained results [12]. Figure 4 shows the vector diagram of the force \(\overline{F}_{m}\) depending on the angular position of the mill relative to the axis \(+d\), which at the qualitative level clearly demonstrates the adequacy of the obtained results. On the basis of the results obtained using the above-mentioned model, force hodographs \(\overline{F}_{m}\), corresponding to a full rotation of the inductor, were constructed (Figure 5). It should be noted that the three dependencies shown in Figure 5a refer to various conditions for conducting a mathematical experiment.