# Experimental Research into the Efficiency of an Electromagnetic Mill
## 5 Mathematical Model of the Grinding Process  
The mathematical model for studying the dynamics of the mix of millstones and WS particles is composed of two connected subsystems. The first one is the subsystem of creating a TF, which approximates the fields of electrodynamic forces and torques affecting the ferromagnetic millstone in a rotating magnetic field described in detail in an earlier study [22]. The second involves the calculation of the trajectories of the millstone and WS particles moving in the WC of the EMM under the influence of aforementioned forces defined in the first subsystem described already in a study reported in [19]. Defining the fields of forces and torques depending on the phase angle of the vector of the resultant magnetomotive force (MMF) of the inductor winding \(\varphi\), length \(r_{m}\) and angle of inclination \(\gamma_{m}\) of the position vector of the center of the mass of each millstone in the working chamber was performed using a two-dimensional finite element method analysis. The force acting on the millstones was calculated using the rule of calculating work with a possible 'virtual' movement (a virtual work method [23]). The derivative of magnetic field energy in a possible 'virtual' movement in any direction was calculated. Those calculations were performed for the layer of finite elements at the border separating the millstones from the environment. The vector of this force is equal to  
\[\overline{F_{s}}=\int\limits_{V}\overline{B}^{T}\frac{\partial\overline{H}}{ \partial s}dV+\int\limits_{V}\frac{\partial}{\partial s}\bigg{(}\int \overline{B}^{T}d\overline{H}\bigg{)}dV, \tag{13}\]  
Figure 4: Functions of ground WS mass regression in relation to: (**a**) grinding duration; (**b**) millstone diameter; (**c**) grinding medium mass; (**d**) WS mass loaded into WC. where \(\overline{F_{s}}\)--force acting in direction s; \(\overline{B}\)--magnetic field induction; \(\frac{\partial\overline{H}}{\partial s}\)--derivative of field intensity over movement s; \(s\)--virtual movement in nodal coordinates (subsequently for x, y); \(V\)--finite element volume. For a two-dimensional task, the electromagnetic torque around axis \(+\)z was defined as:  
\[\overline{M}_{e}=\overline{z}\frac{1}{\mu_{0}}\int\limits_{\
u}\bar{\tau}\times \left[\frac{1}{2}\left(\overline{B}\cdot\overline{B}\right)\
abla s-\left( \overline{B}\cdot\
abla s\right)\overline{B}\right]dV, \tag{14}\]  
Based on (13), (14) the TF with three independent variables \(\varphi,\gamma_{m},r_{m}\) were generated, for modulus of force \(F_{m}(\varphi,\gamma_{m},r_{m})\), acting upon a ferromagnetic millstone in a magnetic field, direction of that force \(\alpha_{F}(\varphi,\gamma_{m},r_{m})\) and torque \(M_{m}(\varphi,\gamma_{m},r_{m})\). The force is applied to the center of mass of the millstone. The torque acts upon the millstone relative to the axis passing through its center of mass perpendicular to the calculation plane. A graph representing those relations in relation to variable \(\gamma_{m}\) for millstones with a diameter of 4 mm is presented in Figure 5. The dimensions of forces and torques in these graphs are expressed in [N/m] and [N/m], which results from the two-dimensional formulation of the field problem solved by FEM analysis. Figure 5a presents the courses representing the variability of the dependence between the force vector modulus \(F_{m}\) and the angular position of the grinding medium for different values of the phase MMF \(\varphi\). The courses show three maxima for the millstones angles close to \(0^{\circ}\), \(120^{\circ}\) and \(240^{\circ}\). The first maximum for the angle of the millstones along the line of the MMF (\(\varphi=0^{\circ}\)) generates the largest value of the force modulus \(F_{m}\), whereas the smallest value \(F_{m}\) is generated by the millstones inclined at an angle of \(216^{\circ}\) in relation to the MMF vector.