## 2 Materials and Methods
### Thermal Analysis  
The rapid increase in the electrical power consumed by the MR clutch (equal to copper losses) with the increased number of discs is the reason why the recommendation for the correct selection of the optimal variant of the device must take into account thermal calculations and the determination of the maximum operating temperature in a steady state. The maximum operating temperature of the clutch working region is important for the assessment of the clutch life, because high temperature may accelerate the aging of the  
Figure 6: (a) Number of turns \(z\) and (b) average length of the turns \(l_{law}\) for individual excitation coils. Figure 7: Drop in internal radius of excitation coil and in its thickness: (a) internal radius of excitation coil \(r_{ci}\) vs. number of discs N; (b) thickness of excitation coil (\(r_{ce}-r_{ci}\)) vs. number of discs N.  
magnetorheological fluid and even-after exceeding the permissible temperature can lead to the degradation of its properties [14]. Heat calculations are carried out according to the power flow diagram presented in Figure 8b. The only heat source taken into account is copper loss: the excitation winding is located in the immediate vicinity of the working region containing the MR fluid; hence, the loss in copper most strongly affects its fluid temperature. Thermal field simulations in a steady state were performed for all 12 variants **V1-V12** in the AGROS2D program [15] (computer program in which magnetic calculations were previously carried out). Table 3 summarized values of the coefficients related to the heating process for the materials used in the clutch construction and for the air. It is worth reminding that the discs of primary and secondary members, cylinder yokes and cover yokes are made of magnetic steel and the shaft, bearings and housing are made of non-magnetic steel. The results of the thermal calculations in the form of spatial temperature distributions for selected representative variants **V1**, **V4**, **V9** and **V12** are presented in Figure 9.  
The next Figure 10 shows an exemplary temperature drop along the axis AB which runs through the point with the highest temperature in the clutch (center of the coil) and through the point with the highest temperature in working region \(\vartheta_{max}\). This is the highest temperature possible for the MR fluid. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline - & **Material** & **\(\mathbf{\kappa}\) [1/K]** & \(\mathbf{\lambda}\) [W/m\(\cdot\)K] & \(\mathbf{C_{h}}\) [W/m\({}^{2}\cdot\)K] & \(\mathbf{c}\) [J/kg\(\cdot\)K] & \(\mathbf{\rho}\) [kg/m\({}^{3}\)] \\ \hline
1 & air & \(3.43\times 10^{-3}\) & \(\lambda_{1}=0.0257\) & \(C_{h1}=10\) & 1005 & 1.205 \\
2 & non-magnetic steel & \(1.44\times 10^{-5}\) & \(\lambda_{2}=16.26\) & - & 502.1 & 8027.2 \\
3 & copper & \(1.66\times 10^{-5}\) & \(\lambda_{3}=372\) & - & 380 & 8800 \\
4 & sealing plastic & \(1.10\times 10^{-4}\) & \(\lambda_{4}=0.25\) & - & 1260 & 1014 \\
5 & magnetic steel & \(1.34\times 10^{-5}\) & \(\lambda_{5}=54\) & - & 465 & 7833 \\
6 & MR fluid & \(6.00\times 10^{-4}\) & \(\lambda_{6}=0.8\) & - & 680 & 3640 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Thermal coefficients for constructional materials and air (where \(\alpha\)—thermal expansion coefficient; \(\lambda\)—thermal conductivity; \(C_{h}\)—heat convection coefficient; \(c\)—heat capacity; \(\rho\)—mass density). Figure 8: Power-flow diagram (**a**) in steady continuous-duty state with no slippage: (**b**) case study considered in the paper (mechanical losses \(P_{fr11}\) and \(P_{fr2}\) are neglected; heat is transferred mainly to surrounding air and because of that it slightly influences temperature in clutch working regions including MR fluid).