# Design Optimization of a Hydrodynamic Brake with an Electrorheological Fluid
## 3 The Mathematical Model of the Hydrodynamic Brake
### Mathematical Model of the Electric Circuit of the Hydrodynamic Brake  
Electrorheological properties of the ER fluids are presented with the use of the following formulas:  
\[\mathrm{\tau_{0}}=\mathrm{a}\mathrm{E}^{2} \tag{27}\]  
\[\mathrm{i_{g}}=\mathrm{b}\mathrm{E}^{2} \tag{28}\]  
wherein:  
\[\mathrm{E}=\mathrm{U}/\mathrm{d} \tag{29}\]  
where \(\mathrm{\tau_{0}}\)--yield stress, \(\mathrm{i_{g}}\)--leakage current density, \(\mathrm{a}\), \(\mathrm{b}\)--coefficients, \(\mathrm{E}\)--the electric field intensity, \(\mathrm{U}\)--voltage, \(\mathrm{d}\)--average distance between blades. The dependence of \(\mathrm{\tau_{0}}\) and \(\mathrm{i_{g}}\) on the temperature \(\mathrm{\Theta}\) is not taken into account, because it is assumed that the HB operates at a constant temperature due to the use of a cooling system. The HB needs to be cooled, because all the energy supplied to the HB is converted to heat. The leakage current flowing between the electrodes is calculated on the basis of the formula:  
\[\mathrm{i_{c}}=\mathrm{S_{e}}\mathrm{i_{g}} \tag{30}\]  
wherein:  
\[\mathrm{S_{e}}=\mathrm{n}\mathrm{\pi}(\mathrm{R}-\mathrm{r_{w}})^{2}/4 \tag{31}\]  
where \(\mathrm{S_{e}}\)--surface of the electrodes, \(\mathrm{n}\)--number of blades, \(\mathrm{r_{w}}\)--inner radius of the rotors, \(\mathrm{R}\)--outer radius of the rotors. In the considered model, the increase of the high voltage \(\mathrm{U}\) applied to the blades causes a decrease in the meridional speed \(\mathrm{c_{m}}\) until it reaches zero. The meridional speed is described by Formula (25). The value of the voltage \(\mathrm{U_{h}}\) for which \(\mathrm{c_{m}}\)= 0 can be calculated on the basis of the equation:  
\[\mathrm{\omega_{1}}^{2}(\mathrm{r_{2}}^{2}-\mathrm{r_{1}}^{2})=\varsigma\tau_{0} \tag{32}\]  
where \(\mathrm{\omega_{1}}\)--angular velocity of the pump rotor, \(\mathrm{r_{1}}\)--the inner radius of the mean streamline, \(\mathrm{r_{2}}\)--the outer radius of the mean streamline, \(\varsigma\)--channel coefficient, \(\tau_{0}\)--yield stress. Having taken into consideration Formulas (27) and (29) in Equation (32), the following is obtained:  
\[\mathrm{U_{h}}=\sqrt{\mathrm{\omega_{1}}^{2}(\mathrm{r_{2}}^{2}-\mathrm{r_{1}} ^{2})\mathrm{\rho d}^{3}/(2 acl)} \tag{33}\]  
where \(\mathrm{\omega_{1}}\)--angular velocity of the pump rotor, \(\mathrm{r_{1}}\)--the inner radius of the mean streamline, \(\mathrm{r_{2}}\)--the outer radius of the mean streamline, a, \(\varsigma\)--coefficients, \(\mathrm{l}\)--length of the channel formed by the blades, \(\mathrm{\rho}\)--working fluid density, \(\mathrm{d}\)--average distance between blades. The maximal power \(\mathrm{P_{e}}\) obtained from the power supply, while stopping the flow of the ER fluid in the brake, is:  
\[\mathrm{P_{emax}}=\mathrm{U_{h}}\mathrm{i_{emax}} \tag{34}\]  
where \(\mathrm{U_{h}}\)--voltage value for which \(\mathrm{c_{m}}\)= 0, \(\mathrm{i_{emax}}\)--leakage current calculated for \(\mathrm{U=U_{h}}\). In order to assess the dynamics of the HB, the torque rise time M (after applying the high voltage to the electrodes \(\mathrm{U_{h}}\)) is taken into consideration. The dynamics of the machine drive system containing the HB is not taken into account due to the fact that the total mass moment of inertia on the input shaft depends not only on the HB structure, but also on the structure of the drive system. The drive system structure, due to the general nature of the considerations, is not specified. The torque rise time for the HB are considered on the basis of the voltage courses in a simplified electrical circuit of the HB, as shown in Figure 4 [45]. The blades of the HB, separated from each other by the ER fluid, form a capacitor whose capacity is \(\mathrm{C_{e}}\). The ER fluid provides the electrical resistance \(\mathrm{R_{e}}\) during the leakage current flow \(\mathrm{i_{c}}\).