# Design Optimization of a Hydrodynamic Brake with an Electrorheological Fluid
## 3 The Mathematical Model of the Hydrodynamic Brake
### Mathematical Model of Hydraulic Interactions  
In order to calculate the transferred torque M, the one-dimensional flow model of a hydrodynamic clutch is used [41; 42]. The torque M for the HB is obtained from the equations of the one-dimensional flow model of a hydrodynamic clutch after assuming that \(\omega_{2}=0\). It is assumed that the flow of the working fluid inside the hydrodynamic clutch is steady, continuous, and takes place along a single line of the mean stream. The radii \(\mathrm{r}_{1}\), \(\mathrm{r}_{2}\), \(\mathrm{r}_{8}\) (Figure 3) determining the position of the mean streamline in the meridional cross-section are calculated on the basis of the following formulas:  
\[\mathrm{r}_{8}=\sqrt{(\mathrm{R}^{2}+\mathrm{r}_{\mathrm{w}}{}^{2})/2} \tag{8}\]  
where \(\mathrm{r}_{\mathrm{w}}\)--inner radius of the rotors, R--outer radius of the rotors. The unit torque dM in relations to the rotation axis of the rotor is only caused by the force dF caused by a change in speed \(\mathrm{c}_{\mathrm{u}}\) of the working fluid particles. The force dF and the unit torque dM are written as:  
\[\mathrm{dF}=\frac{\mathrm{d}(\mathrm{mc}_{\mathrm{u}})}{\mathrm{dt}} \tag{9}\]  
\[\mathrm{dM}=\mathrm{dF}\ \ \mathrm{r}=\frac{\mathrm{d}(\mathrm{mc}_{\mathrm{u}} \ \mathrm{r})}{\mathrm{dt}} \tag{10}\]  
where \(\mathrm{d}/\mathrm{dt}\)--differential operator, \(\mathrm{t}\)--time, m--mass of the fluid particle, \(\mathrm{c}_{\mathrm{u}}\)--lifting speed, r--radius. The mass m of the fluid flowing over time dt can be calculated as:  
\[\mathrm{m}=\rho\ \mathrm{V}=\rho\ \mathrm{Q}\ \mathrm{dt} \tag{11}\]where V--fluid volume, \(t\)--time, Q--flow rate, \(c_{m}\)--meridional speed, A--rotor meridional cross-section. After taking into consideration the dependence (11) in relation to (10), the following is obtained:  
\[dM=\rho\,Q\,dt\,\frac{d(c_{u}\,r)}{dt}=\rho\,Q\,d(c_{u}r) \tag{12}\]  
where the product \(c_{u}r\) is treated as a single variable. Integrating the dependence (12) from point 1 to point 2 positioned on the mean streamline leads to the following equation:  
\[M=\rho\,Q\int\limits_{1}^{2}d(c_{u}\,r)=\rho\,Q[c_{u}\,r]_{1}^{2}\,=\rho Q\,(c _{u2}\,r_{2}-c_{u1}\,r_{1}) \tag{13}\]  
where \(\rho\)--density of the working fluid, Q--flow rate, cu--lifting speed, \(r\)--radius. When the blades of the hydrodynamic clutch are positioned in the planes passing through the rotor axes, then the rotor lifting speed is, respectively:  
\[c_{u1} =\omega_{1}r_{1} \tag{14}\] \[c_{u2} =\omega_{2}r_{2}\]  
where \(\omega_{1}\)--angular velocity of the pump rotor, \(\omega_{2}\)--angular velocity of the turbine rotor, r1--the inner radius of the mean streamline, r2--the outer radius of the mean streamline. For the determined operation conditions of the hydrodynamic clutch:  
\[Q=c_{m}A \tag{15}\]  
where \(c_{m}\)--meridional speed, A--rotor flow field. Considering the fact that for the HB, the angular velocity of the clutch turbine rotor \(\omega_{2}\) equals zero, on the basis of dependences (13)-(15), the following is obtained:  
\[M=\rho\,\,c_{m}A\,\,\,\omega_{1}\,\,r_{2}{}^{2} \tag{16}\]  
where \(\rho\)--working fluid density, \(c_{m}\)--meridional speed, A--rotor flow field, \(\omega_{1}\)--angular velocity of the clutch pump rotor. After taking into consideration the blade thickness, the flow field of the HB is:  
\[\mathrm{dF}=\frac{\mathrm{d}(\mathrm{mc}_{\mathrm{u}})}{\mathrm{dt}} \tag{9}\]0  
where n--number of blades, \(g_{b}\)--blade thickness. In order to calculate the meridional speed \(c_{m}\), the head rise balance is considered, described as:  
\[\mathrm{dF}=\frac{\mathrm{d}(\mathrm{mc}_{\mathrm{u}})}{\mathrm{dt}} \tag{9}\]1  
where \(\Delta p\)--pressure change, \(\rho\)--working fluid density, g--gravitational acceleration. The balance of the head rise has the following form:  
\[\mathrm{dF}=\frac{\mathrm{d}(\mathrm{mc}_{\mathrm{u}})}{\mathrm{dt}} \tag{9}\]2  
where h1--head rise of the pump, hu--impact loss, ht--friction loss.