# Design Optimization of a Hydrodynamic Brake with an Electrorheological Fluid
## 3 The Mathematical Model of the Hydrodynamic Brake  
### The Mathematical Model of the Hydrodynamic Brake Geometry  
The parameters describing the geometry of the brake rotors are as follows: the outer radius of the rotors R, the inner radius of the rotors \(\mathrm{r_{w}}\), and the number of blades n. The radius R is larger than the radius \(\mathrm{r_{w}}\). The blade number n is even, so that in each of the channels formed by the blades, an electric field influences the ER fluid. Moreover, the blades cannot be set too close to each other due to the possibility of electric breakdown occurrence. Thus, the number of the blades should meet the following condition:  
\[\mathrm{n}<\frac{2\,\pi\,\mathrm{r_{w}}}{g_{\mathrm{b}}+h_{\mathrm{min}}} \tag{1}\]  
where \(\mathrm{g_{b}}\)--blade thickness, \(\mathrm{h_{min}}\)--the smallest gap width for which an electric breakdown occurs. The average blade distance \(\mathrm{d}\) is calculated from the formula:  
\[\mathrm{d}=0.5(\mathrm{d_{1}}+\mathrm{d_{2}}) \tag{2}\]  
wherein:  
\[\mathrm{d_{1}}=\left(2\pi\mathrm{r_{w}}/\mathrm{n}\right)-\mathrm{g_{b}} \tag{3}\]  
Figure 3: PHB with the ER fluid: (**a**)—design solution: 1—pump, 2—turbine, 3—casing, 4—slip rings supplying the high voltage, 5—sealing; (**b**)—view of the pump rotor after tests: 1—blade, 2—rotor casing.  
Figure 2: HB geometry for calculations (refer to Table 1).  
where \(\mathrm{d}_{1}\)--the smallest distance between blades, \(\mathrm{r}_{\mathrm{w}}\)--inner radius of the rotors, R--outer radius of the rotors, \(\mathrm{d}_{2}\)--the largest distance between blades, n--number of blades.  
In order to calculate the rotor volume \(\mathrm{V}_{\mathrm{t}}\), the following formula is used:  
\[\mathrm{V}_{\mathrm{t}}=2\pi^{2}\biggl{(}\frac{\mathrm{R}-\mathrm{r}_{\mathrm{w }}}{2}\biggr{)}^{2}\cdot\frac{\mathrm{R}+\mathrm{r}_{\mathrm{w}}}{2} \tag{4}\]  
where R--outer radius of the rotors, \(\mathrm{r}_{\mathrm{w}}\)--inner radius of the rotors.  
The blades volume \(\mathrm{V}_{\mathrm{b}}\) is calculated on the basis of the following formula:  
\[\mathrm{V}_{\mathrm{b}}=\mathrm{n}\cdot\pi\biggl{(}\frac{\mathrm{R}-\mathrm{r} _{\mathrm{w}}}{2}\biggr{)}^{2}\cdot\mathrm{g}_{\mathrm{b}} \tag{5}\]  
where n--number of blades, \(\mathrm{r}_{\mathrm{w}}\)--inner radius of the rotors, R--outer radius of the rotors, \(\mathrm{g}_{\mathrm{b}}\)--blade thickness.  
The volume of the fluid \(\mathrm{V}_{\mathrm{ER}}\) contained in the HB is then:  
\[\mathrm{V}_{\mathrm{ER}}=\mathrm{V}_{\mathrm{t}}-\mathrm{V}_{\mathrm{b}} \tag{6}\]  
where \(\mathrm{V}_{\mathrm{t}}\)--the rotor volume, \(\mathrm{V}_{\mathrm{b}}\)--the blades volume. With the use of the Formulas (4)-(6), it is possible to formulate the weight \(\mathrm{G}\) of the brake as follows:  
\[\mathrm{G}=\mathrm{V}_{\mathrm{ER}}\rho+\mathrm{V}_{\mathrm{b}}\rho_{1}+\alpha \mathrm{V}_{\mathrm{t}}\rho_{2} \tag{7}\]  
where \(\rho\)--density of the working fluid, \(\rho_{1}\)--density of the blade material, \(\rho_{2}\)--density of the casing material, \(\alpha\)--coefficient describing the casing density in relation to the rotor volume.