## 5 Calculation of Losses in FESS
### Mechanical Losses  
Mechanical loss in a FESS is mainly due to windage and bearing friction losses. The former is due to the friction between the rotating flywheel rotor and surrounding gas or air, depending on the type of enclosure. Windage loss is proportional to the cube of speed, and hence, it becomes significant when the rotor speed is increased, as shown in Equation (13) [60]:  
\[P_{W}=\frac{1}{64}C_{M}\rho\omega^{3}\left({D_{ro}}^{5}-{D_{s}}^{5}\right) \tag{13}\]  
where \(C_{M}\) is the torque coefficient, \(\rho\) is the coolant density (\(\mathrm{k_{g}}/\mathrm{m^{3}}\)), \(D_{ro}\) is the rotor outer diameter (m) and \(D_{s}\) is the shaft diameter (m). The torque coefficient is dependent on coolant viscosity, speed and size of the rotating disc. The increase in velocity changes the flow type of a rotating disc in a medium from laminar to turbulent. This transition is determined by Reynolds number \(3\times 10^{5}\), a dimensionless parameter calculated using Equation (16). \[C_{M}=\frac{3.87}{{R_{e}}^{\frac{1}{2}}}R_{e}<3\times 10^{5} \tag{14}\]  
\[C_{M}=\frac{3.87}{{R_{e}}^{\frac{1}{2}}}R_{e}<3\times 10^{5} \tag{15}\]  
\[R_{e}=\frac{\rho\omega{D_{ro}}^{2}}{4\mu} \tag{16}\]  
where \(R_{e}\) is the Reynolds number and \(\mu\) is the dynamic viscosity of the coolant (\(\mathrm{k_{g}}/\mathrm{m\cdot s}\)). \(C_{M}\) is calculated using Equations (14) or (15), depending on the size of the Reynolds number. Predicting bearing torque and power loss of bearing through analytical models is complex and still limited in accuracy in some applications. This is due to the level of difficulty associated with satisfactorily quantifying the relationship between the state of lubricant within the bearing cavity and the applied lubricant conditions. Practical experience shows that increase in amount of lubricant will generally increase the bearing friction, and hence, the torque. Although some percentage of it can be accounted for by some of the lubricant filling the bearing cavity, the relationship of this parameter to the quantity of lubricant supply still needs to be empirically determined [62]. Therefore, empirical methods proposed by bearing manufacturers, such as SKF and NSK, are usually considered for estimating bearing torque and power loss of a FESS. The torque of a rolling bearing consists of two components, one speed dependent \(T_{p}\), the other load dependent \(T_{L}\)[63]:  
\[T_{B}=T_{L}+T_{v} \tag{17}\]  
The load term \(T_{L}\) is determined by bearing type and load:  
\[T_{L}=f_{L}F_{B}D_{p} \tag{18}\]  
where \(F_{B}\) is the load applied on the bearing (N), \(D_{p}\) is the pitch circle diameter of the rolling element (m), and \(f_{L}\) is a factor dependent on bearing design and relative load that can be obtained by  
\[f_{L}=z\big{(}\frac{F_{s}}{C_{s}}\big{)}^{y} \tag{19}\]  
\(F_{s}\) is the static equivalent applied load (N) determined using Equation (20) and \(C_{s}\) is the static load rating of the bearing (N) given in the manufacturer's catalogue, Table 3.  
\[F_{s}=X_{s}F_{r}+Y_{s}F_{a} \tag{20}\]  
where \(F_{r}\) is the radial load (N), \(F_{a}\) is the axial load (N), \(X_{s}\) and \(Y_{s}\) are radial and axial load factors, and \(z\) and \(y\) are load torque factors determined experimentally. These factors are provided by the bearing manufacturers and are given in Table 4.  
The speed dependent component of the bearing torque is determined by oil viscosity and speed of the bearing:  
\[C_{M}=\frac{3.87}{{R_{e}}^{\frac{1}{2}}}R_{e}<3\times 10^{5} \tag{14}\]0  
where \(n\) is the bearing speed (rpm), \(v_{0}\) is the kinematic oil viscosity (centistokes) and \(f_{0}\) is a factor depending the bearing type and mode of lubrication, and is given in \[C_{M}=\frac{3.87}{{R_{e}}^{\frac{1}{2}}}R_{e}<3\times 10^{5} \tag{14}\]1