## 2 Heat Generation and Heat Transfer in a Motorized Spindle  
The motorized spindle is a mechatronic product composed of a stator, a rotor, a spindle, a shell, a water jacket, front bearings, rear bearings, and bearing housing. It is built into an asynchronous alternating current motor, and its structure is as shown in Figure 1. The rotor and spindle are integrated through pressure matching, and the latter is supported by front and rear bearings. The stator of the motor is fixed on a shell through a water jacket. It is well known that heat generation is inevitable owing to high speed of the motorized spindle. The main sources of heat generation are the heating of the motor, and frictional heat from the front and rear bearings. Furthermore, the power supply of the motorized spindle is an inverter that generates higher harmonics, because of which the motor generates additional resistive losses owing to the copper in the stator and rotor, the iron in the stator, and additional stray loss. Moreover, the skin effect in the rotor induces greater losses in the copper at power supplies of high frequency. The losses from these high harmonic voltages and currents result in an increase in motor temperature. This causes thermal deformation in the components of the motorized spindle and affects its dynamic performance. Lubrication and cooling systems are designed to prevent the motorized spindle from overheating. In general, the stator of the motorized spindle is cooled using cooling water pumped through a jacket. Bearings are lubricated with oil and gas, where fine drops of oil are carried by compressed gas into the air gap between the stator and rotor, and the clearance of the bearing. These drops then flow out of the motorized spindle. Therefore, the bearings are lubricated, and the rotor, shaft, and bearings are actively cooled. The above discussion shows that heat transfer in the motorized spindle is complex. The heat transfer mode of the motorized spindle consists mainly of conduction and convection. The equation of energy conservation is as follows:  
\[\rho_{1}C_{p1}\frac{\partial T}{\partial t}+\rho_{2}C_{p2}v\
abla T=\
abla \cdot(k\
abla T)+Q \tag{1}\]  
where \(Q=\frac{P_{tot}}{\
abla}\), \(P_{tot}\) is heat generation, \(V\) is the volume of the heat source, \(\rho_{1}\) is the density of the solid, \(\rho_{2}\) is that of the fluid, \(C_{p1}\) and \(C_{p2}\) are the heat capacities of the solid and the fluid, respectively, \(T\) is the temperature of the motorized spindle, \(v\) is fluid velocity, \(\
abla\) is the Laplace operator, \(k\) is thermal conductivity, and \(Q\) is the rate of heat generation. According to heat transfer theory, the relationship between the coefficient of convective heat transfer and temperature is  
\[q=h\cdot(T_{ext}-T) \tag{2}\]  
Figure 1: Structure of the motorized spindle. where \(q\) is heat flux, \(h\) is the vector of the coefficient of heat transfer, and \(T_{ext}\) is the temperature of the fluid medium. Five processes contribute to heat transfer in the motorized spindle. Between the surface of the spindle shell and air is natural convection heat transfer, with the heat transfer coefficient given by [23]  
\[h_{1}=9.7\ (W/(\mathrm{m}^{\cdot\circ}\mathrm{C})) \tag{3}\]  
Between the gas-gap and the compressed air is the forced convection heat transfer, with the heat transfer coefficient represented as:  
\[h_{2}=\frac{Nu\lambda_{a}}{H} \tag{4}\]  
where \(Nu=0.239(\frac{\delta}{r})^{0.25}Re^{0.5}\), \(Re=\frac{u_{1}H}{r}\), \(u_{1}=(u_{a}^{2}+u_{t}^{2})^{1/2}\), \(u_{1}\) is the speed of airflow in the air gap (m/s), \(u_{a}\) and \(u_{t}\) are the input speed of the compressed air and the speed of rotation of the spindle, respectively, \(r\) is the radius of the outer surface of the rotor (m), \(H\) is the dimensionality of the geometry of the air gap (m), \(Re\) is the Reynolds number, \(Nu\) is the Nusselt number, \(\delta\) is the gap between the stator and the rotor (m), and \(\lambda_{a}\) is the thermal conductivity of air (W/(m\(\cdot\)\({}^{\circ}\)C)).