# Integrated Thermal and Dynamic Analysis of Dry Automotive Clutch Linings
## 3 Thermal Analysis  
In parallel with the dynamic powertrain model, an analytical dry clutch lumped parameter thermal model is developed, based on the principle of conservation of energy. The lumped parameter model assumes that the temperature within all the elements of the clutch assembly is uniform and thus the interfacial temperatures can be derived. The details of the thermal model are reported in [PERSON] et al. [21]. A brief outline of the thermal model is provided here for the sake of completeness. Temperature is calculated for each of the main clutch system components through a balanced thermal energy flow into and out of the system. Thus, the temporal variance of any component's temperature is obtained as:  
\[m_{i}c_{i}\frac{\partial T}{\partial t}=\sum\dot{q} \tag{32}\]  
The modelled clutch system components are the flywheel, fw, the friction disc/lining, fd/fl, and the pressure plate, pp, as shown in Figure 3. The friction disc is considered to have two surfaces and therefore, heat is generated by two sources (contacts), one at each side of the friction disc (i.e., in contact with the flywheel and pressure plate). In Figure 3, heat transfer is through conduction represented by the solid line arrows and through convection by the dotted arrows. Heat transfer through radiation is ignored as the engagement times are very short with the dominant mechanism of heat transfer being conduction [26]. Figure 2: Non-linear clutch clamp load characteristics. As can be seen from Figure 3, the generated heat on each of the friction disc is conducted into the component in contact with the disc (i.e., the flywheel and the pressure plate). After heat is generated and conducted into these main components, it continues to be conducted through other adjacent bodies such as the crankshaft and transmission system components into air. The amount of heat conducted and convected from each of the components is determined by a heat partitioning method as described in [21] in detail. In this approach a portion of the generated heat in the contact between the flywheel and the friction lining is assumed to be transferred to the flywheel, whilst the remainder is transferred to the friction lining material. Similarly, the heat generated between the pressure plate and the friction lining is also partitioned between the two surfaces. The exact values for the portions of generated heat transferred to each component are obtained through considering the thermal resistances in each heat flow direction and the ambient conditions. The frictional power loss produced in the system defines the generated heat in the contact between the mating surfaces. Assuming that all frictional losses in the contact convert to thermal energy and the normal load are applied uniformly, the frictional power loss generated between the mating surfaces can be calculated as [21]:  
\[\dot{q}=Z\frac{2\mu F_{n}\omega r_{m}}{3} \tag{33}\]  
where \(Z\) is the number of frictional interfaces (2 in the current analysis), \(r_{o}\) and \(r_{i}\) are the inner and outer radii of the clutch disc, \(F_{n}\) is the normal contact load, \(\mu\) is the interfacial coefficient of friction of the contacting surfaces, and \(\omega\) is the relative angular velocity between the friction disc and the flywheel/pressure plate. The coefficient of friction in this analysis is considered to vary with the sliding velocity, the applied contact pressure, and the operating temperature.