# Calculation and Verification of the Real-Time Working Characteristics of a Viscous Coupling
## 3 The Temperature Calculation of the VC at Working Course
### The Heat Transfer Calculation of VC Parts  
The VC transmits torque by shearing the silicone oil with the plates. The rotating speed is different from that of the input and the output shafts. Therefore, there is energy loss during the VC's operation. Supposing the torque transmitted when the VC works is \(T_{v}\), the input speed is \(\omega_{1}\), and the output speed is \(\omega_{2}\), the speed difference is:  
\[\Delta\omega=\omega_{1}-\omega_{2} \tag{31}\]  
The loss of power is:  
\[\Delta P=\frac{T_{v}\Delta\omega}{2} \tag{32}\]  
Supposing that the lost power is all converted to heat, the total quantity of heat within \(t\) (the length of working time) is:  
\[Q=\Delta Pt \tag{33}\]  
The heat transfer of each part of the VC can be determined by using the heat-transfer coefficient above. The heat transfer of silicon oil by shell includes three links:  
1. A convectional heat transfer between the silicone oil and the shell's inner wall;
2. A convectional heat conduction between the shell's outer wall and inner wall; and
3. A convectional heat transfer between the outside air and the shell's outer wall. It is same for the heat flow through a series of links under a steady-state condition. The heat flow of the above three are:  
\[Q_{12}=F_{hi}\alpha_{2}(t_{s}-t_{hi}) \tag{34}\]  
\[Q_{12}=\frac{2\pi\gamma_{m}L_{hi}(t_{hi}-t_{ho})}{\ln\left(\frac{t_{ho}}{t_{hi }}\right)} \tag{35}\]  
\[Q_{12}=F_{ho}\alpha_{3}(t_{ho}-t_{A}) \tag{36}\]  
where \(r_{ho},r_{hi}\)--the inside and outside diameter of the shell; and \(F_{ho},F_{hi}\)--the inside and outside surface area of shell. Equation (36) may then be converted to the forms of temperature and pressure, namely:  
\[t_{s}-t_{hi}=\frac{Q_{12}}{F_{hi}\alpha_{2}} \tag{37}\]  
\[t_{hi}-t_{ho}=\frac{Q_{12}}{2\pi\gamma_{m}L_{hi}} \tag{38}\]  
\[t_{ho}-t_{A}=\frac{Q_{12}}{F_{ho}\alpha_{3}} \tag{39}\]  
\[Q_{12}=\frac{t_{s}-t_{A}}{\frac{1}{F_{ho}\alpha_{2}}+\frac{\ln\left(\frac{T_{ h}}{T_{hi}}\right)}{2\pi\gamma_{m}L_{hi}}+\frac{1}{F_{ho}\alpha_{3}}} \tag{40}\]  
The heat transfer of the silicone oil through the left and right end covers includes three links in series:  
1. The convective heat transfer between the silicone oil and the inner wall of the left (right) end cover;
2. The convectional heat conduction between the inner and outer wall of the left (right) end cover; and
3. The convectional heat transfer between the outside air and the outer wall of the left (right) end cover. For the left end cover, the heat flow of the above three are:  
\[\Delta P=\frac{T_{v}\Delta\omega}{2} \tag{32}\]0  
\[\Delta P=\frac{T_{v}\Delta\omega}{2} \tag{32}\]1  
\[\Delta P=\frac{T_{v}\Delta\omega}{2} \tag{32}\]2  
where \(F_{cli},F_{clo}\)--the inside and outside surface area of the left end cover; \(t_{cli},t_{clo}\)--the inside and outside surface temperature of the left end cover; and \(H_{cl}\)--the thickness of the left end cover. Equation (43) may then be converted to the forms of temperature and pressure, namely:  
\[\Delta P=\frac{T_{v}\Delta\omega}{2} \tag{32}\]3  
\[\Delta P=\frac{T_{v}\Delta\omega}{2} \tag{32}\]4  
\[\Delta P=\frac{T_{v}\Delta\omega}{2} \tag{32}\]5  
\[\Delta P=\frac{T_{v}\Delta\omega}{2} \tag{32}\]6  
The above equations are also applicable to the right end cover. The heat transfer of the silicone oil through the shaft includes four links in series:  
1. The convective heat transfer between the silicone oil and the axle inside the shell;
2. The convectional heat conduction between the axle and the surface;
3. The convectional heat conduction between the axle and one outside shell; and
4. The convectional heat transfer between the outside air and the outer wall of the outside shell. When the convectional heat conduction between the axle and the surface is ignored, and then the temperature of each point that is perpendicular to the cross-sectional area of the axle is equal.