# Numerical Analysis via Mixed Inverse Hydrodynamic
## 2 Mathematical Model
### Fluid Mechanics
#### 2.2.4. Temperature Rise in the Sealing Zone  
The friction force between the rubber seal and rod should consider two factors: friction due to fluid viscosity and friction due to asperity contact. Equation (16) shows the shear stress caused by fluid viscosity and shear stress due to surface asperities; the friction force is calculated by integrating them over the stroke length.  
\[\tau_{v}=\frac{h}{2}\frac{\partial p}{\partial x}+\frac{\mu u}{h},\ \ \tau_{c}=fp_{c} \tag{16}\]  
\[F=-\pi Dl\int_{0}^{l}(\tau_{v}+\tau_{c})dx \tag{17}\]  
The equation for heat generation rate \(q_{f}\) due to friction generated while the rod reciprocates is as follows:  
\[q_{f}=(\tau_{v}+\tau_{c})u \tag{18}\]  
In thermal analysis, it is assumed that all of the heat generated in the seal area is transferred to the semi-infinite solid because generally, the rod has a very high thermal conductivity to the seal [6]. Note that if a rubber seal with high thermal conductivity is used, the generated heat must be calculated by dividing the seal and rod. The heat conduction equation of the seal area for a moving heat source can be expressed as [22]:  
\[\begin{array}{l}T_{avg}-T_{a}=1.07\frac{q_{f}l_{c}}{k}\left[\frac{\rho_{c} \tau_{v}\mu l_{c}}{k}\right]^{-\frac{1}{2}}\ \ \ for\ \frac{\rho_{c}\tau_{v}\mu l_{c}}{k}>0.68\\ =0.64\frac{q_{f}l_{c}}{k}In\left[\frac{5k}{\rho_{c}\tau_{v}\mu l_{c}}\ \ \ for\ \frac{\rho_{c}\tau_{v}\mu l_{c}}{k}<0.68\end{array} \tag{19}\]where \(T_{avg}\) is the average temperature in the sealing zone, \(T_{a}\) is the ambient temperature, \(c_{p}\) is the specific heat, \(\rho_{r}\) is the density of the rod, \(k\) is the thermal conductivity of the rod, and \(l_{c}\) is the half-length of \(l\).  
Because the fluid viscosity is strongly temperature-dependent, the temperature of the sealing area is used to re-evaluate the fluid viscosity. The viscosity according to temperature was calculated using the Reynolds temperature-viscosity equation [23].  
\[\mu=\mu_{0}\exp\bigl{(}-a\bigl{(}T_{avg}-T_{0}\bigr{)}\bigr{)} \tag{20}\]  
where \(\mu_{0}\) is the viscosity at \(T_{0}\), \(\mu\) is the viscosity at \(T\), and \(a\) is the viscosity-temperature constant.  
Figure 4 shows a flowchart of the thermal mixed IHL analysis, and the numerical analysis process is as follows. IHL analysis is performed using input variables, such as static contact pressure, rod movement speed, and operating temperature calculated using the FE software. After introducing the boosting zone concept and calculating the oil film thickness at the inflection point and its location, the oil film distribution for the sealing zone section is obtained according to Equation (9), and by substituting the oil film distribution into Equation (14), a distribution can be obtained. The mixed lubrication effect is reflected by moving the entire oil film distribution up and down in parallel; thus, the sum of the supporting force by the oil film and the asperity contact force is equal to the static contact force obtained through the FE analysis. From the obtained new film distribution and pressure distribution, the viscous shear stress and frictional shear stress of the asperities' contact were calculated and, by substituting them into Equations (17) and (18), the surface friction and heat flux at each position can be determined. The average temperature rise of the sealing zone by heat flux was calculated using Equation (19). By substituting the increased \(T_{avg}\) into Equation (20), the updated viscosity in the sealing zone was calculated, and lubrication analysis was performed again with this viscosity. The entire process was repeated until the temperature of the sealing area satisfied thermal equilibrium. As mentioned earlier, this analysis method was assumed to be in a steady state, which is valid when the stroke length is significantly larger than the seal width.