# A Wear Simulation Method for Mechanical Face Seals under Friction Instability Conditions
## 2 Mathematical Model of Mechanical Face Seals under Friction Instability Conditions
### Wear Model  
In this work, the Archard wear model is used to calculate the local wear rate [34]:  
\[V=\frac{K_{W}}{H}F_{W}L, \tag{18}\]  
where \(V\) is the volume loss, \(L\) is the sliding distance, \(K_{W}\) is the dimensionless wear coefficient, \(H\) is the hardness of the stator, and \(F_{W}\) is the normal force. Because the rotor is much harder than the stator, it is assumed that wear only occurs on the stator surface. The wear modulus \(k_{w}\) is defined as \(K_{W}\)/H; therefore, the wear model is rewritten as:  
\[V=k_{w}F_{W}L. \tag{19}\]  
Wear is primarily caused by the asperity contact; therefore, \(F_{W}\) equals \(F_{c}\), and the wear depth \(h_{w}\) is defined as:  
\[h_{w}=k_{w}\frac{F_{c}}{\pi(r_{o}^{2}-r_{i}^{2})}L. \tag{20}\]  
Typically, the wear modulus \(k_{w}\) is determined experimentally for a given lubrication condition. When the shaft of the mechanical face seal is rotating, the lubrication conditions change as the operating conditions change. Therefore, the modified Archard wear equation is used [35]. Scaling factors are introduced to characterize the lubrication conditions. The wear modulus and Archard wear equation are modified based on the wear modulus under dry conditions.  
According to the previous description, the interfacial pressure \(P\) can be expressed as the sum of the asperity contact load \(P_{c}\) and the hydrodynamic load \(P_{f}\):  
\[P=P_{c}+P_{f}. \tag{21}\]  
The scaling factors \(\gamma_{1}\) and \(\gamma_{2}\) are introduced to express the load-sharing ratio of the asperity contact load and the hydrodynamic load, respectively. The scaling factors are defined as:  
\[\gamma_{1}=\frac{P_{c}}{P}, \tag{22}\]  
\[\gamma_{2}=\frac{P_{f}}{P}. \tag{23}\]  
The sum of \(\gamma_{1}\) and \(\gamma_{2}\) equals 1. It is worth noting that time-varying lubrication models need to be considered. Therefore, the time-averaged asperity contact load and time-averaged hydrodynamic load are used to calculate the scaling factors:  
\[\gamma_{1}=\frac{P_{c\_avg}}{P_{avg}}, \tag{24}\]  
\[\gamma_{2}=\frac{P_{f\_avg}}{P_{avg}}. \tag{25}\]  
The time-averaged hydrodynamic load \(P_{f\_avg}\) is defined as:  
\[P_{f\_avg}=\frac{1}{t_{u}}\int_{0}^{t_{u}}P_{f}(t)dt, \tag{26}\]  
where \(P_{f}(t)=\int_{t_{i}}^{t_{o}}2\pi rp_{f}dr/S_{i}\). The time-averaged asperity contact load \(P_{c\_avg}\) is defined as:  
\[P_{c\_avg}=\frac{1}{t_{u}}\int_{0}^{t_{u}}P_{c}(t)dt, \tag{27}\]  
where \(P_{c}(t)=\int_{t_{i}}^{t_{o}}2\pi rp_{c}dr/S_{i}\), \(t_{u}\) is the time span of the simulation, \(S_{i}\) is the seal interface area, \(S_{i}=\pi(r_{o}^{2}-r_{i}^{2})\), and \(P_{avg}=P_{c\_avg}+P_{f\_avg}\).  
Since the wear volume is only related to the asperity contact load, the scaling factor for the asperity contact is used in the Archard wear equation. The modified wear modulus \(k_{\text{m}}\) is defined as \(k_{\text{w}}\gamma_{1}\). Hence, according to the modified Archard wear equation, the wear depth \(h_{\text{w}}\) is defined as:  
\[h_{w}=k_{w}P_{c\_avg}L=k_{w}\gamma_{1}P_{avg}L=k_{m}P_{avg}L, \tag{28}\]  
The derivative of the wear depth with respect to time is defined as the wear time rate \(k_{\text{t}}\), and the derivative of the wear depth with respect to the sliding distance is defined as the wear distance rate \(k_{\text{s}}\). In most cases, the modified wear modulus is assumed to be constant during the iteration cycle. Therefore, the wear time rate and wear distance rate are defined as:  
\[k_{t}=\frac{dh_{w}}{dt}=k_{m}P_{avg}v_{m}, \tag{29}\]  
\[k_{s}=\frac{dh_{w}}{dL}=k_{m}P_{avg}. \tag{30}\]