# Deformation Characteristics of Mechanical Seal Face under Alternating Loads
## 2 Materials and Methods
### Numerical Model  
Considering the complexity of the actual operating conditions of the mechanical seal, some assumptions are made for thermal-mechanical coupling deformation calculation: the temperature field and stress field of the seal ring are axisymmetrically distributed due to the axisymmetric structure; the seal face is in a mixed friction state, and the only heat source is generated from the friction of the seal faces and the heat flow density is evenly distributed on the seal face; the thermal radiation, stirring heat, heat which is taken away by the leaking liquid, and the convection heat transfer of the liquid film between the seal faces are all ignored; and there is no velocity slip on the outer surface of the rotating ring. The differential equation of elastic mechanics of the seal ring is shown as follows [6; 16]:  
\[\frac{1}{2}\iint_{s}r\{\epsilon\}^{T}[D]\{\epsilon\}dS-\int_{l}r\{u\}^{T}\{Q\} dr-\int_{s}r\{\epsilon\}^{T}[D]\{\epsilon_{T}\}dS=0 \tag{1}\]  
where \(\{\epsilon\}\) is the strain component; \([D]\) is the elasticity matrix; \(\{u\}\) is the displacement; \(\{Q\}\) is the pressure acting on the seal rings, including sealed pressure, liquid film pressure, atmospheric pressure, and contact pressure; \(S\) and \(l\) are the computational domains; \(r\) is the radius; \(\{\epsilon_{T}\}\) is the strain component caused by the temperature rise and can be obtained by Formula (2) [6]. \[\{\epsilon_{T}\}=\{\alpha(T-t_{b}),\alpha(T-t_{b}),\alpha(T-t_{b}),0\}^{T} \tag{2}\]  
where \(\alpha\) is thermal expansion coefficient, K\({}^{-1}\); \(T\) is the temperature of the seal ring, \({}^{\circ}\)C; \(t_{b}\) is room temperature, \({}^{\circ}\)C.  
The basic governing equation of transient heat conduction of the seal rings is shown as follows [6; 17]:  
\[\rho c\frac{\partial T}{\partial t}-k\bigg{(}\frac{\partial^{2}T}{\partial x^ {2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2} }\bigg{)}-\rho q=0 \tag{3}\]  
where \(\rho\) is density, kg\(\cdot\)m\({}^{-3}\); \(c\) is the specific heat capacity, J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\); \(t\) is the time, s; \(k\) is the thermal conductivity coefficient, W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\); \(q\) is the heat flux, W\(\cdot\)m\({}^{-2}\). The total heat flux can be obtained by the friction heat of the seal face. The heat flux generated by friction is transferred by conduction to the seal rings entirely. The heat flux in the interface is determined by the friction contact. For the water medium, the fluid film pressure between the seal faces is approximately linear in the radial direction [18]. The  
Figure 1: Geometric model and boundary conditions of the seal rings. friction heat generated on the annular area with a width of \(dr\) at radius \(r\) can be obtained as follows [19]. \[Q_{t}=2\pi fr^{2}\omega\bigg{[}p_{\text{sp}}+p_{o}B+p_{i}(1-B)-\frac{r-r_{o}}{r_ {o}-r_{i}}(p_{o}-p_{i})\bigg{]}dr \tag{4}\]  
where \(Q_{t}\) is friction heat, J;\(f\) is the friction coefficient; \(\omega\) is the rotational angular velocity, rad\(\cdot\)s\({}^{-1}\); \(p_{\text{sp}}\) is spring pressure, Pa; \(B\) is the balance ratio; \(r_{\text{i}}\) is the inner radius, mm; \(r_{\text{o}}\) is the outer radius, mm. The end face of the rotating ring (\(W_{\text{R}}\)) is regarded as the surface on which the friction heat is generated. The contact thermal conductivity with a larger order of magnitude (set as \(10^{3}\)) is defined so that the friction heat is preferentially transmitted between the end faces of the seal rings. To obtain the deformation characteristics of the seal face, the FEM is adopted to discrete the computational domain, and the indirect coupling method is used for the coupling solution. The transient heat transfer numerical model in the research has been validated by the pseudo-real experiment in reference [15].