# Multiphase Conjugate Heat Transfer Analyses on the Assembly Situation of Rotary Shaft Seals
## 2 Materials and Methods
### Governing Equations  
The convective heat transfer in the fluids around the sealing system was described using the Navier-Stokes equations in their conservation form. The heat conduction in the solids of the sealing system and its surroundings was described with the help of the energy equation. Therefore, the following conservation equations of mass, momentum and energy had to be satisfied. The continuity equation  
\[\frac{\partial}{\partial t}\rho=-\
abla\cdot(\,\rho\,\,\mathbf{U}\,) \tag{1}\]  
with the flow velocity vector field \(\mathbf{U}\), the time t and the fluid density \(\rho\) describe the conservation of mass. This equation ensures that the mass entering a system is equal to the mass leaving the system and the accumulation of mass within the system. The first part, the time derivative, describes the loss of mass in the system and the second part, the divergence term, describes the difference between the mass entering and the mass leaving the system. Since the air is assumed to be incompressible, the density is constant and the continuity equation can be simplified to [21]:  
\[0=\
abla\cdot(\,\rho\,\,\mathbf{U}\,). \tag{2}\]  
The momentum equation  
\[\frac{\partial}{\partial t}(\rho\,\,\mathbf{U})=-\
abla\cdot(\,\rho\,\,\mathbf{U}\otimes \mathbf{U}\,)-\
abla p+\
abla\cdot\tau+\mathbf{S}_{M} \tag{3}\]  
where the shear-stress tensor \(\tau\) for Newtonian fluids and the pressure \(p\) are an expression of [PERSON]'s second law of motion and describes the conservation of momentum for moving fluid [22]. The optional source terms \(\mathbf{S}_{\text{M}}\) depend on the application. For example, if buoyancy is considered, the source term  
\[\mathbf{S}_{\text{M}}=\mathbf{S}_{\text{buoy}}=\rho\,g \tag{4}\]  
is added to the momentum equations, where \(g=9.81\) m/s\({}^{2}\) is the gravitational acceleration [23]. The total energy equation is  
\[\frac{\partial}{\partial t}\,\,(\rho\,h_{\text{tot}})-\frac{\partial}{ \partial t}\,\,p+\
abla\cdot(\,\rho\,\,\mathbf{U}\,h_{\text{tot}}\,)=\
abla\cdot( \lambda\
abla\text{T})+\
abla\cdot(\mathbf{U}\cdot\tau)+\mathbf{U}\cdot\mathbf{S}_{\text{ M}}+\mathbf{S}_{\text{heat}} \tag{5}\]  
where the thermal conductivity is \(\lambda\) and the total enthalpy is  
\[h_{\text{tot}}=\text{h}+\frac{1}{2}\,\,\mathbf{U}^{2}. \tag{6}\]  
This states that the total energy of the system is equal to the total work and heat added to the system. The viscous work term \(\
abla\cdot(\mathbf{U}\cdot\tau)\) represents the work due to viscous stresses, and the term \(\mathbf{U}\cdot\mathbf{S}_{\text{M}}\) describes the work due to an external momentum source. The volumetric heat source \(\mathbf{S}_{\text{heat}}\) is optional [23]. In addition to the conservation equations, equations for the determination of the heat transfer must be solved. Heat transfer refers to all processes of transporting energy in the form of heat. The \"internal energy\", i.e., the energy contained in a system, is a state variable. In addition, the energy exists as a conservation variable, as it cannot be created or destroyed, but transformed into another form. This is described by the first law of thermodynamics:  
\[\dot{Q}+\dot{W}=\dot{m}(h_{\text{out}}-h_{\text{in}}), \tag{7}\]  
where \(\dot{Q}\) describes the heat flow across the system boundary and \(\dot{W}\) the working current across the system boundary. For a stationary system, the mass flow \(\dot{m}\) is equal to the incoming mass flow and equal to the outgoing mass flow. The specific enthalpy is divided into the incoming enthalpy \(h_{\text{in}}\) and outgoing enthalpy \(h_{\text{out}}\). As two systems of different temperatures come into contact with each other, heat is exchanged along the negative temperature gradient. The first law of thermodynamics described in Equation (7) does not affect how heat is transferred between multiple systems. Heat can be transferred by conduction, convection or radiation. Thermal radiation is neglected in this work.