## 2 Numerical Calculation of Oil Dispersion Flow in Flowing Air
### Computational Domain and Boundary Conditions  
A three-dimensional numerical simulation was chosen for this study. Numerical calculation was carried out for three different pipe diameters (35, 25 and 15 mm), which were identical to the pipe diameter during the experiment. The pipeline had a length of 6 m with respect to isothermal flow without the need for heat transfer and it consists of only one domain representing air with an oil dispersion. In order to simplify the evaluation results of the analysis, the domain was divided into 30 volumetric subdivisions (0.2 m/element). Each of the areas created for specific geometry and flow velocity was divided by the network into the final number of the tetrahedron volume elements. Modeling the fluid velocity near the pipe wall in the boundary layer was ensured by densifying the mesh at the wall, wherein the thickness of the first element was determined from the selected dimensionless distance value of \(y^{+}=1\). Figure 1 is a view of a part of the mesh formed at the inlet of the pipe for pipe diameter of 35 mm and a flow rate of 5 m\(\cdot\)s\({}^{-1}\), wherein the thickness of the first element at the wall is 0.076 mm. The number of mesh elements after the boundary layer thickness is 12 with an increase of 1.2. With an average mesh size of 5 mm, the total number of elements of the selected geometry is 3.28 mil.  
A numerical simulation of the airflow with an oil dispersion of an average oil particle size of 0.2 \(\mu\)m was from a physical point of view designed as isothermal multi-phase turbulent airflow with a temperature of 10 \({}^{\circ}\)C, using the SST turbulence model. Air with continuous fluid and oil as dispersed fluid morphology was the carrier gas and a buoyancy model with a gravitational acceleration value of 9.81 m\(\cdot\)s\({}^{-2}\) was included in the solution. A multi-phase inhomogeneous flow model with homogeneous turbulence calculation was employed. The domain reference pressure was determined at 1 atm according to the conditions of the experiment. Within the initialization conditions (at time \(\tau=0\) s), the zero velocity of the flowing fluid was defined in the entire volume, as well as zero relative static pressure and zero volume fraction of oil.  
The input boundary condition was set with a defined subsonic air velocity with an oil fraction, based on the set value of the input intensity of the turbulence at the level of 10%. The oil mass flow rate based on experimental knowledge is 4.67\(\cdot\)10\({}^{-4}\) kg\(\cdot\)s\({}^{-1}\). The dispersed oil particles flow at the selected speed according to the boundary condition at the pipe entry, wherein oil volume fraction at the inlet is given by the following formula:  
\[r_{\text{oil}}=\frac{V_{\text{oil}}}{V_{\text{oil}}+V_{\text{air}}}=\frac{4\cdot Q _{\text{m,oil}}}{\pi\cdot\rho_{\text{oil}}d^{2}\cdot v} \tag{12}\]  
where \(r_{\text{oil}}\) is oil volume fraction of oil (1), \(V_{\text{oil}}\)--oil volume (m\({}^{3}\)), \(V_{\text{air}}\)--air volume (m\({}^{3}\)), \(Q_{\text{m,oil}}\)--oil flow rate (kg\(\cdot\)s\({}^{-1}\)), \(\rho_{\text{oil}}\)--oil density (kg\(\cdot\)m\({}^{-3}\)), \(d\)--pipe diameter (mm), \(v\)--airflow velocity (m\(\cdot\)s\({}^{-1}\)). Volume fraction of air is subsequently calculated as (\(1-\mathcal{O}_{\text{oil}}\)). At the outlet of the duct, a boundary condition was specified with an average relative static pressure of 0 Pa by averaging pressure using the Average Over Whole Outlet method. A zero fluid velocity (No Slip Wall) with an absolute surface roughness of 0.03 mm is set on the cylindrical surface of the pipe.