## 2 Materials and Methods
### Numerical Calculations  
Numerical environments use the Finite Volume Method (FVM) [32; 33], based on dividing the volume of the flowing gas into smaller volumes of a shape that depends on the type of numerical mesh used. In each cell of the numerical mesh, it is possible to quantify calculated parameters, such as pressure, velocity, and temperature, for example, without having to solve for the entire model. By subdividing the fluid volume, it is possible to determine flow parameters based on the equilibrium of the volume cell and the neighboring cells. The shape of the chambers of the CAD model does not reproduce the shape of the working chambers of the differential section, as only the nozzle mounted in the threaded sleeve was used during the experimental tests. In order to create the model, two chambers (inlet and outlet) were used, which are halves of spheres with a radius equal to 10 times the diameter of the opening of the largest nozzle (according to Figure 2). The initial pressure conditions were set on the spherical surfaces and the MFR value was numerically determined on these surfaces. The length of the adopted flow channel was equal to 8 mm. Numerical calculations of the issue analyzed in this study were carried out using two numerical environments, SolidWorks Flow Simulation (SW-FS) and Ansys Fluent (A-F). In engineering practice, SolidWorks Flow Simulation is only used for the initial estimation of flow parameter values due to the limited shape of the mesh cells. An orthogonal (hexahedral) mesh consists of cubic cells, perpendicular to the main coordinate system, with a defined degree of density. Wider possibilities in this area are offered by the Ansys Fluent software [34; 35], which allows for the advanced definition of the shape and density of the finite volume mesh. The correct choice of numerical mesh plays a key role in the quality of the results obtained. During numerical calculations using CFDs, global meshes with a declared maximum edge length and shape are used in cross-sections with large flow volumes and local mesh densities in key areas [36], for example narrow channels. Local computational mesh densities allow us to minimize model discretization errors. The parameters of the finite element mesh should be chosen by taking into account the optimization conditions related to the evaluation of the geometrical criteria of the mesh, such as, for example, aspect ratio and skewness [37], as well as simulation time and the number of iterations. By dividing the flow area into a set of discrete cells, it is possible to determine by numerical methods, the parameters of the stream flow, according to the  
Figure 2: Model view: (**a**) dimensions and initial inlet and outlet boundaries, \(\mathrm{d_{n}}=0.75\) mm, nozzle diameter (0.75\(\ldots\).500) mm; (**b**) model prepared in Ansys Fluent with defined inlet and outlet pressure boundaries. assumed boundary conditions, by solving a system of differential equations resulting from the principle of the conservation of the momentum of the flowing fluid and the Navier-Stokes equations [38]. \[\frac{\partial p}{\partial t}+\
abla(pv)=0 \tag{1}\]  
\[p\frac{\partial v}{\partial t}=-\
abla p+\rho g+\mu\
abla^{2}v \tag{2}\]  
Solving such a system of equations allows the velocity and pressure to be evaluated at each model section for geometries of varying complexity. Both numerical environments used provide the ability to consider laminar and turbulent flows for compressible fluids using the RANS model with Favre mass averaging [39]. This approach makes it possible to implement a \(k\)-\(\varepsilon\)[40] or other [41; 42; 43] turbulence models and evaluate the effect of turbulence on the flow parameters after finding the \(\mu_{t}\) values from the kinetic energy of the vortices, \(k\), and the dispersion rate, \(\varepsilon\).