# Aerodynamic Drag Study of the Heat Exchange Equipment with Different Fin Geometries
## 4 Validation  
Solution verification is a very important step in FEM modelling research. To verify the results, it is necessary to compare the actual simulation model with certain and correct results. The peculiarity of the geometry of the objects under study in our work requires a search for scientific works on similar geometric shapes. However, there are practically no works devoted to the aerodynamics of single-finned nozzles, with the exception of studies of the aerodynamic resistance of a bundle of finned heater tubes. For this reason, we had to look for papers that conduct similar studies for similar geometries. An article devoted to the study of the drag coefficients of bullets for air rifles with a wide range of geometries provides a table of coefficient values for 30 typical bullet geometries for air guns. In particular, the drag coefficient for each bullet using 4.5 mm caliber bullets was tested at Ma \(\sim\) 0.57, Re \(\sim\)56,000 and 5.5 mm caliber bullets at Ma \(\sim\) 0.57, Re \(\sim\) 68,000, and the effect on the coefficient aerodynamic drag of detailed geometries of simple cylinders and cones at the same Mach and Reynolds numbers [13].  
We compared the values of the aerodynamic drag coefficients for shapes close to the geometry of our nozzle, and the values turned out to be very close. Conical shapeswith different apex angles (60\({}^{\circ}\), 45\({}^{\circ}\), 30\({}^{\circ}\), 15\({}^{\circ}\), 15\({}^{\circ}\) cut cone) lie in the range between \(C_{X}\) = 1.044 and 0.385 [13]. For 30 bullet shapes with skirts that are close to our geometry, the aerodynamic drag coefficient values are in the range between \(C_{X}\) = 0.358 and 0.777 [13]. The average value of the drag coefficient for our nozzle with orientation (from smaller to larger diameter) for the turbulent regime is \(C_{X}\) = 0.96.  
In addition, based on the graph given by [PERSON] for conical bodies, it appears that the drag coefficient is almost linear with a half-peak angle between 0\({}^{\circ}\) and 90\({}^{\circ}\). This information makes it possible to obtain an equation that approximates the experimental data for the drag coefficient by the angle of the half apex of the cone [28]. The given empirical dependence allows you to analytically calculate the resistance coefficient of the conical element depending on the angle \(\xi\) of its semi-vertex:  
\[C_{X}=0.0112\cdot\xi+0.162 \tag{8}\]  
We used this equation to calculate the drag coefficient; the corresponding angle for our nozzle was chosen to be 60\({}^{\circ}\) (for orientation from smaller to larger diameter), resulting in a value of \(C_{X}\) = 0.834.  
Both verification options provide fairly high convergence, which makes it possible to assert the adequacy of the results obtained.  
Mesh independence was verified to ensure the reliability of the simulation results. A series of research tests were conducted for three fin packing orientations to determine the effect of mesh quality on simulation accuracy. Dimensional resolutions from 3 to 7 were used to create the mesh and calculate the drag coefficient. The number of elements in the aerodynamic channel with a finned nozzle for dimensional resolution 6 was 598,792, and for dimensional resolution 7, the number of elements was 1,707,896. The difference in the average aerodynamic drag coefficient between the results for mesh systems with 598,792 and 1,707,896 elements was less than 2% (Table 1). A further increase in the number of elements is not advisable, since the accuracy of calculations does not increase significantly, and the calculation time increases several times. Therefore, a dimensional resolution of 6 was confirmed and used in this study. The same level of accuracy was obtained for all finned attachment orientations.