# Advanced 2D Computational Fluid Dynamics Model of an External Gear Pump Considering Relief Grooves
## 2 Geometric Model Considering Relief Grooves  
The object of the study is a certain specimen of an external gear pump with a displacement volume of 19 cm\({}^{3}\). The gears teeth number is 12, and their profile is involute. Figure 3 shows a disassembled state of the pump 3D model created by the authors. The creation of this model is described in [3]. In order to model the influence of the relief grooves on the gear pump flow process, it is necessary to create an accurate CAD geometry first. This makes it possible to precisely determine and study the internal geometric volume, which can subsequently be modeled using appropriate 2D shapes. Figure 4 presents a different layout of geometric volume. Figure 4a shows a general view of the entire fluid volume and the plane of symmetry. A section through the plane of symmetry that forms the geometry of a classic 2D model is presented in Figure 4b. The relief grooves in the section parallel to the plane of symmetry are shown in Figure 4c. The challenge is to consider the influence of the grooves shown in Figure 4c, making changes to the section shown in Figure 4b. The relief grooves can be represented in 2D by imaginary radial and tangential channels, which make the same connections at the same moments as the real ones--Figures 5 and 6. The radial channels connect the inter-teeth zone to the tangential channels, which makes the same connection between the inter-teeth zones as the actual grooves. The radial channels rotate with the gears while tangential channels remain stationary. A minimum clearance of 0.01 mm is set between the stationary and rotation walls outside the tangential channels. This value is small enough to prevent leakages and pressure loss but sufficient to ensure finite elements mesh continuity throughout the solution. A similar gap value (0.0092 mm) was used for gear meshing, while the gap above the gear tips was 0.02 mm. The length and width of the radial channels should preferably be minimal so as to not significantly affect the inter-teeth zone dimensions. Two sizes of these grooves were used in  
Figure 4: Geometric volume: (**a**) general view of the fluid volume and the plane of symmetry; (**b**) a section through the plane of symmetry that forms the geometry of a classic 2D model; (**c**) relief grooves in a section parallel to the plane of symmetry. Figure 3: The 3D model of the external gear pump—disassembled state. the present study: 2 \(\times\) 0.25 mm and 0.5 \(\times\) 0.25 mm. The simulations showed that reducing the radial channel length from 2 to 0.5 mm did not affect the characteristic flow rate as a function of the time \(q(t)\) in any way--the differences in the instantaneous and mean values of \(q\) are below 0.02%, so a length of 0.5 mm is preferred. The width of 0.25 mm is chosen so that at each moment of the simulation the channel is meshed by at least two rows of finite elements, thus improving the solution quality. Too small a width runs the risk of an unnatural increase in hydraulic resistance, which is not observed in actual grooves. The length of the tangential channels, as well as the position of their beginning and end, is strictly dependent on the 3D geometry of the studied pump and on the radial channels' width. To find the necessary angles \(\alpha_{1}\)-\(\alpha_{4}\), defining channels begin and end, the drawings shown in Figure 5 were used. Tangential channels are two per gear--Figure 6. The beginning and end of each of them are determined using separate drawings (Figure 5a-d). The angles \(\alpha\) determine the position of the corresponding end relative to the axis passing through the centers of the gears. A careful analysis of the created 3D geometric model showed a gear position, presented in Figure 5d, where the high- and low-pressure zones are connected through the inter-teeth zone. To avoid this, during the tangential channels design, the angle \(\alpha_{4}\) slightly increased from its original value of 7.04\({}^{\circ}\) to the safer value of 8.4\({}^{\circ}\).