# Advanced 2D Computational Fluid Dynamics Model of an External Gear Pump Considering Relief Grooves
## 4 Numerical Results and Discussion  
The main focus of this study is on the volumetric flow rate \(q\), L/min, and pressure \(\Delta p\), bar. The first variable gives information about the flow rate ripple, which can cause undesirable pulsations, noise, and vibrations in the discharge pipeline of the pump, which the authors plan to investigate in the future. The second variable (pressure) well illustrates the benefits of the advanced 2D model. Additionally, the velocity distribution illustrates well the fluid movement through the added channels at different times of the simulation. Of course, the fluid transport through the added channels deforms the velocity field in the inter-teeth zones, so this model is not suitable for velocity analysis near the gears. A comparison between the volumetric flow rate obtained with a classical and an advanced 2D model is shown in Figure 8. The simulation covers a 125\({}^{\circ}\) angle of gear rotation. Because of the established periodicity, it is sufficient to consider an interval of 30\({}^{\circ}\) to study the ongoing effects. The first thirty-degree interval from the beginning of the simulation is skipped to form the velocity field at the inlet and outlet zones, so the next second interval is considered. The initial and final angles of this interval (28.5-58.5\({}^{\circ}\)) are chosen in such a way that they start and end with the average value of the flow rate. Figure 8 illustrates the flow rate-related benefits of adding channels to the 2D model:  
* The new shape of the \(q(t)\) characteristic corresponds well to the classical theory [1]. * The average flow rate increases compared to the classic model from 26.13 to 27.02 L/min, calculated with the actual measured tooth width \(b\) of 30.89 mm. The new result is reliable and eliminates the need for experiment-based correction of \(b\). The increased average flow rate is due to fluid flow through the channels in the direction from the inlet to the outlet, which is explained in detail below. * Along the line 1-2-3, the two models' characteristics \(q(\varphi)\) practically coincides. In this section, the gear mesh zone has a single point of contact (marked with a red circle in Figure 9, position 2), the mesh zone is opened to the surrounding spaces, and the influence of the grooves is minimal. Adding a gap between the fixed and movable walls outside the tangential channels results in a negligible decrease in the maximum value in position 2. * On the 3-4-1 line, the gears mesh zone has two contact points (Figure 9, positions 1 and 3). The connection between the volume trapped by them and the surrounding spaces is made via the added channels. This leads to a large difference in the \(q(\varphi)\) curves compared to the classical 2D model. The fluid flow through the channels is in the direction of the outlet; accordingly, the advanced model curve is located above  
Figure 8: Volumetric flow rate obtained with classical and advanced 2D model. the classical one, which positively affects the average flow rate. The relatively small correction in the \(\alpha_{4}\) angle (shown in Figure 5d) results in a small gap leakage in the vicinity of position 4. Therefore, at position 4, the advanced 2D model flow rate is slightly lower than the classical one. The positions of the gears at each of the points marked in Figure 8 are shown in Figure 9. These positions have the following particularities:  
* **Position 1**. The gears are located symmetrically with respect to the axis connecting their centers of rotation, which passes through the inter-teeth zone of the upper gear and the tooth of the lower gear. After this point, only one point of contact remains in the mesh. * **Position 2**. The single contact point is located on the axis. This is the moment with maximum open inter-teeth zones, minimum fluid return to the inlet, and maximum instantaneous pump flow rate. * **Position 3**. The gears are located symmetrically with respect to the axis, but it passes through a tooth of the upper gear and an inter-teeth zone of the lower gear.