# Advanced 2D Computational Fluid Dynamics Model of an External Gear Pump Considering Relief Grooves
## 4 Numerical Results and Discussion
After this moment, the gears contact at two points, and the trapped fluid between them is transferred back to the inlet, which reduces the instantaneous pump flow rate. The effect is significantly more obvious in the classical 2D model, without channels. * **Position 4**. The two contact points are equidistant from the axis; the trapped volume is not in contact with the channels, but the distance between the radial channels of the trapped volume and the tangential channels is minimal. This position is characterized both by maximum fluid transfer back to the inlet port and by maximum losses via the channels' gaps. Therefore, the instantaneous pump flow rate is minimal. The velocity distribution shown in Figure 9 illustrates the flow through the advanced model channels. Of particular importance is the axis connecting the centers of rotation of the two gears. The role of the channels is greatest when the gears are meshed in two contact points, and the trapped fluid is transferred back to the low-pressure zone. In this case, if the trapped volume is located predominantly to the right of the axis (for example, position 1 in Figure 9), it constantly increases, drawing fluid from the inlet zone. If the trapped volume is located predominantly to the left of the axis (for example, position 3 in Figure 9), it constantly decreases, pushing the fluid from the inter-teeth zone through the channels to the outlet zone. In both cases, there is an additional fluid movement through the channels in the direction from the suction to the discharge port, which leads to a general increase in the average flow rate compared to the classical 2D model. Figure 9: Velocity distribution at four rotation positions of gears. The points of contact are marked with red circles. Some important fluid flow directions are marked with yellow arrows. Figure 10 shows the fluid pressure distribution obtained with consideration of the relief grooves. The pressure in all the inter-teeth zones is equalized with that in the contacting grooves. Figure 10a shows the position where the trapped volume does not contact the surrounding spaces through the added channels, and the pressure in it differs from that of the inlet and outlet zones. This position actually separates the outlet from the inlet. Figure 10b,c show positions very close to position 4, where, however, the pressure in the trapped volume is equalized with that of the outlet or inlet, which is not possible with the classical 2D model. Emphasis is placed on the contact of the teeth with the actual grooves, from where pressure equalization occurs. Figure 11a shows the influence of the rotation frequency \(n\), min\({}^{-1}\), and Figure 11b presents the influence of the output pressure \(\Delta p\) on the current and average values of the pump flow rate \(q\). The curves were obtained by simulations with the advanced 2D model. The average values of the flow rate \(q\) for all investigated operating modes are shown in the figure, as they are further used to validate the obtained results. It can be seen that with a decrease in \(n\) and with an increase in \(\Delta p\), the average value of the flow rate \(q\) decreases. The form and the period of the characteristic \(q(\varphi)\) are preserved and practically do not depend on the rotation frequency and the load. The variation of the average flow rate \(q\) when changing the rotation frequency \(n\) from 1450 to 950 min\({}^{-1}\) is high (\(-\)35.5%) and is proportional to \(n\), while when changing the load from 5 to 150 bar, it is low (\(-\)2.5%) and has a small influence from the working pressure. Practically the same reductions in the average flow rate are obtained with the classic 2D model [3], and the differences between the classical and the advanced 2D model here are less than 1%. Figure 10: Pressure distribution and relief grooves in positions with minimum instantaneous flow rate.