# Advanced 2D Computational Fluid Dynamics Model of an External Gear Pump Considering Relief Grooves
## 1 Introduction and Motivation
Omitting relief grooves causes the following negative effects:  
* Inter-teeth zone pressure obtained using the classical 2D model is not realistic because this zone is, most of the time, connected by such grooves. The pressure in the connected inter-teeth zones is usually equal to that in either the suction or the discharge port. This disadvantage is illustrated in Figure 1, where the relief grooves are superposed on a pressure field obtained with a classical 2D model. It can be seen that in the case of two contact points between the teeth, the pressure in the trapped zone is colored neither orange nor blue but has a much higher value (in red). Also, the inter-teeth zones colored in yellow should have a pressure value equal to that in the outlet zone (in orange). Figure 1: Pressure distribution with a classic 2D model at an output pressure of 150 bar. The relief grooves are superposed, and the unconsidered connections are marked with red circles. * In the presence of two contact points, the fluid volume trapped between them is completely transferred to the low-pressure zone. This reduces the volumetric flow rate below the theoretically and experimentally determined values and increases the pump flow rate ripple. In reality, this closed volume is much shorter than the classical model shows since most of the time (during pump operation), it is connected by the relief grooves to either the high-pressure or the low-pressure zone. Figure 2 shows a technique to compensate for the reduced flow rate \(q\) obtained with a classical 2D model by increasing the gear width \(b\). This technique has been shown in [3] to perform well over a wide range of modes. However, the expected average flow rate value of the studied pump, also shown in Figure 2, at the considered operating mode is nearly 27 L/min, while the classic 2D model gives 26.13 L/min. This disadvantage can be avoided by considering the relief grooves' influence. * The classical 2D model gives an unrealistic volumetric flow rate variation, which cannot be used for further studies of flow and pressure pulsations in the discharge pipeline, causing noise and vibrations. The law shown in Figure 2 is strongly dependent on the number and position of the contact points during the mesh of both gears, while at the same time, it does not depend on the shape and location of the relief grooves, which have a significant influence on it. The literature review showed that relief grooves are considered using 3D models only [14; 26; 27; 28; 29; 30; 41; 42]. This creates certain straits since these models require too much computing resources and/or the use of specialized software that is purchased separately. No attempts were found in the literature to add relief grooves to the significantly simpler and more resource-efficient 2D models, which still have their application in some studies of external gear pumps. This motivates us to prove whether and to what extent it is possible to account for the influence of relief grooves using a 2D model. **The main goal** of the article is to design an advanced 2D CFD model of an existing external gear pump considering the relief grooves' influence on the flow process. This article is organized as follows: Section 2 includes a detailed description of geometric model development; Section 3 presents the numerical results and discussion; Section 4 shows the experimental study and model validation; and in Section 5, some conclusions are given. Figure 2: Flow rate correction by increasing gear face width from 30.89 to 31.5 mm (\(q\)—flow rate; \(b\)—gear width, \(\varphi\)—gears position), as proposed in [3].