# Influence of the Radial Gap on the External Gear Pump Performance
## 2 Design Features and CFD Model
### CFD Model
The general form of the turbulence \(k\)-\(\epsilon\) (kinetic energy) equation is as follows:  
\[\frac{\partial}{\partial t}(\rho k)+\
abla\bigg{(}\partial k\overrightarrow{V }\bigg{)}=\
abla\bigg{[}\bigg{(}\mu+\frac{\mu_{i}}{\sigma_{k}}\bigg{)}\
abla k \bigg{]}+G_{k}+G_{b}-\rho\epsilon-Y_{M}+S_{k}, \tag{8}\]  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\
abla\bigg{(}\partial e \overrightarrow{V}\bigg{)}=\
abla\bigg{[}\bigg{(}\mu+\frac{\mu_{i}}{\sigma_{ k}}\bigg{)}\
abla\epsilon\bigg{]}+C_{1\epsilon}\frac{\epsilon}{K}(G_{k}+C_{3 \epsilon}G_{b})-C_{2\epsilon}\frac{\epsilon^{2}}{K}+S_{k}, \tag{9}\]  
where \(k\) is the kinetic energy, \(\epsilon\) is the kinetic energy dissipation rate, \(\sigma_{k}\) is the turbulent _Prandtl_ number for \(k\), \(G_{k}\) is the generation of turbulence kinetic energy (mean velocity gradients), \(G_{b}\) is the generation of turbulence kinetic energy due to buoyancy, \(\sigma_{\epsilon}\) is the turbulent _Prandtl_ number for \(\epsilon\), \(S_{k}\) is the user-defined source terms, and \(C_{1\epsilon}\), \(C_{2\epsilon}\), \(C_{3\epsilon}\), and \(C_{\mu}\) are the constants. The _Eddy_ viscosity \(\mu_{i}\) can be expressed by combining \(k\) and \(\epsilon\):  
\[\overrightarrow{\
abla}\cdot(\rho\overrightarrow{V})=0. \tag{2}\]0  
The initial condition includes the following: working fluid--hydraulic oil with a density of 890 kg/m\({}^{3}\) and a viscosity of 0.0712 kg/(m-s); a gravity of 9.81 m/s\({}^{2}\); an inlet pressure \(p_{in}\) = 101,325 Pa and an outlet pressure \(p_{out}\) = 10.5 MPa. The inlet and outlet pressures are applied normally to the boundary as shown in Figure 2a, with a turbulent intensity of 5% and a turbulent viscosity ratio of 10%. The rotational frequency \(n\) of the gears, as well as the gap between them and the housing, depends on the simulated operating mode. This is described in the next section. The used initial finite element mesh is shown in \[\overrightarrow{\
abla}\cdot(\rho\overrightarrow{V})=0. \tag{2}\]1 In the inlet and outlet channels, the mesh is built with structured rectangular elements with a size of 0.25 mm, and the element's shape and size are static during the solution. For the rest of the model, the mesh is formed by triangles with a basic element size of 0.14 mm. The mesh is thickened in the gaps, where the element sizes are between 0.0013 mm and 0.08 mm. The mesh around the gears is dynamically changed during the solution and the triangles sizes vary between 0.0007 and 0.32 mm with a maximum mesh skewness of 0.67. The total number of cells during the solution is between 79,821 and 65,401. More details about the model parameters are given in [41; 44]. Each simulation covers a minimum of 40\({}^{\circ}\) of a gear rotation. The first 10 degrees is not used for the result analysis. The average flow rate is calculated for the next interval of 30\({}^{\circ}\). In our previous studies [41; 44], it was found that subsequent 30-degree periods completely repeated at first, which is normal for a gear pump with 12 teeth in steady-state operation. Each simulation included between 1240 and 1720 steps and the angular step was varied over the course of the solution to achieve convergence, in the range of 0.024\({}^{\circ}\) to 0.036\({}^{\circ}\), with a typical value around 0.029\({}^{\circ}\). The simulation research showed that the solution is very sensitive to the choice of the angular step, with values around 0.03\({}^{\circ}\) proving optimal for the used CFD model. For example, when reducing the angular step to twice below these values, the noise in the solution slightly decreased, and the average flow rate slightly increased (by about 0.3%). On the other hand, however, decreasing the angular step led to a proportional increase in the number of required steps, and accordingly, in the total simulation time. At the same time, deviations from the given step values--both decreasing and increasing--led to a poor convergence and more frequent errors and interruptions.