1. Introduction
Aero fuel systems play a crucial role in providing stable and powerful propulsion for engines. As a key component of these systems, aero fuel gear pumps’ performance directly impacts the stability and reliability of engine operation. With the development of aero fuel gear pumps towards higher speeds and greater power, issues such as decreased volumetric efficiency, flow fluctuations, and vibration noise have become more prominent. Among these, the influence of gear teeth number on the performance of aero arc gear pumps is a topic that has received increasing attention.
Previous studies have investigated various aspects related to gear pumps. For example, Zhu et al. analyzed the lubrication characteristics and contact behavior of sliding bearings in gear pumps operating at high speeds and low – medium viscosities. Li Geqiang et al. compared the flow characteristics of double – circular – arc helical gear pumps with those of involute helical gear pumps under the same geometric parameters. However, the research on the influence of the number of gear teeth on the performance of aero arc gear pumps, especially in terms of flow pulsation characteristics in multi – tooth cavities, still needs to be further deepened. This study aims to fill this gap by comprehensively exploring the effects of gear teeth number on aero arc gear pumps.
2. Research Methods
2.1 Establishment of the Computational Model
The aero arc gear pump studied in this paper adopts an arc – involute – arc profile for the gear end – face. Taking a gear pump with 7 teeth as an example, the end – face profile equation is established based on previous research [16]. The rack pressure angle is selected as 14.5° according to reference [17], as this angle is more advantageous. The minimum number of teeth to avoid root cutting at this pressure angle is 6. Thus, computational models of 6 – 9 – tooth arc gear pumps are established.
| Gear Teeth Number (Z) | Helix Angle (β) | Center Distance (A) (mm) | Outer Diameter (mm) | Gear Width (B) (mm) | Import Diameter (Dm) (mm) | Export Diameter (Dcor) (mm) | Radial Clearance (δ) (mm) | Tooth Pitch Angle (τ) |
|---|---|---|---|---|---|---|---|---|
| 6 | 35.45° | 27.20 | 68.16 | 20 | 18 | 15 | 0.02 | \(2\pi/6\) |
| 7 | 32.14° | 28.00 | 68.16 | 20 | 18 | 15 | 0.02 | \(2\pi/7\) |
| 8 | 29.35° | 28.64 | 68.16 | 20 | 18 | 15 | 0.02 | \(2\pi/8\) |
| 9 | 26.97° | 29.16 | 68.16 | 20 | 18 | 15 | 0.02 | \(2\pi/9\) |
As shown in the table above, in order to ensure an axial overlap ratio of 1 and keep the outer diameter of the gear consistent, the helix angle β decreases as the number of gear teeth increases. And with the increase of the number of gear teeth, the number of transition chambers in the pump cavity increases, while the effective volume of the pump cavity decreases.
2.2 Numerical Simulation
2.2.1 Grid Division and Dynamic Grid Model
The entire grid area of the gear pump is divided into inlet, gear cavity, and outlet parts. Considering the small tooth – to – tooth clearance between the two meshing gears and the small radial clearance between the gear and the pump cavity, a quadrilateral structured grid is used for the gear rotation domain. Taking the 7 – tooth arc gear pump as an example, referring to the grid – independence verification method in reference [20], six calculation models with different grid numbers are generated by adjusting the grid division scale.
After verification, the grid number of the 7 – tooth gear pump calculation model is finally determined to be \(1.24×10^{6}\), with 360 layers of grids arranged circumferentially, 60 layers axially, 18 layers in the radial clearance, and 36 layers in the tooth – to – tooth clearance, generating a total of 40 sets of overlapping grids. According to similar grid division scales, the grid numbers of 6 – tooth, 8 – tooth, and 9 – tooth gear pump calculation models are \(1.41×10^{6}\), \(1.36×10^{6}\), and \(1.32×10^{6}\) respectively.
The 输送介质 is aviation kerosene, with physical parameters: \(\rho = 800 kg/m^{3}\), \(\mu=0.007 Pa·s\). The gear pump is set to rotate 10 circles, with each circle divided into 360 steps, and each step rotating 1°. When the relative pressure difference of the monitoring points at the corresponding time steps in adjacent two – cycle periods is less than 5%, it is considered that the periodic requirement is met, and the results of the last rotation are taken for analysis.
2.2.2 Control Equations and Turbulence Models
The fluid model and turbulence model can be directly called in PumpLinx. The turbulence parameters are obtained from the Navier – Stokes equations. The RNG \(k – \varepsilon\) model is used to solve the gear pump cavity. This model uses the renormalization group statistical technique to correct the turbulent viscosity and can handle flows with large curvature, strong rotation, and high strain rate well. The solid wall surface adopts the no – slip wall surface, and the standard wall – function is used in the near – wall region. The contribution of pulsating expansion in compressible turbulence is not considered, and the influence of gravity on the flow field is ignored. Pressure boundary conditions are set at the inlet and outlet.
2.3 Experimental Verification
To verify the accuracy of the numerical simulation method, a 7 – tooth gear pump is used as a prototype for experimental verification. A closed – loop test bench for the arc gear pump is built to test the outlet volume flow rate of the pump under five different working pressures. The test bench is mainly composed of a connection from the oil tank to the pump inlet and a connection from the pump outlet to the oil tank. The test instruments include a Siemens Bade frequency – conversion motor with a power of 7.5kW, a frequency converter S200 – G7.5/P11T4B selected according to the motor configuration, pressure, and flow requirements, a pressure sensor MIK – P300 with a measurement range of 30MPa and an accuracy of 0.5%, a flowmeter LC – A2 – 0.2/AIGFI with a measurement range of \(0.3 – 3m^{2}/h\) and an accuracy class of 0.5, a data collector of type USB3200 with a sampling rate of 500ks/s, a throttle valve with a diameter of DN20, and a safety valve with a diameter of DN10. The test medium is hydraulic oil.
During the test, the inlet pressure of the prototype is fixed at 0. As the motor drives the prototype to operate, the hydraulic oil is transported from the oil tank into the hydraulic pipeline. The pressure control valve is placed on the pump outlet side to adjust the pump outlet pressure. After the system runs stably, the outlet pressure is gradually adjusted, and the pressure and flow values are detected and transmitted to the data collector, and finally displayed on the upper – computer interface after processing.
3. Results and Analysis
3.1 Influence of the Number of Gear Teeth on the Outlet Flow Characteristics of the Arc Gear Pump
The relationship between the outlet flow rate and volumetric efficiency of 6 – 9 – tooth gear pumps under different working conditions is shown in the figure below.
It can be seen from the figure that the number of gear teeth has a significant impact on the pump outlet flow rate. With the increase of the inlet – outlet pressure difference, the outlet flow rate and volumetric efficiency of gear pumps with different numbers of teeth show a linear decrease. This is because as the pressure increases, the pressure gradient in the cavity increases, resulting in an increase in internal leakage. At the same time, the shaft power of the pump decreases, and the average outlet flow rate decreases.
As the number of gear teeth increases, the proportion of the gear volume increases, and the effective volume for transporting fluid in the pump cavity decreases. The theoretical displacement of 6 – 9 – tooth pumps shows a downward trend, but it has a positive effect on improving the volumetric efficiency. When the number of teeth increases to 9, the decline range of the pump’s average outlet flow rate and volumetric efficiency is the smallest.
The flow pulsation coefficient is defined as \(\delta_{Q}=\frac{Q_{max}-Q_{min}}{Q_{ave}}\), and the relationship between the flow pulsation coefficient and the number of gear teeth under different working conditions is shown in the figure below.
The number of gear teeth has a significant impact on the flow pulsation of the pump. In the process of the inlet – outlet pressure difference increasing from 2MPa to 10MPa, the flow pulsation coefficients of gear pumps with different numbers of teeth all show an upward trend. At the same inlet – outlet pressure difference, 7 – tooth and 9 – tooth pumps can maintain a relatively low flow pulsation. When the number of teeth is small, although the theoretical outlet flow rate is large, the number of transition chambers is small, the pressure difference change gradient between adjacent chambers is large, and the leakage is large. When the number of teeth is large, although the flow pulsation coefficient is small, the theoretical outlet flow rate is also small. Therefore, theoretically, there exists an optimal number of teeth under specific working conditions to make the internal flow characteristics of the gear the best.
3.2 Influence of the Number of Gear Teeth on the Internal Flow Characteristics of the Arc Gear Pump
Taking the simulation data of 6 – 9 – tooth pumps under the working conditions of 8MPa and 2000r/min as an example, the influence of the number of gear teeth on the internal flow characteristics and performance of the pump is analyzed. The outlet instantaneous flow characteristics curves of 6 – 9 – tooth gear pumps within one cycle are shown in the figure below.
The outlet instantaneous flow rates of gear pumps with different numbers of teeth show periodic changes, and the number of flow pulsations of the pump outlet is equal to the number of gear teeth. The number of gear teeth has a significant impact on the outlet flow pulsation of the pump. The flow reduction values from the peak to the trough of 7 – tooth and 9 – tooth pumps are relatively small, while those of 6 – tooth and 8 – tooth pumps are relatively large. All 6 – 9 – tooth gear pumps have different degrees of secondary flow pulsation.
When the number of teeth increases to 9, the pulsation characteristic curve gradually changes from an “M” – shaped peak to a “V” – shaped peak, and the pulsation amplitude decreases. The instantaneous flow rate of the arc gear pump is related to the number of teeth Z, helix angle β, and meshing angle φ. As the number of teeth changes, the tooth pitch angle τ changes accordingly, and the pitch circle radius R and base circle radius \(R_{b}\) also change.
The theoretical instantaneous flow rate of the arc gear pump is obtained by the swept – area method. After substituting relevant parameters and integrating, the theoretical instantaneous flow rate of the arc gear pump in different intervals can be obtained. The theoretical instantaneous flow rate pulsation coefficients of 6 – 9 – tooth gear pumps are \(4.50×10^{-5}\), \(3.03×10^{-5}\), \(4.04×10^{-5}\), and \(3.15×10^{-6}\) respectively, which reflects the advantage of the low – flow – pulsation of the arc gear pump from the side. With the increase of the number of teeth, the theoretical instantaneous flow rate pulsation coefficient generally shows a downward trend. However, the numerical simulation instantaneous flow rate pulsation coefficients are all increased to different degrees compared with the theoretical values, which is related to factors such as the extremely high pressure difference between the inlet and outlet, leakage caused by tooth – to – tooth/radial clearances, and the influence of the helix angle.
3.3 Influence of the Number of Gear Teeth on the Internal Pressure Characteristics of the Arc Gear Pump
The central section in the gear width direction is set as the pressure monitoring surface. The schematic diagrams of the internal pressure distribution of 6 – 9 – tooth gear pumps when meshing and rotating τ/4 are shown in the figure below.
With the increase of the number of teeth, the number of serially connected transition chambers in the gear cavity gradually increases. The pressure difference between adjacent independent chambers gradually decreases, and the pressure drop gradient gradually decreases. The pressure distribution in independent chambers is more and more the same when the gear rotates τ/4.
In the gear meshing process, the areas of the high – pressure and low – pressure regions in the pump cavity change periodically. Due to the high inlet – outlet pressure difference, the shear force on the fluid element and the relative movement of the fluid particles increase, and the flow continuity weakens, resulting in a large pressure difference change during fluid transfer. At the inlet end, due to the local vacuum formed by the increase in the variable volume of the bottom cavity hole, vortices are likely to occur below the tooth – to – tooth opening. In addition, the pressure difference between the inlet end and the first rotating 1D/1S control volume is large, and the fluid velocity in the extrusion area is likely to be distorted, forming vortex flows below the left and right gear cavity walls. As the gear rotates, part of the vortex flow moves with the gear rotation, forming vortices in the effective volume between the gear and the cavity wall, and distributing axially, which strengthens the flow instability. With the increase of the number of teeth, the position where the vortices concentrate gradually shifts from near the center of the effective volume to the edge of the cavity wall, which has a positive effect on the stability of the fluid movement in the cavity.
Some of the vortex flows at the inlet end continue to disturb the fluid entering the gear cavity later, causing wear and impact losses on the gear cavity wall, and some leak along the flow path. As the gear continues to rotate, the vortices continue to evolve towards the outlet end. Due to the interaction of the left and right gears, a pair of counter – rotating vortices are formed at the front and rear of the outlet end. Due to the effects of the helix angle and gear width, the flow states and sizes of these counter – rotating vortices at the front and rear of the outlet end are different, and they are asymmetric along the central axis, affecting the unbalance of the fluid pressure on the two gears.
The tooth – to – tooth leakage and radial leakage also affect the internal pressure characteristics of the pump. The leakage between two gears depends on the opening area and the pressure difference between the upper and lower parts of the opening. The number of teeth affects the action line of a pair of meshing helical gears and the change of the opening area. For radial leakage (the radial clearance in this model is set to 0.02mm), the flow here can be laminar or turbulent. The radial leakage model uses a modified leakage equation \(Q = c_{p}A_{k}\sqrt{\frac{2|\Delta P|}{\rho}}\). With the increase of the number of teeth, the \(\Delta P\) between adjacent transition chambers decreases, which can inhibit radial leakage to a certain extent. Therefore, appropriately increasing the number of teeth can effectively suppress the flow instability caused by the pressure difference during the pump operation.
3.4 Influence of the Number of Gear Teeth on the Excitation Force of the Arc Gear
Taking the driven gear as the research object, its radial force and axial force are analyzed. The pulsation amplitude diagrams of the forces on the 6 – 9 – tooth driven gear within one cycle are shown in the figure below.
The radial and axial forces on gears with different numbers of teeth show periodic changes, and the number of gear force pulsations is in a multiple relationship with the number of teeth. As the number of teeth increases, the maximum value of the resultant force F and the maximum value of \(F_{z}\) decrease. The gear force changes are related to gear meshing and the interaction between the gear and the fluid in the cavity. When the number of teeth is small, such as 6 – tooth gear, the number of transition chambers is small, the pressure difference between the transition chamber and the high – pressure chamber at the outlet is large, and the backflow phenomenon is likely to occur, resulting in a reverse acting force on the gear. At the same time, the effective volume in the 6 – tooth gear cavity is large, and the flow separation and secondary flow phenomena are more likely to occur.
Through Matlab wavelet transform of the radial and axial forces, the pulsation spectra of the 6 – 9 – tooth gear’s radial and axial forces within one cycle are obtained, as shown in the figure below.
4. Comprehensive Analysis of the Interaction between Various Factors
4.1 Interaction between Flow, Pressure, and Force Characteristics
The flow, pressure, and force characteristics of the aero arc gear pump are interrelated. The change in the number of gear teeth affects the flow characteristics, which in turn influences the pressure distribution in the pump cavity. As the number of teeth increases, the flow pulsation coefficient decreases, and the flow becomes more stable. This leads to a more uniform pressure distribution in the pump cavity, reducing the pressure difference between adjacent chambers.
The change in pressure distribution further affects the forces acting on the gears. A more stable pressure distribution results in a decrease in the radial and axial forces on the gears, reducing the vibration and noise of the pump. For example, when the number of teeth increases from 6 to 9, the reduction in the pressure difference between adjacent chambers weakens the reverse acting force on the gear caused by backflow, and also reduces the likelihood of flow separation and secondary flow, thereby reducing the forces on the gear.
4.2 Influence of Operating Conditions on the Interaction
Operating conditions such as rotational speed and inlet – outlet pressure difference also play an important role in the interaction between these characteristics. When the rotational speed increases, the theoretical outlet flow rate and volumetric efficiency of the pump increase, but the flow pulsation coefficient may change. High rotational speeds can reduce the time for adjacent teeth to mesh, making the pressure difference change between fluids smaller and the flow more continuous. However, too high a rotational speed may increase the tendency of cavitation in the cavity, affecting the flow quality of the gear pump.
The inlet – outlet pressure difference has a significant impact on the internal leakage and pressure distribution of the pump. A large pressure difference will increase the internal leakage, reducing the volumetric efficiency and affecting the flow characteristics. At the same time, it will also increase the forces on the gears, especially the radial force, which may lead to greater wear and vibration of the pump.
5. Optimization Strategies Based on Research Results
5.1 Tooth Number Optimization
Based on the research results, when designing an aero arc gear pump, an appropriate number of teeth should be selected according to the specific operating conditions. If the pump requires high – pressure operation and low – flow pulsation, 7 – tooth or 9 – tooth gears can be considered. Although the theoretical outlet flow rate of 9 – tooth gears is relatively small, their volumetric efficiency is high, and the flow pulsation coefficient is low, which can meet the requirements of stable flow under high – pressure conditions.
For applications where a large flow rate is required, a smaller number of teeth can be selected on the premise of ensuring a certain volumetric efficiency and flow stability. However, it should be noted that a smaller number of teeth may lead to larger flow pulsation and forces on the gears, so comprehensive consideration is needed.
5.2 Structural Optimization
In addition to optimizing the number of teeth, other structural parameters of the gear pump can also be optimized. For example, the helix angle can be adjusted to improve the flow characteristics and reduce the forces on the gears. A reasonable helix angle can increase the axial overlap ratio, making the meshing of gears more stable and reducing the impact of flow pulsation.
The design of the pump cavity shape can also be optimized. By reducing the sudden change of the flow passage cross – section and improving the smoothness of the flow passage, the flow resistance can be reduced, and the internal flow state of the pump can be improved, which is conducive to reducing the flow pulsation and improving the efficiency of the pump.
5.3 Operating Condition Optimization
Proper adjustment of operating conditions can also improve the performance of the aero arc gear pump. For example, when the pump is operating at a high – pressure and high – flow rate, the rotational speed can be adjusted appropriately to balance the flow characteristics, pressure distribution, and forces on the gears. Reducing the rotational speed slightly can reduce the tendency of cavitation and the forces on the gears, while ensuring a certain flow rate.
Monitoring and controlling the inlet – outlet pressure difference in real – time can also help to optimize the operation of the pump. When the pressure difference is too large, measures can be taken to reduce the load on the pump, such as adjusting the outlet flow rate or using a pressure – regulating device, to prevent excessive internal leakage and damage to the pump components.
6. Conclusion
This study comprehensively investigated the flow pulsation characteristics of aero arc gear pumps with multiple teeth. Through the establishment of computational models, numerical simulations, and experimental verifications, the influence of the number of gear teeth on the flow, pressure, and force characteristics of the pump was analyzed in detail.
The results show that the number of gear teeth has a significant impact on the performance of the aero arc gear pump. With the increase of the number of teeth, the volumetric efficiency generally increases, and the flow pulsation coefficient and the forces on the gears decrease. However, the theoretical outlet flow rate decreases. There exists an optimal number of teeth under specific operating conditions to achieve the best internal flow characteristics of the gear.
The interaction between flow, pressure, and force characteristics is complex, and operating conditions also have a significant impact on this interaction. Based on the research results, optimization strategies for tooth number, structure, and operating conditions are proposed to improve the performance of the aero arc gear pump.
In future research, more in – depth studies can be carried out on the influence of complex working conditions and the coupling effect of multiple factors on the performance of aero arc gear pumps. At the same time, experimental research can be further strengthened to verify and improve the theoretical and simulation results, providing more reliable support for the design and application of aero arc gear pumps.
