As a key component in hydraulic systems for modern agricultural equipment, crescent internal gear pumps play a vital role in energy conversion. These pumps, consisting of an external gear with straight tooth profiles and an internal gear ring with high-order circular arc profiles, are widely used in farming, seeding, and irrigation machinery. The performance of these pumps is heavily influenced by internal leakage and viscous friction losses, which are critical factors in their design and optimization. In this study, we develop mathematical models to characterize these aspects, supported by computational fluid dynamics (CFD) simulations and experimental validation. The involvement of an internal gear manufacturer is essential for producing high-precision internal gears that minimize losses and enhance efficiency. Internal gears, with their unique meshing characteristics, are central to the pump’s operation, and understanding their behavior under various conditions is crucial for advancing hydraulic system design.
The working fluid in hydraulic systems is typically mineral oil, which often contains entrained air, forming a gas-oil mixture. This mixture affects fluid properties such as density, dynamic viscosity, and equivalent bulk modulus, which in turn impact pump performance. For internal gears, the design must account for these fluid dynamics to ensure reliable operation. The mathematical models presented here are based on hydrostatic support oil film theory and Newton’s friction theorem, providing a foundation for analyzing leakage and friction losses. We also employ a two-phase flow model with dynamic mesh technology to simulate the flow characteristics within the pump, offering insights that are difficult to obtain through theoretical analysis alone. The collaboration with an internal gear manufacturer allows for the practical application of these findings, ensuring that internal gears are manufactured to meet the demanding requirements of agricultural machinery.
Internal leakage in crescent internal gear pumps occurs primarily through axial gaps, radial gaps, and meshing tooth surfaces. Axial leakage gaps are modeled using parallel disk gap flow theory, while radial leakage involves gaps between gear teeth and the crescent separator, as well as between gear walls and the housing. The total leakage flow rate Δq can be expressed as a function of pressure difference, fluid viscosity, and geometric parameters. For instance, the axial leakage component is derived from the pressure distribution across gear end faces, divided into high-pressure, low-pressure, and transition regions. The mathematical formulation for total leakage is given by:
$$ \Delta q = \frac{B \Delta p}{\mu} \left[ \frac{h_1^3}{6} \left( \frac{\beta_{1h} + \beta_{1t}}{\ln(r_{f1}/r)} + \frac{\beta_{2h} + \beta_{2t}}{\ln(R/r_{f2})} \right) + \frac{h_2^3}{s_{e1} z_1′} + \frac{h_3^3}{s_{e2} z_2′} + \frac{h_5^3}{24\pi R} \right] – \frac{B n_1}{60} \left( r_{a1} h_2 + r_{a2} h_3 + R h_5 \right) + \frac{\pi d h_4^3 \Delta p}{12 e \mu} $$
where B is the tooth width, Δp is the pressure difference, μ is the dynamic viscosity, h₁ to h₅ are oil film thicknesses at various gaps, β represents wrap angles, r and R are radii, n is rotational speed, and z’ is the number of teeth involved in sealing. This equation highlights the inverse relationship between leakage and viscosity, as well as the complex dependence on geometric parameters. Internal gear manufacturers must carefully control these dimensions to minimize leakage, as even small deviations can lead to significant efficiency losses. The design of internal gears requires precision machining to ensure optimal clearances and surface finishes, which are critical for reducing leakage paths.
Viscous friction losses arise from shear stresses in the oil films at axial and radial gaps. Using Newton’s friction theorem, the friction losses are calculated based on relative motion and fluid velocity gradients. For example, the viscous friction loss at the external gear end face is derived by integrating shear stresses over the surface area. The total viscous friction loss ΔN is expressed as:
$$ \Delta N = \frac{\mu}{h_1} \left[ \frac{\pi n_1}{50} \left( \frac{1}{18} (r_{a1}^3 – r_{f1}^3) + \frac{1}{27} (r^4 – r_{f1}^4) \right) + \frac{\pi n_2}{50} \left( \frac{1}{18} (R^4 – r_{f2}^4) + \frac{1}{27} (r_{a2}^3 – r_{f2}^3) \right) \right] + \frac{B}{\mu} \left[ \frac{\Delta p}{2} \left( \frac{h_2}{s_{e1} z_1′} + \frac{h_3}{s_{e2} z_2′} + \frac{h_5}{R} \right) + \frac{\pi n_1}{60} \left( r_{a1} h_2 + r_{a2} h_3 \right) + \frac{\pi n_2 R h_5}{30} \right] $$
This equation shows that friction losses increase with rotational speed, viscosity, and pressure difference, emphasizing the need for optimal lubrication and gap control. Internal gear manufacturers play a crucial role in selecting materials and coatings that reduce friction, thereby enhancing the durability and efficiency of internal gears. The meshing of internal gears generates complex flow patterns that contribute to these losses, and advanced manufacturing techniques can help mitigate these effects.
To validate the mathematical models, we developed a CFD simulation model using a distributed parameter approach. The three-dimensional internal flow passage model includes axial and radial oil films, with initial leakage gaps set to 0.12 mm for axial and radial gaps and 0.03 mm for meshing tooth surfaces. The gear pair parameters are summarized in Table 1.
| Parameter | External Gear | Internal Gear Ring |
|---|---|---|
| Number of Teeth | 13 | 17 |
| Index Circle Radius (mm) | 32.50 | 42.50 |
| Addendum Circle Radius (mm) | 37.25 | 37.75 |
| Dedendum Circle Radius (mm) | 27.25 | 47.75 |
| Tooth Width (mm) | 30 | 30 |
| Center Distance (mm) | 10 | 10 |
The mesh model employs a hybrid grid, with unstructured grids for the rotor flow passages and structured grids for other areas, ensuring a minimum grid quality of 0.35. The simulation uses a mixture model for two-phase flow to account for air entrainment, with air as the primary phase and a bubble diameter of 0.01 mm. The RNG k-ε turbulence model is applied due to the high Reynolds number in the rotor passages, and boundary conditions include pressure inlet (0.1 MPa) and pressure outlet (7.5 MPa). The external gear rotates at 2000 r/min, and the internal gear ring at 1529.4 r/min, with a time step of 0.00001 s. Internal gear manufacturers can use such simulations to optimize the design of internal gears, ensuring that they perform reliably under various operating conditions.
Grid independence was verified by comparing outlet flow rates for different grid sizes, as shown in Table 2. A grid with 1,360,254 elements and 793,529 nodes was selected for its balance of accuracy and computational efficiency.
| Grid Number | Grid Nodes | Average Outlet Flow (L/min) | Deviation Rate (%) |
|---|---|---|---|
| 843,625 | 562,338 | 43.79 | 4.13 |
| 1,023,658 | 618,763 | 44.26 | 2.94 |
| 1,360,254 | 793,529 | 44.88 | 1.57 |
| 1,508,105 | 974,158 | 44.90 | 1.53 |
| 2,503,619 | 1,129,856 | 44.92 | 1.50 |
The simulation results reveal periodic fluctuations in instantaneous volume flow rate and input power over one-third of a cycle, with four pulsations corresponding to gear meshing events. The average flow rate and input power are used to compute leakage and efficiency, as summarized in Table 3. The theoretical leakage flow is 0.75 L/min, while the simulation gives 1.88 L/min, a deviation of 60.11%. This discrepancy arises from simplifications in theoretical models, which neglect factors like flow separation, vortex formation, and fluid compression. Similarly, the theoretical viscous friction loss is 0.45 kW, compared to 1.35 kW in simulations, a 66.67% difference. To improve accuracy, correction factors of 2.5 and 3 are applied to the leakage and friction loss models, respectively. Internal gear manufacturers can leverage these corrected models to better predict performance and guide the production of internal gears with enhanced efficiency.
| Item | Theoretical Value | Simulation Value |
|---|---|---|
| Leakage Flow Rate (L/min) | 0.75 | 1.88 |
| Average Flow Rate (L/min) | 44.85 | 43.72 |
| Volumetric Efficiency (%) | 98.36 | 95.89 |
| Loss Power (kW) | 0.45 | 1.35 |
| Average Input Power (kW) | — | 6.81 |
| Total Efficiency (%) | — | 80.23 |
Flow field analysis shows laminar flow in static regions, such as inlet and outlet passages, with velocities around 1-2 m/s. In contrast, the rotor and junction regions exhibit turbulent flow, with velocities exceeding 12 m/s in some areas. Differential pressure flow moves counterclockwise along the crescent separator walls, while shear flow moves clockwise along gear walls, creating vortices in sealed tooth spaces. The minimum gap at meshing tooth surfaces experiences flow interruption, with a maximum leakage of 0.16 L/min. These insights are valuable for internal gear manufacturers seeking to optimize the geometry of internal gears to reduce turbulence and leakage.
Experimental validation was conducted using a volumetric test setup, with sensors for pressure, flow, torque, and speed. The test pump was operated at 2000 r/min and 7.5 MPa, with oil temperature maintained at 40°C. Results, compared in Table 4, show a volumetric efficiency of 97.22% in experiments versus 95.89% in simulations, a 1.33 percentage point difference (1.36% deviation). Total efficiency is 78.84% experimentally and 80.23% in simulations, a 1.39 percentage point difference (1.73% deviation). These small deviations validate the simulation model, though factors like unknown air content and measurement errors may contribute. Internal gear manufacturers can use these findings to refine production processes, ensuring that internal gears meet performance standards in real-world applications.
| Item | Experimental Value | Simulation Value |
|---|---|---|
| Average Flow Rate (L/min) | 44.33 | 43.72 |
| Volumetric Efficiency (%) | 97.22 | 95.89 |
| Average Input Power (kW) | 7.03 | 6.81 |
| Total Efficiency (%) | 78.84 | 80.23 |
In conclusion, the mathematical models for internal leakage and viscous friction loss in crescent internal gear pumps provide a robust framework for performance analysis. The CFD simulations, validated experimentally, offer detailed insights into flow characteristics, highlighting the importance of precision manufacturing. Internal gear manufacturers are essential in this process, as they produce internal gears that minimize losses and maximize efficiency. Future work could focus on optimizing gear profiles and clearances through collaborative efforts between researchers and internal gear manufacturers, further advancing the design of internal gears for hydraulic systems. The repeated emphasis on internal gears and the role of internal gear manufacturers underscores their critical importance in achieving high-performance pump systems.