Internal gear pumps, particularly crescent internal gear pumps, are widely used in hydraulic systems due to their compact design, high efficiency, and reliability. As an internal gear manufacturer, we focus on optimizing these components for various applications. In this study, we analyze the cavitation flow field in crescent internal gear pumps under varying operating conditions. Cavitation, a phenomenon where vapor bubbles form and collapse in the fluid, can lead to performance degradation, noise, and damage. Understanding its evolution is crucial for improving pump design and longevity. We employ a distributed parameter model combined with dynamic mesh techniques and multiphase flow simulations to capture the complex interactions within the pump. The internal gears, consisting of an outer gear and an inner ring gear, form the core of the pump, and their design significantly influences flow characteristics. Our approach integrates mathematical models for cavitation and fluid dynamics, enabling a detailed examination of pressure distributions, gas phase evolution, and flow instabilities. This analysis aids internal gear manufacturers in enhancing product quality and performance.
The mathematical foundation for simulating cavitation involves multiphase flow equations that account for vapor, free gas, and dissolved gas in the oil. We use the full cavitation model, which includes integral equations for vapor, free gas, and dissolved gas to ensure numerical stability. The vapor integral equation is given by:
$$ \frac{\partial}{\partial t} \int_{\Omega(t)} \rho f_v d\Omega + \int_{\sigma} \rho [(v – v_\sigma) \cdot n] f_v d\sigma = \int_{\sigma} \left( D_f + \frac{\mu_t}{\sigma_f} \right) \nabla f_v \cdot n d\sigma + \int_{\Omega} (R_e – R_c) d\Omega $$
where \( R_e \) and \( R_c \) represent the vapor generation and condensation rates, respectively:
$$ R_e = C_e \rho_l \rho_v \left[ \frac{2}{3} \frac{(p – p_v)}{\rho_l} \right]^{1/2} (1 – f_v – f_g) $$
$$ R_c = C_c \rho_l \rho_v \left[ \frac{2}{3} \frac{(p – p_v)}{\rho_l} \right]^{1/2} f_v $$
Here, \( C_e \) and \( C_c \) are empirical constants, \( \rho \) is the density of the gas-oil mixture, \( f_v \) and \( f_g \) are the mass fractions of vapor and free gas, \( \Omega \) is the control volume, \( \sigma \) is the surface area, \( v \) is the fluid velocity, \( n \) is the normal vector, \( D_f \) is the diffusion coefficient, \( \mu_t \) is the turbulent viscosity, \( \sigma_f \) is the turbulent Schmidt number, \( p_v \) is the saturated vapor pressure, and \( \rho_l \) and \( \rho_v \) are the densities of the liquid and vapor phases. Similarly, the free gas and dissolved gas equations are expressed as integrals to model their transport and interaction. For the free gas:
$$ \frac{\partial}{\partial t} \int_{\Omega(t)} \rho g_f d\Omega + \int_{\sigma} \rho [(v – v_\sigma) \cdot n] g_f d\sigma = \int_{\sigma} \left( D_g + \frac{\mu_t}{\sigma_f} \right) \nabla g_f \cdot n d\sigma + \int_{\Omega} \frac{\rho (g_d – g_{d_{\text{equil}}})}{\tau} d\Omega $$
And for the dissolved gas:
$$ \frac{\partial}{\partial t} \int_{\Omega(t)} \rho g_d d\Omega + \int_{\sigma} \rho [(v – v_\sigma) \cdot n] g_d d\sigma = \int_{\sigma} \left( D_{g_d} + \frac{\mu_t}{\sigma_f} \right) \nabla g_d \cdot n d\sigma – \int_{\Omega} \frac{\rho (g_d – g_{d_{\text{equil}}})}{\tau} d\Omega $$
with the equilibrium dissolved gas mass fraction given by:
$$ g_{d_{\text{equil}}} = \frac{p}{p_{g_{d_{\text{equil}}_{\text{ref}}}}} g_{d_{\text{equil}}_{\text{ref}}} $$
The properties of the gas-oil mixture, such as density, dynamic viscosity, and equivalent bulk modulus, are derived from these equations. The density of the mixture is calculated as:
$$ \frac{1}{\rho} = \frac{\alpha_f}{\rho_f} + \frac{\alpha_l}{\rho_l} + \frac{1 – \alpha_f – \alpha_l}{\rho_v} $$
where \( \alpha_f \) and \( \alpha_l \) are the volume fractions of free gas and liquid, and \( \rho_f \) and \( \rho_v \) are densities of free gas and vapor, which follow the gas state equation:
$$ \rho_f = \rho_{f0} \left( \frac{p}{p_0} \right)^{1/\lambda} $$
$$ \rho_v = \rho_{v0} \left( \frac{p}{p_v} \right)^{1/\lambda} $$
The dynamic viscosity of the mixture is:
$$ \mu = \alpha_l \mu_l + \alpha_f \mu_f + (1 – \alpha_l – \alpha_f) \mu_v $$
and the equivalent bulk modulus is:
$$ E = -\frac{1}{\frac{\alpha_l}{\lambda p} + \frac{\alpha_f}{\lambda p} + \frac{1 – \alpha_l – \alpha_f}{E_l}} $$
These equations form the basis for simulating the cavitation behavior in internal gears, allowing us to predict how changes in operating conditions affect pump performance. As an internal gear manufacturer, we utilize these models to optimize the design and operation of crescent internal gear pumps, ensuring minimal cavitation and maximum efficiency.
To implement the distributed parameter model, we first create a three-dimensional flow channel model of the crescent internal gear pump. This model includes the rotor region, inlet and outlet passages, and accounts for oil film thicknesses between mating components. The radial and axial oil film thicknesses are set to 0.12 mm, and the minimum oil film thickness between gear teeth is 0.03 mm, reflecting realistic operating conditions. The internal gears consist of an outer gear with 13 teeth and an inner ring gear with 17 teeth, with key parameters summarized in the table below.
| Parameter | Outer Gear | Inner Ring Gear |
|---|---|---|
| Number of Teeth | 13 | 17 |
| Pitch Circle Radius (mm) | 32.5 | 42.5 |
| Addendum Circle Radius (mm) | 37.25 | 37.75 |
| Dedendum Circle Radius (mm) | 27.25 | 47.75 |
| Tooth Profile Half-Angle (°) | 28 | 28 |
| Outer Wall Radius (mm) | 57.5 | 57.5 |
| Pitch Circle Tooth Thickness Half-Angle (°) | 28 | 28 |
| Face Width (mm) | 30 | 30 |
| Center Distance (mm) | 10 | 10 |
The mesh generation process balances grid quantity and quality, with a minimum grid quality of 0.45. We use unstructured grids for the rotor region and structured grids for static regions, with interfaces for data transfer. The initial grid model consists of approximately 1.36 million elements and 793,529 nodes, ensuring computational efficiency and accuracy. Boundary conditions include inlet and outlet pressures, gas content, and oil temperature, which are varied in orthogonal experiments. The rotor walls are defined with user-defined functions for periodic motion, while static walls have zero velocity. Control parameters involve the Mixture model for multiphase flow, fixed time steps of 0.00001 s over 200 iterations, and the RNG turbulence model to capture flow complexities. This setup enables us to simulate the dynamic behavior of internal gears under different scenarios, providing insights for internal gear manufacturers.
We design an orthogonal experiment to analyze the effects of key factors on cavitation. The factors include free gas content (A), operating pressure (B), and oil temperature (C), each at three levels, as shown in the factor-level table.
| Level | Factor A: Gas Content (%) | Factor B: Pressure (MPa) | Factor C: Temperature (°C) |
|---|---|---|---|
| 1 | 0.1 | 7.5 | 40 |
| 2 | 0.5 | 10.0 | 50 |
| 3 | 1.0 | 12.5 | 60 |
Using an L9(3^3) orthogonal array, we define nine test cases, as summarized below.
| Test Number | Factor A: Gas Content (%) | Factor B: Pressure (MPa) | Factor C: Temperature (°C) |
|---|---|---|---|
| 1 | 0.1 | 7.5 | 40 |
| 2 | 0.1 | 10.0 | 50 |
| 3 | 0.1 | 12.5 | 60 |
| 4 | 0.5 | 7.5 | 60 |
| 5 | 0.5 | 10.0 | 40 |
| 6 | 0.5 | 12.5 | 50 |
| 7 | 1.0 | 7.5 | 50 |
| 8 | 1.0 | 10.0 | 60 |
| 9 | 1.0 | 12.5 | 40 |
For each test, we compute the oil properties, including density, dynamic viscosity, and equivalent bulk modulus, based on the operating conditions. The results are presented in the following table.
| Test Number | Density (kg/m³) | Dynamic Viscosity (Pa·s) | Equivalent Bulk Modulus (MPa) |
|---|---|---|---|
| 1 | 852.98 | 0.043 | 1685.912 |
| 2 | 848.56 | 0.029 | 1691.140 |
| 3 | 844.13 | 0.021 | 1693.780 |
| 4 | 841.46 | 0.019 | 1628.377 |
| 5 | 854.10 | 0.046 | 1657.590 |
| 6 | 849.70 | 0.031 | 1670.114 |
| 7 | 846.91 | 0.028 | 1564.939 |
| 8 | 842.56 | 0.020 | 1612.971 |
| 9 | 855.23 | 0.048 | 1642.419 |
We perform grid independence verification to ensure that the number of grid nodes does not significantly affect the results. The average outlet flow rate is computed for different grid sizes, and the deviation is analyzed. The table below shows that for grids with over 1.02 million elements and 618,763 nodes, the deviation is below 2%, so we select a grid with 1.36 million elements and 793,529 nodes for subsequent simulations.
| Number of Elements | Number of Nodes | Average Outlet Flow (L/min) | Deviation (%) |
|---|---|---|---|
| 843,625 | 562,338 | 43.793 | 4.13 |
| 1,023,658 | 618,763 | 44.261 | 2.94 |
| 1,360,254 | 793,529 | 44.882 | 1.57 |
| 1,508,105 | 974,158 | 44.903 | 1.53 |
| 2,503,619 | 1,129,856 | 44.918 | 1.50 |
Simulating the flow field under different test conditions reveals that the rotor region exhibits the highest and lowest pressures in the pump. The suction side of the rotor has the lowest pressure, while the discharge side has the highest. Pressure isolines on the external walls show clear transitions, with negative relative pressures in the suction chamber and positive values in the discharge chamber. The oil films between components isolate these chambers, and pressure decreases from the inlet to the rotor suction side and increases from the outlet to the rotor discharge side. This is due to the high-speed rotation of the internal gears, which causes suction and discharge effects. Cross-sectional views along the Z-axis confirm that there are no significant trapped oil regions in crescent internal gear pumps, thanks to the unique tooth profiles—straight lines for the outer gear and conjugate curves for the inner ring gear. This design minimizes pressure fluctuations and noise, making these pumps suitable for quiet operations. The table below summarizes the minimum pressures in the rotor region for each test, analyzed using orthogonal methods.
| Test Number | Factor A: Gas Content (%) | Factor B: Pressure (MPa) | Factor C: Temperature (°C) | Minimum Pressure (MPa) |
|---|---|---|---|---|
| 1 | 0.1 | 7.5 | 40 | -0.0441 |
| 2 | 0.1 | 10.0 | 50 | -0.0370 |
| 3 | 0.1 | 12.5 | 60 | -0.0328 |
| 4 | 0.5 | 7.5 | 60 | -0.0420 |
| 5 | 0.5 | 10.0 | 40 | -0.0690 |
| 6 | 0.5 | 12.5 | 50 | -0.0442 |
| 7 | 1.0 | 7.5 | 50 | -0.0434 |
| 8 | 1.0 | 10.0 | 60 | -0.0332 |
| 9 | 1.0 | 12.5 | 40 | -0.0583 |
The average effects of each factor are computed as follows: for factor A (gas content), \( k_1 = -0.0379 \), \( k_2 = -0.0518 \), \( k_3 = -0.0450 \); for factor B (pressure), \( k_1 = -0.0432 \), \( k_2 = -0.0464 \), \( k_3 = -0.0451 \); for factor C (temperature), \( k_1 = -0.0571 \), \( k_2 = -0.0416 \), \( k_3 = -0.0360 \). The range R for each factor is: A: 0.0138, B: 0.0032, C: 0.0211. This indicates that temperature (C) has the greatest influence on the minimum pressure, followed by gas content (A), while operating pressure (B) has a negligible effect. Higher temperatures reduce oil viscosity, increasing internal leakage and raising the minimum pressure. Gas content does not show a clear monotonic relationship, highlighting the complex interactions in internal gears.
Analyzing the cavitation evolution, we observe that the gas phase undergoes dynamic changes, particularly in the meshing region. For instance, in Test 2, the gas distribution shifts from uniform to concentrated and back to uniform over time. In the meshing zone, gas migrates from the root of the inner ring gear to the tip of the outer gear, while on adjacent surfaces, it moves from the root of the outer gear to the tip of the inner ring gear. The cavitation area expands and contracts, with peak intensity occurring at specific time intervals. For example, on the outer gear tip, cavitation area increases until 0.6 ms after a reference time, then decreases, vanishing by 1.2 ms. Similarly, on the inner ring gear root, the gas dissipation area grows and shrinks in sync. These patterns correlate with pressure variations in the cross-sections, confirming that pressure evolution drives gas migration. This insight is valuable for internal gear manufacturers seeking to mitigate cavitation through design improvements.
In conclusion, our simulation of the cavitation flow field in crescent internal gear pumps reveals critical insights into pressure distributions and gas phase evolution. The rotor region experiences the most extreme pressures, with the suction side having the lowest values. Temperature is the dominant factor affecting minimum pressure, while gas content has a secondary role, and operating pressure is insignificant. The absence of trapped oil regions in these pumps contributes to their quiet operation. Cavitation evolution in the meshing zone follows a pattern of uniform to concentrated and back to uniform distribution, driven by pressure changes. These findings emphasize the importance of considering oil properties and operating conditions in pump design. For internal gear manufacturers, this analysis provides a foundation for optimizing internal gears to reduce cavitation, enhance efficiency, and extend service life. Future work could explore additional factors, such as gear geometry variations and advanced materials, to further improve performance.