In hydraulic systems, the performance of hydraulic pumps is critical for stability and reliability. Internal gear pumps, known for their compact structure, self-priming capability, and contamination resistance, are widely used in various industries. As an internal gear manufacturer, we focus on optimizing these components, particularly straight line conjugate internal gears, which feature a straight-tooth external gear and a high-order circular-arc internal gear. This design offers advantages in reducing noise and vibration, but its flow characteristics are highly sensitive to oil properties. Understanding how oil attributes like density, viscosity, and effective bulk modulus affect flow pulsation, pressure pulsation, and efficiency is essential for improving pump performance. In this study, we systematically analyze oil properties, develop mathematical models, and employ computational fluid dynamics (CFD) simulations combined with orthogonal experiments to investigate these effects. The findings provide insights for internal gear manufacturers to enhance pump design under variable operating conditions.
Oil properties play a pivotal role in hydraulic systems, often referred to as the “blood” of the system due to their function in energy transmission. Variations in operating conditions, such as pressure, temperature, and air content, fundamentally alter oil characteristics, impacting pump performance. For internal gears, these changes can lead to significant flow and pressure pulsations, affecting overall efficiency and noise levels. We begin by establishing mathematical models for key oil properties, including density, absolute viscosity, and effective bulk modulus, based on empirical formulas. These models help quantify how factors like dissolved air, working pressure, and oil temperature influence oil behavior. For instance, density ($\rho$) is described by the following equation, which accounts for pressure ($p$) and temperature ($T$):
$$ \rho = \frac{\rho_b (1 – \beta T)}{1 – A \log \left( 1 + \frac{p – p_{0\text{eos}}}{B_1 + B_2 T + B_3 T^2 + B_4 T^3} \right)} $$
Here, $\rho_b$ is the density at standard conditions, $\beta$ is the volumetric expansion coefficient, $p_{0\text{eos}}$ is the operating pressure, and $A$, $B_1$ to $B_4$ are parameters related to density. Similarly, absolute viscosity ($\mu$) varies with pressure and temperature, as given by:
$$ \mu = 0.0457 \exp \left\{ 6.58 \left( (1 + 5.1 \times 10^{-9} p)^{2.3 \times 10^{-8}} \left[ \frac{T – 138}{303 – 138} \right]^{-1.16} – 1 \right) \right\} $$
Furthermore, the effective bulk modulus ($K_{\text{ef}}$), which indicates oil compressibility, is modeled using the IFAS approach, considering initial air content ($\alpha$), initial pressure ($p_0$), and the adiabatic index ($k = 1.4$):
$$ K_{\text{ef}} = \frac{(1 – \alpha) \left[ 1 + \frac{m(p – p_0)}{E_{\text{oil}}} \right]^{-1/m} + \alpha \left( \frac{p_0}{p} \right)^{1/k} }{ \frac{1 – \alpha}{E_{\text{oil}}} \left[ 1 + \frac{m(p – p_0)}{E_{\text{oil}}} \right]^{-(m+1)/m} + \frac{\alpha}{k p_0} \left( \frac{p_0}{p} \right)^{(k+1)/k} } $$
where $E_{\text{oil}} = E_{0,T_0} + m p + k_T T$, with $E_{0,T_0}$ as the theoretical bulk modulus at 0°C and 0 Pa, $m = 11.4$, and $k_T = -8$ MPa/°C. The speed of sound ($c_0$) in oil, crucial for CFD simulations, is derived as:
$$ c_0 = \sqrt{ \frac{K_{\text{ef}} – (p – p_0)}{\rho_0} } $$
These equations reveal that variable operating conditions are the essence of oil property changes. For example, as pressure and temperature increase, density initially rises slightly then decreases, while viscosity drops significantly, and the effective bulk modulus increases, with air content having a notable impact on compressibility. This foundation allows us to analyze how these properties affect the flow characteristics of straight line conjugate internal gear pumps, which are critical for internal gear manufacturers aiming to produce efficient and reliable components.
To efficiently study the impact of oil properties, we designed an orthogonal experiment that minimizes the number of tests while providing comprehensive data. The factors considered are air content ($\alpha$), working pressure ($p$), and oil temperature ($\theta$), each at three levels, as shown in the table below. This approach is common in internal gear manufacturer research to optimize parameters without exhaustive testing.
| Level | $\alpha$ (%) | $p$ (10^5 Pa) | $\theta$ (°C) |
|---|---|---|---|
| 1 | 0.1 | 75 | 40 |
| 2 | 0.5 | 100 | 50 |
| 3 | 1.0 | 125 | 60 |
Based on this, we developed nine experimental conditions, detailed in the following table, which guide our CFD simulations and analysis of flow characteristics for internal gears.
| Test No. | $\alpha$ (%) | $p$ (10^5 Pa) | $\theta$ (°C) |
|---|---|---|---|
| 1 | 0.1 | 75 | 40 |
| 2 | 0.1 | 100 | 50 |
| 3 | 0.1 | 125 | 60 |
| 4 | 0.5 | 75 | 60 |
| 5 | 0.5 | 100 | 40 |
| 6 | 0.5 | 125 | 50 |
| 7 | 1.0 | 75 | 50 |
| 8 | 1.0 | 100 | 60 |
| 9 | 1.0 | 125 | 40 |
For the CFD simulations, we constructed a three-dimensional finite element model of the straight line conjugate internal gear pump, which includes critical friction pairs such as the gear end faces and side plates, tooth tips and housing, and meshing tooth surfaces. These pairs are separated by oil films to simulate actual operating conditions, with thicknesses set to 30 μm for gear end faces and tooth tips, and 4 μm for tooth surfaces, based on hydrostatic support theory. The model uses a combination of Cartesian and structured dynamic grids with boundary layers to capture wall flow behavior, and MGI technology for interfaces between dynamic and static regions. The mesh quality was optimized to avoid fragmentation, and the rotor area was refined. Key pump parameters include a displacement of 51.9 mL/r, maximum speed of 2600 r/min, rated outlet pressure of 12.5 MPa, and mineral oil viscosity range of 10–100 mm²/s. Boundary conditions were set as pressure inlet and outlet, with the external gear rotating at 2000 r/min and the internal gear at 1529.4 r/min, reflecting the transmission ratio. The working fluid is 46# mineral oil, and its properties under different test conditions are summarized in the table below, which is vital for internal gear manufacturers to understand material behavior.
| Test No. | $\alpha$ (%) | $p$ (10^5 Pa) | $\theta$ (°C) | $\alpha_m$ (10^{-6}) | $\rho$ (kg/m³) | $\mu$ (Pa·s) | $K_{\text{ef}}$ (GPa) |
|---|---|---|---|---|---|---|---|
| 1 | 0.1 | 75 | 40 | 1.28737 | 852.98 | 0.043 | 1.685912 |
| 2 | 0.1 | 100 | 50 | 1.29407 | 848.56 | 0.029 | 1.691140 |
| 3 | 0.1 | 125 | 60 | 1.30086 | 844.13 | 0.021 | 1.693780 |
| 4 | 0.5 | 75 | 60 | 6.55119 | 841.46 | 0.019 | 1.628377 |
| 5 | 0.5 | 100 | 40 | 6.45424 | 854.10 | 0.046 | 1.657590 |
| 6 | 0.5 | 125 | 50 | 6.48766 | 849.70 | 0.031 | 1.670114 |
| 7 | 1.0 | 75 | 50 | 13.0838 | 846.91 | 0.028 | 1.564939 |
| 8 | 1.0 | 100 | 60 | 13.1514 | 842.56 | 0.020 | 1.612971 |
| 9 | 1.0 | 125 | 40 | 12.9565 | 855.23 | 0.048 | 1.642419 |
We selected the RNG k-ε turbulence model to simulate the complex flow with transient behavior and curved streamlines, and the equilibrium dissolved gas model for cavitation, considering non-condensable gases. Grid independence was verified by comparing results for different mesh sizes; a model with 567,169 nodes showed negligible difference in outlet average flow (less than 0.1%), so it was used for computations. The simulation results for flow pulsation, pressure pulsation, volumetric efficiency, and total efficiency under the nine test conditions are analyzed below, providing valuable data for internal gear manufacturers to refine their designs.
Flow pulsation is a key indicator of instantaneous flow quality in hydraulic pumps. For internal gears, pulsation arises from periodic changes in meshing points, uneven internal leakage, oil compressibility, and trapped fluid phenomena. The instantaneous flow curves for all tests exhibit continuous periodic variations with 13 fluctuations per cycle, corresponding to the 13 meshing events of the gear pair. However, the curves differ due to combined effects of leakage, compression, and trapped flow. The flow pulsation amplitude ($\Delta q$) and pulsation rate ($\delta_q$) are calculated, with the pulsation rate defined as:
$$ \delta_q = \frac{\Delta q}{q_{\text{avg}}} \times 100\% $$
where $q_{\text{avg}}$ is the average flow rate. The results, summarized in the table below, show that Test 2 has the lowest pulsation, while Test 7 has the highest, indicating that oil properties significantly influence flow stability in internal gears.
| Test No. | $\Delta q$ (L/min) | $\delta_q$ (%) |
|---|---|---|
| 1 | 36.3 | 35.9 |
| 2 | 24.8 | 25.6 |
| 3 | 26.9 | 29.5 |
| 4 | 53.6 | 54.9 |
| 5 | 31.5 | 31.4 |
| 6 | 32.9 | 34.2 |
| 7 | 62.3 | 62.2 |
| 8 | 34.8 | 36.3 |
| 9 | 46.5 | 46.8 |
To analyze the influence of factors on flow pulsation rate, we constructed an orthogonal analysis table, calculating the average values ($k_1$, $k_2$, $k_3$) and range ($R$) for each factor level. The results, shown in the table below, indicate that working pressure has the greatest impact, followed by air content, while oil temperature has a negligible effect. This insight is crucial for internal gear manufacturers to prioritize pressure control in design.
| Factor | $k_1$ | $k_2$ | $k_3 | $R$ |
|---|---|---|---|---|
| $\alpha$ | 30.33 | 40.15 | 48.42 | 18.09 |
| $p$ | 50.97 | 31.11 | 36.82 | 19.86 |
| $\theta$ | 38.01 | 40.65 | 40.23 | 2.64 |
The optimal combination for minimizing flow pulsation rate is $\alpha = 0.1\%$, $p = 10.0 \times 10^5$ Pa, and $\theta = 40^\circ$C, yielding a pulsation rate of 23.93%. This highlights the importance of low air content and moderate pressure for internal gears produced by internal gear manufacturers.
Pressure pulsation, closely related to flow pulsation, is another critical aspect. The instantaneous pressure curves also show periodic fluctuations per cycle, with trends matching flow pulsation. The pressure pulsation amplitude ($\Delta p$) and pulsation rate ($\delta_p$) are defined as:
$$ \delta_p = \frac{\Delta p}{p_{\text{avg}}} \times 100\% $$
where $p_{\text{avg}}$ is the average pressure. As summarized in the table below, Test 3 has the lowest pulsation, while Test 4 has the highest rate, demonstrating that pressure pulsation is positively correlated with flow pulsation.
| Test No. | $\Delta p$ (10^5 Pa) | $\delta_p$ (%) |
|---|---|---|
| 1 | 0.30 | 0.40 |
| 2 | 0.18 | 0.18 |
| 3 | 0.18 | 0.14 |
| 4 | 0.53 | 0.71 |
| 5 | 0.47 | 0.47 |
| 6 | 0.61 | 0.49 |
| 7 | 0.52 | 0.69 |
| 8 | 0.57 | 0.57 |
| 9 | 0.67 | 0.53 |
Orthogonal analysis of pressure pulsation rate, presented in the table below, reveals that air content is the most influential factor, followed by working pressure, with oil temperature having minimal impact. The optimal combination for reducing pressure pulsation is $\alpha = 0.1\%$, $p = 12.5 \times 10^5$ Pa, and $\theta = 50^\circ$C, resulting in a pulsation rate of 0.14%.
| Factor | $k_1$ | $k_2$ | $k_3 | $R$ |
|---|---|---|---|---|
| $\alpha$ | 0.243 | 0.556 | 0.598 | 0.355 |
| $p$ | 0.599 | 0.408 | 0.389 | 0.209 |
| $\theta$ | 0.469 | 0.453 | 0.474 | 0.021 |
Efficiency metrics, including volumetric efficiency ($\eta$) and total efficiency ($\eta_z$), are vital for assessing pump performance. Volumetric efficiency accounts for leakage and compression losses, while total efficiency also considers mechanical losses from friction. The results, shown in the table below, indicate that Test 1 has the highest volumetric efficiency, and Test 9 the highest total efficiency, underscoring the trade-offs involved for internal gear manufacturers.
| Test No. | $\eta$ (%) | $\eta_z$ (%) |
|---|---|---|
| 1 | 97.6 | 73.2 |
| 2 | 93.4 | 75.2 |
| 3 | 87.9 | 73.6 |
| 4 | 94.1 | 75.2 |
| 5 | 96.7 | 75.9 |
| 6 | 92.8 | 76.5 |
| 7 | 96.6 | 75.7 |
| 8 | 92.4 | 76.0 |
| 9 | 95.7 | 77.3 |
Orthogonal analysis for volumetric efficiency, summarized in the table below, shows that oil temperature is the dominant factor, followed by working pressure, with air content having the least effect. The optimal combination for high volumetric efficiency is $\alpha = 1.0\%$, $p = 7.5 \times 10^5$ Pa, and $\theta = 40^\circ$C, yielding 97.23%.
| Factor | $k_1$ | $k_2$ | $k_3 | $R$ |
|---|---|---|---|---|
| $\alpha$ | 92.94 | 94.50 | 94.88 | 1.94 |
| $p$ | 96.07 | 94.15 | 92.10 | 3.97 |
| $\theta$ | 96.64 | 94.25 | 91.44 | 5.20 |
For total efficiency, the analysis in the table below indicates that air content is the most influential factor, followed by working pressure, with oil temperature having a minor role. The optimal combination is $\alpha = 1.0\%$, $p = 12.5 \times 10^5$ Pa, and $\theta = 50^\circ$C, achieving 77.52%.
| Factor | $k_1$ | $k_2$ | $k_3 | $R$ |
|---|---|---|---|---|
| $\alpha$ | 73.98 | 75.87 | 76.31 | 2.33 |
| $p$ | 74.67 | 75.68 | 75.82 | 1.02 |
| $\theta$ | 75.48 | 75.77 | 74.92 | 0.84 |
To validate the simulation results, we conducted experiments using a high-pressure positive displacement pump test rig. The setup involved adjusting a proportional pressure-reducing valve to vary load and heat the oil, an electromagnetic relief valve to set working pressure, and a flow meter to measure system flow, with the motor speed fixed at 2000 r/min. We measured output flow and calculated volumetric efficiency under three temperature conditions (40°C, 50°C, 60°C) and three pressure levels (7.5, 10.0, 12.5 × 10^5 Pa), comparing them with simulation data. The results, summarized in the table below, show close agreement, with average flow differences up to 1.9 L/min and volumetric efficiency differences up to 1.8%, confirming the accuracy of our models for internal gear manufacturers.
| Condition | $\theta$ (°C) | $p$ (10^5 Pa) | $\eta_{\text{sim}}$ (%) | $\eta_{\text{exp}}$ (%) | Difference (%) |
|---|---|---|---|---|---|
| 1 | 40 | 7.5 | 97.56 | 95.8 | 1.76 |
| 2 | 40 | 10.0 | 93.40 | 91.6 | 1.80 |
| 3 | 40 | 12.5 | 87.87 | 86.1 | 1.77 |
| 4 | 50 | 7.5 | 94.07 | 92.3 | 1.77 |
| 5 | 50 | 10.0 | 96.68 | 94.9 | 1.78 |
| 6 | 50 | 12.5 | 92.76 | 91.0 | 1.76 |
| 7 | 60 | 7.5 | 96.59 | 94.8 | 1.79 |
| 8 | 60 | 10.0 | 92.38 | 90.6 | 1.78 |
| 9 | 60 | 12.5 | 95.67 | 93.9 | 1.77 |
In conclusion, our study demonstrates that oil properties significantly influence the flow characteristics of straight line conjugate internal gear pumps. Variable operating conditions are the root cause of changes in oil density, viscosity, and effective bulk modulus, with air content notably affecting compressibility. Flow and pressure pulsation rates are positively correlated, and reducing air content while increasing working pressure can minimize pulsation. For efficiency, lowering oil temperature and increasing air content improve volumetric and total efficiency, respectively. These findings provide practical guidance for internal gear manufacturers: maintaining air content around 0.5%, operating at higher pressures near 12.5 MPa, and keeping oil temperatures at 40°C or lower can optimize pump performance. Future work could explore additional factors like gear geometry variations to further enhance the design of internal gears for hydraulic systems.