As an engineer specializing in fluid dynamics, I have conducted extensive research on the internal flow characteristics of internal gear pumps, which are widely used in various industrial applications due to their compact design and efficient performance. Internal gears, as produced by reputable internal gear manufacturers, offer advantages such as reduced noise, lower flow pulsation, and longer service life compared to external gear pumps. In this study, I employed computational fluid dynamics (CFD) to simulate the two-dimensional unsteady flow field within an internal gear pump, focusing on how different operating conditions affect pressure and velocity distributions. The insights gained are crucial for internal gear manufacturers to optimize pump design and enhance performance.
The internal gear pump consists of an outer gear and an inner gear that rotate in the same direction, typically counter-clockwise, with a crescent-shaped separator to manage fluid flow. Key parameters for the pump in this analysis are summarized in Table 1. These parameters are standard in designs from internal gear manufacturers and influence the pump’s efficiency and flow dynamics.
| Parameter | Value |
|---|---|
| Module (mm) | 3 |
| Number of Teeth (Outer Gear) | 13 |
| Number of Teeth (Inner Gear) | 19 |
| Pressure Angle (°) | 22 |
| Modification Coefficient (Outer Gear) | 0.41 |
| Modification Coefficient (Inner Gear) | 0.48 |
To visualize the structure, refer to the following diagram, which illustrates the arrangement of internal gears and the flow path. This configuration is typical in pumps from internal gear manufacturers and highlights the interaction between components.
In my numerical model, I used ANSYS Fluent with dynamic mesh technology to simulate the transient flow. The governing equations for the fluid flow include the mass conservation equation, momentum conservation equation, and the RNG k-ε turbulence model. These equations are fundamental in CFD analyses for internal gears and are expressed as follows:
The mass conservation equation ensures that mass is conserved in the flow domain:
$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{V}) = 0$$
where $\rho$ is the fluid density, $t$ is time, and $\mathbf{V}$ is the velocity vector.
The momentum conservation equation accounts for forces acting on the fluid:
$$\frac{\partial (\rho \mathbf{V})}{\partial t} + \nabla \cdot (\rho \mathbf{V} \mathbf{V}) = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \mathbf{g}$$
where $p$ is pressure, $\boldsymbol{\tau}$ is the viscous stress tensor, and $\mathbf{g}$ is gravitational acceleration.
For turbulence modeling, the RNG k-ε model was selected due to its accuracy in handling rotating flows and high strain rates, which are common in internal gear pumps. The transport equations for turbulent kinetic energy $k$ and dissipation rate $\varepsilon$ are:
$$\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_j} (\rho k u_j) = \frac{\partial}{\partial x_j} \left( \alpha_k \mu_{\text{eff}} \frac{\partial k}{\partial x_j} \right) + G_k + G_b – \rho \varepsilon – Y_M$$
$$\frac{\partial}{\partial t} (\rho \varepsilon) + \frac{\partial}{\partial x_j} (\rho \varepsilon u_j) = \frac{\partial}{\partial x_j} \left( \alpha_\varepsilon \mu_{\text{eff}} \frac{\partial \varepsilon}{\partial x_j} \right) + C_{1\varepsilon} \frac{\varepsilon}{k} (G_k + C_{2\varepsilon} G_b) – C_{1\varepsilon} \rho \frac{\varepsilon^2}{k} – R$$
where $G_k$ and $G_b$ represent turbulence generation terms, $Y_M$ accounts for compressibility effects, and $R$ is an additional term defined as:
$$R = \frac{C_\mu \rho \eta^3 (1 – \eta / \eta_0)}{1 + \beta \eta^3} \frac{\varepsilon^2}{k}$$
with $\eta = S k / \varepsilon$, $\eta_0 = 4.83$, $\beta = 0.012$, and constants $C_\mu = 0.0845$, $C_{1\varepsilon} = 1.42$, $C_{2\varepsilon} = 1.68$, $\alpha_k = 1.0$, $\alpha_\varepsilon = 0.769$. This model is particularly effective for internal gears, as it captures the complex flow patterns in the pump.
For mesh generation, I used Gambit to create an unstructured triangular grid with 210,355 elements. Critical gaps, such as the 0.03 mm minimum clearance at the meshing point and 0.04 mm gaps between gears and the crescent, were refined to ensure accuracy. The boundary conditions included a pressure inlet at 0.15 MPa and a pressure outlet at 10 MPa for baseline cases, with variations up to 25 MPa. The outer and inner gears were defined as rotating walls with rigid motion at speeds ranging from 1000 to 3000 rpm. The fluid properties were set to a density of 960 kg/m³ and a dynamic viscosity of 0.048 N·s/m². The SIMPLE algorithm was employed for pressure-velocity coupling, with discretization schemes including standard for pressure and first-order upwind for momentum, turbulent kinetic energy, and dissipation rate. The time step was 10^{-6} s to maintain numerical stability.
The results from the simulations revealed significant insights into the flow behavior of internal gear pumps. Under a baseline condition of 1000 rpm and 10 MPa outlet pressure, the pressure distribution showed a gradual decrease from the discharge chamber to the suction chamber. Maximum pressures occurred near the gear teeth adjacent to the discharge side due to volume reduction, while minimum pressures appeared near the suction side, indicating potential cavitation risks. However, unlike external gear pumps, the internal gear design with radial holes in the inner gear teeth eliminated trapped oil phenomena, as the meshing zone maintained equal pressure with adjacent chambers. This is a key advantage highlighted by internal gear manufacturers for reducing noise and vibration.
Velocity distributions indicated that leakage flows occurred through gaps between the casing and inner gear, as well as between gears and the crescent, with peak velocities in the crescent gaps. For instance, at 1000 rpm and 10 MPa, the maximum velocity reached approximately 15 m/s in these regions. When the outlet pressure was increased to 25 MPa at the same speed, the leakage velocities increased due to higher pressure differentials, as shown in Table 2. This table summarizes the effects of varying operating conditions on flow characteristics, which internal gear manufacturers can use to optimize clearance designs.
| Condition (RPM / Outlet Pressure) | Max Pressure (MPa) | Min Pressure (MPa) | Max Velocity (m/s) | Notes |
|---|---|---|---|---|
| 1000 / 10 MPa | ~10.5 | -0.2 | 15 | No trapped oil, slight negative pressure in suction |
| 1000 / 25 MPa | ~25.5 | 0.1 | 20 | Reduced negative pressure due to leakage |
| 2000 / 25 MPa | ~26.0 | -0.5 | 25 | Increased negative pressure, risk of cavitation |
| 3000 / 25 MPa | ~26.5 | -0.8 | 30 | Higher pressure differences, need for better inlet design |
As the pump speed increased to 2000 rpm and 25 MPa, negative pressures in the suction chamber became more pronounced, highlighting the importance of adequate inlet pressurization to prevent cavitation. This is critical for internal gear manufacturers to consider in high-speed applications. The pressure distribution along the outer gear circumference exhibited a step-like pattern, with pressure differences between adjacent teeth increasing with speed. For example, at 1000 rpm, the pressure difference was about 2 MPa, while at 3000 rpm, it rose to over 5 MPa. This relationship can be modeled using the equation for pressure gradient in rotating systems:
$$\Delta p = \frac{\rho \omega^2 r^2}{2} + C$$
where $\omega$ is the angular velocity, $r$ is the radius, and $C$ is a constant dependent on pump geometry. This formula helps internal gear manufacturers predict performance under various speeds.
In terms of turbulence, the RNG k-ε model effectively captured the high shear regions near the gear walls. The turbulent kinetic energy peaked in areas with strong velocity gradients, such as the crescent gaps, reaching values up to 0.1 m²/s² at 3000 rpm. This analysis underscores the need for robust material selection in internal gears to withstand dynamic loads. Additionally, the absence of trapped oil in internal gear pumps, due to the radial holes, was consistently observed across all simulations, reducing the risk of pressure spikes and extending component life—a significant benefit for internal gear manufacturers in designing reliable systems.
To further illustrate the impact of speed on performance, I derived a correlation between rotational speed and the pressure difference in the transition zone. Using linear regression on simulation data, the pressure difference $\Delta p_{\text{transition}}$ can be approximated as:
$$\Delta p_{\text{transition}} = k \cdot N + b$$
where $N$ is the speed in rpm, $k$ is a slope coefficient of approximately 0.002 MPa/rpm, and $b$ is an intercept. This empirical relation aids internal gear manufacturers in tailoring pumps for specific operational ranges.
In conclusion, my numerical simulations demonstrate that internal gear pumps offer distinct advantages, such as the elimination of trapped oil phenomena, but require careful design to manage pressure differentials and leakage at high speeds. Internal gear manufacturers should prioritize optimizing gap clearances and inlet conditions to enhance efficiency and durability. Future work could extend this analysis to three-dimensional models or incorporate multiphase flows to better represent real-world conditions. Overall, this study provides valuable insights for advancing the design of internal gears and improving pump performance across industries.