Internal gear pumps are widely used in hydraulic systems due to their compact design, self-priming capability, and smooth operation. As an internal gear manufacturer, we constantly seek innovations to enhance performance, particularly in reducing flow and pressure pulsations, which are common limitations in traditional designs. In this article, I present a comprehensive study on a novel composite internal gear pump, focusing on its parameter optimization and performance analysis. The composite design incorporates multiple sub-pumps to improve flow rates and minimize pulsations, addressing key challenges faced by internal gears in industrial applications. By leveraging mathematical modeling and simulation tools, I aim to demonstrate how this optimized pump can achieve superior performance metrics, including significantly lower pulsation coefficients compared to conventional internal gear pumps.
The composite internal gear pump consists of a central internal gear ring and three smaller external gears arranged symmetrically at 120-degree intervals. This configuration creates three independent sub-pumps that operate in tandem, with their flow outputs叠加 to enhance overall flow while canceling out pulsations. A star-shaped sealing block separates the sub-pumps, ensuring minimal leakage. The internal gears are designed to work in harmony with the external gears, driven by a common shaft, which simplifies the assembly and improves reliability. As an internal gear manufacturer, we prioritize designs that balance efficiency and durability, and this composite approach leverages the inherent strengths of internal gears while mitigating their weaknesses. The working principle involves the rotation of the central internal gear, which meshes with the external gears to draw in and discharge fluid through dedicated ports. The symmetrical layout ensures that radial hydraulic forces are balanced, reducing wear and extending the lifespan of the internal gears.
To model the flow characteristics of the composite internal gear pump, I derived the instantaneous flow rate based on the geometry and kinematics of the internal gears. The flow rate for a traditional internal gear pump is given by the equation: $$ q_v = B \omega_1 \left[ \frac{r_1^2 + r_2^2}{2} + (h_{a1} + h_{a2}) – \frac{1}{2} \left( h_{a1}^2 + h_{a2}^2 \right) \left(1 – \frac{r_1}{r_2}\right) – f^2 \right] $$ where \( B \) is the gear width, \( \omega_1 \) is the angular velocity of the driving gear, \( r_1 \) and \( r_2 \) are the pitch circle radii of the driving gear and internal gear, respectively, \( h_{a1} \) and \( h_{a2} \) are the addendum heights, and \( f \) is the displacement of the meshing point. By substituting gear parameters such as module \( m \), number of teeth \( z_1 \) and \( z_2 \), and pressure angle \( \alpha_n \), this simplifies to: $$ q_v = a – b \phi_2^2 $$ where \( a = \frac{B m^2 \omega_2 (z_2^2 + z_1 z_2 – 2z_1^2)}{2z_1} \) and \( b = \frac{B m^2 \omega_2 (z_2^2 – z_1^2) \cos^2 \alpha_n}{8z_1} \). For the composite pump, the total instantaneous flow is the sum of the flows from the three sub-pumps. When the internal gear teeth count \( z_2 \) is not a multiple of 3, the phase differences between sub-pumps reduce pulsation. The total flow in the interval \( (-\pi/(3z_2), \pi/(3z_2)) \) is: $$ q_{3z_2=k+1} = 3a – \frac{8\pi^2 b}{3z_2^2} – b \phi_2^2 $$ This model highlights the importance of internal gear design in optimizing flow uniformity.
The optimization of the composite internal gear pump aims to minimize flow pulsation and overall volume, which are critical for internal gear manufacturers seeking to enhance product performance. I formulated a multi-objective optimization problem with design variables \( X = [z_1, z_2, B, m]^T \), where \( z_1 \) and \( z_2 \) are the teeth numbers of the driving and internal gears, \( B \) is the gear width, and \( m \) is the module. The first objective function targets the flow pulsation coefficient \( \delta \), defined as: $$ \delta = \frac{q_{\text{max}} – q_{\text{min}}}{q} $$ where \( q_{\text{max}} \), \( q_{\text{min}} \), and \( q \) are the maximum, minimum, and average flow rates, respectively. For \( z_2 = 3k+1 \), this becomes: $$ \delta = \frac{6\pi b}{54a z_2 – 19\pi b} $$ The second objective function minimizes the pump volume, approximated as the sum of the volumes of the internal gear and three external gears: $$ f_2 = \frac{\pi}{4} m^2 B (z_2^2 + 3z_1^2) $$ To handle the multi-objective nature, I normalized the functions and combined them with equal weights: $$ f = 0.5 \frac{f_1 – f_{1\text{min}}}{f_{1\text{max}} – f_{1\text{min}}} + 0.5 \frac{f_2 – f_{2\text{min}}}{f_{2\text{max}} – f_{2\text{min}}} $$ Constraints include tooth number limits to avoid undercutting and ensure proper sealing, flow error within 5%, gear width limitations, and bending strength requirements for the internal gears. Using MATLAB’s fmincon function, I solved this nonlinear optimization problem, resulting in optimal parameters: \( z_1 = 18 \), \( z_2 = 47 \), \( B = 11.2 \, \text{mm} \), and \( m = 3 \, \text{mm} \).
The performance of the optimized composite internal gear pump was evaluated through flow and pressure pulsation analysis. The instantaneous flow rate, calculated using the derived model, shows a maximum of 50.69 L/min, a minimum of 50.48 L/min, and an average of 50.585 L/min at a rated speed of 1500 rpm. This corresponds to a flow pulsation coefficient of 0.41%, which is substantially lower than the 2-5% typical of conventional internal gear pumps. The table below summarizes key flow characteristics:
| Parameter | Value |
|---|---|
| Maximum Flow (L/min) | 50.69 |
| Minimum Flow (L/min) | 50.48 |
| Average Flow (L/min) | 50.585 |
| Flow Pulsation Coefficient | 0.41% |
Pressure pulsation was analyzed using AMEsim software, modeling the pump as a system with a flow source, pressure chamber, and leakage paths. The total leakage flow \( \sum \Delta q_i(t) \) includes radial and axial components. Radial leakage, based on parallel plate gap flow theory, is: $$ \Delta q_r = 3 \left[ \frac{B h_1^3 P_p(t)}{12 \mu S_1 Z_1} + \frac{B h_1^3 P_p(t)}{12 \mu S_2 Z_2} – \frac{B h_1 (v_1 + v_2)}{2} \right] $$ where \( h_1 \) is the radial clearance, \( \mu \) is the dynamic viscosity, \( S_1 \) and \( S_2 \) are tooth tip thicknesses, and \( Z_1 \) and \( Z_2 \) are sealing tooth numbers. Axial leakage, derived from planar gap flow, is: $$ \Delta q_a = 3 \times 2 \times \frac{h_2^3 \theta_1 P_p(t)}{6 \mu \ln(R_{f1}/R_{zf1})} $$ where \( h_2 \) is the axial clearance, \( \theta_1 \) is the equivalent high-pressure angle, and \( R_{f1} \) and \( R_{zf1} \) are radii. The total leakage is \( \sum \Delta q_i = 1.05 (\Delta q_r + \Delta q_a) \). The pressure dynamics follow the continuity equation: $$ q(t) = q_p(t) + \sum \Delta q_i(t) + \frac{V}{K} \frac{\partial P_p(t)}{\partial t} $$ where \( V \) is the working chamber volume and \( K \) is the bulk modulus. Simulation results show a maximum pressure of 10.880 MPa, a minimum of 10.830 MPa, and an average of 10.865 MPa at 10 MPa rated pressure, yielding a pressure pulsation coefficient of 0.46%. This low pulsation underscores the benefits of the composite design for internal gears, as it reduces cyclic stresses and improves fatigue life.
In conclusion, the composite internal gear pump offers a significant advancement in hydraulic pump technology by leveraging multiple sub-pumps to enhance flow and reduce pulsations. The optimization process, focused on minimizing flow pulsation and volume, resulted in parameters that achieve a flow pulsation coefficient of 0.41% and a pressure pulsation coefficient of 0.46%. These values are markedly lower than those of traditional internal gear pumps, demonstrating the efficacy of the composite approach. For internal gear manufacturers, this design not only improves performance but also extends component lifespan through balanced forces and reduced wear. Future work could explore variations in gear geometry or material selection to further optimize these internal gears for specific applications. The integration of such innovations highlights the potential for internal gear pumps to meet increasingly demanding industrial requirements.