In modern hydraulic systems, internal gear pumps play a critical role due to their compact design and efficiency. As an engineer specializing in fluid power systems, I have extensively researched methods to enhance the volumetric efficiency of internal gear pumps under high-pressure conditions. The isolation device, a key component, ensures separation between high and low-pressure zones while compensating for wear. This article delves into the balance mechanism of a novel radial compensation isolation device, incorporating analytical models, optimization techniques, and experimental validation. Throughout this work, insights from internal gear manufacturers have been invaluable in refining designs for durability and performance. Internal gears are central to this discussion, as their geometry directly influences pressure distribution and dynamic behavior.
The isolation device in an internal gear pump typically consists of crescent-shaped elements that separate the high-pressure (ph) and low-pressure (po) chambers. In the proposed design, the device includes an inner crescent block and an outer crescent block, with springs and sealing rods enabling radial compensation. By expanding the high-pressure chamber internally, the pump maintains efficiency even under elevated pressures. For internal gear manufacturers, achieving a balance between compensation forces and hydraulic pressures is crucial to prevent excessive wear or seizure. Internal gears, with their unique meshing characteristics, require precise alignment to minimize leakage and optimize fluid dynamics.
To understand the forces acting on the isolation device, consider the pressure distribution in the transition zone between high and low-pressure regions. Let the angle from the low-pressure boundary point W to any point on the crescent surface be denoted by ζ. The transition zone spans n gear teeth and slots, with each slot corresponding to an angle θ’ and each tooth to θ″. The pressure distribution for a complete slot rotation can be expressed as:
$$ p_{\text{slot}_i} = p_o + \frac{\zeta – (i-1)\theta’}{n\theta”}(p_h – p_o) \quad \text{for } 0 \leq \zeta \leq \theta_{WV} $$
$$ p_{\text{tooth}_i} = p_o + \frac{i}{n}(p_h – p_o) \quad \text{for } 0 \leq \zeta \leq \theta_{WV} $$
$$ p_{UV} = p_h \quad \text{for } \theta_{WV} \leq \zeta \leq \theta_{WU} $$
During rotation, the pressure varies cyclically as teeth and slots alternate. For instance, when rotating through a slot angle θ (0 < θ < θ’), the pressure includes partial slots and teeth, leading to a piecewise linear function. This complexity necessitates a dynamic analysis of hydraulic forces. Internal gear manufacturers must account for these variations to ensure stable operation. Internal gears with optimized tooth profiles can mitigate abrupt pressure changes, enhancing pump longevity.
The hydraulic thrust on the outer crescent block, denoted as F_wd, combines forces from the expanded chamber (F_GH) and the transition zone (F_FG). In Cartesian coordinates, the components are:
$$ F_{wdx} = F_{FG_x} + F_{GH_x} = r_{o2} b \left[ \sum_{i=1}^{n_2} \int_{\alpha_0 + \theta_2 + \theta_2” + (i-1)(\theta_2′ + \theta_2”)}^{\alpha_0 + \theta_2 + (i-1)(\theta_2′ + \theta_2”)} p_{\text{slot}_{1i}} \cos \alpha \, d\alpha + \sum_{i=1}^{n_2-1} \int_{\alpha_0 + \theta_2 + \theta_2” + (i-1)(\theta_2′ + \theta_2”)}^{\alpha_0 + \theta_2 + i(\theta_2′ + \theta_2”)} p_{\text{tooth}_{1i}} \cos \alpha \, d\alpha + \int_{\alpha_0 + \alpha_6}^{\alpha_0 + (n_2-1)(\theta_2′ + \theta_2”)} p_{\text{tooth}_{1n}} \cos \alpha \, d\alpha + \int_{\alpha_0 + \alpha_6}^{\alpha_0 + \alpha_1 + \alpha_2 + \alpha_3 + \alpha_4} p_h \cos \alpha \, d\alpha \right] $$
$$ F_{wdy} = F_{FG_y} + F_{GH_y} = r_{o2} b \left[ \sum_{i=1}^{n_2} \int_{\alpha_0 + \theta_2 + \theta_2” + (i-1)(\theta_2′ + \theta_2”)}^{\alpha_0 + \theta_2 + (i-1)(\theta_2′ + \theta_2”)} p_{\text{slot}_{1i}} \sin \alpha \, d\alpha + \sum_{i=1}^{n_2-1} \int_{\alpha_0 + \theta_2 + \theta_2” + (i-1)(\theta_2′ + \theta_2”)}^{\alpha_0 + \theta_2 + i(\theta_2′ + \theta_2”)} p_{\text{tooth}_{1i}} \sin \alpha \, d\alpha + \int_{\alpha_0 + \alpha_6}^{\alpha_0 + (n_2-1)(\theta_2′ + \theta_2”)} p_{\text{tooth}_{1n}} \sin \alpha \, d\alpha + \int_{\alpha_0 + \alpha_6}^{\alpha_0 + \alpha_1 + \alpha_2 + \alpha_3 + \alpha_4} p_h \sin \alpha \, d\alpha \right] $$
Similarly, for the inner crescent block, the forces F_nd involve integrals over angles β, with contributions from the transition zone and expanded chamber. The compensation forces F_wb and F_nb are derived from high-pressure acting on designated arcs. For the outer crescent, the compensation force is:
$$ F_{wb} = \int_{\alpha_0 + \alpha_1 + \alpha_2}^{\alpha_0 + \alpha_1 + \alpha_2 + \alpha’} p_h (r_{o2} – d) b \, d\alpha $$
And for the inner crescent:
$$ F_{nb} = \int_{\alpha_0 + \alpha_1 + \alpha_2}^{\alpha_0 + \alpha_1 + \alpha_2 + \alpha’} p_h (r_{o2} – d) b \, d\alpha + \int_{\alpha_0 + \alpha_1 + \alpha_2 + \alpha’}^{\alpha_0 + \alpha_1 + \alpha_2 + \alpha’ + \alpha_5} p_h r_{o2} b \, d\alpha $$
Here, α’ and α5 represent the angles defining the compensation cavities. Internal gear manufacturers must carefully select these angles to balance forces. If too large, compensation forces exceed hydraulic ones, causing excessive wear; if too small, leakage increases. Internal gears with precise tolerances are essential for this balance.
Spring forces and pivot reactions also contribute to the overall equilibrium. The spring forces F_wh on the outer crescent include components from multiple springs:
$$ F_{whx} = F_{N1} \cos(\alpha_0 + \alpha_1 + \alpha_2 + \alpha_3) + F_{N2} \cos(\alpha_0 + \alpha_1 + \alpha_2 + \epsilon) + F_{N3} \cos(\alpha_0 + \alpha_1 + \epsilon) $$
$$ F_{why} = F_{N1} \sin(\alpha_0 + \alpha_1 + \alpha_2 + \alpha_3) + F_{N2} \sin(\alpha_0 + \alpha_1 + \alpha_2 + \epsilon) + F_{N3} \sin(\alpha_0 + \alpha_1 + \epsilon) $$
The pivot forces F_EF and F_CD act at an angle λ to the Y-axis, with components:
$$ F_{EFx} = F_{EF} \sin \lambda, \quad F_{EFy} = F_{EF} \cos \lambda $$
$$ F_{CDx} = F_{CD} \sin \lambda, \quad F_{CDy} = F_{CD} \cos \lambda $$
The total forces on the crescents are vector sums of hydraulic, spring, and pivot forces. To achieve dynamic balance, the compensation forces should nearly equal the hydraulic forces over a cycle. This leads to an optimization problem where the objective is to minimize the difference between compensation and hydraulic forces, subject to constraints ensuring non-negative forces and geometric limits.
For design optimization, I formulated the problem using the fmincon tool in MATLAB. The goal was to find optimal angles α’ and α5 that minimize z1 = F_wb – F_w and z2 = F_nb – F_n, with constraints F_w ≥ 0, F_n ≥ 0, and angle bounds. This approach ensures that the isolation device remains balanced throughout operation. Internal gear manufacturers can use such models to prototype designs efficiently. Internal gears with optimized transition zones reduce pressure pulsations, a common issue in high-pressure applications.
In experimental validation, a pump with a theoretical displacement of 50.8 ml/r was tested under input pressure of 0.3 MPa and output pressure of 21.8 MPa at 50°C ± 4°C. The results showed a volumetric efficiency of 94.3% without significant wear. Key parameters are summarized in the table below:
| Parameter | Value | Description |
|---|---|---|
| Module (m) | 3 mm | Gear module |
| Teeth (z1, z2) | 13, 19 | Number of teeth on outer and inner gears |
| Pressure angle | 20° | Gear pressure angle |
| Center distance (e) | 9.253 mm | Distance between gear centers |
| Optimal α’ | 47.1° | Compensation cavity angle for outer crescent |
| Optimal α5 | 22.3° | Additional angle for inner crescent compensation |
| Volumetric efficiency | 94.3% | Measured efficiency under test conditions |
The force variations over a rotation cycle were simulated using MATLAB. The graphs depicted cyclic changes in hydraulic and compensation forces, with compensation forces closely tracking hydraulic ones. This balance is vital for minimizing wear in internal gears. Internal gear manufacturers often conduct similar simulations to predict performance before physical testing. Internal gears produced with advanced machining techniques exhibit lower noise and higher efficiency.
Another critical aspect is the pivot balance. The moments around the pivot point O3 must satisfy:
$$ \sum M_{O3} = -F_{EF} w + F_{CD} u = 0 $$
And the force equilibrium:
$$ F_{O3} = F_{EF} + F_{CD} $$
Solving these equations yields the pivot positions w and u that ensure stability. For the test pump, w = 4.92 and u = 2.13 were derived, contributing to smooth operation. Internal gear manufacturers must consider such geometric constraints in assembly. Internal gears with proper pivot alignment reduce vibration and extend pump life.
In conclusion, the balanced isolation device with radial compensation significantly enhances the performance of internal gear pumps under high pressure. The analytical models for pressure distribution and force analysis provide a foundation for optimization. Through numerical methods and experimental testing, optimal compensation angles were determined, leading to high volumetric efficiency and minimal wear. This research underscores the importance of precise engineering in hydraulic systems. Internal gear manufacturers can leverage these findings to develop more reliable pumps. Internal gears, as core components, benefit from continuous improvement in design and material science. Future work could explore adaptive compensation mechanisms for varying operating conditions.