As an engineer specializing in fluid power systems, I have always been fascinated by the efficiency and reliability of internal gear pumps. These pumps, which utilize internal gears, are widely recognized for their high performance, low noise, and superior self-priming capabilities. They play a crucial role in hydraulic transmission systems, especially when handling viscous or corrosive fluids where other pump types fall short. In this article, I will delve into the precise calculation of displacement for internal gear pumps, focusing on the impact of tooth side clearance and overlap ratio. The insights provided here are essential for internal gear manufacturers aiming to optimize pump design, particularly in achieving accurate flow rates and designing effective side plate unloading slots. Throughout this discussion, I will emphasize the importance of internal gears in these applications, as their geometry directly influences pump performance. Internal gear manufacturers must consider these factors to produce high-quality components that meet demanding industrial standards.
Internal gear pumps operate on the principle of volumetric displacement, where fluid is transported through the meshing of an external and internal gear pair. The displacement calculation is not straightforward due to factors like tooth profile, clearance, and engagement dynamics. Traditional approximate methods often overlook the nuances of tooth side clearance and overlap ratio, leading to inaccuracies. In my experience, a precise approach based on parametric instantaneous flow equations yields more reliable results. This is particularly relevant for internal gear manufacturers who need to ensure consistency in pump output. The following sections will derive the instantaneous flow rate, analyze the oil displacement volume under different clearance conditions, and present exact displacement formulas. I will also include a comparative analysis using a practical example, highlighting how displacement varies with the overlap ratio. To aid visualization, I have included an image that illustrates the meshing of internal gears, which is fundamental to understanding the pump’s operation.
The foundation of displacement calculation lies in deriving the instantaneous flow rate. Consider a pair of involute spur gears—one external and one internal—meshing within the pump. Assuming the fluid is incompressible, with no leakage or component deformation, the instantaneous flow rate can be expressed based on the gear kinematics. Let ω₁ and ω₂ be the angular velocities of the external and internal gears, respectively, and r′₁ and r′₂ their pitch circle radii. The relationship at the pitch point is given by:
$$ ω_1 r’_1 = ω_2 r’_2 $$
This leads to:
$$ d\phi_2 = \frac{r’_1}{r’_2} d\phi_1 $$
where dφ₁ and dφ₂ are the angular displacements. The area swept by the discharge chamber can be divided into contributions from both gears. For the external gear, the differential volume is:
$$ dV_1 = \frac{B}{2} (r_{a1}^2 – r_{n1}^2) d\phi_1 $$
and for the internal gear:
$$ dV_2 = \frac{B}{2} (r_{n2}^2 – r_{a2}^2) d\phi_2 $$
Substituting dφ₂, we get:
$$ dV_2 = \frac{B}{2} \frac{r’_1}{r’_2} (r_{n2}^2 – r_{a2}^2) d\phi_1 $$
The total differential volume is:
$$ dV = dV_1 + dV_2 = \frac{B}{2} \left[ (r_{a1}^2 – r_{n1}^2) + \frac{r’_1}{r’_2} (r_{n2}^2 – r_{a2}^2) \right] d\phi_1 $$
Differentiating with respect to time gives the instantaneous flow rate:
$$ Q_{SH} = \frac{dV}{dt} = \frac{B \omega_1}{2} \left[ (r_{a1}^2 – r_{n1}^2) + \frac{r’_1}{r’_2} (r_{n2}^2 – r_{a2}^2) \right] $$
Here, B is the gear width, rₐ₁ and rₐ₂ are the tip radii, and rₙ₁ and rₙ₂ are the radii at the meshing point. Using geometric identities:
$$ r_{a1}^2 = (r_1 + h_{a1})^2 = r_1^2 + 2r_1 h_{a1} + h_{a1}^2 $$
$$ r_{a2}^2 = (r_2 – h_{a2})^2 = r_2^2 – 2r_2 h_{a2} + h_{a2}^2 $$
where r₁ and r₂ are the pitch radii, and hₐ₁ and hₐ₂ are the addendum heights. The meshing radii can be expressed as:
$$ r_{n1}^2 = r’^2_1 – 2k r’_1 + f^2 $$
$$ r_{n2}^2 = r’^2_2 – 2k r’_2 + f^2 $$
where k is a geometric parameter, and f is the distance from the meshing point to the pitch point. Substituting these into the flow rate equation yields:
$$ Q_{SH} = \frac{B \omega_1}{2} \left[ 2r’_1 (h_{a1} + h_{a2}) + h_{a1}^2 – \frac{r’_1}{r’_2} h_{a2}^2 – \left(1 – \frac{r’_1}{r’_2}\right) f^2 \right] $$
This parametric equation forms the basis for displacement calculation. Internal gear manufacturers must account for these variables to ensure accurate pump performance.
Next, I will analyze the oil displacement volume for one pair of meshing teeth. The volume displaced per tooth pair is integral to the instantaneous flow over one engagement period T:
$$ V_n = \int_T Q_{SH} dt = \frac{B \omega_1}{2} \int_T \left[ 2r’_1 (h_{a1} + h_{a2}) + h_{a1}^2 – \frac{r’_1}{r’_2} h_{a2}^2 – \left(1 – \frac{r’_1}{r’_2}\right) f^2 \right] dt $$
To simplify, we relate dt to df using the base circle radius r_b1 of the external gear:
$$ dt = \frac{df}{r_{b1} \omega_1} $$
The limits of integration depend on the tooth side clearance and overlap ratio ε. I will consider two cases: with and without tooth side clearance.
For internal gears with tooth side clearance, the meshing process involves multiple points. Engagement starts when the external gear tooth tip meets the internal gear at point A, and ends when the previous pair disengages at point F. Due to clearance, the integration limits are from f = -l₁ to f = t_j – l₁, where l₁ = (ε/2) t_j and t_j is the base pitch. Thus:
$$ V_n = \frac{B}{2 r_{b1}} \int_{-\frac{\varepsilon}{2} t_j}^{\frac{2 – \varepsilon}{2} t_j} \left[ 2r’_1 (h_{a1} + h_{a2}) + h_{a1}^2 – \frac{r’_1}{r’_2} h_{a2}^2 – \left(1 – \frac{r’_1}{r’_2}\right) f^2 \right] df $$
Solving this integral gives:
$$ V_n = \frac{B t_j}{2 r_{b1}} \left[ 2r’_1 (h_{a1} + h_{a2}) + h_{a1}^2 – \frac{r’_1}{r’_2} h_{a2}^2 – \left(1 – \frac{r’_1}{r’_2}\right) \frac{(3\varepsilon^2 – 6\varepsilon + 4) t_j^2}{12} \right] $$
When ε = 1, this simplifies to:
$$ V_n = \frac{B t_j}{2 r_{b1}} \left[ 2r’_1 (h_{a1} + h_{a2}) + h_{a1}^2 – \frac{r’_1}{r’_2} h_{a2}^2 – \left(1 – \frac{r’_1}{r’_2}\right) \frac{t_j^2}{12} \right] $$
In contrast, for internal gears without tooth side clearance, the meshing process is symmetric. Engagement starts at f = -l₁ and ends at f = (t_j/2) – l₁. The volume per tooth pair is:
$$ V_n = 2 \times \frac{B}{2 r_{b1}} \int_{-\frac{\varepsilon}{2} t_j}^{\frac{1 – \varepsilon}{2} t_j} \left[ 2r’_1 (h_{a1} + h_{a2}) + h_{a1}^2 – \frac{r’_1}{r’_2} h_{a2}^2 – \left(1 – \frac{r’_1}{r’_2}\right) f^2 \right] df $$
Solving yields:
$$ V_n = \frac{B t_j}{2 r_{b1}} \left[ 2r’_1 (h_{a1} + h_{a2}) + h_{a1}^2 – \frac{r’_1}{r’_2} h_{a2}^2 – \left(1 – \frac{r’_1}{r’_2}\right) \frac{(3\varepsilon^2 – 3\varepsilon + 1) t_j^2}{12} \right] $$
For ε = 1, it becomes:
$$ V_n = \frac{B t_j}{2 r_{b1}} \left[ 2r’_1 (h_{a1} + h_{a2}) + h_{a1}^2 – \frac{r’_1}{r’_2} h_{a2}^2 – \left(1 – \frac{r’_1}{r’_2}\right) \frac{t_j^2}{12} \right] $$
The theoretical displacement per revolution of the external gear (with z₁ teeth) is q = z₁ V_n. Substituting the base pitch t_j = 2π r_b1 / z₁, we derive the exact displacement formulas. For internal gears with tooth side clearance:
$$ q = \pi B \left[ 2r’_1 (h_{a1} + h_{a2}) + h_{a1}^2 – \frac{r’_1}{r’_2} h_{a2}^2 – \left(1 – \frac{r’_1}{r’_2}\right) \frac{(3\varepsilon^2 – 6\varepsilon + 4) t_j^2}{12} \right] $$
Without tooth side clearance:
$$ q = \pi B \left[ 2r’_1 (h_{a1} + h_{a2}) + h_{a1}^2 – \frac{r’_1}{r’_2} h_{a2}^2 – \left(1 – \frac{r’_1}{r’_2}\right) \frac{(3\varepsilon^2 – 3\varepsilon + 1) t_j^2}{12} \right] $$
These equations show that displacement is a quadratic function of ε. For internal gear manufacturers, understanding this relationship is key to designing pumps with desired flow characteristics. The displacement decreases as ε increases, and pumps with tooth side clearance generally have higher displacement.
To illustrate, I will compare displacements for a specific example. Consider an internal gear pump with external gear teeth z₁ = 13, internal gear teeth z₂ = 19, module m = 3 mm, and width B = 41 mm. The base pitch t_j = π m cos(α), where α is the pressure angle (typically 20°). Assuming standard addendum heights, we calculate displacement for various ε values. The results are summarized in the table below.
| Overlap Ratio ε | Displacement with Clearance (ml/rev) | Displacement without Clearance (ml/rev) | Difference Δ (ml/rev) |
|---|---|---|---|
| 1.00 | 18.293 | 18.293 | 0.000 |
| 1.05 | 18.290 | 18.250 | 0.040 |
| 1.10 | 18.284 | 18.204 | 0.080 |
| 1.15 | 18.274 | 18.154 | 0.120 |
| 1.20 | 18.260 | 18.100 | 0.160 |
| 1.25 | 18.242 | 18.042 | 0.200 |
| 1.30 | 18.220 | 17.980 | 0.240 |
The table clearly demonstrates that displacement decreases with increasing ε, and the reduction is more pronounced without tooth side clearance. For internal gear manufacturers, this highlights the importance of controlling clearance and overlap ratio to achieve specific flow rates. In practice, internal gears must be precision-engineered to maintain optimal performance across different operating conditions.
In conclusion, the precise calculation of displacement in internal gear pumps requires careful consideration of tooth geometry, side clearance, and overlap ratio. The derived formulas provide a reliable basis for design and optimization. Internal gear manufacturers should leverage these insights to produce pumps with minimal flow pulsation and high efficiency. As the demand for advanced hydraulic systems grows, the role of internal gears becomes increasingly critical. Future work could explore the effects of tooth profile modifications or thermal expansion on displacement accuracy. Ultimately, a thorough understanding of these principles will benefit internal gear manufacturers in delivering superior products for challenging applications.
Throughout this analysis, I have emphasized the significance of internal gears in pump performance. By integrating theoretical derivations with practical examples, I hope to contribute to the ongoing advancement in fluid power technology. Internal gear manufacturers are encouraged to adopt these precise methods to enhance their design processes and meet the evolving needs of the industry.