Flow Characteristics of Gerotor Pumps With Novel Variable Clearance Designs

Gerotor pumps are widely used in industry because of their many advantages, including simple single-shaft design, low cost, high efficiency, quiet operation, excellent durability, minimal and common parts, packaging flexibility, and single or multi-element arrangements. In the gerotor pump literature, numerous authors have advanced different aspects of gerotor pump design theory, including profile generation and geometric properties [1–9], tooth contact analysis [7,8], a model of meshing clearance [9], and avoidance of undercutting [10,11]. Others have proposed methods for improving and estimating the rotor sealing property or increasing volumetric efficiency [11,12]. Subsequently, Hsieh [13] verified the method of curvature difference to estimate rotor sealing and successfully forecast the variation in Hertz stress on the gerotor tooth profile. Recently, proposed methods for designing the gerotor profile include that by Yang et al. [14–16], who used a deviation function based model to design a gerotor whose pitch curves can be circular or noncircular. Choi et al. [17] further suggested that the tip width of the inner rotor could be controlled by inserting a circular arc curve between the hypocycloid and epicycloid curves. Such insertion eliminates any upper limit on eccentricity. Paffoni et al. [18,19] then proposed a method for deducing pressure and film thickness based on the influence of teeth clearance on tooth number contact in a hydrostatic (gerotor) pump with circular arc profiles. Further research by Gamez-Montero et al. [20–22] developed the gerotor pump package, as well as explored the influence of teeth clearance and relief grooves on the pump's flow characteristics.

The fluid and dynamic analysis models were developed in earlier research [23], which also examined how the span angle design of a rotor profile affects a gerotor pump's dynamics and pumping performance. In these models, the outer rotor profile is formed by epicycloids and a hypocycloid, while the inner rotor profile is obtained by the theory of gearing. When the outer rotors have the same volume, the curvature difference method can be used to show that sealing remains the same across various span angle designs.

When the gerotor pump operating at higher working pressure, due to the tiny clearances exist between inner and outer rotors and between outer rotor and chamber, there may cause shock or collision among pump components that produce large noise, stress fluctuations, and shorten pump life expectancy. To overcome this problem, Daisuke et al. [24] first proposed the inter-rotor clearance between the outer rotor and the inner rotor can be designed as variable. They adopted ellipse curve to modify the addendum profile on the rotors can reach clearance variation between the outer rotor and inner rotor. The purpose mainly reduces the hitting or collision between inner and outer rotors for improving the vibration and noise problem. Then, they also verified the appropriate clearance design can produce a smaller vibration by experiments. However, their design did not consider the relief groove and the design is only an experiment to verify the noise reduction.

For reducing the collisions, vibrations, and keep robust flow characteristics in the gerotor pump, this paper explores ways of better designing the gerotor pump and a novel design is developed by building a fluid analysis model with a relief groove design. Through application of the model to two fixed clearance designs (the Old-0.035 and the Old-0.07 designs that indicate the constant clearance as 0.035 mm and 0.07 mm, respectively) and three different variable clearance designs (the New-Case 1, 2, and 3), this paper then: (1) explores the effects of various fixed clearance sizes on gerotor pump performance and (2) identifies the differences between fixed and variable clearance designs.

As Fig. 1 shows, a gerotor pump consists primarily of a chamber and inner and outer rotors between which there is clearance. In this paper, the clearance varies with the variation in the rotation angle and can thus be denoted as $ɛ(φ1)$⁠.

The coordinate systems for the mathematical model are given in Fig. 2 in which $S1$⁠, $S2$⁠, and $Sf$ with their origins at $O1$⁠, $O2$⁠, and $Of$ are rigidly attached to the outer rotor, inner rotor, and frame, respectively. The relation between the rotation angles $φ2$ and $φ1$ can be represented as

where $N$ is the tooth number of the inner rotor, $N+1$ is the tooth number of the outer rotor, and $m12$ is the angular velocity ratio.

The generated tooth profile of the inner rotor can then be determined by considering Eqs. (4) and (6) simultaneously.

When the inner rotor is used to generate the tooth profile of the outer rotor [11], the coordinate system is as shown in Fig. 3. Here, coordinate systems $Sa$ and $Sb$ with their origins at $Oa$ and $Ob$ are rigidly attached to the inner and outer rotors, respectively, and the relation between rotation angles $θ2$ and $θ1$ is represented as

Finally, the outer rotor can be solved by considering Eqs. (6), (8), and (10) simultaneously.

where $ɛ(0)$ is the clearance specified for the continuity of the moving grid (mesh) in the fluid analysis.

In this paper, the tooth number of the inner and outer rotors is set at eight and nine, respectively. Two fixed clearance (0.035 mm and 0.07 mm) and three variable clearance designs are then used to illustrate the feasibility of the proposed method. For comparative convenience, the outer rotors in all cases have the same volume (33,835 mm³) and share the design parameters listed in Table 1. These pumps sizes refer to the pump used in engine lubrication [22]. The clearance on the different angle positions is as illustrated in Fig. 4, and the five results of clearance are as shown in Fig. 5.

Recently, a new commercial computational fluid dynamics (CFD) package (pumplinx) was developed and demonstrated by comparing simulation and experiment results on varied fluid machinery [26]. Subsequently, Frosina et al. [27,28] presented many confirms to verify the accuracy of simulation of pumplinx package by the experimental results for the gerotor pump. Based on these reasons, pumplinx is used to construct the fluid analysis model.

where $r$ is the position vector from the origin of the coordinates to a specific particle of fluid. $T$ is the sum of the torque on the system.

in which, $C1$ = 1.44, $C2$ = 1.92, Prandtl numbers for the turbulent kinetic energy and the turbulent kinetic energy dissipation rate are $σk$ = 1 and $σɛ$ = 1.3, respectively.

where $μd$ is dynamic viscosity, $ρo$ is density, $Vo$ is outlet velocity, and $Do$ is outlet pipe diameter. $po$ and $pi$ are outlet and inlet pressures, respectively. $qo(ave)$⁠, $qo(max)$⁠, and $qo(min)$ are average flow rate, maximum flow rate, and minimum flow rate, respectively.

The fluid equations are discretized using a finite volume method. The SIMPLE algorithm is used for coupling the pressure and velocity terms. In particular, the numerical method is second-order upwind for discretization of partial differential equations (PDE), conjugate gradient squared (CGS) for momentum equation matrix, and algebraic multigrid (AMG) for pressure equation matrix, and AMG for pressure equation matrix. Besides, time accuracy is first-order discretized. The pressure term is upwind; the viscous term (or called diffusion term) is central difference and advection term (or called convection term) is second-order upwind.

The important assumptions and boundary conditions of the fluid analysis are shown in Table 2. The fluid is set as oil, and the fluid is analyzed using the turbulent, compressibility, and cavitation models.

For the standard k-ε model in the inlet and outlet, the initial conditions in the pumplinx are as below

1. (1)

   Turbulence kinetic energy (⁠$k$⁠) = 0.01 m²/s²

2. (2)

   Turbulence kinetic energy dissipation rate (⁠$ɛ$⁠) = 1 m²/s²

   Turbulence intensity (named $I$⁠) can be calculated as

   $I=23k/Uave=0.08165/Uave$

   (25)

where $Uave$ is average flow velocity.

The fluid analysis model is demonstrated using the Old-0.035 design, which includes relief grooves (see Fig. 6) that mitigate the pressure ripples inside the pump [36]. For a gerotor pump, a moving or sliding methodology should be adopted. Thus, the stationary and moving volumes are meshed separately.

First, the grid size independence test and time step size independence test are included. We get an acceptable accuracy with the maximum grid size and time step size (i.e., the minimum grid amount and minimum time steps amount) to save the required numerical resource. The convergence criterion equals to 0.001. As shown in Table 3, the error is reduced from 320 steps to 480 steps. So, the paper uses 640 time steps per gearing cycle for a more accurate simulation. To be specific, the model of the pump in the Old-0.035 design consists of a total of 130,639 cells (acceptable grid size) and 165,400 nodes, and the time step size is 0.00003125 s. Moreover, this simulation used dynamic grid and mismatched grid interface techniques [26] for algorithm to reduce the calculation time and derive the exact analysis value. The dynamic grid indicates that, when the rotor rotates, the CFD solver performs calculation on the grid automatically (update at each time step), thus, it can calculate the entire variation of the pressure, flow rate, flow velocity, and fluid moment with time. To verify the feasibility of the analytic model, Fig. 7(a) shows the grid and pressure results when the rotor rotates to the four particular times. Besides, Fig. 7(b) shows the flow velocity in X and Y directions at the three positions that the fluid flow through the clearance causing the main leakage problem. This result confirms that the pumplinx model can correctly simulate the flow condition.

As shown in Fig. 8(a), we can see the average flow rate reaches a stable value after 0.04 s. This indicates the flows had reached a steady state after two revolutions of inner rotor. Thus, the following results were obtained based on data from one revolution of the inner rotor after flows had reached a steady state. Figures 8(b) and 8(c) present the average flow rate and instantaneous flow rate, respectively. Comparing the fixed clearances of 0.035 and 0.07 mm shows that the smaller clearance reduced leakage and produced higher flow rates. Hence, although the average clearance in the three variable clearance designs was larger than 0.07 mm, the flow rate of the new-case 1 design fell between the Old-0.035 and Old-0.07 designs, meaning it was higher than that of the Old-0.07 design. Obviously, the geometric profiles of the rotor had an effect on the hydrodynamic forces: in the new-case 1 design, the clearance size ranged between 0.035 and 0.07 mm at 0 deg, 180 deg, and 360 deg.

A comparison of the new-case 2 and new-case 3 designs further shows that clearance in the new-case 2 was symmetrical, with its point at 180 deg, but clearance in the new-case 3 was asymmetrical. For both designs, clearance sizes ranged between 0.035 mm and 0.07 mm except at 0 deg and 360 deg; at all other positions, the clearance was greater than 0.07 mm. According to the flow rate analyses, the flow rate of new-case 2 was higher than that of new-case 3, although because of increased leakage, both flow rates were lower than that of the Old-0.07 design. Based on the results for new-case 1 and new-case 2, however, good flow characteristics were achieved when the wave trough of the clearance curve occurred at 180 deg. Besides, the flow rate irregularity can be used as a performance index related to vibration and noise [17,20], the higher value may produce more vibrations or noises. Obviously, the Old-0.035 design with highest value (see Fig. 8(d)) indicated there is more vibration condition. Particularly, the value of new-case 1 is similar to the Old-0.07 design, but the new-case 1 can greatly reduce hitting or collision opportunity than the Old-0.07 design.

The most turbulent flow situation is shown by the instantaneous Reynolds number and its average, reported in Figs. 9(a) and 9(b), respectively. These results provide additional powerful proof that the small fixed clearance (Old-0.035) achieves the highest Reynolds number, indicating the greatest flow turbulence. This phenomenon, however, is increasingly unsteady in the hydraulic system; the large clearance designs (new-case 2 and new-case 3) have smaller Reynolds numbers and less flow inertia force, leading to a lower flow rate. One aspect worth noting is that in the variable clearance design in new-case 1, the average clearance is larger than 0.07 mm (Old-0.07), but the fluid flow can achieve a larger Reynolds number and flow rate than the Old-0.07 design. So, this new-case 1 design can also enhance the force of the fluid inertia, thereby increasing the flow rate. Nevertheless, its Reynolds number is lower than that of the Old-0.035 design, suggesting that instability in the hydraulic system can be controlled.

In fact, the outlet pressures presented in Fig. 10(a) indicate that the new-case 1 design achieved the highest pressure peak at some positions. In conclusion, the pressure level of new-case 1 design is closed to the Old-0.035 design. In the other designs, on the other hand, when clearance was large, leakage increased, which reduced the pressure peak. According to the nondimensional analysis, the Euler number can be used to indicate the pressure loss property in the fluid flow. The results for the instantaneous Euler number and its average (Figs. 10(b) and 10(c), respectively) show that the new-case 1 design not only achieves relatively lower pressure loss than the Old-0.07 design but that the magnitude of pressure loss is near that of the Old-0.035 design. This significant result again shows the benefit of the new design method. A comparison of the new-case 2 and new-case 3 designs also shows relatively higher pressure losses because of large clearances, so the symmetrical clearance design may be superior to the nonsymmetrical design in terms of pressure loss.

Figure 11 illustrates the hydrodynamic effect of the inner 11(a) and outer rotor 11(b). The inner rotor experienced negative moment, signifying that the inner rotor rotations were affected by fluid resistance. The outer rotor, however, experienced mostly negative with some positive moment, indicating that although the outer rotor rotations were primarily affected by fluid resistance, they were occasionally driven by fluid moment. Figure 11 also reveals that when the clearance was small (i.e., in the case of Old-0.035), the fluid moment curve of the rotor was less stable, particularly when the inner rotor rotated between 120 deg and 240 deg. The flow rate (see Fig. 8) suggests the same result: although a small clearance design reduced leakage, the difference in pressure between the inner and outer rotor volumes increased accordingly, which would reduce the stability of the inner flow field and fluid moment. In addition, a gerotor pump may lose its volumetric efficiency due to cavitation when operating at high rotation speed (especially over 4000 rpm) [26]. In this paper, the gas volume fraction was adopted to estimate the cavitation risk. The gas volume fraction is the ratio of gas volumetric flow rate to the total volumetric flow rate and its results were shown in Fig. 12 for 3000 rpm rotation speed. The Old-0.035 design has highest gas volume fraction in one revolution and may raise the cavitation risk when increasing the rotation speed. Contrasting with the new-case 1 design, it can not only achieve relatively lower gas volume fraction to reduce cavitation risk but also maintain relatively higher volumetric efficiency comparing to the traditional gerotor pump at a high rotation speed (over 4000 rpm).

To clearly show the advantages of the new design, this paper also explores the pressure variation inside the pump. Figure 13 shows the eight survey points, which including R1 to R4 and L1 to L4. These survey positions in Fig. 13 are all stationary. The analysis considers three case studies: one with fixed clearances of 0.035 and no relief groove and two others, the Old-0.035 and new-case 1, which have relief grooves. As Figs. 14(a)–14(h) show, no matter the location, the no-relief groove design not only achieves the maximum pressure peak but also the largest pressure fluctuation. Due to the tiny clearance, the teeth of the inner and outer rotors may collide against each other and generate the largest stress fluctuation and noise. Nevertheless, although the relief groove in the Old-0.035 design can mitigate the pressure ripple, the clearance is small so hitting may still occur between rotors. So, the collision level between the inner and outer rotors mainly depends on the clearance condition. Synthesizing the analytic results for flow rate, fluid moment, and interior pressure confirms that shocks or collisions occasionally occur during rotor rotation, increasing stress fluctuations and reducing rotor life expectancy. The new-case 1 design, however, not only mitigates the pressure peak and pulsation better than the Old-0.035 design but also suppresses collisions between rotors, thereby decreasing stress fluctuations and increasing rotor life expectancy.

This paper has proposed a variable clearance design for gerotor pumps based on a mathematical model of the inner and outer rotors and the fluid analysis model with considering relief grooves for assessing gerotor pump performance.

According to the analytic results, when fixed clearance designs have a small clearance (i.e., 0.035 mm), leakage rates are reduced, thereby yielding high flow rates, but the effects of motor torque and fluid moment may cause to increase collision in the rotors. To mitigate the issue, in designing the curve of variable clearance, the wave trough should be located at 180 deg and have a low clearance value. This design would enhance flow characteristics and substantially reduce shock or collision among gerotor components, thereby yielding low and stable stress. Admittedly, a higher clearance value at the wave trough location (e.g., 0.12 mm) may also reduce stress and fluctuations; however, it would also increase leakage, thus generating poor flow characteristics. In general, the proposed variable clearance design can maintain robust flow characteristics and effectively lengthen the life expectancy of gerotor pumps, while the analytic model can serve as a reference in future gerotor pump development.

The author is grateful to the National Science Council of the R.O.C. for their grant. Part of this work has performed under Contract No. NSC 101-2221-E-150-023.

$c$ =

center distance between the outer rotor and the inner rotor

$cs$ =

surface of control volume

$cv$ =

control volume

$C1$⁠,$C2$ =

turbulence model constant

$Df$ =

diffusivity of vapor mass fraction

$fb$ =

body force (N)

$fg$ =

noncondensable gas mass fraction

$fv$ =

vapor mass fraction

$Gt$ =

turbulent generation term

$h$ =

center distance between the circular-arc and the inner rotor

$k$ =

turbulence kinetic energy

$Mij$ =

coordinate transformation matrix from system $j$ to system i

$n$ =

surface normal

N =

tooth number of the inner rotor

$Oj$ =

origin of the coordinate system, $Sj$⁠, $j=1,2,f$

$p$ =

pressure (Pa)

$Rc$ =

vapor condensation rate

$Re$ =

vapor generation rate

$ri$ =

position vector represented in $Si$⁠, $i=1,2$⁠, $i=a,b$

$Sj$ =

coordinate system $j$ where $j=1,2,f$

$V$ =

velocity vector

$α$ =

profile parameter of the circular-arc

$ɛ$ =

turbulence dissipation

$ɛ(φ1)$ =

function of clearance

$μ$ =

trochoid ratio

$μf$ =

fluid viscosity (Pa·s)

$μt$ =

turbulent viscosity (Pa·s)

$ρ$ =

fluid density (kg/m³)

$ρc$ =

radius of circular-arc

$ρg$ =

gas density (kg/m³)

$ρl$ =

liquid density (kg/m³)

$ρv$ =

vapor density (kg/m³)

$σf$ =

turbulent Schmidt number

$σk,σɛ$ =

turbulence model constant

$τ$ =

stress tensor

$φj$⁠,$θj$ =

rotation angle, $j=1,2$