IOP Conference Series: Earth and Environmental Science A CFD model for orbital gerotor motor To cite this article: H Ding et al 2012 IOP Conf. Ser.: Earth Environ. Sci. 15 062006 View the article online for updates and enhancements. Related content - Cavitation simulation and NPSH prediction of a double suction centrifugal pump P Li, Y F Huang and J Li - Experimental study on the cavitation of vortex diode based on CFD L Jiao, P P Zhang, C N Chen et al. - Numerical study of cavitation flows inside a tubular pumping station X L Tang, W Huang, F J Wang et al. This content was downloaded from IP address 148.251.232.83 on 14/09/2018 at 01:59
A CFD model for orbital gerotor motor H Ding 1, X J Lu 2 and B Jiang 3 1 Simerics Incorporated 1750 112th Ave. NE Ste. A203, Bellevue, 98004, USA 2 Ningbo Zhongyi Hydraulic Motor Co., Ltd. 88 Zhongyi Road, Zhenhai Economic Development Zone, Ningbo, China 3 College of Mechanical Engineering, University of Shanghai for Science and Technology 516 Jun Gong Road, Shanghai, 200093, China hd@simerics.com Abstract. In this paper, a full 3D transient CFD model for orbital gerotor motor is described in detail. One of the key technologies to model such a fluid machine is the mesh treatment for the dynamically changing rotor fluid volume. Based on the geometry and the working mechanism of the orbital gerotor, a moving/deforming mesh algorithm was introduced and implemented in a CFD software package. The test simulations show that the proposed algorithm is accurate, robust, and efficient when applied to industrial orbital gerotor motor designs. Simulation results are presented in the paper and compared with experiment test data. 1. Introduction A gerotor is a positive displacement machine which has an inner gear and an outer gear. For a normal gerotor machine, the inner gear, which is the drive gear, and the driven outer gear rotate around their own fixed centers during operation. Due to their compact design, low cost, and robustness, normal gerotor pumps are widely used in many industrial applications. There is an alternative design, the orbital gerotor, in which the outer gear is stationary, while the inner gear rotates around an orbiting center [1]. The orbital gerotor can be used as a motor to obtain high torque output at low rotation speed with small dimension. In this design, typically a rotating flow distributor is used to maintain proper timing connecting the inlet and the outlet ports to the rotor. CFD models of normal gerotor pumps have been used to improve gerotor designs in many engineering applications for the last decades. In 1997, Jiang and Perng [2] created the first full 3D transient CFD model for a gerotor pump and included a cavitation model. Their model successfully predicted gerotor pump volumetric efficiency loses due to cavitation. Kini et al. [3] coupled CFD simulation with a structural solver to determine deflection of the cover plate in the pump assembly due to variation in internal pressure profiles during operation. Zhang et al. [4] studied the effects of the inlet pressure, tip clearance, porting and the metering groove geometry on pump flow performances and pressure ripples using CFD model. Natchimuthu et al. [5], Ruvalcaba et al. [6] also used CFD to analyze gerotor oil pump flow patterns. Jiang et al. [7] created a 3D CFD model for crescent pumps, a variation of gerotor pumps with a crescent shaped island between the inner and outer gears. In comparison, CFD studies of orbital type of gerotor are rare. Authors of this paper have not found any full 3D CFD model for this type of gerotor in the literature. Because of the difference in motion mechanism, traditional gerotor model cannot be applied directly to orbital gerotor. Modifications in Published under licence by Ltd 1
moving/deforming mesh algorithm as well as modifications in surface velocity assignment, torque and power calculations are necessary. Orbital gerotors are commonly used as motors which have much higher pressure differences and even smaller fluid gaps as compared with normal gerotor pumps. Those two conditions impose big challenges for the flow solver. That could be one of the main reasons why CFD analysis for orbital gerotors is not very popular. 2. Orbital Gerotor Motor Configuration and Simulation Strategy 2.1. Working Principle of an Orbital Gerotor Motor As shown in Figure 1, an orbital gerotor motor has a stationary outer gear and a rotating inner gear. Inner gear has 1 less tooth than the outer gear. During operation, the inner gear rotates and rolls over the outer gear teeth. During the movement, the inner gear center also rotates around the outer gear center in the opposite direction. Each time when the inner gear advances one tooth, the inner gear center already rotates a complete revolution. Therefore the rotation speed of the center is NT in times that of the inner gear rotation speed, where NT in is the number of inner gear teeth. Figure 1.1 to Figure 1.10 show the sequence of gear motion for one complete revolution of the inner gear center. 10 9 8 7 6 2
1 2 3 4 5 Figure 1. Orbital gerotor motor Each cavity between neighboring outer gear teeth, bounded by the inner gear surface, forms a fluid pocket. During the operation, those fluid pockets change shape and volume. When the volume increases, it will draw in fluid. When the volume decreases, it will drive the fluid out. Combined with proper connections with the inlet and the outlet ports, those dynamically changing pockets will move the fluid from the inlet to the outlet while at the same time outputting torque and power to the shaft. Figure 2 shows the complete shape change sequences of one of the pockets when the inner gear advances one tooth over the outer gear. The plots 2.1 to 2.5 show the sequences of the expansion half cycle, and 2.6 to 2.10 show the compression half cycle. Unlike a normal gerotor where the fluid pockets are rotating and the inlet and outlet ports are stationary, for orbiting gerotor, those fluid pockets stay in the same location during the operation. In order to provide proper timing for the connections with the inlet and the outlet, typically there is a rotating distributor to create dynamic bridges between the ports and the rotor. The purpose of the distributor is to connect each pocket to the high pressure inlet during its expansion half cycle, and to the low pressure outlet during its compression half cycle. Typically, the flow distributor rotates at the same speed as the inner gear. Extra caution needs to be taken when creating fluid volumes for the flow distributor and the rotor. It is important to make sure that the initial relative position between the inner gear and the distributor is accurate, otherwise the motor system may not work as expected. 1 2 3 4 5 6 7 8 9 10 Figure 2. Shape and volume change sequence of one fluid pocket 2.2. Instant Center of Rotation Since the inner gear of an orbiting gerotor does not have a fixed rotation axis, calculating the hydraulic torque applied to the inner gear becomes an issue. One way to resolve this issue is to find the instantaneous center of rotation of the inner gear. For a body undergoing planar movement, the instantaneous center of rotation (ICOR) is the point where the velocity is zero at a particular instance of time. At that instance, the body is doing a pure rotation around the ICOR. If the ICOR is known, the hydraulic torque can be calculated as the torque against the ICOR at that moment. 3
Figure 3. Instant center of rotation ICOR of an orbital gerotor inner gear can be found by checking the velocity distribution on the inner rotor. As shown in Figure 4, all the points on the inner gear undergo a composite motion: a) translation with the motion of the gear center, and b) rotation around the gear center with speed in. The inner gear center itself rotates around the outer gear center with the speed of c. As mentioned previously, the relationship between the two rotation speeds is: As shown in figure 4, we can always draw a line (line of symmetry) connecting the inner gear center and the outer gear center at any moment of time. Defining a right-hand coordinate system with the origin at the inner gear center, the y axis along the symmetry line, and the x axis in a direction perpendicular to the y axis enables the velocity of the inner gear center in x and y directions to be defined as: where Ec is the eccentricity of the inner gear, or the distance between the inner gear center and the outer gear center. For any point on inner gear with coordinates (x, y), the velocity components for rotation around the inner gear center are; and the combined velocities are: (1) (2) (3) (4) (5) (6) (7) 4
From equation (6) and (7), it is clear that at the point (0, ), both velocity components equal zero. Therefore, that point corresponds to the coordinates of the instant center of rotation. Since the line of symmetry rotates around the outer gear center at the speed of c, it is very straight forward to calculate ICOR during the simulation. 2.3. Mesh Solution Similarly, the motion of the inner gear boundary can be determined through the composite motion of the rotation around the inner gear center plus the translation of the inner gear center. The shape of the fluid volume for the rotor is then properly defined. Meshing of moving/deforming fluid domains in a positive displacement (PD) fluid machine is always very challenging. As a typical PD machine, gerotor motor has many dynamic fluid gaps with very small clearances, down to several microns. Those gaps have a strong influence on machine s performance including flow leakage and volumetric efficiency, flow and pressure ripple, pressure lock, cavitation and erosion, and torque and power. Therefore they have to be modeled accurately. Many generic moving mesh solutions, for example the immersed boundary method, have difficulties in modeling such dynamic gaps. So far, the most successful solution for creating a gerotor rotor mesh is the structured moving/sliding mesh approach commonly used in normal gerotor pump simulations (Jiang and Perng [2]). This approach is also adapted in this study. In the structured moving/sliding mesh approach, the fluid volume of the rotor chamber is separated from the other parts of the fluid domain. Topologically, the rotor volume is similar to a ring, and an initial structured mesh can be easily created for that kind of shape. The rotor mesh will be connected to other fluid volumes through sliding interfaces. When the inner gear surface moves to a new position, the mesh on the surface of the inner gear does not simply move with the inner gear surface. Instead, the mesh slides on the inner gear surface while make the necessary adjustments to conform to the new clearance between the inner gear surface and the outer gear surface. Simultaneously, the interface connections between the rotor volume and other fluid volumes are updated. Figure 3 shows a typical structured mesh for a gerotor rotor volume. Figure 4. Gerotor rotor structured mesh 2.4. Implementation The proposed orbital gerotor model was implemented in the commercial CFD package PumpLinx as a new template. A template in PumpLinx provides two main functionalities: 1) It creates the initial rotor mesh, and controls mesh moving /deformation of the rotor and other dynamic fluid volumes during the simulation; and 2) It provides special setup and post processing options for that specific 5
fluid machine. With the help of the template, user can setup a complete orbital gerotor motor in less than 30 minutes starting from proper CAD geometry output. One can refer to Ding et al. [8] for a more detailed description of the software. 3. CFD Solver and Governing Equations The CFD package used in this study solves conservation equations of mass and momentum using a finite volume approach. Those conservation laws can be written in integral representation as (8) (9) The standard k two-equation model (Launder & Spalding [9]) is used to account for turbulence, (10) (11) The cavitation model included in the software describes the cavitation vapor distribution using the following formulation (Singhal et al., [10]) (12) where is the diffusivity of the vapor mass fraction and f is the turbulent Schmidt number. The effects of liquid vapor, non-condensable gas (typically air), and liquid compressibility are all accounted for in the model. The final density calculation for the mixture is done by This software package has been successfully used in CFD simulations for many different types of positive displacement machines including: swash plate piston pump [11], gerotor pump [8], external gear pump [12], crescent pump [7], and variable displacement vane pump [13]. 4. Gerotor Motor Test Case An industrial orbital gerotor motor was used to demonstrate the proposed CFD model. Figure 5 is the solid model of the motor. This motor has two ports, port A and port B. The inner gear and flow distributor can also rotate in both directions without mechanical adjustment. The flow and rotation directions are determined by which port is connected to the high pressure fluid and which port is connected to the low pressure fluid. The one connected to the high pressure fluid becomes the inlet and the rotation direction will also change accordingly. (13) 6
Figure 5. Solid model of an orbital gerotor motor The fluid domain was subtracted from CAD geometry and divided into several volumes and meshed separately (Figure 6). Except for the rotor part which was created with structured mesh, all other fluid volumes were meshed with unstructured binary tree mesh. The special moving/sliding mesh of rotor volume and the rotation of flow distributor volume were automatically processed by the template, and the rest of the fluid volumes stayed stationary during the simulation. Those independent volumes were connected through sliding interfaces during simulation. A total of 360,000 cells was used in this model. Figure 6. Fluid volumes with mesh The working fluid used in the model is the high performance anti-wear hydraulic fluid HM46. The properties of HM46 are listed in Table 1. Determined based on the information provided by motor manufacturer, operating conditions used in simulation are also listed in table 1. 7
Table 1. Fluid properties and operating conditions Density (kg/m 3 ) 879 Viscosity (PaS) 0.04 Rotation speed (RPM) 100 Inlet pressure (MPa) 1 Outlet pressure (MPa) 16 5. Simulation Results and Discussion Figure 7 shows the pressure distribution of high pressure inlet, low pressure outlet, and the flow distributor. The magenta color indicates high pressure and the blue color indicates low pressure, with an overall pressure range from 0 to 18 MPa. Figure 7. Pressure distribution on inlet/outlet ports and flow distributor The flow distributor for this motor has a total of 16 shoe shaped connectors to be connected to the rotor fluid pockets. Eight of the connectors connect to the low pressure outlet, and the other eight connect to the high pressure inlet. The connectors are arranged alternately and rotate at the same speed as the inner gear to create the proper timing of the connections. Figure 8 shows the simulation results at 4 different moments. In the picture, surfaces are colored by pressure with red representing high pressure, and blue representing low pressure, with an overall range from 0 to 20 MPa. Small spheres in those pictures are massless particles used to visualize the flow field. The white lines extruding from the particles show the direction and magnitude of the velocity of each particle. One can see that the red particles, coming from the high pressure inlet, are drawn into the rotor. And the blue particles, after the pockets connect to the low pressure port, are driven away from the rotor towards the outlet. 8
Figure 8. Pressure distribution and particle tracing Figures 9 to 12 plot the time history of the pressure in one of the fluid pocket; the mass flow rate; the power applied to the inner gear, and the torque applied to the inner gear. These curves correspond to a 100 RPM rotation speed for one complete revolution of the inner gear. The horizontal axis for these plots is the rotation angle of the inner gear. Figure 9. Pressure in a fluid pocket Figure 10. Mass flow rate The plots show that the solution has a clear periodical pattern except in the first couple of time steps. The pattern repeats itself every time the inner gear advances one tooth. This means that, under the current simulation conditions, one only needs to solve 2 to 3 inner gear teeth rotation, or 90 to 135 degree of the inner gear rotation, to have a complete set of flow characteristics of the motor. The transient simulation time to model one gear tooth rotation for these simulation conditions is about 35 minutes on a quad-core single CPU 2.2GHZ I7 2720QM Laptop Computer. 9
Figure 11. Hydraulic power Figure 12. Torque Experimental test samples provided by the manufacturer have rotational speeds ranging from 103 to 117RPM, and pressure differences ranging from 15 to 17 MPa. For this type of motor, the flow rate is a linear function of the rotation speed, and the torque is a linear function of the pressure difference. In order to have a fair comparison, the test flow rates are linearly converted to 100 RPM, and the test torques are linearly converted to15 MPa pressure difference. The converted volume flow rate and output torque of 41 test samples are plotted in figure 13 and 14 against the CFD simulation results. The horizontal axis of the two plots is test sample number. The plots show that the CFD flow rate prediction matches very well with the test data. The predicted torque is about 12% higher than the test results. Since torque measured in the experiment is the final output torque from the motor, it has mechanical and friction loses that are not accounted for in CFD results. This could be the main reason for the discrepancy in CFD torque prediction. Figure 13. Comparison of predicted and test flow Figure 14. Comparison of predicted and test rate torque Figures 15 and 16 plot the flow rate and power vs. rotation speed respectively. As expected, both the flow rate and the power are linearly increasing with the rotation speed. 10
Figure 15. Flow rate vs. rotation speed Figure 16. Power vs. rotation speed Figure 17 plots the torque vs. the rotational speed. From this plot, one can see that the torque of orbital gerotor motor is not a strong function of rotational speed. However the torque does decrease slightly when the rotational speed increases. Figure 17. Torque vs. rotation speed 6. Conclusions By analyzing the working mechanism of orbital gerotor motors, a CFD model for such fluid machine was developed and implemented as a new template in the CFD software PumpLinx. Simulation for a production motor shows that the present computational model is accurate and efficient. It s also found that the flow solver used in the current study is very robust in handling very high mesh aspect ratios and very small dynamic leakage gaps. With the demonstrated speed, robustness, and accuracy, this model can be used as a high fidelity design tool in the design process or as a diagnosis tool for orbital gerotor motors. Nomenclature c Inner gear center C 1 Turbulence model constant C 2 Turbulence model constant C c Cavitation model constant C e Cavitation model constant C Turbulence model constant D f Diffusivity of vapor mass fraction Ec Inner gear eccentricity f Body force (N) f v Vapor mass fraction t S' ij U u u' v v' vx, vy x, y Time Strain tensor Initial velocity Velocity component (m/s) Component of v' Velocity vector Turbulent fluctuation velocity Velocity in x, y direction Coordinates Turbulence dissipation 11
f g G t ICOR in k L M NT n p Q R c R e RPM Non-condensable gas mass fraction Turbulent generation term Instant center of rotation Inner gear Turbulence kinetic energy Length Mass flow rate (Kg/s) Number of gear teeth Surface normal Pressure (Pa) Flow rate (m 3 /h) Vapor condensation rate Vapor generation rate Revolution per minute t g l v k l f Fluid viscosity (Pa-s) Turbulent viscosity (Pa-s) Fluid density (kg/m 3 ) Gas density (kg/m 3 ) Liquid density (kg/m 3 ) Vapor density (kg/m 3 ) Surface of control volume Turbulence model constant Surface tension Turbulence model constant Turbulent Schmidt number Stress tensor Control volume Rotation speed References [1] Ivantysyn J and Ivantysnova M 2003 Hydrostatic Pumps and Motors (New Delhi : Tech Books International) [2] Jiang Y and Perng C 1997 An Efficient 3D Transient Computational Model for Vane Oil Pump and Gerotor Oil Pump Simulations SAE Technical Paper 970841 [3] Kini S, Mapara N, Thoms R and Chang P 2005 Numerical Simulation of Cover Plate Deflection in the Gerotor Pump SAE Technical Paper 2005-01-1917 [4] Zhang D, Perng C and Laverty M 2006 Gerotor Oil Pump Performance and Flow/Pressure Ripple Study SAE Technical Paper 2006-01-0359 [5] Natchimuthu K, Sureshkumar J and Ganesan V 2010 CFD Analysis of Flow through a Gerotor Oil Pump SAE Technical Paper 2010-01-1111 [6] Ruvalcaba M A and Hu X Gerotor Fuel Pump Performance and Leakage Study ASME 2011 Int. Mechanical Engineering Congress & Exposition (Denver, Colorado, USA, 2011) [7] Jiang Y, Furmanczyk M, Lowry S and Zhang D et al. 2008 A Three-Dimensional Design Tool for Crescent Oil Pumps SAE Technical Paper 2008-01-0003 [8] Ding H, Visser F C, Jiang Y and Furmanczyk M 2011 J. Fluids Eng. Trans ASME 133(1) 011101 [9] Launder B E and Spalding D B 1974 Comput. Methods Appl. Mech. Eng. 3 269-289 [10] Singhal A K, Athavale M M, Li H Y and Jiang Y 2002 J. Fluids Eng. Trans ASME 124(3) 617-624. [11] Meincke O and Rahmfeld R 2008 6th Int. Fluid Power Conf. (Dresden, 1-2 April 2008) 485-99 [12] Heisler A, Moskwa J and Fronczak F 2009 The Design of Low-Inertia, High-Speed External Gear Pump/Motors for Hydrostatic Dynamometer Systems SAE Technical Paper 2009-01- 1117. [13] Wang D, Ding H, Jiang Y and Xiang X 2012 Numerical Modeling of Vane Oil Pump with Variable Displacement SAE Technical Paper 2012-01-0637. 12