IOP Conference Series Earth and Environmental Science
OPEN ACCESS
A CFD model for orbital gerotor motorTo cite this article H Ding et al 2012 IOP Conf Ser Earth Environ Sci 15 062006
View the article online for updates and enhancements
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A CFD model for orbital gerotor motor
H Ding1 X J Lu
2 and B Jiang
3
1Simerics Incorporated
1750 112th Ave NE Ste A203 Bellevue 98004 USA 2Ningbo Zhongyi Hydraulic Motor Co Ltd
88 Zhongyi Road Zhenhai Economic Development Zone Ningbo China 3College of Mechanical Engineering University of Shanghai for Science and
Technology 516 Jun Gong Road Shanghai 200093 China
hdsimericscom
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 movingdeforming 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
Published under licence by IOP Publishing Ltd 1
movingdeforming 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
21 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 NTin times
that of the inner gear rotation speed where NTin is the number of inner gear teeth Figure 11 to Figure
110 show the sequence of gear motion for one complete revolution of the inner gear center
6
7
8
9
10
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
2
Figure 1 Orbital gerotor motor
Each cavity between neighboring outer gear teeth bounded by the inner gear surface forms a fluid
ldquopocketrdquo 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 21 to 25 show the sequences of the expansion half
cycle and 26 to 210 show the compression half cycle
Unlike a normal gerotor where the fluid ldquopocketsrdquo are rotating and the inlet and outlet ports are
stationary for orbiting gerotor those fluid ldquopocketsrdquo 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
Figure 2 Shape and volume change sequence of one fluid pocket
22 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
1
2 3 4 5
1
2
3
4
5
6
7
8
9
10
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
(1)
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
(2)
(3)
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
(4)
(5)
and the combined velocities are
(6)
(7)
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
23 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 movingdeforming 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 machinersquos
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 movingsliding mesh approach commonly used in normal gerotor pump simulations
(Jiang and Perng [2]) This approach is also adapted in this study
In the structured movingsliding 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 ldquoslidesrdquo 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
24 Implementation
The proposed orbital gerotor model was implemented in the commercial CFD package PumpLinxreg
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 amp 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
(13)
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 movingsliding
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 360000 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
7
Table 1 Fluid properties and operating conditions
Density (kgm3) 879
Viscosity (PaS) 004
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 inletoutlet 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 22GHZ I7 2720QM Laptop Computer
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
A CFD model for orbital gerotor motor
H Ding1 X J Lu
2 and B Jiang
3
1Simerics Incorporated
1750 112th Ave NE Ste A203 Bellevue 98004 USA 2Ningbo Zhongyi Hydraulic Motor Co Ltd
88 Zhongyi Road Zhenhai Economic Development Zone Ningbo China 3College of Mechanical Engineering University of Shanghai for Science and
Technology 516 Jun Gong Road Shanghai 200093 China
hdsimericscom
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 movingdeforming 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
Published under licence by IOP Publishing Ltd 1
movingdeforming 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
21 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 NTin times
that of the inner gear rotation speed where NTin is the number of inner gear teeth Figure 11 to Figure
110 show the sequence of gear motion for one complete revolution of the inner gear center
6
7
8
9
10
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
2
Figure 1 Orbital gerotor motor
Each cavity between neighboring outer gear teeth bounded by the inner gear surface forms a fluid
ldquopocketrdquo 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 21 to 25 show the sequences of the expansion half
cycle and 26 to 210 show the compression half cycle
Unlike a normal gerotor where the fluid ldquopocketsrdquo are rotating and the inlet and outlet ports are
stationary for orbiting gerotor those fluid ldquopocketsrdquo 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
Figure 2 Shape and volume change sequence of one fluid pocket
22 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
1
2 3 4 5
1
2
3
4
5
6
7
8
9
10
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
(1)
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
(2)
(3)
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
(4)
(5)
and the combined velocities are
(6)
(7)
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
23 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 movingdeforming 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 machinersquos
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 movingsliding mesh approach commonly used in normal gerotor pump simulations
(Jiang and Perng [2]) This approach is also adapted in this study
In the structured movingsliding 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 ldquoslidesrdquo 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
24 Implementation
The proposed orbital gerotor model was implemented in the commercial CFD package PumpLinxreg
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 amp 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
(13)
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 movingsliding
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 360000 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
7
Table 1 Fluid properties and operating conditions
Density (kgm3) 879
Viscosity (PaS) 004
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 inletoutlet 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 22GHZ I7 2720QM Laptop Computer
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
movingdeforming 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
21 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 NTin times
that of the inner gear rotation speed where NTin is the number of inner gear teeth Figure 11 to Figure
110 show the sequence of gear motion for one complete revolution of the inner gear center
6
7
8
9
10
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
2
Figure 1 Orbital gerotor motor
Each cavity between neighboring outer gear teeth bounded by the inner gear surface forms a fluid
ldquopocketrdquo 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 21 to 25 show the sequences of the expansion half
cycle and 26 to 210 show the compression half cycle
Unlike a normal gerotor where the fluid ldquopocketsrdquo are rotating and the inlet and outlet ports are
stationary for orbiting gerotor those fluid ldquopocketsrdquo 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
Figure 2 Shape and volume change sequence of one fluid pocket
22 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
1
2 3 4 5
1
2
3
4
5
6
7
8
9
10
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
(1)
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
(2)
(3)
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
(4)
(5)
and the combined velocities are
(6)
(7)
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
23 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 movingdeforming 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 machinersquos
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 movingsliding mesh approach commonly used in normal gerotor pump simulations
(Jiang and Perng [2]) This approach is also adapted in this study
In the structured movingsliding 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 ldquoslidesrdquo 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
24 Implementation
The proposed orbital gerotor model was implemented in the commercial CFD package PumpLinxreg
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 amp 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
(13)
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 movingsliding
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 360000 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
7
Table 1 Fluid properties and operating conditions
Density (kgm3) 879
Viscosity (PaS) 004
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 inletoutlet 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 22GHZ I7 2720QM Laptop Computer
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
Figure 1 Orbital gerotor motor
Each cavity between neighboring outer gear teeth bounded by the inner gear surface forms a fluid
ldquopocketrdquo 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 21 to 25 show the sequences of the expansion half
cycle and 26 to 210 show the compression half cycle
Unlike a normal gerotor where the fluid ldquopocketsrdquo are rotating and the inlet and outlet ports are
stationary for orbiting gerotor those fluid ldquopocketsrdquo 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
Figure 2 Shape and volume change sequence of one fluid pocket
22 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
1
2 3 4 5
1
2
3
4
5
6
7
8
9
10
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
(1)
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
(2)
(3)
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
(4)
(5)
and the combined velocities are
(6)
(7)
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
23 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 movingdeforming 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 machinersquos
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 movingsliding mesh approach commonly used in normal gerotor pump simulations
(Jiang and Perng [2]) This approach is also adapted in this study
In the structured movingsliding 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 ldquoslidesrdquo 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
24 Implementation
The proposed orbital gerotor model was implemented in the commercial CFD package PumpLinxreg
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 amp 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
(13)
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 movingsliding
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 360000 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
7
Table 1 Fluid properties and operating conditions
Density (kgm3) 879
Viscosity (PaS) 004
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 inletoutlet 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 22GHZ I7 2720QM Laptop Computer
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
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
(1)
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
(2)
(3)
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
(4)
(5)
and the combined velocities are
(6)
(7)
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
23 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 movingdeforming 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 machinersquos
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 movingsliding mesh approach commonly used in normal gerotor pump simulations
(Jiang and Perng [2]) This approach is also adapted in this study
In the structured movingsliding 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 ldquoslidesrdquo 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
24 Implementation
The proposed orbital gerotor model was implemented in the commercial CFD package PumpLinxreg
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 amp 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
(13)
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 movingsliding
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 360000 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
7
Table 1 Fluid properties and operating conditions
Density (kgm3) 879
Viscosity (PaS) 004
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 inletoutlet 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 22GHZ I7 2720QM Laptop Computer
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
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
23 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 movingdeforming 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 machinersquos
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 movingsliding mesh approach commonly used in normal gerotor pump simulations
(Jiang and Perng [2]) This approach is also adapted in this study
In the structured movingsliding 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 ldquoslidesrdquo 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
24 Implementation
The proposed orbital gerotor model was implemented in the commercial CFD package PumpLinxreg
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 amp 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
(13)
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 movingsliding
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 360000 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
7
Table 1 Fluid properties and operating conditions
Density (kgm3) 879
Viscosity (PaS) 004
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 inletoutlet 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 22GHZ I7 2720QM Laptop Computer
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
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 amp 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
(13)
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 movingsliding
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 360000 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
7
Table 1 Fluid properties and operating conditions
Density (kgm3) 879
Viscosity (PaS) 004
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 inletoutlet 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 22GHZ I7 2720QM Laptop Computer
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
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 movingsliding
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 360000 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
7
Table 1 Fluid properties and operating conditions
Density (kgm3) 879
Viscosity (PaS) 004
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 inletoutlet 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 22GHZ I7 2720QM Laptop Computer
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
Table 1 Fluid properties and operating conditions
Density (kgm3) 879
Viscosity (PaS) 004
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 inletoutlet 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 22GHZ I7 2720QM Laptop Computer
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
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 22GHZ I7 2720QM Laptop Computer
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
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
rate
Figure 14 Comparison of predicted and test
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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
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 Itrsquos 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
C1
C2
Cc
Ce
C
Df
Ec
f
fv
Inner gear center
Turbulence model constant
Turbulence model constant
Cavitation model constant
Cavitation model constant
Turbulence model constant
Diffusivity of vapor mass fraction
Inner gear eccentricity
Body force (N)
Vapor mass fraction
t
Sij
U
u
u
v
v vx vy x y
Time
Strain tensor
Initial velocity
Velocity component (ms)
Component of v
Velocity vector
Turbulent fluctuation velocity
Velocity in x y direction
Coordinates
Turbulence dissipation
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
11
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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 FlowPressure
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 amp 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 ndash 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 ndash 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 PumpMotors 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
26th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 15 (2012) 062006 doi1010881755-1315156062006
12
fg
Gt
ICOR
in
k
L
M
NT
n p
Q
Rc
Re
RPM
Non-condensable gas mass fraction
Turbulent generation term
Instant center of rotation
Inner gear
Turbulence kinetic energy
Length
Mass flow rate (Kgs)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3h)
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 (kgm3)
Gas density (kgm3)
Liquid density (kgm3)
Vapor density (kgm3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress 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
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