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A CFD model for orbital gerotor motor
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2012 IOP Conf. Ser.: Earth Environ. Sci. 15 062006
(http://iopscience.iop.org/1755-1315/15/6/062006)
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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
[email protected]
Abstract. In this paper, a full 3D transient CFD model for
orbital gerotor motor is described in
detail. One of the key technologies to model such a fluid
machine is the mesh treatment for the
dynamically changing rotor fluid volume. Based on the geometry
and the working mechanism
of the orbital gerotor, a moving/deforming mesh algorithm was
introduced and implemented in
a CFD software package. The test simulations show that the
proposed algorithm is accurate,
robust, and efficient when applied to industrial orbital gerotor
motor designs. Simulation
results are presented in the paper and compared with experiment
test data.
1. Introduction
A gerotor is a positive displacement machine which has an inner
gear and an outer gear. For a normal
gerotor machine, the inner gear, which is the drive gear, and
the driven outer gear rotate around their
own fixed centers during operation. Due to their compact design,
low cost, and robustness, normal
gerotor pumps are widely used in many industrial applications.
There is an alternative design, the
orbital gerotor, in which the outer gear is stationary, while
the inner gear rotates around an orbiting
center [1]. The orbital gerotor can be used as a motor to obtain
high torque output at low rotation
speed with small dimension. In this design, typically a rotating
flow distributor is used to maintain
proper timing connecting the inlet and the outlet ports to the
rotor.
CFD models of normal gerotor pumps have been used to improve
gerotor designs in many
engineering applications for the last decades. In 1997, Jiang
and Perng [2] created the first full 3D
transient CFD model for a gerotor pump and included a cavitation
model. Their model successfully
predicted gerotor pump volumetric efficiency loses due to
cavitation. Kini et al. [3] coupled CFD
simulation with a structural solver to determine deflection of
the cover plate in the pump assembly due
to variation in internal pressure profiles during operation.
Zhang et al. [4] studied the effects of the
inlet pressure, tip clearance, porting and the metering groove
geometry on pump flow performances
and pressure ripples using CFD model. Natchimuthu et al. [5],
Ruvalcaba et al. [6] also used CFD to
analyze gerotor oil pump flow patterns. Jiang et al. [7] created
a 3D CFD model for crescent pumps, a
variation of gerotor pumps with a crescent shaped island between
the inner and outer gears.
In comparison, CFD studies of orbital type of gerotor are rare.
Authors of this paper have not found
any full 3D CFD model for this type of gerotor in the
literature. Because of the difference in motion
mechanism, traditional gerotor model cannot be applied directly
to orbital gerotor. Modifications in
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moving/deforming mesh algorithm as well as modifications in
surface velocity assignment, torque and
power calculations are necessary. Orbital gerotors are commonly
used as motors which have much
higher pressure differences and even smaller fluid gaps as
compared with normal gerotor pumps.
Those two conditions impose big challenges for the flow solver.
That could be one of the main reasons
why CFD analysis for orbital gerotors is not very popular.
2. Orbital Gerotor Motor Configuration and Simulation
Strategy
2.1. Working Principle of an Orbital Gerotor Motor
As shown in Figure 1, an orbital gerotor motor has a stationary
outer gear and a rotating inner gear.
Inner gear has 1 less tooth than the outer gear. During
operation, the inner gear rotates and rolls over
the outer gear teeth. During the movement, the inner gear center
also rotates around the outer gear
center in the opposite direction. Each time when the inner gear
advances one tooth, the inner gear
center already rotates a complete revolution. Therefore the
rotation speed of the center is NTin times
that of the inner gear rotation speed, where NTin is the number
of inner gear teeth. Figure 1.1 to Figure
1.10 show the sequence of gear motion for one complete
revolution of the inner gear center.
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Figure 1. Orbital gerotor motor
Each cavity between neighboring outer gear teeth, bounded by the
inner gear surface, forms a fluid
pocket. During the operation, those fluid pockets change shape
and volume. When the volume increases, it will draw in fluid. When
the volume decreases, it will drive the fluid out. Combined
with
proper connections with the inlet and the outlet ports, those
dynamically changing pockets will move
the fluid from the inlet to the outlet while at the same time
outputting torque and power to the shaft.
Figure 2 shows the complete shape change sequences of one of the
pockets when the inner gear
advances one tooth over the outer gear. The plots 2.1 to 2.5
show the sequences of the expansion half
cycle, and 2.6 to 2.10 show the compression half cycle.
Unlike a normal gerotor where the fluid pockets are rotating and
the inlet and outlet ports are stationary, for orbiting gerotor,
those fluid pockets stay in the same location during the operation.
In order to provide proper timing for the connections with the
inlet and the outlet, typically there is a
rotating distributor to create dynamic bridges between the ports
and the rotor. The purpose of the
distributor is to connect each pocket to the high pressure inlet
during its expansion half cycle, and to
the low pressure outlet during its compression half cycle.
Typically, the flow distributor rotates at the
same speed as the inner gear. Extra caution needs to be taken
when creating fluid volumes for the flow
distributor and the rotor. It is important to make sure that the
initial relative position between the inner
gear and the distributor is accurate, otherwise the motor system
may not work as expected.
Figure 2. Shape and volume change sequence of one fluid
pocket
2.2. Instant Center of Rotation
Since the inner gear of an orbiting gerotor does not have a
fixed rotation axis, calculating the hydraulic
torque applied to the inner gear becomes an issue. One way to
resolve this issue is to find the
instantaneous center of rotation of the inner gear. For a body
undergoing planar movement, the
instantaneous center of rotation (ICOR) is the point where the
velocity is zero at a particular instance
of time. At that instance, the body is doing a pure rotation
around the ICOR. If the ICOR is known, the
hydraulic torque can be calculated as the torque against the
ICOR at that moment.
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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)
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From equation (6) and (7), it is clear that at the point (0, ),
both velocity components equal
zero. Therefore, that point corresponds to the coordinates of
the instant center of rotation. Since the
line of symmetry rotates around the outer gear center at the
speed of c, it is very straight forward to calculate ICOR during
the simulation.
2.3. Mesh Solution
Similarly, the motion of the inner gear boundary can be
determined through the composite motion of
the rotation around the inner gear center plus the translation
of the inner gear center. The shape of the
fluid volume for the rotor is then properly defined.
Meshing of moving/deforming fluid domains in a positive
displacement (PD) fluid machine is
always very challenging. As a typical PD machine, gerotor motor
has many dynamic fluid gaps with
very small clearances, down to several microns. Those gaps have
a strong influence on machines performance including flow leakage
and volumetric efficiency, flow and pressure ripple, pressure
lock,
cavitation and erosion, and torque and power. Therefore they
have to be modeled accurately. Many
generic moving mesh solutions, for example the immersed boundary
method, have difficulties in
modeling such dynamic gaps. So far, the most successful solution
for creating a gerotor rotor mesh is
the structured moving/sliding mesh approach commonly used in
normal gerotor pump simulations
(Jiang and Perng [2]). This approach is also adapted in this
study.
In the structured moving/sliding mesh approach, the fluid volume
of the rotor chamber is separated
from the other parts of the fluid domain. Topologically, the
rotor volume is similar to a ring, and an
initial structured mesh can be easily created for that kind of
shape. The rotor mesh will be connected
to other fluid volumes through sliding interfaces. When the
inner gear surface moves to a new position,
the mesh on the surface of the inner gear does not simply move
with the inner gear surface. Instead,
the mesh slides on the inner gear surface while make the
necessary adjustments to conform to the new clearance between the
inner gear surface and the outer gear surface. Simultaneously, the
interface
connections between the rotor volume and other fluid volumes are
updated. Figure 3 shows a typical
structured mesh for a gerotor rotor volume.
Figure 4. Gerotor rotor structured mesh
2.4. Implementation
The proposed orbital gerotor model was implemented in the
commercial CFD package PumpLinx
as
a new template. A template in PumpLinx provides two main
functionalities: 1) It creates the initial
rotor mesh, and controls mesh moving /deformation of the rotor
and other dynamic fluid volumes
during the simulation; and 2) It provides special setup and post
processing options for that specific
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fluid machine. With the help of the template, user can setup a
complete orbital gerotor motor in less
than 30 minutes starting from proper CAD geometry output. One
can refer to Ding et al. [8] for a more
detailed description of the software.
3. CFD Solver and Governing Equations
The CFD package used in this study solves conservation equations
of mass and momentum using a
finite volume approach. Those conservation laws can be written
in integral representation as
(8)
(9)
The standard k two-equation model (Launder & Spalding [9])
is used to account for turbulence,
(10)
(11)
The cavitation model included in the software describes the
cavitation vapor distribution using the
following formulation (Singhal et al., [10])
(12)
where is the diffusivity of the vapor mass fraction and f is the
turbulent Schmidt number. The effects of liquid vapor,
non-condensable gas (typically air), and liquid compressibility are
all accounted for in
the model. The final density calculation for the mixture is done
by
(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.
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Figure 5. Solid model of an orbital gerotor motor
The fluid domain was subtracted from CAD geometry and divided
into several volumes and
meshed separately (Figure 6). Except for the rotor part which
was created with structured mesh, all
other fluid volumes were meshed with unstructured binary tree
mesh. The special moving/sliding
mesh of rotor volume and the rotation of flow distributor volume
were automatically processed by the
template, and the rest of the fluid volumes stayed stationary
during the simulation. Those independent
volumes were connected through sliding interfaces during
simulation. A total of 360,000 cells was
used in this model.
Figure 6. Fluid volumes with mesh
The working fluid used in the model is the high performance
anti-wear hydraulic fluid HM46. The
properties of HM46 are listed in Table 1. Determined based on
the information provided by motor
manufacturer, operating conditions used in simulation are also
listed in table 1.
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Table 1. Fluid properties and operating conditions
Density (kg/m3) 879
Viscosity (PaS) 0.04
Rotation speed (RPM) 100
Inlet pressure (MPa) 1
Outlet pressure (MPa) 16
5. Simulation Results and Discussion
Figure 7 shows the pressure distribution of high pressure inlet,
low pressure outlet, and the flow
distributor. The magenta color indicates high pressure and the
blue color indicates low pressure, with
an overall pressure range from 0 to 18 MPa.
Figure 7. Pressure distribution on inlet/outlet ports and flow
distributor
The flow distributor for this motor has a total of 16 shoe
shaped connectors to be connected to the
rotor fluid pockets. Eight of the connectors connect to the low
pressure outlet, and the other eight
connect to the high pressure inlet. The connectors are arranged
alternately and rotate at the same speed
as the inner gear to create the proper timing of the
connections.
Figure 8 shows the simulation results at 4 different moments. In
the picture, surfaces are colored by
pressure with red representing high pressure, and blue
representing low pressure, with an overall range
from 0 to 20 MPa. Small spheres in those pictures are massless
particles used to visualize the flow
field. The white lines extruding from the particles show the
direction and magnitude of the velocity of
each particle. One can see that the red particles, coming from
the high pressure inlet, are drawn into
the rotor. And the blue particles, after the pockets connect to
the low pressure port, are driven away
from the rotor towards the outlet.
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Figure 8. Pressure distribution and particle tracing
Figures 9 to 12 plot the time history of the pressure in one of
the fluid pocket; the mass flow rate;
the power applied to the inner gear, and the torque applied to
the inner gear. These curves correspond
to a 100 RPM rotation speed for one complete revolution of the
inner gear. The horizontal axis for
these plots is the rotation angle of the inner gear.
Figure 9. Pressure in a fluid pocket
Figure 10. Mass flow rate
The plots show that the solution has a clear periodical pattern
except in the first couple of time
steps. The pattern repeats itself every time the inner gear
advances one tooth. This means that, under
the current simulation conditions, one only needs to solve 2 to
3 inner gear teeth rotation, or 90 to 135
degree of the inner gear rotation, to have a complete set of
flow characteristics of the motor. The
transient simulation time to model one gear tooth rotation for
these simulation conditions is about 35
minutes on a quad-core single CPU 2.2GHZ I7 2720QM Laptop
Computer.
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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.
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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. Its 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 CDf 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
S'ij
U
u
u'
v
v' vx, vy x, y
Time
Strain tensor
Initial velocity
Velocity component (m/s)
Component of v'
Velocity vector
Turbulent fluctuation velocity
Velocity in x, y direction
Coordinates
Turbulence dissipation
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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 (Kg/s)
Number of gear teeth
Surface normal
Pressure (Pa)
Flow rate (m3/h)
Vapor condensation rate
Vapor generation rate
Revolution per minute
t g l v k l f
Fluid viscosity (Pa-s)
Turbulent viscosity (Pa-s)
Fluid density (kg/m3)
Gas density (kg/m3)
Liquid density (kg/m3)
Vapor density (kg/m3)
Surface of control volume
Turbulence model constant
Surface tension
Turbulence model constant
Turbulent Schmidt numberStress tensor
Control volume
Rotation speed
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PublishingIOP Conf. Series: Earth and Environmental Science 15
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