Principal Positive Displacement Machines 1st Edition 2019

Positive Displacement Machines 1st Edition 2019

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POSITIVE
DISPLACEMENT
MACHINES
Modern Design
Innovations and Tools
Edited By

IBRAHIM A. SULTAN
TRUONG H. PHUNG

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Contributors
Mohammad Suliman Abuhaiba
The Islamic University of Gaza, Mechanical Engineering Department, Gaza, Palestine
Mattia Battarra
University of Ferrara, Department of Engineering, Ferrara, Italy
Adrian Clenci
University of Pitești, Department Automobiles and Transport, Pitești, Romania; Cnam
Paris, Lab. of Molecular Chemistry and Engineering of Chemical and Energetic Processes,
Paris, France
Giorgio Dalpiaz
University of Ferrara, Department of Engineering, Ferrara, Italy
Georges Descombes
Cnam Paris, Lab. of Molecular Chemistry and Engineering of Chemical and Energetic
Processes, Paris, France
Mohamed A. Elgamil
Mechanical Design and Production Department, Faculty of Engineering, Cairo University,
Giza, Egypt
Muhammad Imran
Mechanical Engineering and Design, School of Engineering and Applied Science, Aston
University, Birmingham, United Kingdom
Saad A. Kassem
Mechanical Design and Production Department, Faculty of Engineering, Cairo University,
Giza, Egypt
Ahmed Kovacevic
City, University of London, Centre for Compressor Technology, London, United Kingdom
Kui Lu
School of Science, Engineering and IT, Federation University Australia, Ballarat, VIC,
Australia
Franc Majdic
Laboratory for Fluid Power and Controls, Faculty of Mechanical Engineering, University of
Ljubljana, Ljubljana, Slovenia
Javier Martı́nez
Politecnico di Milano, Dipartimento di Energia, Milano, Italy
Yonggang Meng
State Key Laboratory of Tribology, Tsinghua University, Beijing, China
Jintai Mi
SINOPEC Research Institute of Petroleum Engineering, Beijing, China

xi

xii

Contributors

Emiliano Mucchi
University of Ferrara, Department of Engineering, Ferrara, Italy
Kim Tiow Ooi
Nanyang Technological University, School of Mechanical and Aerospace Engineering,
Singapore, Singapore
Truong H. Phung
School of Science, Engineering and IT, Federation University Australia, Ballarat, VIC,
Australia
Michele Pinelli
University of Ferrara, Department of Engineering, Ferrara, Italy
Sham Rane
University of Oxford, Department of Engineering Science, Oxford, United Kingdom
Bryce Shaffer
Air Squared, Engineering, Broomfield, CO, United States
Ian K. Smith
City, University of London, Centre for Compressor Technology, London, United Kingdom
Ian James Spark
School of Science, Engineering and IT, Federation University Australia, Churchill; Centre
for Informatics and Applied Optimisation, Federation University Australia, Mount Helen,
VIC, Australia
Nikola Stosic
City, University of London, Centre for Compressor Technology, London, United Kingdom
Ervin Strmcnik
Laboratory for Fluid Power and Controls, Faculty of Mechanical Engineering, University of
Ljubljana, Ljubljana, Slovenia
Alison Subiantoro
The University of Auckland, Department of Mechanical Engineering, Auckland, New
Zealand
Ibrahim A. Sultan
School of Science, Engineering and IT, Federation University Australia, Ballarat, VIC,
Australia
Alessio Suman
University of Ferrara, Department of Engineering, Ferrara, Italy
Muhammad Usman
Centre of Advanced Powertrain and Fuels, Department of Mechanical, Aerospace, & Civil
Engineering, Brunel University London, London, United Kingdom

Introduction
Ibrahim A. Sultan, Truong H. Phung

School of Science, Engineering and IT, Federation University Australia, Ballarat, VIC, Australia

Fluid processing machinery are broadly classified into as distinct categories,
rotodynamic machines and positive displacement machines. The workings
of a rotodynamic machine is dominated by the exchange of kinetic energy
between the fluid and a disk-like impeller that spins at a controlled rotational
speed inside the machine. These machines are characterized by high nonintermitted flowrates as their speeds often fall in the range of 5000–50,000 rpm
in order for the energy transfer process to be effective. These high speed
levels are facilitated by the mechanical simplicity of the machine structure
which is mainly a balanced disk rotating about a stationary pivot inside a
cylindrical cavity. The other parts of the machines which steer the flow
in specific directions or control its velocity levels in relation to pressure
are usually stationary and their performance will depend on their geometric
characteristics and the accuracy of their manufacturing processes. However,
most applications, especially in gas and petrochemical industry, decentralized energy production, manufacturing, and control engineering require
low to medium flowrates. For these applications, a positive displacement
fluid processing machine would be a more preferable option.
For the purpose of the current volume, it suffices to describe the positive
displacement machine as a device that has one or more confined chambers
whose volume varies in a controlled fashion with the angle rotated by a
crankshaft. The chambers expand and shrink in order to either extract torque or impart pressure energy to the fluid entrapped before this fluid is
released into a discharge manifold. Therefore, the process results in an
intermittent-type flow and features the exchange of potential, rather than
kinetic, energy transfer. As such, the shaft speed is expected to be much
lower than the speed levels present in a rotodynamic machine. In fact, for
a positive displacement machine, lower shaft speed can result in better performance provided that leakage is effectively controlled.
The first five chapters of this volume present a brief account of some creative thinking in the area of positive displacement machines. As these chapters show, innovation in the design process of a positive displacement

xiii

xiv

Introduction

machine lies in creating the geometric construct which produces fluid processing chambers with as much simplicity as possible. The mechanical interface (i.e. piston or rotor) which falls between the fluid and the machine is
usually driven by a linkage designed to impart a desired motion pattern to
that interface. A good example is available in the conventional reciprocating
engine which is driven by the well-known slider-crank linkage. The linkage
drives a piston in a sliding fashion up and down the engine cylinder in accordance to a set mathematical formula. Whereas the chamber volume is controlled by means of a sliding action, the machine is referred to as a
“reciprocating machine”, which is one class of the positive displacement
machines. Chapters 6 and 7 in this volume present sample work in the area
of reciprocating machinery. The other class (i.e. rotary positive displacement
machinery) is represented by the scroll machines, the various screw machine
embodiments, the gear pumps and many other innovative designs. These
machines feature geometric constructs in which the chamber volume is varied by means of a “rotor”. Chapters 3–5 offer work related to this class of
positive displacement machinery.
The broad opportunity for innovation in the positive displacement
domain does come at a cost as the designer of these machines has to consider
the following aspects:
• High levels of manufacturing accuracy are required to machine certain
rotor profiles with three-dimensional intricate shapes
• Prevention of metal-to-metal interference may require special geometric
treatment in some designs
• Mechanical constructs in positive displacement machines are likely to feature sliding friction with associated losses and heat generation
• Leakage to and from the working chamber is a major challenge to consider
in positive displacement designs
• The linkage which drives the fluid–machine interface is likely to undergo
some severe dynamic forces and may require special balancing
arrangement
• Valves used to control the flow to and from a positive displacement
machine are subject to impact loading and may require sophisticated control systems to operate
• Special storage arrangements may be required to dampen the effects of
intermittent flow on subsequent machinery.
• The intermittent nature of flow in positive displacement applications
interacts with the dynamic characteristics of the discharge manifold
pipelines.

Introduction

xv

Despite the design challenges presented by positive displacement machines,
the current resources available to developers make it possible to overcome
these challenges and even breakthrough to new uncharted concepts in that
domain. In relation to analysis, designers currently have at their disposal such
tools as three-dimensional (3D) geometric modelling, Computational Fluid
Dynamics (CFD) and Finite Element Methods (FEM). Also, current
advances in the areas of manufacturing and material technology present great
opportunities to creating highly accurate intricate shapes at continuously
reducing cost. All these newly introduced resources combined with growing
fields of applications have resulted in a current push in the developmental
research effort in the domain of positive displacement machines. The current volume is meant to capture a glimpse of this push and presents a decent
sample of research work being conducted around the world in that domain.
The content of the volume has been structured in a fashion which highlights
the three main fields of enquiry which dominate the effort in the domain of
positive displacement machinery. These fields of enquiry are
• geometric innovation and creativity (Chapters 1–5),
• mechanism synthesis and analysis (Chapters 6–10), and
• mathematical and simulation tools used for the design process (Chapters
11–15).
The volume is divided into three parts where each part is dedicated to a field
of enquiry. The chapters in each part are contributed by researchers with
track records in their areas of interest. It is the sincere hope of the editors
that all these technically minded readers whose interest is mechanical engineering will benefit from reading the invaluable contributions presented in
the book by many authors. Current experts and new students alike in the
area of positive displacement machines will find chapters whose content will
satisfy their technical curiosity and present concepts with potential for
expansion into future research endeavours.

Contents
Contributors
Introduction

xi
xiii

Part 1 Geometric innovation in positive displacement
machines

1

1. The orbital displacer: Implications and applications

3

Ian James Spark, Kui Lu
Introduction
Swept volume to total volume ratio
Working volume as a function of crank angle
Minimum geometrically necessary clearance volume (and maximum
compression ratio) as a function of displacer geometry
Valveless port timing
Support of the orbiting piston and vanes
Balancing of the orbital piston and vanes
Sealing of the working volume
Frictional losses and lubrication
Combustion in orbital displacers
Cooling of orbital engines
Manufacturing processes and materials
Conclusions
Acknowledgements
References

2. Cardiac action pumps and motors

3
7
9
12
14
23
29
29
31
32
33
33
34
34
34

37

Mohamed A. Elgamil, Saad A. Kassem
Introduction
Cardiac action pumps and motors mechanism
Cardiac action pumps layout and principle of operation
Pumps model
Numerical simulation of pump performance
Pump steady state characteristics
Effect of pump geometric volume variation on its flow rate pulsations
Pump transient response

37
38
40
46
48
49
51
52

v

vi

Contents

Acknowledgements
References
Further reading

3. Geometric design of the limaçon machine

61
61
61

63

Ibrahim A. Sultan, Truong H. Phung
Introduction
Limaçon drives
Geometric equations for the limaçon-to-limaçon machine
The circolimaçon machine
The limaçon-to-circular machine
Conclusions
References
Further reading

4. The geometrical problem in the positive displacement
scroll machines

63
67
71
79
86
91
92
92

95

Bryce Shaffer
Introduction
Step 1: Base scroll curve
Step 2: The mating curve
Step 3: Reflection
Step 4: Tip geometry
Step 5: Area calculation
Leakage
Discussion
References

5. Screw compressors and expanders

95
96
98
99
100
104
109
113
113

115

Nikola Stosic, Ian K. Smith, Ahmed Kovacevic
Introduction
Mode of operation
Rotor profiles and their definition
The screw compressor process and its mathematical modeling
Three-dimensional flow modeling in screw compressors
Comparison between modeling and experimental results
Advances in screw compressor development
Screw expanders
Conclusion
References
Further reading

115
116
117
123
131
132
134
137
138
138
138

Contents

vii

Part 2 Mechanism analysis for positive displacement
machines

143

6. Improving torque performance in reciprocating compressors
via asymmetric stroke characteristics

145

Ibrahim A. Sultan, Truong H. Phung
Introduction
A mathematical model for the compressor cycle
Torque performance in reciprocating compressor
Example alternative compressor drive
Design example
Discussions and conclusions
References

7. Variable compression ratio at internal combustion engine

145
147
151
153
157
159
161

163

Adrian Clenci, Georges Descombes
Introduction
A summary of different VCR concepts
Case study: Some aspects concerning the geometry of the hinged
VCR engine
Conclusions
References

8. Dynamic modelling and design of bent axis pumps

163
167
169
183
183

185

Mohammad Suliman Abuhaiba
Introduction
Literature review
The geometric and kinematic model
Numerical verification of the theoretical model
Description of ADAMS model
Numerical solution for the yoke displacement mechanism
Conclusion
References

9. Numerical analyses of THD lubrication and dynamics
of rolling piston and bearings in a rotary compressor

187
190
192
204
204
218
222
222

225

Yonggang Meng, Jintai Mi
Introduction
Dynamics and lubrication models
Numerical calculation

229
234
244

viii

Contents

Calculation results and discussion
Discussion
References

10. Dynamic characteristics of rolling piston machines

247
259
260

263

Alison Subiantoro, Kim Tiow Ooi
Introduction
Geometry
Thermodynamics
Forces and torques
Internal leakage
Lubrication
Optimization
Recent developments
Discussion
References

Part 3 Tools for analysis: Mathematical modelling
and CFD
11. Mathematical modelling for positive displacement expanders

263
265
267
271
279
281
281
282
285
288

291
293

Muhammad Imran, Muhammad Usman
Introduction
Scroll expanders
Piston expanders
Screw expanders
Vane expanders
Performance comparison
Concluding remarks
References
Further reading

12. Mesh handling for the CFD simulation of external gear pumps

296
296
307
317
329
339
341
341
343

345

Javier Martínez
Introduction
EGP working principle
Numerical simulation of external gear pumps
Computational fluid dynamics
Discussion
References

345
346
349
351
364
366

Contents

13. Combining lumped parameter modelling and CFD analysis
for the pressure ripple estimation of tandem gear pumps

ix

369

Emiliano Mucchi, Mattia Battarra, Alessio Suman, Michele Pinelli,
Giorgio Dalpiaz
Introduction
Mechanical system description
Tandem pump modelling
Results and validation
Concluding remarks
References

14. Analytical, numerical and experimental approaches to
improving the performance indices of an orbital
hydraulic motor

369
372
374
387
394
395

399

Ervin Strmcnik, Franc Majdic
Introduction
Research item: Orbital hydraulic motor
Analytical approach
Numerical approach
Experimental approach
Results and discussion
Conclusion
Acknowledgement
References

15. Modelling of twin-screw machines by use of CFD

400
403
404
407
411
415
420
421
421

423

Ahmed Kovacevic, Sham Rane, Nikola Stosic
Introduction
Multiphase modelling in CFD
Grid generation for oil injected screw rotors
Case study CFD model of oil injected compressor
Development trends and future work
Conclusions
References
Index

423
428
429
433
439
440
440
443

PART 1

Geometric innovation in
positive displacement
machines

CHAPTER 1

The orbital displacer: Implications
and applications
Ian
James Sparka,b, Kui Luc
a

School of Science, Engineering and IT, Federation University Australia, Churchill, VIC, Australia
Centre for Informatics and Applied Optimisation, Federation University Australia, Mount Helen, VIC,
Australia
c
School of Science, Engineering and IT, Federation University Australia, Ballarat, VIC, Australia
b

Contents
Introduction
Swept volume to total volume ratio
Working volume as a function of crank angle
Minimum geometrically necessary clearance volume (and maximum compression
ratio) as a function of displacer geometry
Valveless port timing
Support of the orbiting piston and vanes
Balancing of the orbital piston and vanes
Sealing of the working volume
Frictional losses and lubrication
Combustion in orbital displacers
Cooling of orbital engines
Manufacturing processes and materials
Conclusions
Acknowledgements
References

3
7
9
12
14
23
29
29
31
32
33
33
34
34
34

Introduction
In 1972 the orbital engine invented by Sarich (1970, 1973) was widely publicized in Australia. As a result of this publicity two mechanical engineering
students at the Gippsland Institute of Advanced Education decided to make
the orbital engine the subject matter of their final year project in 1973. Since
publicity indicated that Sarich was concentrating his efforts on the development of a four-stroke orbital engine, the students (Joe Rosin and Robert
Sincich) and the first author decided to concentrate their efforts on the
design of an orbital two-stroke engine.
Positive Displacement Machines
https://doi.org/10.1016/B978-0-12-816998-8.00001-7

© 2019 Elsevier Inc.
All rights reserved.

3

4

Positive Displacement Machines

One obvious disadvantage of the orbital displacer invented by Sarich
(1970, 1973) as the basis of a two-stroke engine was that as it did not have
a crankcase, crankcase compression could not be used to force the fuel/air
mixture into the combustion cavity. In order to overcome this problem, the
students and the author conceived the idea of turning the orbital displacer as
conceived by Sarich (1970, 1973) inside out (i.e. everting it) and then combining the Sarich displacer and the everted displacer to form a hybrid orbital
displacer with a much higher swept volume/total volume ratio than could
be achieved by either displacer alone (Spark, Rossin, & Sincich, 1973a,
1973b, 1973c, 1973d). The initial attraction of the hybrid orbital displacer
was that the normal displacer could be used to precompress the fuel/air mixture while the everted displacer could be used to produce power by means of
the two-stroke cycle.
It was also discovered that the orbital motion of the piston allowed great
flexibility in port timing. In short, asymmetric port timing comes naturally
to the orbital displacer, whereas this can only be achieved in a reciprocating
displacer with considerable difficulty. See ‘Valveless port timing’ section.
Further, by developing the concept of asymmetric port timing, it was
found that the orbital displacer could be used as the basis of valveless pneumatic compressors and motors and hydraulic pumps and motors. It could
also be used as the basis of valveless Rankine (i.e. steam) engines. A steam
engine with three cut-offs for both forward and reverse is described below
in ‘Valveless port timing’ section.
The variants of the orbital displacer alluded to above are especially suitable as the subject matter of student projects since the students (and supervising staff ) are forced to go back to first principles and exercise their
imaginations in order to solve many of the associated problems.
It is not generally realized that Wankel (1963) attempted to classify all
possible forms of two-stroke rotary displacers before he concentrated his
development effort on the four stroke Wankel engine as we know it today.
Fig. 1 shows a displacer which except for the motion of the vanes is equivalent to Sarich’s orbital displacer as shown in Fig. 2 (Sarich, 1970, 1973).
Fig. 3 shows a displacer which except for the motion of the vanes is
equivalent to the everted orbital displacer of Spark et al. (1973a, 1973b,
1973c, 1973d) shown in Fig. 4.
In the displacers foreshadowed by Wankel the vanes both slide and
rotate, whereas in the displacers of Sarich (1970, 1973) and Spark et al.
(1973a, 1973b, 1973c, 1973d) the vanes execute simple harmonic motion.
However, this difference in vane motion is not trivial as it greatly influences
the stresses acting on the vanes and the ease with which they can be sealed.

Stationary housing

Orbiting piston

Fig. 1 Orbital displacer foreshadowed by Wankel (1963).

TDC
Orbiting
piston

Vane

BDC

Stationary housing

Fig. 2 Sarich orbital displacer.

6

Positive Displacement Machines

Stationary
core

Master vane
Vane

Orbiting piston

Fig. 3 Everted orbital displacer foreshadowed by Wankel (1963).

TDC

Stationary
core

BDC

Orbiting piston

Fig. 4 Everted orbital displacer with oscillating vanes.

The orbital displacer: Implications and applications

7

Fig. 5 (after Wankel, 1963) shows a displacer wherein a single vane reciprocates in a stationary external housing in a manner similar to the movement
of the four vanes shown in Sarich’s orbital displacer (Fig. 2).
Similarly Fig. 6 depicts a displacer where a single vane reciprocates in a
stationary internal housing similar to the motion of the four vanes shown in
the everted orbital displacer of Spark et al. (1973a, 1973b, 1973c, 1973d)
(Fig. 4).
The essential difference between the displacers foreshadowed by Wankel
(1963) and those invented by Sarich (1970, 1973) and Spark et al. (1973a,
1973b, 1973c, 1973d) is that as only one vane is used in the displacers foreshadowed by Wankel (1963), the total constant working volume is divided
into two variable working volumes by means of arctuate contact between
the cylindrical orbiting piston and the cylindrical internal or external housings. Sarich (1970, 1973) and Spark et al. (1973a, 1973b, 1973c, 1973d) have
overcome the problem of sealing the line of arctuate contact by using two or
more vanes—thus making arctuate contact unnecessary.

Swept volume to total volume ratio
One advantage of rotary displacers is that they do not use connecting rods. In
reciprocating displacers, the “big end” of the connecting rod executes circular motion, whereas the ‘small end’ executes approximately simple

Stationary housing

Vane

Orbiting piston

Fig. 5 Orbital displacer foreshadowed by Wankel (1963).

8

Positive Displacement Machines

Orbiting piston

Vane

Stationary
core

Fig. 6 Everted orbital displacer foreshadowed by Wankel (1963).

harmonic motion. This complex motion must be accommodated in the
housing (or crankcase) that connects the cylinders to the crankshaft main
bearings. This accommodation of the connecting rods generally ensures that
the total volume of a reciprocating displacer is significantly greater than that
of a rotary displacer of the same displacement. This generally leads to the
rotary displacer having a higher weight to displacement ratio.
Everting the Sarich orbital displacer (1970, 1973) places the orbiting piston out-board of the stationary internal core. This will generally decrease the
displacement to total volume ratio as the size of the orbiting piston will be
increased while the displacement will be decreased. However, if the Sarich
orbital displacer is combined with the everted orbital displacer to form a
hybrid orbital displacer, both the inside and outside of the orbiting piston
can be used. The inside of the orbiting piston forms a moving boundary
of the everted orbital displacers while the outside of the orbiting piston forms
a moving boundary of a Sarich orbital displacer. As a single orbiting piston is
used in both the Sarich and the everted displacer, the displacement to total
volume of the hybrid orbital displacer would be greater than that of either
displacer alone.
The Sarich and everted orbital displacers could be more or less independent. For example, one displacer could be used as a prime mover while the

The orbital displacer: Implications and applications

9

Fig. 7 Hybrid orbital displacer. Outer (Sarich) orbital displacer ¼ precompressor. Inner
(everted) orbital displacer ¼ two-stroke engine.

other could be used as a pump or compressor. Alternatively, the two displacers could be connected in series to form a two-stage pneumatic motor or
compressor. Alternatively, the two displacers could be connected in series so
that the Sarich displacer acts as a precompressor for the everted displacer
which is used as a two-stroke engine. Since the displacement of the Sarich
orbital displacer must be greater than that of the everted orbital displacer, a
super charging effect is inevitable. See Fig. 7.

Working volume as a function of crank angle
With respect to each working volume the area of the orbiting piston effectively decreases and increases with distance from the top dead center (TDC)
position for the Sarich and everted displacers, respectively. Fig. 8 shows the
working volumes of the Sarich orbital displacer (Fig. 8A) and the everted
orbital displacer (Fig. 8B).

10

Positive Displacement Machines

b

b

BDC
r
θ
b

TDC

Orbiting
piston

r

θ

b′

Stationary
core

b

TDC

f

BDC
Orbiting
piston

f

Stationary
core

f

f
b′

(A)

(B)

Fig. 8 Dimensions of working chamber of orbital displacer. (A) Sarich and (B) Everted.

In these figures the working volume is assumed to be symmetrical with
respect to the TDC–BDC plane, where r is the throw of the orbital motion,
b is the distance perpendicular to the direction of oscillation of the vane from
the point of contact of the vane and the piston to the TDC–BDC plane,
when the piston is in the TDC position, w is the width of the piston (which
is perpendicular to the page), 2ϕ is the angle between the directions of oscillation of the two vanes, Vo is the minimum everted clearance volume when
ϕ is 45 degrees, and θ is the clockwise rotation of the crankshaft from the
TDC position. Note that the shape of the top of the piston between the vane
pads does not affect the magnitude on the swept volume.
The volume of the working cavity of the Sarich displacer VS is given by;
θ
Vs ¼ Vos + 4wr cosϕsin 2 ðb + r sin ϕcosθÞ
(1)
2
where Vos is the clearance volume of the Sarich displacer,
The volume of the working cavity of the everted orbital displacer is given
by;
θ
Ve ¼ Voe + 4wr cosϕsin 2 ðb  r sinϕcos θÞ
(2)
2
where Voe is the clearance volume of the everted orbital displacer,
The displacement of the Sarich orbital displacer Vds is the maximum Vs
(when θ ¼ 180°, so sin θ2 ¼ 1, and cos θ ¼ 1), minus the minimum Vs
(when θ ¼ 0°, so sin θ2 ¼ 0). Therefore:
Vds ¼ 4wr cos ϕðb  r sin ϕÞ

(3)

The orbital displacer: Implications and applications

11

Similarly, the displacement of the everted orbital displacer vde is
Vde ¼ 4wr cos ϕðb + r sin ϕÞ

(4)

Fig. 9 shows the ratio of the instantaneous volume of the working cavity
to its displacement plotted against sin 2 θ2 when ϕ is 45 degrees.
The advantage of this plot is that it emphasizes nonsinusoidal variation
since a sinusoidal variation will plot as a straight line. It can be seen that
the volume of the working cavity of the Sarich orbital displacer increases
more rapidly from the TDC position than the working volume of the
everted orbital displacer. This difference can be explained in terms of the
effective area of the orbiting piston decreasing and increasing from the
TDC position for the Sarich displacer and the everted orbital displacer,
respectively.
Note that due to the cosine term the swept volume for both the Sarich
and everted orbital displacers will decrease as ϕ increases. In theory, they
become zero when ϕ ¼ 90 degrees. This problem can be overcome by redefining the shape of the working volume. For example, b could be replaced
with b0 which is the distance between the end of each vane in contact with
the piston pad at the TDC configuration for the working cavity and the axis
of displacer, where the lines of oscillation of all vanes intersect. The advantage of b0 is it is indicative of the diameter of the approximately cylindrical
displacer. Since b ¼ b0 tan ϕ  2t , Eqs. (3), (4) become

Fig. 9

Volume of orbital working chamber
displacement

VS sin 2 θ2 :

12

Positive Displacement Machines



t
Vds ¼ 4wr cosϕ b0 tanϕ   r sinϕ
2

(5)

and



t
Vde ¼ 4wr cos ϕ b0 tan ϕ  + r sinϕ
(6)
2
where t is the thickness of the vane.
Note that the shape of the surface of the inner housing between the leading
and trailing vanes, and the shape of the crown of the orbiting piston between
the vane pads does not influence the variable volume of the working cavity.
However, these shapes will affect both the geometrically necessary clearance
volume and the minimum geometrically necessary clearance volume.

Minimum geometrically necessary clearance volume
(and maximum compression ratio) as a function of
displacer geometry
In orbital displacers, there is generally a clearance volume required simply to
enable the orbital piston to orbit without making contact with the external
housing, in the case of the Sarich displacer, or the internal core, in the case of
the everted orbital displacer.
First let us consider the case where the vane pads are extended until they
meet on the TDC–BDC plane. In this case, the geometrically necessary
clearance volume for the Sarich orbital displacer is

Vos ¼ 2w br  r 2 sinϕ + wr 2 ðϕ  sin ϕcos ϕÞ
 wr 2
ð2ϕ  sin2ϕÞ
(7)
¼ 2w br  r 2 sin ϕ +
2
For the everted orbital displacer the geometrically necessary clearance
volume is;
Voe ¼ 2br ð1  cosϕÞ  wr 2 ð1  cos ϕÞ2 tan ϕ

(8)

The geometrically necessary clearance volume is very sensitive to the
shape of the orbiting piston. In general, this clearance volume will be minimized if the area of the piston that is perpendicular to the TDC-BDC plane
is maximized. Flat pads must be provided on the orbiting piston to allow the
ends of the oscillating vanes to oscillate on these pads without fluid leakage.
The minimum length of these pads must be the sum of the stroke of the
crankshaft and the thickness of the vanes. See Fig. 10A.

BDC
E

F
H

TDC
B

r

C

q

f f

D
TDC
G

G
E

f

r

B

C

F

BDC

A

A

D

f

f

(A)
(II)

Maximum compression ratio

Sarich
45

700

40

600

35
500

30
25

400

20

300

15

200

10
100

5
0

0
0

20

40

(B)

60

80

100

Minimum clearance volume, (mm3)

(I)

Minimum clearance
volume
Maximum compression
ratio

Phi, (degrees)

400

45

350

40

300

35
250

30
25

200

20

150

15

100

10
50

5
0
−20

(C)

0

20

40

60

Phi, (degrees)

Fig. 10 See figure legend on next page

80

0
100

Minimum clearance volume, (mm3)

Maximum compression ratio

Everted
50

Maximum compression
ratio
Minimum clearance
volume

14

Positive Displacement Machines

The minimum geometrically necessary clearance volume for the Sarich
orbital displacer Vos (see Fig. 10A I) is given by
Vos_ min ¼ 2wr 2 ð1  cosϕÞ + wr 2 ðϕ  sinϕcos ϕÞ
wr 2
ð2ϕ  sin2ϕÞ
(9)
2
The minimum geometrically necessary clearance volume for the everted
orbital displacer Voe (see Fig. 10A II) is given by
¼ 2wr 2 ð1  cos ϕÞ +

Voe_ min ¼ 2wr 2 ð1  sinϕÞð1  cos ϕÞ +

wr 2 ð1  cosϕ Þ2
tan ϕ

(10)

The equations above show that the geometrical necessary clearance volume can be reduced to zero if the angle ϕ between the line of oscillation of
the two vanes is zero for both the Sarich displacer and the everted orbital
displacer.
The maximum compression ratio can be calculated by dividing the displacement by the minimum clearance volume and adding one. Fig. 10B
plots the minimum clearance volume and the maximum compression ratio
against the angle between the two vanes for the Sarich orbital displacer. Fig.
10C plots the minimum clearance volume and maximum compression ratio
for the everted orbital displacers.

Valveless port timing
It is not generally realized that the orbiting motion of the piston in an orbital
displacer makes any desired (two stroke) port timing possible. Fig. 11 depicts
a hybrid orbital displacer working as a double acting pump where slots are
machined in the side of the piston that are parallel to the TDC/BDC line.
These slots are used to open inlet ports in the side plate when the volume of
the working cavity is increasing and outlet ports when the volume of the
working cavity is decreasing. This pump would work equally well as a
double acting hydraulic motor. Note that the displacement of the everted
Fig. 10 (A) Location of minimum geometrically necessary clearance volume. (I) Sarich,
(II) Everted. (B) The minimum clearance volume and the maximum compression ratio
against the angle between the two vanes (Sarich). (C) The minimum clearance
volume and the maximum compression ratio against the angle between the two
vanes (Everted).

The orbital displacer: Implications and applications

i = inlet port
o = outlet port

15

Orbiting
piston

TDC
o
i
i

BDC
o

Stationary
internal
core

o
i
i

o

Stationary
external
housing

Fig. 11 Double acting orbital pump (or motor).

displacer has been increased by decreasing the distance between the
inner and outer vanes. This would increase the geometrically necessary
clearance volume, but this would have little effect on the pumping of
incompressible fluids.
The valve action of the slotted piston is quantitatively explained in Figs
12–14. Fig. 12 depicts a slot NOPQ in the piston of an everted orbital displacer. Rotation of edge of slot Δθ in the direction of rotation of the crankshaft will cause the opening and/or closing of a corresponding port in the
side plate to be delayed.
Fig. 13 depicts the variation of the volume of the working cavity with
crankshaft rotation past TDC.

16

Positive Displacement Machines

BDC
q
O

F

P K

L

G
E

TDC

H

J

N

M

Stationary core
Q

Orbiting piston

Δq

Vane

Fig. 12 ‘Valving’ slot in orbiting piston.

Volume

ΔV

V0

q
V≈V0+ΔVsin2(–)
2

q
TDC

90°

BDC

270°

TDC

Fig. 13 Variation of volume of working chamber.

Fig. 14 shows the relative locations of the edges of the slot and the ports
in the end plate varies with crankshaft rotation. It can be seen that the position of maximum opening of the inlet port is 90 + Δθ, whereas the duration
of the port opening is determined by the position of the orbiting active edge
AE of the slot relative to the stationary active edge of the corresponding port.

The orbital displacer: Implications and applications

17

X
EFGH open
when XAB > X4
X4

XAB = X1 + rsin(q +Dq)
X4 = GH

X3

X2
X1

XCD = X3 + rsin(q +Dq)
X1 = JK
JKLM open
when XCD < X1

q

Fig. 14 Position of edge of valving slot.

The correct angle of the edges of the slot can be easily deduced from the
desired port timing. Assuming the piston is orbiting in a clockwise direction,
mark the desired opening and closing angles for both the inlet and outlet
ports relative to the TDC/BDC line. See Fig. 15.
The chord connecting the opening and closing of the inlet port yields the
required angle for the edge of the slot which opens the inlet port. Similarly,
the chord connecting the opening and closing of the exhaust port yields the
required angle of the other edge of the slot which opens the exhaust port.
Note that the maximum distance between the circular arc and the chord
gives the maximum opening of the ports if the radius of the timing diagram
is the throw of the crankshaft. The slots in the end plates need to be moved
sideways to get them to open at the correct crankshaft angle.
Note that the depth of the orbiting piston must be at least four times the
throw of the crankshaft in order to enable the orbital motion of the piston to
effect valveless port timing without undesirable leakage paths. To increase
the effective depth of the piston, ears can be located adjacent to the end plate.
As these ears will decrease the compression ratio of the displacer, their thickness could be limited to 10% of the width of the piston. The effective depth
of the orbiting piston could be further increased by interposing port plates

18

Positive Displacement Machines

Outlet closes
TDC = 0
Maximum
315°
opening
of outlet port

270°

45°
Inlet opens

Outlet opens
90°
112.5°
Maximum
opening
of inlet port

ae = Active edge of port
AE = Active edge of piston ‘slot’

Orbiting piston

AE
AE

ae

Inlet port

Outlet port
ae

BDC = 180
Inlet closes

Fig. 15 Determination of angle of valving slots from desired timing diagram.

between the orbiting piston and the end plates. These port plates could also
form slots to accommodate the edges of the sliding vanes. One advantage of
separate port plates is that they only have to accommodate the minor diameter of the crank eccentric. Another advantage is that the port timing could
be changed by simply replacing the port plates and the slots in the piston.
Fig. 16A shows the desired pressure–volume diagram for one working
volume of an orbital displacer pumping a compressible fluid. Fig. 16B shows
the crank angle–volume diagram for the same displacer cavity where flow
reversals are avoided. At TDC, the outlet port has just closed and the inlet
port is still closed. As the crankshaft rotates to 45 degrees the volume of the
working cavity increases and the pressure drops from the outlet manifold
pressure to the inlet manifold pressure. At this point, the inlet port opens
so that compressible fluid flows into the expanding working cavity. At
180 degrees the inlet port closes. The compressible fluid is now compressed
until its pressure increases to that of the outlet manifold at 270 degrees when

The orbital displacer: Implications and applications

19

External delivery pressure

Pressure

Inlet pressure

(A)

V0

V1

Volume

V2

V3

0°

90°

112.5°

Inlet port open

Max opening
at 112.5°

Δq = 22.5°

180°

270°

315°

Outlet port open

Max opening
at 315°
Δq = 45°

(B)

45°

360°

Fig. 16 Pressure and crank angle versus working volume for compressor.

the outlet port opens. The fluid is then pumped from the working cavity
until the outlet port closes at 360 degrees. The cycle then repeats.
Fig. 17 shows the shape and location of suitable inlet and outlet ports
required to achieve the above cycle. The orbiting piston is in the TDC position for the working chamber shown.
Fig. 18 shows one working cavity of a Rankine (steam) engine with three
cut-offs for the inlet port for both forward and reverse motion. The cut-offs
are 45 degrees, 90 degrees and 180 degrees. This requires three inlet
ports in the end plate to effect forward (CW) rotation and three inlet ports
for reverse (ACW) motion. Note that the inlet ports for reverse motion act
as outlet ports for forward motion, and vice versa. Corresponding slots in the
side of the orbiting piston connect the working cavity to the ports. The flow
of steam to the various ports is controlled by a spool valve. In Fig. 18, the
spool valve is shown in the null (zero steam flow) position.

20

Positive Displacement Machines

Outlet open
C

0°

Outlet

45°
Inlet open

BD

18

TD

270°

C

90°
112.5°

AE

AE

AE = active edge
AE
AE
Inlet

Fig. 17 Arrangement of valving slot and inlet and outlet ports for compressor of Fig. 16.

TDC

HP
Steam
Reverse

Forward

ACW

CW
0/−180 0/−90 0/−45

0/45 0/90 0/180

Condensor

Fig. 18 Arrangement of valving slot and inlet and outlet ports for Rankine (steam)
engine with three forward and reverse cut-offs.

The orbital displacer: Implications and applications

21

The underlying principles for positioning the stationary ports in the end
plate and the orbiting slots in the side of the piston are as follows:
• Approximately rectangular ports in the end plate or opened by approximately rectangular ports in the side of the orbiting piston.
• The active edges of the ports and associated slots are always parallel.
• The crank angle at which the maximum opening of a port occurs is equal
to the angle of the active edge to the TDC–BDC line—90 degrees.
• The ratio of port opening to crank throw is 1  cos Δθ
2 where Δθ is the
angular duration of the required port opening.
• The slots in the orbiting piston must not intrude into the adjacent working
cavities
• The slots on the orbiting piston must not open the wrong ports
• The latter two problems become harder to avoid as the ratio of piston
throw to piston size increases.
If the maximum desired port opening is close to BDC, then the edge of the
piston can be used to achieve the desired port timing, so no slot is required in
the side of the orbiting piston.
Fig. 19 shows an orbital two-stroke engine model aircraft engine
designed by Hancorne (1979) with a precompression piston which acts as
a supercharger, and a power piston. The eccentric cams supporting the

IP = inlet port
OP = outlet port
TP = transfer port
E = exhaust port
PP = port plate
EP = end plate
CP = centre plate

EP

PP

CP

PP

PP

PP

EP

TDC

IP

TP
Pre-compression
piston

BDC

E

OP
Power piston

OP
IP

TP

BDC

Fig. 19 Orbital model aero engine. Section through TDC/BDC line.

E

22

Positive Displacement Machines

two pistons are coaxial but 180 degrees out of phase. Fig. 19 is a section
through the TDC–BDC line of two of the precompression cavities and
two of the power cavities.
Fig. 20 is a section through opposite vanes.
Fig. 21 is a section through the inlet slots of the precompression piston
looking towards the power piston. The inlet port opens and closes 0 degrees
and 210 degrees after TDC, respectively.
Fig. 22 is a section through the outlet slots of the precompression piston
looking away from the power piston. The outlet port opens and closes 180
degrees and 290 degrees after TDC, respectively.
Fig. 23 is a section through the transfer edge of the power piston looking
away from the precompression piston. The transfer port opens and closes
150 degrees and 240 degrees after TDC, respectively.
Fig. 24 is a section through the exhaust edge of the power piston looking
towards the precompression piston. The exhaust port opens and closes 120
degrees and 210 degrees after TDC, respectively.
Fig. 25 shows an exploded view of the precompression piston and the
power piston (with ears), and the associated vanes.
Fig. 26 shows an exploded view of the two-stroke engine. Note, only a
quarter of the required housings and port plates are shown. The ports in the
port plates are not shown.

Vane
Vane
Piston
Piston

Piston
Piston
Vane
Vane

Fig. 20 Orbital model aero engine. Section through vanes.

The orbital displacer: Implications and applications

23

IP= inlet port

IP

IP

IP

IP

Fig. 21 Precompression piston. Section through in inlet slots.

Note that the length and thereby the area of the inlet and outlet ports can
be increased by locating them in opposing port plates. Their length can be
further increased if ears on the face of the piston are used to open and close
the ports.

Support of the orbiting piston and vanes
In order to prevent jamming of the sliding components, it is essential that the
orbital piston is prevented from rotating. This can be achieved by supporting
the orbiting piston on three or more eccentrics (provided their axes do not
lie in the same plane). This was the method originally used by Sarich (1970,
1973), where the piston was supported by a main crankshaft eccentric and
multiple stabilizing eccentrics. As the piston tended to expand more than the
end plate when the displacer was operating as an internal combustion
engine, the stabilizing eccentrics tended to tighten up leading to bearing failure. To overcome this problem multiple eccentrics were used to support a

OP = outlet port
Orbiting
piston
Stabilising
plate

OP

OP

OP

OP

Fig. 22 Precompression piston. Section through outlet slots.

TP = Transfer port

Orbiting
piston

TP

Stabilising
plate

Fig. 23 Power piston. Section through transfer end.

The orbital displacer: Implications and applications

25

EP = Exhaust port

EP

EP

EP

EP

Fig. 24 Power Piston. Section through Exhaust end.

stabilizing plate where a tongue on the stabilizing plate engaged a groove in
the orbiting piston. This system enabled the piston to expand radially without increasing the load on the eccentric bearings. The orbital motion was
enforced by a main crankshaft eccentric (Sarich, 1975, 1976a).
Alternatively, the orbiting piston can be supported by a single crankshaft
eccentric combined with a stabilizing plate based on an Oldham coupling.
See Fig. 27. This plate is imposed between one end plate and the side of the
adjacent orbital piston. Tongues on one side of the plate slidably engage with
slots in the end plate. These slots could be continuations of the grooves that
slidably engage the vane legs. The other side of the stabilizing plate has tongues, which are aligned at right angles to the tongues that slidably engage the
groove in the end plate. The former tongues slidably engage with slots in the
end of the piston. This arrangement enables the stabilizing plate to execute
simple harmonic motion (SHM) relative to the end plate, and the piston to
execute simply harmonic motion relative to the stabilizing plate. These two
SHMs can add to allow the piston to orbit around a circular path, while

26

Positive Displacement Machines

Fig. 25 Precompression and power pistons.

preventing its rotation. The stabilizing plate must contain an elongated
hole to allow it to oscillate relative to the crankshaft. The side of the piston
must contain a recess that enables the stabilizing plate to oscillate relative to
the piston.
The orbiting pistons of everted and hybrid displacers could best be supported by multiple eccentrics. If the eccentrics are connected by means of an
idler gear, then only two eccentrics would be required. The idler gear could
be used to deliver or extract power from the orbiting piston. It could also be
used for speed reduction of the power input or output. Three or more
unconnected eccentrics could be used as shown in Fig. 28.
The vanes can be forced to execute SHM by slidably connecting them to
the orbiting piston by means of lugs on the ends of the vane legs which slide
in grooves in the sides of the orbiting piston (which are at right angles to the
direction of oscillation of the vanes). The vane legs slide in groves in the end
plates and are attached to the vane proper in the course of assembling the
displacer.

The orbital displacer: Implications and applications

Fig. 26 Exploded view of orbital model aero engine.

27

Fig. 27 Piston stabilizing plate based on Oldham coupling.

E

T2

T2

T3

T3

T1

E

E

Fig. 28 Orbital two-stroke engine with three precompression and three power
chambers.

The orbital displacer: Implications and applications

29

One of the factors that distinguishes the Sarich orbital engine from other
less successful vane engines is that the vanes of the orbital engine are positively supported on three of their four edges—as opposed to being cantilevered from slots in the housing. Supporting opposite edges of the vanes in
grooves in the end plates is possible because the vanes only execute SHM
relative to these end plates.
In the case of the hybrid displacer, the vanes and vane legs encompass the
hybrid orbital piston, thus making grooves in the sides of the piston unnecessary. See Figs 7 and 11. The crankshaft and stabilizing eccentrics can be
balanced by traditional methods.

Balancing of the orbital piston and vanes
One advantage of the orbital engine is that there is no secondary out of balance. Static and dynamic balance can be easily achieved by means of adding
counter weights at both ends of the crankshaft. The question arises as to how
the reciprocating vanes can be balanced. If the direction of reciprocation is
uniformly distributed around the crankshaft, and all the vanes have identical
mass, then the reciprocation of all the vanes is equivalent to rotation of a mass
equal to half the mass of the vanes. Hence, for the purpose of balancing the
vanes, half their mass should be added to the mass of the associated piston, in
order to deduce the mass and location of the counter weights required to
produce static and dynamic balance of the displacer.
When the lines of action of the vanes exhibit less than three-fold symmetry, either dummy vanes can be added to restore the necessary axial symmetry, or balance weights on a counter-rotating shaft can be used.
Note that the stabilizing plate generally oscillates in common with two
diametrically opposite vanes. In this case the stabilizing plate can be balanced
by adjusting the weight of the associated vanes so that the weight of each of
these vanes plus half the weight of the stabilizing plate is the same as the
weight of a standard vane. Indeed, the stabilizing plate could be attached
to the vane leg that slides in the same groove.

Sealing of the working volume
Whereas the working volume of a reciprocating displacer is bounded by two
surfaces, the working volume of the orbital displacer is bounded by four surfaces. These are the stationary housing or core (and end plates), the face of
the orbiting piston and the sides of two vanes. The piston orbits relative to

30

Positive Displacement Machines

the stationary housing or core (and end plates). The vanes execute SHM relative to both the stationary housing or core (and end plates), and the orbiting
piston. Effective sealing must be achieved between the following sliding
couples: The piston sides and the end plates, the piston faces and the end
of the vanes; the vane legs and the grooves in the end plates; the side of
the vanes and slots in the housing or core; the inner face of the vane leg
and the side of the piston. Sarich (1974, 1976b) has proposed a suitable sealing system. An exploded view of an alternative suitable sealing system is
shown in Fig. 29.
Fig. 30 shows a section through the centre of an assembled vane. In this
case the seals are rectangular prisms pushed against their mating surface by
wave springs.
It is anticipated that sealing of a 6.0 cc model aircraft engine shown above
will be less of a problem since effective sealing can be achieved by lapping
together interacting components, and then maintaining the resultant small
clearances by balancing the thermal expansion of these components. It
should be noted that many model aircraft engines of comparable displacement run without the benefit of piston rings.
Since the orientation of the components that move relative to one
another, remains constant, they will tend to make contact over an area rather

Fig. 29 Exploded view of sealing possible system.

The orbital displacer: Implications and applications

31

WS = wave spring
SE = sealing element

Vane leg

Key

WS
WS
SE

SE

WS

SE

Piston

Fig. 30 Sealing system. Section through centre of vane.

than along a line, thus reducing the compressive stresses acting in the
contact zone.

Frictional losses and lubrication
Since the piston orbits but does not rotate, all positions on the piston
describe a circle whose radius is the throw of the crank. Consequently, rubbing speeds are much lower than in rotary engines where the piston also
rotates, such as the Wankel engine. These low rubbing speeds lead to
low frictional losses of power.
There appear to be no lubrication problems unique to the orbital displacer where pressurized oil systems can be used. Since “crankcase
compression” is not used in the orbital two-stroke engines cited above, a
total loss oil system (with its “sooty” exhaust) is not required.

32

Positive Displacement Machines

Combustion in orbital displacers
Although it is difficult to design an orbital displacer where the ratio of the
surface area of the working cavity to the displacement is as low as that of a
reciprocating internal combustion engine, this ratio will generally be much
lower than that pertaining to the Wankel engine.
Note that the centre of the working volume moves in a roughly elliptical
orbit, whereas it moves straight up and down in a reciprocating engine. This
should cause an inherent swirl of the fuel/air mixture which could be used to
facilitate combustion. See Fig. 31.
Furthermore, if the clearance between the crown of the piston and the
housing is minimized, a very high gas velocity across the piston crown can be
achieved. Just before TDC most of the working volume is located adjacent
to the leading vane. Just after TDC most of the working volume is located
adjacent to the trailing vane. See Fig. 32. If a shallow groove in the piston
crown links the leading vane pad to the trailing vain pad, most of the fuel/air
mixture will be forced to flow through this groove around TDC. If one or
more spark plugs were located in the housing above this groove, all the fuel/

Fig. 31 Schematic of swirl in orbital power chamber.

Fig. 32 Squish in orbital power chamber.

The orbital displacer: Implications and applications

33

air mixture could be squirted past these discharging spark plugs around
TDC—thus making the combustion process less dependent on the rate of
flame propagation.

Cooling of orbital engines
In the (single rotor) Wankel engine, the combustion chambers travel past a
single inlet port, single sparkplug and single exhaust port. In this case, the
heat flow to the housing is nonuniform, thus complicating the thermal
expansion and cooling problems.
In an orbital engine, the combustion chambers tend to be uniformly distributed around the axis of their crankshaft. In general, if there are N vanes
there will be N combustion chambers, each with its own sparkplug and inlet
and exhaust ports. In this case, the heat flow to the housing should show
radial symmetry, thus simplifying the thermal expansion and cooling
problems.
Since the orbiting piston will generally float between the end plates and
does not touch the housing, its cooling is a potential problem. A similar
problem occurs in the Wankel engine. Forced convection will be required
to cool the orbiting piston. In small engines air circulation may suffice,
whereas in larger engines oil cooling of the piston may be required.
Cooling holes could be drilled through the piston parallel to the axes of
the eccentrics, which intermittently align with cooling holes in the end
plates. Air could be drawn through these holes by a fan attached to one
or more of the eccentrics. Alternatively, oil could be pumped through much
smaller holes through the orbiting piston.

Manufacturing processes and materials
In theory, the cost of manufacturing an orbital displacer could be less than
that pertaining to a reciprocating displacer. First, with the exception of the
sealing elements, there are fewer components required—especially in the
case of orbital pumps and motors, where valveless port timing can be used
to eliminate the need for valves.
Furthermore, little machining is required. In the external housing or
internal core, only the vane slots and end faces need to be machined. On
the end plates, only the inner faces, the grooves for the vanes, and the bores
for the crankshaft need to be machined. On the piston, only the end faces,
the pads for the vanes, and the eccentric bore need to be machined. Most

34

Positive Displacement Machines

surfaces of the vanes will need to be machined, as will the crankshaft eccentric and stabilizing plate.
The 6 cc model aircraft engine was designed to be manufactured from an
Al-17% Si alloy which combines lightness with wear resistance.

Conclusions
The orbital displacer originally conceived by Ralph Sarich may have greater
significance than its application as a four-stroke internal combustion engine.
Although the automobile engine market is very large, it is also the hardest to
penetrate due to the high degree of sophistication of the design and manufacture of reciprocating engines and the large number of often conflicting
requirements that must be satisfied. It may be more rewarding to first penetrate a smaller, less competitive market and then use this as a base from
which to expand into larger more difficult markets.
Asymmetric port timing comes naturally to orbital displacers. This valveless port timing can be used to advantage in orbital two-stroke engines,
hydraulic pumps and motors, pneumatic compressors and motors, and Rankine engines.
Engineering students have found orbital displacer projects stimulating in
the originality required, as they generally had to go back to first principles to
solve problems.

Acknowledgements
The authors would like to thank the staff and students of the Gippsland School of Engineering
(now part of Federation University Australia) for their contribution to this paper.

References
Hancorne, J. A. (1979). Student project report. Gippsland Institute of Advanced Education. Private Communication.
Sarich, T. R. (1970). An improved rotary motor. Australian patent 467415, filing date 6 July
1970.
Sarich, T. R. (1973). Improved vane type internal combustion engines. Australian patent 477125,
filing date 16 January 1973.
Sarich, T. R. (1974). Gas seal for vane type internal combustion engine. (US patent 3938916, filing
date 16 January 1974).
Sarich, T. R. (1975). Improved orbital engine with stabilising plate. In Australian patent
491267, filing date 3 February 1975.
Sarich, T. R. (1976a). Orbital engine with stabilizing plate. US patent 4037997, filing date 3
February 1976a.

The orbital displacer: Implications and applications

35

Sarich, T. R. (1976b). Vane type orbital engine. US patent 4079083, filing date 3 February
1976b.
Spark, I. J., Rossin, J. M., & Sincich, R. (1973a). Improved orbital displacers. Australian patent
487540, filing date 2 July 1973.
Spark, I. J., Rossin, J. M., & Sincich, R. (1973b). Orbital displacer. US patent 487540, filing
date 2 July 1973.
Spark, I. J., Rossin, J. M., & Sincich, R. (1973c). Improvements in rotary positive-displacement
machines. UK patent 1480137, filing date 2 July 1973.
Spark, I. J., Rossin, J. M., & Sincich, R. (1973d). Orbital displacers. Canadian patent 1043267,
filing date 2 July 1973.
Wankel, F. (1963). Rotary piston machines. London: Iliffe Books.

CHAPTER 2

Cardiac action pumps and motors
Mohamed A. Elgamil, Saad A. Kassem

Mechanical Design and Production Department, Faculty of Engineering, Cairo University, Giza, Egypt

Contents
Introduction
Cardiac action pumps and motors mechanism
Cardiac action pumps layout and principle of operation
Pumps model
Numerical simulation of pump performance
Pump steady state characteristics
Effect of pump geometric volume variation on its flow rate pulsations
Pump transient response
Pump response during geometric volume step decrease
Pump response during geometric volume step increase
Acknowledgements
References
Further reading

37
38
40
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48
49
51
52
52
56
61
61
61

Introduction
Among the mechanical power transmission systems, hydraulic systems enjoy
being one of the highest power to weight/volume ratio because the oil used
in these systems can be pressurized to high values with acceptable deformations. In many applications, they have no replacement to meet the requirements of high forces and torques. For precise position, speed and/or force
control of mechanical systems, they provide drive and control of high speed
of response and accuracy. This is because the pressurized oil response to control is fast, since inertia of the oil is small compared with that of the mechanical components. But to generate and control hydraulic power, mechanical
systems incorporating components of considerable inertia are generally
nowadays used, which would have an adverse effect on the speed of
response. On the other hand, the competency of these systems is challenged
by their cost, safety considerations and their low efficiency that results whenever throttling is used for controlling the hydraulic power.
Positive Displacement Machines
https://doi.org/10.1016/B978-0-12-816998-8.00002-9

© 2019 Elsevier Inc.
All rights reserved.

37

38

Positive Displacement Machines

Cardiac action pumps and motors, presented in this chapter, are now
under development to be used in hydraulic systems for possible improvement of the system speed of response and efficiency. These pumps and
motors fall in the category of digital variable geometric volume hydraulic
machines. In these units the control of the geometric volume is carried out
through manipulating small volumes of oil using low cost fast switching
valves rather than the inertial mechanical systems, which would improve
the speed of response of the units and the hydraulic systems. A cardiac
action pump/motor consists of several heads of compact size and high
geometric volume. The compact size of the heads allows constructing a
compact cardiac action pump with several pumping heads. A group of
these heads in a pump can be interconnected to drive and control an actuator. Consequently, several groups of interconnected pumping heads in
one pump can be utilized to drive and independently control several actuators, with each of these groups driving one actuator in an efficient and
independent pump-controlled actuator mode. Thus, the throttling
actions needed for control purposes would be eliminated, which would
enhance the efficiency of the hydraulic systems incorporating several
actuators.

Cardiac action pumps and motors mechanism
A cardiac action machine consists of one or more heads, where two separate
chambers are formed in each head inside and outside a group of followers
that completely encircle a cam. The inner chamber of each head is confined
by the cam, the followers and side plates, while the outer chamber is formed
between the followers, the housing and the side plates. With the cam rotation, the followers move outwards and inwards with respect to the cam centre of rotation, causing the outer chamber volume to decrease and increase
while the inner chamber volume is increasing and decreasing. In the case of a
pump, the inner chamber can be used as a pumping chamber in which liquid
is sucked during its volume increase and discharged during its decrease. The
motion of the followers and the mechanism of suction and discharge in this
case resemble that of a heart. For this reason, the unit is called the cardiac
action pump.
Fig. 1 shows a configuration of a machine head, when the followers (3)
are fitted with roller bearings (4). These followers completely encircle a cam
(2) and separate the inside chamber (7) from the outside one (8). The cam (2)
is fitted to the machine shaft (1). Another chamber (9) is formed between

Cardiac action pumps and motors

39

6
5
9

8
7

1
4
3
2
10

Fig. 1 Example of a configuration for a head.

each two adjacent followers and is to be connected to the tank or to either of
the inner or outer chambers. The outward and inward motions of the followers are guided using guides (6) fitted with the rollers (5). Roller bearings
(4) are used in this case to minimize friction between the cam and the followers, while rollers (5) are used to minimize friction between the followers
and their guides.
Fig. 1 shows the case when the followers are fully contracted, i.e. the
inside chamber (7) is at its minimum volume and the outside chamber (8)
is of maximum volume. Fig. 2 shows the case when the followers are fully
extended. The projected areas of the followers in their direction of motion
are A1, A2 and A3 in the chambers (7), (8) and (9), respectively. These areas
are subjected to different pressures according to the unit type and the functionality. The area A2 is generally selected equal to A1 + 2A3.
As the cam (2) rotates, the followers move outwards and inwards radially
with respect to the cam centre of rotation. This causes the volumes of the
inside chamber (7) and the chamber (9) to increase during followers’ outward motion and the volume of the outside chamber (8) to decrease, and
vice versa during the followers’ inward motion.
In the case of a pump, the inside chamber (7) can be used for liquid suction and discharge while chamber (8) can be used for control purposes.

40

Positive Displacement Machines

A3
8

A2
A1

7

A3
9

Fig. 2 The head at the fully expanded followers’ position.

Cardiac action pumps layout and principle of operation
Fig. 3 shows layout of a cardiac action pump with two pumping heads,
namely the heads (a) and (b). Each of the shown two pumping heads consists
of three followers (3) surrounding a 3-rise cam (2). The void space between
the cam, the followers, and two side plates (not shown) forms the pumping
chamber (4) of the pumping head, where the suction and delivery of
the hydraulic oil occurs. The outer chamber (5) surrounding the followers
of the pumping head is called in this case the control chamber. The cams
(2) of the two pumping heads are mounted on the pump drive shaft (1). They
are out of phase, with a phase shift angle ϕ that depends on the number of the
pumping heads and the number of rises of each follower per one cam revolution. It can be shown that the value of ϕ is given by:
360
(1)
nz
where n is the number of cam rises per revolution and z is the number of the
pumping heads. For the pump shown in Fig. 3, ϕ ¼ 60°.
During the cam rotation, the followers of one of the pumping heads, e.g.
head (a), are pushed outwards during a part of a revolution. This causes the
volume of the pumping chamber of this head to increase to suck oil from the
tank (6) via the suction valve (7). The outward motion of these followers
ϕ¼

41

Cardiac action pumps and motors

X
± V

10
V± V

9
5
4

V

3
2
1
(b)

(a)

8
7

Suction

6

D ischarge

Fig. 3 Cardiac action pump principle of operation.

decreases the volume of the control chamber in head (a) and pushes control
oil to the control chamber of the head (b). The transferred oil to head
(b) pushes the followers of this pumping head, through the area A2 (see
Fig. 2), inwards towards its cam, causing the delivery of oil from head
(b) to the pump delivery port through valve (8). During the subsequent part
of the shaft rotation, suction in head (b) takes place and delivery of head
(a) occurs in the same way. This means that the followers are pushed outwards by means of the cam, and an external force is required to push them
towards the cam during their inward stroke. This external force would result
from the pressure of the control chamber oil only, which acts on the external
surfaces of the followers. But additional means, such as springs, should be
provided in the pumping head for pushing the followers inwards whenever
the control oil pressure is not enough to realize the delivery stroke of the
pumping head, this might be necessary for pump start up.
The pump geometric volume can be controlled by controlling the volume of the oil exchanged between the control chambers of the pumping
heads. This can be achieved using a control valve (10). When the control
valve connects the control chambers to the tank for a short period of time,
some control oil is released to the tank. With a control oil volume release of
value ΔV, and when the followers of one of the pumping heads are at their
maximum outward position, the followers of the other head would not

42

Positive Displacement Machines

reach their ultimate inward position, causing a decrease in the pump flow
rate of value that depends upon ΔV. With this release of oil, the control
oil volume reaching each pumping head, namely the effective control oil
volume Ve, is given by V  ΔV, where V is the maximum possible effective
control oil volume giving the maximum geometric volume of the pump.
The pump geometric volume, on the other hand, can be increased by adding
a certain amount of high-pressure oil to the control chambers, through valve
(10), in order to increase Ve. This can be realized either internally, as shown
in Fig. 3, or externally from a high-pressure line through the port X. The
change in Ve is repeated a number of times equal to the number of the
pumping heads z multiplied by the number of cam rises n per one pump
drive shaft rotation. The pump geometric volume is thus changed according
to this multiple, and a small change in Ve is consequently sufficient to cause a
considerable change in the pump geometric volume.
The opening period of valve (10) for decreasing or increasing the pump
geometric volume depends on the required change in this volume and hence
the required change in Ve, and the duration of Ve change depends on the
control valve size and the value of its supply pressure. At zero geometric volume, when the released oil volume equals to V, and hence Ve becomes zero,
the followers remain nearly stationary at their most outward position during
pump rotation.
The control oil pressure required for the pumping head delivery action
depends on the value of the load pressure. Control oil pressure can be
allowed to be less than the delivery pressure when the buckets (9), shown
in Fig. 2, are connected to the tank to be at zero pressure. In this case the
area A2 of each follower external surface subjected to the control oil pressure
is larger than the follower internal surface area A1 subjected to the load pressure. This allows to use the pump own delivery flow for internal piloting
purposes. For the pump start up, however, and when the delivery pressure
is zero, a priming spring action is required for the pumping head to perform a
delivery stroke.
As seen, in the cardiac action pump the control of the volume of control
oil replaces the heavy mechanical control mechanisms used to control the
geometric volumes of the currently available pumps, which would result
in sensible improvement in the pump dynamic response in comparison with
the currently available pumps.
In this type of pumps, and for improved balancing, flow uniformity and
other considerations, each pumping head would preferably perform more
than one pumping cycle per one shaft revolution. In the pump configuration
shown in Fig. 3, each pumping head performs three pumping cycles during

Cardiac action pumps and motors

43

each shaft rotation. For a pump running for example at 1800 rpm, the
operating frequency for each of the suction and delivery valves would be
90 Hz. The suction and delivery valves of the currently available pumps
operate generally at frequencies up to 35 Hz and are therefore not suitable
for the cardiac action pumps. A new type of these valves has been developed.
It is shown schematically in Fig. 4. This type of valves would have low inertia and high flow gain in order to cope with the requirements of high operation frequency and discharge.

6
2
3
A

A

5

7

Pumping chamber side
2

4
Suction valve

q

q

1
4

Delivery valve
3
Port plate side

Fig. 4 Suction and delivery valves.

44

Positive Displacement Machines

The figure shows that the set consists of two strip ring valving elements
(2) and (3) that cover a set of suction and delivery openings (5) and (6)
formed in a valve plate (1). A set of springs (4) pushes the valving elements
on the valve plate. The strip ring lift away from the valve plate causes a high
flow rate (q) to pass through the ring edges as shown in the figure. The suction valve is made larger than the delivery valve for the former to have a
higher flow gain. Seats (7) for springs (4) are formed in the elements (2)
and (3). These seats are also used to guide the motion of the elements.
Fig. 5 shows another configuration for a pump, where swiveling followers (1) are used instead of the linearly moving ones. Guidance of the
oscillating followers is simpler than the guidance of the linearly moving ones.
Besides, the interacting force between the cam and a follower during the
suction stroke is reduced due to carrying part of the control chamber pressure force on the follower’s pivot. In this configuration, the rolling bearings
are replaced by pads of special shapes (2), as shown in the figure.
Fig. 6 shows an exploded view for a proposed pumping group consisting
of two pumping heads with swiveling followers, in a pump with several

Fig. 6 Exploded view of a pumping group consisting of two pumping heads.

Fig. 5 Swiveling followers and special pads.

Cardiac action pumps and motors

45

pumping heads. Each of these pumping heads is with swiveling followers
and specially shaped pads, as shown in Fig. 5.
The pumping group has two separating plates (1) that separate the pumping chambers of this group from the surrounding ones. The separating plate
(1), the followers (2), the pads (8) and the side plate (3) form a pumping head.
The followers and the pads move between the plates (1) and (3).
Plate (3) contains housing for a suction valve element (7), and its supporting small wave springs (12). Side plate (3) provides flow paths from the
pumping chamber to both the delivery and suction valves. The delivery
and suction valving elements (6) and (7) rest on the valve plate (4) and cover
its openings. The port plate (5) services both of the pumping heads. It contains a housing for the delivery valve element (6) and its supporting wave
springs (12). It collects the delivery flow of each head and leads it to the
pump outlet port. It also connects both heads suction valves to the suction
port of the pump. Spacers (9) might be used to prevent seizing the followers’
motion between the side and the separating plates.
A design for a 40 cc/rev geometric volume five-head cardiac action
pump with swiveling followers has been proposed and is shown in Fig. 7.
In this design, the five cams of the five heads are milled on the shaft. The
partial section shows how plates (3) of each pumping group connect the
delivery flow to the main delivery port in the pump casing. The pump outer
dimensions are seen to be 170  170  187 mm, and its weight is nearly
30 kg. Compared to other pumps with the same geometric volume, the cardiac action pump is relatively more compact and of lower weight.
It is to be noted that the cardiac action pump can also be constructed as a
fixed displacement one. In this case a compensating check valve is to be used
to connect the pump delivery line to the control chambers in order to compensate for any control oil leakage and/or compressibility.
Another configuration for the unit head operating mechanism and the
pump or motor control is shown in Fig. 8. In this configuration the roller
bearings are installed on the shaft instead of the followers. The operation and
control of the shown configuration are also different. The followers’ inner
chamber is connected with the tank, and the outer chamber of the head is
connected either to the unit high pressure port or to its low pressure one via
the shown control valve. Each head in this configuration is controlled separately from the other heads by its own directional control valve. By this
way, if the valve solenoids are energized in timing with the shaft angle,
the head can serve either as a pump, or as a motor in both directions of
rotation.

46

Positive Displacement Machines

Fig. 7 Partial section in a 40 cc/rev geometric volume cardiac action pump.

If used as a pump, external means such as springs should be used to push
the followers inwards during the suction stroke. If used as a motor, and continuously connected with the high-pressure line at the shown fully contracted position, firm stoppage can be obtained without additional brakes.

Pumps model
Using a SimulationX software package, a physical simulation model for cardiac action pumps can be obtained. Fig. 9 shows this model for a pump with
five pumping heads, when priming springs are used for startup and probable
other requirements.

Cardiac action pumps and motors

Fig. 8 Alternative unit head configuration.

Fig. 9 Physical simulation model for a pump with five pumping heads.

47

48

Positive Displacement Machines

In this model a standard hydraulic differential cylinder model (1) represents both the pumping and control chambers of each pumping head. The
rod side chamber of the cylinder represents the pumping chamber while the
piston side chamber represents the control chamber. The mass of the followers and pads of each pumping head is represented in this model by a mass
element m (4) connected to the piston rod, and the pump priming spring and
the damping effects are represented by a spring damper model (5). All the
control chambers of the pumping heads are connected together, and they
are represented in the model by a volume element Vc (6). Each pumping
chamber is connected either to the tank or to the pump delivery line through
a suction valve (7) and a delivery valve (8), respectively. A cam disk model
(9) driving a piston is used to represent the cam-follower motion. Each cam
is assumed, in this case, to be of 3-rises per revolution. According to Eq. (1),
the five cams should be mounted on the pump drive shaft at 24° phase shift.
The pump is driven by the motor (10). The separation that develops
between the cam and the followers, when the pump geometric volume is
less than its maximum, is modeled using an end stop model (11). The
end stops used in the pump model are assumed to be rigid end stops with
an impact of the plastic type. To prevent the reverse flow from the load
(12) to the pump, a check valve (13) is used. A 3/3 proportional directional
control valve (14) is used to connect the control chamber either to the tank
or to the pilot pressure line for a specified short period when the pump geometric volume is to be decreased or increased, respectively. The proportional valve is assumed to have identical edges, with linear flow-spool
displacement characteristics, and has zero lap conditions.

Numerical simulation of pump performance
Simulation runs are carried out using the developed model, for a designed
pump with oscillating followers and five pumping heads, and of 104 cm3/rev
nominal geometric volume. This pump was found to be represented by a
model with the following parameters. The diameters of the pistons and piston rods of the differential cylinders (1) have been found to be 39 and
20 mm, respectively, in view of the area ratio of the followers’ outer and
inner surfaces, and the maximum stroke is limited to 8 mm. The length
of each cylinder is assumed to be 10 mm, with the remaining 2 mm accounting for the control oil minimum volume. The actual maximum geometric
volume of the pump represented by this model is 105.6 cm3/rev. The value
of the mass element m has been determined such that it yields inertial effect

Cardiac action pumps and motors

49

equivalent to the moment of inertia of the followers about their pivots. This
yielded m equal to 50 g. The priming spring has been selected to be of stiffness that equals 10,000 N/m, while a damping coefficient of 400 Ns/m
representing the various damping sources affecting the followers’ motion
is assumed. The maximum separation between the end stops ends is equal
to 8 mm. The oil volume in the volume element Vc at pump zero geometric
volume is taken 650 cm3, while the pump oil is assumed to be HLP 46 at
40°C temperature. The check valves in this model are assumed to have a
nominal flow rate of 50 L/min at a pressure drop of 0.01 MPa, with laminar
flow characteristics. The control valve (14) is selected from standard available
valves (Rexroth, Bosch Group, 2005a). The chosen valve for simulation is
of a nominal flow rate 45 L/min at full input signal and 3.5 MPa pressure
drop at each control edge. The flow rate in such a valve would change
by 0.45 L/min per each 1% change in input signal. In other words, it has
a gain of 0.4 L/min/% change in input signal, at this pressure drop. The
dynamic characteristics of the valves are neglected.

Pump steady state characteristics
With the above given parameters, a simulation run has been carried out
when the cam profile is selected so as to generate a follower displacement
x in the form:
x ¼ X  absð sin ðωt + ϕÞÞ

(2)

where X is the follower maximum displacement, ω is the cam angular velocity, and ϕ is the phase shift angle. The simulation run has been carried out
assuming that the pump rotational speed is 1450 rpm and the pump delivery
pressure is zero.
The obtained results are shown in Fig. 10 that shows the variation of each
of the pump delivery flow rate, the control chamber pressure, and the volume Ve of exchanged control oil between the heads, with time.
The control chamber pressure Pc is seen in Fig. 10C to attain an unacceptable high value amounting to 3 MPa despite that the pump delivery pressure is
only zero. This is attributed to the decrease of the effective control oil volume
by only 5% at 15% of the delivery cycle of each pumping head, as can be seen
in Fig. 10B. The change in Ve in this case would lead to 15 pressure pulses
per one shaft revolution at the no load condition. The pump delivery
flow rate, as seen in Fig. 10D, is highly fluctuating with an average of
152 L/min and coefficient of fluctuation of discharge equal to about 50%.

Positive Displacement Machines

100
90
80
70
60
50
40
30
20
10
0

Delivery flow rate (I/min)

200

0

50

(A)

100

250
150
200
Cam angle (deg)

300

Control pressure Pc (MPa)

2

1

(C)

0

2

4

6
Time (ms)

8

160
140
120
100
80
0

2

4

6
Time (ms)

8

10

12

0

2

4

6
Time (ms)

8

10

12

(B)

3

0

180

60

350

10

12

Control oil volume change DVe (cm3)

Follower displacement (%)

50

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

(D)

Fig. 10 Pump performance for the cam profile given by Eq. (2).

The shown undesirable characteristics are a result of using this cam profile,
which means that the cam profile plays an important role regarding the performance of this type of pumps. Any cam profile that renders characteristics
similar to those presented would not be suitable for the cardiac action pumps.
To overcome this behaviour, the profile of the cams has been modified
to render the followers’ displacements pattern shown in Fig. 11A, which is
represented by the following equation:
1
x ¼  X  ð1  cos ð2ωt + ϕÞÞ
(3)
2
With this cam profile, the variations of the effective control volume Ve,
the control pressure Pc, and the pump flow rate with time are shown in
Fig. 11.
The control pressure Pc is seen in this case to vary between 0.31 and
0.38 MPa only at pump zero delivery pressure, which is practically acceptable. This is because with this cam profile, Ve is seen in Fig. 11D to remain
nearly constant during the steady state operation. The control pressure Pc
would consequently assume the nearly constant values required only to
overcome the inertia, damping, and compressibility effects.
The pump flow rate pulsations in this case, as can be seen in Fig. 11B,
amount to about 6% only.

100
90
80
70
60
50
40
30
20
10
0

Delivery flow (I/min)

160
120
100
80
60
40
0
60

(A)

120
180
240
Cam angle (deg)

300

360

4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0

2

4

6
Time (ms)

8

0

2

4

6
Time (ms)

8

10

12

0

2

4

6
Time (ms)

8

10

12

(B)

× 10–1
Control pressure Pc (MPa)

140

20
0

(C)

51

180

10

12

Control oil volume change DVe (cm3)

Follower displacement (%)

Cardiac action pumps and motors

0.25
0.2
0.15
0.1
0.05
0

(D)

Fig. 11 Effect of modified cam profile on pump performance.

Effect of pump geometric volume variation on its flow rate
pulsations
At the maximum geometric volume of a cardiac action pump the suction
and delivery periods of each head are equal, and each is 50% of the total
pumping cycle time as can be seen in Fig. 11A. Varying the pump geometric
volume is carried out by controlling the value of effective control volume
Ve, which manipulates the suction period over the total pumping cycle time,
and affects the pump flow rate pattern. Decreasing Ve shortens the suction
period and consequently decreases the amount of oil sucked per head during
one pumping cycle. In this case, the followers follow the cam during parts of
the cam rotation only.
To show the effect of Ve decrease on the flow rate pulsations, simulation
runs have been carried out, using the pump model, at different values of Ve
ranging between 0% and 100% of the maximum value of Ve that equals
24 cm3. At each value of Ve the displacement of the followers and the pump
geometric volume were determined. The variation of a follower displacement with time at each pump geometric volume is shown in Fig. 12.
The figure shows that the suction period of each pumping head decreases
with the pump geometric volume decrease. At a geometric volume 50%
of the maximum value, the suction period duration is seen to be 3.4 ms

52

Positive Displacement Machines

Fig. 12 Displacements of a follower at different pump geometric volumes.

which is about 25% of the total pumping cycle time amounting to 13.4 ms.
Fig. 13 shows the variation of both the pump geometric volume and the
suction period of any pumping head with the variation of the value of
the effective control volume.
Fig. 14 shows the variation of the pump flow rate with time at different
values of the pump geometric volume. With the decrease of the pump
geometric volume, the pump flow rate pulsations are seen to increase.
This requires a special accumulator, e.g. the accumulator presented in
(Mostafa, 2015), to handle these pulsations.
Fig. 14 shows also that at geometric volumes less than a certain value, the
delivery flow rate becomes zero for certain durations during pump rotation.
These durations depend upon the value of the reduced geometric volume,
the cam profile, as well as the number of the pumping heads. Fig. 15 shows
the effect of the number of pumping heads on these durations when the
pump cam profile is as presented by Eq. (3).

Pump transient response
Pump response during geometric volume step decrease
The pump geometric volume can be decreased by connecting the control
chamber to the tank for a certain period of time through the control valve,

Cardiac action pumps and motors

53

Fig. 13 Variation of suction period and pump geometric volume with the variation of
effective control volume.

to release a certain amount of Ve that depends on the required change in the
pump geometric volume. The duration of this release depends on the control valve size, and the values of the control chamber pressure and the load
pressure. The relation between these two pressures depends on the ratio
between the followers’ internal and external surface areas. Since this area
ratio is less than unity, the required control chamber pressure would generally be less than the load pressure. Calculations carried out for the followers’
area ratio, at all the positions assumed by the followers, showed that this area
ratio is nearly constant at all positions.
The pump response has been numerically simulated when the pump
geometric volume is decreased from its maximum value to zero. The
response was estimated at different load pressures while using a control valve
with a nominal flow rate of 45 L/min at 3.5 MPa pressure drop per control
edge, henceforth referred to as valve (1). The results are presented in Fig. 16,
in which the pump response is seen to be linear throughout most of the time,
and nonlinear during the last part only. The response time of the step geometric volume decrease is seen to be 25 ms at the pump delivery pressure of
20 MPa and 40 ms at 5 MPa pump delivery pressure.
Fig. 17 shows the effect of load pressure and valve size on the pump step
decrease response time. The pump response has been numerically simulated
in this case when three different valves are considered. The nominal flow
rates of these valves, henceforth referred to as valves (1), (2), and (3) are

160

Delivery flow (I/min)

180

140
120
100
80
60
40

140
120
100
80
60
40
20

20

0

0
0

2

4

6

8

10

12

0

2

4

8

10

12

10

12

Pump Displacement 25%

Pump Displacement 50%
200

200

180

180

160

160

Delivery flow (I/min)

Delivery flow (I/min)

6

Time (ms)

Time (ms)

140
120
100
80
60
40

140
120
100
80
60
40
20

20

0
0

2

4

6

8

10

12

Time (ms)

Fig. 14 Pump delivery flow rate at different pump geometric volumes.

0

2

4

6

Time (ms)

8

Positive Displacement Machines

Delivery flow (I/min)

Pump Displacement 75%
200

160

0

54

Pump Displacement 100%

180

Cardiac action pumps and motors

55

Fig. 15 Durations of pump zero delivery flow.

Fig. 16 Pump response during step decrease of pump geometric volume with valve (1).

45, 60, and 80 L/min at 3.5 MPa pressure drop per control edge, respectively. The results show that with the valve size and/or load pressure increase
the response time decreases. For the largest valve, the step decrease response
time of the pump is 12 ms at 20 MPa pump delivery pressure, while it is 18 ms

56

Positive Displacement Machines

Fig. 17 Effect of load pressure and valve size on response time during geometric
volume decrease.

at 5 MPa delivery pressure, while for the smallest valve it decreases from
36 ms at 5 MPa load pressure to 20 ms at the higher load pressure of 20 MPa.
The effect of the damping coefficient on the pump speed of response was
tested, and the obtained results showed that the damping coefficient has negligible effect on the pump speed of response.

Pump response during geometric volume step increase
Increasing the pump geometric volume is achieved by increasing Ve using
the control valve, which is to be supplied with high pressure oil either internally or externally as shown in Fig. 3.
Internal piloting
Internal piloting is performed by connecting the pump delivery line to the
inlet pressure port of the control valve. At the pump conditions of zero geometric volume and zero load pressure, pump priming springs should be used
to push the followers towards the cam. With the inward motion of the followers, some delivery oil is to be pushed through the control valve to the
control chamber for the pumping actions of the pumping heads to start.
During the early stages of the response, the speed of response depends on
the stiffness of the priming springs. After the followers reach positions where

Cardiac action pumps and motors

57

they are in control by the cam, the pump speed of response would depend on
the difference between the load and control chamber pressures, as well as on
the control valve size.
The pump response has been numerically simulated when the pump
geometric volume is increased from zero to its maximum value. The
response has been estimated at different load pressures while using the valve
(1), and priming springs of stiffness 5000 N/m are used. The results are presented in Fig. 18, which shows that the pump response is nonlinear at the
early stages of response and becomes linear throughout the remaining
period. The geometric volume step increase response time is seen to be
62 ms at 20 MPa delivery pressure, while it is 88 ms at 5 MPa delivery
pressure.
Other simulation runs have been carried out to determine the effect of
the load pressure and control valve size on the response of the pump during
its geometric volume step increase. The obtained results are presented in
Fig. 19 and show that with the valve size increase and/or load pressure
decrease, the response time is decreased. With the largest valve, the geometric volume step increase response time is 40 ms at 20 MPa load pressure,
while it is 48 ms at 5 MPa.
The pump step geometric volume increase response has also been simulated numerically at different values of the stiffness of the priming springs.

Fig. 18 Pump response during its geometric volume step increase using internal
piloting and control valve (1).

58

Positive Displacement Machines

Fig. 19 Effect of load pressure and valve size on geometric volume step increase
response time.

The response has been determined at the same load pressure of 10 MPa when
the valve (1) is used. The pump geometric volume in the different cases is
increased from zero to maximum, and the pump is internally piloted. The
simulation results are shown in Fig. 20, which shows that with the priming
spring stiffness increase the response time is slightly decreased. The step-up
response time of the pump is seen to be 66 ms at 15000 N/m spring stiffness,
and it increases to 73 ms when the stiffness is reduced three times to be
5000 N/m.
As seen, a 10% increase in the step-up response time results from reducing the priming spring stiffness three times that clarifies the minor effect of
this stiffness on the speed of response.
External piloting
In this case the pump control chamber is to be supplied with a certain volume of pressurized oil from an external pressure source via the control valve,
to increase the pump geometric volume. The priming springs are no longer
needed, and the pump speed of response depends on the control valve size
and the pressure difference across it.
The pump response is numerically simulated in this case when the pump
geometric volume is increased from zero to the maximum value using valves
(1), (2) and (3) and an external pilot oil source of pressure either 10, 15 or

Cardiac action pumps and motors

59

Fig. 20 Effect of priming spring stiffness on pump speed of response during step
increase of geometric volume using internal piloting.

Fig. 21 Pump response during geometric volume step increase using external piloting
and valve (1).

20 MPa. The obtained results, when valve (1) is used, are presented in
Fig. 21. The pump response, as depicted in the figure, is seen to be slow
during the early stages of response and turns to be fast and linear throughout
the remaining period. The response time is seen to depend on the value of

60

Positive Displacement Machines

the pilot pressure and decreases from 48 ms at 10 MPa pilot pressure to 22 ms
at 20 MPa pilot pressure.
Fig. 22 shows the effect of the pilot pressure and valve size on the pump
step increase response time. The presented results show that with the valve
size and/or pilot pressure increase the response time decreases. For the largest valve, the geometric volume step increase response time is 11 ms at
20 MPa pump pilot pressure, while it is 24 ms at 5 MPa.
The obtained pump response times during geometric volume decrease
and/or increase are generally small when compared with those of the currently available pumps. For a specific comparison, the step response times of
a cardiac action pump of geometric volume 71 cm3/rev and equipped with
the smallest valve (1) have been calculated and compared with those of a
swash plate pump of the same geometric volume when the load pressure
on each one is 10 MPa. The response time during the geometric volume step
increase of the cardiac action pump has been found to be 54 ms when using
internal piloting, and 34 ms when external pilot pressure source of 10 MPa is
used. The swash plate pump response time under the same conditions is
60 ms. During the step geometric volume decrease, the determined response
time of the cardiac action pump is 15 ms while that of the swash plate is 55 ms
(Rexroth, Bosch Group, 2005b). This shows how fast is the cardiac
action pump.

Fig. 22 Effect of pilot pressure and valve size on step up response time.

Cardiac action pumps and motors

61

Acknowledgements
The authors would like to thank the Egyptian Science and Technology Development Fund
(STDF) for funding the research through the project “Development of a Variable Geometric
Volume Positive Displacement Pump and New Hydraulic Servovalve”, Grant no. 2466. The
authors also appreciate Prof. Dr. Ing. J€
urgen WEBER and Eng. Khaled Mostafa for their
valuable contributions in this work.

References
Mostafa, K. G. (2015). Development of a variable geometric volume cardiac action hydraulic pump. M.
Sc. Thesis. Faculty of Engineering, Cairo University.
Rexroth, Bosch Group (2005a). Hydraulic components for industrial applications, RE 00112_Part
3 proportional, high-response and servo-valves.
Rexroth, Bosch Group (2005b). Hydraulic components for industrial applications, RE 00112_Part
1: Hydraulic pumps and motors.

Further reading
Elgamil, M. A., Mostafa, K. G., B€
ugener, N., Kassem, S., & Weber, J. (2014). In Potentials and
challenges of a new variable geometric positive displacement pump. The 9th JFPS International
Symposium on Fluid Power Matsue, Japan, Oct/28/2014–Oct/31/2014.
Elgamil, M., Mostafa, K., El-Husseiny, M., & Kassem, S. (2015). Design aspects of a cardiac
action hydraulic pump. In ASME. Fluid Power Systems Technology, ASME/BATH 2015
Symposium on Fluid Power and Motion Control. https://doi.org/10.1115/FPMC20159557. V001T01A029.
Elgamil, M. A., Zeyada, Y. F., & Kassem, S. A. (2003). In Design aspects of a new type of hydraulic pumps. The 8th Scandinavian international conference, SICFP03, May 7–9, Tampre,
Finland.

CHAPTER 3

Geometric design of the limaçon
machine
Ibrahim A. Sultan, Truong H. Phung

School of Science, Engineering and IT, Federation University Australia, Ballarat, VIC, Australia

Contents
Introduction
Limaçon drives
Geometric equations for the limaçon-to-limaçon machine
Profiles of the machine housing and rotor
Volumetric relationship for the limaçon-to-limaçon machine
Housing and rotor interference
The circolimaçon machine
Housing-to-apex clearance
Rotor design for the circolimaçon machine
Volumetric equations for the circolimaçon machine
The limaçon-to-circular machine
Housing and rotor interference of the circular-to-limaçon machine
Volumetric relationship for the limaçon-to-circular machine
Conclusions
References
Further reading

63
67
71
71
72
74
79
80
82
84
86
86
90
91
92
92

Introduction
Positive displacement machines have acquired particular popularity with the
current push for sustainable energy production and consumption. It is currently not acceptable to waste energy available in high-pressure gasses
throttled or vented to atmosphere in industrial processes (Samuel, 2007;
Shu, Yu, Tian, Wei, & Liang, 2014) or blown out of cylinders in large diesel
engines (Amann, 1987; Panesar, 2015; Karvountzis-Kontakiotis, Pesiridis,
Zhao, Alshammari, et al., 2017). For such applications, a suitably sized
positive displacement machine would be employed to harvest energy from
these gases and transform it into useful forms. Turbines that are traditionally
employed in massive power plants cannot be used for energy harvesting
Positive Displacement Machines
https://doi.org/10.1016/B978-0-12-816998-8.00003-0

© 2019 Elsevier Inc.
All rights reserved.

63

64

Positive Displacement Machines

applications due to their inability to handle neither two-phase vapors
(Groniewsky, Gy€
orke, & Imre, 2017) nor flow occurring at small rates.
For these applications, a gas expander, which is a positive displacement
machine is usually employed. In relation to pressure creation, positive
displacement pumps and compressors are commonly employed to impart
pressure to respectively gaseous and liquid fluids flowing at rates which
are required by most industries. This chapter presents the limaçon rotary
positive displacement machine, which can be employed for energy production (i.e., as a gas expander) and energy consumption (i.e., as a compressor or
a pump) with equal ease and effectiveness. The limaçon machine (Sultan,
2005) offers a straightforward technology, which has occurred to many
creative minds as evident by patents which can be traced back to the 19th
century. However, it is unfortunate that such technology did not make it
to mass production due to the lack of mathematical understanding of their
geometric characteristics and unavailability of manufacturing tools required
to machine their chambers. Recently, however, with technical papers published on the topic, industry take-up of the technology has started to emerge.
A good example of this take up is shown by the patent of Nystrom (2016).
In the linkage shown in Fig. 1, a rigid crank, cm, drives a rigid output
link, p1p2, that is simultaneously capable of performing sliding and rotational
Pa
t

h

of

p

1

an

d

p

2

Base circle

p1

Y

Inst. centre
of p1p2

p2

I

w

m

L

m

q
X

ng

idi

p1

L

r
2q

Sl

m

2w

c

=

=

o

p2

Fig. 1 A linkage to produce limaçon motion.

Geometric design of the limaçon machine

65

motions. This is an inversion of the well-known slider-crank mechanism,
except for this inversion, the sliding joint, o, is attached to the circular
trajec