Mechanical Engineering: Limacon-Driven Compressor Modeling Project

This project focuses on the modeling of a limacon-driven reciprocating compressor, a design approach that utilizes the limacon curve geometry to drive the piston, offering potential efficiency gains compared to traditional crankshaft-based systems. The research investigates the geometric properties of the limacon curve, its kinematic background, and its application in reciprocating compressor design, addressing the challenges of crankshaft design, including robustness and fatigue performance. The project includes a literature review, research methodology, and a timeline, aiming to demonstrate the advantages of the limacon-driven system. The study also explores the operational characteristics of reciprocating compressors, including indicator diagrams and thermodynamic analysis, highlighting the importance of instantaneous angular speed (IAS) for fault identification. The project emphasizes the potential for improved performance and reduced maintenance costs through the adoption of this alternative design.

Abstract:
A reciprocating compressor or a piston compressor is a positive displacement machine that
uses a piston driven by the crankshaft to deliver the gas at a high pressure rate. They have a
special attention in the industries but the system is highly expensive and most critical. The work
of the crankshaft is to convert the reciprocating displacement of the shaft to the rotary
movement. The system is highly complex with the four link mechanism. The robustness and the
fatigue performance of the system have to be considered while designing the system as they
could experience a maximum number of load cycles during their operation. The work mainly
concentrates on the design approach of the reciprocating compressor for their better
manufacture so that they could have a better performance. The research proposal deals with
the Limacon-driven reciprocating compressor that drives the reciprocating compressor with the
limacon curve geometry. The geometry of the limacon of Pascal is a plane curve, which have
certain characteristic that could be used in the application to handle the fluid. The curve has
not drawn enough attention towards the current science investigation rather few trials were
conducted in the past century. By using the limacon curve driven technology in the mechanical
linkage we could obtain higher efficiency than the normal crankshaft operated reciprocating
compressor. The mechanical gadgets should be bought to test application to make all the
specification incontrovertible.

Table of contents
1. INTRODUCTION………………………………………………………………………………3
2. PROBLEM DESCRIPTION …………………………………………………………………5
3. LIMACON GEOMETRY……………………………………………………………………..5
4. LITERATURE REVIEW ……………………………………………………………………5
5. RESEARCH METHDOLOGY AND SCIENTIFC WORK…………………………….10
6. SCOPE OF RESEARCH……………………………………………………………………….13
7. TIMELINE AND RESOURCES……………………………………………………………..15
8. CONCLUSION…………………………………………………………………………………..15
9. REFERENCING………………………………………………………………………………… 16

Introduction:
Reciprocating compressors has been widely used in many mechanical systems. They could able
to handle huge variations in the discharge as well as in the suction conditions and are very
simple in their operation (AlThobiani & Ball, 2014). Their operating range could vary widely with
the minimum power per machine volume under the combination of their flexibility.
Reciprocating compressors is the only machine that could give maximum pressure ratios. For
the Thermodynamic analysis of the reciprocating compressors various modeling methods have
been developed. Based on the time and their crank angle, this could be widely classified as the
global model as well as the differential model. It is essential to develop a simulation with the
mathematical model that has several advantages that are as follows: 1) For the real-time
machine that are under test study, we could obtain the effective fault signatures. 2) We could
able to determine the performance of the reciprocating compressor with the same operating
condition. Hence, developing the detection features for reciprocating compressors is utmost
important due to these advantages. It is necessary to study the instantaneous angular speed
(IAS) of the crankshaft for fault identification. IAS is the important factor to be studied with the
pressure measurement since IAS measurement is non-intrusive in comparison. It is more likely
related with the machine dynamics with the low noise contamination compared with
conventional structural vibration and airborne acoustics. Hence IAS could provide accurate
result and the diagnosis could be easily interpreted. It is not necessary to provide a periodic
calibration and the IAS measurement encoder is relatively cheaper (Adair, Qvale & Pearson,
1972). The measurements could be compared accurately with various time periods with various
sensors. Wide study has been carried out recently with the IAS measurement.
Figure 1: Indicator diagram of the reciprocating compressor

Figure 2: Temperature entropy diagram of the compression process (Stouffs et al., 2001)
Crankshaft has a complex geometry that could convert the reciprocating motion of the
component into rotary motion. This operation could be carried out in four bar slider crank
phases and is highly possible with the other two components such as the piston and the
connecting rod (Becerra, Jimenez, Torres, Sanchez, & Carvajal, 2011). Since the rotation is the
most practical and the applicable input for the other devices, it is necessary to convert the
machine output in the rotary motion. Since the combustion is provided by the combustion
chamber of the engine, the linear displacement will not be smooth that could lead to
unexpected shock and when the output of the engine is given as an input to other machine it
could cause severe damage.
Figure 1 represents the indicator diagram or the watt diagram of the reciprocating compressor
operation that shows the variation of the pressure with the varying piston positions at the
cylinder. In the diagram volume 1 is said to be the high volume state and the volume 3 is said to
be the minimum volume state whereas points 2 and 4 are conventionally defined such as p2 D
p3 and p4 D p1. The operation of the piston is mentioned by four phases: 1) The piston
compresses the gas from the cylinder which is carried out in phase 1-2. 2) The piston will
discharge the gas to the plenum from the cylinder in the phase 2-3. Simultaneously, the
discharge plenum pressure p will vary from the in-cylinder pressure that happens due to the
pressure drop from the discharge valve the non-instantaneous valve motion and the valve
spring. There will be a minimum pressure variation inside the cylinder during the discharge
process when compared to the pressure variation in the compression process. 3) The gas from
the clearance volume V3 will expand during the process 3-4. 4) Finally the cylinder will be filled
with the gas during the phase 4-1 from the suction plenum and the pressure inside the cylinder
will be slightly different from the suction plenum pressure psuc. This is due to the pressure drop
through the non-instantaneous valve motion, valve spring and the suction valve. Moreover, the

pressure variations inside the cylinder will be very large in the compression process rather than
the suction process and the Work done by the piston during one cycle is denoted as
Wind= -∮ p dV -------------------------------------------- (eqn 1)
Where,
Wind is the indicated work (Joule)
P denotes the pressure (pa)
This is assumed only when the pressure recorded by the indicator diagram is same as the
pressure in piston face.
Problem Description:
Crankshaft experiences a huge number of load cycle that tends to consider the performance
and the robustness of the component during the design process. There is a huge challenge that
has been faced while designing the system since it is required to promote the equipment with
less weight and with the minimum cost. The system should also satisfy high durability with the
fatigue strength. By promoting this implementation in the machine, the engine could provide
maximum output power with maximum fuel efficiency.
Crankshaft is composed of the ball bearing at one side and the journal on the other side. The
ball bearing is fitted in such a way to perform the movement of the rotation about it s main
axis. The movement of the crankshaft will be constrained by the bearing surface that faces at
180 degrees from the load direction (Todescat, Fagotti, Prata, & Ferreira, 1992). This constrain
should be defined by the fixed semicircular surface as wide as ball bearing width. The opposite
side of the crankshaft is named as the journal bearing. The semicircular edge is free to move
along the direction of the central axis and it is mounted at the side of the load at the lower end
which is perpendicular to the central axis. There is a uniform pressure when the connecting rod
is fitted to the load at 120 degree from the contact area. There is a uniform loading distribution
in the crankshaft due to the interaction of the connecting rod. The design of the crankshaft is
the most challenging one among all the production industry. The challenge occurs while
designing the crankshaft with the minimum weight with the less cost and serving high
robustness. Crankshaft must be physically powerful and sufficient to lead the downward force
of the power stroke without unnecessary bending. Hence the lifetime of the internal
combustion engine highly relay on the construction of the crankshaft since the power of the
engine hits the shaft from one place to another.
Limacon Geometry:
The operation of the crankshaft in the reciprocating compressor could be replace by the
Limacon geometry. The geometric properties of limacon are that their curves are traced by a

point since it consists of features such as rotational motion, sliding and pole. As soon it sets in
motion, the centre point of the chord is restrained by a fixed circle radius which is considered as
base circle and also fixed point of the chord. The fixed pole should be in line with the base circle
such a way that the chord is divided into two equal length sections. Two co-ordinate systems
can be employed to obtain geometric properties. As soon as the crank performs a rotational
displacement, the angle rotated by the chord is determined. Generally, the chord length
commands of the global size of the limacon as well as the geometric characteristics of the
dimples and loops. As discussed by (Costa et al, 1999), the aspect ratio should be maintained
below 0.25 in order to obtain single-looped, dimple-free limacon. This also satisfies limacon
fluid processing machines.
The intersection point of the curves reveals the mirror-range reflections of its rotor lenticular
section. Most of the curve depicts lower range portion of limacon. These curves share same
pole as well as chord with rotor as horizontal position. The lower base circle represents upper
portion of the rotor while the upper base circle depicts the lower portion of the rotor. As a
result, a rigid plane motion of the rotor is formed with base circle and pole.
Furthermore, reliable running conditions can be enhanced by lessening the small radical
clearance of the rotor. In-order to enhance the design process, rotational motion of the rotor
should not intrude the rotor-housing. This is done by considering slopes of the tangent of the
vectors of the rotor-housing and apices as well as the contact points.
The housing limacon should be lower when compared with the rotor limacon climb. This is
done by selecting the equal or similar indexed rotor-surface should be arranged geometrically.
The reason is that radius of curvature vectors should be perpendicular in tangent vectors.
Kinematic Background:
The modification of the Nicodemes choncoidograph gives a limacon curve in a way where a
circle is obtained by the base of the conchoids and the poles of the conchoids falls on the
circumference. A simple mechanism or a motion produced by the limacon technology is shown
in the figure given below. Any point at the chord (p1p2) seen from the rotating point m at a
distance L forms a limacon. The point m falls on the chord and it is a necessary condition that
the chord should always pass through the pole point o. The uniform rotation is been performed
at the point m with the radius r on the base circle. The two points on the chord at the same
distance on either side of the rotating point, m, fall on the same limacon (Sultan, 2006).

Figure 3: A limacon mechanism employed for a compression–expansion machine
Figure 4: A mechanism to produce the limacon motion
In a wide range of industries, reciprocating compressor plays a vital role and it could be used in
the refineries, pipelines in the gas transmission as well as in the petrochemical plants, etc. This
major purpose is due to the high pressure ratio achievement. Certain types of centrifugal
compressors use RC three times higher than the normal that arise the maintenance cost higher
than the normal one (Griffith & Flanagan, 2001). The high temperature and the pressure
condition, potential failure is a normal case in this high complex and huge component structure
that could affect the production process and leads to the stoppage of RC. The faults could occur
frequently in the RC discharge and suction valves. According to (Leonard, 1996), 50% of cases
are related to the faults that occurred in the valves and 36% of cases is due to the shut down
process of the compressor. For enhancing the safety and minimizing the maintenance cost it is
necessary to consider the accurateness and reliable fault diagnosis in the RC valves (AlThobiani
& Ball, 2014).

Literature Survey:
The authors describe an optimization procedure based on gradient to obtain dimensions of a
particular piston trajectory and its methods and procedures are discussed in a detail manner.
An excellent work about effective optimization techniques as well as methods including
algorithms was introduced in the field of mechanism design was provided by Gabriele (1993).
However, Gabriele optimization techniques were outdated but recently updated optimization
techniques was presented by Hansen (2009). These optimization techniques are constrained by
geometric properties and sometimes physical properties too, where they transforms into
numerical structure. But these numerical structure leads to instability when they are exposed
to iterative synthesis based on gradient procedures. This instability was proved by Cossalter et
al. (1992), Minnar et al. (2001) and also Hansen (2002). The research work by these authors also
presented innovative, reliable as well as efficient solutions and methodologies to encounter the
difficulty experienced because of numerical singularities. But some research work of some
other authors replaced gradient-based optimization techniques and methods with stochastic
based techniques and methods. One such work is presented by Cabrera et al (2002), in which
the research methodology is based on genetic algorithms to encounter synthesis challenges.
And Samili et al (2005) research work was based on tabu search and its simulation of numerical
consisted of multi-stage reciprocating compressor. The working of the reciprocating
compressor was replicated and trained under different conditions in-order to enhance and
develop the diagnostic characteristics as well as the features for predictive monitoring. The
steps involved in simulation development and its mathematical operations as follows: speed–
torque characteristics of an induction motor, cylinder pressure variation, crankshaft rotational
motion, flow characteristics through valves and vibration of the valve plates. Basically, five
different steps were created for designing valve leakage as well as valve spring deterioration.
The valve spring deterioration was successfully overcome in this simulation carried out by
MATLAB environment. Matlab code consisted of numerical solution and efficient present of
result. Elhaj et al (2006) presented an innovative solution for mechanism synthesis. The work
employed a differential thermodynamic system which can determined compressor
performance indices of a selected piston trajectory. A loss function is built to enhance indices
performances and then treated with an optimization procedure which is derived from SPSA -
Simultaneous Perturbation Stochastic Approximation. The work done by Spall (1992) showed
that Simultaneous Perturbation Stochastic Approximation technique is reliable, efficient and
well-recognized and well suited for intricate optimization application. Such similar work was
also presented by Kothandaraman and Rotea (2005) where an efficient stochastic optimisation
algorithm was employed and the stochastic optimisation algorithm used the characteristics of
the specific piston trajectory as parameters design in which length of the stroke as well as
diameters of the bore were kept constant during the optimization process such a way that the
speed of the crank and its inlet and discharge pressures were maintained. By employing

thermodynamic framework, leakage can be avoided as stated by Dagilis et al. (2007) optimized
reciprocating compressor driven employing conventional slider-crank mechanism and
employed reciprocating compressor in stroke bore ratio and in turn enhances parameter design
such a way which increases performance of the index which is the ratio of cooling capacity of
the consumption of the power. As discussed by authors Leki and Kok (2008) emphasis the
importance of heat transfer nature process and the authors report the establishment of a
highly advanced test rig devoted for this purpose.
Catto and Prata (2000) presented numerical outcome that is evident from the notion of the
temperature-time gradient of the compressed gas which affects the transfer of the heat process
between the cylinder wall and gas occurring in the temperature. The importance of the work is
highlighted in temperature-gradient optimization procedure which enhances the cooling
system and decreasing the huge amount of the heat produced in the compression stroke
caused because of the increased consumption of the power, thereby, reliability is satisfied.
Stouffs et al (2001) captures mathematically equations and applies it in the thermodynamic
field for reciprocating compressors. These mathematical equations are proven experimentally
and can be employed for future research purposes. A detailed study of the working of the
compressors are examined and analysed by Rigola (2002) and later Rigola et al (2004) carried
out experimental work based on the previous work. Such similar work was also examined by
Haping (2005) but the author mainly focused on the workings of the compressor valves. The
reason for authors Rigola and Haping to conduct experimental work leads to better and clear
understanding of the dynamical and thermal valves which is most important for compressor
performance and reliability. A comprehensive work by Esupin and Santos (2007) which focus
on the mechanical aspects of reciprocating compressors and fluid film lubrication in the
crankshaft bearings. Moon and Cho (2005) examine the effects of the oil film found in the
underlining of the piston-cylinder connection. Nieter and Singh (1984), investigated the
performance of these compressors and piston cylinder interaction of various thermodynamic as
well as mechanical aspects of the piston cylinder interaction as well as compressor efficiency
and pressure pulsations. Peng et al. (2002) work reports on thermodynamic framework for an
effective rotary compressor design. Sultan (2005) and Sultan (2006) emphasized on the limac
applications as well as compression-expansion applications. (Meng, Liu & Liu, 2011) presented
a three-dimensional technique of a diesel engine crankshaft was built by employing the PRO/E
software. Employing ANSYS analysis tool, it presents high stress region predominately focusing
on the knuckles of the crank arm as well as the crank arm and connecting rod which is weak
and may be easily broken. (Yingkui & Zhibo, 2011) examined the crankshaft model and crank
throw which was built employing Pro/ENGINEER software and later imported into ANSYS

software. The crankshaft deformation occurred due to lower frequency and crankshaft
deformation leads to bending of the crankshaft. Generally increased deformation occurs at the
interconnection of the link between the important bearing journal, crankpin and crank cheeks.
(Zhou, Cai, Zhang, Cheng, 2009) focused on the stress formed due to the cconcentration of the
static analysis presented in the crankshaft frame.
Research Methodology:
In this research the better performance could be achieved by inserting the Limacon technology
into the reciprocating compressor. The volume action with the torque and the pressure
characteristics has been described in the graph shown below that states about the limacon
driven into the reciprocating compressor.
Figure 5: Torque–u spectrum for a limacon compressor (Sultan, 1999)
Figure 6: A PV-diagram for a limacon compressor (Sultan, 1999)

The calculation of the volumetric effectiveness of the reciprocating compressor is as follows:
The piston direction is downwards when the pressure p3 goes up to p4 with the mass m3 of gas
expands inside with the clearance volume V3. The cylinder permits only fresh gas to enter from
the top dead centre to bottom dead centre of the movement of the piston stroke. The gas
flowing through the compressor over one cycle is
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (i)
C is defined as the clearance factor that is termed as the ratio of the clearance volume V3 to the
cylinder swept volume VC = V1 -V3. Hence the mass of gas over one cycle is defined as,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ii)
The condition p1 < psuc is referred as the pressure loss through the suction valve and the
condition T1 >Tsuc is said to be the heat transfer from the suction pipe, which leads to the
situation of v1 > vsuc (the suction volume of the plenum is said to be lower than that of the
specific volume of the gas in the cylinder. This leads to the equation (iii) that could be derived
from the equation (ii), which is given below: (Stouffs et al, 2001)
. . . . . . . . . . . . . . . . . . . . . . . . . . . .(iii)
The indicated work efficiency can be calculated as
Where,
Wm=specific work does this suit the project?
dis = discharge

s=suction section (Stouffs et al, 2001)
The device to perform limacon was proposed by (Artobolevsky, 1964)The centre point of the
chord denoted as m is restricted to move towards the stationary circle with the radius r. This
stationary circle is referred as the limacon base circle that is also fixed centrode to the chord of
the limacon. For the limacon motion to be performed the stationary pole of the limacon o
should also fall on the same base circle. The centre point m divides the chord as p1m and mp2.
They are of equal length L performing the same limacon motion.
My scope of the proposed work is to insert the limacon technology into the reciprocating
compressor and try to Increase the performance of the reciprocating compressor with the help
of the piston trajectory action.
For obtaining the geometric models for the compressor to perform the limacon motion, the co-
ordinates XY and XrYr is mentioned. XY is the stationary frame sufficient for the pole (system)
and XrYr is a frame fixed to the chord rigidly and makes a movement with the centre point as
shown in the figure 4. θ is the angle measure due to the rotation of the chord from the right
side of X to Xr and Ѱ is the angle measured from the Y direction.
Through the geometric and the mathematical formulation of the limacon curve we could be
able to track the motion and the torque of the piston. The crank angle Ѱ will be twice the rotor
angle θ since the base circle contains both the point m and the pole point o. The sliding
distance S could be expressed by the following formula:
S = 2r sinθ………………………………………………………………………………………………………………………(eqn 1)
Where,
r is the radius of the base circle
The radial distance Rh of position p1 with respect to the pole, is given as follows:
Rh = 2r sin θ + L ………………………………………………………………………………………………………….(eqn 2)
Where,
L is half the chord length.
The x and y coordinates of p1 can be expressed as follows: (Sultan, 2006)
x = r sin 2θ + L cos θ…………………………………………………………………………………………………….(eqn 3)
y = r - r cos 2θ + L sin θ………………………………………………………………………………………………..(eqn 4)
The basic mechanism for the varying elective changes has made the limacon mechanism
sufficient for handling the liquid. The demonstration could be done in the components in the