Problem Description

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. 

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. 

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 p_suc. 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

Limacon Geometry

W_ind= –   ——————————————– (eqn 1)

Where,

W_ind 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. 

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. 

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. 

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)  

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). 

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. 

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. 

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  

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,  

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)   

Where,

W_m=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 ? 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 ? 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 R_h of position p1 with respect to the pole, is given as follows:

R_h = 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 form of schematic that does not need the operational activity of the plan. In every activity the limacon movement of the component is analyzed especially the chord, middle point (m), the shaft (o), the base circle as well as the instantaneous centre (l). To make the complete limacon movement, the middle point should drop on the base circle continously and the chord pivots at the precise speed half of m approximately to the centre of the circle. The momentary middle of the chord tends to fall on the base circle that is diametrically opposite to the rotor geometric middle (m).

The calculation of the volumetric effectiveness of the Limacon reciprocating compressor will be as follows:

The volumetric relationship could be calculated by the area bounded by the housing of the limacon, which is shown in the figure given below. The figure shows the infinitesimal area denoted as δA bounded by the small arc of the limacon contour as well as two radial rays that represents the limacon chord at the positions θ and θδ. The infinitesimal area is denoted as 

Where, R_h is defined in the equation 2.

Normally we use crankshaft for the reciprocating compressor to give a rotational movement. The scope of this research is to enhance the efficiency with the fatigue strength with the help of Limacon drive. Figure 7 shows the analysis of the actuation gases that is compressed with the varying stroke time on the piston, which is injected with the certain temperature on the piston stroke. The gas could vary based on their own physical characteristic with their adiabatic coefficients. The adiabatic coefficients of propane is said to vary about 1.13 and that of ammonia is said to be 1.31. The slow-speed long-stroke reciprocating compressors has a high cooling stage when compared to the adiabatic working process while analyzing the efficiency and it is said to have a temperature decrease of more than 100k with adiabatic coefficient close to 1.3. Further decreasing the temperature to a lower temperature could decrease the adiabatic coefficient value.  The external cooling is the another pacing factor that depends on the flow rate of the fluid around the compressor, the characteristic of the cooling medium with its parameters and supplementary fining 

Dependence of the average injected temperature T on the piston stroke S for propane: (Yusha et al., 2015)

• 1 – α=2000 W/m2*?, N=2 s;
• 2 – α=20 W/m2*?, N=2 s;
• 3 – α=2000 W/m2*?, N=0.5 s;
• 4 – α=20 W/m2*?, N=0.5 s;
• 5 – adiabatic compression 

The research also deals with the Global model sensitivity parameter, which is also one of the problem faced by the normal reciprocating compressors that is sufficiently eradicated with the improvement in the flow rate as well as in the efficiency. By using the values mentioned in the table of (Stouffs, Tazerout, & Wauters, 2001), the compressor is simulated with the values ranging from psuc = 100 kPa and Tsuc = 293.5 K to pdis = 700 kPa and the following inferences are made:

• Volumetric effectiveness: ε∗v = 0.573,
• Specific work: w∗m = 385 kJ·kg−1,
• Indicated efficiency: η∗ind = 0.568.   

From the obtained result, the relative deviation by considering the minimum and maximum value of each parameter is obtained in the Table 1. Additionally, the values of the remaining parameter stay unchanged. However, it is not necessary to take the whole parameter into account since here we are going to deal only with the single compressor. Following considerations are taken into account from the above table: 1) the most influencing parameter is said to be the temperature 2) in practical cases these parameters are easily estimated 3) For the compressor simulation performance, parameter f and the pressure parameters βsuc and βdis also plays the significant role 4) f is the most difficult parameter to be evaluated and the simulation result does not depend on the numerical value of the four other parameters. The global model relays on the five main parameters and the other four parameters should have an appropriate value in several practical cases that is secondarily important. The value of wall temperature €w is sufficiently small that is sufficiently note worthy. Contradictory, it is said that higher the wall temperature significantly promotes a better performance in the reciprocating compressor. Residual mass fraction has an induced cooling when the wall temperature is reduced. This could lead to the two damaging effect:  the residual mass is larger and the loss due to the residual mass reversed cycle is also larger. These negative impacts should be considered and it is the most important factor rather than saving the work with the cooling system during the process of compression. Hence inference from the table 1 says that it is better to cool the suction pipe rather than concentrating the cooling process in the walls of the cylinder.   

Conclusion

The above represented work about limacon-driven reciprocating compressor helps machines and compressor to perform better than the technology’s we are having now. It is been argued and done lot of work about the compressors but the reciprocating compressor with the limacon technology would be a new invention for the engineering world. It is expected that Limacon-based technology would gain a higher interest in the industrial community. The above taken calculations and the graphs are supposed to be assumption for the limacon-driven reciprocating compressor. 

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