mathematical modeling of compressors

Mathematics Modelling and Applied Computing. ISSN 0973-6093 Volume 1 Number 1 (2010), pp. 81–96 © Research India Publications http://www.ripublication.com/mmac.htm

Mathematical Modeling and Simulation of Reciprocating Compressors – A Review of Literature 1*

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Kumar Subramanian , L R Ganapathy Subramanian , Benny Joseph and 4 Venkatesan Jayaraman 1

Research Scholar, School of Mechanical Engineering, SRM University, India, (Corresponding author). Lecturer, International College of Engineering and Management, Muscat, Oman 2 Professor, Aerospace Engineering, SRM University, India 3 Professor, International College of Engineering and Management, Muscat, Oman 4 Assistant Professor, Department of Mechanical Engineering, Sri Venkateswara College of Engineering, Sri Perumbudur, India

Abstract

Mathematical modeling is the process of designing a model of a real complex system and conducting experiments in it for understanding its behavior. It is an effective tool in either designing or in decision making. Further due to the higher levels of developments occurred in computational techniques, a computer simulation is possible based on mathematical models. It helps us to enable the complicated problems to be solved with minimum number of assumptions. The aim of the present work is to identify an effective mechanism which includes numerous couplings involved in developing an effective computation model such as cylinder kinematics, thermodynamic modeling, fluid flow modeling, heat transfer modeling, valve dynamics, valve impact mechanisms, mufflers behavior etc… In fact creation of a model which is easy to understand, easy to detect errors in the process of building a model and easy to compute a solution is a frisking fri sking challenge as it involves knowledge integration of various core areas like mechanics and design of valves, heat transfer, fluid dynamics and programming. A fully fledged mathematical model for reciprocating compressor simulations is not yet be produced, but numerous attempts made by developing mechanisms focusing on dissemination of above information independently. This paper reviews the works carried out so for in the interdependent models produced by various researchers in time to time and hinting about the way the model development has to be taken care of.

Kumar Subramanian et al Key words: Mathematical modeling, simulation, reciprocating compressors, valve dynamics, compressor performance characteristics.

air

Introduction Since then the introduction of controlled flow of fluid as the input for many machinery operations, the artificial cooling of objects by removing heat from an enclosed space (refrigeration) and conditioning of air for human comfort, many researchers over the past 50 years focused on reciprocating compressors as their major area of research. Reciprocating compressors mostly employing either air or refrigerant as a working medium, therefore the compressors are classified as air compressors and refrigerant compressors. Though the working principles of both are same, the interactions from engineering point of view are significantly different in terms of ambient conditions and working fluids. Air is closely approximated as an ideal fluid and refrigerants are considered as real fluids. Institutes like Purdue university have a compressor research group which contributed almost 50% of the developments in studying compressors. Organizations like Gas Machinery Research Center which having 102 member companies [42] and European Forum of Reciprocating Compressors having 8 member companies [43] carried out specific research tasks in the forms of projects resulting in technical papers and reports. Mathematical Model Building a mathematical model for any real complex system based on the underlying the scientific concepts is useful as it simplifies the lead time involved in the design process. Irrespective of the system involved, the general approach is common and listed below: (1) Identify the problem, define the terms in the problem and draw diagrams where appropriate. (2) Start with a simple model, stating the assumptions. (3) Identify the important variables and determine how they are relating to each other. (4) Develop formidable mathematical equations with proven scientific background expressing the relationship between variables. (5) Solve the above set of equations for reasonable outcome. (6) Refine the above process by removing assumption one by one until to get a model closer to real world observations. The following figure 1 shows the flow chart for mathematical modeling. The flow chart is abstract one and can be modified depending upon the nature of the physical problem.

Mathematical Modeling and Simulation of Reciprocating Compressors

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Figure 1: Mathematical Modelling. Limitations Mathematical modeling is too difficult, if the physical situation is too complex or physical models formed are mathematically intractable. But these limitations can be overlooked by enabling the use of fast computing machines. Reciprocating Compressors Compressor is a machine providing fluids at high pressure. The required work for pressure conversion is delivered externally. Reciprocating compressors are used in chemical industries, foundries, breweries, process plants, refrigeration and air conditioning equipments and automotive systems. These are positive displacement machines where they increase pressure by reducing the volume. This means that they are taking in successive volumes of fluids, which are confined within a closed space, and elevating it to a higher pressure by means of a piston which is displacing agent. Single stage and multi stage are commercially available and single stage develops pressure in the range of 1-9 bar and multistage up to even 100 bars. Load reduction is achieved by unloading individual cylinders. Typically, throttling the suction pressure to the cylinder or bypassing air either within or outside the compressor is the technique. Capacity control is achieved by varying speed in engine driven units through fuel flow control. These type of compressors are available either as air cooled or water cooled in lubricated and non lubricated configurations. Operations These compressors consists of a crankshaft driven by either gas or engine attached to a connecting rod, which transfer the rotary motion of the crank shaft to the reciprocating motion of a piston. The piston compresses the air to increase its pressure. Air enters the cylinder through a suction valve at suction pressure and is compressed to reach desired delivery pressure. At delivery pressure, the air is

Kumar Subramanian et al

discharged out of the compressor through delivery valve. Figure2 shows the sectional view of the compressor.

Figure 2: Sectional view of a typical compressor.

The operation of an ideal compressor is simple and the performance analysis has been formulated easily by the following pressure volume diagram shown in figure 2[54]. The entire cycle of operations consists of four continuous processes. Starting at point 1, air is compressed from 1 to 3 following isothermal compression(1 to 2 represents isentropic compression) where air at atmospheric pressure is compressed to reach delivery pressure, 3 to b following isobaric discharge process where is discharge to discharge plenum or chamber, b to a following isothermal expansion where air trapped inside the clearance volume at discharge pressure expands till its pressure reduction to atmospheric, and a to 1 following isobaric suction process where air from atmosphere is drawn inside the cylinder. Minimum theoretical work is required if compression and expansion following isothermal process as the area under the PV diagram is the work done. On the other hand, work required is maximum, if compression and expansion following reversible adiabatic process. So, the actual working process for compression and expansion are suppose to be in between the above to be and must be a polytropic process with index of compression/expansion between 1.0 to 1.4. These conditions discussed are only applicable to ideal condition where there is no valve or heat transfer. However, these values may rarely reach infinity also (more than 1.4) due to accountability of heat transfer.


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Figure 3: Pressure – Volume diagram: Ideal cycle without clearance (Indicator diagram).

The re-modified ideal cycle by accounting the clearance volume provided between the piston face and head to avoid collision of piston on the cylinder head, the following indicator diagram is obtained.

Figure 4: Pressure – Volume diagram: Ideal cycle with clearance volume.

Normally the clearance provided is represented as a percentage of stroke volume available. This will introduce a term called effective volume available for during expansion [21]. This will reduce the capacity of the compressor. Therefore, theoretically the capacity of the compressor will be decided based on clearance volume. Higher the fixed clearance, the lower the volumetric efficiency and volumetric efficiency can be calculated as the percentage of stroke that can be filled with suction gas and it is the cylinder capacity. The efficiency of compression related with volumetric efficiency through the piston velocity and volumetric efficiency does


not have an effect of valve open timings and normally discharge volumetric efficiency is typically two-third of the suction volumetric efficiency [28]. The actual compressor indicator diagram differs from the above due to the effects of valve opening and closing delays, inertia of valves and approximately looks like the below.

Figure 5: Pressure – Volume Diagram : Actual Cycle.

Mathematical Modelling of Reciprocating Compressors Global model depicting the behavior of reciprocating piston compressor is a challenging task which is yet to be formalized. Numerous attempts so far made in this direction, leads to the formation of global models separately for hermetic r efrigeration compressors and they were tested successfully. However these models are still lacking the characteristic phenomenon in terms of valve dynamics as in most of the cases, valve dynamics is modeled with single degree of freedom vibratory system subjected to harmonic forcing. Most global models used experimentally proven coefficients for correcting flow area, impact mechanisms, sound generation mechanisms etc. are only holding good for specific compressor models. On the contrary, mathematical models must reflect the behavior of real system modeled based on pure mathematical modeling without inputting the experimentally proven coefficients which is not yet fully described. The following discussion is based on various independent models described by various researchers. This study carried out to illustrate the necessity of development of global mathematical model for reciprocating compressors incorporating kinematics and dynamics of piston crank assembly, dynamics of valves, fluid flow interactions and heat transfer.


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Simplified Models Winnady Eric et.al[1], presented a simplified model of an open-type refrigeration reciprocating compressor which integrates the separate models for valve flow, cylinder heat transfer, bearing losses, noise proposed by Qvale E B et.al (1972)[2]. The model was developed based on experimental results and it is a simple steady-state model. The model predicted the ambient losses, but the compression was considered as reversible adiabatic process, which in reality is a continuously varying polytropic process which does not obey any polytropic at any instant of time due heat transfer from cylinder surface. However his integrated model predicted mass flow rate, shaft power, exhaust temperature and ambient losses and it needs to be integrated into a global simulation of compression cycle. It addresses the need of a global simulation compressor model. A phenomenological model was developed to reproduce the compressor efficiency and the volumetric efficiency as a function of set of parameters obtained by correlations of standard characterization of compressor performance data and used for predicting the extreme operating conditions like extreme temperature or very lower loads.[34] Thermodynamic models Thermodynamic modeling of compressors can be divided into two major categories based on the works carried out by various researchers so far. One is empirical approach based on empirical polynomials based on laboratory experiments [19-20] and the other is the theoretical approach which is the modeling of compression or expansion as isentropic[58-59]. Modeling the thermodynamic behavior of a compressor in an appropriate manner could solve almost of the problems associated with mathematical modeling. Pascal Stouffs presented a global model for the thermodynamic analysis of reciprocating compressors based on in-cylinder pressure ratio, discharge –suction pressure ratio, isentropic temperature ratios, dimensionless work terms for estimating specific work, estimating indicated efficiency etc. and concluded that in-cylinder residual mass fraction and the wall to fluid heat transfer influences the performance of the compressors.[6] Though the model was a global one, the variables are considered independent of time. There is a need to have a thermodynamic time dependent model which is a part of compressor overall system model. Similar type of work carried out by Cavallini A on development of global model for analyzing thermal behavior of hermetic reciprocating compressors [7]. Si Yung Sun developed a new method of computing thermodynamic behavior and simulated all the working processes for instantaneous values of thermodynamic parameters such as pressure, temperature, mass and enthalpy [8]. He used crank rotation as the basis for his calculation. It appeared like a good way of approaching the basic problem of globalized mathematical model with minimum assumptions. But the model proposed predicts the pressure pulsations in discharge conditions, without considering the real valve dynamics and the effect of muffler.

Kumar Subramanian et al Flow Models Flow models are grouped into either steady state or dynamic models depending upon the degree of complexity. The models developed based on first law of thermodynamics (Steady flow energy equations) are called as steady flow models [7,34]. But these models cannot capture the dynamic behavior of the system. These models considered the suction and discharge masses are equal, but it in real, compressors are having transient flow. Dynamic models are models which are depicting the transient conditions, that is unequal suction and discharge mass flow rates. A compressor dynamic model developed for on-off cycling operations of a hermetic reciprocating compressor was developed. This model is good enough in accounting the influence of suction and discharge mass flow rates and the electrical power and claimed to be good for different reciprocating compressor designs.[46]. The models has assumptions like the refrigerant mixture is thermally homogeneous and the effects of suction and discharge mufflers on the heat exchange and pressure pulsations are not accounted for emphasis, the need of a model which will account this. Valve Dynamics Models Valve is the vital and critical component of the compressor which controls the flow and the appropriate valve modeling will solve major problems causing sound and vibrations. Around 36% of the failures occurred in compressors are due to valves [24]. Various categories of valves are used in compressor like spring valves, plate valves, bob valves, ring valves, poppet valves, disc valves and reed valves [26]. Though the functioning is similar, the operating behaviors for various types are inherently different. Most present day compressors are using reed valves due to their simplicity, modular design, easy manufacturability and geometric advantage. The valve must be highly elastic and endurable as most high pressure compressors operating at the range above 3000 rpm. Ideally, valve should open early and quickly by a minimum pressure difference between cylinder pressure and head pressure, and also it should remain open without fluttering till the end of the suction or delivery process, and it should create a minimal impact on valve seat and also it should close with minimal impact. A poorly designed valve will create valve flutter, which in turn will cause multiple impacts and also will reduces the effective lift area and hence efficiency. Delayed valve closing is another effect of poor valve design, that could reduce the valve life since it is highly associated with slamming of valve against a seat, which will reduce the flow area, augment back flow and reduces the overall performance of the compressor considerably[16,23] . Most of the research carried out on compressors focusing mainly on thermodynamics and flow parts. The valve dynamics is one of the key areas where not much research been carried out. The very first work on dynamics of reed valve traces back to the doctoral research by Micheal Costagliola at Massachusetts Institute of Technology [5]. He concluded that all the automatic valves of spring loaded type acts in the same manner and the flow area is more important rather than the valve dynamics. He further inferred that dynamics of valves are influential, only if the flow area is large, otherwise its effect is not accountable to the losses. In contrast, the


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effective flow area depends on the linear behavior of valve motion, hence valve dynamics is quite important in deciding the performance of a compressor. Subsequently most of the works carried out on valve dynamics later by considering the valve as single degree of freedom vibration systems. Derek Woollatt presented a technical paper on reciprocating compressor valve design by considering valve selection and reliability as the primary criteria. His valve [9] dynamics program however, is much suitable for spring valves [3]. Rigola carried out numerical experiments over a fluid flow through the reed valve using computational fluid dynamics. He introduced a new unstructured and parallel object oriented CFD code using multi-dimensional explicit finite volume fractional step based algorithm and for turbulence modeling he used extended Yoshizawa nonequilibrium fixed parameter sub grid scale model. The solver used is Fourier Schur decomposition solver. However, he simplified the flow through valve as a flow through a simplified geometry of an axial hole plus a radial diffuser. He coined a methodology based on effective flow area that relates the actual mass flow rate with an ideal one per unit flow area by assuming isentropic contraction process and defined effective force area as the ratio of net force on valve and force on assumption of constant pressure drop. The output produced by him as pressure maps evolution and [10] velocity map profiles. Enzo Giacomelli found that pulsations generated by the compressor and piping systems specially in large compressors can have a critical impact on valve behavior and proposed computational fluid dynamics models for calculation of displacement, pressure drop and pressure pulsation based coefficients. Spring valves Helical compression springs are used as self acting valves in reciprocating compressors. These springs are subjected to dynamic and impact loading when opening and closing events. Moreover, their performance is crucial for any compressor, and their design and selection will influence the performance of the compressor. Spring selection based on stresses computed as a function o f plate impact velocity was developed using conventional theories based on work done [27]. These valves are essentially modeled as single degree of freedom vibratory system[50]. Poppet valves In the earliest compressor, poppet valves have been used and the developments made on them brought plate and ring type valves. The materials used initially is limited to steel and bronze due their weight and impact forces. Now a days valves are using metallic materials or thermoplastics depending upon the speed [57]. Nowadays PEEK (PolyEtherEtherKetone) a high performance engineering plastic is used due to high temperature resistance without permanent loss in physical properties. Reed valves Reed valves are simplest of the valves used so far, yet they are low cost and efficient. Reed valve is a thin strip of metal used a flapper to cover holes which govern the flow during suction and discharge. It is fastened at one end and free at the other end. By the application of gas forces, it will travel and create a flow area to gases to escape in and


out. Reeds are made of carbon steel or stainless steel. The carbon steel is martensitic and has the composition C 0.37, Si 0.39,Mn 0.6, Cr 13.5, Mo 1, P 0.02, S 0.002 and for stainless steel C 0.38, Si 0.4, Mn 0.55, Cr 13.5, Mo 1 and it is heat treated to martensitic condition. This is due to the requirement of higher fatigue strength(number of cycles the valve oscillates is quite high)[22]. The optimum design of a reed valve typically based on machine dynamic analysis [23]. The first work on reed valve dynamics traced back to 1950s,a doctoral thesis submitted by Costagliola. He suggested two different models, one is to consider the valve as a spring, and the other is to consider as a beam. [15]. Though reed valve mechanism is treated as the motion of cantilever beam and the displacement is calculated using the mode superposition of higher influencing natural frequencies that is the first five, it is also an approximate. Mathematically it can be treated as plate vibration in its first mode and shell vibration from its second mode onwards as the shape of the valve will change. Interestingly this structural analysis considering the valve vibration will not help us to understand the valve behavior fully, because valve is in continuous contact with fluid with pressure variations on both sides and during the valve motion also, fluid particles flowing over the valve will create thermal boundary convection. So it is necessary to study the behavior of valve not only based on the structural point of view, look into fluid structure interaction point of view [9-10, 25]. The behavior of reed valves either in compressors or in internal combustion engines are same and lot of research carried out on engine valves can also be accounted for the development of reed valve dynamics. Use of finite element solvers coupled with computational fluid dynamics programs is basis for fluid structure interaction problems computationally. [51-53].Fluid structure interaction analysis of reed valve was presented by coupling the user defined finite element model with computational fluid dynamic models and the finite element packages like ADINA (Advanced dynamic incremental non-linear dynamics), ANSYS – CFD Flow are employed with such a provisions. However, for a global mathematical modeling, interfacing with these packages would be difficult and cumbersome task. [55-56]. Piston Dynamics Models Piston is the primary component which imparts energy to the system in all reciprocating compressors, therefore piston dynamics plays a vital role, in deciding the performance of any compressors. Piston dynamics in reciprocating compressor or engine are almost similar. In the literatures analyzed in both the compressors and engines, the first work was traced back on theoretical attempt to explore piston lubrication through an analytical model explaining the piston skirt friction [29]. A numerical model on piston slap in diesel engines exploring the effects of arbitrary skirt surfaces and pin eccentricity was developed and successively, a model has been developed based on four stroke engines by considering piston motion, lubrication and friction taken into account of surface waviness, roughness, surface profile, bulk deformation, thermal distortion and proved the influence of hydrodynamic lubrication on piston impact and frictional loss.[30-31] . A numerical study of instantaneous heat transfer during compression and expansion in piston cylinder geometry was carried out based on model developed earlier on internal combustion engine piston dynamics


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[32] and subsequently a dynamic model for piston lubrication for small reciprocating compressors was developed based on Newton-Raphson procedure [33]. For the concern of the modeling of reciprocating compressor, specially look in to the effect of heat transfer between the piston and gases in contact with the piston surface, the proper hydrodynamic lubrication will certainly reduces the effect of eddy currents developed and an eddy viscosity turbulence models based on large eddy simulation [9] can be modified based the piston dynamics. Effect of Polytropic Index of compression and expansion: One of the major areas of concern in the modeling of the combined heat and mass transfer is the prediction of polytropic index. As far as the compressor is concerned, the theoretical work done(least work required per cycle) will be calculated by considering isothermal compression and expansion which is no way possible and the maximum work is required when the compression and expansion is reversible adiabatic or isentropic. For an actual compressor cycle, both compression and expansion are polytropic with index varying between 1 to 1.4. This is true by only without considering the effect of heat transfer. For an example, during the initial part of the expansion, the gas temperature is higher the wall temperature of the cylinder therefore heat will transfer from gas to cylinder wall, but after sometime, the gas temperature will fall below the wall temperature and wall will transfer heat to gas. By accounting these effects, the index of expansion or compression will be continuously varying one. So for our convenience, modeling with the fixed value of index of compression will leads to unpredictable changes in the other parameter values. This is one area where the researchers are not in a position to identify the specific formulation. The sensitivity analysis of polytropic index on compressor performance was carried out by Campell & Company (2009) and they produced a model for corrected polytropic index based on statistically averaging the values like weighted average value, based on discharge temperature and pressure, suction temperature and pressure, average temperature and pressure, etc. and studied the performance characteristics like adiabatic head pressure, temperature at any point of time etc. for deviation in polytropic index. However, engineering conclusion regarding the variation of polytropic index for the sensitive compressor performance was not made. Heat transfer Modeling Modeling cylinder heat transfer process and coupled with the cylinder processes are quite important for determining the actual performance of the compressor. Lot research work has been carried to determine the in-cylinder heat transfer processes of reciprocating machines.[35-40] They can be considered as a reference for developing the heat transfer modeling for reciprocating compressor also as functionally both are working as same. Summary of the work provides a vital information about the convective heat transfer correlations. The simplified modeling of cylinder heat transfer process can be formulated by considering the behavior turbulent flow of gases in pipes and over plates by idealizing cylinder as a pipe and valves as plates. This is useful in calculating instantaneous heat transfer coefficients based on Nusselt number,


Prandtl number and Reynolds number. Experimental investigations of heat transfer in high pressure reciprocating gas compressor followed by 95% confidence model was developed for investigating heat transfer characteristics of two stage gas compressor and correlations for finding out heat transfer coefficients were presented. [41]. Lot of analogous research work on reciprocating engines, all proved that the heat transfer from the cylinder walls to the fluid is an important process in deciding the overall performance because it decreases the cylinder temperature and pressure which will affect the indicated efficiency and decreasing the work transferred on the piston in each cycle.[44-45]. Many researchers in the recent past, used forced convection with internal flow models for depicting the cylinder heat transfer towards the fluid inside and it appears to be in agreement with the real process involved[47- 49] . Expressions based on the above are good to be considered as a part of global model. Many researchers used computational fluid dynamics as a tool to explore the in-cylinder heat transfer characteristics, but in global mathematical modeling, it is very difficult to include CFD models, as the calculations are too complex to be a part of mathematical modeling.

Computer Simulation Computer simulation is a computer program that can simulates an abstract model that can be studied and analyzed further. They are useful to estimate the performance of a complex mechanical system. A mathematical model basically aimed for predicting the behavior of a system with a set of initial conditions for the parameters involved can be converted into a complete computer program where we can play around the parameters to gain insight knowledge of a robust system. For engineering models, the usual way is to develop an algorithm which can be coded as a program. Higher level languages like C++, Java, Ch Language are called as object oriented programming languages which are having higher level flexibility due to their object orientation in nature. They can be successfully integrated with Excel like office programs or with higher end finite element packages like ANSYS, Nastran etc to analyse their behavior. Otherway around is mathematical packages like Matlab, Mathematica where we can develop a program and with least effort we can get the performance the characteristics. Many of the computer simulations developed based on the mathematical models are done with the help of Fortran 90 which is widely considered as an engineering programming language. Some of them were developed with the help of C language environment, a universal language which is a successful alternate to Fortran 90. Also attempts were made to effectively integrate the object oriented programming techniques which will reduce the level of complexities of interlinking the various independent and interdependent models.


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Conclusions Mathematical modeling of reciprocating compressors considering the various factors affecting the performance of it is reviewed. Based on the papers reviewed, the research needs to be focused on the global mathematical model which should include the complex behaviors like heat transfer, valve dynamics, piston dynamics, flow characteristics, thermodynamics. The level of modeling requires a compromise between the resources like computational methodology, feasible mathematical models, coupling between various models. The expected challenge in this process is the effective experimentation to check the validity of the model over the range of compressors with the variations at small crank angles.

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mathematical modeling of compressors

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