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Improving torque performance in reciprocating compressors via asymmetric stroke characteristics
CHAPTER 6
Improving torque performance in reciprocating compressors via asymmetric stroke characteristics Ibrahim A. Sultan, Truong H. Phung School of Science, Engineering and IT, Federation University Australia, Ballarat, VIC, Australia
Contents Introduction A mathematical model for the compressor cycle Torque performance in reciprocating compressor Example alternative compressor drive Design example Discussions and conclusions References
145 147 151 153 157 159 161
Introduction Reciprocating compressors present a well-established technology, which offers a superior sealing performance that enables them to achieve high delivery pressures at low flow rates, as many industries require. Indeed, this class of compressors gains more popularity every day as new investments are directed towards gas transmission pipelines, petrochemical plants, refineries and many other industries and manufacturing ventures. As such, market analysis reports tend to assign almost half the positive displacement compressor market to reciprocating compressors and the other half to the rotary type machines. This is also reflected in the body of literature available on the topic of positive displacement compressors. Whilst some papers have been published to offer rotary novel alternatives to the reciprocating designs, most papers are written to discuss the mechanical and thermal performance of reciprocating compressors. Examples of the rotary alternatives are found in the work by Rukanskis (2017) on the rotary vane or the paper by Tarel’nik, Konoplyanchenko, Kosenko, and Martsinkovskii (2017) on screw compressors. Other rotary machines such as the scroll Positive Displacement Machines https://doi.org/10.1016/B978-0-12-816998-8.00006-6
© 2019 Elsevier Inc. All rights reserved.
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compressors have been discussed in the excellent work by Chen, Braun, and Groll (2004) and the seminal paper by Sultan (2005) on the limac¸on machine. Ooi (2005) offers a great insight into the workings of the rolling piston compressor. Despite their popularity, reciprocating compressor are costly to maintain as Almasi (2009) explains and as such their reliability is of particular importance as discussed by Bassetto, Neto, and de Souza (2009). As part of the effort to ensure reliable delivery by reciprocating compressors, a good number of papers have been published on their dynamical performance where forces acting on various members were calculated under the assumptions of manufacturing errors and severe operating conditions emitting from heat, pressure, sliding friction and valve vibration. A good example is found in the paper by Xiao et al. (2017) who studied the effects of joint clearance and stroke line offset on the dynamic behaviour of the piston motion. On the other hand, Tsuji et al. (2010) modeled the dynamical forces in the compressor drive and used these to estimate the frictional mechanical losses in reciprocating compressors. The method of Finite Element Analysis (FEA) was employed by Cho and Moon (2005) to study the pressure distribution in the oil film around the piston. The major compressor components which influence the overall performance are the inlet and discharge valves which are subject to severe service conditions under the effect of speed and interaction with gasses, lubrication oil and impact loading from surroundings. As such, the objectives of research effort conducted on the valves are often two-fold, to improve overall reliability and optimize thermodynamic performance. Bhakta, Dhar, Bahadur, Angadi, and Dey (2013) investigated the effects of valve flutter on compressor efficiency with the help of modeling, CFD simulation and experimental work. An interesting work was published by Tang et al. (2013) to model the valve dynamics with and without the use of actuating mechanisms. Besides the mechanical aspects, researchers also employed mathematical modeling to describe the thermodynamic characteristics of the reciprocating compressor in order to understand how design parameters impact various performances indices. For example, Ndiaye and Bernier (2010) developed a mathematical model which could predict both the transient and steady state performance of compressors with a level of accuracy. Indeed, cooling is a factor which impacts compressor thermal performance considerably. Thermodynamically, the best performance of a reciprocating compressor corresponds to the hypothetical isothermal
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compression process. Researchers in this specific domain endeavor to realize conditions as close as possible to this process. Kremer, Barbosa, and Deschamps (2012) present a good example of this effort as they offer a good insight into the cooling performance in reciprocating compressor and present and elaborate mathematical model to describe the process of oil atomization in the work medium. On the other hand, Coney et al. (2002) present a great effort to obtain near isothermal performance by injecting water in the compression cylinder and separating this afterwards from the resulting two-phase flow. This chapter is intended to show that it is possible to obtain a considerable improvement in compressor performance and reliability by modifying the design of the compressor drive. Instead of the traditional slider-crank mechanism, which has been used as a drive since the compressor industry started, it is now possible to replace this drive by more sophisticated ones which offer more flexible stroke design. Such replacement has been made possible by the modern manufacturing methods and analysis tools which are currently at the disposal of industry and research bodies. Earlier work in the domain of proposing new linkages to drive positive displacement machines can be found in the work by Ooi and Wan (2000) who presented a proposal to drive the compressor piston directly from the back using a camlike mechanism. Also, Mimmi and Pennacchi (2001) proposed a pump drive that produces four variable-length strokes per crank revolution. On the other hand, Sultan and Kalim (2011) presented a five-bar geared linkage which can be used to drive a reciprocating compressor instead of the conventional slider-crank mechanism. The next section will present a mathematical model for the compressor operation.
A mathematical model for the compressor cycle Fig. 1 illustrates some important geometric and thermodynamic characteristics of a reciprocating compressor. To facilitate reading of the mathematical model presented below, the following list of variable definitions is provided: θ θo X Xmin and Xmax
the angular position of the compressor crankshaft as measured from a given datum the value of θ at which the cycle starts the linear position of the piston as measured from a given extreme stationary location the minimum and maximum values of X, respectively.
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Fig. 1 Reciprocating compressor geometry and fluid properties. Pi, Ti, and respectively, pressure, temperature, and density of fluid in the inlet manifold. ρi Pc, Tc and respectively, pressure, temperature and density of fluid in the working ρc chamber. Po, To and Respectively, pressure, temperature and density of fluid in the outlet ρo manifold. area of the inlet valve opening Ai Ao area of the discharge valve opening mi mass admitted to the control volume mo mass discharged discharge coefficient at the inlet valve Cdi Cdo discharge coefficient at the discharge valve
By virtue of the mechanical linkage employed to drive the compressor, the crank angular position, θ, determines the instantaneous position, X, of the piston as follows: X ¼ X ðθÞ : θ 2 ½θo , θo + 2π
(1)
The exact mathematical formulation of the functional relationship symbolized in Eq. (1) depends on the kinematical characteristics of the compressor drive. However, without loss of generality, it is possible to assume the maximum value of X occurs at the start of the compressor work cycle; i.e. Xmax ¼ X(θo). As such, the instantaneous volume, Vc, of the compressor working chamber can be calculated as follows: Vc ¼ Vo + Ap ðXmax X Þ
(2)
where Vo is the minimum value of the volume, and Ap is the cylinder crosssectional area. The rate, Vc0 , at which the volume changes with respect to the crank angle is given as, Vc0 ¼ Ap
dX dθ
(3)
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During the compressor operation, the conservation of mass in the control volume is expressed by the following differential equation, dρc 1 dV c ðm_ i m_ o Þ ωρc ¼ (4) dθ ωV c dθ where ( _ ) signifies differentiation with respect to time and ω is the angular velocity of the crank. The inlet mass flow rate can be calculated by the following expressions, 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 1 Pi Pc Pc > > < ρc Ai Cdi if > rc Pi 2 ρi ρc m_ i ¼ (5) > p ffiffiffiffiffiffiffiffiffiffi ffi P > c > : ρc Ai Cdi γRTc if rc Pi where R is the gas constant for the work medium, γ is its specific heat ratio, and rc is the critical pressure ratio. The outlet mass flow rate can be calculated as follows: 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > 1 P P Po > c o > < ρo Ao Cdo if > rc Pc 2 ρc ρo m_ o ¼ (6) > p ffiffiffiffiffiffiffiffiffiffi ffi P > o > ρ Ao Cd o γRTo : if rc o Pc The inlet and discharge valves control the flow to and from the cylinder. With the help of these valves, the sliding motion of the piston results in the chamber pressure, Pc, changing in response to variations in the volume in order for the compressor to perform its intended function. Assuming adiabatic process, the energy balance in the control volume is given by, dH i ¼ dH o + dU c + Pc dVc
(7)
where Hi and Ho signify the enthalpies flowing in and out of the control volume respectively and Uc refers to the internal energy of the fluid inside the control volume. The energy balance equation can be manipulated to various equivalent forms (e.g., Chen et al. (2004) and Ooi (2005)). In this chapter, the following simplified differential form of the energy balance equation is adopted for the analysis (Sultan & Kalim, 2011), dP c γ dV c RðTi m_ i Tc m_ o Þ ωPc ¼ (8) dθ ωV c dθ where Tc is calculated from the Equation of State at every step.
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To calculate the areas of the valve openings indicated in Eqs (5) and (6), the simplified dynamic model presented by Sultan and Kalim (2011) is adopted here (Fig. 2). To calculate the instantaneous valve areas (i.e., Ai and Ao featured in Eqs (5) and (6)), the approach followed Sultan and Kalim (2011) is adopted here with the below definitions: Yv δv Lv Mv Kv Cv Kseat and Cseat Kstop and Cstop
the instantaneous spring deflection, which signifies the valve position initial deflection in the valve spring maximum allowable spring deflection, which signifies the valve lift equivalent mass of the valve and attachments the stiffness of the valve spring equivalent damping coefficient in the valve guides equivalent stiffness and damping coefficient of the valve seat equivalent stiffness and damping coefficient of the valve stop
Based on these definitions, the expressed as follows: 8 < fv1 2 dY v 1 fv2 2 ¼ Mv ω2 : dθ fv3
valve equation of motion can be for δv Yv Lv + δv for Yv < δv for Yv > δv + Lv
(9)
where the various functions in [9] are given as follows: dYv Kv Y v fv1 ¼ ΔPv Av Cv ω dθ dYv (10) fv2 ¼ ΔPv Av ðCv + Cseat Þω ðKv + Kseat ÞYv + Kseat δv dθ dYv fv3 ¼ ΔPv Av Cv + Cstop ω Kv + Kstop Yv + Kstop ðδv + Lv Þ dθ where ΔPv is the pressure differential across the valve and Av is the valve area exposed to that pressure differential. The instantaneous area of the valve port
Fig. 2 Modeling the dynamic behaviour of the compressor valves.
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(i.e., Ai or Ao given in Eqs 5 and 6) which is available for fluid flow, is calculated using the following expression which is simplified from the formula presented by Tuymer and Machu (2001); vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 u A¼u (11) 2 2 ! u 1 1 t + 0:85πDv Yv Av where Dv is the diameter of the valve seat. The complex multi-physical model which spans from Eqs. (2) to Eq. (11) can be solved in a cyclical fashion from θ ¼ θo to θ ¼ θo + 2π until the thermodynamic properties inside the control volume at the start of the cycleare sufficiently close to their corresponding values at the end of the cycle.
Torque performance in reciprocating compressor Thermodynamically, the infinitesimal work, dw, required to drive the compressor is given as follows: dw ¼ Pc dVc
(12)
Mechanically, the work required at the crankshaft to drive the compressor is provided via the torque, τ, where the relationship between the torque and the work is given as follows: dw ¼ τdθ
(13)
As such, considering the comparative nature of this chapter, we overlook the effects of mechanical losses and surrounding pressure, the shaft torque can be calculated from the following expression, τ ¼ Pc Vc0
(14)
A typical torque curve is shown in Fig. 3 where the variable torque acting on the crankshaft is plotted against its angular position. This graph is of special importance to the design of the compressor mechanical components. The mean torque, τ, acting on the crank is calculated as follows: θo + ð 2π
τdθ τ¼
θo
2π
(15)
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Fig. 3 A typical torque against crank angle curve for a reciprocating compressor.
The area that falls above the mean torque line, τ, and below the torque curve, τ, signifies the maximum energy variation, ΔEmax, in the compressor flywheel which can be calculated as follows: θð2
ΔEmax ¼
τdθ ðθ2 θ1 Þτ
(16)
θ1
where θ1 and θ2 are the angles at which τ intersects τ. The value of ΔEmax determines the size of the flywheel as per the following equation, If ¼ Cf ΔEmax
(17)
where If is the flywheel mass moment of inertia and the value of Cf depends on the speed levels stipulated at the crankshaft. Moreover, the torque characteristics also impact the size of the mechanical components of the compressor. For stress calculations, the alternating torque, τa, is used together with the midrange torque, τm. The alternating torque is calculated as follows: τ τ max min τa ¼ (18) 2 The midrange torque is calculated as follows: τmax + τmin (19) 2 where τmax is the peak value of the τ curve and τmin is its minimum value. τm ¼
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Stress-based design for fatigue loading is discussed in a number of textbooks, e.g. Budynas and Nisbott (2008). The design process employs the loads and the part dimensions to calculate various stresses acting at points of concern on a machine component. For the purpose of the current discussion, it suffices to point out that the factor of safety, nf, calculated at a point of interest may boil down to the following generalized form; nf ¼
1 Ka τ a + Km τ m
(20)
where Ka and Km signify mathematical expressions which feature the material properties and the component geometric characteristics. Based on the above discussion, it would be favorable to modify the characteristics of the piston strokes such that the extreme values of the torque curve, τmin and τmax, are curtailed. This would be achieved by modifying the timing of the strokes such that the extreme values of the rate Vc0 featured in Eq. (14) are reduced. This reduction is useful particularly towards the end of the compression stroke where the chamber pressure has already grown well above the inlet value. The modified torque curve would also feature a decreased ΔEmax which would subsequently result in a reduced flywheel size as per Eq. (17). Moreover, conducting compression over an extended portion of the cycle time would present favorable conditions for the cooling system to pick up more heat from the compressed fluid and thus reduce the temperature levels in the systems.
Example alternative compressor drive Conventionally, reciprocating compressors are driven by the well-known slider-crank linkage which is shown in Fig. 4. The position of the piston as measured from the crank pivot along the stroke line is given as, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 (21) X ¼ r cos θ + L 1 sinθ L
Fig. 4 The slider-crank mechanism.
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where r is the crank length, L is the length of the connecting rod and the angle θ is measured from the stroke line to the crank as shown in Fig. 4. Figs 5 and 6, respectively, depict Vc and Vc0 against the crank angle, θ. For the conventional linkage, these relationships are symmetric about θ ¼ π
Fig. 5 A typical symmetric Vc θ curve produced by the conventional drive.
Fig. 6 A typical symmetric Vc0 θ curve produced by the conventional drive.
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which confirms the symmetry of the stroke characteristics offered by the use of that linkage. It is possible to modify the workings of the conventional slider-crank compressor by offsetting the sliding line, a certain distance, below the pivot of the input crank. However, this does not provide an effective handle to control the various geometric characteristics of the piston stroke in a way which yields the desired compressor performance. A more versatile compressor drive is shown in Fig. 7 below. As shown in the figure, the piston is made to slide along a line which falls a distance h below the pivot o and makes an angle ϕ with the x direction. The position, X, of the piston along the axis of sliding is given by the following equations, X ¼ ð2r sinθ + L Þcos ðθ ϕÞ + hsin ϕ + Lc cosσ
(22)
where Lc is the length of the connecting rod and σ is its angle with the sliding direction. The angle σ can be obtained from the following relationship, ð2r sinθ + L Þ sin ðθ ϕÞ + h cos ϕ ¼ Lc sinσ
(23)
Solving (22) and (23) together leads to the following expression for Xp, X ¼ ð2r sinθ + L Þ cos ðθ ϕÞ + h sinϕ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð2r sinθ + L Þ sin ðθ ϕÞ + hcos ϕ 2 + Lc 1 Lc
Fig. 7 An alternative mechanical linkage to drive the compressor.
(24)
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The maximum and minimum values, Xmin and Xmax, of X are found by calculating the roots of the following equation and substituting these roots in Eq. (23), dX ¼0 (25) dθ The piston stroke, S, can, be found from the following equation, S ¼ Xmax Xmin
(26)
As suggested by Sultan and Kalim (2011), the torque performance of the drive is quantified by ητ, which is given as follows: ητ ¼ 1
τmax τ τmax + τ
(27)
The purpose of Eq. (27) is to judge how close τmax is to the mean torque, τ. A small difference between the two torque values suggests a smoother operation with a reduced flywheel size and smaller stresses in the various components of the drive. The design process which is used to find the optimum dimensions of the drive is given in the next section. In the design process, the user sets such parameters as, the pressure levels at the inlet and outlet manifolds, the shaft speed, the required stroke. The process features the use of an optimization procedure and results in the dimensions of the linkage found as shown in Fig. 7. The design vector, Ζ, which is employed to optimize compressor drive is given as follows: 2 3 r 6 L 7 6 7 6 7 Ζ ¼ 6 Lc 7 (28) 6 7 4 h 5 ϕ The cost function, Fc(Z), which is required to be minimized for optimum performance is given as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fc ðZ Þ ¼ wτ ð1 ητ Þ2 + ws ð1 ηs Þ2 (29) where ηs is factor which determines the accuracy of the obtained stroke as given below: ηs ¼
Sk Sr
(30)
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where Sr is the required stroke and Sk is the stroke obtained at iteration step number k. In Eq. (29), wτ and ws signify weighting factors used to reflect the relative importance of the parameters used in Eq. (29) for a specific design problem. The optimization problem can now be presented as follows: min
Zmin ZZmax
Fc ðZ Þ
(31)
where the entries of the constraint vectors, Zmin and Zmax, are set by the user to meet the requirements of a given problem. The iterative search for the optimized solution is conducted using randomized cosine functions as follows: Zik ¼ Zik1 + Δki cos ψ ki π (32) where the various variables are explained as follows: Zki is element number i in the Z vector at iteration step number k, Δki signifies the difference between the current value of Zk1 and its near i constraint. This variable is calculated as follows: ( Δki
¼
Zimax Zik1 if Zimax Zik1 Zik1 Zimin k1 Zi Zimin otherwise
(33)
The value of ψ ki is assigned randomly between 0 and 1 at every iteration for each entry number i of the Z vector. The solution, Zk, is accepted only if Fc(Zk) < Fc(Zk1), otherwise, the iteration is repeated for a set number of times.
Design example For the example featured here, the following parameters are set. • Shaft speed ¼ 800 rpm • Po ¼ 8 bar, Pi ¼ 1 bar, and Ti ¼ 303 K • Sr ¼ 100 mm The iterative procedure explained above resulted in reducing the cost function from 63.3 to 3.4 as shown in Fig. 8. The stroke at the start of iterations was 413.9 mm and this had to be iteratively reduced to 101.3 mm at the end of the iterations. As such, the stroke error dominated the optimization calculations.
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Fig. 8 Reduction of cost function with iterations.
The thermodynamic model detailed above was used to investigate the performance of two compressors, one driven by the optimum compressor and one driven by the conventional drive. The maximum energy variation, ΔEmax, in the flywheel was found to be 251 J in the conventional drive and 226.7 J for the optimum drive. This will result in about 10% in the flywheel mass. Also, the maximum torque is 360 Nm and minimum torque is 113 Nm in the conventional drive. The corresponding values in the new drive are 273.3 Nm and 87.7 Nm respectively. As such, the torque values used for stress calculations in the conventional drive are as follows: τm ¼ 123:5 Nm and τa ¼ 236:5 Nm: For the optimum drive, the corresponding values are given as τm ¼ 92.5 Nm (i.e. 25% reduction) and τa ¼ 180.5 Nm (which offers 23.7% reduction). The new torque levels would make it possible to reduce the sizes of various parts in a way which motivates more sustainable use of material and manufacturing resources at the design end. The following graphs, Figs 9–12, offer an insight into how various process parameters vary during the work cycle for both the conventional and the optimum compressors.
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Fig. 9 Pressure against angle of conventional and optimum compressors.
Fig. 10 Torque against angle of conventional and optimum compressors.
Discussions and conclusions Despite their immense popularity, reciprocating compressors still use the traditional slider-crank drives they have always used since the early years of compressed air industry. The current push for more sustainable use of material and resources combined with the modern manufacturing and analytical tools
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Fig. 11 The profile of chamber volume.
Fig. 12 The rate of change of chamber volume.
should motivate the search for a modern more sophisticated drives more capable of improving compressor performance. In this chapter, a candidate linkage has been proposed and utilized in a mathematical model to describe the workings of the compressor. An optimization technique has been employed to obtain the dimensions of the linkage which would achieve an improved torque curve for the compressor without impacting the required pressure or mass flow rate performance. The model demonstrated takes into consideration the
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dynamics of the compressor valves in form of a nonlinear single degree of freedom vibrating system. The example presented proved the validity of the proposed linkage and its ability to reduce the flywheel size by 10% and the stresses in components by more than 23%. These reductions would result in a more reliable operations and it would be possible to reduce part size and achieve savings in relation to material consumption and manufacturing cost. Moreover, due to the fact that the compression stroke occurs over a longer period of time, it is expected that the cooling performance is expected to be improved resulting in improved thermal efficiency. However, the heat transfer analysis was not included in the mathematical model featured in the chapter. Moreover, the modified stroke is likely to impact the lubrication characteristics of the compressor. Future studies in this domain will feature the inclusion of the heat transfer analysis and lubrication modeling in the mathematical presentation of the compressor in order for these phenomena to be better understood in asymmetric stroke situations.
References Almasi, A. (2009). Advanced Technologies in Reciprocating Compressor with respect to performance and reliability. In 5th International Advanced Technologies Symposium. Bassetto, I., Neto, A., & de Souza, G. (2009). Reliability analysis in reciprocating compressors for refrigeration systems. HVAC&R Research, 15(1), 137–150. https://doi.org/ 10.1080/10789669.2009.10390829. Bhakta, A., Dhar, S., Bahadur, V., Angadi, S., & Dey, S. (2013). A valve design methodology for improved reciprocating compressor performance. In International Compressor Engineering Conference (Paper 2077). Budynas, R. G., & Nisbott, J. K. (2008). Shigley’s mechanical engineering design (8th ed.). Singapore: McGraw Hill. Chen, Y., Braun, J., & Groll, E. (2004). Modelling of hermetic scroll compressors: Model development. HVAC&R Research, 10(2), 129–152. https://doi.org/10.1080/ 10789669.2004.10391096. Cho, J., & Moon, S. (2005). A numerical analysis of the interaction between the piston oil film and the component deformation in a reciprocating compressor. Tribology International, 38(5), 459–468. https://doi.org/10.1016/j.triboint.2004.10.002. Coney, M. W., Stephenson, P., Malmgren, A., Linnemann, C., & Morgan, R. E., Richards, R. A., Abdalla, H. (2002). Development of a reciprocating compressor using water injection to achieve quasi-isothermal compression. In International Compressor Engineering Conference (Paper 1508). Kremer, R., Barbosa, J., Jr., & Deschamps, C. (2012). Cooling of a reciprocating compressor through oil atomization in the cylinder. HVAC&R Research, 18(3), 481–499. https:// doi.org/10.1080/10789669.2012.646571. Mimmi, G., & Pennacchi, P. (2001). A special type of crank mechanism with variable stroke. Journal of Mechanical Design, 123(3), 468–472. https://doi.org/10.1115/1.1376397. Ndiaye, D., & Bernier, M. (2010). Dynamic model of a hermetic reciprocating compressor in on–off cycling operation (abbreviation: Compressor dynamic model). Applied Thermal Engineering, 30(8–9), 792–799. https://doi.org/10.1016/j.applthermaleng.2009.12.007.
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Ooi, K. (2005). Design optimization of a rolling piston compressor for refrigerators. Applied Thermal Engineering, 25(5–6), 813–829. https://doi.org/10.1016/j.applthermaleng. 2004.07.017. Ooi, K., & Wan, T. (2000). A rotaprocating compressor. In International Compressor Engineering Conference (Paper 1461). Rukanskis, M. (2017). Vane friction and Wear influence on rotary vane compressor efficiency and operation: research and analysis review. Agricultural Engineering, 49, 1–12. Sultan, I. (2005). The Limac¸on of Pascal: Mechanical generation and utilization for fluid processing. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 219(8), 813–822. https://doi.org/10.1243/095440605x31698. Sultan, I., & Kalim, A. (2011). Improving reciprocating compressor performance using a hybrid two-level optimisation approach. Engineering Computations, 28(5), 616–636. https://doi.org/10.1108/02644401111141046. Tang, B., Zhao, Y., Li, L., Wang, L., Liu, G., Yang, Q., et al. (2013). Dynamic characteristics of suction valves for reciprocating compressor with stepless capacity control system. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, 228(2), 104–114. https://doi.org/10.1177/0954408913477784. Tarel’nik, V., Konoplyanchenko, E., Kosenko, P., & Martsinkovskii, V. (2017). Problems and solutions in renovation of the rotors of screw compressors by combined technologies. Chemical and Petroleum Engineering, 53(7–8), 540–546. https://doi.org/10.1007/ s10556-017-0378-7. Tsuji, T., Ishii, N., Anami, K., Sawai, K., Hiwata, A., Morimoto, T., et al. (2010). Fundamental optimal performance design guidelines for off-set type reciprocating compressors to maximize mechanical efficiency. In International Compressor Engineering Conference, Paper 2009. Tuymer, W. J., & Machu, E. H. (2001). Compressor valves. In P. C. Hanlon (Ed.), Compressor Handbook (pp. 20.1–20.29). USA: McGraw-Hill. Xiao, S., Liu, S., Cheng, S., Xue, X., Song, M., & Sun, X. (2017). Dynamic analysis of reciprocating compressor with clearance and subsidence. Journal of Vibroengineering, 19(7), 5061–5085. https://doi.org/10.21595/jve.2017.18771.
Improving torque performance in reciprocating compressors via asymmetric stroke characteristics Ibrahim A. Sultan, Truong H. Phung School of Science, Engineering and IT, Federation University Australia, Ballarat, VIC, Australia
Contents Introduction A mathematical model for the compressor cycle Torque performance in reciprocating compressor Example alternative compressor drive Design example Discussions and conclusions References
145 147 151 153 157 159 161
Introduction Reciprocating compressors present a well-established technology, which offers a superior sealing performance that enables them to achieve high delivery pressures at low flow rates, as many industries require. Indeed, this class of compressors gains more popularity every day as new investments are directed towards gas transmission pipelines, petrochemical plants, refineries and many other industries and manufacturing ventures. As such, market analysis reports tend to assign almost half the positive displacement compressor market to reciprocating compressors and the other half to the rotary type machines. This is also reflected in the body of literature available on the topic of positive displacement compressors. Whilst some papers have been published to offer rotary novel alternatives to the reciprocating designs, most papers are written to discuss the mechanical and thermal performance of reciprocating compressors. Examples of the rotary alternatives are found in the work by Rukanskis (2017) on the rotary vane or the paper by Tarel’nik, Konoplyanchenko, Kosenko, and Martsinkovskii (2017) on screw compressors. Other rotary machines such as the scroll Positive Displacement Machines https://doi.org/10.1016/B978-0-12-816998-8.00006-6
© 2019 Elsevier Inc. All rights reserved.
145
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compressors have been discussed in the excellent work by Chen, Braun, and Groll (2004) and the seminal paper by Sultan (2005) on the limac¸on machine. Ooi (2005) offers a great insight into the workings of the rolling piston compressor. Despite their popularity, reciprocating compressor are costly to maintain as Almasi (2009) explains and as such their reliability is of particular importance as discussed by Bassetto, Neto, and de Souza (2009). As part of the effort to ensure reliable delivery by reciprocating compressors, a good number of papers have been published on their dynamical performance where forces acting on various members were calculated under the assumptions of manufacturing errors and severe operating conditions emitting from heat, pressure, sliding friction and valve vibration. A good example is found in the paper by Xiao et al. (2017) who studied the effects of joint clearance and stroke line offset on the dynamic behaviour of the piston motion. On the other hand, Tsuji et al. (2010) modeled the dynamical forces in the compressor drive and used these to estimate the frictional mechanical losses in reciprocating compressors. The method of Finite Element Analysis (FEA) was employed by Cho and Moon (2005) to study the pressure distribution in the oil film around the piston. The major compressor components which influence the overall performance are the inlet and discharge valves which are subject to severe service conditions under the effect of speed and interaction with gasses, lubrication oil and impact loading from surroundings. As such, the objectives of research effort conducted on the valves are often two-fold, to improve overall reliability and optimize thermodynamic performance. Bhakta, Dhar, Bahadur, Angadi, and Dey (2013) investigated the effects of valve flutter on compressor efficiency with the help of modeling, CFD simulation and experimental work. An interesting work was published by Tang et al. (2013) to model the valve dynamics with and without the use of actuating mechanisms. Besides the mechanical aspects, researchers also employed mathematical modeling to describe the thermodynamic characteristics of the reciprocating compressor in order to understand how design parameters impact various performances indices. For example, Ndiaye and Bernier (2010) developed a mathematical model which could predict both the transient and steady state performance of compressors with a level of accuracy. Indeed, cooling is a factor which impacts compressor thermal performance considerably. Thermodynamically, the best performance of a reciprocating compressor corresponds to the hypothetical isothermal
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compression process. Researchers in this specific domain endeavor to realize conditions as close as possible to this process. Kremer, Barbosa, and Deschamps (2012) present a good example of this effort as they offer a good insight into the cooling performance in reciprocating compressor and present and elaborate mathematical model to describe the process of oil atomization in the work medium. On the other hand, Coney et al. (2002) present a great effort to obtain near isothermal performance by injecting water in the compression cylinder and separating this afterwards from the resulting two-phase flow. This chapter is intended to show that it is possible to obtain a considerable improvement in compressor performance and reliability by modifying the design of the compressor drive. Instead of the traditional slider-crank mechanism, which has been used as a drive since the compressor industry started, it is now possible to replace this drive by more sophisticated ones which offer more flexible stroke design. Such replacement has been made possible by the modern manufacturing methods and analysis tools which are currently at the disposal of industry and research bodies. Earlier work in the domain of proposing new linkages to drive positive displacement machines can be found in the work by Ooi and Wan (2000) who presented a proposal to drive the compressor piston directly from the back using a camlike mechanism. Also, Mimmi and Pennacchi (2001) proposed a pump drive that produces four variable-length strokes per crank revolution. On the other hand, Sultan and Kalim (2011) presented a five-bar geared linkage which can be used to drive a reciprocating compressor instead of the conventional slider-crank mechanism. The next section will present a mathematical model for the compressor operation.
A mathematical model for the compressor cycle Fig. 1 illustrates some important geometric and thermodynamic characteristics of a reciprocating compressor. To facilitate reading of the mathematical model presented below, the following list of variable definitions is provided: θ θo X Xmin and Xmax
the angular position of the compressor crankshaft as measured from a given datum the value of θ at which the cycle starts the linear position of the piston as measured from a given extreme stationary location the minimum and maximum values of X, respectively.
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Fig. 1 Reciprocating compressor geometry and fluid properties. Pi, Ti, and respectively, pressure, temperature, and density of fluid in the inlet manifold. ρi Pc, Tc and respectively, pressure, temperature and density of fluid in the working ρc chamber. Po, To and Respectively, pressure, temperature and density of fluid in the outlet ρo manifold. area of the inlet valve opening Ai Ao area of the discharge valve opening mi mass admitted to the control volume mo mass discharged discharge coefficient at the inlet valve Cdi Cdo discharge coefficient at the discharge valve
By virtue of the mechanical linkage employed to drive the compressor, the crank angular position, θ, determines the instantaneous position, X, of the piston as follows: X ¼ X ðθÞ : θ 2 ½θo , θo + 2π
(1)
The exact mathematical formulation of the functional relationship symbolized in Eq. (1) depends on the kinematical characteristics of the compressor drive. However, without loss of generality, it is possible to assume the maximum value of X occurs at the start of the compressor work cycle; i.e. Xmax ¼ X(θo). As such, the instantaneous volume, Vc, of the compressor working chamber can be calculated as follows: Vc ¼ Vo + Ap ðXmax X Þ
(2)
where Vo is the minimum value of the volume, and Ap is the cylinder crosssectional area. The rate, Vc0 , at which the volume changes with respect to the crank angle is given as, Vc0 ¼ Ap
dX dθ
(3)
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During the compressor operation, the conservation of mass in the control volume is expressed by the following differential equation, dρc 1 dV c ðm_ i m_ o Þ ωρc ¼ (4) dθ ωV c dθ where ( _ ) signifies differentiation with respect to time and ω is the angular velocity of the crank. The inlet mass flow rate can be calculated by the following expressions, 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 1 Pi Pc Pc > > < ρc Ai Cdi if > rc Pi 2 ρi ρc m_ i ¼ (5) > p ffiffiffiffiffiffiffiffiffiffi ffi P > c > : ρc Ai Cdi γRTc if rc Pi where R is the gas constant for the work medium, γ is its specific heat ratio, and rc is the critical pressure ratio. The outlet mass flow rate can be calculated as follows: 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > 1 P P Po > c o > < ρo Ao Cdo if > rc Pc 2 ρc ρo m_ o ¼ (6) > p ffiffiffiffiffiffiffiffiffiffi ffi P > o > ρ Ao Cd o γRTo : if rc o Pc The inlet and discharge valves control the flow to and from the cylinder. With the help of these valves, the sliding motion of the piston results in the chamber pressure, Pc, changing in response to variations in the volume in order for the compressor to perform its intended function. Assuming adiabatic process, the energy balance in the control volume is given by, dH i ¼ dH o + dU c + Pc dVc
(7)
where Hi and Ho signify the enthalpies flowing in and out of the control volume respectively and Uc refers to the internal energy of the fluid inside the control volume. The energy balance equation can be manipulated to various equivalent forms (e.g., Chen et al. (2004) and Ooi (2005)). In this chapter, the following simplified differential form of the energy balance equation is adopted for the analysis (Sultan & Kalim, 2011), dP c γ dV c RðTi m_ i Tc m_ o Þ ωPc ¼ (8) dθ ωV c dθ where Tc is calculated from the Equation of State at every step.
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To calculate the areas of the valve openings indicated in Eqs (5) and (6), the simplified dynamic model presented by Sultan and Kalim (2011) is adopted here (Fig. 2). To calculate the instantaneous valve areas (i.e., Ai and Ao featured in Eqs (5) and (6)), the approach followed Sultan and Kalim (2011) is adopted here with the below definitions: Yv δv Lv Mv Kv Cv Kseat and Cseat Kstop and Cstop
the instantaneous spring deflection, which signifies the valve position initial deflection in the valve spring maximum allowable spring deflection, which signifies the valve lift equivalent mass of the valve and attachments the stiffness of the valve spring equivalent damping coefficient in the valve guides equivalent stiffness and damping coefficient of the valve seat equivalent stiffness and damping coefficient of the valve stop
Based on these definitions, the expressed as follows: 8 < fv1 2 dY v 1 fv2 2 ¼ Mv ω2 : dθ fv3
valve equation of motion can be for δv Yv Lv + δv for Yv < δv for Yv > δv + Lv
(9)
where the various functions in [9] are given as follows: dYv Kv Y v fv1 ¼ ΔPv Av Cv ω dθ dYv (10) fv2 ¼ ΔPv Av ðCv + Cseat Þω ðKv + Kseat ÞYv + Kseat δv dθ dYv fv3 ¼ ΔPv Av Cv + Cstop ω Kv + Kstop Yv + Kstop ðδv + Lv Þ dθ where ΔPv is the pressure differential across the valve and Av is the valve area exposed to that pressure differential. The instantaneous area of the valve port
Fig. 2 Modeling the dynamic behaviour of the compressor valves.
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(i.e., Ai or Ao given in Eqs 5 and 6) which is available for fluid flow, is calculated using the following expression which is simplified from the formula presented by Tuymer and Machu (2001); vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 u A¼u (11) 2 2 ! u 1 1 t + 0:85πDv Yv Av where Dv is the diameter of the valve seat. The complex multi-physical model which spans from Eqs. (2) to Eq. (11) can be solved in a cyclical fashion from θ ¼ θo to θ ¼ θo + 2π until the thermodynamic properties inside the control volume at the start of the cycleare sufficiently close to their corresponding values at the end of the cycle.
Torque performance in reciprocating compressor Thermodynamically, the infinitesimal work, dw, required to drive the compressor is given as follows: dw ¼ Pc dVc
(12)
Mechanically, the work required at the crankshaft to drive the compressor is provided via the torque, τ, where the relationship between the torque and the work is given as follows: dw ¼ τdθ
(13)
As such, considering the comparative nature of this chapter, we overlook the effects of mechanical losses and surrounding pressure, the shaft torque can be calculated from the following expression, τ ¼ Pc Vc0
(14)
A typical torque curve is shown in Fig. 3 where the variable torque acting on the crankshaft is plotted against its angular position. This graph is of special importance to the design of the compressor mechanical components. The mean torque, τ, acting on the crank is calculated as follows: θo + ð 2π
τdθ τ¼
θo
2π
(15)
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Fig. 3 A typical torque against crank angle curve for a reciprocating compressor.
The area that falls above the mean torque line, τ, and below the torque curve, τ, signifies the maximum energy variation, ΔEmax, in the compressor flywheel which can be calculated as follows: θð2
ΔEmax ¼
τdθ ðθ2 θ1 Þτ
(16)
θ1
where θ1 and θ2 are the angles at which τ intersects τ. The value of ΔEmax determines the size of the flywheel as per the following equation, If ¼ Cf ΔEmax
(17)
where If is the flywheel mass moment of inertia and the value of Cf depends on the speed levels stipulated at the crankshaft. Moreover, the torque characteristics also impact the size of the mechanical components of the compressor. For stress calculations, the alternating torque, τa, is used together with the midrange torque, τm. The alternating torque is calculated as follows: τ τ max min τa ¼ (18) 2 The midrange torque is calculated as follows: τmax + τmin (19) 2 where τmax is the peak value of the τ curve and τmin is its minimum value. τm ¼
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Stress-based design for fatigue loading is discussed in a number of textbooks, e.g. Budynas and Nisbott (2008). The design process employs the loads and the part dimensions to calculate various stresses acting at points of concern on a machine component. For the purpose of the current discussion, it suffices to point out that the factor of safety, nf, calculated at a point of interest may boil down to the following generalized form; nf ¼
1 Ka τ a + Km τ m
(20)
where Ka and Km signify mathematical expressions which feature the material properties and the component geometric characteristics. Based on the above discussion, it would be favorable to modify the characteristics of the piston strokes such that the extreme values of the torque curve, τmin and τmax, are curtailed. This would be achieved by modifying the timing of the strokes such that the extreme values of the rate Vc0 featured in Eq. (14) are reduced. This reduction is useful particularly towards the end of the compression stroke where the chamber pressure has already grown well above the inlet value. The modified torque curve would also feature a decreased ΔEmax which would subsequently result in a reduced flywheel size as per Eq. (17). Moreover, conducting compression over an extended portion of the cycle time would present favorable conditions for the cooling system to pick up more heat from the compressed fluid and thus reduce the temperature levels in the systems.
Example alternative compressor drive Conventionally, reciprocating compressors are driven by the well-known slider-crank linkage which is shown in Fig. 4. The position of the piston as measured from the crank pivot along the stroke line is given as, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 (21) X ¼ r cos θ + L 1 sinθ L
Fig. 4 The slider-crank mechanism.
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where r is the crank length, L is the length of the connecting rod and the angle θ is measured from the stroke line to the crank as shown in Fig. 4. Figs 5 and 6, respectively, depict Vc and Vc0 against the crank angle, θ. For the conventional linkage, these relationships are symmetric about θ ¼ π
Fig. 5 A typical symmetric Vc θ curve produced by the conventional drive.
Fig. 6 A typical symmetric Vc0 θ curve produced by the conventional drive.
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which confirms the symmetry of the stroke characteristics offered by the use of that linkage. It is possible to modify the workings of the conventional slider-crank compressor by offsetting the sliding line, a certain distance, below the pivot of the input crank. However, this does not provide an effective handle to control the various geometric characteristics of the piston stroke in a way which yields the desired compressor performance. A more versatile compressor drive is shown in Fig. 7 below. As shown in the figure, the piston is made to slide along a line which falls a distance h below the pivot o and makes an angle ϕ with the x direction. The position, X, of the piston along the axis of sliding is given by the following equations, X ¼ ð2r sinθ + L Þcos ðθ ϕÞ + hsin ϕ + Lc cosσ
(22)
where Lc is the length of the connecting rod and σ is its angle with the sliding direction. The angle σ can be obtained from the following relationship, ð2r sinθ + L Þ sin ðθ ϕÞ + h cos ϕ ¼ Lc sinσ
(23)
Solving (22) and (23) together leads to the following expression for Xp, X ¼ ð2r sinθ + L Þ cos ðθ ϕÞ + h sinϕ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð2r sinθ + L Þ sin ðθ ϕÞ + hcos ϕ 2 + Lc 1 Lc
Fig. 7 An alternative mechanical linkage to drive the compressor.
(24)
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The maximum and minimum values, Xmin and Xmax, of X are found by calculating the roots of the following equation and substituting these roots in Eq. (23), dX ¼0 (25) dθ The piston stroke, S, can, be found from the following equation, S ¼ Xmax Xmin
(26)
As suggested by Sultan and Kalim (2011), the torque performance of the drive is quantified by ητ, which is given as follows: ητ ¼ 1
τmax τ τmax + τ
(27)
The purpose of Eq. (27) is to judge how close τmax is to the mean torque, τ. A small difference between the two torque values suggests a smoother operation with a reduced flywheel size and smaller stresses in the various components of the drive. The design process which is used to find the optimum dimensions of the drive is given in the next section. In the design process, the user sets such parameters as, the pressure levels at the inlet and outlet manifolds, the shaft speed, the required stroke. The process features the use of an optimization procedure and results in the dimensions of the linkage found as shown in Fig. 7. The design vector, Ζ, which is employed to optimize compressor drive is given as follows: 2 3 r 6 L 7 6 7 6 7 Ζ ¼ 6 Lc 7 (28) 6 7 4 h 5 ϕ The cost function, Fc(Z), which is required to be minimized for optimum performance is given as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fc ðZ Þ ¼ wτ ð1 ητ Þ2 + ws ð1 ηs Þ2 (29) where ηs is factor which determines the accuracy of the obtained stroke as given below: ηs ¼
Sk Sr
(30)
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where Sr is the required stroke and Sk is the stroke obtained at iteration step number k. In Eq. (29), wτ and ws signify weighting factors used to reflect the relative importance of the parameters used in Eq. (29) for a specific design problem. The optimization problem can now be presented as follows: min
Zmin ZZmax
Fc ðZ Þ
(31)
where the entries of the constraint vectors, Zmin and Zmax, are set by the user to meet the requirements of a given problem. The iterative search for the optimized solution is conducted using randomized cosine functions as follows: Zik ¼ Zik1 + Δki cos ψ ki π (32) where the various variables are explained as follows: Zki is element number i in the Z vector at iteration step number k, Δki signifies the difference between the current value of Zk1 and its near i constraint. This variable is calculated as follows: ( Δki
¼
Zimax Zik1 if Zimax Zik1 Zik1 Zimin k1 Zi Zimin otherwise
(33)
The value of ψ ki is assigned randomly between 0 and 1 at every iteration for each entry number i of the Z vector. The solution, Zk, is accepted only if Fc(Zk) < Fc(Zk1), otherwise, the iteration is repeated for a set number of times.
Design example For the example featured here, the following parameters are set. • Shaft speed ¼ 800 rpm • Po ¼ 8 bar, Pi ¼ 1 bar, and Ti ¼ 303 K • Sr ¼ 100 mm The iterative procedure explained above resulted in reducing the cost function from 63.3 to 3.4 as shown in Fig. 8. The stroke at the start of iterations was 413.9 mm and this had to be iteratively reduced to 101.3 mm at the end of the iterations. As such, the stroke error dominated the optimization calculations.
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Fig. 8 Reduction of cost function with iterations.
The thermodynamic model detailed above was used to investigate the performance of two compressors, one driven by the optimum compressor and one driven by the conventional drive. The maximum energy variation, ΔEmax, in the flywheel was found to be 251 J in the conventional drive and 226.7 J for the optimum drive. This will result in about 10% in the flywheel mass. Also, the maximum torque is 360 Nm and minimum torque is 113 Nm in the conventional drive. The corresponding values in the new drive are 273.3 Nm and 87.7 Nm respectively. As such, the torque values used for stress calculations in the conventional drive are as follows: τm ¼ 123:5 Nm and τa ¼ 236:5 Nm: For the optimum drive, the corresponding values are given as τm ¼ 92.5 Nm (i.e. 25% reduction) and τa ¼ 180.5 Nm (which offers 23.7% reduction). The new torque levels would make it possible to reduce the sizes of various parts in a way which motivates more sustainable use of material and manufacturing resources at the design end. The following graphs, Figs 9–12, offer an insight into how various process parameters vary during the work cycle for both the conventional and the optimum compressors.
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Fig. 9 Pressure against angle of conventional and optimum compressors.
Fig. 10 Torque against angle of conventional and optimum compressors.
Discussions and conclusions Despite their immense popularity, reciprocating compressors still use the traditional slider-crank drives they have always used since the early years of compressed air industry. The current push for more sustainable use of material and resources combined with the modern manufacturing and analytical tools
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Fig. 11 The profile of chamber volume.
Fig. 12 The rate of change of chamber volume.
should motivate the search for a modern more sophisticated drives more capable of improving compressor performance. In this chapter, a candidate linkage has been proposed and utilized in a mathematical model to describe the workings of the compressor. An optimization technique has been employed to obtain the dimensions of the linkage which would achieve an improved torque curve for the compressor without impacting the required pressure or mass flow rate performance. The model demonstrated takes into consideration the
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dynamics of the compressor valves in form of a nonlinear single degree of freedom vibrating system. The example presented proved the validity of the proposed linkage and its ability to reduce the flywheel size by 10% and the stresses in components by more than 23%. These reductions would result in a more reliable operations and it would be possible to reduce part size and achieve savings in relation to material consumption and manufacturing cost. Moreover, due to the fact that the compression stroke occurs over a longer period of time, it is expected that the cooling performance is expected to be improved resulting in improved thermal efficiency. However, the heat transfer analysis was not included in the mathematical model featured in the chapter. Moreover, the modified stroke is likely to impact the lubrication characteristics of the compressor. Future studies in this domain will feature the inclusion of the heat transfer analysis and lubrication modeling in the mathematical presentation of the compressor in order for these phenomena to be better understood in asymmetric stroke situations.
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