Copyright © 2010 KSAE 1229−9138/2010/052−03
International Journal of Automotive Technology, Vol. 11, No. 3, pp. 317−322 (2010)
DOI 10.1007/s12239−010−0039−8
EXPERIMENTAL VALIDATION OF A MATHEMATICAL MODEL OF A REED-VALVE RECIPROCATING AIR COMPRESSOR FROM AN AUTOMOTIVE-BRAKING SYSTEM J. VENKATESAN , G. NAGARAJAN , R. V. SEENIRAJ and R. MURUGAN 1)*
2)
2)
1)
Department of Mechanical Engineering, Sri Venkateswara College of Engineering, Chennai 602 105, India Department of Mechanical Engineering, College of Engineering, Anna University, Chennai 600 025, India
1)
2)
(Received 25 November 2008; Revised 28 October 2009)
ABSTRACT−Mathematical simulation is the process of designing a model of a real system and then conducting experiments with the simulation to understand the system’s behavior. Mathematical simulation is widely used for investigating and designing compressors, and with a minimal number of simplifying assumptions, mathematical models can be used in conjunction with modern computing tools to solve complicated problems. A considerable amount of previous research has focused on the mathematical modeling of reciprocating air compressors used in automotive braking. The aim of the present work was to experimentally validate the mathematical model for such compressors. We present a simplified and effective mathematical model for estimating compressor performance, and this model can easily be executed using personal computers. Parameters such as compressor speed, discharge pressure and clearance volume were evaluated in terms of their effect on the thermodynamic behavior of compressors. The model can predict cylinder pressure, cylinder volume, cylinder temperature, valve lift and resultant torque at different crank angles; it can also predict the free air delivered and the indicated power of the compressor. Therefore, the model has been validated using experimental results. KEY WORDS : Resultant torque, Indicated power, Peak pressure, Free air delivered (FAD), Volumetric efficiency
NOMENCLATURE A D L T T l r
: mass of air discharged out through the delivery valve, kg S : valve lift (distance between valve plate and valve), m n : ratio of connecting rod length to cylinder diameter F : force acting on the crank, N F : net force acting on the delivery valve, N F : net force acting on the piston, N F : force due to initial compression of valve, N ω : natural frequency of valve, rad/s B : factor accounting the instantaneous change of specific volume, m /kg ρ : density of air, kg/m ζ : damping factor m : instantaneous mass, kg Q : heat transfer to actuating medium, J α (θ ) : heat transfer coefficient, W/m-K Exp : experimental Pre : predicted m
: dross sectional area of cylinder, m : diameter of cylinder, m : stroke length, m : temperature of air at particular crank angle, K : torque on the crankshaft, Nm : length of connecting rod, m : crank radius (L/2), m : crank angle, deg : angular velocity of the crank, rad/s : compressor speed, rpm : pressure of air an instant, Pa : volume of air inside the cylinder, m : number of ports (openings in the compressor head on suction and delivery sides) : young’s modulus of valve material, N/m : stiffness of the suction valve, m/N : area moment of inertia of valve, m : distance of point of application of force from fixed end, m : specific heat at constant volume, J/kg-K : mass of air in the cylinder, kg : mass of air flowing through the suction valve, kg 2
c
r
c
θ ω
N p V Z
c
d
p
si
n
3
3
3
E k I x
2
s
d
4
d
Cv m m s
SUBSCRIPTS s d e i o op
*Corresponding author. e-mail:
[email protected] 317
: suction : delivery : effective : inlet : outlet : other processes
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J. VENKATESAN, G. NAGARAJAN, R. V. SEENIRAJ and R. MURUGAN
1. INTRODUCTION A reciprocating compressor consists of a crankshaft (driven by a gas engine, electric motor, or turbine) attached to a connecting rod, which transfers the rotary motion of the crankshaft to the piston. The piston travels back and forth in a cylinder. Air enters the cylinder through a suction valve at suction pressure, and the piston compresses the air to reach the desired discharge pressure. When the air reaches the desired pressure, it is then discharged through a discharge valve. The desired discharge pressure can be reached through utilization of either a single or doubleacting cylinder. In a double-acting cylinder, compression takes place at both the head-end and the crank-end of the cylinder. The cylinder can be designed to accommodate any pressure or capacity, thus making the reciprocating compressor the most popular type used in the automobile and gas industries. Therefore, it is important to construct an accurate mathematical model that can predict the behavior of these compressor systems. Building a mathematical model (Venkatesan ., 2007; Lawson and McLaren, 1984; Tian ., 2005) for any project may be a challenging, yet interesting, task. To build such models, a thorough understanding of the relevant underlying scientific concepts is necessary, and a mentor with expertise in the project is invaluable. It is also best to work as part of a team that can provide more brainstorming power. In industry and engineering, it is common practice for a team of people to work together toward building a model, and the individual team members bring different areas of expertise to the project. Once the model has been developed and applied to the problem, the resulting model solution must be analyzed and checked for accuracy. This process may require modifying the model to obtain a et al
et al
reasonable outcome. This refining process should continue until a model that agrees as closely as possible with realworld observation is obtained.
2. MODEL FORMATION The physical dimensions of the reciprocating compressor are shown in Figure 1. The model was based on the following thermodynamic equations (Venkatesan ., 2007; Lawson and McLaren, 1984). et al
Suction dQ mRT dV dms ------ + ----------- ------ + -------- ( CvT – RCvTs )− ------- =0 mCv= dT dt V dt dt dt
(1)
Compression and reexpansion mRT dV dQ ------ + ----------- ------ − ------- =0 mCv dT dt V dt dt
(2)
Discharge mRT dV dm-d ( RCvTd – CvT )− dQ ------- =0 ------ + ----------- ------ + -------mCv dT dt dt V dt dt
(3)
The governing equation for determining the instantaneous cylinder pressure was expressed as the following: pv RT
------ =1+
∑ Bi T ρi n
(4)
( )
i=1
The second term in equation (4) accounts for the compressibility of air. Because single-stage compressors are designed for limited pressure ratios, the second term can be neglected for analysis purposes. The governing equation for determining the mass flow was expressed as the following: dmi dm-o dm ------- = -------- − -------− dθ dθ dθ
∑ dmdθ
(5)
op ----------
The third term in equation (5) indicates losses by various processes (e.g., leakage loss, etc). The governing equation for determining the working volume (Venkatesan ., 2007; Francis ., 1965) was expressed as the following: et al
dV L------ =± A c -dθ 2
sin
n sinθ cosθ θ + -----------------------------2 2 1 – n sin θ
et al
(6)
The resultant torque (Tr) was calculated using the following expression Tr = F p r
Figure 1. Schematic diagram showing the physical dimensions of the reciprocating compressor.
sin
2 sin θ θ + -----------------------------------2 2 2 ( l c / r ) − sin θ
(7)
2.1. Indicated Power (IP) Because all of the processes do not follow a particular thermodynamic law, it was not advisable to use readily
EXPERIMENTAL VALIDATION OF A MATHEMATICAL MODEL OF A REED-VALVE RECIPROCATING
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Figure 5. Suction reed in the closed position. Figure 2. Integration method for estimating the indicated power. available equations for determining the indicated power during a suction or discharge process. Figure 2 illustrates the integration method used to estimate the indicated power. The following general and effective model was used for estimating IP during any incremental change in crank angle (Venkatesan ., 2007). et al
IPθ =IPθ
-1 +
pθ-1 + pθ N ( Vθ -1 – Vθ ) ⎛ -------------------⎞ ⎛ ------ ⎞ ⎝ ⎠ ⎝ 60 ⎠ 2
(8)
2.2. Discharge Process The deflection of the delivery reed was calculated from the following expression (Werner, 2007; Arne, 1974) Sd= F----d---x---d--(--l--d--–----x---d--)---(---4---l--d--–----x---d-)12 E I d l d 3
3
(9)
2.3. Suction Process Using effective valve dynamics (Kazutaka and Susuma, 1980; Stif Helmer Joergensen, 1980) the following expression can be written: F –F Ss=⎛⎝ ----s----------si-⎞⎠ ks J s
(10)
Figure 6. Suction reed in the full open position. where Js =
Js
is a factor given by the following:
ω ω
2
⎛ ⎛ 1 – ⎛ --------⎞ ⎞ ⎝ ns ⎠ ⎠ ⎝⎝
2
⎛ ⎝
+ 2
ξ ⎛⎝ ω ⎞⎠⎞⎠ ω ns --------
⎞ ⎠
(11)
3. EXPERIMENTAL DETAILS A reed-valve reciprocating compressor used in the braking system of heavy passenger vehicles and trucks was tested using sophisticated test rig (Figure 7). The compressor was powered by an electric motor. The pressure inside the cylinder was measured by a piezo-electric pressure transducer, and the data was stored using a data acquisition system. The compressor speed was controlled by a speed pot in the control panel, and it was cooled by a fan. The compressor was connected to a 50 liter reservoir, and the pressure was maintained by using a governor valve.
Figure 3. Delivery reed in the closed position.
Figure 4. Delivery reed in the full-open position.
2
Figure 7. Experimental setup.
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J. VENKATESAN, G. NAGARAJAN, R. V. SEENIRAJ and R. MURUGAN
Compressor details: Bore diameter (D) 66.67 mm Crank radius (r) 23 mm Connecting rod length (l ) 70 mm Suction reed lift (h ) 2.2 mm Delivery reed lift (h ) 1.8 mm Mass of reciprocating parts (m ) 0.245 kg Clearance volume (V ) 6.67 cc Discharge pressure (p ) 5 to 9 bar (abs) Diameter of suction port (d ) 11 mm Diameter of delivery port (d ) 11 mm Distance (x ) 59 mm Distance (x ) 26,5 mm Effective length of suction reed (l ) 71 mm Effective length of delivery reed (l ) 45.5 mm Mass of suction valve (m ) 7g Mass of delivery valve (m ) 2g Number of suction ports (Z ) 4 Number of delivery ports (Z ) 2 c
s
d
rec
c
Figure 8. Pressure-volume diagram (speed=3000 rpm).
d
os
od
s
d
s
d
sv
dv
s
d
Figure 9. Pressure-crank angle diagram (speed=3000 rpm).
4. RESULTS AND DISCUSSION In an ideal compressor, the suction pressure and the discharge pressure are constant because the cylinder diameter is assumed to be equal to the suction/discharge port diameter. In an actual compressor, the port diameter is less than the cylinder diameter. Therefore, during the suction process, the volume displaced by the piston is greater than the volume of air entering the cylinder during a particular time interval. The net effect is a decrease in the suction pressure to a level below that of an ideal compressor. Similarly, during the discharge process, the volume displaced by the piston is greater than the volume of air discharged through the discharge port. The net effect is an increase in the cylinder pressure to a level above the discharge pressure. Due to excess peak pressure during the discharge process, the indicated power of the compressor is
Figure 10. Valve lift-crank angle diagram (speed=3000 rpm). always greater than the ideal indicated power for a particular amount of free air delivered (FAD). Compressor capacity is generally expressed in terms of FAD, which is defined as the volume of air delivered by the compressor when the condition (the temperature and the pressure) of air is reduced to the intake condition. The compressor’s
Table 1. Performance of the compressor at different discharge pressures (N=3000 rpm). Results 6 bar 6 bar 7 bar 7 bar 8 bar Pre Exp Pre Exp Pre ↓ Peak pressure (bar) 8.09 8.05 9.20 9.08 10.23 Free air delivered (lpm) 305.7 303 294 300 282.7 Volumetric efficiency (%) 63.5 63.1 61.0 62.5 58.7 Shaft power (W) 2284 2250 2206 2230 2300
8 bar Exp 10.37 276 57.5 2394
9 bar Pre 11.81 264.7 54.9 2534
9 bar Exp 11.42 264 55.0 2542
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Figure 11. Torque-crank angle diagram (speed=3000 rpm). Figure 13. Volumetric efficiency-discharge pressure diagram (speed=3000 rpm).
Figure 12. Free air delivered-discharge pressure diagram (speed=3000 rpm). volumetric efficiency is mainly dependent on the suction pressure. The effect of reduced suction pressure is to significantly reduce the volumetric efficiency. Here, the aforementioned model was tested using different discharge pressures and compressor speeds. The simulated results were very close to the experimental results, which indicated the accuracy of the model. 4.1. Sensitivity Analysis The clearance volume was increased to 8.31 cc in an existing 160 cc air-cooled compressor, and the system’s performance was tested at different speeds and delivery pressures. The sensitivity of the developed model was tested using the experimental results from the modified compressor. Table 2 summarizes the results obtained from experiments and from simulations using the developed model. The increase in clearance volume caused a decrease in the volumetric efficiency and the FAD. Both the experi-
Figure 14. Shaft power-discharge pressure diagram (speed =3000 rpm). mental and the predicted results indicate that the volumetric efficiency and the FAD were each reduced when the clearance volume was increased from 6.67 cc to 8.31 cc. In the modified compressor, the deviation of the predicted value from the actual value was about 6% for peak pressure, 8% for both FAD and volumetric efficiency, and 5% for shaft power. The predicted values are slightly higher than the values from the actual compressor, but they are still acceptably close to the expected level. Based on previous work on compressor design, it has been shown that clearance volumes ranging from 2.5 to 4.5% of the stroke volume give better performance (Venkatesan ., 2007; Werner, 1980; Lawson and McLaren, 1984). In our modified compressor, the clearance volume was 5.2%, which was the primary cause of the large observed deviations.
Table 2. Performance of the modified compressor at different discharge pressures (N=3000 rpm). Results 6 bar 6 bar 7 bar 7 bar 8 bar 8 bar Pre Exp Pre Exp Pre Exp ↓ Peak pressure (bar) 7.61 8.10 8.92 9.15 9.92 10.19 Free air delivered (lpm) 186 195 158 173 144 156 Volumetric efficiency (%) 48.1 50.6 41.2 44.9 37.5 40.5 Shaft power (W) 1677 1704 1816 1716 1775 1826
et al
9 bar Pre 11.61 127 32.9 1785
9 bar Exp 11.27 137 35.7 1850
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5. CONCLUSION
The model presented here predicts fluctuations in pressure during the suction and discharge processes of a reciprocating compressor. It also predicts valve fluttering during suction and discharge at all delivery pressures. The simulated results from the model are comparable with the experimental results. Using this model, it is possible to compute volumetric efficiency, free air delivered, indicated power, cylinder air pressure, cylinder air temperature, resultant torque and mass of air imported or discharged per cycle. It is also possible to determine these values after varying either the operating parameters (e.g., speed, discharge pressure, etc.) or the physical parameters (e.g., clearance volume, crank radius, connecting rod length and cylinder diameter). The model can be used for theoretical analysis of single-stage, single-cylinder reciprocating air compressors with a disc valve. The development of this model was based on the previous research and technical resources available from the compressor-design field. The constants used in the development of the model were based on the available experimental results and on information from previous research in the compressor-design field. Simple assumptions were made in the development of the model, and these assumptions could be varied or omitted depending on the operating parameters and physical conditions of the compressor. Finally, the effectiveness of the developed model was very much dependent on the “usage of suitable constants” in the model (e.g., coefficient of discharge, index of compression, etc). REFERENCES
Arne, M. B. (1974). Computer simulation of valve dynamics
as an aid to design. Norwegian Institute of Technology Proc. Int. Conf. Compressor Technology, Purdue Univer-
sity, West Lafayette, Indiana, USA. Francis, L. S., LaiSing, T. and Francis, T. (1965). Mechanical Vibrations. CBS Distributors. Delhi. India. Kazutaka, S. and Susuma, N. (1980). Practical method for analysis and estimation of reciprocating hermetic compressor performance. Hitachi Ltd, Japan - Proc. Int. Conf. Compressor Technology, Purdue University, West Lafayette, Indiana, USA. Lawson, S. and McLaren, R. J. L. (1984). An approach to computer modeling of reciprocating compressors. prestoold limited. U.K, Proc. 1984 Purdue Compressor Technology Conf., Purdue University, West Lafayette, Indiana, USA. Stif Helmer, J. and Danfoss, N. (1980). Transient valve plate vibrations. Proc. Int. Conf. Compressor Technology, Purdue University, West Lafayette, Indiana, USA. Tian, C., Liao, Y. and Li, X. (2005). A Mathematical model of variable displacement swash plate compressor for automotive air conditioning system. Int. J. Refrigeration 29, 2, 270−280. Venkatesan, J., Nagarajan, G., Seeniraj, R. V. and Sampath, S. (2007). Mathematical model for theoretical investigation of a disc valve reciprocating air compressor of automotive braking system. Int. J. Applied Mathematical Analysis and Applications 2, 1-2, 209−227, Serial Publications, New Delhi, India. Werner, S. (1980). Design and Mechanics of Compressor Valves. Ray. W. Herrick Laboratories. School of Mechanical Engineering. Purdue University. West Lafayette. Indiana. USA.
