Kothmann, R. E.

Squarer, D.

English

Purdue E-Pubs

Purdue UniversityPurdue e-PubsInternational Compressor Engineering Conference School of Mechanical Engineering1972Digital Computer Simulation of a ReciprocatingCompressor-A Simplified AnalysisD. SquarerWestinghouse Electric CorporationR. E. KothmannWestinghouse Electric CorporationFollow this and additional works at: https://docs.lib.purdue.edu/icecThis document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at https://engineering.purdue.edu/Herrick/Events/orderlit.htmlSquarer, D. and Kothmann, R. E., "Digital Computer Simulation of a Reciprocating Compressor-A Simplified Analysis" (1972).International Compressor Engineering Conference. Paper 83.https://docs.lib.purdue.edu/icec/83DIGITAL COMPUTER SIMULATION OF A RECIPROCATING COMPRESSOR -- A SIMPLIFIED ANALYSIS D. Squarer Westinghouse Research Laboratories Churchill Borough, Pennsylvania INTRODUCTION Although the reciprocating compressor has been in existence for the past few decades no mathematical modeling of a small reciprocating compressor was published until 1958.1 Evidently, the lack of intensive work in this area may be attributed partly to the unavailability of large and fast digital computers which are a prerequisite to any such study. Considerable effort has been devoted to this problem in the past five years and a few excellent publications mostly in the form of a Ph.D. dissertation are available today in the open literature (e.g. reference 2). An exact analytical treatment of a reciprocating compressor is very involved and should include such aspects as: (1) heat and mass transfer between the various parts of the compressor, viz. compression cylinder, suction and discharge lines, motor and case, (2) dynamics of the piston - connecting rod - crank system, (3) suction valve dynamics and discharge valve dynamics, (4) electric motor, (5) transient phenomena (e.g. reflected waves in the discharge line). Only rarely such an exact analysis, which treats all the above aspEcts simultaneously, is pursued. Instead, some simplifying assumptions which are supported by experimental evidence are made in order to save both an analysis effort and computer time. BASIC ASSUMPTIONS One of the most difficult problems in a recipro-cating compressor analysis is the valve dynamics. Furthermore, even when a rigorous valve dynamics analysis is followed, some "unexplained" behavior of discharge valves in particular is observed. 2 On occasions it may therefore be desirable to by-pass the valve analysis all together, provided of course that we are not interested in the valves themselves. Although such a simplified analysis has obvious shortcomings, it may still yield valuable information as will be shown below. The valve dynamics analysis has been bypassed in our analysis by making the following four assumptions (see Figure 1): 1. The discharge valve opens when the pressure in the cylinder exceeds the discharge pressure, p0 , by Llp . 2. rRe suction valve opens when the suction pres-sure, ps, exceeds the cylinder pressure, by Llps. 502 R. E. Kothmann Westinghouse Research Laboratories Churchill Borough, Pennsylvania 3. Once the suction valve opens it remains fully open until o radians after bottom dead center (BDC). s 4. Once the discharge valve opens it remains fully ,open until o0 radians after top dead center (TDC). Available experimental evidence2 • 3 indicate that the values of LlpD is a few tens of psi (~ 40 psi) whereas Llp is a few psi <~ 4 psi). The assumed value of Ll~D has little effect on the timing of the discharge valve opening due to the steep slope of the pressure vs. crank angle diagram near point C. However, the assumed value of Llps has a larger effect on the timing of the suction valve opening due to the mild slope of the p.~ssure vs. crank angle diagram near point F. Experimental evidence indicate that a ~ 0.2 to 0.3 radians and aD ~ 0.1 to 0.2 radians.s An additional assumption was made in the heat trans-fer analysis that the thermal impedance of the cylinder wall is negligible and hence its tempera-ture is equal to the oil temperature in the case which is known. RECIPROCATING COMPRESSOR CYCLE Figure 1 is a schematic representation of the com-pressor cylinder pressure vs. crank angle which our model attempts to simulate. It describes the fol-lowing events: Interval AB - suction valve is open; interval BC - compression, both valves are closed; interval CE - discharge valve is open; interval EF - expansion, both valves are closed; interval FA - suction valve is open; point B -suction valve closure; point C - discharge valve opening; point E - discharge valve closure; point F - suction valve opening. ANALYTICAL REPRESENTATION Application of the rate form of the first law of thermodynamics for an open system to the control volume which is bounded by the cylinder walls and the piston yields, T T n R V T W -cTw-3Cv+k/ws v v w w (1) where T is the gas temperature, 0 is the heat flux, W is the gas mass, V is the gas volume, k ~ C /C p v is the ratio between constant pressure and constant volume specific heats, R is the universal gas con-stant, J = 778 ft-lbf/Btu, the subscripts D and s designate discharge and suction respectively and a dot above a symbo:j.. designates the time rate form of a variable, e.g. Q = dQ/dt. In deriving Equation (1) we have used the equation of state for an ideal gas and we have neglected the change in the poten-tial and kinetic energies. The last term in Equation (1) is the rate of change of mass within the cylinder and is given by W/W = -WD/W during the discharge period and by W/W = W /W during the suction period. s In order to solve Equation (1) we have to substitute proper expressions for V, W, and Q. The displace-ment volume V can be found from geometrical consid-erations and for the system shown in Figure 2 is given by: V = XcAp = [Leo+ R1(1- cos8) + R2(1- cos¢)) Ap (2) where A is the piston cross sectional area, X is the pis£on position with respect to the valve plate, Leo is the clearance distance between the piston face and the valve plate at TDC, R1 is the crank length, R2 is the connecting rod length, e = wt is the crank angle, ¢ is the angle between the connect-ing rod and the cylinder axis, w is the angular speed of the crank and t is the time. The time rate of change of V is simply given by V = A X • The mass flow rate W can be computed by thePf§llowing equation which describes the mass flow rate of a compressible ideal fluid through a constriction, w KYAdsyus (3) /1 - (Ad /A 7 s us where K is a discharge coefficient, Y is a compress-ibility factor, A is the constriction area, g is the acceleration due to gravity, p is the gas pressure and the subscripts us and ds designate upstream and downstream respectively. The density y may be found from the equation of state, y £..... = w RT V (4) The maximum mass flow rate occurs when the velocity through the constriction has reached the sonic limit. This condition may be found from the follow-ing relationship, r c k pds = (-2-)~ pus k + 1 When the critical pressure ratio r is reached, Equation (3) is modified as follow~: w ~ p (1 - r ) ; " us c yus /1 - (Ad 1 A ) s us (5) (6) 503 The rate of heat transfer from the cylinder wall t<' the gas is given by Q = A(8) U(8) (Tw - T) (7) where A(6) is the heat transfer area (a function of the crank angle) through which heat is transferred with a coefficient U(8) and the subscript w desig-nate the cylinder wall. The gas pressure in the cylinder can be computed directly from the equation of state of the particu-lar gas when V, Wand T are known. All the gas properties such as density, enthalpy, viscosity, etc. can be computed from published data. In addition to the above analysis we have also evaluated the pressure drop and heat transfer through the suction and discharge tubes assuming a steady rather than pulsating flow. This assumption is justified by considering the magnitude of the pulsating volume relatively to the volumes of the suction muffler or the discharge muffler or the discharge tube. NUMERICAL SOLUTION A predictor-corrector scheme was used to obtain a simultaneous solution of the three rate equations for W, Q, and T which in turn was used in the pres-sure computations. The volume V was evaluated directly from Equation (2). The solution was started by assuming initial conditions for the temperature, pressure and mass flow and was pursued until the computed values at the end of the cycle agreed to within the desired accuracy with the . assumed initial conditions. The mass flow rate, m, was evaluated by, 1 2tA-A where the subscripts on the time t refer to the points shown in Figure 1. RESULTS (8) Some typical model predictions are displayed in Figures 3-5. Figure 3 shows the cylinder gas pres-sure. Figure 4 shows the gas mass in the cylinder and Figure 5 shows the p-V diagram. Comparison between these figures and available experimental data (e.g. references 2 and 3) shows that the model prediction is in general agreement with experimen-tal data and with a more rigorous analysis.2 The cylinder gas pressure (Figure 3) is character-ized by a pressure "overshoot" before the opening of the discharge valve, by the pressure drop immediately after the discharge valve closure and by the quick pressure recovery after the suction valve opening<~ 77°). The gas mass in the cylin-der (Figure 4) and the volumetric efficiency depend on the time of closure of the discharge valve. The earlier the discharge valve closes the less mass will remain in the cylinder when re-expansion starts and a higher volumetric efficiency will be obtained. The two constant mass "levels" in Figure 4 are a direct indication of the compressor pumping capacity, the further apart they are the higher the compressor capacity. The p-V or the indicator diagram which is shown in Figure 5 clearly demonstrates the deviation from an ideal compression-expansion cycle. CONCLUSIONS The simplified analysis presented herein does adequately simulate the gas pressure, temperature, volume and mass flow rate under.various pressure ratios. Consequently, it may be used reliably to investigate the effect of the following factors on the compressor performance: heat transfer during compression on the maximum temperature, clearance volume, valve areas, timing of closure and opening of the valves, pressure ratio, connecting rod-crank geometry, crank angular speed, etc. Such informa-tion is of utmost importance in a compressor design. REFERENCES 1. Brunner, W., "Simulation of a Reciprocating Com-pressor on an Electronic Analog Computer", ASME Paper Number 58-A-146. Presented at the ASME Annual Meeting, New York, N.Y., November 30-December 5, 1958. 2. Gatecliff, G. W., "A Digital Simulation of a Reciprocating Her~etic Compressor Including Comparisons with Experiments", Ph.D. Thesis, The University of Michigan, 1969. 3. Wambsganss, M. W., Jr., "Mathematical Modeling and Design Evaluation of High-Speed Reciprocat-ing Compressors, Ph.D. Thesis, Purdue University, 1966. . Clearance::::: 1.5% 280 t.P0 ::::: 40 PSIA 240 LlP s::::: 4 PSIA o5 = 17.2° ::.; (/') 200 6 -6° a... D-a} P5 =17.7PSIA ,_ :::J "" 160 P0 = 214.7 PSIA "" ~ a... m = 22.2 LBM/HR "" "" 120 ('_!) 80 40 Crank L.. Q} = c D I 6D f--E---P D ~t-~~-------1--~~~~~~Ps <...) BDC TDC BDC Crank Angle. degrees Fig. !-Schematic representation of one cycle -Dp ~ L ==Clearance Piston~ : co Connecting Rod l;l c Fig. 2-Schematic of the crank-connecting roo-piston system -80 -40 0 40 80 120 160 200 Crank Angle, degrees Fig. 3-Gas pressure in the cylinder 504 ~ ~ 0.10 Clearance== 1.5% fi.P D == 40 PSIA C"t"\ 0 -X 0.08 fi.P S::::: 4 PSIA 65 =17.2° "" "" ~ "" 0.06 6 ::::6° D c3 0.04 0.02 P5 :::: 17.7 PSIA PD:::: 214.7 PSIA m= 22.2 LBM/HR 0. 00 L..____j _ __L _ ___..L _ __.....L _ _.J__...,L.__--1...-_...!....__.....____.. -200 -160 -120 -80 -40 0 40 80 120 160 200 280.00 240.00 <( Vi 200.00 c... ai s 160.00 "" 1!l ~ 120.00 (g 80.00 40.00 Crank Angle, degrees Fig. 4-Gas mass in the cylinder Clearance== 1.5% f\.PD:::: 40 PSIA f\.P S :::: 4 PSIA 6 s = 17.2° 6D = 6o PS:::: 17.7 PSIA P0 :::: 214.7 PSIA m= 22.2 LBM/HR 0. 00 L.___.J _ ____l_---l._----l..._---L..._--L._...J....__...l.------.J 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Gas Volume x 103, tt3 Fig. 5-P-V diagram 505 

Digital Computer Simulation of a Reciprocating Compressor-A Simplified Analysis

https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1082&amp;context=icec

Digital Computer Simulation of a Reciprocating Compressor-A Simplified Analysis

Abstract

Similar works

Full text

Available Versions

Purdue E-Pubs