Digital Computer Simulation of a Reciprocating Compressor-A Simplified Analysis

Purdue University Purdue e-pubs International Compressor Engineering Conference School of Mechanical Engineering 1972 Digital Computer Simulation of a Reciprocating Compressor-A Simplified Analysis D. Squarer Westinghouse Electric Corporation R. E. Kothmann Westinghouse Electric Corporation Follo this and additional orks at: http://docs.lib.purdue.edu/icec Squarer, D. and Kothmann, R. E., "Digital Computer Simulation of a Reciprocating Compressor-A Simplified Analysis" (1972). International Compressor Engineering Conference. Paper 83. http://docs.lib.purdue.edu/icec/83 This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional 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.html

DIGITAL COMPUTER SIMULATION OF A RECIPROCATING COMPRESSOR -- A SIMPLIFIED ANALYSIS D. Squarer Westinghouse Research Laboratories Churchill Borough, Pennsylvania R. E. Kothmann Westinghouse Research Laboratories Churchill Borough, Pennsylvania INTRODUCTION Although the reciprocating compressor has been in existence for the past fe decades no mathematical modeling of a small reciprocating compressor as published until 1958.1 Evidently, the lack of intensive ork in this area may be attributed partly to the unavailability of large and fast digital computers hich are a prerequisite to any such study. Considerable effort has been devoted to this problem in the past five years and a fe 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 beteen 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 aves in the discharge line). Only rarely such an exact analysis, hich treats all the above aspects simultaneously, is pursued. Instead, some simplifying assumptions hich 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 reciprocating compressor analysis is the valve dynamics. Furthermore, even hen a rigorous valve dynamics analysis is folloed, some "unexplained" behavior of discharge valves in particular is observed. 2 On occasions it may therefore be desirable to bypass the valve analysis all together, provided of course that e are not interested in the valves themselves. Although such a simplified analysis has obvious shortcomings, it may still yield valuable information as ill be shon belo. The valve dynamics analysis has been bypassed in our analysis by making the folloing four assumptions (see Figure 1): 1. The discharge valve opens hen the pressure in the cylinder exceeds the discharge pressure, p 0, by Llp. 2. rre suction valve opens hen the suction pressure, ps, exceeds the cylinder pressure, by Llps. 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 o 0 radians after top dead center (TDC). Available experimental evidence 2 3 indicate that the values of LlpD is a fe tens of psi ( 40 psi) hereas Llp is a fe psi < 4 psi). The assumed value of LlD 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. Hoever, 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 as made in the heat transfer analysis that the thermal impedance of the cylinder all is negligible and hence its temperature is equal to the oil temperature in the case hich is knon. RECIPROCATING COMPRESSOR CYCLE Figure 1 is a schematic representation of the compressor cylinder pressure vs. crank angle hich our model attempts to simulate. It describes the folloing 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 la of thermodynamics for an open system to the control volume hich is bounded by the cylinder alls and the piston yields, T T n R V T W -ct-3cv+k/s v v here 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 (1) 502

is the ratio beteen constant pressure and constant volume specific heats, R is the universal gas constant, 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) e have used the equation of state for an ideal gas and e have neglected the change in the potential and kinetic energies. The last term in Equation (1) is the rate of change of mass ithin 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) e have to substitute proper expressions for V, W, and Q. The displacement volume V can be found from geometrical considerations and for the system shon in Figure 2 is given by: V = XcAp = [Leo+ R 1 (1- cos8) + R 2 (1- cos )) Ap here A is the piston cross sectional area, X is the pis on position ith respect to the valve plate, Leo is the clearance distance beteen the piston face and the valve plate at TDC, R 1 is the crank length, R 2 is the connecting rod length, e = t is the crank angle, is the angle beteen the connecting rod and the cylinder axis, 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 flo rate W can be computed by thepf lloing equation hich describes the mass flo rate of a compressible ideal fluid through a constriction, KYAdsyus /1 - (Ad /A 7 s us here K is a discharge coefficient, Y is a compressibility 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 donstream respectively. The density y may be found from the equation of state, y... = RT V The maximum mass flo rate occurs hen the velocity through the constriction has reached the sonic limit. This condition may be found from the folloing relationship, pds = (-2-) r c k pus k + 1 When the critical pressure ratio r is reached, Equation (3) is modified as follo: p (1 - r ) /1 - (Ad 1 A ) s us ; " us c yus (2) (3) (4) (5) (6) The rate of heat transfer from the cylinder all t<' the gas is given by Q = A(8) U(8) (T - T) (7) here A(6) is the heat transfer area (a function of the crank angle) through hich heat is transferred ith a coefficient U(8) and the subscript designate the cylinder all. The gas pressure in the cylinder can be computed directly from the equation of state of the particular gas hen V, Wand T are knon. All the gas properties such as density, enthalpy, viscosity, etc. can be computed from published data. In addition to the above analysis e have also evaluated the pressure drop and heat transfer through the suction and discharge tubes assuming a steady rather than pulsating flo. 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 as used to obtain a simultaneous solution of the three rate equations for W, Q, and T hich in turn as used in the pressure computations. The volume V as evaluated directly from Equation (2). The solution as started by assuming initial conditions for the temperature, pressure and mass flo and as pursued until the computed values at the end of the cycle agreed to ithin the desired accuracy ith the. assumed initial conditions. The mass flo rate, m, as evaluated by, 1 2tA-A here the subscripts on the time t refer to the points shon in Figure 1. RESULTS Some typical model predictions are displayed in Figures 3-5. Figure 3 shos the cylinder gas pressure. Figure 4 shos the gas mass in the cylinder and Figure 5 shos the p-v diagram. Comparison beteen these figures and available experimental data (e.g. references 2 and 3) shos that the model prediction is in general agreement ith experimental data and ith a more rigorous analysis.2 The cylinder gas pressure (Figure 3) is characterized 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 cylinder (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 ill remain in the cylinder hen re-expansion starts and a higher volumetric efficiency ill be obtained. The to constant mass "levels" in (8) 503

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 hich is shon in Figure 5 clearly demonstrates the deviation from an ideal compression-expansion cycle. c D I 6D f- Ē---P D CONCLUSIONS The simplified analysis presented herein does adequately simulate the gas pressure, temperature, volume and mass flo rate under.various pressure ratios. Consequently, it may be used reliably to investigate the effect of the folloing 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 information is of utmost importance in a compressor design. REFERENCES 1. Brunner, W., "Simulation of a Reciprocating Compressor on an Electronic Analog Computer", ASME Paper Number 58-A-146. Presented at the ASME Annual Meeting, Ne York, N.Y., November 30- December 5, 1958. 2. Gatecliff, G. W., "A Digital Simulation of a Reciprocating Heretic Compressor Including Comparisons ith Experiments", Ph.D. Thesis, The University of Michigan, 1969. Crank L.. = Q} t--------1--ps <...) BDC TDC BDC Crank Angle. degrees Fig.!-Schematic representation of one cycle Piston Connecting Rod l;l -Dp L ==Clearance : co Fig. 2-Schematic of the crank-connecting roo-piston system c 3. Wambsganss, M. W., Jr., "Mathematical Modeling and Design Evaluation of High-Speed Reciprocating Compressors, Ph.D. Thesis, Purdue University, 1966.. ::.; (/') Clearance::::: 1.5% 280 t.p 0 ::::: 40 PSIA 240 200 LlP s::::: 4 PSIA o 5 = 17.2 6-6 a... D-,_ a} P 5 =17.7PSIA :::J 160 "" P 0 = 214.7 PSIA a... m = 22.2 LBM/HR "" 120 ('_!) 80 40-80 -40 0 40 80 120 160 200 Crank Angle, degrees Fig. 3-Gas pressure in the cylinder 504

C"t"\ 0.10 0 -X 0.08 "" Clearance== 1.5% fi.p D == 40 PSIA fi.p S::::: 4 PSIA 6 5 =17.2 "" 0.06 6 ::::6 c3 0.04 0.02 D P 5 :::: 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 Crank Angle, degrees Fig. 4-Gas mass in the cylinder <( 280.00 240.00 Vi 200.00 c... ai s 160.00 "" 1!l 120.00 (g 80.00 Clearance== 1.5% f\.pd:::: 40 PSIA f\.p S :::: 4 PSIA 6 s = 17.2 6D = 6o PS:::: 17.7 PSIA P 0 :::: 214.7 PSIA m= 22.2 LBM/HR 40.00 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 10 3, tt 3 Fig. 5-P-V diagram 505