Review of Simple Mathematical Models of Valves in Reciprocating Compressors

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1 Prde University Prde e-pbs nternational Compressor Engineering Conference School of Mechanical Engineering 197 Review of Simple Mathematical Models of Valves in Reciprocating Compressors.F. T. MacLaren University of Strathclyde Follow this and additional works at: MacLaren,.F. T., "Review of Simple Mathematical Models of Valves in Reciprocating Compressors" (197). nternational Compressor Engineering Conference. Paper 8. This docment has been made available throgh Prde e-pbs, a service of the Prde University Libraries. Please contact epbs@prde.ed for additional information. Complete proceedings may be acqired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at Herrick/Events/orderlit.html

2 A REVEW OF SMPLE MATHEMATCAL MODELS OF VALVES N RECPROCATNG COMPRESSORS ohn F. T. Maclaren, Reader in Thermodynamics, University of Strothclyde, Glasgow, U.K. NTRODUCTON Since 195 a nmber of relatively simple mathematical models have been developed which describe a compressor, its valves and the working flid. n general the analysis yields two non-linear differential eqations which relate the many parameters involved: a "flow" eqation which relates the pressre difference across the valve to mass flow rate and valve opening and a "dynamic" eqation which describes the valve movement. Simltaneos soltion of the eqations by iterative methods yields pressre difference across the valve and valve displacement as fnctions of crankangle and sbseqently allows the constrction of a conventional pressre-volme diagram. To describe the complex physical sitation in mathematical form reqires that a nmber of assmptions be made. t is sally assmed that infi'nite volme receivers exist at inlet and discharge and hence that pressres Pi and Pd, Fig. 1, remain constant: this simplification permits the stdy of a compressor and its valves in isolation from the associated receivers and pipework system. The vale of any sch model depends on its ability to describe actal events with sfficient accracy. Hence, assessment of the validity of the model reqires that accrate experimental data be available from a compressor for the prpose of comparison. This paper reviews briefly a nmber of investigations directed to this end. SEQUENCE OF EVENTS N THE CYCLE A mathematical model of a compressor and its valves mst accont for the series of events in a cycle. This series of events may be designated (a) re-exponsion, (b) sction, (c) compression, (d) discharge. A starting point mst be chosen and one sch is point, Fig. 1. nitially it is assmed that at point the cylinder pressre is eqal to the (constant) discharge pressre, Pd, and that the piston is at inner dead centre. Event (a): The clearance gas expands from point to point 1 where the sction valve begins to open. f there is valve spring preloading or if significant oil stiction exists at the valve seat, valve opening is delayed and the clearance gas will contine to expand in the cylinder to a pressre less than the (constant) sction pressre, P;. Event (b): At commencement of the sction process, near point 1, as the sction valve is opening, both the flow eqation and the dynamic eqation are solved simltaneosly. When the valve reaches the stop, and remains there, the dynamic eqation is not applicable and the flow eqation alone is sed to compte the pressre difference across the valve. Later in the stroke, de to the redction in piston velocity, the pressre difference across the valve redces and the drag force on the valve thereby redces. When this force becomes less then the valve spring force the valve will begin to close and dring valve closre both the flow and dynamic eqations are applicable. However, when the valve reaches the stop it may bonce, or, if the pressre difference across the valve is insfficient to hold it open against the force exerted by the valve spring(s), the valve may fltter. Sch fltter may contine throghot the sction process. Zero coefficient of restittion de to, sey, the damping effect of oil on the valve stop, wold mean zero valve bonce, yet fltter cold still occr. f valve bonce or fltter occrs, both eqations are applicable. The valve may fltter dring the closing phase and may bonce after impact on its seat. At piston reversal at oter dead centre the sction valve may or may not be closed and the cylinder pressre may be greater or less than the sction line pressre. Ths nder certain circmstances "blow-by" can occr before the valve is finally seated. Event (c): The gas is compressed from point to point 3 where the discharge valve begins to open. tf there is valve spring preloading or if significant oil stiction exists at the valve sect, valve opening is delayed and the gas in the cylinder will contine to be compressed to a pressre greater than the. (constant) discharge pressre, Pd. 18

3 Event (d): After commencement of the dlscharge process near point 3, the appropriate flow and dynamic eqations for rhe -discharge valve are solved dring opening, bonce, Fltter and closre: if the valve remains open at rest against the stop, only the flow eqation is applicable. At piston reversal at inner dead centre, the discharge valve may or may not be closed and the cylinder pressre may or may not be eqal to the discharge pressre so the comptation is contined till the valve is finally closed. This ends the seqence of events bt the finish of the cycle may not be at the initial point. After re-expansion from the finish of the cycle to point 1', Fig. 1, comparison may be made with point 1. f the two p6ints, 1' and 1, do not coincide then the original assmptions abot conditions at point were in error. f the discrep::mcy is considered appreciable the comptations for the cycle may be repeated sing a revised starting point. Hence soltion of the eqations yields the displacement of each valve and the pressre in the cylinder dring the cycle. REVEW OF SMPLE MATHEMATCAL MODELS Costagliola (1) at MT. in 195 prodced the first worthwhile mathematical model of a reciprocating compressor and its valves. The analysis of valve dynamics was the primary concern. Corresponding experimental work was condcted with a 3! in bore x 4 in stroke single cylinder air compressor, fitted with flexing reed "feather" type valves, in the speed range 9-18 revmin. The soltion of the non-linear differential eqations by graphical methods was too t_edios for the model to be of interest as an indstrial design tool. Many theoretical pressre and valve displacement diagrams were calclated, and althogh these were not shown sperimposed on experimental diagrams, it was claimed that the model was "essentially correct". Widespread se of digital compters has allowed later investigators to solve the eqations rdpidly and has allowed the basic model to be refined and extended. Virtally all the models now in existence ore based to some degree on the pioneer analysis by Costagliola. Wambsganss and Cohen () of Prde University in 1967 developed a similar model and made comparison with experimental records for a! hp, 36 rev'min hermetically sealed compressor fitted with reed type valves and pmping air or R1. Fig. shows the extent of the correlation obtained l:ietween the analytical and experimental records. This correlation may be jdged to be good when allowance is made for the difficlties inherent in instrmenting sch a small compressor and its valves. (From a similar exercise Maclaren and Kerr (1 ) of Strathclyde University, conclded that an analytical model cold provide qalitative reslts mch more rapidly and cheaply than on extensive experimental programme. t shold not be inferred from this statement that sch models will replace all development testing). n the investigations at Prde University (), attention was paid to the details of the dynamics of reed valves and several degrees of freedom were allowed. (n most other stdies it has been assmed that the valve had only a single degree of freedom). Amongst the conclsions drawn were (a) a single degree of freedom approximation was notsfficient to represent the valve reed dynamics in a high speed compressor (b) a condition of valve "stiction" existed which cold have a large effectot low vales of compressor pressre ratio and (c) a damping term in the dynamic eqation for the valves was considered to be important. Borisoglebski and Kzmin (3) in Rssia in 1965 combined the simltaneos flow and dynamic eqations into a single non-linear differential eqation as Costagliolo (1) did in his doctoral thesis. The varios geometric and operating dimensions were arranged into a small nmber of lmped dimensionless parameters which were evalated from nomograms. The soltion of the single general eqation was by an iterative process sing the Rnge-Ktta procedre. Fig. 3 shows the extent of correlation achieved between analytical cmd exoerimental reslts for a mlti-ring sction valve in a for stageaircompressor. Upfold (4) of the University of New Soth Wales in 1967 stdied the behavior of a ring-plate sction valve in one cylinder (5 in bore x 4 in stroke) of an eight cylinder doble V type air compressor. Oscillogroms of valve displacement were compared with those predicted by an analytical model similar to Costagliola's bt extended to accont for heat transfer and valve damping. Upfold measred the coefficient of restittion of the ring plate valves by sing a high speed photography techniqe. Upfold sed the model to examine the effect of changes in varios dimensionless lmped parameters on compressor performance criteria, in particlar their effect on valve impact velocity. From Fig. 4 (Test 719) it cold be claimed that the correlation between analytical and experimental reslts was good. However, the valve dis1placement diagram when fltter was present, as in Test 73, indicated that this model wold be seriosly in error if sed to predict the point of final closre of the sction valve, impact on the valve seat, or "blow-by" loss. n this case the experimental diagram was the more credible, sggesting errors in the complex compter programorerrors in the vales of empirical coefficients sed. Troversari and Lacitignola (5) in taly in 197 constrcted a model based on those by Costagliola (1) and Maclaren and Kerr (6). Modifications were mode to accont for damping de to the pnematic type of valve stop sed with the mlti-ring plate valves. As in most models, discharge, drag and damping coefficients were assmed to have constant vales. Provision was also made for si.mlating delay in valve opening de to oil stiction effects. Fig. 5 (a) compares analytical and experimental records for a discharge vqlve monted at the inner end of a doble-acting cylinder. The pressre plsations within the cylinder were qite small. Fig. 5 (b) shows this valve when it failed to reach its permitted lift and that severe valve fltter and corresponding large pressre flctations ensed. This nsteady sitation creates a severe test of 181

4 of a model and the correlation between analytically predicted and experimental reslts appeares to be good. Fig. 5( c) is the record of a sction valve for the inner end of the doble-acting cylinder when late valve closre occrred. Fig. 5 (d) shows the comptation for a discharge valve when opening was assmed to be delayed by 15 crankangle degrees. As a reslt the valve impact velocity at the stop increased from 1 4 m/s to m/s. (Sch a delay is nlikely to be de only to oil stiction: the rate of increase of pressre difference across a discharge valve is so large that oil stiction shold not delay discharge valve opening by this amont.) The analytical comptation of valve displacement (Fig. 5) ended at piston reversal. n the seqence of events in the cycle otlined in Fig. 1, the behavior of one valve as it affects the commencement of opening of the other valve is acconted for. t wold appear that the model of Traversari and Lactignola was not sfficiently complete to accont for sch valve interaction. Traversari and Lactignola conclded that the differences between analytical and experimental reslts were de to (a) pressre plsations in the sction and discharge piping, (b) delay in opening and closing on accont of valve stiction (c) error in the selection of vales of empirical coefficients and the assmption that they were independent of valve lift, (d) experimental errors. Taber and Blomsa (7} in Holland in 1971 examined the seqence of events throgh a complete cycle. The model inclded a simple simlation of inlet and discharge pipework. The sction and discharge valves stdied were mlti-finger reeds placed circmferentially rond a cylinder, 16 mm bore x 11 mm stroke. Althogh comparison was made between analytical and experimental records when p.mping R, Fig. 6, shows thecomparison when pmping air at two compressor speeds. The increased fltter of the sction valve at the lower speed is apparent. t cold be conclded that this model described valve behavior adeqately for many practical design prposes. Maclaren and Kerr (8) of the University of Strathclyde in 197 described a model developed from that of Costagliola. Comparison was made between theoretical and experimental reslts for a single stage, single-acting single cylinder air compressor, 6 in bore x 4 in stroke, fitted with :i in o.d. spring loaded single ring-plate valves at both sction and discharge. The test series was condcted with a ronge of speed rev/min and compressor pressre ratio Fig. 7 (a} shows that at the lowest speed there was considerable fltter of the sction valve. Again the presence of valve fltter provided a severe test of the analytical model to predict valve behavior. The lowest speed was near to the compressor rated speed and the sction valve displacement diagram sggests that the valve design shold be modified to redce the effect on valve spring ife of the excessive nmber of spring compressions per cycle. (The matter of sitable valve spring characteristics was discssed in a paper by Steindel (9} in Poland in 1964}. The experimentally measred valve displacement diagram at A in Fig. 7 (a) sggests that the valve was partially sticking on its gide, perhaps de to neven departre from the stop. Non-niform circmferential valve displacement may also accont for the sction valve chatter arond valve closre at 18. t may be that the valve displacement transdcer, sensing only one point of the valve, was recording a dying spinning penny action rcther than a niform valve bonce on the seat. Neither the partial sticking nor the neven seating of the valve plate was simlated in the mathematical model. Dring the experiments designed to assess the general validity of the model the sction valve operated withot significant flow restriction at compressor inlet, the inlet filter and pipework having been removed. Hence the area of the sction "loops" in Fig. 7 (a) and (b) corre$ponds with the sction loop in Fig. 1. (the lower shaded area) i.e. the sction plenm chamber pressre did not vary significantly from the atmospheric pressre Pi. However, this simple sitation cold not be created experimentally for the discharge valve. i.e. an arrangement cold not be mode to maintain Pd constant. The cylinder pressre (X), Fig. 7 Fig. 7(b}, the plenm chamber pressre (Y), and the pressre difference across the discharge valve (Z = X - Y) were each measred separately. Accrate records were difficlt to obtain and an element of ncertainty was involved in fixing the datm for the experimental trcces. Hence it was not claimed that the experimental records of pressre difference (Z) were sfficiently accrate for meaningfl comparison with the analytically predicted pressre difference (Z)'. t was apparent, however, that the plenm chamber pressre Y varied significantly and that the assmption in the analytical model that Pd remained constant was qestionable. Nevertheless, the correlation between theoretical and experimental reslts for both valves was considered to be sfficient for many design prposes: for example, to estimate the loss of volmetric efficiency de to sction valve throttling, the power consmption de to the "pls-loading" by the valves, changes in valve impact velocities at seat and stop with alteration of compressor speed, pressre ratio, valve lift, valve spring stiffness and preloading. Other similar analytical models have been constrcted by manfactrers bt do not appear to have been described in pblished literatre. EMPRCAL COEFFCENTS All the models referred to above are semi-analytical: several empirical coefficients have been inclded and generally these were evalated by experiment. Coefficients to accont for valve bonce (restittion) damping and stiction cold, in principle, be estimated by fitting analytical reslts from the mathematical model to accrate experimental reslts from a compressor bt this wold be difficlt to achieve in practice. 18

5 Pblished vales of coefficient of restittion vary from. (measred by Upfold (4) for a metallic ring-plate valve) to.365 (qoted by Borisoglebski (3) from a Rssian sorce). n the athor's experience an accrate vale of this coefficient need not be known when compting valve displacement. However, impact velocity and the coefficient of restittion are very relevant to valve plate srface stressing and valve spring srging and hence to valve life. Vales of damping coefficients were assmed and acconted for by Wambsganss and Cohen () who.stated that damping was a significant effect in high speed compressors fitted with flexing light-weight reed valves. Traversari and La tignola (5) also had to accont for this effect de to the signific-::nt pnematic damping incorporated in their ring valves. Maclaren and Kerr (1) showed, Fig. 8, the compted delay in the opening of a reed sction valve de to an assmed vale of oil stiction force. This diagram shows the reslting change of peak pressre drop across the valve and illstrates that at low evaporating pressres there is a possibility that a sction valve may open in the oil free condition bt fail to do so when oil is present at the seat. (The same vales of oil stiction force estimated from the sction valve were applied to the discharge valve and fond to have negligible delaying effect on the commencement of opening of the discharge valve). mportant empirical coefficients are those relating to gas flow, e.g. the coefficient of discharge for a valve and its associated passages and the coefficient of pressre drag reslting from the flow over the moving valve element. Sch coefficients have sally been determined for a particlar geometry by steady flow tests within or otwith the compressor. Althogh most investigators have treated flow coefficients as constants, these coefficients are fnctions of several parameters, principally the instantaneos vale of valve lift. Wambsganss and Cohen (), Ucer (11) and Kerr (13) expressed these coefficients as fanctions of valve lift. Fragmentary information relating to these coefficients is avail able in many pblications. To srvey the information wold be a difficlt task since athors seldom define flly the coefficients sed. For the same reason, too mch shold not be made of the difference between nmerical vales qoted since each may be valid within its particlar definition. A welcome bt perhaps wishfl otcome of a review wo.ld be to find that the flow coefficients, expressed in dimensionless form, sch as sed by Davis (1). did not vary mch from one type of valve to another. f this were so, the mathematical model cold be sed with some confidence prior to any manfactre for a compter oh:led design stdy of a proposed compressor and its valves. CONCLUSONS Mathematical models of a reciprocating compressor and its valves now exist in several contries. n general the validity of these models has been adeqately established for a particlar design by making comparison between analytical and experimental reslts. A cheap procedre is now available for the rapid stdy of some aspects of a design or the ikely effects of proposed modifications to a design. This shold permit more rational design and shorter experimental development programs. All models have been semi-analytical, containing a nmber of empirical coefficients. Most investigators have devoted considerable effort to evalating these coefficients by experiment. t is sggested that a comprehensive review of the considerable fragmentary data available is reqired and that the significance of each coefficient be more flly assessed before time and expense is allocated to frther experimental stdy of them. n the majority of models available to date, the effect of pressre variations de to the plsating natre of the flow in the system adjoining the valves has been neglected. These plsations can be of considerable amplitde and models shold now be developed to inclde a simlation of the gas flow pattern in the pipe-work of the system. To implement this proposal wold constitte a major extension of the stdy of compressors and valves and wold reqire considerable compter capacity. De accont wold have to be taken of the interaction between plsation effects and valve behavior. Many more bondary co,nditions wold hove to be satisfied.' REFERENCES 1. Costagliola, M. "The Theory of Spring-Loaded valves for Reciprocating Compressors". App.Mech., Dec. 195, 17, 4, p.415. "Dynamics of a Reed Type Valve'' D.Sc. Thesis, M.l. T., Wombsganss, M. W. & Cohen, R. "Dynamics of a Reciprocating Compressor with Atomatic Reed Valves" Proc. X lnt. Congr. Refrig., Madri::l, 1967, paper no Borisoglebski, A & Kzmin, R. W. "K raschet protsessov sayvamiya i negnetaniya porshnevych khampressorov" (Calclation of sction and compression processes in reciprocating compressors) Khimisheskoe Neftyonae Mashinostroenie nil, 1965, p Upfold, R. W., "A Stdy of Unsteady Flow in a Reciprocating Compressor" Ph.D. Thesis, University of New Soth Wales, Traversari, A. & Lacitignola, P. "Use and Calclation of Ring Type Valves for Reciprocating Compressors", Oanderni Pignone Hose ornal No. 16, Sept , Maclaren.. F. T. & Kerr, S. V. "Analysis of Valve Behavior in Reciprocating Compressors" Proc. X lp1 Cong. Refrig., Madrid 1967, paper no

6 7. Taber, S. & Blomsa, E.C. "Theoretical and Experimental nvestigation of Valve Movement and lnslotiormry Gas Flow in a Reciprocating Compressor" Proc. Xllllnt. Cong. Refrig., Washington, 1971, tcper rn.3.14 B. Maclaren,.F.T. & Kerr, S.V.!'An Analytical and Experimental Stdy of Self-acting Valves in a Reciprocating Air Compressor".Mech.E. Conference, "ndstrial Reciprocating and Rotary Compressor Design and Operational Problems" paper no. 3, London, Steindel, S. "Wplyw charakierystyki sprezyn no dzialanie zaworow plytkowych i no wskazniki ekonomiczne sprezarki". (The effect of spring characteristics on disc valve operation and compressor performance). Archivm Bdowy Maszyn V n, 1964, p Maclaren,.F.T. & Kerr, S.V. "Atomatic Reed Valves in Hermetic Compressors" 1.1. R. Commission!!!r Prage 1969, Annexe Ucer, A.S. "Unsteady Flow in Reciprocating Compressor Systems" Ph.D. Thesis, UMLS.T., Davis, H. "Effects of Reciprocating Compressor Valve Design on Performance and Reliability".Mech. E. Conference "ndstrial Reciprocating and Rotary COmpressor Design and Operational Problems". Paper No., London Kerr, S. V. "Stdy of AtoPatic Valves in Reciprocating Gas Compressors" Ph.D. Thesis, University of Strathclyde, 197 E".ZZl Stop Discharge valve displacement Zl Seat (Not to scale) 14 lbo 1 1 ::z " ::::; 1- :;;: w ::> w _j. (/) 5 3 -to -- TEO1ETC.t.L EXF'E111MENTAL. 7 3b qo SO C?/>,K ANGl-E (DEGEES) Stop= Volme Sction valve displacemew HE?METC CO.-tPii?E'lSO<'; Y4 hp ; 3bOO l<'ev/m ; R\; SUCTON P1ESSURE <;'-7L8F/; COMPESSO fi'ato ; iioe( n 1-1 OSCH>,l<'GE VALVE FRST MODE DNAPG FACTO<' 8 SECOD MODE DAAPNG FACTOR - FL\11 FORCE ON VALVE 1 \.BF SUCTO VAl-VE 1 5LBF FG_. PRESSURE- VOLUME AND VALVE DSPLACEMENT DAGRAMS. i=g. THEORETCAL-E(f'Ei?AENTAL Coi1RLATON : WAM8SCANSS AND COHEN. 184

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