Computation of a Cooperation between Reciprocating Compressor and Complex Piping System Including Mufflers

Transcription

1 Purdue University Purdue e-pubs International Compressor Engineering Conference School of Mechanical Engineering 198 Computation of a Cooperation between Reciprocating Compressor and Complex Piping System Including Mufflers W. J. Jungowski J. R. Piechna Follow this and additional works at: Jungowski, W. J. and Piechna, J. R., "Computation of a Cooperation between Reciprocating Compressor and Complex Piping System Including Mufflers" (198). International Compressor Engineering Conference. Paper 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 Herrick/Events/orderlit.html

2 COMPUTATION OF A COOPERATION BETWEEN RECIPROCATING COMPRESSOR AND COMPLEX PIPING SYSTEM INCLUDING MUFFLERS Wiktor M. Jungowski, Janusz R. Piechna Warsaw Technical University Poland ABSTRACT Proposed scheme of computation in a general outline is following. In the first stage by use of a method based on the wave theory pressure spectra for the whole installation are predicted. As a boundary condition a standard expansion of piston velocity into Fourier series is used. Then by the inverse Fourier transformation pressure in the outlet section of the compressor muffler is computed. This pressure now becames a boundary condition in the method of numerical simulation. This simulation comprises only a small part of the installation, i.e. between the compressor and the outlet of a muffler. Such a simulation is well known, but usually the influence of the rest of pipeline is neglected. This simplification is not always a proper one. INTRODUCTION The main engineering problem of reciprocating compressor plant design lies in junction of a compressor ~ith a technological installation in such a way which simultanously fulfils all technological requirements, lightens the influence of pipeline pressure pulsation on the run of the compressor and excludes mechanical resonance. Chemical installations are always rather complicated. They are composed of many volumes like irrigation coolers, separators, dampers and also many branches and conections. Designer needs two basic information i.e. about v. ork of compressor ( pol'der consumption, volumetric efficiency, valve plate impact velocity etc.) and about the pipeline (pulsating pressure spectrum at basic points of the system) This problem may be solved by means of electroacoustical an~logy using special analog computers [ 1], ;~hich are not everywhere available. Digital computers however, are now easy of acces,thus a numerical method should be also taken into account. There are many well known methods of numerical computation of nonstationary flow. The most exact is a method of cbaracteristics [ 2 J or Lax-Wendroff method. But when the installation is complicated or contains some volumes the solution loses the accuracy. So in the complicated cases simpler methods acheave almost the same accuracy. The problem of accuracy is less connected with a method but rather with transmission of particulars of a real installation into a model. OBJECT OF INVESTIGATION A simple one-cylinder compressor joined to installation shown in Fig. 1 was an object of investigation. For simplicity suction valve chamber was opened directly to the atmosphere. Geometry of the pipeline was not optimised. The stand was s~pposed to be a test bed for chekino methods of predicting a pulsating flow. In Fig. 2 a computed damping characteristic {transmission loss ) of the applied attenuator has been shown. WAVE THEORY In 1966 Kuhlman [ 3] has used a solution of wave equation for predicting the damping characteristics of complicated pipelines. In H' 67 Chen [ 4] presented a graphical method for an improved solution with frictional terms. In 197 Abe, Fuiik2wa and Ito [::] proposed a matrix method for solution of a set of equations. This method made possible the construction of an almost automaticaly operated elastic program of computation. Ve have used a solution obtained by Chen [ 4] and have built a numerical program based on the method proposed by Abe, Fuj ilt,arw and I to [ 5 J The following set of kno~n solutions of wave equations 389

3 ((i-)au)wx) ( {i-;,uo)wx) u(x) ::A e Oo+ Uo - Be Oo- Uo ((i- >.u )WX) ( (i- ),u )Wx) a(x) =A e ao+uo. +Be Oo-Uo has been used with different boundary conditions, e.g. volumes, branches and abrupt chanoe in cross section. As an excitation function Fourier series expanssion of piston velocity, during time interval of valve opening, has been used. Pressure spectra at some points of the investiqated installation have been calculated-and measured. In Fig. 3 spectra corresponding to the valve chamber (point 1 in Fig. 1) are shown. Predicted and measured values of higher harmonics(- 2) exhibit a significant discrepancy. This results from oversimplified boundary conditions (piston velocity) and threedimensional phenomena in a damper at high frequencies. The spectrum computed for the part downstream of the muffler (a closed end of the branch, point 2 in Fig. 1) shows a better agreement with experimental data ( Fig. 4 ) From our experience with this method comes out that it is effective particularly in the case of a complicated configuration. The method, however, has two serious disadvantages: monlinear elements of the installation need an iteration procedure and it seems not possible to take directly into account a dynamics of the valving system. But the influence of the leakage or opening delay time have been investigated using the wave theory [ 6 J Elson and Soedel [. 7 J h~ve applied a wave theory w1th 1terat1ve procedure for predicting the interaction of valve with pulsating flow in long duct. METHDO OF SIMULATION For computation of an interaction between a valve and pulsating flow the method of numerical simulation may be used. A set of ordinary differential and algebraic equations describj_ng the work of ell components of the system must be collected. For the nonstationary flow in a pipe of constant cross section, the following equations have been used:!j p =.16 (!:!!:!_ + "A u2) LlX df 2d!J u = 1..sliL LlX.foCi2 dt It is an lumped parametar approach to simplified, equations of mot:i.on, conti-: nuity and state with density change be1ng neglected (low Mach number) and convective derivatives of pressure and velocity omited. Similar equations in linerised form but in differen~ ~aye have been used l::;y Grover [ 8] and t.uszczych [ 9] For the simulation of a thermodynamic process in a cylinder a polytropic compression has been used:!!.& = n Pc ( Q _ Q. _ dv) dt v 5 tt dt The leakage was not considered and the heat transfer through the walls has been included in the polytropic exponent. The change of the cylinder volume has been des c r i be d by : V = Vst((1-cos wt) +r/1/4(1-cos2wt))/2+ Vm For describing flo~ through the valves the following simple equations have been.applied: for suci i,., - and for dischar~e where h is a valve plate lift. The discharge valve ~as considered as hawing one degree of freedom: The force acting on the valve plate is a product of pressure difference!j p across the valve times an effective force area A The coefficients A, oc 5 and o<:.d h~ve been determined exberimentally by steady flow tests and then assumed constant. A solution of such a set of equations might be performed by an analog computer [ ], but in this case a digital computer with a symulating language MIMIC has been used. If the method of eimul8tion is joined to the method of characteristics it m~y be noticed that (tor moderate pressure amplitudes ) the results are almost the same, but the required time for programing is sign;ficantly shortened and computer time bonsumption enlarged. The method of simulation is very effective but only for not too complicated systems. PROPOSED MIXED METHOD The proposed method comes out from an idea that a strong interaction between valve and puls~_flo~'l depends primarily on the flow between the compressor and the 39

4 damper. The valve plate oscillation may cau~e the flow pulsation With high frequencies only. The period of oscillation is shorter than the time of valve opening. But the influence of the rest of installation on the pressure pulsation in a muffler not always can be neglected. In the mixed method both previously described methods {a wave theory method and a method of simulation) in two steps have been used. First pressure spectra, for some points of the installation, by the wave theory, have been obtained. Then by the invers Fourier transformation a pressure pulsation in the muffler has been found. This pressure as a boundary condition in the method of numerial simulation is applied. This simulation comprises only a small part of the installation between the compressor and the muffler. Eventualy the Fourier transformation of flow through the valve may be used as an improved boundary condition in the wave theory method applied again. In Fig. 5 and 6 pressure pulsation at point 1 has been shown. The curve of small amplitude in FiQ. B corresponds to a partial simulation of the system, based on the common asumption that the pressure pulsation downstream of a muffler may be neglected. The smooth curve comes from wave theory, so without taking into account walve motion. The agreement with an experimental curve is rather poor. Fig. 6 shows resul~ts obtained by: full simulation of the installation, by proposed mixed method and by experiment. Full simulation and experiment agree very well. Generally mi 7 xed method shows greater discrepancy than full simulation but much smaller than ~ave theory or partial simulation. Also the information about operation of valves was secured. In the case of this simole geometrical configuration a computer' time consumption by full simulation is more than 5% higher th~n by mixed method. The time. increases rapidly ~hen the installation becomes more complicated. The applied muffler was not very effective and this has strenqthend the influence of the downstream par~ of the installation on pressure pulsation. In the case of a pro 7 per muffler the influence may be less significant. COI'<CLUSION Method of simulation considered above is very simple and no~ many simulating languages as MIMIC or CSMPL are available. Eut for simplicity in programing must be paid with great computer time consumption, so application of this method is limitted to simple installations only. Method based on wave theory is fast and effective in the case of complicated configurations but no information about the operation of valving system is obtained. Proposed scheme gives full information and the accuracy of the mixed method seems sufficient for engineering applications. NOTATION a speed of sound a pulsation of speed of sound A area c damping.. coefficient d pipe diameter h valve lift --.. k spring coefficient L lenghtof the valve slit m valve plate mass n polytropic exponent p pressure Q rate of flov, through the valve t time u particle velocj.ty u pulsation of particle velocity v cylinder volume X distance c< flo\1\: coefficient )\ friction coefficient jj density w rotational velocity SUBSCRIPTS c cylinder s suction d discharge v valve o mean value m dead space st static REFERENCES 1. ~.Nimitz "Pulsation and Vibration", Pipe Line Industry 1S6S~ August pp.36-?~ and Sptember pp J.F.T.MacLaren, A.B.Tramschek, A.Sanjines, O.F.Pastrana "A Comparison of Numerical Solutions of the Unsteady Flow Equations Applied to Reciprocating Compressor Systems", Journal Mechanical Engineering Science Vol.17 No.~ P.Kuhlmann "Berechnung von Schwingungen in den Rohrleitungen von Kolbenverdichtern", VDI - Forschungsheft 1866, ~lo.516, pp Y.N.Chen "Calculation of Gas Due to Simultaneous Excitations in Reciprocating Compressor Piping Systems with Allowance for Frictional Effekt and Temperature Change in the Flo~", Journal of Sound and Vibrations 1967,5/2/,pp

5 5. T.Abe,T.Fujikav,a,S.Ito "A Calculating Method of Pulsation in a Pi~ing System", Bulletin of the ::JSI<IE Vol.13 No.59,197 e. Vi.::Jungowski,::J.Piechna ''\'.'pj:yw parametr6~ spr@±arki tlokokej na widmo pulsacji ci~nienia" Influence of Operating Conditions of a Reciprocating Compressor on the Spectra of Pressure Pulsation ZN AGH:~o.72S' J.P.Elson, W.Soedel "Simulation of the Interaction of Compressor Valves with Acoustic Back Pressures in Long Discharge Lines" Journal of Sound and Vibration 34/2/,pp S.S.Grover "Analysis of Pressure Pulsations in Reciprocating Compressor Piping Syste~s", Journal of Engineering for Industry 1966,888. pp s. M.~uszczycki "Aproximate Calculation Method for the Run of Changes of Pulsation Pressure in Reciprocating Compressors Pipelines" Proc.2nd Compressor Technology Conference, Purdue University, July 1974, pp J.Piechna "Analogowy model spr~tarki t~okowej" Analog Model of the Reciprocating Compressor ABM,Z J.Brablik "Computer Simulation of the Working Process in the Cylinder of a Reciprocating Compressor with Piping System" Proc.2nd Compressor Technology Conference, Purdue University, July 1?74,pp T 2 ldbi Fig. 2 Dampin2 characteristic of the muffler (transmission loss) 3~ % d8 2 i- o computed I measured P. = w3jy..2 o m - 2 '? 5 15 Harmonic number n 2 Fig. 1 Scheme of installation Fig. 3 Spectra of pressure in valve chamber (point 1 in Fig. 1) 392

6 3..., % Tb computed d8 I measured 2 I'". P. = 3 N2 o m ~ > > l c f Harmonic number n Fig. 4 Spectra of pressure at closed end of the branch (point 2 in Fig. 1) _. "' ~IE2 S?" '- d:~b Q.l... =:1 16 ~- <1.1 '- Q. ~ 1:4 -~ & m~asured wave theory... simplified simulation '& Crank angle.)'; (deg) 3 36 Fig. 5 Cylinder pressure p, discharge valve chamber pressure pv, dischafoe valve flow rate Qd, valve lift h 393

7 2.4 -measured ---- simulation mixed method!jt o.:...: Ql ~ 2.1.._ cz., -Q 2. ~ u ~ 19 ~ 1. ~ 5.c:: u a -~ PJJ Crank angle 24 ( deg) 3 36 Fig. 6 Cylinder pressure p, discharge valve chamber pressure p, dischafge valve flo~ rate ~d' valve lift v h 394