Analytical Model of an Oil-Free Screw Compressor

The use of analytical modelling is a signifi cant way in reducing of experimental costs in re search and development. An analytical model of an oilfree screw compressor is described and u sed to study the effects of rotation speed, compre ssion losses and difference of built-in and exter nal pres• sure ratio on compression process. The in fluence of gas pulsation in the discharge piping upon the working process of the screw compres sor is included. INTRODUCTION With the development of the computer art a nalytical models are asserting themselves in a n ever increasing measure. By them costly experi ments can be replaced to a certain extent. Among positive displacement compressors the modelli ng technique has been most highly developed o n reciprocating compressors. Their present .s tandard permits the course of the compressio n and the motion of the valves to be comprehensi vely treated and also the effect of the connected piping to be included, Dependences thus calc ulated are usually the basis for optimalization of the planned equipment. The good experiences gained from the mod elling of reciprocating compressors were the inc entive for the creation of a similar model of an oil-free screw compressor. The concept o f the analytical model corresponds to screw com pressors which the CKD Praha, Compress or Division , has been manufacturlng al ready for a number of years (Fig.l) in the licenc e of SVENSKA ROTOR MAS KINER A. B. Thei r active parts are two rotors engaging with ea ch other the mutual rotation of which is synch ronized by a set of gears. The rotors, toget her with the compressor shell, form four paire d chambers which are the working spaces. T he cycle of suction, compression and discharg e takes its course, with these chambers, wi th a phase displacement of one quarter of a re volution and causes that the fourth harmonic component, referred to also as chamber frequ ency, manifests itself in both the noise spectrum and the vibration in a significant way. 356 Discounting the possibility that vibration b y the chamber frequency may be caused by conta ct of the rotors such a case would lead very q uickly to a breakdown the most probable cau se of vibration is pulsation in the discharge p iping. Similarly as· with piston compressors its le vel will depend· on the dimensions of the piping and will, moreover, be markedly aifected by th e pressure in the chamber at the moment the discharge port is opened. These facts are we ll known and have been experimentally prove d many times. Experimental research becomes, however, very costly when one endeavours to find an optim um design or optimum oper11ting conditions for a given case. A suitable means to rationalize this work is the modelling technique. However, it is only rarely that an analytical model can be produced which takes all substantial influence s into account. Only some of the dependence s can be studied, depending on the purpose whic h the model is intended to serve, all others whic h aifect the process under observation in a smaller measure being disregarded. The proposed analytical model of a screw compre-sor was built with the aim of observing the working process of the compression in conjunc tion with the dynamic processes in the discharg e piping. The relatively uniform behaviour of the suction of a screw compressor and thus also i ts small effect on the dynamics of the gas in the suc tion piping were the reason why the dynamic phen omena in the suction piping were not considered. Fig. 2 shows the diagram of a screw comp ressor with a discharge piping which leads into a noise damper. The diagram represents a model t he individual parts of which will be analytically d escribed below. COMPRESSION SPACE The compression space consists of four pa ired chambers the actio61 of which is mutually d isplaced in phase by 90 . A change of the volum e of one paired chamber can be expressed by m eans of the model shown in Fig. 3. The actual w orking space of the chamber is replaced by an equ al volume in the shape of an oblique prism in wh ich a partition is moving. The volume of the wor king space determined in th1s way changes from Ot.t to oc. It and its change is protted against angle ol , also in Fig. 3. With screw compressors the compression process is affected by untightnesses to such an extent that, for purposes of the calculation of the compression, it cannot be disregarded. The calculation is made by steps and one calculating step corresponds to time interval Lit in which the main rotor turns through angle L1 0(. Thus td;;: ~01. The volume quantity of gas which escapes from the chamber to the suction space in time interval L1 f is determined by the relation I t1 Uj = ( 'ls ~s) at· A-M where r at-(] A= ~of 2 :, R rt·lt-e'sp;s )~ The resultant pressure in the chamber in the course of the compression will be, after the main motor has turned through angle t1 ex and after time Lit p. _ p. ( Vi )~. t+f .t Vi-LIVt"+LIUi where Vi= f(cx.,j and Ll Vi in interval o c2 0 <i~ ~~ .dVi = K, (0<1.."-o<.,).tJ 0t L1 Vi = K 2 L1 oc L1Vi = K., (o<¥-o<i)ilcx.. K., and K2 are constants corresponding to the dimensions of the rotors. The expression "/~ repr~s.ents th~ total untightness of the chamber, coeff1c1ent ~ 1s the specific untightness referred to the area of the face of the main rotor. The penetration of the gas through the untightnesses into suction space has the consequence that, in the suction space, gas of suction temperature Ts ·mixes with gas the temperatures of which correspond to the temperatures in the individual chambers. Gas which has escaped in time At will have temperature Tt_· and will occupy, on the suction side, space ( . ) ae. -1 JUU. "" _b_ '!A-Lit t~s Ps The resultant temperature on the suction side will be determined by the relation ( ,·=# ") e"=N 11 T, /1. ~ vT -1 LIVt: ·Ts +r Ju,:·Ti s p:.f t=l s = ------~~--~----~~-----where N= 360 t1cx. 9s Vr 357 Untightness coefficient :f is an important factor which affects the course of the compression and may therefore be considered one of t:P.e criteria of the quality of the compressor. However, the determination of it is difficult, for the clearances measured while the machine is at rest need not correspond to the operating values. A method which permits at least an approximate appraisal of the untightness during operation consists in a comparison of the measured and calculated values of the delivered gas and of the discharge temperature of it. The quantity of gas escaped from the working space to the suction space depends not only on the untightnesses but also on time. Therefore the course of the compression will depend also on the speed of the machine. The connection of the working space with the discharge starts as the discharge port is being opened and takes its course within a range of angles («2 -D<o}< O<i< 01."' . , Fig. 4. In an ideal case there should be an equality of pressures in the working space of the compressor and in the discharge chamber at the moment of opening. Since such a case is an exception rather than the rule the analytical model reckons, in the initial phase of the opening of the discharge port, with a gradual equalization of pressures, The volume quantity of gas which escapes from the chamber to the discharge in interval Ll t will be I .1ai = ( :;i) ~8-Llt where