CENTRIFUGAL COMPRESSORS SURGE SIMULATION

Transcription

1 U.P.B. Si. Bull., Series D, Vol. 75, Iss., 1 ISSN CENTRIFUGAL COMPRESSORS SURGE SIMULATION Virgil STANCIU 1, Emilian POPOVICI Various theoretial results were determined modifying the losing time of a entrifugal ompressor evauation valve. Hereby, it an be observed that, for short losing times, the system evolutions are similar with those found for the instantaneous vane s losure. While the losing time inreases, the system behavior grows toward a normal one espeially if the losing time overpasses too muh the period of an potential periodi proess. We determine some small osillations at the beginning of the proess, after whih the flow is slowly dereasing to zero, without anything spetaular. In the presented variants it an be observed that in the first moment, the omputed knots number and, obvious, the analyze time, have a great influene on the system osillations desription. A large number of knots results when the flow osillations number tends to infinite. It is obvious that the values of this vane s losing time ould not go down below partiular values, beause the analysis would lose the relevane in these ases. Not the same thing ould be said about the knots number, whih is reommended to be as high as possible. An interesting analysis of this phenomenon ould be made studying the ompressor system behavior in the ase of a finite evauation vane s losing time. Elaborating different theoretial models desribed in literature, the authors present an original mathematial model of simulation for the entrifugal ompressors surge. Keywords: entrifugal ompressor, evauation valve, flow osillations, losing time 1. Introdution Until now, only few generalities were defined in the speialized literature. Starting from these general equations, an analytial model was established for studying the kineti ompressors surge (aial and entrifugal). The analytial simulation was performed using this model. A flow harateristi was aquired on a benh through a ompressor test, whih ontains the variation for the adiabati effiieny ompression proess urves, together with the ompression degree variation ones.. Geometrial data of the ompression system One an onsider the priniple shema of a ompressor system, used effetively in a entrifugal ompressor test, like in Fig. 1, [1], []. 1 Professor, Aerospatiale Engineering Faulty, University POLITEHNICA of Buharest, Romania PhD, S.C. Blue Air Transport Aerian S.A., Buharest, Romania, popovii.emilian@yahoo.om

2 44 Virgil Staniu, Emilian Popovii M Mult. L 5 P, T, m V R.S. L 7 Fig. 1. The priniple shema of a ompressor system The main omponents of the installation were marked in the figure M, engine for ompressor drive; C, entrifugal ompressor; CA,.. air intake dut; R. A., air intake reservoir from the intake hamber; RS.., air outlet reservoir; 1, emission valve;, evauation valve; L l, the length of the onnetion pipes; Pl, T l, the air pressure and temperature in the ompressor system harateristi setions. Taking into aount that the analyti epressions of those urves are neessary in the net omputations, we tabular etrated the oordinates of few essential points for the distributions of π, η, based on the available images. The data are ontained in Table no. 1 and Table no.. Table no. 1 m red -1 1 π m red -1 1 η L 4 C L PT,, m CA L 1 V R.A. P, T, m L L 6 1 P, T, m Table no.

3 The entrifugal ompressors surge simulation 45 η % -1 π 1 1 m red 9% 8% 7% 6% Fig.. The distribution ofπ and η funtion of m red m red Hereinafter, few geometrial data are presented, whih are used in the net omputations V = 9.4 m A =.685 m For the two urves approimation, a polynomial funtion will be used in the form, [] ah 1 1 ah ah y = ah (1) 1!!! where

4 46 Virgil Staniu, Emilian Popovii H = 1 H1 = h ( )( 1) H = h H = h n 1 h = n 1 a = y 1 a1 = y1 y =Δ y =Δ y a 1 Pratially, the tables with differenes are presented below: Table no. for the ompression degree and Table no. 4 for the effiieny. 1 4 m -1 1 red y ( π ) Δ 1 y Δ y Δ y Δ 4 y -.45 Δ 5 y Table no. 1 4 m -1 1 red η = y Δ 1 y Δ y Δ y. -.1 Δ 4 y -.4 Table no. 4

5 The entrifugal ompressors surge simulation 47 Based on those values, the oeffiients a 1 of the two polynomials whih approimate the ompressor performane at n =.9 are determined, [4]. Replaing and elaborating those polynomials, we get the epressions of the ompression degree 4 π =.5.165m red m red +.15m red.1875m red () respetively for effiieny 4 η = m.4 1 m m m red red red red. both equations being valid for m [, ] red. The oeffiients omputation from the fundamental equations Given the geometrial and gasodynamial harateristis, one may pass at the equations oeffiients omputation whih defines the surge proess or, generally, the non stationary flow through the ompression system, [5]. where R = L / A = () R = L / A = 9.6 S = krt / V T = 88K, V = 9.4m, S = 11.7, k = , k = 777, B = 7.7,.675 S = 7.7 T y B =, 5 C = , B = 57.74, Regarding the other quantities whih appear D = 6.47, Q =.9, P =., M =.9, J = 6.47, M =.418, M M The following relations may be written π = m red +.75m red.75m red (4) ( 88.9y )( π 1) T ( y) = I

6 48 Virgil Staniu, Emilian Popovii 4. The problems determination 4.1. The omplete problem The differential omplete system whih haraterizes the non stationary flow in the ompression system was established in the anterior paragraphs. This system will be partiularized in the onrete ase of the disussed installation, [6]. If we onsider F( y) = f ( y) dy+, where = onstant of integration (5) y G( z, y) = k+ k z y H(, y) =q(, y) q(, y) E(, y, z) = h1( y)( y z) + h( y)( y ) where m = y f red ( y) π = π R + R 1 (, ) = + q y B p BT y πρ h1 ( y) = θ R1+ R S ( y) h ( y) = θ R1+ R then, the system may be written = H(, y) y = u F( y) z = G( z, y) u = E(, y, z) with the initial onditions at t <, at the stopping moment y = m red = kg/ s z = = u = F y (6)

7 The entrifugal ompressors surge simulation 49 respetively at the starting moment y = = z = u = Replaing the onstant terms, the variables from the system get the following forms π f ( y) =1 (7) π +.64 g ( y, ) =.675 p T ( y ) where (.86 π 1 ) T y = y I I y = y+.4y +.84y.1667 y and where 4 π h1 ( y) = π +.64 T ( y) h ( y) =.8 π +.64 = 7.7 S y T y G( z, y) y = z E, y, z = h y y z + h y y 1 H(, y) q(, y) 1 y = 4.. The problem in the ase of the omplete losure of the outlet valve and the inlet aomplishment from an unlimited spae In this ase, the system beomes, [7] y = u1f1( y) u 1 =E1( y) where π 5 π F1 ( y) = = = 487.6π R 9.6 and where the funtion E y = g y y 1 1 (8)

8 5 Virgil Staniu, Emilian Popovii where where with the initial onditions y = u = F y ( y) S g y = = 1.76 T y 1 R = 88 + ( 88.9 ) T y y π 1 I 4.. The simplified problem of the evauation valve instantaneous losure In this ase, the equivalent differential system beomes y y = u α y (9) u =y with the initial onditions y, u. Considering that y m y = red ω = ω a1 α = Rω where a 1 would be the oeffiient of y in the approimation by a three degree parabola of the ompression degree π, this means π = a + a1y+ ay + ay Taking only the part for π, with y >, one aquires a 1 =.5. Therefore, α = / ω. Regarding the pulsationω, it may be established using the Horvath relation, []: S ω = =.448 rad / s (1) R At this pulsation orresponds a frequeny ν =.74 Hz and a period of the proess T =.674 s.

9 The entrifugal ompressors surge simulation 51 Therefore, replaing α = the system may be written y y = u y (11) u =y with the onditions y, u = F1( y). The previous system may lead to the Van der Pol equation y y 1 y + y = (1) 5. The main ompression system variables omputation Resolving the system, one gets the funtions y( t), z( t), ( t ). These allow establishing the variation laws in time for the air pressures and temperatures in the main setions, [8] p Ks z = p A p1 = p 45.86y p = pπ y 1 (1) p = p + Kd Kd = 9.7 T y, K s = Therefore p = p.8z (14) where p = p+ 9.7 T y The temperatures are given by the relations T1 = 88.9y T.86 = T ( π 1 ) / I T = T +.9y 6. The surge problem solution Consequently to the anterior hapter, one will attempt to establish the analytial solution of the differential equations system. Firstly, one resolves the simplest problem, whih leads to the Van der Pol equation, [9].

10 5 Virgil Staniu, Emilian Popovii d = y α dt dy = dt in Cartesian oordinates. The limit onditions are written =, y = y 5 and the onstant term α = Generally, a Van der Pol equation has the form, [1] + k 1 + = Bsinωt (16) where k > and ω and B are onstants The Van der Pol equation study In the ase of the omplete evauation vane losure, one gets a Van der Pol type differential equation 5 y y 1 y + y = (17) dy 1 dy y y = = = d ω dt ω The equation may be generated starting from the first degree differential equation system y = u α ( y /) y (18) u =y with the initial onditions y yu = α y 5 y =, α = One arrives at the equation known form from literature, if one hanges the variable terms notations [11]. y, u y, T t. (15)

11 The entrifugal ompressors surge simulation 5 It may be written d y α ( /) = dt dy = dt with the initial onditions = = y = y = 6α = = = y 5 (19) Fig.. Graphial representation of the Van der Pol equation solutions 6.. The analytial method of the Taylor development One notes the funtions from the right member of the equations by f y, = y () 5 g ( y, ) = Developing the two unknown funtions in Taylor series [5], one gets: = t (1) y = y t t t () t = + + ( ) !!

12 54 Virgil Staniu, Emilian Popovii t t y() t = y + y + y ( ) !! The oeffiients ompute is done taking into aount that = = 5 y = y = 6α = and replaing in the equations system = y = But f f = + y y g g y = + y y Taking into aount that f = f = 1 y g = 1 g = y one gets ( 5 = ) + y y = Therefore, the oeffiients beome = y = It passes at the three degree derivative = ( 1 ) + y y = 5 5 () ()

13 The entrifugal ompressors surge simulation 55 The oeffiients beome = y = We are eeuting now the four degree derivative IV = ( 1 ) IV y = Replaing, one aquires IV 5 5 = ( 8) = IV 5 y = It is omputed the five degree derivative V 5 = ( + ) IV IV ( 1 ) + + y V IV y = Doing the omputations, one aquires V 5 5 = ( + ) IV IV ( 1 ) + y V IV y = V = IV IV ( 1 ) + y V IV y = For t = one aquires V = V y = The si degree derivative omputation is done hereby (4)

14 56 Virgil Staniu, Emilian Popovii ( IV ) ( IV V ) V y VI 5 5 = VI V y = At t = it results VI 5 =9.5 1 VI y = But, if we stop here with the oeffiients omputation, the two developments beome 4 t t t () ( 5 t = + t ) + ( ) t t 5 ( ) + ( ) t t t 5 y() t = t( ) ( ) t t ( 5.186) + ( ) 1 7 Going bak to the old variable terms or notations, one aquires yt = T T 167.5T T T (5) u ( T) = T T T T.9945 T Taking into aount that T = ωt =.488t by substitution, it results y( t) = t t t t t u1 ( t) = t t t t t where t is inserted in seonds.

15 The entrifugal ompressors surge simulation 57 If we aept t in hundredths of a seond[ t s ], the funtions beome y( t) = ts ts ts ts ts (6) u ( t) = ts ts t t t One determined the funtion y f ( t) s s = of the air flow, the pressures and temperatures time dependent on the ompression system may be immediately established, on the base of the anterior determined relations. 7. Conlusions The mathematial model we appealed to in general onditions and whih was singularized for some partiular onrete situations lead us to Van der Pol type equation. Numerial simulations of this equation graphially generated a suggestive image of the surge phenomenon from whih its yli behavior obviously results, as well as the fators whih singularize the evolution or involution of this undesired aerodynami phenomenon. R E F E R E N C E S [1] V. Pimsner,, V.Staniu, C.Tătăranu, Teoria şi onstruţia sistemelor de propulsie (The propulsion systems theory and onstrution), Editura Institutului Politehni Buureşti, 1984; [] V. Staniu, Teoria şi onstruţia sistemelor de propulsie. Calulul arateristiilor (The propulsion systems theory and onstrution. The harateristis omputation), Editura Institutului Politehni Buureşti, 1985; [] M.H. Vavra, Aerothermodynamis and flow in turbomahines, John Willey, New York, 196; [4] V. Silivestru, R. Petu, V. Vilag, C. Mandiş, D. Robesu, Mathematial modeling and numerial simulation of proess inside a turboengine, Sientifi Bulletin edited by The University POLITEHNICA of Buharest, Series D, Vol. 71, iss., 9; [5] A.G. Corbett, R.L. Elder, Mathematial modeling of ompressor stability in steady and unsteady flow onditions in unsteady phenomena in turbomahinery, AGARD CP 177, 1976; [6] M. Boşoianu, F.Zăgănesu, V. Staniu, Modelarea pompajului rotitor în ompresoarele aiale polietajate (The multi stage aial ompressors rotating surge modeling), a XXVIII-a Sesiune de Comuniări Ştiinţifie u Partiipare Internaţională, 1 otombrie 1999; [7] V. Staniu, E. Popovii, G. Stroe, Pompajul ompresorului entrifug (The kineti ompressor surge), Editura Printeh, Buureşti, 7; [8] J. Breugelmans, Flow measurements in a rotating stall ell, Von Karman Institute Leture Series, Unsteady Flow in Turbomahines, Bruelles, Belgium, 1979;

16 58 Virgil Staniu, Emilian Popovii [9] Stall and surge in aial flow ompressors, VKI Leture Series, 199-, jan. 199; [1] St. Ştefan, Euaţiile meaniii fluidelor (The hydrokinetis equations), Editura ATM, Buureşti, 1996; [11] H. Takata, Y. Tsukuda, Study of the meanism of stall margin improvement of asing treatment, ASME Paper 75-GT-1, ASME Gas Turbine Conferene.