The 20th International Conference on Hydraulics and Pneumatics, Prague, September 29 – October 1, 2008 1 Centrifugal Compressor Dynamics and Software System for Surge Control NEVRLÝ, Josef 1 , MAREK, Jiří 2 , VARGOVČÍK, Luboš 2 & OLDŘICH, Jiří 4 1 Prof., RNDr., Ing., Ph.D., ÚK FSI VUT, Technická 2, 616 69 Brno,
[email protected], http://www.fme.vutbr.cz 2 Ing., Ph.D., UNIS, a. s., Jundrovská 33, 624 00 Brno,
[email protected] 3 Ing., Ph.D., UNIS, a. s., Jundrovská 33, 624 00 Brno,
[email protected] 4 Ing., Ph.D., ČKD NOVÉ ENERGO, Klecakova 1947, 190 02 Prague 9
[email protected] Abstract: Centrifugal compressor surge dynamics and software system for centrifugal compressors surge control inside the safe working area but in the near proximity of control (or backup- or surge) line based upon ČKD Nové Energo research are described in the paper. At small mass flows the performance of a compression system is limited by the occurrence aerodynamic flow instabilities, which can lead to catastrophic failure of the compressor due to mechanical and thermal loads. Recent innovations in control technology have made it possible to supply surge control systems which are capable of coping with rapid flow fluctuations and process gas variations. A compression system model developed by Greitzer is mentioned in the paper. The oscillations in the system are modeled in a manner analogous to those in a Helmholtz resonator. It is supposed that all the kinetic energy of the oscillations is associated with the motion of the fluid in the compressor and ducts. The suggested system of radial compressors surge control consists of parts of - measuring of technical data of the compressor (compressor operating point) - computation of a distance of the operating point from the “anti-surge” action - control parameter setting - setting of approximation parameters for calculation of the “anti surge” action (input of all approximation parameters of approximating polynomials and correcting relations) - computation of a working space of the compressor (a row of arranged couples – values of flow and pressure in the compressor outlet) - computation of a working characteristic of the compressor (a row of arranged couples – values of flow and pressure in the compressor outlet for a given compressor operating point) – depicting of the compressor working regime Suggested software for surge control systems provides protection against surge together with more efficient compressor operation, especially in process applications that involve variations in the gas mixture and temperature. Key words: surge, compressor, rotating stall, control, model 1 Introduction 1.1 Basic operation of radial compressors Within the framework of an introduction, a part of radial compressors theory introduced in [3] can be mentioned. Fig. 1 Scheme of compressor rotor wheel, velocities diagrams [3] Theoretical compression in one stage of compressor equipped by unlimited number of blades is given by the relation = A · t p ) . . ( . 1 1 2 2 u u t c u c u h ÷ = A · p p [Pa] where o cos . c c u = is tangential component of absolute velocity. Theoretical specific energy is | | ) ( ) ( ) ( 2 1 2 2 2 1 2 1 2 2 2 1 2 2 w w u u c c h t ÷ + ÷ + ÷ = A · [J.kg -1 ] From velocity triangles results 2 2 2 2 | tg c u c m u ÷ = , 2 2 2 sin . o c c m = and the equation for theoretical specific energy can be written as Q tg F u u tg c u u h m t · · ÷ = | | . | \ | ÷ = A · 2 2 2 2 2 2 2 2 2 | | [J.kg -1 ], 2 F [m 2 ] - flow area at impeller outlet. Energy input of compressor is G h P t t · · A = [W], G [kg s -1 ] – mass flow. Torque moment according Euler equation for volume gas flow Q [m 3 s -1 ] is ) . . ( . 1 1 2 2 u u c r c r Q M ÷ = p [N m] at angle velocity ], [s 30 / -1 n t e = n is machine speed (revolutions per minute) [s -1 ]. Tangential velocities follow from following expressions: = ], [ms -1 1 1 n D u t ] [ms -1 2 2 n D u t = ], [ms cos -1 1 1 1 o c c u = ] [ms cos -1 2 2 2 o c c u = Theoretical impeller input is e . M P t = [W] Isentropic efficiency is h h s s A A = n Coefficient of contraction [3] at outlet of the impeller is 2 2 2 2 sin 1 | t u · · · · ÷ = b D F z l Where z 2 is number of vanes of impeller, F l is area of square cut of vane and b 2 is output with of the impeller: 2 2 2 2 2 u D Q b · · · · = m u t [m], here 2 2 p G Q = [m 3 s -1 ] and 2 m is flow coefficient 2 2 2 2 2 2 u F Q u c m · = = m Output angle of impeller vane 2 | is usually in range 0 0 65 ; 20 . 1.2 Rotating stall and surge Various types of instabilities are encountered in compression systems, e.g. combustion induced instabilities or aero-elastic instabilities such as flutter. This paper is restricted to aerodynamic flow instabilities in centrifugal compressors: rotating stall and surge. A review of several types of instabilities that are found in turbomachines can be found in Greitzer [2]. Rotating stall Rotating stall is a two-dimensional, local instability phenomenon in which one or more local regions of stagnant flow, so-called stall cells, rotate around the circumference of the compressor (see Fig. 2). Fig. 2 Rotating stall cells [10] Depending on their size, these cells usually have a constant rotational speed between 20 and 80% of the rotor speed: larger cells rotate at a lower rotational speed than small cells [2]. In this flow regime, the annulus-averaged compressor mass flow is steady, but circumferentially nonuniform. In a compressor map, the occurrence of rotating stall is seen as a rapid movement from the unstalled characteristic (1) to a point on the stalled characteristic (2), as shown in Fig. 3. Rotating stall induces large vibratory stresses in the blading and can result in a large drop in performance and efficiency. Fig. 3 Rotating stall in compressor map [10] Surge Surge is characterized by large amplitude fluctuations of the pressure and by unsteady, but circumferentially uniform, annulus-averaged mass flow. This essentially one-dimensional instability affects the compression system as a whole and results in a limit cycle oscillation in the compressor map. Figure 4 shows a pressure trace for a compressor system, which was initially operated in a steady operating point. Frequencies of several types of aerodynamic flow instabilities can be seen in Fig. 5. Figure 6 shows a typical example of a deep surge cycle, which is associated with reverse flow over part of the cycle. The frequency of oscillations during deep surge is normally well below the Helmholtz frequency as it is set by the filling (4→1) and emptying period (2→3) of the plenum. It is noted that Pinsley et al. (1991), [8] have shown that mild surge transforms into other types of surge by throttling the compressor to lower mass flows. Fig. 4 Surge initiation measured at the compressor outlet [10] Fig. 5 Frequencies of several types of aerodynamic flow instabilities [10] Fig. 6 Compressor map with deep surge cycle [10] 2 Centrifugal Compressor Surge Modeling The modeling of the compression system with a lumped parameter approach is widely accepted in literature, and is useful in the stabilization of a rotating stall and surge by means of active control. It forms a basis for models dealing with compressibility in the compressor. Greitzer [1] developed a nonlinear mathematical model of transient compression system behavior because it was not understood what fundamental mechanisms are responsible for determining the mode of instability cause by surge occurring at centrifugal compressors. The compressor and its ducting are replaced (Fig. 7) by an actuator disc, to account for the pressure rises due to the compressor, and a constant area pipe with a certain length, to account for the dynamics of the gas in the compressor duct. Fig. 7 Scheme of equivalent compression system used in analysis [1] The rate of change of the mass flow in the compressor duct, represented by the axial velocity, c xc , is related to the pressure difference across the duct, o p p p p ÷ = A , and the pressure rise across the compressor, o c c p p p ÷ = A . Using relation c o p x c p p p dt dc L c A + ÷ = ) ( p and similar relation describes the flow in the throttle duct: T o p x T p p p dt dc L T A + ÷ = ) ( p where o T T p p p ÷ = A represents the pressure rise across the throttle. After further derivation (see e.g. [1]) Greitzer’s resulting equations of the dynamics of the system are: ) ( ' c c c c p p L A dt dm A ÷ A ÷ = ) ( ' T c T T p p L A dt dm A ÷ A ÷ ÷ = ) ( ' ' 2 T c p p m m V a dt dp + = where Δp is the pressure rise across the duct, the loss (viscous) terms, and the contribution of rotation in case of compressor, are assumed to react quasi-steady to mass flow, changes are represented by the steady-state characteristics, c p A , T p A ; a is speed of sound. The first and second equation are the one-dimensional “incompressible” momentum equations for the compressor and throttle duct, respectively, the third equation expresses the conservation of mass in the plenum in which an isoentropic compression is as supposed. 3 Surge control of the compressor 4 RSA 32 3.1 Software Configuration – Function of the Program Components Suggested compressor control system was implemented [4], [9] in a realized project in Novokujbyschevsk in Russia. As far a program equipment is concerned, it consists from a program for direct control implemented into a programmable controller PLC (client SCADA) and from application securing user interface created in environs Win CC. Mutual linkages of program components and brief description of their functions are introduced in Fig 8. 3.2 Application of user interface (Win CC) – client DDE The application of the user interface solves following tasks: 1. user interface of compressor control 2. setting of control parameters 3. setting of approximation parameters for computation of anti surge action 4. computation of operating space of the compressor 5. computation of operating characteristic of the compressor Fig. 8 Relations of program components of the system PLC client SCADA Win CC server SCADA client SCADA client DDE 1. measuring of compressor technological data – operation point 2. computation of operation point distance 1. user interface of compressor control 2. setting of control parameters 3. setting of approximation parameters for computation of anti surge action 4. computation of operating space of the compressor 5. computation of operating characteristic of the compressor Surge VS server DDE user interface – depiction of compressor operation regime In connection with solution of the problem of anti surge action, the control parameter is relative distance of action-line from anti surge line. It is necessary to secure access of all approximation polynomial parameters and correction relations into control process. Control coefficients: - approximation polynomial coefficients of reference characteristic - correction polynomial coefficients of flow parameter concerning molar hydrogen fraction - correction coefficients of compressor output pressure concerning molar hydrogen fraction - approximation coefficients of compression factor product and universal gas constant at suction state - approximation coefficients of compression factor product and universal gas constant at standard state - approximation polynomial coefficients of dependence of isoentropic coefficient on the molar hydrogen fraction Fig. 9 User interface – dependence of discharge pressure on volume flow Operating area of compressor function is delimitated by lower limiting performance characteristic (60 % of output), upper limiting performance characteristic (110 % of output) and anti surge limit line. Fig. 10 Movement of compressor operating point P Q Anti Surge Line P 2 wp Q wp P 2 as Q as WP start and stop of automatic refresh of depiction operating area of compressor function anti surge limit line stop of server activity setting of anti surge operation performance characteristic actual operation point suction pressure p1 compressor discharge compressor volume flow Q The actual compressor operating point must be situated on the performance characteristic approximation. The actual compressor operating point trajectory must not cross anti surge line. This condition is expressed by following condition: as wp as wp Q Q P P 2 2 2 2 > . s . 4 Results Suggested control algorithm and created software ensure stable operation of compressor due to the fact that the actual operating point was kept on the performance characteristic approximation and did not cross anti surge line at a gas of given composition. The compressor characteristic transformed into dimensionless co-ordinates was used to this purpose. Values of correction coefficients were dependent on discharge coefficient and on compressed gas composition. Control action occurs when the operating point coincides with the anti surge intervention point. 5 Conclusions In this paper, the modeling and control of surge in a centrifugal compression system is investigated. The aim of this study is the development of feedback software control that stabilize surge: suggested software for surge control systems provides protection against surge together with more efficient compressor operation, especially in process applications that involve variations in the gas mixture and temperature. Further fields worth investigating are the integration and implementation of active surge control strategies on further existing systems as well as the modeling and control of instabilities in transonic machines. 6 References [1] GREITZER E. M. Surge and rotating stall in axial flow compressors. Part I: Theoretical compression system model. ASME, Journal of Engineering for Power, 98(2), April 1976, pp. 191-198. [2] GREITZER E.M. The stability of pumping systems - the 1980 Freeman scholar lecture. Transactions of the ASME, Journal of Fluids Engineering, 103(2), June 1981, pp.193–242. [3] CHLUMSKÝ V., LIŠKA A. Kompresory (Compressors). Praha: SNTL, Alfa, 1982. 196 p. [4] MAREK J. “Anti surge” řízení kompresoru 4 RSA 32 (Surge Control of the Compressor 4 RSA 32). Internal report, Brno, UNIS, 2003. [5] MEULEMAN C. H. J. Measurement and Unsteady Flow Modeling of Centrifugal Compressor Surge. TU Eindhoven, The Netherlands, ISBN 90-386-2564-2, 2002. [6] NEVRLÝ J. Modelování pneumatických systémů (Pneumatic Systems Modeling). Brno: CERM, 2003. 183 p. ISBN 80-7204-300-5. [7] OLDŘICH, J. Antipompážní regulace turbokompresoru na základě jeho matematického modelu (Turbocompressor Surge Control Based on its Mathematical Model), Internal report , Prague, ČKD NOVÉ ENERGO, a.s., 2003. [8] PINNSLEY, J., GUENETTE, G., EPSTEIN, A., GREITZER, E. Active stabilization of centrifugal compressor surge. ASME J. Turbomachinery, 1991, 113(4), 723–732. [9] VARGOVČÍK, L. Sistěma upravlenija kompressorov Novokujbyševsk (Compressor Control System Novokujbysevsk). Softwareový projekt NKRK-02-P-001-00 (Software Project NKRK-02-P-001-00). Brno, UNIS, 2003. [10] WILLEMS F. P. T. Modeling and Bounded Feedback Stabilization of Centrifugal Compressor Surge. Technische Universiteit Eindhoven, 2000, ISBN 90-386-2931-1. Informace pro zpracovatele příspěvku (nepublikují se): Použitý formát souboru s příspěvkem: (např. Microsoft Word 2000) Údaje jednotlivých autorů pro zpracování autorského rejstříku: Pořadí Příjmení Jméno Název organizace a stát Email 1 NEVRLÝ 1 Josef VUT, Brno, Czech Republic
[email protected] 2 MAREK 2 Jiří UNIS, Brno, Czech Republic
[email protected] 3 VARGOVČÍK 3 Luboš UNIS, Brno, Czech Republic
[email protected] 4 OLDŘICH 4 Jiří ČKD NOVÉ ENERGO, Prague, Czech Republic
[email protected] c2u c2 cos 2 [ms-1 ] Theoretical impeller input is Pt M . h Isentropic efficiency is s s h [W] . a part of radial compressors theory introduced in [3] can be mentioned. (r2 .c1u ) [Pa] where cu c. n is machine speed (revolutions per minute) [s-1 ].kg-1].flow area at impeller outlet. 1 Scheme of compressor rotor wheel. G [kg s-1 ] – mass flow.1 Introduction 1.kg-1] 2 From velocity triangles results c c2u u2 2 m . velocities diagrams [3] Theoretical compression in one stage of compressor equipped by unlimited number of blades is given by the relation pt ht .c 2u r1 . (u 2 .c 2u u1 . Fig. Torque moment according Euler equation for volume gas flow Q [m3 s-1 ] is M Q. c2 m c2 .sin 2 tg 2 and the equation for theoretical specific energy can be written as c u2 2 ht u2 u2 2 m u2 Q [J.c1u ) [N m] at angle velocity n / 30 [s -1 ]. tg 2 F2 tg 2 Energy input of compressor is Pt htG [W]. F2 [m2] . Tangential velocities follow from following expressions: u1 D1n [ms-1 ]. 1 2 2 2 Theoretical specific energy is ht (c 2 c12 ) (u 2 u12 ) ( w12 w2 ) [J.1 Basic operation of radial compressors Within the framework of an introduction. cos is tangential component of absolute velocity. u2 D2 n [ms-1 ] c1u c1 cos 1 [ms-1 ]. local instability phenomenon in which one or more local regions of stagnant flow. A review of several types of instabilities that are found in turbomachines can be found in Greitzer [2]. Fig. rotate around the circumference of the compressor (see Fig. here Q2 [m3 s-1 ] 2 D2 u2 2 and 2 is flow coefficient c Q2 2 2m u2 F2 u 2 Output angle of impeller vane 2 is usually in range 20 0 . 65 0 . but circumferentially nonuniform. 1. Fig. Fl is area of square cut of vane and b2 is output with of the impeller: G Q2 b2 [m]. 2).Coefficient of contraction [3] at outlet of the impeller is z 2 Fl 1 D2 b2 sin 2 Where z2 is number of vanes of impeller. 3. e.g.2 Rotating stall and surge Various types of instabilities are encountered in compression systems. Rotating stall induces large vibratory stresses in the blading and can result in a large drop in performance and efficiency. In a compressor map. so-called stall cells. the annulus-averaged compressor mass flow is steady. 3 Rotating stall in compressor map [10] . combustion induced instabilities or aero-elastic instabilities such as flutter. Rotating stall Rotating stall is a two-dimensional. as shown in Fig. the occurrence of rotating stall is seen as a rapid movement from the unstalled characteristic (1) to a point on the stalled characteristic (2). these cells usually have a constant rotational speed between 20 and 80% of the rotor speed: larger cells rotate at a lower rotational speed than small cells [2]. 2 Rotating stall cells [10] Depending on their size. In this flow regime. This paper is restricted to aerodynamic flow instabilities in centrifugal compressors: rotating stall and surge. which was initially operated in a steady operating point. 5 Frequencies of several types of aerodynamic flow instabilities [10] . 5. 4 Surge initiation measured at the compressor outlet [10] Fig. Figure 4 shows a pressure trace for a compressor system. (1991). [8] have shown that mild surge transforms into other types of surge by throttling the compressor to lower mass flows. Frequencies of several types of aerodynamic flow instabilities can be seen in Fig. Figure 6 shows a typical example of a deep surge cycle. The frequency of oscillations during deep surge is normally well below the Helmholtz frequency as it is set by the filling (4→1) and emptying period (2→3) of the plenum. It is noted that Pinsley et al. but circumferentially uniform.Surge Surge is characterized by large amplitude fluctuations of the pressure and by unsteady. Fig. which is associated with reverse flow over part of the cycle. annulus-averaged mass flow. This essentially one-dimensional instability affects the compression system as a whole and results in a limit cycle oscillation in the compressor map. 7 Scheme of equivalent compression system used in analysis [1] The rate of change of the mass flow in the compressor duct. Fig. [1]) Greitzer’s resulting equations of the dynamics of the system are: dmc' A c (p pc ) dt Lc .g. and is useful in the stabilization of a rotating stall and surge by means of active control. 7) by an actuator disc.Fig. and the pressure rise across the compressor. cxc . represented by the axial velocity. is related to the pressure difference across the duct. 6 Compressor map with deep surge cycle [10] 2 Centrifugal Compressor Surge Modeling The modeling of the compression system with a lumped parameter approach is widely accepted in literature. Using relation dc xc Lc ( p p po ) pc dt and similar relation describes the flow in the throttle duct: dc xT LT ( p p po ) pT dt where pT pT po represents the pressure rise across the throttle. The compressor and its ducting are replaced (Fig. pc pc po . to account for the pressure rises due to the compressor. and a constant area pipe with a certain length. Greitzer [1] developed a nonlinear mathematical model of transient compression system behavior because it was not understood what fundamental mechanisms are responsible for determining the mode of instability cause by surge occurring at centrifugal compressors. p p p po . It forms a basis for models dealing with compressibility in the compressor. After further derivation (see e. to account for the dynamics of the gas in the compressor duct. [9] in a realized project in Novokujbyschevsk in Russia. are assumed to react quasi-steady to mass flow.' dmT A T ( p pT ) dt Lc dp p a 2 ' ' (mc mT ) dt V p where Δp is the pressure rise across the duct. computation of operation point distance client SCADA Win CC client DDE server DDE 1. As far a program equipment is concerned. user interface of compressor control 2. Mutual linkages of program components and brief description of their functions are introduced in Fig 8. 2. The application of the user interface solves following tasks: user interface of compressor control setting of control parameters setting of approximation parameters for computation of anti surge action computation of operating space of the compressor computation of operating characteristic of the compressor client SCADA PLC server SCADA 1. the third equation expresses the conservation of mass in the plenum in which an isoentropic compression is as supposed. 3. 4. respectively. the loss (viscous) terms. setting of control parameters 3. computation of operating characteristic of the compressor user interface – depiction of compressor operation regime Surge VS Fig. and the contribution of rotation in case of compressor. 3 Surge control of the compressor 4 RSA 32 3. 5. pc . The first and second equation are the one-dimensional “incompressible” momentum equations for the compressor and throttle duct. pT . a is speed of sound. 8 Relations of program components of the system .1 Software Configuration – Function of the Program Components Suggested compressor control system was implemented [4]. it consists from a program for direct control implemented into a programmable controller PLC (client SCADA) and from application securing user interface created in environs Win CC. 3.2 Application of user interface (Win CC) – client DDE 1. changes are represented by the steady-state characteristics. computation of operating space of the compressor 5. setting of approximation parameters for computation of anti surge action 4. measuring of compressor technological data – operation point 2. approximation polynomial coefficients of dependence of isoentropic coefficient on the molar hydrogen fraction actual operation point suction pressure p1 setting of anti surge operation performance characteristic start and stop of automatic refresh of depiction anti surge limit line compressor discharge operating area of compressor function compressor volume flow Q stop of server activity Fig. upper limiting performance characteristic (110 % of output) and anti surge limit line. It is necessary to secure access of all approximation polynomial parameters and correction relations into control process. P P2 as P2 wp Anti Line Surge WP Q as Q wp Q Fig.correction coefficients of compressor output pressure concerning molar hydrogen fraction .In connection with solution of the problem of anti surge action. 9 User interface – dependence of discharge pressure on volume flow Operating area of compressor function is delimitated by lower limiting performance characteristic (60 % of output).approximation polynomial coefficients of reference characteristic . Control coefficients: . 10 Movement of compressor operating point .approximation coefficients of compression factor product and universal gas constant at suction state . the control parameter is relative distance of action-line from anti surge line.approximation coefficients of compression factor product and universal gas constant at standard state .correction polynomial coefficients of flow parameter concerning molar hydrogen fraction . s. Active stabilization of centrifugal compressor surge. 4 Results Suggested control algorithm and created software ensure stable operation of compressor due to the fact that the actual operating point was kept on the performance characteristic approximation and did not cross anti surge line at a gas of given composition. [10] WILLEMS F.. The actual compressor operating point trajectory must not cross anti surge line. Kompresory (Compressors). 191-198. The aim of this study is the development of feedback software control that stabilize surge: suggested software for surge control systems provides protection against surge together with more efficient compressor operation. the modeling and control of surge in a centrifugal compression system is investigated. 98(2). Brno: CERM. M. P. UNIS. Part I: Theoretical compression system model. ISBN 90-386-2931-1.. Softwareový projekt NKRK-02-P-001-00 (Software Project NKRK-02-P-001-00). Surge and rotating stall in axial flow compressors. 2003. GUENETTE. Further fields worth investigating are the integration and implementation of active surge control strategies on further existing systems as well as the modeling and control of instabilities in transonic machines. pp. 183 p. [4] MAREK J. J. [6] NEVRLÝ J. Prague. 113(4). Journal of Engineering for Power. [7] OLDŘICH. Turbomachinery. J. A. June 1981. Praha: SNTL. pp. Antipompážní regulace turbokompresoru na základě jeho matematického modelu (Turbocompressor Surge Control Based on its Mathematical Model). 2003. L. [8] PINNSLEY. 2000. ČKD NOVÉ ENERGO. 196 p. 103(2). Internal report . . J. especially in process applications that involve variations in the gas mixture and temperature. Brno. TU Eindhoven. April 1976. The Netherlands. “Anti surge” řízení kompresoru 4 RSA 32 (Surge Control of the Compressor 4 RSA 32). [3] CHLUMSKÝ V. Control action occurs when the operating point coincides with the anti surge intervention point. E. 2003. ASME. Brno... [9] VARGOVČÍK. Modelování pneumatických systémů (Pneumatic Systems Modeling). 6 References [1] GREITZER E. 1991. Sistěma upravlenija kompressorov Novokujbyševsk (Compressor Control System Novokujbysevsk). [2] GREITZER E. H. ISBN 80-7204-300-5. Modeling and Bounded Feedback Stabilization of Centrifugal Compressor Surge. Measurement and Unsteady Flow Modeling of Centrifugal Compressor Surge. EPSTEIN. [5] MEULEMAN C. 5 Conclusions In this paper. Technische Universiteit Eindhoven. ISBN 90-386-2564-2. G.193–242. Values of correction coefficients were dependent on discharge coefficient and on compressed gas composition. 2003. This condition is expressed by following condition: P2 wp P2 as Q2 wp Q2 as . 723–732.the 1980 Freeman scholar lecture. 1982. Journal of Fluids Engineering. GREITZER. The stability of pumping systems . ASME J.M. a. UNIS. Internal report. Transactions of the ASME.. LIŠKA A. 2002. The compressor characteristic transformed into dimensionless co-ordinates was used to this purpose. Alfa.The actual compressor operating point must be situated on the performance characteristic approximation. T. Prague.cz jmarek@unis. Czech Republic UNIS. Brno.Informace pro zpracovatele příspěvku (nepublikují se): Použitý formát souboru s příspěvkem: Údaje jednotlivých autorů pro zpracování autorského rejstříku: Pořadí 1 2 3 4 Příjmení NEVRLÝ1 MAREK2 VARGOVČÍK3 OLDŘICH4 Jméno Josef Jiří Luboš Jiří Název organizace a stát
[email protected] jiri.cz vargovcik@unis. Brno.cz . Brno. Czech Republic UNIS. Czech Republic ČKD NOVÉ ENERGO. Microsoft Word 2000) Email nevrly@fme. Czech Republic (např.