ISSN 1068-7998, Russian Aeronautics (Iz.VUZ), 2011, Vol. 54, No. 2, pp. 217–222. © Allerton Press, Inc., 2011. Original Russian Text © S.S. Evgen’ev, V.A. Futin, 2011, published in Izvestiya VUZ. Aviatsionnaya Tekhnika, 2011, No. 2, pp. 69–72.
AIRCRAFT AND ROCKET ENGINE DESIGN AND DEVELOPMENT
Influence of Geometry and Operating Conditions of a Centrifugal Compressor Stage on the Radial Gas Force S. S. Evgen’eva and V. A Futinb a
Tupolev State Technical University, Kazan, Russia b ZAO NIIturbocompressor, Kazan, Russia Received February 14, 2011
Abstract—The influence of geometry and operating conditions of the centrifugal compressor stage on the radial gas force is determined on the basis of the theoretical method and calculation program using experimental boundary conditions. DOI: 10.3103/S1068799811020188 Keywords: centrifugal compressor, impeller, radial force, rotor, critical rotational speed.
Centrifugal compressor stages with an open diffuser (OD) and output devices in the form of a lateral internal scroll (IS) or annular chamber (AC) are widely used as end ones in the centrifugal and mixedflow compressors and in aviation and industrial turbo-expanders. The use of internal scrolls or annular chambers considerably reduces the radial dimensions, which is important for high pressure turbomachines. The scheme of the end compressor stage with the open diffuser and the internal scroll or annular chamber is shown in Fig. 1. The IS flow passage is formed by the outer constant radius rext , the inner radius rint ( θ ) varying by the angle of turn θ and the chamber width Bch . An outlet pipe can have a radial or tangential direction (is shown by a dot line). The AC flow passage is formed by two constant radii rext , rint and the chamber width Bch .
(a)
(b)
Fig. 1. The scheme of the end compressor stage with the internal scroll or annular chamber: (a) in the radial plane; (b) the stage meridian section.
The internal scrolls being considered are designed according to the estimated flowrate, while the chamber sections are selected according to the law rcu = const ( cu is the circumferential component of the flow absolute velocity). The annular chambers have a constant cross-section, coinciding with the last section of the profiled (at the expense of rint ( θ ) ) output device of the IS type. 217
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The stream in the IS and AC type output devices with the radial or tangential pipes exerts an appreciable influence on the flow structure downstream of the impeller at the radius r2 [1]. It is connected with absence of geometric symmetry of these devices relative to the impeller axis. As a result, there occurs the velocity and pressure field nonuniformity along the circle at the impeller outlet, that is mostly expressed in the off-design stage operating conditions. The nonuniformity mentioned is a source of the radial gas force, which affects the end impeller stage with the open diffuser. At considerable pressures of the compound compression the radial forces in the end centrifugal stage can be substantial, comparable with the whole rotor weight, and must be taken into account when designing reference units and studying the critical rotational speed of the rotor. The aim of this paper is to determine the effect of the geometry and operating conditions of the end centrifugal compressor stage on the radial gas force and the action of the latter on the critical rotational speed of the rotor. For the analysis we use the familiar calculation method of the radial gas force, affecting the impeller of the centrifugal compressor stage [2]. The method is based on the initial momentum equations in projections on the x and y axes in the plane normal to the rotation axes. As a result of transforming the equations, we obtained the expressions in the dimensionless form for the radial force components on the x and y axes, respectively [2]: 2π 1 1 ⎡ ⎤ Rx = −0.5b2′ ∫ ⎢ ∫ ( p2 − p0 ) + ε 2 cr22 dz cos θ − ∫ ε 2 cr 2 cu 2 dz sin θ ⎥ d θ; 0 ⎣0 0 ⎦
(1)
2π 1 1 ⎡ ⎤ Ry = −0.5b2′ ∫ ⎢ ∫ ( p2 − p0 ) + ε 2 cr22 dz sin θ + ∫ ε 2 cr 2 cu 2 dz cos θ ⎥ d θ, 0 ⎣0 0 ⎦
(2)
(
)
(
)
where b2′ = b2′ D2 , b2′ is the impeller width on the outer diameter D2 including the disk thickness; p2 , p0 is the gas pressure at the impeller outlet and inlet; ε2 = ρ2 ρ0 , ρ2 , ρ0 are gas densities at the impeller outlet and inlet; cr 2 = cr 2 u2 , cr 2 , u2 are the radial component of the flow absolute velocity and the impeller circumferential velocity at radial distance r2 ; cu 2 = cu 2 u2 , cu 2 is the circumferential component of the impeller
flow absolute velocity at r2 ;
(
)
(
)
p = p ρ0u22 ;
z = z b2′ ,
z is the impeller axial coordinate;
R = R ρ0u22 D22 . The radial force and its direction are determined by the expressions
R = Rx2 + Ry2 ;
(3)
θ R = arctan ( Ry Rx ) ;
(4)
R = Rρ0u22 D22 .
(5)
To calculate the radial force by Eqs. (1) and (2), we used the known boundary conditions in the form of the experimental relationships ( p2 − p0 ) = f ( z, θ ) , cr 2 ( θ ) , cu 2 ( θ ) . They are obtained in testing the typical model stages with the internal scroll and annular chamber and with the impeller, having blade angles at the radial distance r2 βbl 2 = 45 − 62° and the relative width of the impeller flow passage at r2 b2 = b2 D2 = 0.025 − 0.05 [1], and also the impeller with the parameters βbl 2 = 38° and b2 = 0,05 [3]
and the impeller with βbl 2 = 32° and b2 = 0.05 [4]. These boundary conditions are presented in [2] in the convenient form and can be applied when observing the equality of the basic similarity criteria: rext = rext r 2 , r4 = r4 r2 , Bch = Bch D2 , b2 = b3 b2 ( b3 is the open diffuser width), Mu = u2 a0 (the Mach
number), ϕ2 = ϕ2 ϕ2 est ( ϕ2 = cr 2 = с r 2 u2 , ϕ2 est = сr 2 est u2 , cr 2 est is the calculated radial component of the absolute flow velocity at r2 ) , Ω is the stage reactivity extent.
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The analysis of the experimental data, obtained at Mu = 0.5 − 0.7 by different authors, shows: 1. The pressure nonuniformity is practically the same along the circle downstream of the impeller for the end stage with the internal scroll and annular chamber at the flowrates, equal to the estimated one and more than it. When the flowrates are less than the estimated one, the pressure nonuniformity downstream of the impeller in the stage with the annular chamber decreases, whereas that with internal scroll grows [3]. 2. The variation of the basic geometric AC relationships in the range rext = 1.71 − 1.94,
Bch = 0.35 − 0.45 practically has no effect on the pressure nonuniformity along the circle downstream of the impeller [1, 5]. 3. The principal influence on the pressure nonuniformity along the circle downstream of the impeller is exerted by the stage reactivity extent. Its decrease, that is, the increase of the velocity level downstream of the impeller and the subsequent transformation into static pressure with great losses in the open diffuser and further, in the annular chamber or internal scroll, leads to the nonuniformity growth [1, 4]. Figure 2 shows the influence of the stage reactivity extent Ω on the average in the width b2′ pressure nonuniformity along the circle downstream of the impeller in the form Δp = p2 max − p2 min for the stages different in geometry.
Fig. 2. The influence of Ω on Δp along the circle downstream of the impeller: (A) b2 = 0.05; (B) b2 = 0.066; (C) b2 = 0.025.
The points on the curves in Fig. 2 denote the geometry features of the considered stages: (1) the impeller with βbl 2 = 62° and b2 = 0.05 [1]; (2) the impeller with βbl 2 = 38° and b2 = 0.05 [3]; (3) the impeller with βbl 2 = 32° and b2 = 0.05 [4]; (4) the impeller with βbl 2 = 45° and b2 = 0.068 [6]; (5) the impeller with βbl 2 = 34° and b2 = 0.066 [6]; (6) the impeller with βbl 2 = 45° and b2 = 0.025 [1].
The conditions at the flowrate were chosen estimated ( ϕ2 = 1 and Mu = 0.5 − 0.7 ) , since in this case,
the use of the internal scroll or annular chamber gives practically the similar pressure nonuniformity downstream of the impeller. We observe an appreciable increase of the quantity Δp with decrease of the stage reactivity extent Ω and with the increase of the impeller flow passage relative width b2 . Figure 3 presents the influence of the operating conditions of the end stages of different types on the dimensionless radial force. Table 1 shows the stage parameters corresponding to the curves in Fig. 3. As seen from Fig. 3, the end stages with the internal scroll (curves 1 and 2) have a minimal radial force in the estimated flow conditions ( ϕ2 = 1) . In the off-design conditions ( ϕ2 < 1; ϕ2 > 1) the radial force grows. In the end stages with the annular chamber there takes place a continuous reduction of the radial force with the flowrate decrease (curves 3−5). As the stage reactivity extent Ω increases, the radial forces are considerably reduced, it is seen from the comparison of curves 1 and 2, as well as curves 3−5. RUSSIAN AERONAUTICS
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Fig. 3. The influence of operating conditions on the radial gas forces in different types of stages (designation in Table 1).
Table 1 Curve number 1 2 3 4 5
Stage type IS NZL[4] IS KTSKU-4 [1] AC BDKA-2 [6] Ural AC GPA-16 Volga AC GPA-16
r4
rext
Bch
βbl 2 , deg
b2
b2
1.60 1.65 1.59 1.45 1.53
1.73 1.78 1.69 1.63 1.68
… … 0.34 0.30 0.32
32 45 45 67 34
0.050 0.040 0.066 0.042 0.066
0.066 0.054 0.084 0.054 0.079
Ω 0.72 0.65 0.66 0.60 0.692
2r2 , m
0.305 0.24 0.38 0.87 0.75
Using Fig. 2 and 3, it is possible to determine the radial gas forces in the stages with open diffuser and internal scroll or annular chamber, following the equality of the listed similarity criteria and without applying to a PC program [2]. The influence of the radial gas forces is particularly great in the turbo-machines with cantilever impeller position. For example, Fig. 4 shows the radial forces in the BDKA-2 turbo-expander, being used for processing the accompanying gas. The technical data of this turbo-expander are presented in [7]. It has the impellers disposed in cantilever of the centripetal expander ( D2 = 0.35 m ) and the centrifugal
compressor ( D2 = 0.38 m ) with the open diffuser and annular chamber, the rotational speed of the rotor is n = 12 000 rev/min, the rotor weight is 641 N, the distance between the supports is 0.284 m, and the diameters of bearing journals is 0.075 m. The parameters of the compressor stage, which is part of BDKA-2, are shown in Table 1. The calculations of the radial forces for the compressor stage are performed for small ( ϕ2 = 0.83 ) , medium ( ϕ2 = 1.24 ) , and maximal ( ϕ2 = 1.54 ) flowrates. To make them more pictorial, the vectors of the radial forces R1 , R2 , R3 corresponding to the flowrate are constructed relative to the outlet tangential pipe of the annular chamber and the location of five segments (position 1) of the journal slider bearing. As it is seen from Fig. 4, the radial forces can be disposed differently against the journal bearing segments and, as is known [8], correspondingly influence its stiffness C and damping coefficients K.
Fig. 4. The scheme of the radial gas force effect on the compressor impeller of the BDKA-2 turbo-expander. RUSSIAN AERONAUTICS
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The influence of the radial gas force in the compressor stage of the BDKA-2 turbo-expander on the first ( ncr ) critical rotational speed of the whole rotor is determined by the calculation according to the PC program, which is described in [8], for absolutely rigid and pliable supports. The calculation results are shown in Table 2. Table 2 Influence factor
Characteristic of support
Without regard of radial force
ncr , rev min
25050
With regard of radial force, 25000
ϕ2 = 0.83; R1 = 340 N; θ R = 20°
With regard of radial force,
Absolutely rigid
ϕ2 = 1.24; R2 = 531 N; θ R = 64°
20600
With regard of radial force, 18800
ϕ2 = 1.54; R3 = 638 N; θ R = 70°
Pliable: C1 = 1.39 × 10 7 N m ,
With regard of radial force, ϕ2 = 1.24; R2 = 531 N; θ R = 64°
K1 = 3.36 × 10 4 N ⋅ s m;
14200
C2 = 4.4 × 10 N m; 7
K 2 = 8.67 × 10 4 N ⋅ s m
The stiffness and damping coefficients of the support at the side of the expander impeller are denoted as C1 and K1 , while at the side of the compressor impeller they are denoted as C2 and K 2 . For the slider bearings these coefficients are anisotropic (that is, their values along the vertical and transverse axes are not equal). However, for the segmental slider bearings of the turbo-expander being considered they are taken as averaged ones per revolution of the rotor. It is allowable for engineering problems, since a noticeable anisotropic influence is shown only in the transcritical area of the rotational speeds. Table 2 shows the evident effect of the radial force quantity and direction on the critical rotation speed of the rotor ncr . The radial force increase with the growth of the flowrate through the compressor stage (increase of ϕ2 ) promotes the decrease of the quantity ncr of the two- cantilever rotor, especially, for the rotor with the pliable supports. The examined data considerably complement the available information and is recommended for practical application. REFERENCES 1. Evgenev, S.S. and Kokhanov, S.G., The Calculation of Aerodynamic Forces Affecting the Centrifugal Compressor Rotor, Preprint of Kazan State Technical Univ., Kazan, 2002, no. 02P4, p. 58. 2. Evgen’ev S.S., Zalyaev, R.R., and Futin, V.A. The Method of Calculating the Radial Gas Force Affecting the Centrifugal Compressor Impeller, Trudy 14-ogo mezhdunarodnoi nauchno-tekhnicheskoi konferentsii po kompressornoi tekhnike (Int. Sc.-Tech. Conf. on Compressor Equipment), 2007, vol. 2, Kazan: Slovo, pp. 237–247. 3. Stolyarskii, M.T., The Research Results of the Output Devices with Open Diffuser and Lateral Collecting Chamber for the Superchargers of the Natural Gas Transportation and High-Pressure Centrifugal Compressors, Trudy TKTI, issue 77, Leningrad: Izd. NPO TsKTI, 1967, pp. 62–81. RUSSIAN AERONAUTICS
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4. Den, G.N., Mekhanika potoka v tsentrobezhnyh kompressorah (Flow Mechanics in Centrifugal Compressors), Leningrad: Mashinostroenie, 1973. 5. Miftakhov, A.A. Aerodinamika vyhodnyh ustroystv turbokompressorov (Aerodynamics of Turbo-Compressor Output Devices), Moscow: Mashinostroenie, 1999. 6. Evgenev S.S. and Futin V.A., The Pressure Distribution along the Circle Near Disks of the Closed Type Impellers of Centrifugal Compressors End Stages, Kompressornaya Tehnika i Pnevmatika, 2004, no. 8. pp. 28–30. 7. Evgenev, S.S., Petrosyan, G.G., Sidorov, V.P., et al., Block Turbo- Expander for the Gas Processing, Materialy mezhdunarodnogo seminara “Gazovye turbiny”(Proc. Int. Seminar on Gas Turbines), Kazan, 1989, pp. 185–191. 8. Zalyaev, R.R., Evgenev, S.S., and Khamidullin, I.V. The Calculation of Amplitude-Frequency Characteristics of the Turbo-Expander Rotor on Slider Bearings with Constant Slants, Preprint of Kazan State Technical Univ., Kazan, 2004, no. 04P1, p. 67.
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