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3.22 CHAPTER THREE r 2 r 1 i r 1 r 1 o 1 α β 1 θ 1 w θ 1 c 1 u 1 w m 1 c 1 c m 1 2 c c 2 2 α θ2 u w β 2 1 w θ 2 c v s FIGURE 3.22 Velocity triangles for a centrifugal stage. ation, conventionally termed slip, can be observed, so that the relative velocity on discharge from the impeller is not aligned with the direction of the blade. The head coefficient for a centrifugal compressor can be expressed as follows: h Ϫ hUCϪ UC UC cos ␣ Ϫ UC cos ␣ 02 00 2 ␽ 21 ␽ 122 a 11 1 ␶ ϭϭ ϭ (3.34) 22 2 UU U 22 2 The dependency between the structural angle b 2 and t can be expressed in explicit form by introducing the quantity: CQ 2m 2 ␾ ϭϭ (3.35) 2 U ␲ bDU 2222 called flow coefficient at the impeller discharge section. A quantity ␴ , termed slip factor, which takes account of the imperfect guiding action of the impeller, is also introduced; it may be defined as: V S ␴ ϭ 1 Ϫ (3.36) U 2 where the term V S represents the tangential velocity defect associated with the slip effect. Utilizing these definitions and hypothesizing inlet guide vane conditions null (C q1 ϭ 0), Eq. (3.32) is rewritten as: COMPRESSOR PERFORMANCE—DYNAMIC 3.23 τ 2 β >0 φ 2 σ (forward sweep) (sweep) β =0 2 (back sweep) β <0 2 FIGURE 3.23 Loading coefficient vs. flow coefficient for a centrif- ugal stage. C ␪ 2 ␶ ϭϭ ␴ Ϫ ␾ tg ␤ (3.37) 2 b2 U 2 The above equation is illustrated in Fig. 3.23, which is the equivalent of the one already given for axial machines. For centrifugal compressors, geometries with structural angles b b2 greater than zero (i.e., blades turned in the same direction as that of rotation) are not utilized insofar as they generate high pressure drops. Radial blades or those turned in the direction opposite that of rotation up to b b2 values of about Ϫ60 degrees are normally used in common applications. 3.4.3 Conventional Representation of Pressure Drop in Compressors Pressure drops in compressors are conventionally divided into two main categories. 1. Pressure drop due to friction 2. Pressure drop due to incidence These two phenomena are discussed in the following paragraphs. Pressure Drop Due to Friction. These are dissipation terms associated with fric- tion phenomena between the walls of the ports of the machine (both rotor and stationary) and fluid flowing through it. In general, the flow in compressors is characterized by turbulence, so it can be considered that the energy dissipated is proportional, in first approximation, to the square of the fluid velocity and thus to the square of the volume flow in inlet conditions. This energy is not transferred to the fluid under the form of potential energy, but only under the form of heat. 3.24 CHAPTER THREE ii incidence los s constant mass flow rate FIGURE 3.24 Typical airfoil losses distribution as a function of incidence. Accordingly, indicating by W A the work per mass unit associated with dissipation due to friction, we may write: 2 W ϭ kQ (3.38) AA00 where k A represents a suitable constant that takes account of the specific fluid- dynamic characteristics of the stage in question. In dimensionless terms this can be expressed as: 2 W ϭ k ␾ (3.39) AA1 Pressure Drop Due to Incidence. With reference to the classic studies on bi- dimensional wing contours, it should be recalled that the pressure drop of a generic contour, stated in relation to the incidence, follows a trend of the type shown in Fig. 3.24. This distribution can be approximated with a parabolic law that presents a minimum point at a certain incidence i*. Although the behavior described here applies, strictly speaking, to wing contours alone, it may be extended with reasonable accuracy to the blading of centrifugal machines as well. It is thus possible to define, for a generic turbomachine blading, whether stationary or rotary, an optimum incidence condition, at which the pressure loss phenomena deriving from incidence are minimum. This optimum incidence value depends on the geometry of the blade and on the speed triangle immediately upstream of the blade leading edge. When the speed of rotation and the geometry have been assigned, the speed triangle and the incidences depend only on the volume flow of the processed fluid. Pressure drop due to incidence may therefore be expressed in the form: 2 W ϭ k (Q Ϫ Q*) ϩ k (3.40) IU00 00 0 This equation can be expressed in dimensionless form: COMPRESSOR PERFORMANCE—DYNAMIC 3.25 φ τ 1 C INCIDENCE LOSSES A' A B C' B' FRICTION LOSSES FIGURE 3.25 Losses breakdown as a function of the flow coefficient. 2 WyI ϭ k ( ␾ Ϫ ␾ *) ϩ k (3.41) U 11 0 The constants k 0 and k 1 are once again associated to the particular problem con- sidered. Overall Pressure Drop. In going on to consider the overall pressure drop of a compressor stage, note that in general both of the above-mentioned contributions will be present. The overall pressure loss can thus be represented as: 22 W ϭ W ϩ W ϭ kQ ϩ k (Q Ϫ Q*) ϩ k (3.42) TIAA00 U 00 00 0 or in dimensionless form: 22 W ϭ k ␾ ϩ k ( ␾ Ϫ ␾ *) ϩ k (3.43) TA1 U 11 0 The above equation can be given significant graphic interpretation, as shown in Fig. 3.25. Curve A represents the evaluated relationship between flow coefficient and head, evaluated taking account of the effective deviation phenomena that occur in a real blading. Curve A thus represents all of the energy per mass unit that is transferred to the fluid and that is thus theoretically available to be converted under the form of pressure. This quantity is diminished by the dissipation associated with pressure drop due to friction (curve B) and pressure drop due to incidence (curve C), both expressed by parabolic equations. Point C1 thus expresses the work which is ef- fectively contained in the fluid under the form of potential energy and kinetic energy for an assigned flow coefficient ƒ 1 . From this analysis, it can be stated that, due to the shapes of the various curves considered, the quantity defined above tends to present a maximum in coincidence with a clearly determined value of ƒ 1 . 3.26 CHAPTER THREE Other Pressure Drop Contributions. The description given in the preceding par- agraphs of subsection 3.4.3 provides a substantially correct illustration of the main effects linked to pressure drop in a generic compressor and their influence on overall performance. However, real compressors present dissipation effects, which do not fall within the simplified scheme of pressure drops due to incidence and pressure drops due to friction. In these cases, it is often necessary to classify the pressure drop contributions through detailed reference to the physical modes in which dissipation takes place. For centrifugal machines the main effects of additional pressure drop are linked to the presence of the blade tip and casing recess (in open machines), to ventilation phenomena between the rotating and stationary surfaces in the spaces between hubs and diaphragms, and to the presence of end seals and interstage seals. Further pressure drops can be attributed to the presence of separation areas in the impeller. For axial machines, the representation of pressure drop is slightly different to take account of the different aerodynamic phenomena involved. One possible dis- tinction could be the following: • Contour pressure drop. This is pressure drop deriving from the presence of boundary layers which develop along the blade surfaces. It can be estimated through the methods used for calculating turbulent boundary layers. • Endwall pressure drop. Pressure drop of this kind depends on the presence of localized limit states on the casing surface or the compressor rotor. These effects are usually evaluated through experimental correlations. • Pressure drop due to impact. This term indicates phenomena of the dissipation type linked to the generation of impact waves and to consequent production of entropy. In general these consist of leading edge impact waves and port impact waves, depending on the place where these effects occur. These phenomena tend to involve all types of compressors, both axial and centrifugal, with the exception of the totally subsonic ones. • Pressure drop due to mixing. This consists of irreversibility associated with transition between a non-uniform fluid-dynamic state, linked for example to local separation effects, and a uniform condition. These phenomena take place in the regions downstream of the stator or rotor blade arrays and are estimated through experimental correlations. In spite of the physical diversity of the pressure drop contributions involved, the qualitative considerations on the overall pressure drop curves, presented in subsec- tion 3.4.3 in the paragraph entitled Overall Pressure Drop, remain valid in a general sense for both axial and centrifugal compressors. 3.4.4 Operating Curve Limits: Surge and Choking The operating curves of the stages, both centrifugal and axial, present limits to the flow ranges that can be processed by the stage itself or by the machine of which COMPRESSOR PERFORMANCE—DYNAMIC 3.27 it is a part. These limits are established by two separate phenomena, called surge and choking, described below. Surge. The term ‘‘surge’’ indicates a phenomenon of instability which takes place at low flow values and which involves an entire system including not only the compressor, but also the group of components traversed by the fluid upstream and downstream of it. The term ‘‘separation’’ indicates a condition in which the bound- ary layer in proximity to a solid wall presents areas of inversion of the direction of velocity and in which the streamlines tend to detach from the wall. Separation is in general a phenomenon linked to the presence of ‘‘adverse’’ pressure gradients in respect to the main direction of motion, which means that the pressure to which a fluid particle is subjected becomes increasingly higher as the particle proceeds along a streamline. The term ‘‘stall,’’ referring to a turbomachine stage, describes a situation in which, due to low flow values, the stage pressure ratio or the head do not vary in a stable manner with the flow rate. Stall in a stage is generally caused by important separation phenomena in one or more of its components. Surge is characterized by intense and rapid flow and pressure fluctuation throughout the system and is generally associated with stall involving one or more compressor stages. This phenomenon is generally accompanied by strong noise and violent vibrations which can severely damage the machines involved. Experience has shown that surge is particularly likely to occur in compressors operating in conditions where the Q-H curve of the machine has a positive slope. Less severe instability can moreover take place also in proximity to areas of null slope. This depends on the presence of rotary stall, defined as the condition in which multiple separation cells are generated which rotate at a fraction of the angular velocity of the compressor. Surge prevention is effected through experimental tests in which pressure pul- sation at low flow rates is measured on the individual stages. On this basis, it is possible to identify the flow values at which stable operation of the stage is guar- anteed. A knowledge of the operating limits of each stage can then be used to evaluate the corresponding operating limits of the machine as a whole. Choking. Assume that a stage of assigned geometry is operating at a fixed speed of rotation and the flow rate of the processed fluid is increasing. A condition will ultimately be reached at which, in coincidence with a port, the fluid reaches sonic conditions. In this situation, termed ‘‘choking,’’ no further increase in flow rate will be possible and there will be a rapid, abrupt decrease in the performance of the stage. The occurrence of choking depends not only on the geometry and operating conditions of the stage, but also on the thermodynamic properties of the fluid. In this regard, choking can be particularly limiting for machines operating with fluids of high molecular weight, such as coolants. Many types of compressors, including industrial process compressors, normally operate in conditions quite far from those of choking. For these machines, the 3.28 CHAPTER THREE FIGURE 3.26 Non-dimensional performance curves for a stage. maximum flow limit is frequently defined as the flow corresponding to a prescribed reduction in efficiency in respect to the peak value. 3.4.5 Performance of Stages The discussion contained in the previous paragraphs provides the necessary ele- ments for understanding and interpreting the global performance of a generic stage and the manner in which it is usually represented through suitable diagrams. This subject is further discussed in the next two paragraphs. Dimensionless Representation of Performance. A possible dimensionless pre- sentation of stage performance can be effected as shown in Fig. 3.26. The inter- pretation of the various parameters utilized is the one given by the definitions provided above. The dimensionless representation is such that once the design values for the flow coefficients and the Mach number have been established, the behavior expressed by the curves is independent of the actual size of the stage. Dimensional Representation of Performance. The dimensionless performance of the stage being known, it is possible to obtain a representation in dimensional form with the use of equations given in (3.4) to (3.17). One possible description of this type is given in Fig. 3.27. The conditions of the gas on discharge from the stage can be evaluated once the gas properties and the stage inlet conditions, defined by the pressure p 00 and the temperature T 00 , have been specified. In cases where the behavior of the gas can be diagrammed through the perfect gas model, we will have for instance: COMPRESSOR PERFORMANCE—DYNAMIC 3.29 FIGURE 3.27 Dimensional performance curves for a stage. ( ␥ / ␥ Ϫ 1) ␩ P H p P 04 ␭ ϭϭ1 ϩ P ␥ ΄΅ 00 ␩ ZRT P 00 ␥ Ϫ 1 ␥ Ϫ 1/ ␥␩ P T ϭ T ␭ 04 04 P 04 ␳ ϭ (3.44) 04 ZRT 404 In cases where the perfect gas model is not applicable, it becomes necessary to apply an equation of state for real gases, for example, of the type defined by the Benedict-Webb-Rubin-Starling model. 3.5 MULTISTAGE COMPRESSORS 3.5.1 General Information The pressure ratio obtainable with a simple single-stage compressor is normally limited by constraints of both aerodynamic and structural type. In the field of centrifugal compressors for aeronautic applications, unitary pres- sure ratios of about 12 have been obtained. In industrial applications the values are much lower, usually not exceeding the limit of three. The unitary pressure ratios of the centrifugal stages are limited mainly by the maximum tip speed allowable in relation to the structural integrity requirements of the rotor and thus of the material of which it is built. For axial compressors, the maximum unitary pressure ratio obtained in advanced compressors for aeronautic applications is about 2.5. In this case, the unitary pres- sure ratio is constrained essentially by limitations of the aerodynamic type linked 3.30 CHAPTER THREE to the need to keep the work transferred to the fluid within acceptable limits so as to avoid stall. In all situations where the pressure ratio exceeds the maximum unitary value for the particular type of compressor in question it becomes necessary to recur to a multistage arrangement with two or more stages arranged in series in a repetitive configuration. The methods employed are analyzed here, with determination of the operating curves of a generic multistage compressor, taking into consideration the problems involved in the coupling of the various stages in both design and off- design conditions. 3.5.2 Multistage Compressor Operating Curves In selecting the stages that make up the complete machine, an obvious consideration is that each of them should be utilized in conditions of maximum efficiency. The efficiency of a stage is maximum in the design condition identified by a given value of the flow coefficient ƒ 1 , a value which decreases progressively in moving away from this condition. In designing a multistage compressor, each individual stage must be utilized around the design condition, accepting a performance slightly lower than that of design, since it is impossible, in practice, to size the individual stage for each specific design condition relevant to the complete compressor. It thus becomes necessary to establish suitable operating conditions, different from those of design, at which the efficiency of each individual stage is satisfactory while margins are provided as regards stall and choking. Determination of the global compressor curves requires knowledge of the per- formance curves of each of its individual stages. In the case of a multistage cen- trifugal compressor which will be examined below, the performance of the indi- vidual stage can be represented by the following parameters: (ƒ 1 ) ϭ* i design flow coefficient of nth stage ϭ * ␳ 01 ␾ ͩͪ 1 ␳ ) 06 i design flow coefficient of nth stage corrected for variation in density between inlet and discharge t , ϭ* M u head coefficient corresponding to and to M u ϭ M * ␳ 01 ␾ * ͩͪ 1 u ␳ 06 i ϭh* * PMu ,D2 polytropic efficiency for (ƒ 1 ) i ϭ (ƒ 1 ) and for M u ϭ M correspond-** iu ing to a given reference diameter The mode in which the performance of a stage varies around design conditions must also be specified. This can be done utilizing curves that describe the behavior of the head coefficient and the efficiency in relation to independent parameters. A possible general form of this representation is: COMPRESSOR PERFORMANCE—DYNAMIC 3.31 ␩ ␾␾ P 11 ϭ ƒ 1 ϩ ƒ (3.45)a ͩͪ ͩͪͲͫ ͩ ͪͬ 12 ␩ * ␾ * ␾ * P 11 ii * ␶ ␳␳ P 01 01 ϭ ƒ ␾␾ (3.45)b ͩͪ ͫͩ ͪͲͩ ͪͬ 31 1 ␶ * ␳␳ P 06 06 iii in which the functions ƒ 1 , ƒ 2 and ƒ 3 express dependencies that can be made explicit through experimental tests where the performance of each individual stage is mea- sured for a particular set of operating conditions. The above-mentioned curves are also associated to suitable constraints that represent the operating limits for the stage in question relevant to choking and surge, also determined through testing. Calculation proceeds from the first stage, first evaluating the dimensionless par- ameters ƒ 1 and M u relevant to a generic operating condition defined by volume flow rate, Q and speed of rotation N (e.g., for the design condition). The conditions on outlet from the first stage are then calculated utilizing equations of the type (3.44) and introducing various corrections to take account of the effects of the Reynolds number. The subsequent stages are then calculated in sequence, ultimately deter- mining the compressor discharge conditions. For off-design conditions, the volume flow rate and speed of rotation are varied in parametric manner to obtain the performance levels relevant to a prescribed set of operating conditions. Through calculation it is also possible to verify the con- ditions corresponding to the operating limits of the compressor and to identify the stages responsible for any surge or choking. If the working gases cannot be rep- resented through the perfect gas diagram it will be necessary to use a real gas model to calculate the thermodynamic state on inlet to and discharge from each stage. A typical complete compressor map, evaluated for different speeds of rotation, is shown in Fig. 3.28. 3.5.3 Effect of Variation in Flow Rate on Stage Coupling In evaluating the behavior of a multistage compressor, changes in the operating conditions of the individual stages consequent to variations in flow rate should be examined. For this purpose we may consider Fig. 3.29, which shows the Q curve for all of the stages of a multistage compressor at design speed of rotation. Assume that the first stage operates at its own design flow rate Q 1 . In this condition, the density of the fluid on discharge from the stage is known and it is possible to evaluate the volume flow rate Q 2 for the second stage, which is hypothesized as being that of design. If the flow rate Q 1 is decreased by a quantity DQ 1 , the first stage will then operate at a pressure ratio higher than in the preceding situation. In this case, it can be seen that the density of the fluid on inlet to the second stage is increased, [...]... The equations for pressure and temperature are again of the type: 2 p 02 /p2 ϭ (1 ϩ (g Ϫ 1)M2 /2) (␥/(␥Ϫ1)) T 02 /T2 ϭ (1 ϩ (g Ϫ 1) M /2) 2 2 in which: (3.64) 3.40 CHAPTER THREE 2 C2 ϭ (C m2 ϩ C 2 )1 /2 q2 (3.65) M2 ϭ C2 /(␥ R T2) 1 /2 The isentropic efficiency of the impeller can be defined as follows: ␩S,ROT ϭ ͩ ͪ ͩ ͪ p 02 p00 (␥Ϫ1 / ␥) Ϫ1 T 02 Ϫ1 T00 (3.66) The utilization of a value correlated for thermodynamic... to section 2, shown in Fig 3.34 The velocity U2 can be obtained through the simple kinematic equation: U2 ϭ 2 p r2 N (3.57) The meridian component of the absolute velocity of the fluid is calculated here too through the flow continuity equation: Cm2 ϭ m/r2 A2 with A2 ϭ 2 p r2 b2 The tangential component Cq2 is given by: (3.58) COMPRESSOR PERFORMANCE—DYNAMIC c 2 3.39 w 2 vs u2 c2 w1 c m1 2 2 r FIGURE... also: W1 ϭ ((U1o Ϫ Cq1 )2 ϩ Cm 12) 1 /2 U1o ϭ 2 p r1o N The local absolute Mach number is given by: (3.53) 3.38 CHAPTER THREE cθ1 wθ1 u1 c1 w1 α1 c m1 β 1 FIGURE 3.33 Impeller inlet velocity triangle M1o ϭ C1 /(g R T2)1 /2 (3.54) Pressure and temperature are linked to the Mach number through the following equations: p01 /p1 ϭ (1 ϩ (g Ϫ 1) M2 /2) (␥/(␥Ϫ1)) 1o T01 /T1 ϭ (1 ϩ (g Ϫ 1) M2 /2) 1o (3.55) A further... molecular weight) with respect to the real one, hence an extensive use of similitude laws is done 3.46 CHAPTER THREE c3 δ3 α3 c1 α u1 c θ1 1 β1 c c m3 m1 i1 s ta to r w θ1 w1 c θ3 l u1 c2 c 2 i2 ro to r α 2 c m2 2 2 u2 w 2 w2 FIGURE 3.36 Velocity triangles for an axial stage Single Stage Testing This type of test is normally performed using scale models and closed or open loop facilities handling air...3. 32 CHAPTER THREE FIGURE 3 .28 FIGURE 3 .29 Performance map for a multistage compressor Effect of mass flow rate in a multistage compressor COMPRESSOR PERFORMANCE—DYNAMIC 3.33 so that the volume flow rate of the second stage is decreased by a quantity DQ2 Ͼ DQ1 in respect to the value Q2 All this shows that flow perturbation tends to ‘‘amplify’’... proposed by Wiesner: ␴ϭ1Ϫ ͙cos( ␤b2) Z 0.7 (3.61) valid for R1i /R2 Ͻ e Ϫ8.16(cos( ␤b2))/Z Application of the Euler equation for turbomachines produces: Dh0 ϭ U2Cq2 Ϫ U1Cq1 (3. 62) and thus the increment in total temperature, assuming that secondary energy contributions deriving from the effects of recirculation, friction, etc., can be ignored, is given by: Dh0 ϭ (U2Cq2 Ϫ U1Cq1)/CP (3.63) The equations... exit velocity triangle Cq2 ϭ U2 ϩ Cm2 tan(␤b2) Ϫ Vs (3.59) where ␤b2 is the structural angle of the blade at the discharge section and Vs represents tangential speed defect associated with the slip factor s: VS ϭ U2(1 Ϫ s) (3.60) Numerous correlations between slip factor and rotor geometry, obtained both theoretically and experimentally, are available for an estimation of Cq2 to be used in design problems... where A1 ϭ cD (r1o2 Ϫ r1i2) cD ϭ blockage factor due to presence of the blades The tangential component of the absolute velocity Cq1 depends on whether or not inlet guide vanes are utilized In the absence of vanes, we will have Cq1 ϭ 0 Consequently, it is possible to resolve the rotor inlet speed triangle, illustrated in Fig 3.33, through the following equations: C1 ϭ (Cm 12 ϩ Ct 12) 1 /2 (3. 52) and also: W1... is a change in flow rate DQ2 Ͼ DQ1, resulting in an ‘‘amplification’’ effect capable of determining final choking in the last stage These considerations show that in a multistage compressor where the stages have been correctly coupled, compressor stall and possible surge are always determined by stall in the final stage due to diminution in its volume flow rate In the same way, compressor choking is determined... stiffness (Fig 3. 42) The mode shapes are important because they indicate the relative vibration amplitude at each station along the rotor If relative amplitudes at the bearings are low a high unbalance producing considerable deflection in some sections of the shaft will cause very 3. 52 CHAPTER THREE FIGURE 3.41 FIGURE 3. 42 Map of lateral critical speeds Typical rotor response diagram COMPRESSOR PERFORMANCE—DYNAMIC . equation: C ϭ m/r (3.58) m22A2 with A 2 ϭ 2 pr 2 b 2 . The tangential component C q2 is given by: COMPRESSOR PERFORMANCE—DYNAMIC 3.39 m1 2 c c 2 2 α 2 u w β 2 1 w 2 c v s r FIGURE 3.34 Impeller. centrifugal compressor can be expressed as follows: h Ϫ hUCϪ UC UC cos ␣ Ϫ UC cos ␣ 02 00 2 ␽ 21 ␽ 122 a 11 1 ␶ ϭϭ ϭ (3.34) 22 2 UU U 22 2 The dependency between the structural angle b 2 and t can. UC )/C (3.63) 02q21q1 P The equations for pressure and temperature are again of the type: 2( ␥ /( ␥ Ϫ 1)) p /p ϭ (1 ϩ (g Ϫ 1)M /2) 02 2 2 (3.64) 2 T /T ϭ (1 ϩ (g Ϫ 1) M /2) 02 2 2 in which: 3.40