8.11 Multistage Compressor Starting Problem

The annulus flow area in a multistage compressor is designed based on the calculated density rise along the axis of the machine operating at the design point. The assumption of constant axial velocity, for example, implies that the channel area shrinks inversely proportional to the density rise, to conserve mass, namely,

(8.199)

This behavior is schematically shown in Figure 8.73.

In the starting phase of a compressor, the mass flow rate is initially small, which means a higher loading for the front stages. The picture for the aft high-pressure stages is just the opposite of the front low-pressure stages. Initially, the compressor does not develop the density rise that it was designed to produce, hence with lower than design densities in the aft stages the axial velocity is increased to satisfy the continuity equation. Higher than the design axial velocity leads to a lower loading of the aft stages. Actually the aft stages would be windmilling and their blades operating in a separated flow. A comparison of the velocity triangles in the starting phase and the design point will help us understand the (starting) problem in a high-pressure compressor. Figure 8.75 shows the velocity triangles for the front and aft stages of a compressor. Note that the flow coefficient is lower than the design value for the front stages, and the flow coefficient is higher for the aft stages. We remember from the compressor performance map that with a reduction of flow coefficient the compressor can go into stall for a given shaft speed. With front stages not producing the design density ratio, the axial flow in the aft stages accelerates to conserve mass. Hence, the two ends of the compressor see the opposite flow fields and are subjected to opposite flow coefficients, that is, lower and higher than the design, respectively. However, the tendency for both front and aft stages is to stall in the starting phase of a high-pressure compressor.

The loading mismatch between the front and aft stages in the starting phase of a high-pressure axial compressor may be solved in several ways. Here we introduce three distinct approaches to the starting problem.

Proposition 1: Split shaft (or multispool shaft system)

Let the aft stages operate at a higher rotational speed than the front stages. This proposition aims at matching the relative flow angle to the rotor blades in the aft stages. Pratt & Whitney spearheaded the development of the split shaft concept in the United States in the 1950s. Today, all modern high-pressure compressors employ a two-to-three spool or shaft configuration to alleviate the problem of starting and improve compressor efficiency and stability. This method primarily attacks the aft stages, which would have been windmilling without a higher shaft speed, as in a single-shaft gas turbine engine. Rolls-Royce is the only engine company that has manufactured a production three-spool commercial engine, known as the RB 211, with several successful derivatives.

Proposition 2: Variable stators

This proposition aims at adjusting the absolute flow angle (α) through a stator variable setting to improve the mismatch between the flow and the rotor relative flow angle β. GE-Aircraft Engines spearheaded the development of high-pressure compressors with the variable stator approach in the United States, also in 1950s. Today, modern high-pressure compressors use variable stators for the front several stages to help with the starting/off-design operation efficiency. This method primarily attacks the front stages. Although a variable stator benefits all stages of a compressor, its use is limited to the front few (say six) low-pressure stages as the sealing of hot gases through the variable stator seal poses an operational and maintenance problem on the high-pressure end of the compressor.

Proposition 3: Intercompressor bleed

The windmilling operation of the aft stages is the result of high axial velocities through those stages. To cut down on excessive axial velocities, we may bleed off some mass flow in the intermediate stages of the compressor, the so-called intercompressor bleed. Therefore in the starting phase, bleed ports need to be opened to help the high-pressure end of the compressor to operate properly. This proposition represents a relatively low-cost method of starting compressors that does not employ either a split shaft or the variable stators. Stationary gas turbine power plants employ the intermediate bleed solution as a method of starting the compressor. This method attacks the aft as well as the front stages, as the bleed ports are located in the middle. Therefore, mass withdrawal causes a reduction in the axial velocity to the aft stages as well as lowering the backpressure for the front stages, hence increasing the flow speed in the machine. The ability to tailor the flow through the engine to improve overall efficiency, component stability, and provide cooling to accessories and engine components all speak in favor of intercompressor bleed. In fact, all modern gas turbine engines today employ intercompressor bleed/flow control. These methods are summarized in Figure 8.75.

8.12 The Effect of Inlet Flow Condition on Compressor Performance

Aircraft gas turbine engines operate downstream of an air intake system. The level of distortion that an inlet creates at the compressor face affects the performance and the stability of the compressor. First, what do we mean by distortion? In simple terms, distortion represents nonuniformity in the flow. The nonuniformity in total pressure as in boundary layers and wakes, the nonuniformity in temperature as in gun gas ingestion or thrust reverser flow ingestion and the nonuniformity in density, as created by hot gas ingestion are some of the different types of distortion. The common feature of all different types of distortion is found in their destabilizing impact on the compressor performance. This means that all distortions reduce the stability margin of a compressor or fan, potentially to the level of compressor stall or the engine surge. The types of distortion are

1. Total pressure distortion p_t(r, θ)
2. Total temperature distortion T_t(r, θ)
3. Flow angle distortion α(r, θ)
4. Secondary flow – swirl at the engine face C_θ1(r, θ)
5. Entropy distortion s(r, θ)
6. Combinations of some or all of the above distortions.

The most common type of inlet distortion is the total pressure distortion that is caused by separated boundary layers in the inlet. Under normal operating conditions, the boundary layers in the inlet are well behaved and remain attached. However, if the boundary layer management system in a supersonic inlet, as in the bleed system, fails to react to an abrupt change in the flight operation (potentially due to a rapid combat maneuver), the flowfield at the engine face will contain large patches of low-energy, low-momentum flow that could cause flow separation in the front stage(s) of the fan or compressor. In describing the total pressure distortion and its impact on compressor performance, we divide the spatial extent of the spoiled flow according to its radial and circumferential extent, as shown in Figure 8.76.

These inlet total pressure distortion patterns may be simulated by installing screens of varying porosity upstream of the compressor or fan (typically one diameter upstream) in propulsion system ground test facilities. The distortion patterns generated by screens in a test set up in ground facilities represent the “steady-state” component of the distortion and thus lack the “dynamic” or transitory nature of the distortion encountered in real flight environment. The full description of distortion requires both the steady-state and the dynamic components, as in the study of turbulent flow requiring a mean and an rms level of the fluctuation.

The results of a NASA-Glenn 10 × 10 supersonic wind tunnel study on the response of a simple turbojet engine (J-85) to steady-state inlet total pressure distortion are shown in Figure 8.77 (from Povolny et al., 1970).

The normal operating line, the undistorted stall boundary, and two corrected shaft speeds of 100% and 93% design are shown in dashed lines. Four solid lines correspond to the stall boundaries of the four distortion patterns simulated at the compressor face via screens. From having the least to the most impact on the stall margin deterioration, we identify the culprits as (1) radial hub, (2) radial tip, (3) circumferential hub, and (4) the full-span circumferential distortion, respectively. We also note that the full-span distortion at the 100% corrected speed operates at the stall boundary, that is, zero-stall margin! Further research identified a critical circumferential extent of the spoiled sector that causes the maximum loss in the stall pressure ratio of a compressor is at nearly 60°, as evidenced in Figure 8.78 (from Povolny et al., 1970).

The loss in stall pressure ratio with circumferential inlet distortion reaches to ∼10% at 100% corrected speed, as shown in Figure 8.78. At higher shaft speeds, the incidence angle in the spoiled sector is larger than at the lower shaft speed, thus the trend of higher loss of the stalling pressure ratio at higher shaft speeds becomes evident using simple velocity triangle arguments.

The temperature distortion also leads to a reduction in stall margin. In general, static temperature distortion in a flow brings about density nonuniformity, which creates a nonuniform velocity field. Consequently, it is impossible to create a static temperature distortion without creating other forms of nonuniformity, for example, density, velocity, total pressure, in the flow. To quantify the impact of a spatial temperature distortion on engine stall behavior, NASA researchers have conducted experiments with representative data shown in Figure 8.79 (from Povolny et al., 1970). The undistorted operating line, stall limit, and different shaft speeds are shown in dashed lines. Data points corresponding to the effect of temperature distortions of 45–120°F on the stall behavior of a variable geometry turbofan engine high-pressure compressor are plotted in solid lines. The circumferential extents of the temperature distortions were 90 and 100° in different tests. A 100°F distortion of ∼90–100° circumferential extent is seen to stall the high-pressure compressor operating at its 90% corrected flow.

The inadequacy of steady-state distortion simulation in a wind tunnel is best seen in Figure 8.80 showing F-111 flight test data of compressor stall with TF-30 engine. A steady-state distortion parameter K_D is graphed at different corrected airflows through the engine, and a band of maximum allowable distortion based on previous wind tunnel tests is shown to miss the mark of in-flight compressor stall data by a large margin (from Seddon and Goldsmith, 1985). The culprit is identified as the dynamic distortion.

The steady-state distortion parameter K_D expresses a weighted average of engine face total pressure distortion pattern (circumferential and radial) recorded by an engine face rake system composed of several radial pitot tube measurements. For a mathematical description of K_D and other distortion parameters, Hercock and Williams (1974) may be consulted. In addition, Farr and Schumacher (1974) present evaluation methods of dynamic distortion in F-15 aircraft. Schweikhand and Montoya (1974) treat research instrumentation and operational aspects in YF-12. These references are recommanded for further reading.

8.13 Isometric and Cutaway Views of Axial-Flow Compressor Hardware

To meet the real world, engineers need to spend time with the real hardware. Isometric and cutaway views of the hardware provide for a three-dimensional feel but may not be a substitute for the cutaway of a real engine (or its components) in the laboratory. The real engine, albeit old and surplus, shows real fasteners, components assembly, manufacturing tolerances; some operational degradation, for example, wear, on the rotor blade tips, shows labyrinth seals between rotating and stationary parts; it shows signs of erosion in the compressor and turbine and perhaps corrosion in the hot section. A visual inspection and a feel of the compressor and turbine blades show the turbine blades receive heavy deposits from the combustor (and its fuel additives) and give a new meaning to “surface roughness, ” which will not feel “hydraulically smooth.”

Having pointed out some of the advantages of real hardware, let us examine some isometric and cutaway views of axial-flow compressors, designed, built, tested, and flown on aircraft in this section. Figure 8.81 is taken from a manual by Pratt & Whitney Aircraft (1980), whereas Figures 8.82–8.84 are taken from Rolls-Royce’s The Jet Engine (2005). Figure 8.81 shows a two-stage fan and the low-pressure compressor rotor. Mid-span shrouds prevent the first bending mode of the first fan rotor blades. The anti-icing air valve and the actuator point to an operational need of the aircraft (i.e., flight under icing condition) and demands on the compressor air.

8.14 Compressor Design Parameters and Principles

In this section, we present design guidelines that are useful in the preliminary design of axial-flow compressors. The approach to turbomachinery design in textbooks is neither unique nor exact. In general, we use the lessons learned from our predecessors and the invaluable contributions of NASA and open literature, which often come from academia and industry. One possible approach to compressor design is outlined in steps that are particularly useful for students who want to learn and practice their turbomachinery design skills.

We propose to design an axial-flow compressor for a design-point mass flow rate and pressure ratio. Here, we first review the steps and in the next section we design an axial compressor at the pitchline. We start the design process by choosing

1. Axial Mach number at the compressor/fan face is M_z1 ∼0.5
2. Flow area at the compressor face is that sizes the engine and thus establishes the mass flow rate
3. Typical hub-to-tip radius ratio, r_h1/r_t1 is ∼0.4, a little less or more (may be up to ∼0.5 or as low as ∼0.3). The relative tip Mach number M_r1 establishes the tip radius
4. 
   • M_T1, tip tangential Mach number, that is, (ωr_t)/a₁ is in excess of sonic
   • M_r1 is supersonic, ∼1.2 to ∼1.5 (remember that M_z ∼0.5)
5. Thickness-to-chord ratio varies from ∼10% in the subsonic hub region to ∼3% in the supersonic tip region, linearly varying in between
6. The Reynolds number based on chord has to be > 300, 000 (in relative frame), at altitude. This establishes the minimum chord length for turbulent boundary layer at all altitudes, the other parameter contributing to the chord length is the blade aspect ratio and bending stresses. Often chord length is several times bigger than that required for the 300, 000 Reynolds number at altitude
7. Centrifugal stress at the blade root (i.e., the hub) is calculated based on the simple formula
   (8.200)

   

   The centrifugal force is the integral of mv²/r, which is mω²r, that is,

   (8.201)

   where ρ_blade is blade material density and A(r) is the blade cross-sectional area as a function of span.

   (8.202)
   (8.203)
   (8.204)

   The blade area distribution along the span, A_b(r)/A_h, is known as taper and is often approximated to be a linear function of the span. Therefore, it may be written as

   (8.205)

   We may substitute A_b(r)/A in the integral and proceed to integrate; however, a customary approximation is often introduced that replaces the variable r by the pitchline radius r_m. The result is

   (8.206)

   The taper ratio A_t/A_r is ∼0.8–1.0. Therefore, the ratio of centrifugal stress to the material density is related to the square of the angular speed, the taper ratio, and the flow area, A = 2πr_m(r_t − r_h). Equation 8.206 is the basis of the so-called AN² rule, where the RHS is related to the size (i.e., A) of the machine and the square of the angular speed (i.e., N²), and the LHS is related to material property known as specific strength.

   The material parameter of interest in a rotor is the creep ruputure strength, which identifies the maximum tensile stress tolerated by the material for a given period of time at a specified operating temperature. Based on the 80% value of the allowable 0.2% creep in 1000 h for aluminum alloys and the 50% value of the allowable 0.1% creep in 1000 h for other materials, Mattingly, Heiser, and Pratt (2002) have graphed Figures 8.86 and 8.87. They show the allowable stress and the allowable specific strength of different engine materials as a function of temperature

8. The solidity at the pitchline σ_m ∼ 1 to 2
9. The degree of reaction at the pitchline may be chosen to be °R_m ∼ 0.5 or 0.6 for subsonic sections and considerably higher (e.g., 0.8) for the supersonic portion of the blade. This choice specifies the C_{θ2, m}. We can use this information along with Euler turbine equation to establish the rotor-specific work at the pitchline as well as the stagnation temperature rise, that is, we get T_t2m. The choice of °R is less critical than D-factor or de Haller criterion
10. The D-factor at the pitchline should be ≤0.5 or 0.55, often a quick check on the acceptable level of diffusion is made using the de Haller criterion that limits diffusion to a level corresponding to W₂/W₁ ≥ 0.72
11. The choice of the vortex design, that is, C_θ(r), establishes the degree of reaction and D-factor at other radii, which we examine closely. It is acceptable for the degree of reaction to deviate from 0.5, but we do not want it to become negative in a compressor (i.e., it then behaves like a turbine!). The D-factor has to remain below the critical value of ∼0.5 (or 0.55). We may iterate on the vortex design choice in order to get these parameters right
12. The polytropic efficiency is assumed to be ∼0.90, that is, e_c∼0.90, which in combination with T_t ratio gives p_t ratio, that is, we get total pressure at the pitchline at station 2, downstream of the rotor
13. To get the flow parameters downstream of the stator, we preserve T_t and calculate p_t3 based on an assumed ϖ_s of ∼0.04–0.05; also the idea of repeated stage gives us the α₃
14. We assume the axial velocity remains constant throughout the compressor (at the design radius, i.e., the pitchline). This is a preliminary design choice
15. Bending stresses on a cantilevered blade, under aerodynamic loading, is estimated (by Kerrebrock, 1992) to be
    (8.207)

    Bending stress is, therefore, proportional to gas pressure, flow coefficient at the tip, stage total temperature ratio, and to the square of tip radius-to-thickness ratio. Bending stress is also inversely proportional to solidity

16. Thermal stresses are often small in axial-flow compressors and fans; however, they are significant in the turbine section. Also, the aft stages of high-pressure compressors in modern gas turbine engines operate at high gas temperatures, such as T_gas ∼850–900 K, which require special attention to thermal strains and tip clearance, as well as total stress (i.e., the sum of centrifugal, bending, and thermal) at the hub
17. Blades as cantilevered structures attain diverse mode shapes, such as first bending, second bending, first torsional, coupled first bending and torsional, as well as higher modes of vibration. Each mode shape has its own natural frequency ω_n. The effect of blade rotation is to stiffen the structure and thus raise these natural frequencies. The shaft and the discs also exhibit their natural vibrational mode shapes and frequencies. To avoid resonance between these frequencies and the shaft rotational speed (and its multiples), a frequency diagram, known as Campbell diagram (Figure 8.87), is generated to examine possible match points between these frequencies inherent in the compressor, that is, in the natural modes and rotor shaft frequency. Kerrebrock (1992), Wilson and Korakianitis (1998), and Mattingley, Heiser, and Pratt (2002) provide a detailed account of engine structure.

8.14.1 Blade Design – Blade Selection

For blade design, we need to estimate the following angles:

1. Incidence angle i
2. Deviation angle

The incidence angle is estimated at the location of minimum loss, from cascade data that is known as i_optimum. Typically, optimum incidence angle is ∼−5° to +5°.

Iteration loop

The deviation angle is first estimated from the simple Carter’s rule:

(8.208)

which is based on the net flow turning and the coefficient “m” in Carter’s formula is taken to be 0.25.

Also, based on the inlet and exit flow angles, we may use the (NACA 65-series) cascade correlation data of Mellor (Appendix I) to identify a suitable 65-series cascade geometry that gives an adequate stall margin. Note that the front and aft stages of a multistage compressor or fan face different stall challenges; for example, the front stages are more susceptible to positive stall and the aft stages are more susceptible to negative stall. Therefore, we may choose the design point to be farther away from the two stall boundaries for front and aft stages. Figure 8.88 is a definition sketch for the design point selection.

The optimum incidence angle is defined (by Mellor) to be at the point of lowest profile loss, although some authors prefer the optimum incidence to correspond to the angle where lift-to-drag ratio is at its maximum. Here, we adopt Mellor’s definition, due to its ease of use. The cascade loss data, as shown in Figure 8.22, can be used to arrive at the i_opt.

Johnsen and Bullock (1965) in NASA SP-36 outline optimum incidence estimation based on cascade data correlations for different profile types and profile thicknesses. This reference may be used for more accurate estimation of the incidence angle. From the inlet flow and an estimate of the optimum incidence angle, we calculate the leading-edge angle on the mean camber line (MCL), κ₁, and from the exit flow angle and deviation we calculate the blade angle at the T.E.

The iteration loop on the deviation angle may be initiated at this point, since we have calculated a camber angle and we may now use the expanded version of the Carter’s rule:

Although we have introduced a possible iteration loop on the deviation angle, in the preliminary stage of the compressor design it is acceptable to use the simple formula of Carter for deviation.

The supersonic section is designed based on

• Double-circular arc (DCA) blades
• J-profile design
• Multiple-circular arc design (MCA).

These designs are described in Cumpsty (1989), Schobeiri (2004), and Wilson and Korakianitis (1998).

8.14.2 Compressor Annulus Design

The annulus flow area shrinks inversely proportional to the rising fluid density for a constant axial velocity, that is,

We can calculate density rise per stage (at the pitchline) from the pressure and temperature rise and the use of the perfect gas law. This is an approximate method, but is adequate for the preliminary design purposes.

Common practices are

• Keep the casing, that is, the tip, radius constant, therefore the information about the annulus area gives the hub radius
• Keep the hub radius constant and thus calculate the tip radius from the annulus area
• Keep the pitchline radius constant, which then shrinks the tip and hub equally, and we can calculate those from the annulus area
• Keep the axial gap between the rotor and stator blade rows to approximately 0.23–0.25 c_z (i.e., about one quarter axial chord length gap between blade rows).

8.14.3 Compressor Stall Margin

Compressor stall margin may be established based on the use of Koch equivalent diffuser model of the compressor blade passage. This is rather laborious, but produces good results. We start with the equivalent enthalpy rise coefficient C_h (for notation, see section 8.10).

Note that the difference between U₁ and U₂ across the rotor (in the numerator) is due to streamline shift in the radial direction. We further correct for Reynolds number, tip clearance gap, and the axial blade spacing other than the nominal values that are used in Koch’s stall margin correlations. The correction factors are in Figures 8.64–8.66. Finally, we correct the adjusted enthalpy rise coefficient for the effect of stagger and the velocity vector diagram (wake effect) to arrive at the effective static enthalpy rise coefficient according to

The critical stalling pressure rise is shown in Figure 8.73 as “stall line” or the case of 0% stall margin.

Guidelines on the Range of Compressor Parameters
Parameter	Range of values	Typical value
Flow coefficient	0.3 ≤ ≤ 0.9	0.6
D-Factor	D ≤ 0.6	0.45
Axial Mach number Mz	0.3 ≤ Mz ≤ 0.6	0.55
Tip Tangential Mach Number, MT	1.0–1.5	1.3
Degree of reaction	0.1 ≤ °R ≤ 0.90	0.5 (for M < 1)
Reynolds number based on chord	300, 000 ≤ Rec	>500, 000
Tip relative Mach number (1st Rotor)	(M1r)tip ≤ 1.7	1.3–1.5
Stage average solidity	1.0 ≤ σ ≤ 2.0	1.4
Stage average aspect ratio	1.0 ≤ AR ≤ 4.0	<2.0
Polytropic efficiency	0.85 ≤ ec ≤ 0.92	0.90
Hub rotational speed	ωrh ≤ 380 m/s	300 m/s
Tip rotational speed	ωrt∼450—550 m/s	500 m/s
Loading coefficient	0.2 ≤ ψ ≤ 0.5	0.35
DCA blade (range)	0.8 ≤ M ≤ 1.2	Same
NACA-65 series (range)	M ≤ 0.8	Same
De Haller criterion	W2/W1 ≥ 0.72	0.75
Blade leading-edge radius	rL.E. ∼5–10% of tmax	5% tmax
Compressor pressure ratio per spool	πc < 20	up to 20
Axial gap between blade rows	0.23 cz to 0.25 cz	0.25 cz
Aspect ratio, fan	∼2–5	<1.5
Aspect ratio, compressor	∼1–4	∼2
Taper ratio	∼0.8–1.0	0.8

8.15 Summary

Axial-flow compressors are highly evolved machines that provide efficient mechanical compression (of the gas) in a gas turbine engine. The mechanical work is delivered to the medium by a set of rotating blade rows. The rotor blades impart swirl to the flow, thereby, increasing the total pressure of the fluid. In between the rotor blade rows are the stator blade rows that remove the swirl from the fluid, thereby increasing the static pressure of the fluid. The combination of one rotor row and one stator blade row is called a compressor stage. Since the primary flow direction is along the axis of the machine, staging the axial-flow compressor is easy. The flow in an axial-flow compressor is exposed to an adverse pressure gradient environment, that is, a climbing pressure hill that tends to stall the boundary layer. Therefore, the phenomenon of stall is experienced in a compressor. The stalled compressor flow is inherently unsteady and thus may cause a system-wide instability (between the compressor and combustor) known as surge.

In analyzing the flow in the rotor, it is most convenient to use the rotating frame of reference that spins with the rotor, which is known as the relative frame of reference. The flow in stator blade row is best analyzed by a stationary observer (i.e., a frame of reference that is fixed with respect to the casing) known as the absolute frame of reference. The velocity vectors in the two frames of reference form a triangle, described by

The Euler turbine equation is known as the fundamental equation in turbomachinery that relates the change of angular momentum Δ (rC_θ) across a rotor blade row to the rotor specific work via

(8.9a)

If we express the jump in swirl across a rotor in terms of the (rotor) inlet absolute and exit relative flow angles (that remain constant over a wide operating range of the compressor), we deduce from Equation 8.25a that

The significance of this equation is the appearance of U² in front of the bracket on the RHS, which indicates that the total temperature rise across a rotor is proportional to the square ofthe wheel speed U². Therefore a high blade Mach number is desirable, which is exactly the argument for the development of transonic compressors. A rule of thumb is that the blade tangential Mach number at the tip is slightly supersonic, M_T ∼ 1.2–1.3. The structural limitations currently limit the blade tip Mach number to ∼1.5 in conventional (subsonic throughflow) compressors. A high strength-to-weight ratio material, such as titanium, is desirable for fan blade construction.

Since we have to limit the level and extent of the flow diffusion in a compressor blade row, we define a diffusion factor D for the rotor and stator blade rows according to

The experience shows that diffusion factor is to be limited to ∼.6 about the mid-span of a blade row and to ∼0.4 near the hub or the tip. The lower diffusion imposed on the hub and the tip is due to a complex hub boundary layer/corner vortex formation and the tip clearance flows that dominate the two ends of a blade row. We may view the diffusion factor as a blade-row-specific figure of merit. We define a stage-based figure of merit as well, which is called the degree of reaction °R,

Degree of reaction is the ratio of the rise of the static enthalpy across the rotor to that of the stage. In essence, it speaks to the rotor’s share of the static pressure rise to that of the stator. Therefore, a 50% degree of reaction stage equally divides the burden of the pressure rise to the rotor and the stator. Although 50% sounds desirable for equal burden, since the rotor blades spin their boundary layers are more stable and thus can withstand higher static pressure rise than the stationary stator blades. Consequently, a 60% degree of reaction may be more optimal than the 50%.

Cascade aerodynamics provides a rich background for a two-dimensional (subsonic) blade design. The minimum-loss incidence angle i_opt and the cascade (total pressure) loss bucket are used for the preliminary 2D design of compressor blade sections as well as identifying the positive and negative stall boundaries. The limited supersonic cascade data make the preliminary design of the supersonic sections of transonic fans less grounded. The general rules are based in gas dynamics, which indicate the selection of the thinnest possible sections in order to minimize the wave drag. The thinnest (structurally feasible) section is ∼3% thick and the subsonic root is often ∼10% thick.

Three-dimensional design of blades, in the preliminary stage, is achieved by the so-called vortex design approach. Here we introduce a catalog of swirl profiles, such as free-vortex, solid-body rotation, and others that will be anchored at the pitchline and then give us a swirl profile along the span. We applied radial equilibrium theory to calculate an axial velocity profile. Three-dimensional losses are due to secondary flows, tip clearance, and the blade junction corner vortex. The unsteady flow losses are related to upstream wake chopping (by the downstream blade row) and the subsequent vortex shedding in the wake. Losses due to compressibility are minimized by thin profile designs that form the sections of swept blades.

A single-spool compressor is of limited capability in high-pressure compressor applications. The primary reason for that is the differing rotational requirement of the front versus the aft stages. This is the crux of the classical starting problem for a high-pressure ratio compressor. Multistage compressors are often driven by different shafts in order to spin the high-pressure stages faster than the low-pressure compressor. Variable stators in the front stages are always adjustable to help with the starting problem as well as the off-design operation of the compressor. A fraction of airflow in the compressor is bled at the intermediate and exit sections to cool the HPT, the casing, and the exhaust nozzle. The cooling fraction is ∼10–15% in modern engines.

As an upper bound, multispool, high-pressure ratio compressors can achieve a pressure ratio of ∼45–50. The exit temperature of ∼900 K is the limit of current materials for an uncooled compressor, which limit the compressor pressure ratio. The advances in computational fluid dynamics, parallel processing, and computer memory have elevated the design of compressors to be based on flow physics and with less reliance on empiricism and cascade data. The result has been the appearance of high-efficiency unconventional transonic blades and stages with forward and aft swept blades and lean. The advances in integrated manufacturing technology that involve super plastic forming and diffusion bonding (SPF/DB) are used for the manufacturing of modern (composite) wide-chord fan blades. New materials and manufacturing technology, e.g., BLISK, offer weight savings in compressors with the subsequent improvement on the engine thrust-to-weight ratio.

Dixon [7], Cheng et al. [2], Prince, Wisler and Hilvers [41], Whitcomb and Clark [53] are recommended for additional reading.

References

1. Bullock, R.O. and Finger, H.B., “Compressor Surge Investigated by NACA, ” SAE Journal, Vol. 59, September 1951, pp. 42–45.
2. Cheng, P., Prell, M.E., Greitzer, E.M., and Tan, C.S., “Effects of Compressor Hub Treatment on Stator Stall and Pressure Rise, ” AIAA Journal, Vol. 21, No. 7, July 1984.
3. Carter, A.D.S., “The Axial Compressor, ” in Gas Turbine Principles and Practice, Ed., Cox, H.R., Newnes Ltd., London, UK, 1955.
4. Cumpsty, N., Compressor Aerodynamics, Cambridge University Press, UK, 1989.
5. Cumpsty, N., Jet Propulsion, Cambridge University Press, Cambridge, UK, 1997.
6. De Haller, P., “Das Verhalten von Tragfluegelgittern in Axialverdichtern und im Windkanal, ” Brenstoff und Waermekraft, Vol. 5, 1953.
7. Dixon, S.L., Fluid Mechanics, Thermodynamics of Turbomachinery, 2nd edition, Pergamon Press, Oxford, UK, 1975.
8. Donaldson, C., Du, P., and Lange, R.H., “Study of Pressure Rise Across Shock Waves Required to Separate Laminar and Turbulent Boundary Layers, ” NACA Technical Note 2770, 1952.
9. Emmons, H.W., Pearson, C.E., and Grant, H.P., “Compressor Surge and Stall Propagation, ” Transactions of the ASME, Vol. 79, May 1955, pp. 455–467.
10. Farr, A.P. and Schumacher, G.A., “System for Evaluation of F-15 Inlet Dynamic Distortion, ” Paper in Instrumentation for Airbreathing Propulsion, Progress in Astronautics and Aeronautics, Vol. 34, Eds. Fuhs, A.E. and Kingery, M., MIT Press, Cambridge, MA, 1974.
11. Garvin, R.V., Starting Something Big: The Commercial Emergence of GE Aircraft Engines, AIAA, Inc., Reston, VA, 1998.
12. Greitzer, E.M., “Surge and Rotating Stall in Axial Flow Compressors, ” ASME Journal of Engineering for Power, Vol. 98, No. 2, 1976, p. 190.
13. Gostelow, J.P., Cascade Aerodynamics, Pergamon Press, Oxford, UK, 1984.
14. Hawthorne, W.R., “Secondary Circulation in Fluid Flow, ” Proceedings of Royal Society, London, Vol. 206, 374, 1951.
15. Hechert, H., Steinert, W., and Lehmann, K., “Comparison of Controlled Diffusion Airfoils with Conventional NACA-65 Airfoils Developed for Stator Blade Application in a Multistage Axial Compressor, ” Transactions ofthe ASME, Journal ofEngineering for Gas Turbines and Power, Vol. 107, April 1985, pp. 494–498.
16. Hercock, R.G. and Williams, D.D., “Distortion-Induced Engine Instability: Aerodynamic Response, ” AGARD, LS72-Paper No. 3, 1974.
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20. Horlock, J.H., Axial Flow Compressors, Krieger Publishing Company, Huntington, NY, 1973.
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22. Howell, A.R., “Fluid Dynamics of Axial Compressors”, Proceedings of Institution of Mechanical Engineers, London, Vol. 153, 1945.
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24. Kerrebrock, J.L., “Flow in Transonic Compressors, ” AIAA Journal, Vol. 19, No. 1, 1981, pp. 4–19.
25. Kerrebrock, J.L., Aircraft Engines and Gas Turbines, 2nd edition, MIT Press, Cambridge, MA, 1992.
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27. Kerrebrock, J.L. and Mikolajczak, A.A., “Intra-Stator Transport of Rotor Wakes and Its Effect on Compressor Performance, ” ASME Journal of Engineering for Power, October 1970, p. 359.
28. Kline, S.J., Abbott, D.E., and Fox, R.W., “Optimum Design of Straight-Walled Diffusers, ” Journal of Basic Engineering, Vol. 81, series, D, No. 3, September 1959, pp. 321–331.
29. Koch, C.C., “Stalling Pressure Rise Capability of Axial Flow Compressor Stages, ” Transactions of the ASME, Journal of Engineering for Power, Vol. 103, October 1981, pp. 645–656.
30. Kotidis, P.A. and Epstein, A.H., “Unsteady Radial Transport in a Transonic Compressor Stage, ” Transactions of ASME, Journal of Turbomachinery, Vol. 113, April 1991, pp. 207–218.
31. Lieblein, S., Schwenk, F.D., and Broderick, R.L., “Diffusion Factor for Estimating Losses and Limiting Blade loadings in Axial-Flow Compressor Blade Element, ” NACA RM E53D01, June 1953.
32. Lieblein, S., “Loss and Stall Analysis of Compressor Cascades, ” Transactions of the ASME, Journal of Basic Engineering, September 1959, pp. 387–400.
33. Lieblein, S., “Experimental Flow in Two-Dimensional Cascades, ” in Aerodynamic Design of Axial Flow Compressors, NASA SP-36, 1965.
34. Lieblein, S. and Roudebush, W.H., “Theoretical Loss Correlation for Low-Speed Two-Dimensional Cascade Flow, ” NACA TN 3662, 1956.
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36. Mattingly, J.D., Heiser, W.H., and Pratt, D.T., Aircraft Engine Design, 2nd edition, AIAA Education Series, AIAA, Reston, VA, 2002.
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Problems

• 8.1 The absolute flow at the pitchline to a compressor rotor has a coswirl with C_θ1 = 78 m/s. The exit flow from the rotor has a positive swirl, C_θ2 = 172 m/s. The pitchline radius is at r_m = 0.6 m and the rotor angular speed is ω = 5220 rpm. Calculate the specific work at the pitchline and the rotor torque per unit mass flow rate.
• 8.2 An axial-flow compressor stage has a pitchline radius of r_m = 0.6 m. The rotational speed of the rotor at pitchline is U_m = 256 m/s. The absolute inlet flow to the rotor is described by C_zm = 155 m/s and C_θ1m = 28 m/s. Assuming that the stage degree of reaction at pitchline is °R_m = 0.50, α₃ = α₁, and C_zm remains constant, calculate
  1. rotor angular speed ω in rpm
  2. rotor exit swirl C_θ2m
  3. rotor specific work at pitchline, w_cm
  4. relative velocity vector at the rotor exit
  5. rotor and stator torques per unit mass flow rate
  6. stage loading parameter at pitchline, ψ_m
  7. flow coefficient ϕ_m
• 8.3 A rotor blade row is cut at pitchline, r_m. The velocity vectors at the inlet and exit of the rotor are shown.

  Assuming that U_m1 = U_m2 = 210 m/s and C_z1 = C_z2 = 175 m/s, ρ₁ = 1 g/m³ β₂ = −25°, and ϖ_r = 0.03, calculate

  1. W_θ1 and W_θ2
  2. W_1m and W_2m
  3. D-factor D_rm
  4. Circulation Γ_m
  5. rotor lift at pitchline per unit span
  6. lift coefficient at pitchline
  7. rotor-specific work at r_m
  8. loading coefficient ψ_m
  9. degree of reaction °R_m

  

• 8.4 A rotor blade row at the hub radius is shown. The rotor total pressure loss coefficient at this radius is ϖ_rm = 0.04.
  1. How much deceleration is allowed in the rotor under de Haller criterion? i.e., What is the minimum W₂?
  2. What is the static pressure rise coefficient, assuming incompressible flow and W_2min from de Haller criterion?
  3. Compare the C_p in part (b) to the Arbitrary C_pmax shown in Figure 8.30 at the hub.

  

• 8.5 A compressor stage develops a pressure ratio of π_s = 1.6. Its polytropic efficiency is e_c = 0.90. Calculate the stage total temperature ratio τ_s and compressor stage adiabatic efficiency η_s. Assume γ = 1.4.
• 8.6 An axial-flow compressor stage is shown at the pitch-line. Assuming
  
  1. w_c (in kJ/kg)
  2. T_t3/T_t1
  3. M_2r
  4. p_t2/p_t1
  5. p_t3/p_t2
  6. η_s
  7. °R_m

  

• 8.7 The flow at the entrance to an axial-flow compressor rotor has zero preswirl and an axial velocity of 175 m/s. The shaft angular speed is 5000 rpm. If at a radius of 0.5 m, the rotor exit flow has zero relative swirl, calculate at this radius
  1. rotor specific work w_c in kJ/kg
  2. degree of reaction °R
• 8.8 The absolute flow angle at the inlet of a stator blade in a compressor is α₂ = 45°, as shown. The absolute total pressure and temperature in station 2 are p_t2 = 150 kPa and T_t2 = 300 K, respectively. The total pressure loss coefficient for this section of the the stator blade is ϖ_s = 0.02. Assuming the axial velocity remains constant and gas properties are γ = 1.4 and c_p = 1.004 kJ/kg · K, calculate
  1. entrance Mach number M₂
  2. exit total pressure p_t3
  3. exit Mach number M₃
  4. stator torque for a mass flow rate of = 100 kg/s
  5. static pressure rise, Δp = p₃ − p₂
  6. static temperature rise ΔT = T₃ − T₂
  7. entropy rise (s₃ − s₂)/R

  Note that the radius of this cut (section) is at r ≅ 0.5 m from the axis of rotation, as shown in the diagram.

  

• 8.9 An axial-flow compressor with four stages is shown. Assuming a repeated stage design, with constant throughflow speed, C_z = 150 m/s, and 50% degree of reaction at the pitch-line with zero preswirl, calculate
  1. rotor specific work at the pitchline (kJ/kg)
  2. stage pressure ratio for an η_s = 0.90
  3. compressor pressure ratio π_c
  4. shaft power (in MW) for a mass flow rate of = 100 kg/s
  5. D-factor for the first rotor at the pitchline for σ_r = 2.0

  

• 8.10 An axial-flow compressor stage is designed on the principle of constant through flow speed. The flow at the entrance to the rotor has 100 m/s of positive swirl and 180 m/s of axial velocity. Assuming we are at the pitchline radius r_m = 0.5 m, where the rotor rotational speed is U_m = 230 m/s, the degree of reaction °R_m = 0.5, the radial shift in the streamtube is negligible, i.e., r_1m ≈ r_2m ≈ r_3m, and also assuming a repeated stage design principle is implemented, calculate
  1. α_1m and β_1m
  2. α_2m and β_2m
  3. rotor specific work at the pitchline w_cm in kJ/kg
  4. stator torque at the pitchline per unit mass flow rate,
• 8.11 In an axial-flow compressor test rig with no inlet guide vanes, a 1-m diameter fan rotor blade spins with a sonic tip speed, i.e., U_tip/a₁ = 1.0. If the speed of sound in the laboratory is a₀ = 300 m/s, and the axial velocity to the fan is C_z1 = 150 m/s, calculate the fan rotational speed ω in rpm.
  
• 8.12 An axial-flow compressor rotor has an angular velocity of ω = 5000 rpm. The flow entering the compressor rotor has zero preswirl and an axial velocity of C_z1 = 150 m/s. Assuming the axial velocity is constant throughout the stage, and the rotor specific work at the radius r = 0.5 m is w_c = 62 kJ/kg (γ = 1.4 and R = 287 J/kg · K) calculate
  1. stage degree of reaction, °R, at this radius
  2. total pressure ratio across the rotor, p_t2/p_t1, at this radius, assuming a polytropic efficiency of 90% and T₁ = 20 °C.
• 8.13 An axial-flow compressor rotor at the pitchline has a radius of r_m = 0.35 m. The shaft rotational speed is ω = 5000 rpm. The inlet flow to the rotor has zero preswirl and the axial velocity is C_z1 = C_z2 = C_z3 = 175 m/s. The rotor has a 50% degree of reaction at the pitchline. The stage adiabatic efficiency is nearly equal to the polytropic efficiency η_s ≅ e_c = 0.92. Assuming the inlet total temperature is T_t1 = 288 K and c_p = 1.004 kJ/kg · K, calculate
  1. rotor specific work at r = r_m
  2. stage loading ψ at r = r_m
  3. flow coefficient at r = r_m
  4. rotor relative Mach number at the pitchline, M_{1r, m}
  5. stage total pressure ratio at the pitchline
• 8.14 For the multistage compressor, as shown, calculate
  1. compressor adiabatic efficiency η_c
  2. shaft power
  3. exit total temperature T_t3
  4. average π_s for ten stages, i.e., N = 10 Assume γ = 1.4 and c_p = 1004 J/kg · K.

  

• 8.15 A compressor test rig operates in a laboratory where θ₂ = 0.95 and δ₂ = 0.95. The mass flow rate is measured at the compressor face to be = 100 kg/s and the shaft power is measured to be ℘_c = 50 MW. Assuming compressor poly-tropic efficiency is e_c = 0.90, calculate
  1. compressor (total) pressure ratio π_c
  2. compressor corrected mass flow rate
  3. compressor adiabatic efficiency η_c
• 8.16 A compressor adiabatic efficiency is measured to be η_c = 0.85 for a compressor total pressure ratio of π_c = 20. What is the “small-stage” efficiency for this compressor?
• 8.17 A compressor has a polytropic efficiency of e_c = 0.92 and a pressure ratio, π_c = 25 for an inlet condition of T_t1 = 520 °R and c_p = 0.24 BTU/lbm · °R, calculate
  1. exit total temperature T_t2
  2. compressor adiabatic efficiency η_c
  3. compressor specific work w_c
  4. shaft power ℘_s for a 100 lbm/s flow rate
• 8.18 A multistage compressor develops a total pressure ratio π_c = 25, and is designed with eight identical (i.e., “repeated”) stages. The compressor polytropic efficiency is e_c = 0.92. Calculate
  1. average stage total pressure ratio π_s
  2. stage adiabatic efficiency η_s
  3. compressor total temperature ratio τ_c
• 8.19 Plot compressor adiabatic efficiency η_c versus π_c ranging from 1.0 to 50, for the following polytropic efficiencies e_c = 0.95, 0.90, and 0.85 as the running parameter.
• 8.20 A rotor blade row has a solidity of 1.0 at its pitchline. The absolute flow enters the rotor with no preswirl at C_z1m = 500 fps, and the rotor rotational speed is U_m = 1200 fps (at the pitchline). If the rotor exit flow angle at r_m is β_2m = −30°, calculate
  1. exit swirl velocities W_θ2m and C_θ2m
  2. blade torque (per unit mass flow rate) at the pitchline assuming r_m = 1.0 ft
  3. the nondimensional total temperature rise across the rotor ΔT_t/T_t1
  4. exit speed of sound a₂ assuming inlet speed of sound is a₁ = 1100 fps
  5. exit absolute Mach number M_2m
  6. inlet static pressure p₁ for an inlet total pressure of p_t1 = 14.7 psia
  7. exit total pressure, assuming rotor adiabatic efficiency is 0.9
  8. rotor static pressure rise,
  9. shaft rotational speed ω (rpm)
  10. rotor torque at r_m per unit mass flow rate
  11. the axial force on the blades at r_m, assuming the mean chord c_m = 4 in.
  12. tangential force on the blade at r_m
  13. sectional lift-to-drag ratio, L′/D′
  14. rotor specific work at r_m
• 8.21 Calculate the Reynolds number based on chord at the pitchline for the rotor blade described in Problem 8.20, assuming fluid coefficient of viscosity is μ = 1.8 × 10⁻⁵ kg/m · s. Compare the Reynolds number that you calculate to the upper critical Reynolds number in a compressor.
• 8.22 Calculate the circulation at the pitchline for the rotor blade row described in Problem 8.20. Also calculate the fraction of “ideal” lift that was destroyed by the total pressure losses in the blade row. Is there any indication of shock losses at the pitchline?
• 8.23 A rotor blade row has a hub-to-tip radius ratio of 0.5, solidity at the pitchline of 1.0, the axial velocity is 160 m/s, and zero preswirl. The mean section has a design diffusion factor of D_m = 0.5. Calculate and plot where appropriate
  1. exit swirl at the pitchline assuming the shaft rpm of 6000 and r_m = 1.0 ft (0.3 m)
  2. downstream swirl distribution C_θ2 (r) assuming a freevortex design rotor
  3. the radial distribution of degree of reaction °R along the blade span
  4. radial distribution of diffusion factor D_r(r).
• 8.24 An axial-flow compressor stage is downstream of an IGV that turns the flow 15° in the direction of the rotor rotation, as shown. The axial velocity component remains constant throughout the stage at C_z = 150 m/s. The rotor rotational speed is ω = 3000 rpm and the pitchline radius is r_m = 0.5 m. The rotor relative exit flow angle is β₂ = −15°. The static temperature and pressure of air upstream of the rotor are T₁ = 20°C and p₁ = 10⁵ Pa, respectively. Assuming c_p = 1.004 kJ/kg · K and γ = 1.4, calculate
  1. relative Mach number to the rotor, M_1r
  2. absolute total temperature T_t1
  3. rotor specific work w_c in kJ/kg
  4. total temperature downstream of the rotor, T_t2
  5. relative Mach number downstream of the rotor, M_2r
  6. stage total pressure ratio at the pitchline radius for e_c = 0.92
  7. stage degree of reaction °R_m at the pitchline

  

• 8.25 A compressor stage with an inlet guide vane is shown at its pitchline radius r_m = 0.5 m. The rotor angular speed is ω = 4000 rpm. The axial velocity is constant throughout at C_z = 150 m/s and the IGV imparts a preswirl of 75 m/s in the direction of rotor rotation, as shown. Assuming the inlet flow to IGV has p_t0 = 100 kPa and T_t0 = 25°C, calculate
  1. T₀, M₀, p₀

     Assuming the IGV has a total pressure loss coefficient of ϖ_IGV = 0.02, calculate

  2. p_t1, T_t1, T₁, M₁, p₁, M_1r and p_t1r

     Knowing that the compressor stage has a degree of reaction of °R = 0.5 at the pitchline, calculate

  3. C_θ2, T_t2, T₂, M₂, M_2r

     For a rotor total pressure loss coefficient of _r = 0.03 at the pitchline, calculate

  4. p_t2

     Assume: γ = 1.4 and c_p = 1004 J/kg · K throughout the stage.

  

• 8.26 Apply Euler turbine equation to a streamtube that enters a compressor rotor blade row at r = 2.5 ft with zero preswirl and exits the row at r = 2.7 ft and attains 1000 ft/s of swirl velocity in the absolute frame. Assume that the shaft rotational speed is 5000 rpm.
• 8.27 Assume that we can analyze a 3D compressor rotor by stacking up its 2D flowfield, what is known as the “strip theory.” Now, let us consider one such section. The inlet flow approaches the rotor blade at β₁ = −45°. The relative exit flow angle is β₂ = −30° for a net 15° turning. The solidity of the rotor at this section is 1.5. Identify the most suitable NACA 65-series profile for this section that could produce the largest positive stall tolerance. Estimate the stagger angle that the blades need to be set at this section. To solve this problem, you have to use the cascade data. What is the safe operating range of the incidence angle (or equivalently the inlet flow angles), in degrees, for this section?
• 8.28 A simple method to establish the annulus geometry in a multistage compressor is to assume a constant throughflow (i.e., axial) speed Cz. Calculate
  1. the density ratio ρ₃/ρ₂
  2. the exit-to-inlet area ratio A₃/A₂

  

  for

  

  Also, if the hub radius is constant and for a mass flow rate of 100 kg/s and the inlet total pressure of p_t2 = 100 kPa, calculate the tip-to-tip radius ratio (r₃/r₂)_tip

• 8.29 Consider a compressor stage with no inlet guide vane. The hub-to-tip radius ratio for the rotor is r_h1/r_t1 = 0.5 and the mass flow rate (of air) in the compressor is 100 kg/s, at the standard sea level condition, p_t1 = 100 kPa and T_t1 = 288 K. For an axial Mach number of M_z = 0.5, calculate
  1. the rotor hub and casing radii r_h1 and r_t1 (in meters) To achieve a relative tip Mach number of (M_1r)_tip = 1.4, calculate
  2. the rotor rotational speed ω (in radians per second) For a design pitchline degree of reaction of °R_m = 0.5, and constant axial velocity, calculate
  3. the rotor exit swirl at the pitchline radius C_θ2m (in m/s)
  4. the total temperature T_t2m (in K) using Euler equation
  5. the total pressure ρ_t2m (in kPa), assuming ϖ_rm = 0.005
  6. the fluid density ρ_2m in kg/m³

  Now, assume that we chose a free-vortex design for the rotor.

  

  Calculate

  1. the degree ofreaction forthe rotorhub and tip °R_h and °R_t
  2. the diffusion factor at the pitchline D_m
  3. the hub radius in station 2, r_h2, from continuity equation.
• 8.30 A compressor rotor velocity triangles at the pitchline are shown. The compressor hub-to-tip radius ratio is r_h/r_t = 0.4. The rotor solidity at the pitchline is σ_m = 1.0. Assuming constant axial velocity across the rotor at pitchline, calculate
  1. solidity at the tip for c_t = c_m
  2. solidity at the hub for c_h = c_m
  3. degree of reaction °R_m
  4. diffusion factor D_m
  5. de Haller criterion
  6. rotor loading coefficient ψ_m
  7. flow coefficient ϕ_m

     Assuming that the rotor has a free-vortex design, calculate

  8. diffusion factor at the tip D_t
  9. the degree of reaction at the hub, °R_h
  10. the axial velocity distribution downstream of the rotor (assuming radial equilibrium)

  

• 8.31 The flow coefficient to a rotor at pitchline is ϕ_m = 0.8, its loading coefficient is C_m = 1.0. The inlet flow to the rotor has zero swirl in the absolute frame of reference. Assuming axial velocity C_zm = constant across the rotor, calculate
  1. the relative inlet flow angle β_1m
  2. the relative exit flow angle β_2m
  3. the degree of reaction °R_m
• 8.32 The end wall boundary layers and a rotor blade are shown. The hub radius is r_h = 0.5 m and the shaft speed is ω = 4000 rpm. The rotor inlet flow has zero preswirl, i.e., α₁ = 0. The rotor blade is twisted to meet the flow at zero incidence within the boundary layer. Calculate the blade stagger angle at the hub and its change within the boundary layer if δ_{bl, h} = 10 cm and it has a 1/7th power law profile. C_z(y)/C_zm = (y/δ)^1/7 and C_zm = 150 m/s. [Assume blade stagger angle is ≈ relative flow angle, β₁]

  

• 8.33 A multistage compressor has 12 repeated stages. Each stage produce a constant total temperature rise of ΔT_t = 25°C. The inlet total temperature is T_t1 = 288 K, and stage adiabatic efficiency is η_s = 0.90 and is assumed constant for the 12 stages. Calculate and graph the stage total pressure ratio for all 12 stages. What is the compressor overall total pressure ratio?
• 8.34 A compressor stage has 37 rotor blades and 41 stator blades. The shaft rotational speed is 5000 rpm. Calculate
  1. the rotor blade passing frequency as seen by the stator blades
  2. the stator blade passing frequency as seen by the rotor blades
• 8.35 The absolute flow to a compressor rotor has a coswirl with α₁ = 15°. The exit flow from the rotor has an absolute flow angle α₂= 35°. The pitchline radius is at r_m = 0.6 m and the rotor angular speed is ω = 5220 rpm. Assuming the axial velocity is C_zm = 150 m/s and is constant across the rotor, calculate
  1. the specific work at the pitchline
  2. the rotor torque per unit mass flow rate
  3. the degree of reaction
• 8.36 A rotor blade row has a hub-to-tip radius ratio of 0.4, solidity at the pitchline of 1.2, the axial Mach number of 0.6, and zero preswirl. The mean section has a design diffusion factor of D_m = 0.5. Assuming a₁ = 330 m/s, calculate and plot where appropriate
  1. exit swirl at the pitchline assuming the shaft rpm of 5000 and r_m = 0.4 m
  2. downstream swirl distribution C_θ2(r), assuming a solid-body rotation vortex design rotor
  3. the radial distribution of degree of reaction °R(r), along the blade span
  4. radial distribution of diffusion factor D_r(r).
• 8.37 Bending stresses on a cantilevered blade, under aerodynamic loading, is estimated (by Kerrebrock, 1992) to be
  

  Estimate the ratio of bending stress to inlet pressure (σ_bending/p) in a rotor with axial velocity 165 m/s, the rotor tip radius is r_t = 0.75 m, the mean solidity of 1.5, angular speed is ω = 4000 rpm. The maximum blade thickness is t_max = 1 cm. The stage total pressure ratio is 1.6 and the stage adiabatic efficiency is 0.88.

• 8.38 Centrifugal stress is proportional to AN² as we discussed in the compressor design section. It follows
  

  for a linear taper ratio. For a titanium rotor blade of r_h = 0.4 m, r_t = 0.8 m, and taper ratio A_t/A_h = 0.8, calculate the acceptable shaft speed ω if the allowable strength-to-weight ratio is ∼9 ksi/slug/ft³, which is equivalent to

  allowable creep rupture     strength required strength

  120, 560 m2/s2 (or ∼120 kPa/kg/m3) 68, 326 m2/s2 (or ∼68 kPa/kg/m3)

• 8.39 A rotor section has de Haller criterion W₂/W₁ = 0.75. Assuming the axial velocity remains constant across the rotor and the velocity triangles are as shown, calculate the corresponding D-factor and degree of reaction for the rotor section.

  

• 8.40 A portion of a compressor map surrounding the design point is shown.

  Calculate this compressor’s stall margin.

  

• 8.41 A compressor stage at the pitchline, r_m = 0.5 m, is shown. The inlet flow to the rotor has a preswirl with α₁ = 22°. The axial velocity is C_z1 = C_z2 = C_z3 = 170 m/s, i.e. constant throughout the stage. The stage is of repeated design, with α₃ = α₁. Rotor and stator solidities at pitchline are 1.2 and 1.0 respectively. The rotor inlet relative velocity at pitchline is sonic, i.e., W₁ = a₁ and the rotor relative exit velocity, following de Haller criterion, is W₂ = 0.75 W₁.

  

  Calculate:

  1. shaft rotational speed, ω, in rpm
  2. rotor exit (absolute) swirl, C_θ2 in m/s
  3. stage degree of reaction, °R_m
  4. rotor D-factor, D_r
  5. stage total temperature ratio
  6. stage total pressure ratio for e_c = 0.9
• 8.42 An axial-flow compressor stage at its pitchline radius (r_m = 0.50 m) has zero preswirl, an axial velocity of 150 m/s. Its degree of reaction is °R_m = 0.50, and the shaft angular speed is 5, 200 rpm. Assuming constant throughflow speed, i.e., C_z = const., and repeated stage design at the pitchline, calculate:

  

  1. relative Mach number at rotor inlet, M_1r
  2. rotor exit relative velocity, W₂ (m/s)
  3. static temperature downstream of the rotor, T₂ (K)
  4. absolute Mach number downstream of the rotor, M₂
  5. absolute Mach number downstream of the stator, M₃
  6. stage total temperature ratio, τ_s
  7. stage total pressure ratio if η_s = 0.90
• 8.43 The relative flow across a compressor rotor is shown. The rotor rotational speed is U = 300 m/s. The axial velocity is C_z = 165 m/s and it remains constant across the rotor. Assuming the rotor solidity is σ_r = 0.8, calculate:
  1. de Haller criterion
  2. rotor degree of reaction (for a repeated stage)
  3. rotor specific work (in kJ/kg)
  4. rotor D-factor

  

• 8.44 A multistage compressor develops a total pressure ratio of π_c = 35 with a polytropic efficiency of e_c = 0.90. The air mass flow rate through the compressor is = 200 kg/s. Assuming γ and c_p remain constant, calculate:
  1. compressor shaft power, ℘_c, in MW
  2. flow area in 2, i.e., A₂ in m²
  3. density of air in station 3, ρ₃, in kg/m³

     [note: the axial velocity at the compressor exit, V₃ = V₂]

  4. The nondimensional entropy rise across the compressor, Δs/R

  

• 8.45 A cylindrical cut of the rotor in an axial flow compressor is shown. The rotor uses constant-axial velocity design, C_z1 = C_z2. Assuming diffusion in the rotor, based on de Haller criterion is described by W₂/W₁ = 0.75, calculate:
  1. relative flow angle at the rotor inlet, β₁ in degrees
  2. relative flow angle at the rotor exit, β₂ in degrees
  3. rotor exit swirl, C_θ2, in m/s
  4. stage loading parameter, ψ

  

• 8.46 An axial-flow compressor stage at its picthline radius (r_m = 0.40 m) has zero preswirl, an axial velocity of 175 m/s, a degree of reaction of °R_m = 0.65, with the shaft angular speed of 6450 rpm. Assuming constant throughflow speed, i.e., C_z = const., and a repeated-stage design at the pitchline, calculate:
  1. relative Mach number at rotor inlet, M_1r
  2. rotor exit relative velocity, W₂ (m/s)
  3. static temperature downstream of the rotor, T₂ inK
  4. absolute Mach number downstream of the rotor, M₂
  5. absolute Mach number downstream of the stator, M₃
  6. stage total temperature ratio, τ_s
  7. stage total pressure ratio if η_s = 0.88

  

• 8.47 A multistage compressor has a total pressure ratio of π_c = 15, a polytropic efficiency of e_c = 0.90 and the inlet condition: C₂ = C_z2 = 160 m/s, T_t2 = 288 K, p_t2 = 100 kPa. Assuming constant gas properties γ = 1.4 and c_p = 1004 J/kgK and C_z remains constant with compressor exit purely in the axial direction (i.e., no exit swirl), calculate:
  1. compressor exit total temperature, T_t3 in K
  2. compressor inlet Mach number, M₂
  3. compressor exit Mach number, M₃ = M_z3
  4. compressor Density ratio, ρ₃/ρ₂
  5. compressor area ratio, A₃/A₂

  

• 8.48 An axial-flow compressor stage is shown at its pitchline radius (r_m = 0.30 m). The axial velocity is 175 m/s and is constant. The rotor exit relative flow angle is β₂ = −15°. Assuming the inlet preswirl exists with α₁ = +10° and shaft angular speed is 7600 rpm, calculate:
  1. absolute Mach number at the inlet, M₁
  2. relative Mach number at the inlet, M_1r
  3. static pressure at the inlet, p₁ (kPa)
  4. total temperature at the inlet, T_t1 (K)
  5. rotor exit absolute swirl, C_θ2 (m/s)
  6. stage degree of reaction, °R_m (assume repeated stage)
  7. rotor specific work at pitchline, w_m in kJ/kg
  8. static temperature downstream of the rotor, T₂ (K)
  9. absolute Mach number downstream of the rotor, M₂
  10. speed of sound downstream of the stator, a₃ (m/s)
  11. absolute Mach number downstream of the stator, M₃

  

• 8.49 An axial-flow compressor rotor has a tip relative Mach number (at the inlet) of 1.24, as shown. The tip radius is r_1t = 0.73 m and the inlet speed of sound is a₁ = 337 m/s. The rotor inlet flow has zero preswirl and has an axial Mach number of M_z1 0.5. The rotor exit has a relative velocity, W_2tip = 320 m/s. Assuming C_z is constant, calculate:
  1. the shaft angular speed, ω in rpm
  2. the rotor specific work, w_c, at the tip in kJ/kg
  3. rotor exit absolute Mach number, M₂

  

• 8.50 An axial flow compressor stage at r = 0.4 m is shown. The inlet flow is purely axial with C_z1 = 150 m/s, which remains constant across the rotor and stator. The rotor solidity at this radius is σ_r = 1.5. The rotor angular speed is ω = 7400 rpm. The stage degree of reaction at this radius is °R = 0.75. The rotor total pressure loss parameter (in relative frame) is ϖ_r = 0.05.

  The gas properties are: γ = 1.4 and R = 287 J/kgK.

  Calculate:

  1. absolute swirl at rotor exit, C_θ2 in m/s
  2. rotor exit (absolute) total temperature, T_t2 in K
  3. static temperature at the rotor exit, T₂ in K
  4. rotor exit absolute Mach number, M₂
  5. rotor exit relative Mach number, M_2r
  6. relative dynamic pressure at rotor inlet, q_1r, in kPa
  7. inlet relative total pressure, p_t1r, in kPa
  8. static pressure at rotor exit, p₂, in kPa
  9. rotor exit (absolute) total pressure, p_t2, in kPa
  10. rotor D-factor, D_r, at this radius

  

• 8.51 The pitchline radius of a compressor rotor is at r_m = 0.3 m. The degree of reaction is °R_m = 0.75. The axial velocity to the rotor is C_z1 = 175 m/s = constant across the blade row. The flow to the rotor has zero preswirl and the rotor angular speed is ω = 6500 rpm. The solidity of the rotor at the pitchline is σ_m = 1.0. Calculate:
  1. absolute swirl velocity downstream of the rotor, i.e., C_θ2m, in m/s
  2. relative swirl upstream and downstream of the rotor, i.e., W_θ1 and W_θ2 in m/s
  3. de Haller parameter, W₂/W₁
  4. diffusion factor at the pitchline radius, D_rm
• 8.52 An axial-flow compressor rotor at the pitchline has a radius of r_m = 0.5 m. The shaft rotational speed is ω = 6000 rpm. The inlet flow to the rotor has zero preswirl and the axial velocity is constant, with C_z = 150 m/s. The stage has a 80% degree-of-reaction at the pitchline where the solidity is σ_m = 1.2. The stage adiabatic efficiency is equal to the polytropic efficiency, e_c = 0.92.

  Assuming that the inlet total temperature is T_t1 = 288 K, γ = 1.4 and c_p = 1.004 kJ/kg.K, calculate:

  1. rotor specific work at r = r_m in kJ/kg
  2. stage loading, ψ, at r = r_m
  3. flow coefficient, , at r = r_m
  4. rotor relative Mach number at the pitchline, M_{1r, m}
  5. stage total pressure ratio at the pitchline
  6. rotor diffusion factor at the pitchline
  7. is the de Haller criterion satisfied?
• 8.53 A compressor stage at its pitchline radius (r_m = 0.25) has °R_m = 0.73. The axial velocity is constant at C_zm = 150 m/s. The flow entering the compressor is swirl free, i.e., α₁ = 0. The flow exiting the stage at pitchline is also swirl free, i.e., α₃ = 0. The compressor angular speed is ω = 7500 rpm. The speed of sound at the entrance to the stage is a₁ = 330 m/s and rotor solidity at the pitchline radius is σ_rm = 0.80. Assuming the gas constant is R = 287 J/kg·K and γ = 1.4, calculate:
  1. absolute swirl downstream of the rotor, C_θ2m, in m/s
  2. rotor specific work at pitchline radius, w_c, in kJ/kg
  3. total temperature downstream of the rotor, T_t2m, in K
  4. de Haller criterion for the rotor at pitchline
  5. the rotor diffusion factor at the pitchline radius, D_rm

  

• 8.54 An axial-flow compressor has no IGV. Its inlet conditions are: p_t1 = 100 kPa, T_t1 = 288 K and the mass flow rate . The compressor pressure ratio is π_c = 15 and the polytropic efficiency is e_c = 0.90. The axial flow in the compressor is designed to be constant at C_z = 166 m/s. Assuming the gas constant is R = 287 J/kg · K and γ = 1.4, calculate:
  1. absolute Mach number at the inlet to the compressor, M₁
  2. compressor shaft power, ℘_c, in MW
  3. compressor inlet flow area, A₁ in m²
  4. exit Mach number, M₂ (assume the exit flow is swirl free)
  5. compressor exit area, A₂ in m²