Unit 3: Axial Flow Compressors

ASE202 — Propulsion-I 8 min read

I. Orientation — Energy Transfer in Compressors

An axial-flow compressor raises the stagnation pressure of a continuously flowing gas through alternating rows of rotating blades and stationary vanes. The rotor transfers shaft work to the gas according to Euler’s turbomachinery equation; the rotor and stator then diffuse the flow to convert kinetic energy into static pressure.

A. Governing Framework

The analysis rests on conservation laws applied to a steady-flow compressor stage.

Stage arrangement: One stage normally consists of a rotor followed by a stator.
- The rotor adds stagnation enthalpy and angular momentum.
- The stator removes swirl and converts velocity head into static pressure.
Velocity convention: Absolute velocity is (\mathbf C), blade velocity is (\mathbf U), and rotor-relative velocity is (\mathbf W).
TEXT
```
  W = C - U
```
Station numbering: Subscript (1) denotes rotor inlet, (2) rotor exit, and (3) stator exit.
Ideal assumptions: Elementary analysis commonly assumes steady, adiabatic flow; uniform axial velocity; constant mean radius; negligible radial velocity; and perfect-gas behavior.
Performance objective: High pressure ratio and efficiency must be obtained without excessive diffusion, blade loading, choking, rotating stall, or surge.

II. Elementary Compressor Theory — Stage Work and Pressure Rise

A. Elementary theory of axial flow compressor

Elementary theory treats a blade row through its mean-radius velocity triangles and applies angular-momentum conservation to calculate stage work.

Euler work equation: For a rotor operating at constant blade speed,
TEXT
```
  Δh₀ = U(Cθ₂ − Cθ₁)
```
Here, (\Delta h0) is rotor stagnation-enthalpy rise in J/kg, (U) is blade speed in m/s, and (C{\theta1},C_{\theta2}) are inlet and exit tangential absolute-velocity components.
Power requirement:
TEXT
```
  P = ṁΔh₀
```
(P) is ideal power in W and (\dot m) is mass flow rate in kg/s.
Temperature rise: For constant specific heat,
TEXT
```
  ΔT₀ = Δh₀/cp
```
(\Delta T_0) is stagnation-temperature rise and (c_p) is specific heat at constant pressure.
Ideal pressure ratio: For an isentropic compression,
TEXT
```
  p₀₂/p₀₁ = (T₀₂/T₀₁)^[γ/(γ − 1)]
```
(p_0) and (T_0) are stagnation pressure and temperature, while (\gamma) is the specific-heat ratio.
Actual compression: Compressor isentropic efficiency is
TEXT
```
  ηc = (T₀₂s − T₀₁)/(T₀₂ − T₀₁)
```
(\etac) is efficiency, (T{02s}) is the isentropic exit temperature, and (T_{02}) is the actual exit temperature.
Multistaging: Since one axial stage develops a modest pressure ratio, aircraft compressors use successive stages; annulus area usually decreases as density rises.

III. Flow Kinematics — Blade-Row Geometry

A. Velocity triangles

Velocity triangles relate absolute, relative, and blade velocities and therefore determine work input, incidence, turning, and diffusion.

Components: Absolute velocity is resolved into axial velocity (Ca) and whirl velocity (C\theta); blade speed is
TEXT
```
  U = ωr
```
where (\omega) is angular velocity in rad/s and (r) is radius.
Air angles: If (\alpha) is the absolute-flow angle measured from the axial direction,
TEXT
```
  Cθ = Ca tan α
```
Relative angles: Defining (\beta) from the axial direction toward the direction opposite blade motion,
TEXT
```
  tan β = (U − Cθ)/Ca
```
Rotor work in angular form: With constant (C_a),
TEXT
```
  Δh₀ = UCa(tan α₂ − tan α₁)
```
Thus greater change in absolute-flow angle produces greater specific work.
Rotor triangle: The rotor turns the relative flow from (\beta_1) to (\beta2), increasing (C\theta) in the direction of rotation.
Stator triangle: Since (U=0) for the stator, no shaft work occurs; the stator reduces whirl and prepares the inlet angle for the next rotor.
Design significance: Blade metal angles differ from fluid angles because inlet incidence and exit deviation arise from viscosity, boundary layers, and finite blade spacing.

IV. Stage Loading — Distribution of Static Enthalpy Rise

A. Degree of reaction

Degree of reaction measures the fraction of a stage’s static-enthalpy rise occurring in the rotor.

Definition:
TEXT
```
  R = (static-enthalpy rise in rotor)/(static-enthalpy rise in stage)
```
(R=0) represents an impulse-type rotor, while (R=1) places all ideal static-pressure rise in the rotor.
Constant-radius result: For constant (U), constant (C_a), and equal stage-inlet and stage-exit absolute speeds,
TEXT
```
  R = 1 − (Cθ₁ + Cθ₂)/(2U)
    = 1 − [Ca(tan α₁ + tan α₂)]/(2U)
```
Fifty-percent reaction: At (R=0.5), rotor and stator ideally share the static-enthalpy rise equally. The velocity triangles become symmetric under the appropriate repeating-stage angle convention.
Influence on diffusion:
1. Low reaction: The rotor performs work mainly by changing whirl, while the stator carries more diffusion.
2. High reaction: More rotor diffusion occurs, increasing sensitivity to rotor boundary-layer separation.
Practical selection: Values near (0.5) provide balanced rotor–stator loading, although front and rear stages may use different reactions to satisfy annulus and matching requirements.

V. Radial Flow Structure — Departure from Mean-Line Theory

A. Three dimensional flow

Real compressor flow varies in axial, circumferential, and radial directions, so mean-radius triangles cannot represent the entire blade span.

Radial equilibrium: Swirling flow requires a radial pressure gradient:
TEXT
```
  ∂p/∂r = ρCθ²/r
```
Here, (p) is static pressure and (\rho) is density; higher swirl therefore raises pressure toward the casing.
Spanwise blade speed: Because (U=\omega r), tip sections travel faster and experience different relative velocities and Mach numbers from hub sections.
Secondary flow: End-wall boundary layers encounter cross-passage pressure gradients, producing passage vortices and additional total-pressure loss.
Tip leakage: Pressure difference across a rotor drives flow through the blade-tip clearance, forming a leakage vortex that reduces work and stall margin.
Annulus effects: Hub and casing curvature, streamline contraction, and density increase create radial and axial velocity variations.
Design treatment: Through-flow methods establish radial distributions, blade-to-blade methods resolve cascade flow, and three-dimensional computational fluid dynamics captures shocks, leakage, separation, and secondary vortices.
Limitations: Radial-equilibrium models are efficient preliminary tools but cannot independently predict viscous loss or strongly separated flow.

VI. Spanwise Design — Vortex Laws and Flow Angles

A. Air angle distributions for free vortex and constant reaction designs

Air-angle distributions specify how rotor and stator blade angles must twist from hub to tip to satisfy selected radial work and reaction laws.

Free-vortex design:
- Vortex law:
  TEXT
```
     rCθ = K
```
  (K) is a constant for a given station, so (C_\theta=K/r).
- Constant-work property: If (C_{\theta1}=K1/r) and (C{\theta2}=K_2/r),
  TEXT
```
     Δh₀ = ω(K₂ − K₁)
```
  The ideal rotor work is therefore uniform along the span.
- Angle distributions:
  TEXT
```
     tan α = K/(rCa)
     tan β = [ωr − K/r]/Ca
```
  Absolute angle generally decreases toward the tip, while relative angle changes strongly because (U) increases with radius.
Constant-reaction design:
- Reaction condition:
  TEXT
```
     Cθ₁ + Cθ₂ = 2ωr(1 − R)
```
  (R) is prescribed uniformly over the span.
- With constant spanwise work (H):
  TEXT
```
     Cθ₂ − Cθ₁ = H/(ωr)
     Cθ₁ = ωr(1 − R) − H/(2ωr)
     Cθ₂ = ωr(1 − R) + H/(2ωr)
```
  (H) denotes stagnation-enthalpy rise per unit mass.
- Air angles: Each absolute angle follows (\tan\alphai=C{\theta i}/C_a), and each relative angle follows (\tan\betai=(\omega r-C{\theta i})/C_a), where (i=1,2).
- Comparison: Free-vortex design naturally gives constant work but varying reaction; constant-reaction design needs combined forced-vortex and free-vortex terms and usually produces different blade twist.

VII. Aerodynamic Geometry — From Triangles to Blades

A. Compressor blade design

Compressor blade design converts required velocity turning and diffusion into an efficient, mechanically viable cascade geometry.

Cascade parameters: Chord (c), pitch (s), and solidity
TEXT
```
  σ = c/s
```
control blade loading; increased solidity generally permits greater turning but adds wetted-area friction.
Blade angles: The leading-edge metal angle is selected for design incidence, while trailing-edge geometry accounts for flow deviation.
Airfoil choice: Camber establishes turning, thickness supplies structural strength, and leading-edge radius affects off-design incidence tolerance.
Diffusion control: Excessive deceleration causes boundary-layer separation. A common rotor diffusion measure is
TEXT
```
  D = 1 − W₂/W₁ + ΔWθ/(2σW₁)
```
(D) is diffusion factor, (W_1,W2) are relative-speed magnitudes, and (\Delta W\theta) is tangential relative-velocity change.
Spanwise stacking: Sections are twisted to match radial air angles and stacked about a chosen axis; sweep, lean, and end-wall contouring can reduce shock and secondary-flow losses.
Mechanical constraints: Centrifugal stress, vibration, flutter, foreign-object tolerance, tip clearance, and manufacturing limits must accompany aerodynamic optimization.

VIII. Operating Maps — Stable Range and Efficiency

A. Centrifugal and axial compressor performance characteristics

Compressor characteristics relate pressure rise, mass flow, speed, efficiency, and stability over the operating envelope.

Map coordinates: Pressure ratio is plotted against corrected mass flow at lines of corrected rotational speed:
TEXT
```
  ṁcorr = ṁ√(T₀₁/Tref)/(p₀₁/pref)
  Ncorr = N/√(T₀₁/Tref)
```
(T{ref}) and (p{ref}) are reference conditions, and (N) is rotational speed.
Boundaries: Choke limits high flow when a passage reaches sonic conditions; rotating stall and surge limit low flow. Surge is a system-wide oscillation, whereas rotating stall consists of circumferentially propagating stalled cells.
Axial compressors: They provide high mass flow and small frontal area but have a relatively narrow stable range and strong stage-matching sensitivity. Variable inlet guide vanes, variable stators, and bleed valves improve starting and part-speed operation.
Centrifugal compressors: The impeller produces large work per stage,
TEXT
```
  Δh₀ = U₂Cθ₂ − U₁Cθ₁
```
but slip, impeller diffusion, and diffuser losses reduce ideal pressure rise. They usually offer wider stability and higher single-stage pressure ratio at greater frontal area.
Efficiency islands: Highest efficiency occurs near the design flow and speed; incidence, leakage, shock, friction, and recirculation lower efficiency away from this region.
Nondimensional comparison: Flow coefficient (\phi=C_m/U) and loading coefficient (\psi=\Delta h_0/U^2) compare geometrically different machines, where (C_m) is meridional velocity.