| name | axial-compressor |
| description | Axial compressor design — Euler work equation, degree of reaction, velocity triangles, stage loading/flow coefficient, stall/surge, polytropic efficiency, annulus sizing, blade profiles, NASA SP-36. |
| metadata | {"priority":7,"promptSignals":{"phrases":["axial compressor","compressor stage design","velocity triangle compressor","stage loading","compressor stall","compressor blade design"],"minScore":3}} |
Axial Compressor Design — Complete Skill
Euler Turbomachinery Equation
Stage work:
W_stage = U₂ C_θ2 - U₁ C_θ1 [J/kg; U = blade speed; C_θ = absolute tangential velocity; subscripts 1=inlet, 2=exit of rotor]
Stagnation enthalpy rise:
Δh₀ = W_stage = U(C_θ2 - C_θ1) = U ΔC_θ [J/kg; for constant radius U₁ = U₂ = U]
Stage pressure ratio:
π_stage = (1 + Δh₀ × η_p / (c_p T₀₁))^(γ/(γ-1)) [η_p = polytropic efficiency; T₀₁ = inlet stagnation T]
Velocity Triangles
Convention:
C = absolute velocity; W = relative velocity; U = blade speed
C_θ, W_θ = tangential components; C_a = W_a = axial velocity (radially constant in simplified 2D)
Velocity triangle (inlet to rotor, blade station 1):
W_θ1 = C_θ1 - U [relative swirl at rotor inlet; for zero inlet swirl C_θ1 = 0]
W₁ = √(C_a² + W_θ1²); β₁ = atan(W_θ1 / C_a) [relative flow angle]
Velocity triangle (exit from rotor, blade station 2):
C_θ2 = W_θ2 + U
W₂ = √(C_a² + W_θ2²); β₂ = atan(W_θ2 / C_a)
Stage work in terms of angles:
ΔC_θ = C_a (tan α₂ - tan α₁) = C_a (tan β₁ - tan β₂) [α = absolute angle; β = relative angle]
Non-Dimensional Parameters
Stage loading coefficient ψ:
ψ = Δh₀ / U² = ΔC_θ / U = C_a/U × (tan β₁ - tan β₂)
Flow coefficient φ:
φ = C_a / U [typical 0.4–0.8 for subsonic stages]
Degree of reaction R:
R = static enthalpy rise in rotor / total stage enthalpy rise
R = 1 - (C_θ1 + C_θ2) / (2U) = 1 - ψ/2 - φ(tan α₁ + tan α₂)/2
Practical designs: R = 0.5 (symmetric; equal loading on rotor and stator)
For R = 0.5: α₁ = β₂; α₂ = β₁; blade shapes are mirror images
Design point target (subsonic single stage):
ψ = 0.3–0.5; φ = 0.5–0.7; R = 0.5; efficiency η_s = 0.88–0.92
Blade Loading and Diffusion Factor
Diffusion factor D (Lieblein):
D = 1 - W₂/W₁ + |ΔW_θ| / (2σ W₁)
σ = c/s = chord-to-spacing (solidity); typically σ = 1.0–1.5
Design limit:
D ≤ 0.45–0.50 (rotor); D ≤ 0.40–0.45 (stator) — above this, blade boundary layer separates → stall
Equivalent diffusion factor (de Haller number):
de Haller = W₂/W₁ ≥ 0.72 (for attached flow; = 1 - 0.28 ≈ 0.72)
Annulus Design
Continuity equation:
ṁ = ρ₁ C_a1 A₁ = ρ₂ C_a2 A₂ [mass flow; A = annular area]
A = π (r_tip² - r_hub²) = π(1 - (r_hub/r_tip)²) r_tip²
Hub-to-tip ratio (htr):
At inlet: htr = r_hub/r_tip; typical 0.4–0.7
Low htr: high mass flow but high 3D effects; high htr: more uniform flow
Mean radius and blade speed:
r_mean = (r_hub + r_tip)/2; U_mean = ω × r_mean [m/s]
Tip speed: U_tip = ω × r_tip ≤ 400 m/s (subsonic); 350–500 m/s (transonic front stage)
Aspect ratio (span/chord):
AR = h / c; typical 1.0–3.0 (modern low AR preferred for better stall margin)
Mach Number Effects
Relative inlet Mach number:
M_rel,1 = W₁ / a₁ [a₁ = sound speed at rotor inlet]
Subsonic stage: M_rel,1 < 0.8; transonic: M_rel,1 = 0.9–1.3
Tip Mach number:
M_tip = U_tip / a₀ [reference to inlet sound speed]
M_tip < 1.0 → subsonic; M_tip 1.2–1.5 → transonic (most modern first stages)
Shock losses (transonic):
η_s decreases ~1.5–3% per 0.1 increase in M_tip above 1.0
Stall and Surge
Rotating stall: local separation of flow → blade cells rotate at 50–70% of rotor speed
Surge: full annulus flow breakdown → periodic flow reversal; damaging to compressor
Stall margin (SM):
SM = [(π_stall - π_design) / π_design] × 100% [%]
Acceptable SM ≥ 15% (aircraft engines); 10% (industrial compressors)
Stall inception:
Modal stall: long wavelength disturbance; builds slowly; good warning
Spike stall: short wavelength spike; fast; difficult to detect
Active surge control:
Bleed valves; inlet guide vane modulation; active magnetic bearing damping
Polytropic and Isentropic Efficiency
Polytropic efficiency η_p:
η_p = [(γ-1)/γ] × (ln π_stage) / (ln T₀₂/T₀₁) [for gas stages]
Isentropic efficiency η_is:
η_is = (T₀₂_is - T₀₁) / (T₀₂_actual - T₀₁) = (π^((γ-1)/γ) - 1) / (T₀₂/T₀₁ - 1)
Typical multistage compressor:
η_p = 0.88–0.92; η_is = 0.85–0.90 (n-stage builds up difference)
Compressor Map
Parameters:
- x-axis: corrected mass flow ṁ_c = ṁ√(T₀₁/T_ref) / (P₀₁/P_ref)
- y-axis: pressure ratio π
- Lines of constant corrected speed N_c = N/√(T₀₁/T_ref)
- Surge line: left limit; choke line: right limit (vertical)
Operating line (engine + nozzle match):
Turbine/nozzle area dictates back pressure → operating line slope ≈ constant β line
Standards and References
| Reference | Scope |
|---|
| NASA SP-36 | Aerodynamic design of axial-flow compressors |
| AGARD-AG-10 | Axial flow turbines and compressors |
| Dixon & Hall | "Fluid Mechanics and Thermodynamics of Turbomachinery" |
| ASME PTC-10 | Performance test code for compressors |
Output
Provide: stage loading ψ, flow coefficient φ, degree of reaction R, blade speed U [m/s], axial velocity C_a [m/s], velocity triangle angles α₁, α₂, β₁, β₂ [°], diffusion factor D [rotor and stator], tip and hub-to-tip ratio, stage pressure ratio π, stage efficiency η_s [%], number of stages N for total pressure ratio, stall margin SM [%], and reference (NASA SP-36).
