turbomachinery

name	turbomachinery
description	Turbomachinery — Euler turbine equation, velocity triangles, centrifugal compressors, axial turbines/compressors, specific speed, cavitation, pump selection, turbine efficiency.
metadata	{"priority":7,"promptSignals":{"phrases":["turbomachinery","turbine","compressor","centrifugal pump","axial flow","impeller","specific speed","cavitation","NPSH","velocity triangle"],"minScore":3}}

Turbomachinery — Complete Skill

Euler Turbomachinery Equation

Fundamental Energy Transfer

W_s = U₂V_u2 - U₁V_u1 [J/kg] W_s = shaft work per unit mass (+ for pump/compressor, - for turbine) U = blade tip speed = r×ω [m/s] V_u = absolute tangential (whirl) velocity component [m/s]

Power: P = ṁ × (U₂V_u2 - U₁V_u1)

Head (pump): H = (U₂V_u2 - U₁V_u1) / g [m]

Velocity Triangles

Triangle Notation

At any radius: V = absolute velocity, U = blade speed, W = relative velocity Vector: V = U + W

Components: V_a = axial velocity (flow velocity) V_u = tangential (whirl) component V_r = radial component

Inlet (1) and Exit (2): α = angle between V and axial direction β = angle between W and axial direction (blade angle)

tan(α) = V_u / V_a tan(β) = (V_u - U) / V_a = W_u / V_a

Centrifugal Pump/Compressor Velocity Triangles

Radial machine: V_a1 = V_a2 (approximately) Zero inlet whirl (V_u1 = 0): W_s = U₂V_u2 V_u2 = U₂ - V_a2/tan(β₂)

Theoretical head (zero inlet whirl): H_th = U₂²/g - U₂V_a2/(g×tan(β₂))

Slip factor σ = V_u2,actual / V_u2,ideal σ ≈ 1 - π/z (Stanitz, z = number of blades, backward-swept) Typical: σ = 0.85-0.95

Centrifugal Pump/Compressor

Performance

Flow rate: Q = V_a × (2πr₂b₂) [m³/s] b₂ = impeller width at exit [m] Tip speed: U₂ = π D₂ N/60

Pressure rise: ΔP = ρ × η_h × U₂ × V_u2 [Pa]

Efficiency: η_hydraulic = H_actual / H_theoretical (losses: friction, recirculation) η_mechanical = shaft power in - bearing losses η_volumetric = useful Q / total Q through impeller η_total = η_h × η_m × η_v

Centrifugal Compressor (Compressible Flow)

Total pressure ratio: π_c = P₀₂/P₀₁ = [1 + η_c × U₂²/(c_p×T₀₁)]^(γ/(γ-1)) c_p = 1005 J/kgK (air), γ = 1.4 (air)

Pressure coefficient: ψ = ΔP_total / (½ρU₂²) = 2η_h σ (typical 0.4-0.7)

Flow coefficient: φ = V_a2 / U₂ (typical 0.2-0.5)

Axial Flow Machines

Stage Work (Axial Compressor/Turbine)

W_s = U(V_u2 - V_u1) = U × ΔV_u

Axial compressor stage: Degree of reaction: R = 1 - (V_u1 + V_u2)/(2U) = (β₂ - β₁)/(α₂ - α₁ + β₂ - β₁) 50% reaction: symmetric velocity triangles, β₁ = α₂, β₂ = α₁

Axial turbine: Stage loading: Ψ = W_s/U² = ΔV_u/U Flow coefficient: Φ = V_a/U R = 1 - Ψ/2 (50% reaction nozzle and rotor symmetric)

De Haller Criterion (Compressor)

W₂/W₁ ≥ 0.72 (relative velocity deceleration limit, otherwise stall) Maximum pressure rise per stage limited by diffusion

Dimensionless Parameters & Specific Speed

Specific Speed (Pump)

N_s = N × Q^(1/2) / H^(3/4) [rpm, m³/s, m — SI form] N_s,US = N × GPM^(1/2) / H_ft^(3/4) [US customary]

N_s (SI) ranges: N_s < 20: radial/centrifugal (high head, low flow) 20 < N_s < 80: mixed flow N_s > 80: axial flow (low head, high flow)

Specific Speed (Turbine)

N_sp = N × P^(1/2) / H^(5/4) [rpm, kW, m]

Turbine selection by N_sp: Pelton wheel: N_sp < 30 (very high head, low flow) Francis turbine: 30-300 (medium head) Kaplan turbine: 300-900 (low head, high flow)

Similarity Laws (Affinity Laws)

At constant speed (varying diameter D): Q ∝ D³, H ∝ D², P ∝ D⁵

At constant diameter (varying speed N): Q ∝ N, H ∝ N², P ∝ N³

Pump curves: H-Q, η-Q, P-Q — use to find operating point with system curve

Cavitation

NPSH (Net Positive Suction Head)

NPSH_A = (P_s - P_v) / (ρg) + V_s²/(2g) [m] P_s = absolute suction pressure [Pa] P_v = vapor pressure at fluid temperature [Pa]

System NPSH_A: NPSH_A = P_atm/(ρg) ± z_s - h_L - P_v/(ρg) z_s = suction head (+ if pump above liquid, - if below) h_L = suction line friction losses [m]

Rule: NPSH_A > NPSH_R + safety margin (1-3 m) NPSH_R = required by pump (from pump curve)

Thoma cavitation number: σ = NPSH / H ≥ σ_critical (from pump specific speed) σ_critical ≈ N_s^(4/3) / 6.33 × 10⁻⁵ (US specific speed)

Cavitation Prevention

• Lower pump elevation (increase z_s negative value)
• Increase suction pipe diameter (reduce h_L)
• Cool fluid (reduce P_v)
• Select pump with lower NPSH_R
• Use inducer (pre-swirl to lower NPSH_R)

Gas Turbine Cycle

Brayton Cycle (Ideal)

η_Brayton = 1 - 1/r_p^((γ-1)/γ) r_p = P₂/P₁ = pressure ratio

Specific work: w_net = c_p(T₃ - T₄) - c_p(T₂ - T₁) T₂/T₁ = r_p^((γ-1)/γ) (ideal compression) T₄/T₃ = (1/r_p)^((γ-1)/γ) (ideal expansion)

With real efficiencies: w_compressor = c_p(T₂ - T₁) / η_c w_turbine = c_p(T₃ - T₄) × η_t

Optimum pressure ratio for max specific work: r_p,opt = (T₃/T₁)^(γ/(2(γ-1)))

Steam Turbine

Impulse vs. Reaction

Impulse (0% reaction): all pressure drop in nozzle; blade speed ratio u/C₁ ≈ 0.5 Reaction (50%): pressure drops in both nozzle and blade

Stage efficiency (impulse): η_blade = 2φ²(cos α₁ - φ)(1 + ψcos(β₁-β₂)/cos α₁) φ = blade speed ratio = U/C₁

Curtis stage (velocity compounded): 2 rows of moving blades with guide vanes; u/C₁ ≈ 0.25; handles high ΔP per stage

Pump Selection Guide

N_s (SI)	Type	η_max	Application
< 15	Radial centrifugal	70-80%	High pressure, low flow
15-40	Standard centrifugal	75-88%	General service
40-100	Mixed flow	80-88%	Medium head
> 100	Axial flow	82-88%	Low head, high flow

Output

Provide: velocity triangle diagram (α, β, U, V, W at inlet/exit), W_s [J/kg], H [m] or ΔP [bar], N_s, NPSH_A vs. NPSH_R comparison, efficiency η_total [%], operating point on H-Q curve, cavitation risk assessment.