propeller-design

name	propeller-design
description	Propeller design — blade element momentum theory (BEM), thrust/torque coefficients, Betz limit, propulsive efficiency, cavitation, ship propeller open-water diagrams (K_T/K_Q), pitch ratio, NACA series.
metadata	{"priority":7,"promptSignals":{"phrases":["propeller design","blade element theory","BEM propeller","thrust coefficient","propulsive efficiency","ship propeller","K_T K_Q diagram"],"minScore":3}}

Propeller Design — Complete Skill

Fundamental Relationships

Thrust (actuator disk theory): T = ṁ × (V₂ - V_∞) = ρ A_disk (V_∞ + v_i) × 2v_i

v_i = induced velocity at disk; V_∞ = freestream velocity V₂ = 2(V_∞ + v_i) = far wake velocity

Power required: P = T × (V_∞ + v_i)

Ideal (Froude/actuator disk) efficiency: η_ideal = T × V_∞ / P = 2/(1 + V₂/V_∞) = 1/(1 + v_i/V_∞)

Betz limit (wind turbines — maximum power extraction): C_P,max = 16/27 = 0.593 at a = v_i/V_∞ = 1/3

Dimensionless Coefficients

Thrust coefficient: K_T = T / (ρ n² D⁴)

Torque coefficient: K_Q = Q / (ρ n² D⁵)

Power coefficient (wind): C_P = P / (½ ρ V_∞³ A)

Advance ratio: J = V_a / (n D) [V_a = advance velocity; n = rev/s; D = diameter]

Open water efficiency: η_O = K_T × J / (2π × K_Q)

K_T, K_Q, η_O plotted vs. J = open-water diagram (propeller characteristic)

Design point: select J where η_O is maximum → optimal pitch ratio

Blade Element Momentum Theory (BEM)

Strip Theory

Blade divided into radial elements dr Each element treated as airfoil section; local velocity = vector sum of axial + rotational

Local velocity: V_local = √((V_a(1-a))² + (Ω r (1+a'))²) a = axial induction factor; a' = rotational induction factor

Angle of attack: α = φ - β (inflow angle minus pitch angle) tan(φ) = V_a(1-a) / (Ω r (1+a'))

Local forces: dL = ½ ρ V_local² c × C_L(α) × dr dD = ½ ρ V_local² c × C_D(α) × dr

Thrust and torque contribution: dT = N_b × (dL cosφ - dD sinφ) [N_b = number of blades] dQ = N_b × r × (dL sinφ + dD cosφ)

BEM iteration: Initialize a = a' = 0; compute φ, α; get C_L, C_D; update a and a'; repeat until convergence

Prandtl Tip Loss Factor

F = (2/π) arccos(exp(-f)) f = N_b/2 × (R - r) / (r sinφ) [correction for finite number of blades]

Apply F to momentum equations: replaces perfect actuator disk

Marine Propeller — Wageningen B-Series

Polynomial form for K_T and K_Q: K_T = Σ C_s,T × J^s × (P/D)^t × A_E/A_O^u × Z^v K_Q = Σ C_s,Q × (similar terms)

Z = number of blades; P/D = pitch ratio; A_E/A_O = expanded area ratio

B-series typical range: Z: 2–7 blades; P/D: 0.5–1.4; A_E/A_O: 0.30–1.05; J: 0–1.5

Optimum pitch ratio (maximum η_O): From K_T/K_Q diagram at desired J → find P/D that maximizes η_O

Ship Propeller Open-Water Diagram Analysis

Thrust deduction factor t: T = R_hull / (1-t) [t typically 0.05–0.20 for single screw]

Wake fraction w: V_a = V_ship × (1-w) [w = 0.15–0.40 for full-form ships]

Propulsive efficiency: η_D = η_O × η_R × η_H

η_R = relative rotative efficiency (≈ 0.95–1.05) η_H = hull efficiency = (1-t)/(1-w)

Thrust matching: T_propeller = R_hull / (1-t) at operating advance ratio J = V_a/(n D)

Cavitation Check

Cavitation number: σ = (P_∞ - P_v) / (½ ρ V²)

Burrill chart cavitation criterion: τ_c = T / (A_p × ½ ρ V_R²) [thrust loading at 0.7R]

For no cavitation: τ_c ≤ τ_limit(σ) from Burrill chart If cavitation expected: use supercavitating sections or reduce P/D

Minimum expanded area ratio (Keller formula): A_E/A_O ≥ (1.3 + 0.3 Z) × T / ((P + 0.1) × D²) + K K = 0.1 for sing. screw; 0.2 for twin screw (P in atm absolute)

Blade Sections

Marine (NACA/Ogival)

NACA 65 series: standard for propellers; good cavitation performance Modified sections: leading edge radius, camber, thickness optimized for cavitation

Wind Turbine Airfoils

NREL S-series: S809, S814, designed for wind turbine conditions Δc_p/Δα — milder stall behavior; thick roots (30–40%c) for structural strength Flatback airfoils at root: thick trailing edge; improved structural efficiency

Wind Turbine Aerodynamic Design

Design tip speed ratio: λ_design = Ω R / V_∞ (optimal: λ = 6–10 for modern HAWT)

Chord distribution (Betz optimum): c(r) = 8πrV_∞² a(1-a) / (N_b V_local² C_L)

Twist distribution: β(r) = φ(r) - α_design (optimum angle of attack along blade)

Maximum C_P at BEM design point: C_P,BEM ≈ 0.50–0.52 for well-designed 3-blade turbine (including losses)

Propeller Materials

Marine: Nickel-aluminum bronze (NAB, CuAl10Ni5Fe5); ASTM B148 alloy 954 E = 120 GPa; S_y = 240 MPa; excellent corrosion resistance Wind turbine: GFRP (glass-epoxy) or CFRP (carbon-epoxy) spar caps + sandwich shell Large (> 70m) → CFRP spar caps required for stiffness; flap-edgewise natural frequency away from 1P, 3P

Noise (Marine Propeller)

Blade passing frequency: f_BPF = Z × n [Hz; n = rev/s] Singing: if blade natural frequency coincides with shedding frequency → add trailing edge bevel Duct propeller (Kort nozzle): improves thrust in bollard/low-speed; reduces cavitation noise

Output

Provide: K_T and K_Q at design J, open-water efficiency η_O, thrust T [N or kN], torque Q [N·m], blade chord c(r) and twist β(r) distribution, cavitation criterion check (Burrill or σ vs. σ_i), propulsive efficiency η_D (ship) or C_P (wind), blade material and stress at root, number of blades Z and pitch ratio P/D.