impeller-design

name	impeller-design
description	Centrifugal impeller design — Euler turbomachinery equation, specific speed, blade angle, meridional velocity, slip factor, NPSH, cavitation, diffuser matching, volute design, ANSYS CFX, impeller-diffuser interaction, shroud clearance, balancing.
metadata	{"priority":7,"promptSignals":{"phrases":["impeller design","centrifugal impeller","pump impeller","impeller blade","specific speed impeller","NPSH impeller"],"minScore":3}}

Centrifugal Impeller Design — Complete Skill

Euler Turbomachinery Equation

Euler head: H_E = (U₂ × C_u2 - U₁ × C_u1) / g [m; H_E = Euler head; U = blade tip speed; C_u = tangential (whirl) component of absolute velocity; g = 9.81 m/s²]

For zero pre-swirl (C_u1 = 0, radial inlet): H_E = U₂ × C_u2 / g = U₂² × (1 - φ₂ / tan(β₂)) / g φ₂ = C_r2 / U₂ [flow coefficient at exit; C_r2 = radial velocity at impeller exit]

Blade tip speed: U₂ = π × D₂ × N / 60 [m/s; D₂ = impeller outer diameter [m]; N = rotational speed [rpm]]

Hydraulic efficiency: η_h = H_actual / H_E [typically 0.85–0.92 for well-designed centrifugal impellers]

Pump head: H = η_h × H_E [m]

Specific Speed

Dimensional specific speed: N_s = N × √Q / H^(3/4) [rpm, m³/s, m; OR: N_s = N [rpm] × √Q [gpm] / H [ft]^(3/4) for US units] N_s_SI = N × √Q / (g × H)^(3/4) [dimensionless specific speed; N in rad/s; Q in m³/s; g×H in m²/s²]

Impeller type by specific speed:

N_s (rpm, m³/s, m)	N_s (dimensionless)	Impeller Type
< 500	< 0.25	Radial (centrifugal); high H, low Q
500–2,000	0.25–1.0	Mixed flow; moderate H and Q
2,000–5,000	1.0–2.5	Axial (propeller); low H, high Q

Suction specific speed: S = N × √Q / NPSH_r^(3/4) [S < 9,000 for good cavitation resistance (US units)]

Blade Geometry

Blade Angles

Exit blade angle β₂ (measured from tangential): β₂ = 15°–35° → backward-swept blades (most common; stable; better efficiency) β₂ = 90° → radial blades (no H-Q drooping; used for industrial fans) β₂ = 90°–135° → forward-swept (unstable H-Q curve; risk of surge; rarely used)

Velocity triangle at exit (without slip): C_r2 = Q / (π × D₂ × b₂ × C_c) [b₂ = impeller exit width; C_c = contraction factor ≈ 0.85–0.95] W_u2 = U₂ - C_r2 / tan(β₂) [relative tangential velocity] C_u2 = U₂ - C_r2 / tan(β₂) [absolute tangential velocity for ideal case]

Inlet blade angle β₁ (no shock condition): tan(β₁) = C_r1 / (U₁ - C_u1) [for zero pre-swirl C_u1=0: tan(β₁) = C_r1 / U₁] U₁ = π × D₁ × N / 60 [D₁ = impeller inlet diameter]

Slip Factor

Slip factor (σ) corrects for finite blade count: σ = C_u2_actual / C_u2_ideal [σ < 1; typically 0.85–0.95]

Stodola slip factor: σ = 1 - (π × sin β₂) / z [z = number of blades; β₂ = exit blade angle]

Wiesner formula (more accurate): σ = 1 - (√sin β₂) / z^0.7 [preferred for engineering design]

Corrected head: H_E_actual = σ × U₂² × (1 - φ₂ × cot β₂) / g

Number of blades selection: z = 6–9 for backward-swept impellers z = 5–7 for mixed flow More blades → higher slip factor → better head but more friction losses; optimal at z = 6–8

Meridional Velocity Distribution

Continuity equation: Q = C_r × A_meridional = C_r × π × D × b [constant flow through impeller] C_r2/C_r1 = (D₁ × b₁) / (D₂ × b₂) [if uniform meridional velocity; not realistic]

Width at exit: b₂ = Q / (π × D₂ × C_r2 × C_c) [C_c = blockage correction for blade thickness]

Eye diameter (impeller inlet): D_eye = D₁ for mixed-flow; eye sized to give C_r1 = 2–4 m/s (low velocity reduces NPSH required) D_shroud / D_hub = 1.5–2.5 (diameter ratio at eye)

Hub diameter: D_hub ≥ 1.2 × D_shaft (minimum clearance for stress and assembly)

NPSH and Cavitation

NPSH Required (NPSH_r)

Available NPSH (system): NPSH_a = (P_atm - P_vapor) / (ρ × g) + V_s² / (2g) - Z_s [m; V_s = inlet pipe velocity; Z_s = static head above pump; P_vapor = vapor pressure]

Required NPSH (pump): NPSH_r from manufacturer curve; must satisfy: NPSH_a > NPSH_r + safety margin (0.5–1 m minimum)

Thoma cavitation parameter: σ_c = NPSH_r / H [σ_c < 0.1 indicates good cavitation resistance]

Critical cavitation inception: Cavitation starts at suction side of blade where local pressure minimum occurs: P_local = P_inlet - ρ × (C_blade^2 - C_inlet^2) / 2 Cavitation when P_local < P_vapor(T)

Cavitation types:

• Suction recirculation: at low flow (< 50% BEP)
• Discharge recirculation: at very low flow
• Vaporization (classical): at low NPSH

Design for Cavitation Resistance

Leading edge geometry: Thin leading edge (< 3 mm); smooth radius; NACA profile minimizes local acceleration → lower NPSH_r Double-curvature (3D) impeller inlet: reduces secondary flow → better cavitation performance

Surface finish: Ra ≤ 0.8 μm on pressure surfaces → delays cavitation initiation

Impeller Structural Design

Centrifugal stress (rotating disk): σ_r = (3 + ν) / 8 × ρ × ω² × (R_o² + R_i² - R_i²R_o²/r² - r²) [MPa; for solid disk; ρ = density; ω = angular velocity; ν = Poisson's ratio]

Maximum stress (at bore for solid disk): σ_max = (3 + ν) / 4 × ρ × ω² × R_o² [at bore r = 0]

Hoop stress at rim: σ_θ = ρ × U₂² [MPa; at tip radius; dominates; must check against material yield/safety factor]

Material selection by tip speed: U₂ ≤ 300 m/s: cast iron (pumps); gray iron ASTM A48 U₂ ≤ 400 m/s: cast stainless steel ASTM A351 CF8M; cast aluminum U₂ ≤ 500 m/s: forged stainless 17-4 PH; wrought aluminum 2024-T4 U₂ > 500 m/s: titanium Ti-6Al-4V; Inconel 718

Safety factor for burst speed: n_burst = √(σ_y / σ_rim) ≥ 1.5 (minimum); 2.0 for pumps with personnel exposure

Diffuser and Volute Design

Vaneless Diffuser

Radius ratio: r₃/r₂ = 1.3–1.8 (vaneless diffuser exit/impeller exit) Velocity in vaneless diffuser: C_u × r = constant (angular momentum conservation) C_r × b × r = constant (continuity)

Vaned Diffuser

Diffuser blade angle at inlet: α₃ = arctan(C_r3 / C_u3) ≈ 15°–25° Blade setting 2°–5° higher than flow angle for positive incidence control

Volute

Volute throat area: A_throat = Q / C_throat [C_throat ≈ 0.8 × C₃ for good efficiency] Volute width b_v = b₂ + 2 × (diffuser width increment) Volute constant: K_v = r × C_u = U₂ × r₂ × σ (constant angular momentum in volute)

Volute area distribution: A(θ) = A_throat × (θ/360°) [linear growth with angle θ] Cut-water (tongue) clearance: 5–7% of impeller radius (reduce rotor-stator interaction)

Impeller-Diffuser Interaction (Noise and Vibration)

Blade passing frequency: f_BPF = z × N / 60 [Hz; z = blade count; N = rpm]

Interaction noise: occurs when impeller blade count and diffuser vane count share common factors Optimal: impeller z and diffuser vanes n_v with GCD(z, n_v) = 1 (coprime) Example: z = 6 impeller blades + n_v = 7 diffuser vanes (no common factor)

Clearance effect: larger gap (r₃/r₂ > 1.1) reduces pressure fluctuations and noise

Computational Design (CFD)

ANSYS CFX/Fluent turbomachinery approach:

• Rotating domain: impeller (MRF — Multiple Reference Frame for steady; sliding mesh for transient)
• k-ω SST turbulence model: recommended for adverse pressure gradient (diffuser)
• Total pressure inlet BC; mass flow outlet BC
• Y+ = 1–5 for wall-resolved k-ω SST; Y+ = 30–100 for wall functions
• Mesh: 1–5 million cells for single passage; periodic boundaries

Performance prediction: Head coefficient: ψ = g × H / U₂² [target 0.5–0.7] Flow coefficient: φ = Q / (π/4 × D₂² × U₂) [target 0.1–0.3] Power coefficient: λ = P / (ρ × U₂³ × D₂²)

Balancing

Static balance: residual imbalance U_r ≤ G_balance_grade × m / ω [ISO 1940] Dynamic balance: ISO 1940 G2.5 for industrial pumps; G1.0 for precision applications Balance planes: overhung impeller — single plane; for shaft assembly — two-plane dynamic balance

Standards

Standard	Scope
ISO 9906	Rotodynamic pumps — hydraulic performance acceptance tests
ANSI/HI 1.6	Centrifugal pump tests
ASME PTC 8.2	Centrifugal pump performance
ISO 5198	Precision testing of centrifugal pumps
ISO 1940	Mechanical vibration — balance quality requirements
API 610	Centrifugal pumps for petroleum/petrochemical
API 685	Sealless centrifugal pumps

Output

Provide: impeller type (radial/mixed/axial) based on N_s, design point (Q [m³/s], H [m], N [rpm]), impeller diameter D₂ [mm] and eye diameter D₁ [mm], tip speed U₂ [m/s], blade exit angle β₂ [°], number of blades z, slip factor σ (Wiesner), Euler head H_E [m], hydraulic efficiency η_h, impeller width b₂ [mm], NPSH_r [m] and cavitation parameter σ_c, centrifugal hoop stress at tip [MPa] vs. material σ_y, burst speed safety factor n_burst, blade passing frequency f_BPF [Hz], diffuser geometry (type; radius ratio), volute throat area [mm²], and applicable standard (ISO 9906, API 610, ISO 1940).