labyrinth-seal

name	labyrinth-seal
description	Labyrinth seal design — leakage flow analysis, Carry-Martin equation, tooth geometry, clearance, pressure drop per tooth, straight/stepped/interlocked configurations, honeycomb stator interaction, rotordynamic effects, gas turbine applications, ASME PTC 19.1 leakage measurement.
metadata	{"priority":7,"promptSignals":{"phrases":["labyrinth seal","labyrinth leakage","seal leakage calculation","turbine labyrinth","labyrinth seal design","interstage seal"],"minScore":3}}

Labyrinth Seal Design — Complete Skill

Principle and Types

Labyrinth seal: non-contact seal using a series of teeth creating alternating flow expansion/compression chambers; reduces leakage without rubbing Leakage mechanism: gas enters at high pressure → expands through tooth clearance (kinetic energy) → dissipates kinetic energy in cavity → process repeats for each tooth → total pressure reduction across N teeth

Seal configurations:

• Straight through: teeth on one member; constant radius; simplest; highest leakage
• Stepped: different radii for different tooth groups; seals in two radial locations; better than straight
• Interlocked: teeth alternate from rotor and stator; smallest per-tooth leakage; best performance
• Staggered: teeth offset axially so vortices don't carry kinetic energy → improved dissipation

Leakage Flow Equations

Carry-Martin Formula (Most Common)

Leakage mass flow: ṁ = μ × C_D × A × √(P₁² - P₂²) / (n^0.5 × C_f × √(T₁ × R_gas)) [kg/s; basic form]

More complete form: ṁ = C_D × A_tooth × P_up / √(T_up × R × n) × √(1 - (P_down/P_up)²) [for choked: use P_cr = P_up × (2/(γ+1))^(γ/(γ-1))]

Variables: A_tooth = π × D_tooth × c [clearance annular area; D_tooth = sealing diameter; c = radial clearance] P_up, P_down = upstream and downstream pressures [Pa] T_up = upstream temperature [K] R_gas = specific gas constant [J/(kg·K)] n = number of teeth C_D = discharge coefficient ≈ 0.67–0.85 (depends on tooth geometry)

Egli Equation (Simpler Engineering Form)

ṁ_labyrinth = α × A × (P₁/√T₁) × √(1/(n)) × Φ(P₁/P₂)

Pressure function Φ: Φ = √[(P₁² - P₂²) / P₁²] = √[1 - (P₂/P₁)²] [for subsonic flow] If P₂/P₁ < 0.5 (approximately choked): Φ = √[(γ+1)/2]^[(γ+1)/(γ-1)] / γ^0.5 (Fanno choked flow)

Discharge coefficient α: Sharp-edge (machined) tooth: α ≈ 0.707 (ideal thin orifice) Rounded tooth tip: α = 0.75–0.85 Knife-edge (thin, sharp): α = 0.61–0.70

Stepped Seal Carry Factor

Carry-over coefficient (K_c): fraction of kinetic energy carried into next tooth gap vs. fully dissipated in cavity K_c = 0 → perfect dissipation; K_c = 1 → no dissipation Straight seal (wide-gap cavities): K_c = 0.6–0.9 (poor dissipation; nearly as bad as single orifice) Stepped/interlocked: K_c = 0.2–0.4 (better dissipation)

Effective number of teeth: n_eff = n / (1 + K_c × (n-1) / n) [replace n with n_eff in Egli; accounts for carry-over]

Tooth Geometry

Standard Geometry

Tooth height (radial): h_tooth = 3–8 mm for large turbomachinery; 1–3 mm for small machines Tooth tip width: t_tip = 0.3–1.0 mm (thinner → sharper → lower C_D) Pitch (axial spacing): p = 4–10 mm; wider spacing → better cavity dissipation Cavity depth (axial): L_cav = 5–15 mm; deeper → better vortex formation → better dissipation Clearance: c = 0.15–0.6 mm (nominal cold clearance); thermal + centrifugal growth must maintain c > 0 (no rub)

Clearance calculation: c_hot = c_cold + ΔR_thermal_rotor - ΔR_thermal_stator + ΔR_centrifugal [all in mm] ΔR_thermal = α × R × ΔT [α = thermal expansion coefficient; R = radius; ΔT = temperature rise] ΔR_centrifugal = (ρ × ω² × R²) / (2E) × (3+ν)/(1-ν) × R_tip [for thin disk; approximate]

Honeycomb Stator

Honeycomb lining (stator side): Replaces smooth stator surface with hexagonal honeycomb cells Purpose: increases surface roughness → turbulence → better kinetic energy dissipation in cavity Cell size: 1.6–3.2 mm (1/16"–1/8"); depth: 3–8 mm Material: Hastelloy X brazed to stator; abradable (sacrificial in rub events)

Leakage reduction vs. smooth stator: Honeycomb reduces leakage by 15–35% vs. smooth stator (same clearance) due to increased carry-over dissipation Effective α_honeycomb = 0.85–1.0 × α_smooth (slightly different discharge, but better dissipation = lower K_c)

Abradable coatings alternative: AlSi-polyester, Metco 2043: soft enough to allow rotor rub without damage; maintains tight clearance Temperature limit: Metco 601 (800°C); more advanced MCrAlY-based for hot stages

Leakage Calculation — Worked Example

Conditions: Gas: air; γ = 1.4; R_gas = 287 J/(kg·K) P_up = 10 bar = 10⁶ Pa; P_down = 5 bar = 5×10⁵ Pa T_up = 600 K; n = 6 teeth; D = 0.3 m (300 mm diameter); c = 0.3 mm = 3×10⁻⁴ m; α = 0.7

Annular area: A = π × D × c = π × 0.3 × 3×10⁻⁴ = 2.83×10⁻⁴ m²

Pressure ratio: P₂/P₁ = 0.5; approximate choked conditions at high n Φ: √(1 - 0.5²) = √0.75 = 0.866

Egli flow: ṁ = 0.7 × 2.83×10⁻⁴ × (10⁶ / √600) / √6 × 0.866 ṁ = 0.7 × 2.83×10⁻⁴ × 40,825 / 2.449 × 0.866 ṁ = 0.7 × 2.83×10⁻⁴ × 14,450 = 2.86 × 10⁻³ kg/s ≈ 2.9 g/s

Check vs. total stage mass flow: 2.9 g/s / (total ṁ) = leakage fraction; target < 1% for good efficiency

Rotordynamic Effects

Cross-Coupled Stiffness

Destabilizing cross-coupled stiffness from labyrinth seals: k (cross-coupled) = α × (ṁ × U_mean) / (2 × clearance) × (P_up - P_down) / P_up [approximate] k_cross > 0 → forward whirl excitation → potential instability for flexible rotors

Alford force (aeroelastic instability): F_Alford = β × T × (ε/c) [β = Alford coefficient ≈ 1–3; T = stage work; ε = eccentricity; c = clearance] Destabilizes rotor if exceeds bearing damping

Mitigation: Swirl cancellation: inject anti-swirl flow into labyrinth (reduces circumferential velocity → reduces k_cross) Honeycomb seals: reduce k_cross by 50–70% vs. smooth stator Hole-pattern seals: 5–8 mm diameter holes in stator; similar to honeycomb in rotordynamic effect

Stability Criterion

Logarithmic decrement: δ = π × (2 × system_damping - N × C_eff) / √(ω_n² - γ²) [simplified; N = number of seals; C_eff = effective cross-coupling] Rotor stable if δ > 0 (net damping positive); threshold above cross-coupling stiffness

API 617 (compressors) rotordynamic stability requirement: Δ = -2 × k_xy / (ω_n × D_bearing) > 0 (simplified log decrement) Full rotordynamic analysis: Newkirk-Jones factor; eigenvalue analysis with damped natural frequency

Gas Turbine Application

High-pressure compressor (HPC) interstage seals: Multiple straight-through labyrinth seals at disk rim; 4–8 teeth; clearance 0.2–0.4 mm Sealing air leakage 0.5–2% of core mass flow → direct SFC (specific fuel consumption) penalty

HPT (high-pressure turbine) over-tip seal: Knife-edge rotor tips against abradable stator; tight clearance (0.2–0.4 mm at operating) 1% clearance increase → ~0.5% turbine efficiency loss → careful thermal modeling

Rim seal (disk-to-nozzle): Prevents hot gas ingestion into wheel space; double-rim double-seal configuration Cooling air purge flow required: C_w = ṁ_purge / (μ × Ω × r²) > 0.12–0.15 for adequate sealing (C_w = non-dimensional flow)

Standards

Standard	Scope
ASME PTC 19.1	Flow measurement uncertainty (for leakage test)
API 617	Axial and centrifugal compressors (seal requirements)
API 614	Seal system (lube oil and buffering)
AGMA 9001	Seals for enclosed gear drives
MIL-L-23699	Turbine lubricant compatibility (seal material)

Output

Provide: seal type (straight/stepped/interlocked), seal diameter D [mm], number of teeth n, nominal clearance c [mm] (cold) and hot clearance [mm] with thermal/centrifugal correction, upstream pressure P_up [bar] and temperature T_up [K], downstream pressure P_down [bar], discharge coefficient C_D (tooth geometry), leakage mass flow ṁ [g/s] from Egli equation, leakage fraction [%] of main flow, carry-over coefficient K_c and effective n_eff, honeycomb stator? (yes/no; leakage reduction [%]), cross-coupled stiffness k_cross [N/mm] from Alford, rotordynamic stability check (log decrement δ > 0?), anti-swirl injection required? (yes/no), and applicable standard (ASME PTC 19.1, API 617).