Straight and Stepped Labyrinth Seal Leakage Calculator

Estimate steady ideal-gas leakage through straight-through or stepped labyrinth seals using pressure, gas properties, fin count and annular clearance.

Predicted gas leakage

—

kg/s

Mass flow— g/s

Pressure ratio, r—

Critical ratio—

Area used— m²

Mass-flow function—

Coefficient product—

Enter valid inputs and calculate.

How to use

Select Straight — Martin for a straight-through labyrinth or Stepped — Zimmermann–Wolff for a stepped geometry.
Enter upstream total and downstream static pressures as absolute kPa, with downstream pressure below upstream pressure.
Enter upstream total temperature in kelvin, the gas-specific constant, heat-capacity ratio and integer fin count.
Enter clearance area directly in mm², or select diameter plus radial clearance to calculate the exact annular area. In stepped mode, also enter independently validated C_d and k_s.
Select Calculate to update leakage and diagnostics. Reset restores the air example and recalculates.
Read mass flow first. Review pressure ratio, critical ratio, area and warnings; correct any red message before using a value.
Do not use the model when real-gas, transient, rotating-cavity, heat-transfer, rub, eccentricity or safety-critical effects require a validated seal analysis or test.

Equations used

Definitions: p₀ is absolute upstream total pressure (Pa); pₙ is absolute downstream static pressure (Pa); T₀ is upstream total temperature (K); R is the specific gas constant (J/(kg·K)); n is the dimensionless integer fin count; A is annular flow area (m²); m is predicted mass flow (kg/s); r is the downstream-to-upstream absolute-pressure ratio (dimensionless); Q is the normalized mass-flow function (√K·s/m); rcrit is the ideal-gas critical pressure ratio (dimensionless); γ is the heat-capacity ratio (dimensionless); kₛ is the stepped-geometry correction (dimensionless); C_d is the discharge coefficient (dimensionless); Ainput is directly entered flow area (mm²); D is seal diameter (m); and c is radial clearance (m).

r = pₙ / p₀

A = Ainput × 10⁻⁶

A = (π/4)[(D + 2c)² − D²]

m = A p₀ √{(1 − r²) / [R T₀ (n − ln(r))]}

For stepped mode, the coefficient-explicit equation is:

m = kₛ C_d A p₀ √{(1 − r²) / [R T₀ (n − ln(r))]}

m_g/s = 1000m

Q = m√T₀/(A p₀) ; r_crit = [2/(γ + 1)]^{γ/(γ − 1)}

Entered pressures in kPa are multiplied by 1000 before substitution. Geometry inputs D and c are entered in mm and multiplied by 10⁻³ before the exact-annulus equation. Positive mass flow is defined from upstream to downstream. The model assumes steady one-dimensional ideal-gas flow, constant T₀ and equal throttling stages. A ratio at or below rcrit triggers a warning but does not alter the cited empirical equation.

The stepped coefficients must represent the applicable clearance-to-land ratio, step height, pitch and flow regime. Published broad discharge-coefficient ranges are not a substitute for design-specific validation. This screening model excludes rotation, swirl, Reynolds corrections, heat transfer, real-gas behavior, rubs, eccentricity and transients.