Steam & Gas Turbines, Centrifugal Compressors & Martin Labyrinth Seal Leakage
Turbomachinery Labyrinth Seal Martin Leakage Formula
Calculate Annular Tooth-Tip Clearance Area (mm²), Martin Compressible Expansion-Throttling Factor, Parasitic Gas Leakage Mass Flow (kg/s), and Hourly Leakage Loss (kg/h).
How 12 non-contacting knife-edge labyrinth teeth choke high-pressure turbine leakage—and why a rotor rub doubles steam loss!
At `10,000+` RPM and `620 K`, contacting shaft seals burn up instantly—so steam turbines, aviation gas turbines, and centrifugal compressors rely on non-contacting **Knife-Edge Labyrinth Seals** of Shaft Diameter `D` (mm), Radial Tip Clearance `c` (`~0.25–0.45` mm -> Annular Gap Area `A_cl = PI() * D * c`), and `N` sequential throttling cavities (`N = 8–20` teeth)! Each knife-edge converts static pressure head into a high-velocity wall jet whose kinetic energy dissipates as turbulent eddies inside the stepped pocket (`Fanno-like` isothermal enthalpy conservation `T_0 = const`). H. M. Martin derived the classic **Compressible Labyrinth Leakage Equation** fromUpstream `P_1` to Downstream `P_2` (`r_p = P_2 / P_1`): **`dot_m = C_d * A_cl * (P_1 / SQRT(R_gas * T_1)) * SQRT((1 - r_p^2) / (N - LN(r_p)))`** (`kg/s`)!
Verify on a 12-Tooth Stepped-Stator Compressor Balance-Drum Labyrinth—Shaft_Diameter `D` = 160.0 mm (B2), Cold/Hot_Radial_Clearance `c` = 0.35 mm (B3 -> Annular Clearance Area = `175.93` mm² in E2), Upstream_Pressure `P_1` = 18.0 bar(a) (`1.8` MPa, B4), Balance-Line_Downstream `P_2` = 3.0 bar(a) (`r_p = 1/6`, B5), Inlet_Gas_Temp `T_1` = 620.0 K (B6), Specific_Gas_Constant `R_g` = 287.05 J/(kg·K) (B7), Knife_Teeth `N` = 12 (B8), and Stepped_Tooth_Discharge `C_d` = 0.72 (B9): Martin Expansion Factor equals `0.2655` (E3), throttling Parasitic Leakage to `0.1435` kg/s (E4 = `516.6` kg/h in E5)!
| Martin Labyrinth Seal KPI (D=160 mm, c=0.35 mm, 18 -> 3 bar(a), 620 K, N=12 Teeth) | Formula | Verified Output |
|---|---|---|
| Annular Gap Area A_cl (mm²) | PI() * Shaft_D_mm * Radial_c_mm | 175.93 mm² (376.99 mm² @ worn 0.75 mm!) |
| Martin Expansion Factor | SQRT((1 - (P2/P1)^2) / (N - LN(P2/P1))) | 0.2655 |
| Parasitic Leakage Flow (kg/s) | Cd * A_m2 * (P1_Pa/SQRT(R*T1)) * Factor | 0.1435 kg/s (0.3075 kg/s @ worn 0.75 mm!) |
| Hourly Parasitic Loss (kg/h) | Leakage_Mass_Flow_kg_s * 3600 | 516.6 kg/h (1,107.0 kg/h = +114% loss!) |
H. M. Martin Multi-Tooth Compressible Labyrinth Leakage Formula
Validate `Shaft_Diam D > 10` mm, `Radial_Clearance c` in `(0.02, 5.0]` mm, `0 < P_2 < P_1`, `Temp T_1 > 100` K, `R_gas > 50` J/(kg·K), integer `Teeth N >= 2`, and `C_d` in `[0.4, 1.0]`!
=LET(dm_m, B2 / 1000, cl_m, B3 / 1000, pu_p, B4 * 100000, pr_r, B5 / B4, tk_g, B6, rg_j, B7, nt_c, B8, cd_f, B9, ac_m, PI() * dm_m * cl_m, mf_t, SQRT((1 - (pr_r ^ 2)) / (nt_c - LN(pr_r))), ms_k, cd_f * ac_m * (pu_p / SQRT(rg_j * tk_g)) * mf_t, IF(OR(B2 <= 10, B3 <= 0.02, B3 > 5, B5 <= 0, B4 <= B5, tk_g <= 100, rg_j <= 50, nt_c < 2, cd_f < 0.4, cd_f > 1), "Invalid Labyrinth Input", ROUND(ms_k * 3600, 1)))Critical-Speed Rotor Rub Worn Clearance (c = 0.75 mm) Efficiency Loss Check (Set Clearance B3 = 0.75 mm)
Let a startup critical-speed bow mushroom the aluminum/abradable knife tips from `0.35` mm out to `0.75` mm (`B3 = 0.75`): Annular Gap Area E2 widens to `376.99` mm², Parasitic Leakage E4 jumps `+114%` to `0.3075` kg/s (`1,107.0` kg/h in E5)—costing `~1%–2%` of stage polytropic efficiency until turnaround re-stripping!
=IF(ISTEXT(E5), E5, LET(ac_m, PI() * (B2 / 1000) * (B3 / 1000), pr_r, B5 / B4, mf_t, SQRT((1 - (pr_r ^ 2)) / (B8 - LN(pr_r))), ROUND(B9 * ac_m * ((B4 * 100000) / SQRT(B7 * B6)) * mf_t, 4)))Try the example
Use the original values or change them to test your understanding.