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Stepped Labyrinth Seal Equation and Calculator

Fluids Flow Design and Engineering

Stepped Labyrinth Seal using Zimmermann and Wolff Equation and Calculator

A step can be incorporated into a labyrinth seal design to prevent carryover of kinetic energy from occurring. This provides improved sealing characteristics in comparison to straight labyrinth seals. However, radial space must be available in machines where sealing is required to accommodate a stepped seal. In addition, axial pressure gradients acting on step faces can cause undesirable axial loads. The geometric form of the seal can be defined to enhance the dissipation of kinetic energy. Such stepped seals have been applied to gas turbine engine applications.

Studies on stepped labyrinth seals have been reported by a number of researchers, including Stocker (1976), Stocker et al. (1977), Vermes (1961), Benckert and Wachter (1979), Stepanoff (1931), Isaacson (1957) and Yamada (1962); a variety of flow models have been developed for stepped labyrinth seals, as indicated in Table 1. The detailed flow structure of flow in stepped labyrinth configurations has been studied by Rhode et al. (1997a,b).

Figure 1 Stepped Labyrinth Seal

Preview Stepped Labyrinth Seal Calculator

In the absence of a validated database or code, the model proposed by Zimmermann and Wolff (1998) is suggested for stepped labyrinths, based on the following equation:

Eq. 1, mass flow

${\displaystyle m=k_s\cdot <!--\; ·\; -->C_d\cdot <!--\; ·\; -->A\cdot <!--\; ·\; -->p_{t0}\sqrt{\frac{1-<!--\; -\; -->(p_n/p_{t0}{)}^2}{R\cdot <!--\; ·\; -->T_{t0}\cdot <!--\; ·\; -->[n+ln(p_{t0}/p_n)]}}}$

where

m = mass flow rate (kg/s)
A = annular clearance area of the seal (m²)
p_t0 = upstream total pressure (Pa)
p_n = downstream static pressure (Pa)
R = characteristic gas constant (J / kg-K)
T_t0 = upstream total temperature (K)
k_s = correction factor for C_d
C_d = discharge coefficient
n = number of labyrinth fins (-)

Figure 1 C_d correction factor vs c/t

Figure 2, Discharge coefficients for a stepped labyrinth seal (single fin).

c = seal clearance (m)
t = fin-tip land width (m)

Typical C_d ranges of values of the discharge coefficient

0.53 ≤ Cd ≤ 0.64 for c/t = 0.909
0.46 ≤ Cd ≤ 0.57 fo c/t = 1.515
0.46 ≤ Cd ≤ 0.53 for c/t = 2.424

${\displaystyle }$

Zimmermann and Wolff (1998) presented proprietary data for the influence of step height A minimum at H / l_pitch = 0.075 is recommended if practicable.

Mechanical Design Engineering Handbook
Peter R. N. Childs
2014

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