The BFM models rely on classical Neumann's equation model to characterize the flow through a labyrinth tooth. Presently, a CFD analysis quantifies the effects of tip clearance (Cr) and operating conditions on the prediction of LS mass flow, and then derives an updated kinetic energy carry-over coefficient (µ1i) to improve the accuracy of Neumann's leakage equation. The analysis selects a 14 teeth on stator LS with clearance Cr=(1/733)D and operating at nominal supply pressure Pin=73 bar and discharge pressure Pout=51 bar, and at a rotor speed of 12 krpm. The CFD produces flow fields for LSs with a clearance varying from 80% to 200% of the nominal Cr, a gas supply pressure from 60 bar to 100 bar, and the pressure ratio (PR=Pout/Pin) ranging from 0.40 to 0.85. The numerous predictions deliver the mass flow as well as the bulk-flow velocities and cavity pressures. µ1i increases with respect to the seal radial clearance (Cr). µ1i shows a parabolic correlation with PR. The coefficient µ1i is only sensitive to the pressure ratio and not to the magnitude of either the supply or discharge pressures. Lastly, for use with Neumann's leakage model, the CFD predictions produce an updated µ1i, a function of the seal geometry and the PR condition. Integration of the new µ1i correlation into a BFM code improves its accuracy to predict LS mass flow rate, a 19% difference against test data reduces to within 6%.
Labyrinth gas seals (LSs) commonly used in turbomachines reduce secondary flow leakage. Conventional see-through labyrinth seal designs include either all teeth-on-stator (TOS) or all teeth-on-rotor (TOR). Experience shows that an interlocking labyrinth seal (ILS), with teeth on both stator and rotor, reduces gas leakage by up to 30% compared to the conventional see-through designs. However, field data for ILS rotordynamic characteristics are still vague and scarce in the literature. This work presents flow predictions for an ILS and a TOS LS, both seals share identical design features, namely radial clearance Cr = 0.2 mm, rotor diameter D = 150 mm, tooth pitch Li = 3.75 mm, and tooth height B = 3 mm. Air enters the seal at supply pressure Pin = 3.8, 6.9 bar (absolute) and temperature of 25 °C. The ratio of gas exit pressure to supply pressure ranges from 0.5 to 0.8, and the rotor speed is fixed at 10 krpm (surface speed of 79 m/s). The analysis implements a computational fluid dynamics (CFD) method with a multi-frequency-orbit rotor whirl model. The CFD predicted mass flow rate for the ILS is ∼ 21% lower than that of the TOS LS, thus making the ILS a more efficient choice. Integration of the dynamic pressure fields in the seal cavities, obtained for excitation frequency (ω) ranging from 12% to 168% of rotor speed (sub and super synchronous whirl), allows an accurate estimation of the seal dynamic force coefficients. For all the considered operating conditions, at low frequency range, the TOS LS shows a negative direct stiffness (K < 0), frequency independent; whereas the ILS has K > 0 that increases with both frequency and supply pressure. For both seals, the magnitude of K decreases when the exit pressure/inlet pressure ratio increases. On the other hand, the cross-coupled stiffness (k) from both seals is frequency dependent, its magnitude increases with gas supply pressure, and k for the ILS is more sensitive to a change in the exit/inlet pressure ratio. Notably, k turns negative for subsynchronous frequencies below rotor speed (Ω) for both the TOS LS and the ILS. The direct damping (C) for the TOS LS remains constant for ω > ½ Ω and has a larger magnitude than the damping for the ILS over the frequency range up to 1.5 Ω. An increase in exit/inlet pressure ratio decreases the direct damping for both seals. The effective damping coefficient, Ceff = (C-k/ω), whenever positive aids to damp vibrations, whereas Ceff < 0 is a potential source for an instability. For frequencies ω/Ω < 1.3, Ceff for the TOS LS is higher in magnitude than that for the ILS. From a rotordynamics point of view, the ILS is not a sound selection albeit it reduces leakage. Comparison of the CFD predicted force coefficients against those from a bulk flow model demonstrates that the later simple model delivers poor results, often contradictory and largely indifferent to the type of seal, ILS or TOS LS. In addition, CFD model predictions are benchmarked against experimental dynamic force coefficients for two TOS LSs published by Ertas et al. (2012, “Rotordynamic Force Coefficients for Three Types of Annular Gas Seals With Inlet Preswirl and High Differential Pressure Ratio,” ASME J. Eng. Gas Turbines Power, 134(4), pp. 04250301–04250312) and Vannini et al. (2014, “Labyrinth Seal and Pocket Damper Seal High Pressure Rotordynamic Test Data,” ASME J. Eng. Gas Turbines Power, 136(2), pp. 022501–022509.)
In straight-through centrifugal pumps, a grooved seal acts as a balance piston to equilibrate the full pressure rise across the pump. As the groove pattern breaks the development of fluid swirl, this seal type offers lesser leakage and lower cross-coupled stiffnesses than a similar size and clearance annular seal. Bulk-flow models predict expediently the static and dynamic force characteristics of annular seals; however they lack accuracy for grooved seals. Computational fluid dynamics (CFD) methods give more accurate results, but are not computationally efficient. This paper presents a modified bulk-flow model to predict the rotordynamic force coefficients of shallow depth circumferentially grooved liquid seals with an accuracy comparable to a CFD solution but with a simulation time of bulk-flow analyses. The procedure utilizes the results of CFD to evaluate the bulk flow velocity field and the friction factors for a 73 grooves annular seal (depth/clearance dg/ Cr = 0.98 and length/diameter L/D = 0.9) operating under various sets of axial pressure drop and rotor speed. In a groove, the flow divides into a jet through the film land and a strong recirculation zone. The penetration angle (α), specifying the streamline separation in the groove cavity, is a function of the operating conditions; an increase in rotor speed or a lower pressure difference increases α. This angle plays a prominent role to evaluate the stator friction factor and has a marked influence on the seal direct stiffness. In the bulk-flow code the friction factor model (f = nRem) is modified with the CFD extracted penetration angle (α) to account for the flow separation in the groove cavity. The flow rate predicted by the modified bulk-flow code shows good agreement with a measured result (6% difference). A perturbation of the flow field is performed on the bulk-flow equations to evaluate the reaction forces on the rotor surface. Compared to the rotordynamic force coefficients derived from the CFD results, the modified bulk-flow code predicts rotordynamic force coefficients within 10%, except that the cross-coupled damping coefficient is over-predicted up to 14%. An example test seal with a few grooves (L/D = 0.5, dg/Cr = 2.5) serves to further validate the predictions of the modified bulk-flow model. Compared to the original bulk-flow analysis, the current method shows a significant improvement in the predicted rotordynamic force coefficients, the direct stiffness and damping coefficients in particular.
Gas labyrinth seals (LS) restrict secondary flows (leakage) in turbomachinery and their impact on the efficiency and rotordynamic stability of high-pressure compressors and steam turbines can hardly be overstated. Among seal types, the interlocking labyrinth seal (ILS), having teeth on both the rotor and the stator, is able to reduce leakage up to 30% compared to other LSs with either all teeth on the rotor (TOR) or all teeth on the stator. This paper introduces a revamped facility to test gas seals for their rotordynamic performance and presents measurements of the leakage and cavity pressures in a five teeth ILS. The seal with overall length/diameter L/D = 0.3 and small tip clearance Cr/D = 0.00133 is supplied with air at T = 298 K and increasing inlet pressure Pin = 0.3–1.3 MPa, while the exit pressure/inlet pressure ratio PR = Pout/Pin is set to range from 0.3 to 0.8. The rotor speed varies from null to 10 krpm (79 m/s max. surface speed). During the tests, instrumentation records the seal mass flow (m˙) and static pressure in each cavity. In parallel, a bulk-flow model (BFM) and a computational fluid dynamics (CFD) analysis predict the flow field and deliver the same performance characteristics, namely leakage and cavity pressures. Both measurements and predictions agree closely (within 5%) and demonstrate that the seal mass flow rate is independent of rotor speed. A modified flow factor Φ¯=m˙T/(PinD1−PR2) characterizes best the seal mass flow with a unique magnitude for all pressure conditions, Pin and PR.
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