Theoretical Leakage Equations Towards Liquid-Phase Flow in the Straight-Through Labyrinth Seal

Labyrinth seals are widely applied in turbomachinery for gas and liquid sealing. A series of labyrinth seal leakage equations so far have been proposed for compressible gas and few equations for incompressible liquid. Based on the flow conserving governing equations, this paper originally presents semi-empirical analytic equations of the leakage flow rate and tooth-clearance pressure for liquid-phase flow in the straight-through labyrinth seal. The equations indicate that the leakage and pressure are closely related to the inlet pressure, outlet pressure, seal geometrical parameters, and four empirical coefficients, while no relation to the temperature and compressibility effects compared to the common gas equations. The empirical coefficients include the velocity compensation coefficient, friction coefficient, jet contraction coefficient, and resistance coefficient. Particularly, the velocity compensation coefficient is determined through an optimization by the genetic algorithm, while others are referred from previous research. Ultimately, taking the sealing of deeply subcooled liquid nitrogen within the spindle of the cryogenic cooling machine tool as a case, the accuracy of proposed equations is evaluated under various pressure ratios and geometry conditions using the numerical approach, whose numerical model has been validated by the experimental data in the literature. The results show that errors between calculation and simulation are generally within the limit of ±5%, except for the pressure values at the first two teeth. This work provides a theoretical basis for further studies on the liquid leakage equations in other labyrinth seal types.