**1. Introduction**

Labyrinth seals are widely used in various types of fluid-flow machines, such as steam turbines, gas turbines, and compressors. Labyrinth seals enable reducing the leakage of working medium between two elements that are non-contacting. Concern for the protection of the natural environment imposes current trends to achieve greater efficiency of flow machines and flowing systems. High parameters of machines operation are required, which means that seals work in the environment characterized by high temperatures and large pressure drop. Labyrinth seals have particular impact on the efficiency of high powergenerating machines. In the paper [1], the impact of the leakage in the internal gland of the steam turbine 13K215 on the power loss was analyzed. It was showed that the power loss resulting from the steam leakage through the internal seal of nominal geometry had achieved approx. 1 MW, and through the worn out seal it could achieve 2 MW. Therefore, the problem of leakage minimization is of great importance. The paper [2] presented the numerical analysis of fluid flow in the two-teeth straight through over-bandage seal in the axial turbine stage, with particular attention to the impact of the leakage on the main flow. Authors of this paper showed that the leakage had a considerable impact on the agitation loss and secondary flows generating serious energy losses in the turbine stage. It was proved in the paper [3] that 1% increase in the seal-tooth clearance height causes a significant decrease in performance and efficiency of the multistage axial compressor. Labyrinth seals also have a great influence on the efficiency of reciprocating machines [4–6].

**Citation:** Joachimiak, D. Novel Method of the Seal Aerodynamic Design to Reduce Leakage by Matching the Seal Geometry to Flow Conditions. *Energies* **2021**, *14*, 7880. https://doi.org/10.3390/en14237880

Academic Editor: Artur Bartosik

Received: 6 October 2021 Accepted: 18 November 2021 Published: 24 November 2021

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Various calculation methods for estimating the leakage rate in the labyrinth seal are presented in the scientific literature. The first calculation model describing the gas flow in the straight through seal was developed by Martin [7]. Egli [8] and other researchers, Hodkinson [9], Zimmermann and Wolff [10], had included in their calculation models the phenomenon of kinetic energy carry over as well as the result of flow contraction. Those models were next developed by Neumann [11] and Scharrer [12], among others, who proposed the method for calculating the seal leakage tooth by tooth. There are new one-dimensional models appearing in the literature [13,14], enabling the determination of supercritical CO2 leakage. The most advanced of these models provide for such features of the seal geometry as the clearance height, chamber length, and tooth thickness. These methods have different accuracy which depends on coefficients being determined experimentally. Applying the above-mentioned models to analyze the seal geometry in order to optimize it with respect to the leakage rate would be subjected to great error, which was proved in the paper [15].

Usage of the CFD flow analysis enables better observation of physical and flow phenomena, which in turn enables designing seals of higher leak tightness. In papers [16–20], the impact of the seal geometric parameters, such as the pitch or the chamber size, on the leakage rate was described. The degree of sealing wear and tear has a considerable impact on the failure of the seal integrity [21–23]. Slight changes in the dimensions of the clearance influence the leakage value and the flow coefficients, which is presented in the paper [24]. A seal with staggered helical teeth, which is characterized by reduced leakage, is presented in the paper [25]. In the paper [26], the reduction of leakage is achieved by slight changes in the chamber geometry, which were disturbing the gas jet flowing with high velocity. Surrogate models of optimization based on a set of input data, which features had been specified parametrically, obtained from CFD calculations, are discussed in the paper [27]. Radial clearance height and teeth spacing were variable parameters. The leakage minimization and decreasing total increase of enthalpy resulting from the friction were taken as the optimization criterion. The paper [28] presents the method for optimizing the straight through sealing with the use of CFD by changing the teeth inclination angle and spacing. The paper [29] presents an optimization model taking into account such design variables as the seal clearance, teeth width, teeth height, pitch, and teeth backward and forward expansion angle. The problem of optimizing the geometry of seal chambers is considered in the paper [30]. Semi-empirical model and parametric CFD analysis were used for this purpose. In the paper [15], the impact of the increasing number of teeth for constant length of the straight through seal on the leakage rate is investigated. Data presented therein indicate that there is a specific range of the seal pitch length in the straight through seal for which the minimum leakage is obtained. The increase of the number of seal teeth results in the decrease of pitch length and reduction of chambers size. It was proved in the paper [9] that the application of too many teeth leads to the effect of increasing leakage in the seal. This is the result of fading gas expansion in chambers of reducing size. Then the flow character becomes similar to that in the slot seal [31]. In this paper a new method for designing the geometry of the labyrinth seal is described. The inspiration for research work presented herein was to develop a relatively simple in-use method for designing such seal geometry which is based on the observation of flow phenomena intensifying leakage. So far, seals with equal chamber dimensions or with sequentially repeated chambers were designed.

The approach to designing seals by matching the seal geometry to flow conditions, and particularly based on the analysis of the gas kinetic energy distribution along the seal length, has not yet been presented in scientific literature. The method for labyrinth seal design, presented in this paper, enables quick obtaining of results. It enables determination of many dimensions of the geometry in one calculation step. Application of this method results in obtaining the geometry with a lower leakage than the geometry with the same external dimensions and teeth being spaced evenly (variant A) or the geometry with slightly increased chambers height (variant B). This problem belongs to the group of geometry-related inverse problems [32,33].
