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Labyrinth seal: A labyrinth seal consists of several annular sealing teeth arranged in sequence around the rotating shaft; gaps and expansion chambers are formed between these teeth. As the fluid to be sealed passes through these intricate gaps, a throttling effect occurs, which helps to prevent leakage. Due to the gap between the rotor and the casing in a labyrinth seal, there is no solid contact, thus lubrication is not required, and thermal expansion is allowed; this makes it suitable for applications involving high temperatures, high pressures, and high rotational speeds. This type of seal is widely used for sealing the shaft ends and stages in turbines, gas turbines, compressors, and blowers, as well as as a preliminary seal for other dynamic seals. 1. Sealing mechanism of labyrinth seals: The mechanism by which fluid encounters resistance as it passes through the labyrinth, resulting in a reduction in its flow rate, is known as the “labyrinth effect”. For liquids, there are fluid dynamics effects, including hydrodynamic friction effects and flow convergence effects ； For gases, there are also thermodynamic effects, namely the heat conversion that occurs in the labyrinth as a result of compression or expansion ； In addition, there is also the “ventilation effect,” among others. The labyrinth effect is a combined result of these effects; therefore, the mechanism of labyrinth sealing is very complex. (1) Frictional effect: As the leaking fluid flows through the labyrinth, the friction resulting from the viscosity of the liquid slows down the flow rate and reduces the leakage volume. In simple terms, the friction along the flow path and the local losses in the fluid constitute the drag effect; the former is related to the length and cross-sectional shape of the channel, while the latter is related to the number of bends and the geometry of the labyrinth. Generally, when the flow channel is long, has sharp bends, and sharp corners, the resistance is high, the pressure drop loss is significant, and the leakage amount decreases. (2) Beam contraction effect: As the fluid passes through the labyrinthine openings, it contracts due to inertia, resulting in a reduction of the cross-section of the flow beam. Let the area of the orifice be A; then the minimum area of the converged flow jet is CcA, where Cc is the contraction coefficient. At the same time, the velocity of the gas after passing through the orifice also changes. Assuming that the flow velocity under ideal conditions is u1, the actual flow velocity is smaller than u1. Let Cd be the velocity coefficient; then the actual flow velocity is u1 = Cd·u1. Therefore, the flow rate through the orifice will be equal to q = Cc·Cd·A·u1, where Cc·Cd = α (the flow coefficient). The flow coefficient at the labyrinth seal opening is related to the shape of the gap, the shape of the tooth tips, and the roughness of the wall surface. For incompressible fluids, it is also related to the Reynolds number ； For compressible fluids, it is also related to the pressure ratio and Mach number. At the same time, it also affects the flow state before the seam. Therefore, in labyrinths with complex configurations, the flow coefficient of one opening cannot be taken as that of all openings. According to the tests, the flow coefficient is lower for the first stage, while it is higher for the slits in stages subsequent to the second stage; generally, the flow coefficient is taken as 1. However, the flow coefficient of serrated teeth is less than 1, around 0.7, while that of rounded teeth is close to 1; usually α is taken as 1, which results in an overestimated leakage amount. (3) Thermodynamic effects: An ideal labyrinth flow channel model is composed of a series of annular tooth gaps and inter-tooth cavities. The flow of gas as it passes through each tooth gap and inter-tooth cavity can be described as follows: at the entrance to the gap, the gas has conditions of p0, T0, and zero velocity; the closer the gas gets to the entrance, the more the flow contracts and accelerates, and shortly after reaching the narrowest part of the gap, the flow attains its maximum velocity ； Upon entering the cavity, the flow velocity cross-section suddenly expands, resulting in strong vortices forming within the cavity. From an energy perspective, before and after the gap, the pressure energy of the airflow is converted into kinetic energy. At the same time, as the temperature drops (the heat enthalpy value h decreases), when the gas enters the annular chamber between the two teeth at high speed, its volume expands suddenly, resulting in intense vortices. As a result of eddy friction, the vast majority of the kinetic energy of the airflow is converted into thermal energy, which is absorbed by the airflow within the chamber and raises its temperature. The enthalpy then returns to a value close to that before entering the gap; only a small portion of the kinetic energy continues to move into the next gap at residual speed, and this process repeats step by step. (4) Ventilation effect: In an ideal labyrinth, it is assumed that the kinetic energy of the airflow passing through the gaps in the expansion chamber is entirely converted into heat energy. In other words, it is assumed that the asymptotic velocity at the next seam is zero, but this holds only when the expansion chamber is particularly wide and long. In a typical straight-through maze, since the airflow passing through the slit can only spread in one direction, the conversion of this kinetic energy into thermal energy cannot take place effectively within the expansion chamber. On the side with smooth walls, the speed of some of the gas does not decrease or decreases only slightly, and it flows directly over the tops of the teeth toward the low-pressure side. This phenomenon of passing by in such a manner is known as the \"air leakage effect\". 2. Structural types of labyrinth seals: Depending on the structure of the sealing teeth, labyrinth seals are divided into two main categories: sealing sheets and sealing rings. The sealing sheet has a compact structure; during operation, when it comes into contact with the casing, it can bend to both sides, reducing friction, and it is also easy to replace. The sealing ring is composed of 6 to 8 fan-shaped pieces, which are installed in the housing and the rotating shaft; spring plates are used to press each piece of the ring against the housing, with a pressing force of around 60 to 100 N. When the shaft comes into contact with the gear ring, the gear ring springs back automatically to avoid friction. This type has larger dimensions and is more complex to manufacture; once the teeth wear out, the entire sealing ring must be replaced, which is why it is not as widely used as the sealing ring design. 3. Calculation of leakage in an ideal labyrinth Given the following conditions: (1) The leaking gas is an ideal gas, and the Joule-Thomson effect is not taken into account; that is, the enthalpy of the gas depends only on temperature ； (2) Assume that the maze is a series of consecutive slits, with the expansion chamber between two adjacent slits being large enough ； (3) The flow through the slit acts as an adiabatic cyclic expansion; here a flow coefficient α is used ； (4) The flow velocity energy after passing through the slit is completely restored to a constant temperature in the expansion chamber due to isobaric conditions; therefore, the velocity just before each slit approaches 0, meaning no air leakage occurs. 4. Characteristics of the through-type labyrinth: Since it is easier to machine grooves or teeth of various shapes on the surface of the shaft than inside a hole, holes are often left with smooth surfaces, and together with a shaft having grooves or teeth, they form a labyrinth. This is the through-type labyrinth. Due to its ease of fabrication, the through-type labyrinth is the most widely used. However, the through-type labyrinth suffers from air leakage, with a leakage rate higher than that of an ideal labyrinth. Factors affecting labyrinth characteristics: (1) The effect of teeth. Tests conducted abroad have shown that, at a constant pitch, the greater the number of teeth, the less leakage there is. When the tooth pitch is changed, the larger the tooth pitch, the more sharply the leakage rate decreases, and it also helps to reduce the impact of air leakage. (2) Effect of the expansion chamber. Experimental studies on the impact of expansion chamber depth have been conducted abroad, with the conclusion that a shallower expansion chamber is beneficial for reducing leakage. Based on the observation of the flow conditions in the expansion chamber, it is considered that the vortices in the shallow expansion chamber are unstable. Since vortices can quickly deplete energy, the asymptotic velocity in the expansion chamber decreases, thereby reducing leakage. (3) The influence of the auxiliary chamber. The so-called “auxiliary chamber” refers to an additional groove made in the smooth surface of a straight-through labyrinth; once such a groove is created, the flow pattern within the labyrinth changes significantly immediately. Tests have shown that, as long as the position of the auxiliary chamber is appropriate, the rate of reduction in leakage is quite significant. 5. Clearance of labyrinth gas seals: Except in special cases, labyrinth gas seals are generally used in turbomachines such as turbines and gas turbines. Its radial clearance should be selected based on the following factors: bearing clearance, manufacturing tolerances and assembly errors, deformation of the components (such as casting shrinkage and out-of-roundness), rotor deflection, as well as the amplitude at the critical rotational frequency, thermal expansion and the resulting deformations. In many cases, the effect of thermal expansion is most prominent. Therefore, the changes in the dimensions of individual components during startup and shutdown, as well as the relative displacement of these components, must be estimated in advance. Static and dynamic finite element methods can be used to determine the law of thermal expansion over time, thereby identifying the critical conditions and the appropriate size of the gap. Key points in labyrinth seal design. Summarizing the experience accumulated in labyrinth seal design, the following key points can be identified: (1) Try to convert the kinetic energy of the airflow into thermal energy, so that the residual velocity does not enter the next gap. An appropriate distance should be maintained between the teeth, or high-low teeth can be used to force a change in the direction of the airflow. The tooth spacing is generally 5 to 9 mm. (2) The sealing teeth should be made as thin as possible and have sharp corners. The tip thickness should be less than 0.5 mm; when it occasionally comes into contact with the shaft during operation, the tip wears out first and loses contact, thereby preventing local overheating of the shaft due to friction and avoiding accidents. (3) Due to the large leakage rate of labyrinth seals, care must be taken to prevent environmental contamination when sealing flammable, explosive, or toxic gases. An inflatable labyrinth seal is used, with an inert gas introduced into the gap; the pressure of this gas is slightly higher than that of the gas to be sealed ； If the medium does not allow air inclusion, a vacuum-type labyrinth seal can be used.