Labyrinth seal design for space applications

Labyrinth seal design for space applications Josef Pouzar a,* , David Kostal a , Lars-G¨ oran Westerberg b , Erik Nyberg c , Ivan Krupka a a Faculty of Mechanical Engineering, Brno University of Technology, Brno, 61669, Czech Republic b Division of Fluid and Experimental Mechanics, Luleå University of Technology, SE-971 87, Luleå, Sweden c Division of Machine Elements, Luleå University of Technology, SE-971 87, Luleå, Sweden ARTICLE INFO Handling Editor: Prof. L.G. Hultman Keywords: Vacuum evaporation Molecular flow Labyrinth seals Contamination Liquid lubricants Space tribology ABSTRACT Labyrinth seals, extensively used in space applications, serve to prevent the loss of liquid lubricants and shield satellite subsystems from contamination. These seals are essential for the reliable functioning of bearings and for protecting satellite subsystems from contamination. This study compares analytical predictions of lubricant loss against experimental measurements and computer simulations to optimize labyrinth seal configurations. Analytical models tend to overestimate mass loss by 5–8 times compared to experimental data, indicating limited reliability for complex seal geometries. Simulations using MolFlow+and COMSOL Multiphysics align closely with experimental results, providing accurate mass loss predictions. Key findings highlight that labyrinth length, width, and surface roughness are critical factors in minimizing evaporative mass loss. Notably, stepped labyrinth seals with relief grooves and optimized step positioning effectively reduce molecular beaming effects and improve sealing performance compared to straight geometries. Effective sealing not only reduces mission failures but also helps to minimize space debris, thereby promoting safer satellite missions. Abbreviations and Symbols bLabyrinth gap width [mm] P i Inlet pressure [Pa] d, d 1,2 Diameter of annular seal [mm] P o Outlet pressure [Pa] ESTL European Space Tribology Laboratory QFlow rate [mbar⋅mm 3 /s] ETR Evaporation Test Rig Q m Mass loss [g/s] Kn Knudsen number [−]rCircular arc radius [mm] L, L 1,2,3 Labyrinth path length [mm] Ra Average roughness [ μ m] lRecess depth [mm] Sa Arithmetic mean height [ μ m] λMean free path [mm] Sdr Developed interfacial area ratio [%] MLubricant molar mass [g/mol] Sq Squared mean height [ μ m] N 0 , N Number of molecules (inlet, outlet) [−] TAbsolute temperature [K] PFPE Perfluoropolyether TP Transmission probability [−] PFluid vapor pressure [torr] vAverage molecular velocity [m/s] 1. Introduction Space technology operates under demanding conditions, where mechanical components often lack redundancy due to weight and size limitations, thus posing a risk of satellite failure if malfunctions occur during a mission. Tribological failures, resulting from inadequate liquid lubrication, such as lubricant creep [1,2], thermo-catalytic degradation [3], and vacuum evaporation [4–6], are primary contributors to mission failures. There is a growing potential for utilizing greases and base oils in space applications, coupled with a need to extend their service life [7]. However, their utility is limited by extreme temperatures and vacuum evaporation, which pose risks of breakdown and contamination [5,8,9]. High temperatures can induce molecular decomposition, while low temperatures lead to undesirable viscosity increases [5]. Vacuum evaporation occurs when the ambient vacuum pressure reaches the substance vapor pressure [10–12], exacerbating challenges by causing contact drying and potential contamination [8,9], ultimately jeopardizing subsystem performance or leading to complete failure. This concern is compounded by the inclusion of delicate optical elements in scientific missions [13]. In space applications, mechanisms containing liquid lubricants are partially enclosed [14], with lubricants shielded from the open environment using non-contact labyrinth seals. These seals incorporate narrow pathways strategically positioned between inner and outer * Corresponding author. E-mail address: [email protected] (J. Pouzar). Contents lists available at ScienceDirect Vacuum journal homepage: www.elsevier.com/locate/vacuum https://doi.org/10.1016/j.vacuum.2024.113882 Received 12 August 2024; Received in revised form 30 October 2024; Accepted 24 November 2024 Vacuum 232 (2025) 113882 Available online 26 November 2024 0042-207X/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). rotating parts of mechanisms [12,14,15]. The thickness of the labyrinth gap is a critical parameter, balancing safe mechanism operation against flow restriction. Current research overlooks the individual effects of labyrinth geometry and surface topography optimization on reducing liquid lubricant loss [12,16,17]. Further investigation is needed to fully understand and optimize these crucial components. This study examines the performance of labyrinth seals in vacuum environments, aiming to understand and enhance their effectiveness. By integrating analytical models, simulation, and experimental validation, the research offers comprehensive insights into labyrinth seal behavior under varying conditions. Additionally, the study explores the impact of surface topography on lubricant loss and addresses the molecular beaming effect in narrow passages [18,19]. These findings are crucial for optimizing labyrinth seal design to minimize lubricant loss and contamination. By preventing tribological failures and contamination associated with liquid lubricant evaporation, the likelihood of mission failures is significantly reduced, thus decreasing economic and environmental production impacts and mitigating orbital pollution—a threat to current and future space ventures [20]. 2. Material and methods 2.1. Experimental measurements Experimental measurements are conducted to obtain real-world data regarding the performance of the labyrinth seals. These empirical results serve as a crucial benchmark for validating the accuracy of the analytical and simulation outcomes and assessing the practical applicability of the modelled parameters. 2.1.1. Evaporation test rig For the experimental measurement of lubricant loss due to vacuum evaporation, the modified Evaporation Test Rig (ETR) equipment [6] is employed; see Fig. 1, developed specifically for continuous measurement of mass loss resulting from evaporation under vacuum conditions. The setup is based on a balance-scale mechanism, comprising a pivoted scale beam with a labyrinth seal platform on one side and counterweights on the opposite side. The system is equipped with sharp wedges, allowing single-axis rotation, and is balanced on a support beam within a vacuum chamber. A capacitive proximity sensor, fixed to the chamber flange, measures the displacement of the scale pointer during lubricant evaporation. This displacement is used to calculate mass loss over time, accounting for variations in pressure and temperature. The ETR operates under controlled vacuum conditions with externally applied heating. The experimental testing begins after temperature stabilization, which is achieved by maintaining constant thermo-vacuum conditions for 24 h. The conversion ratio between the proximity sensor’s displacement signal and mass loss is 10.0 μ m per 1.0 mg, with a weighing precision of ±0.04 mg under thermo-vacuum conditions. The labyrinth seal platform is modular, facilitating the testing of various labyrinth seal geometries. It consists of a main pan attached to the ETR, housing the liquid lubricant reservoir, and accommodating various labyrinth configurations (Fig. 2) for testing under high vacuum and elevated temperatures. All components of the labyrinth seal platform (main pan, labyrinth seals) are made of aluminum alloy 6061. 2.1.2. Test sample The Evaporation test rig setup ensures precise evaluation of labyrinth seal performance. The chosen geometries for experimental testing are selected to be comparable with the simulation and analytical models. Three specific geometries in Fig. 3 are implemented during the experimental testing, with dimensions specified in Table 1. In addition to the overall geometries, the “LONG”labyrinth seal was prepared in two variations, each with different surface roughness, to allow further evaluation (refer to Chapter 3.2). All labyrinth seal samples were tested with the same oil to ensure consistency in assessing their effectiveness. In this study, a low-outgassing oil, FOMBLIN Y LVAC 25/6 [35], was selected for experimental testing due to its vacuum compatibility; see Table 2. The lubricant test sample is applied to the annulus reservoir of the labyrinth seal test rig pan (Fig. 2). The amount of lubricant applied must be sufficient to achieve an optimal film height covering the entire reservoir area to prevent local drying. However, excessive lubricant may lead to undesired creep through the seal, potentially compromising the experimental evaluation. Prior to experimentation, a series of pre-tests were conducted to assess the thermal conditions of the setup and evaluate creep intensity. 2.1.3. Experimental procedure During the experimental measurements, only one specific labyrinth seal geometry can be evaluated at a time. First, the sample oil is injected into the labyrinth seal reservoir in an amount specified in Table 2. The chosen labyrinth seal geometry (SHORT, LONG, or STEP) is then mounted onto the reservoir using screws, ensuring a secure fit to maintain the labyrinth gap conditions throughout the test. This labyrinth assembly is subsequently attached to the Evaporation Test Rig (ETR), and experimental testing under vacuum conditions is initiated. To facilitate the vacuum evaporation of the oil, experiments are conducted at elevated temperatures. Increased temperatures raise the oil’s vapor pressure, promoting its evaporation. As the oil evaporates, the ETR balance system compensates for the mass loss by tilting. This tilt is measured by a proximity sensor, which translates the tilt data into mass loss using calibrated parameters. All subsequent experiments are conducted under consistent conditions, including a heating temperature of 170 ◦C, a setpoint temperature for the labyrinth seals of 90 ◦C (±5◦C), a setpoint vacuum pressure of 2E-5 mbar, and a duration inside the vacuum chamber of 24 h. These controlled parameters ensure the reproducibility of the results. 2.2. Analytical model In our experiments, evaporated lubricant molecules travel in the molecular flow regime, characterized by extremely low gas densities and long mean free paths between molecules, as described by the Knudsen number [21,22]. Kn =λ L,(1) Fig. 1. Evaporation test rig (ETR) with labyrinth seal platform. J. Pouzar et al. Vacuum 232 (2025) 113882 2 where λis the mean free path of the gas molecules, and Lis a characteristic length of the system, such as the chamber diameter or length [16, 23]. Lower Knudsen numbers (Kn <0.01) indicate viscous or continuum flow, where intermolecular collisions dominate. As the Knudsen number increases above 0.5, intermolecular collisions become rare, and surface interactions prevail [16,23,24]. Under the vacuum conditions used during these experiments, with pressures around 2E−5 mbar, and temperatures around 81 ◦C the Knudsen number (Kn) ranges from approximately 11 to 13. This range places the system firmly in the free molecular flow regime (Kn >0.5), where molecular movement is largely influenced by interactions with the chamber surfaces rather than collisions between molecules [16,23,24]. Within the labyrinth, the molecules temporarily adhere to surfaces before dispersing, losing their previous direction and velocity with each interaction [23]. Evaluating evaporation rates of liquid lubricants in space requires analytical and experimental methods [12,15,25], with analytical models serving as vital tools for predicting lubricant loss due to vacuum evaporation [15,26–28]. To calculate the flow rate or leakage through labyrinth seals analytically, the labyrinth original geometry configuration must undergo a conversion to a linearized state, wherein its diameter, gap width and length dimensions are equivalently represented [15]. This linearized labyrinth representation facilitates the calculation of the flow rate using a combination of pressure gradient, molecule transmission probability through the labyrinth, and average molecular velocity according to Q=(Pi−Po) π dbv (4+1,5 b).(2) Based on a detailed analysis of experimental data from ESTL, a refinement to Eq. (2)has been proposed [12] to offer the most accurate correlation with real-world tests currently available, serving as an expression for the flow of molecules. Importantly, this refinement maintains consistency with the previous approach regarding the process of simplifying analytical computations [15], such that the overall methodology remains coherent and practical for implementation in relevant applications Qm=0,0436 • P π db(M T)0,5 1+0,375 •L b .(3) The ESTL model presented in Eq. (3)serves as the most reliable analytical instruments for predicting the quantity of lubricant molecules leaking through the labyrinth seal. However, these analytical models exhibit inherent limitations [12,15,27] mainly due to their simplified geometrical representations and possibly due to lubricant thermal history and inaccurate vapor pressure determination [17,29,30]. Comparison with experimentally measured results indicates overestimation of the analytical evaporation rates prediction by an order of magnitude [11,12]. Manufacturers typically determine vapor pressure using the Knudsen effusion method, which lacks standardization and potentially yields inaccuracies [11,31]. Proposed correction factors aim to align analytical predictions with real-world scenarios, relying on experimental data [11,29]. Despite widespread adoption, analytical models frequently fall short in accurately predicting evaporated molecule flow rates [6,12]. Addressing these disparities may involve employing numerical simulation tools, such as Monte Carlo random sampling functions [30,32,33]. Monte Carlo simulation stands as a robust computational technique crucial for acquiring numerical approximations of molecular flow behavior within complex systems or processes [19,34]. For predicting the mass of lubricant evaporation through specific labyrinth seal geometries, the ESTL model presented in Eq. (3)is typically utilized, despite its demonstrated lack of accuracy. This analytical model will be employed to compare the predicted evaporation rates with simulation and experimental measurements. This comparative approach aims to highlight the discrepancies between the analytical predictions and the more precise data obtained from simulations and experimental tests, thereby facilitating improvements in the predictive accuracy of lubricant evaporation rates for space applications. 2.3. Modeling and simulation This phase focuses on developing mathematical models to accurately simulate molecular flow within labyrinth seal geometries. The input Fig. 2. Labyrinth seal platform setup and assembly breakdown. J. Pouzar et al. Vacuum 232 (2025) 113882 3 data for the simulations and analytical models are derived from boundary conditions measured during the experimental phase. These include the temperature of the labyrinth seal, the vacuum chamber pressure, and the parameters of the evaporated oil; see Table 2. Since the pressure inside the labyrinth seal reservoir (at the labyrinth seal inlet) cannot be measured directly, it is assumed to be equal to the oil vapor pressure in the simulation models. Through these simulations, critical parameters influencing seal performance are identified and comprehensively described. This serves as the foundation for understanding the dependence between various labyrinth geometry factors and their impact on seal effectiveness in preventing molecular flow. To simulate space conditions and molecular flow regimes within labyrinth seal geometries, two representative software tools are utilized. 2.3.1. MolFlow+ MolFlow +[36] is employed for molecular flow simulation and evaluation of labyrinth seal geometry parameters. It is primarily used for pressure distribution calculations within complex geometries under ultra-high vacuum conditions [36,37]. The software employs Monte Carlo simulation techniques to accurately model particle behavior in rarefied gas environments, making it particularly suitable for analyzing molecular flow. 2.3.2. COMSOL multiphysics For the purpose of comparing simulation results, COMSOL Multiphysics v.6.2 software [38] was integrated into the research methodology. The software accommodates various study types, encompassing stationary and time-dependent (transient) studies, as well as linear and nonlinear studies. In this research, two distinct modules are employed: a) Molecular Flow Module: Simulating kinetic gas flows, this module offers specialized physics interfaces ideal for vacuum systems simulation. It is designed to address kinetic gas flows with Knudsen numbers greater than 0.1, encompassing both the transitional and free molecular flow interfaces. In our experiments, the Knudsen numbers exceeded 11, making this module highly appropriate for our application. This method computes flow by integrating fluxes from all visible surfaces, with dependencies confined to surface variables. b) Particle tracing module: This module provides a versatile tool for tracking particles through various geometries under different forces. The mathematical particle tracing interface allows for flexible formulation of particle motion using Lagrangian or Hamiltonian approaches, ideal for free molecular flow simulations. Special variables offer insights into particle populations and statistical data, aiding in analysis. This module computes quantities like particle count and transmission probability, facilitating the visualization and analysis of particle trajectories. Fig. 3. Labyrinth seal geometries and their experimental setup (SHORT, LONG, STEP). Table 1 Labyrinth seal geometry dimensions. Labyrinth Width Diameter Length [mm] SHORT b =0.5 d =86.5 L =1.5 LONG b =0.5 d =86.5 L =10 STEP b =0.5 d 1 =86.5 L 1 =8.25   d 2 =92.5 L 2 =3.00    L 3 =1.75 Table 2 Oil test sample properties [35]. Lubricant name FOMBLIN Y LVAC 25/6 Lubricant type PFPE Vapor pressure (mbar) 6e-8 (25 ◦C) 6e-5 (100 ◦C) Kinematic viscosity (cSt) 276 (20 ◦C) Viscosity index 113 Molecular weight (g/mol) 3300 Density (g/cm 3 ) 1.90 Applied oil sample: mass (g) 1.8–2 volume (ml) 0.95–1.05 J. Pouzar et al. Vacuum 232 (2025) 113882 4 3. Results The results encompass analytical, experimental, and simulation approaches, which are compared and collectively evaluated. The main evaluation subject is a set of three labyrinth seals (Table 1), featuring precisely manufactured and measured geometries utilized in both analytical and simulation models. For experimental testing, the modified ETR measuring device [6] is employed, along with the labyrinth seal assembly, to continuously measure lubricant mass loss resulting from vacuum evaporation. Subsequent research extends to evaluating the overall geometry’s impact on the labyrinth seal’s effectiveness, as well as exploring the influence of surface roughness and molecular beaming effects occurring in narrow corridors within the molecular flow regime. 3.1. Labyrinth comparison The experimental section initially compares three labyrinth geometries (Fig. 3) through multiple independent measurements. The measured results obtained from the ETR are depicted as displacement signals, which are then converted into lubricant mass loss using calibration ratios. As a result, the measurement outcomes portray a continuous signal of lubricant mass loss over a specified time period for all three types of labyrinth seal geometries; Fig. 4. To further validate the measured mass loss, the experimental setup is weighed both before and after the experiment. The results clearly indicate that more complex geometries exhibit superior sealing capabilities, aligning closely with the expected behavior according to the research [12,14,27]. The temperature and pressure of experimental measurements were recorded and subsequently incorporated into the analytical and simulation models to achieve close correlation. The analytical approach utilized the ESTL model (Eq. (3)), while both the MolFlow+and COMSOL Multiphysics software were employed for simulation. A comparison of the results obtained from all three approaches is shown in Fig. 5 and reveals that the analytical approach yields significantly greater mass loss, potentially due to simplifications inherent in the analytical model. Conversely, the simulation approach closely aligns with experimentally acquired results. When comparing mass loss and relative leak rate, using the experimental results as a reference (see Table 3), the analytical model overestimates the leak rate 5–8 times, while simulations deviate by only 29 % at most. These findings indicate that simulation models offer a better fit for complex geometries, though they may be less accurate for shorter labyrinth gaps. Expanding upon the previous findings, a design of experiment analysis was undertaken to comprehensively explore the principal influencing parameters of labyrinth seals. To facilitate a meaningful comparison of labyrinth parameters, the evaluation of transmission probability (TP) is employed, indicating the ratio of molecules traversing from the inlet to the outlet [23,39]. TP =N N0 ,(4) where Nis the number of molecules at the outlet, and N0is the number of molecules at the inlet. This probability, essential for conductance calculations, offers a convenient metric for comparing labyrinth geometry parameters. Notably, the primary influencing parameters encompass the width of the labyrinth gap and the length of the corridor, as depicted in Fig. 6. These results align with the assumptions derived from previous research. According to the results shown in Fig. 4, the labyrinth seal with a stepped geometry is the most effective sealing solution, largely due to its extended corridor length, identified as the primary influencing parameter in Fig. 6. The longer corridor increases the probability of molecular interactions with the seal surfaces before molecules reach the outlet. Additionally, the stepped structure introduces multiple surfaces at various angles, which increases the likelihood of molecules reflecting off these surfaces rather than passing directly through the seal. This reflective behavior disrupts the direct flow of molecules, significantly reducing transmission probability and enhancing overall sealing performance. The combination of an elongated path and multiple reflection points within the stepped geometry creates a more efficient barrier to molecular flow compared to simpler seal designs. 3.2. Surface roughness In addition to considering the overall geometry of the labyrinth seal, surface topography must also be taken into account, as it can significantly impact conductance results. Current knowledge lacks evidence of surface roughness evaluation in labyrinth seals for space applications. Therefore, a series of experimental tests were conducted to underscore the importance of the surface roughness parameter. Two LONG geometry labyrinth seals produced using a turning machining process and with different surface finishes - one untreated (ROUGH) and the other polished (SMOOTH) - were compared (Table 4). The inner and outer surface roughness of the labyrinth seal was assessed using a Contour GTX 3D optical profilometer (Fig. 7), and the data was subsequently incorporated into the COMSOL Multiphysics simulation environment for molecular flow analysis, considering real surface topography. The analytical model does not include a surface texture parameter; therefore, it will not be compared further with other approaches. The comparison between experimental measurements and simulations Fig. 4. Experimental measurements comparison of mass loss for various labyrinth seal geometries. Fig. 5. Comparison of analytical, experimental and simulation approaches of evaporation mass loss. J. Pouzar et al. Vacuum 232 (2025) 113882 5 reveals a significant influence of surface roughness on the mass loss of evaporated lubricant molecules; see Fig. 8. The intensity of mass loss varies by 14.4 % in experimental measurements and 14.6 % in simulations for the different surface roughness. The results presented have been unified to ensure comparability across the same seal geometries, with one being subjected to a polishing process. Based on the experimental measurements, the sealing efficiency of the SMOOTH labyrinth seal decreased by 14.4 % due to its lower surface roughness, resulting in an overall lubricant loss that was 0.55 mg/h higher compared to the ROUGH labyrinth seal, which featured a higher surface roughness. The increased roughness in the ROUGH seal likely enhanced the interaction between the seal surfaces and the evaporating lubricant molecules, contributing to better sealing performance. Surfaces with higher area roughness exhibit improved sealing effectiveness which can be interpreted in two ways: 1) Increased surface roughness results in approximately a 1.7 % larger area for molecular adhesion (refer to the Sdr parameter in Table 4). 2) Higher surface roughness introduces geometrically complex passages, impeding molecule propagation. 3.3. Molecular beaming effect One of the factors that has not been considered in the optimization of labyrinth seal geometry design is the phenomenon of molecular beaming effect, which occurs in long and narrow tubes. When the impingement rate for the tube facets in the normal direction is higher than for the parallel facets, the angular distribution of molecular velocities is no longer cosine-like, resulting in more molecules traveling along the labyrinth corridor path [19,40]. This beaming effect is always present and should be taken into account in the optimization of labyrinth geometry as it may influence the overall mass loss of lubricant molecules. To mitigate the molecular beaming effect, it is recommended to employ more intricate geometries that redirect the flow of molecules, preventing them from traveling in a straight line. An effective geometry involves a stepped configuration, altering the direction of molecular flow by 90◦. Careful consideration should be given to the placement of the step within the labyrinth to minimize the impact of the molecular beaming effect. To achieve this, a simulation of the original stepped labyrinth seal, used in the experimental measurements, was performed using COMSOL Multiphysics software. Adjustments were made to the step position along the entire length of the labyrinth by performing a parametric sweep across the 10 mm width of the labyrinth in 0.5 mm increments. The results in Table 5 indicate the favorable placement of the step in the middle of the labyrinth seal (Fig. 9), effectively reducing the growing influence of the molecular beaming effect. Given the existence of the molecular beaming effect and the necessity for a stepped geometry in labyrinth seals to effectively diminish its impact, further research into the geometry of the step corner has been conducted. In the corner where the molecular beam strikes the surface and disperses the molecules, the corner geometry is essential for disrupting the spread of molecules through the labyrinth seal. Conventionally, the classical geometry of a manufactured labyrinth seal lacks corner optimization, and the corner is rounded due to the manufacturing process. Consequently, this geometrical shape increases the likelihood of molecules propagating further, necessitating its elimination. To address this issue, a series of local geometrical shapes for the corner were simulated using COMSOL Multiphysics (Fig. 10), and their impact in terms of mass loss reduction was assessed; see Table 6. Based on these findings, it is recommended to incorporate corner shaping into the manufacturing process, such as implementing corner relief grooves. These geometric modifications, designed to be produced using conventional machining processes, contribute to a slight reduction in mass loss while potentially yielding significant lubricant savings over the satellite’s lifetime. 4. Discussion The study aims to explore the fundamental aspects of labyrinth seals used in space mechanisms, focusing on understanding the behavior of liquid lubricant molecules that evaporate in vacuum conditions. To understand how lubricant molecules leak through these seals, we use three methods: experimental testing, analytical analysis, and computer simulation. Table 3 Mass flow evaluation for specific labyrinths using diverse approaches. Approach SHORT LONG STEP [mg/h] relative leak rate [mg/h] relative leak rate [mg/h] relative leak rate Experiment 4.350 1 3.41 1 2.73 1 Analytical 32.81 7,54 19.27 5,65 16.69 6,11 Simulation MolFlow+5.39 1,24 3.24 0,95 2.33 0,85 COMSOL 5.63 1,29 3.58 1,05 2.72 1,00 Fig. 6. Identification of key labyrinth geometry parameters impacting transmission probability (TP). Table 4 Surface roughness analysis for inner and outer components of labyrinth seals. Surface roughness ROUGH SMOOTH Inner Outer Inner Outer Ra [ μ m] a 3.88 1.30 0.13 0.56 Sa [ μ m] 3.70 1.30 0.14 0.56 Sq [ μ m] 4.34 1.60 0.16 0.67 Sdr [ μ m] 1.99 % 5.56 % 0.01 % 4.14 % a Average Ra parameter in the direction of highlighted mid-planes shown in Fig. 7. J. Pouzar et al. Vacuum 232 (2025) 113882 6 Experimental testing is crucial for comparing the outcomes of the analytical and simulation approaches. The main inaccuracies arise from the experimental measurements, so each measurement must undergo a calibration procedure before the actual test. Calibration focuses primarily on temperature distribution measurement, the conversion ratio between proximity sensor distance change and actual weight loss, and achievable vacuum conditions. The conversion ratio calibration for the evaporation test rig is performed before each labyrinth seal modification. The detailed process for this calibration is described in previous research [6]. Fig. 7. Surface roughness analysis for inner and outer components of labyrinth seal LONG geometries using a 3D optical profilometer. Fig. 8. Experimental and simulation investigation of surface roughness impact on labyrinth seal performance. Table 5 Labyrinth corner geometries and their impact on evaporated lubricant loss. Step position (see Fig. 9) Mass loss [mg/h] Loss rate a 10 % 0.665 +5.19 % 25 % 0.643 +1.62 % 50 % 0.633 – 75 % 0.640 +1.24 % 90 % 0.662 +4.71 % a Relative to the reference step position at the midpoint (50 %) of the labyrinth seal. Fig. 9. Mass loss variation in labyrinth STEP geometry influenced by molecular beaming effect on step position. J. Pouzar et al. Vacuum 232 (2025) 113882 7 Temperature is the most critical parameter influencing the evaporation process, as it directly affects the substance’s vapor pressure [12, 41]. During the experiments, it is not possible to measure the exact temperature of the oil sample inside the labyrinth seal test rig. To address this, we perform a thermal calibration procedure before experimental testing, where reference temperatures are measured at specific locations (see. Fig. 1) and at the main pan of the labyrinth seal platform (see Fig. 11). The measured temperature difference of approximately 20 ◦C between the labyrinth seal and the support beam holding the evaporation test rig is primarily due to the differing mechanisms of heat transfer affecting each component. The support beam is directly connected to the heated vacuum chamber, which allows for efficient heat conduction. In contrast, the labyrinth seal experiences heating predominantly through radiation. The presence of very limited conductive paths makes radiative heat transfer the dominant mechanism, which is less efficient compared to conduction. It was experimentally verified in a dedicated test with additional thermocouples that the lubricant surface temperature is equal to the reservoir temperature and the labyrinth seal gap surface, due to convection effects. Despite initial thermal calibration, the oil surface temperature remains estimated, as thermocouples cannot be placed in the pan during evaporation tests without compromising weighing precision. The test rig for the labyrinth seals and the oil are maintained at a temperature of 92.0 (±0.3) ◦C, which is dependent on the geometry. Once the substance starts to evaporate at specific temperatures and pressures, it cools down as a result of the evaporation process. The temperature subsequently stabilizes at a new value depending on the evaporation rate of the cooling liquid and the external heating. This derived temperature introduces the primary inaccuracy in both the analytical and simulation approaches. Nonetheless, during the experiments with various labyrinth seals, the thermal conditions remained consistent, with a maximum deviation of only 1 ◦C; see Fig. 12. The vapor pressure of the liquid lubricant is estimated using the Clausius-Clapeyron approximation [17,42], making it entirely dependent on the known vapor pressures provided by the manufacturer. However, the Knudsen effusion method used by manufacturers for vapor pressure determination lacks standardization. Consequently, the vapor pressure values in the oil datasheet might be inaccurately determined, leading to discrepancies in experimental and other results [11,31]. To address this issue, it is advisable to test the oil’s vapor pressure using the Knudsen effusion method prior to the experimental measurements. However, in our experiments, vapor pressures were sourced from the datasheet, which could potentially influence the results obtained through analytical and simulation approaches. Although the exact deviation in vapor pressure measurement is unknown, the simulation results align well with the experimental data. The pressure of the vacuum chamber is measured outside the labyrinth seal test rig as it can’t be directly measured inside the labyrinth seal. Therefore, the exact pressure in the lubricant reservoir at the inlet to the labyrinth seal is assumed to be equal to the vapor pressure of the liquid lubricant. The pressure inside the vacuum chamber was monitored during the experiments and remained constant at 2E-5 (±1E-6) mbar. At the set oil temperatures, the oil’s vapor pressure is estimated to be 3.3E-5 (±7E-7) mbar, slightly higher than the chamber’s vacuum pressure. While this difference is small, the primary indicator of the evaporation process is the measured change in distance, which follows a mostly linear trend and only occurs once oil evaporation begins. Other significant disruptive factors in the experimental Fig. 10. Local geometries in the corner of labyrinth seals with stepped geometry. Table 6 Labyrinth corner geometries and their impact on evaporated lubricant loss. Corner geometry Characteristic parameter Loss reduction Classical – – Circular arc r =b 3.8 % Relief groove type G a 3.9 % Dead end l =2b 4.3 % a ISO 18388:2016. Fig. 11. Thermal calibration analysis for labyrinth seal test rig experiments. Fig. 12. Temperatures of support beam during the experiments for various labyrinth seals. J. Pouzar et al. Vacuum 232 (2025) 113882 8 measurements stem from the design of the labyrinth seal test rig. Unwanted movement of the lubricant in the test rig may occur during sample handling due to the absence of a porous reservoir. Additionally, temperature and pressure variations over time during the experiment can lead to lubricant creep. The oil may creep along the surface, through the labyrinth seal gap, and even through the tightening contact surfaces. This could lead to increased weight loss due to the greater oil surface area, thus increasing evaporation intensity. During the calibration and experimental testing, lubricant creep was addressed through visual inspections before and after the experiments; however, it was not observed at all. In actual space applications, labyrinth seals incorporate anti-creep barrier surface coatings with low surface tension [43] to prevent lubricant creep through the seal. However, these films were not used during the experiments because visual inspections did not detect any creep. Additionally, using barrier films could have distorted the results, especially when measuring the effect of labyrinth seal surface roughness. The inaccuracies of the analytical evaporation model for labyrinth seals stem from several simplifications. The model simplifies the complex geometry of the labyrinth seal, thereby overlooking the behavior of molecules in the molecular flow regime and the emergence of the molecular beaming effect. Another simplification involves approximating the labyrinth transmission probability, which should ideally be calculated differently for each type of geometry. These simplifications cause the analytical model to be less accurate in predicting lubricant loss during the vacuum evaporation process, with accuracy varying based on the seal’s geometry; see Fig. 5. Discrepancies between simulation models and experimental measurements may arise mainly from the real oil temperature and assumption of a uniform temperature distribution across the seal. Other limitations of the simulation model stem from surface roughness generation, mesh finesse, pressure distribution, and the numerical model used. Despite these challenges, the solid simulation model remains the closest to experimental measurements. Therefore, it is recommended to simulate each labyrinth seal assembly with an optimized simulation model and calibrated initial conditions to accurately predict lubricant mass loss over time. 5. Conclusion The vacuum evaporation intensity of space oils through labyrinth seals was investigated using experimental, analytical, and simulation approaches. The study underscores the need to refine analytical models, as they tend to overestimate mass loss by 5–8 times compared to experimental results. It also demonstrates a strong correlation between simulation results and experimental data. The simulation tools MolFlow+and COMSOL Multiphysics are recommended for accurately predicting oil evaporative mass loss. Key findings regarding the influence of labyrinth seal geometry on molecular flow include: •Stepped labyrinth seal configurations provide enhanced sealing performance. A comparison of two labyrinth seals—one featuring a straight, long corridor and the other a stepped configuration—reveals that the stepped seal, which is 30 % longer, should theoretically provide a 16 % improvement in sealing effectiveness due to its increased length. However, the stepped design actually achieved a 20 % improvement, indicating additional benefits from the stepped geometry itself. •Molecular beaming effects within labyrinth seals contribute to oil mass loss. This effect can be significantly reduced by using a stepped configuration with the step positioned at the midpoint of the seal, which best distributes the molecular beaming effect. Conversely, placing the step near the inlet or outlet can negatively impact sealing performance by increasing the mass loss rate by up to 5 %. •Surface roughness impacts molecular transmission probability. A comparison between two seals, with average surface roughness values of Ra 2.6 and Ra 0.3, shows that the seal with greater roughness achieves a 14.4 % improvement in sealing performance over the smoother seal. •Local geometrical adjustments, such as adding relief grooves, can enhance sealing efficiency. Depending on the specific modification, sealing effectiveness can improve by up to 4 %. CRediT authorship contribution statement Josef Pouzar: Writing –original draft, Visualization, Validation, Software, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. David Kostal: Writing –review &editing, Supervision, Methodology, Conceptualization. Lars-G¨ oran Westerberg: Writing –review &editing, Supervision, Conceptualization. Erik Nyberg: Writing –review & editing, Methodology, Conceptualization. Ivan Krupka: Supervision, Resources, Funding acquisition. Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Josef Pouzar reports financial support was provided by European Space Agency. Ivan Krupka reports financial support was provided by European Union. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This research was supported by the activity “Effect of local geometrical changes and polarization of labyrinth seal surfaces on the evaporation rate of liquid lubricants in space applications”, funded as a Discovery element contract 4000139889 by the European Space Agency; and by the project “Mechanical Engineering of Biological and Bio-inspired Systems”, funded as project No. CZ.02.01.01/00/22_008/ 0004634 by Programme Johannes Amos Comenius, call Excellent Research. 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