Proceedings M2D2022 - 9th International Conference on Mechanics and Materials in Design Funchal/Portugal 26-30 June 2022, Editors J.F. Silva Gomes and S.A. Meguid, pp.1035-1050 https://paginas.fe.up.pt/~m2d/proceedings_m2d2022/ - 1035 - PAPER REF: 19120 COMPUTATIONAL FLUID DYNAMICS OF THE LUBRICANT FLOW IN THE SEALING GAP OF ROTARY SHAFT SEALS Jeremias Grün(*), Simon Feldmeth, Frank Bauer University of Stuttgart - Institute of Machine Components, Germany (*)Email: jeremias.gruen@ima.uni-stuttgart.de ABSTRACT The lubricant flow in the sealing gap significantly affects the sealing behavior of rotary shaft seals. Computational fluid dynamics (CFD) are applied here to analyze these fluid flows. A decisive input parameter is the sealing gap height respectively the lubricant film thickness. The film thickness is estimated on the basis of several analytical elastohydrodynamic lubrication (EHL) equations. The results for sealing characteristics such as the pumping rate and the friction torque derived from the numerical analyses are compared and discussed with the results obtained from test rig studies. This allows the validity of the applied methods to be verified. Keywords: tribology, rotary shaft seals, computational fluid dynamics (CFD), elastohydro- dynamic lubrication (EHL), film thickness prediction. INTRODUCTION Rotary shaft seals have been used for decades to seal shaft passages on a wide variety of industrial aggregates. The complex tribological system rotary shaft seal shown in Figure 1 involves the sealing ring itself, the sealing counterface (shaft) and the fluid to be sealed (lubricant). Under operation, the sealing gap filled with lubricant builds up in the contact area between the sealing edge and the shaft. An overview of the fundamentals of tribology and the function of rotary shaft seals can be found in (Bauer, 2021b, 2021a). The modeling and simulation of rotary shaft seals comprises a complex, coupled fluid-structure interaction problem with a number of submodels on multiple scales. There are various approaches available for this purpose. For instance, (Stakenborg, 1988; Stakenborg et al., 1990) investigated the radial deformation and contact pressure distribution based on ideal two-dimensional models of rotary shaft seals applying finite element analysis (FEA). (Wenk et al., 2016; Grün et al., 2021a, 2021b) extended this approach to three-dimensional models and investigated the contact behavior of rotary shaft seals in FEA considering measured surface data. EHL simulations were generally used to investigate the flow in the sealing gap of rotary shaft seals. A review of theoretical aspects especially with respect to EHL modeling can be found in (Salant, 2010). For this purpose, especially the Reynolds equations are evaluated, whereby the inertia and volume forces are neglected. For the analysis of micro structured sealing lips, (Keller et al., 2020) evaluated the complete Navier-Stokes equations in CFD. Furthermore, there is a lack of studies of the lubricant flow in the sealing gap of rotary shaft seals using CFD. This paper presents an approach to simulate the lubricant flow in the sealing gap of rotary shaft seals applying CFD. The sealing edge geometry initially deformed during mounting and operation and the sealing gap height serve as input for the modeling. The deformed sealing edge geometry can be determined by FEA following (Grün et al., 2021a, 2021b). Analytical EHL Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1036 - film thickness equations summarized in (Wheeler et al., 2016; Marian et al., 2020) provide an estimate of the sealing gap height and thus the radial deformation of the sealing edge surface at different operating points. Fig. 1 − Elastomer sealing ring in the sealing system rotary shaft seal. Computational Domain A decisive step in modeling and simulation of the flow in the sealing gap is the definition of the computational domain. In the present study, FE simulations provide the input for the subsequent CFD analyses. A detailed description of the FE modeling and simulation of rotary shaft seals is given in (Grün et al., 2021a, 2021b). In this approach, a segment of the sealing ring is considered. After mounting on the shaft and applying the spring load, the shaft starts to rotate, distorting the sealing edge surface. Figure 2 illustrates the modeling process in this study. In addition to the deformed sealing edge surface, the previous FEA provides the structural mechanical normal force F. Together with the minimum gap height h_0, this yields the fluid domain for the subsequent discretization. Fig. 2 − Principle process of modeling. Figure 3 summarizes the FE modeling and visualizes the FEA results. In contrast to (Grün et al., 2021a, 2021b), the present study considers a synthetic sealing edge roughness similar to the approach of (Wennehorst and Poll, 2017). (Wang and Schipper, 2020) adopt a similar approach to approximate roughnesses. The surface roughness can be described via ���, �� � Δ sin 2 � �� � � Δ� sin �2 � ��� � (1) where � and �� are the effective wavelength in circumferential direction � and in axial direction �. According to (Griffiths, 2001), the effective wavelengths are given by � ,� � 2 � ����,� , (2) Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1037 - where � is the root mean square slope in the circumferential direction � and �� is the root mean square slope in the axial direction �. �� is the root mean square roughness of the sealing edge surface. The amplitudes can be determined according to (Baglin, 1986) via Δ � Δ� � √2 ��. (3) Roughness measurements on sealing edge surface similar to (Grün et al., 2021b) yield a root mean square roughness of �� � 1.66 μm and a root mean square roughness slope in circumferential � and axial � direction of � � 87.8° and � � 89.7° as input data for surface roughness modeling. Fig. 3 − Illustration of the FE modeling and simulation. After modeling and solving the structural mechanics of the rotary shaft seal, the inputs required for CFD can be determined in FE post-processing. These include the coordinates of the surface nodes ℎ��, �� of the deformed sealing edge surface and the values for the contact pressure *��, �� at the corresponding nodes in the contact area. Starting from the shaft surface, all surface nodes of the sealing edge are exported with the pressure data up to a height of ℎ��, �� + 5 -m. The computational domain considered in the CFD, thus results as: ℎ./0��, �� � ℎ��, �� � ℎ1, (4) where ℎ1 is the minimum gap height respectively the lubricant film thickness. Analytical EHL equations serve to estimate the gap height ℎ1 in this study. Material Properties The lubricant data and the material parameters are essential for determining the sealing gap height via film thickness equations and for the definition of the fluid in CFD. Operating parameters such as temperature significantly affect the lubricant in the sealing gap of rotary shaft seals. (Feldmeth et al., 2016) presented a method for estimating the contact temperature Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1038 - of rotary shaft seals. This allows the temperature in the sealing gap to be estimated for given operating conditions, friction conditions, fluid data and process-related settings. The pressure-viscosity coefficient 23 and the dynamic viscosity 4 are considered temperature- dependent in the following. In the experimental studies to validate the numerical results, an additive-free mineral oil (FVA 3 reference oil, ISO VG 100) specified in (Forschungsvereinigung Antriebstechnik e. V., 1985) is used. The dynamic viscosity can be described as a function of temperature via 4�5� � 461 78 61°9:;9<; <= 3, (5) according to (Forschungsvereinigung Antriebstechnik e. V., 1985), whereby 2 � >1 ? >@ 5 � >A 5A. (6) The temperature dependence of the pressure-viscosity coefficient is given by 23�5� � ?0.555 ln�5� � 4.0095, (6) which results from curve fitting based on the data sheet values from (Forschungsvereinigung Antriebstechnik e. V., 1985). Figure 4 and Figure 5 show the temperature dependence of the dynamic viscosity 4 and the pressure viscosity coefficient 23. Fig. 4 − Dynamic viscosity vs. temperature. Fig. 5 − Pressure viscosity coefficient vs. temperature. In the numerical analyses in this study, the rotary shaft seal is numerically investigated at six operating points. The lubricant temperature in the oil sump is 5EFGH � 80 °C. The shaft has a diameter of J � 80 mm. Table Table1 lists the applied lubricant data at the considered operating points. The contact temperature 5KLM is estimated from (Feldmeth et al., 2016). Based on the contact temperature, the dynamic viscosity 4 and the pressure viscosity coefficients 23 in the sealing gap can be calculated via equations (5) – (7). Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1039 - Table 1 − Lubricant parameters at different operating conditions. rotational speed N/�min:@� circumferential velocity P/�m s:@� contact temperature 5KLM/�°C� dynamic viscosity 4/�Pa s� pressure viscosity coefficient 23/�Pa:@� 500 2.1 82.4 0.0147 1.56 ⋅ 10:T 1000 4.2 89.6 0.0118 1.51 ⋅ 10:T 2000 8.4 103.8 0.0081 1.43 ⋅ 10:T 3000 12.6 117.3 0.0059 1.37 ⋅ 10:T 4000 16.8 130.2 0.0044 1.31 ⋅ 10:T 5000 20.9 142.5 0.0036 1.26 ⋅ 10:T Film Thickness Equations In the present study, six different analytical expressions provide the basis for estimating the sealing gap height. The gap height ℎ1 assumed in the following is the central lubricant film thickness ℎK. For this purpose, (Dowson and Higginson, 1960; Dowson et al., 1962) introduced the material parameter V � 23 W′, (8) the velocity parameter Y � Z [\]^�, (9) and the load parameter _ � `\] ^�a, (10) with the pressure-viscosity coefficient 23, the dynamic viscosity 4 and the hydrodynamic effective velocity �[ . The normal load b results from the pressure distribution after mounting shown in Figure 3. The reduced Young’s modulus is given by 1W′ � 12 �1 ? c@AW@ � 1 ? cAAWA �. (11) The radii of curvature in circumferential direction � and axial direction � result according to (Karami et al., 1987) via d ,� � 1√2 �� �� ,�2 � �. (12) Based on experimental results obtained by tribological tests and numerical results, (Hamrock and Dowson, 1977) determined the central lubricant film thickness ℎKd � 2.69 Y1.ef V1.6g _:1.1ef h1 ? 0.61 7:1.f6 ^�^��i.jkl. (13) For low elastic moduli (Hamrock and Dowson, 1978) modified the equation to ℎKd � 7.32 Y1.em _:1.AA �1 ? 0.72 7:1.AT ^�^���. (14) Based on the parameters given in (8) - (10), (Moes, 1965) introduced the modified load parameter n � _ �2 Y� ok, (15) and the viscosity parameter p � V �2 Y�qk. (16) (Evans and Snidle, 1981) took this up and suggested the analytical expression for central lubricant film thickness ℎKd �2 Y�1.6 � 1.7 n:1.1Ae p1.m1. (17) Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1040 - (Chittenden et al., 1985) extended previous approaches and introduced the central lubricant film thickness ℎKd � 4.31 Y1.eT V1.mr _:1.1fg s1 ? 7:@.Ag ^�^��aot. (18) (Nijenbanning et al., 1994) applied a multilevel solver to estimate the central lubricant film thickness ℎKd �2 Y�1.6 � s�uvw gA � �uxw:m � u11:m�:gT�A yg � �uxw:T � u11:T�:yTt @y (19) by curve fitting. They distinguished the asymptotic regimes rigid isoviscous uvw � 145 1 � 0.796 z@m@6�:@6f z:@ n:A, (20) elastic isoviscous uxw � 3.18 1 � 0.006 ln�z� � 0.63 zmf�:@m@6 z: @@6 n: A@6, (21) rigid piezoviscous uv{ � 1.29 �1 � 0.691 z�:Ag pAg, (22) and elastic piezoviscous ux{ � 1.48 1 � 0.006 ln�z� � 0.63 zmf�: fA1 z: @Am n: @@A pgm, (23) where z � d d� (24) is the ratio of the reduced radii of curvature and the parameters | � 1.5 1 � 7:@.A}~�}��� (25) and u11 � 1.8 z:@ (26) ensure a smooth transition between the asymptotic solutions. (Masjedi and Khonsari, 2015) solved the modified Reynolds equation, the surface deformation and elasto-plastic asperity contact equations simultaneously to analyze the film thickness and asperity load ratio. Based on curve fitting to the results, the central lubricant film thickness is given by ℎKd � 3.672 Y1.eeg �i.ia� V1.61A �i.ijk _:1.1m6 �i.q� �1 ? 0.573 7:1.fm ��. (27) Figure 6 shows the calculated lubricant film thicknesses and gap heights at the defined operating points using the models presented. The lubricant data from Table 1 and the structural parameters from Table 2 serve as input data. The structural parameters are assumed to be independent of temperature. The equations according to (Chittenden et al., 1985; Nijenbanning et al., 1994; Masjedi and Khonsari, 2015) provide comparable lubricant film thicknesses. (Evans and Snidle, 1981) yields comparatively high and (Hamrock and Dowson, 1977) the lowest lubricant film thicknesses Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1041 - . Table 2 − Structural parameters. Young’s modulus of the seal W@ 8.04 ⋅ 10e Pa Poisson's ratio of the seal c@ 0.5 ? Young’s modulus of the shaft WA 210 ⋅ 10e Pa Poisson's ratio of the shaft cA 0.3 ? root mean square roughness (sealing edge surface) �� 1.66 ⋅ 10:e m root mean square roughness slope in circumferential direction � � 1.53 ? root mean square roughness slope in axial direction � �� 1.56 ? effective wavelength in circumferential direction � � 6.81 ⋅ 10:e m effective wavelength in axial direction � �� 6.66 ⋅ 10:e m radius of curvature in circumferential direction � d 0.50 ⋅ 10:e m radius of curvature in circumferential direction � d� 0.48 ⋅ 10:e m normal force (0.1°-segment) b 0.0054 N Fig. 6 − Lubricant film thickness vs. speed. Meshing A Script serves for automatic discretization of the lubricant film in the sealing gap. It generates a structured computational grid consisting of hexahedral cells. The principle flow of the script is shown in Figure 7. The main function addresses various subroutines during the run. The input for the meshing algorithm consists of an initial surface, described by node coordinates and the sealing gap height ℎ1. The three N � N� matrices �, � and � define the surface coordinates. Based on the input of the sample points in � and � direction N and N�, the initial surface mesh is interpolated (refineMesh.m). The number of nodes in � direction is defined by the number of cell layers N�. The function getNodes.m is used to set up the node matrix ���⋅��⋅���<@��g containing all node coordinates of the computational grid. Input data are the three interpolated surface coordinate matrices �, � and � and the number of cell layers N� in � direction. The node matrix is constructed in three nested loops as shown in Algorithm 1. Each loop represents one dimension. Starting from the origin, a row of node coordinates in the � direction is determined in the inner loop. At the end of each row, the � coordinate of the next row is determined in the middle loop Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1042 - and the inner loop are executed again. Once the coordinates of one layer are determined, the index is increased in the outer loop and the next layer is built up again by the middle and inner loops. Based on the node matrix �, the algorithm constructs the hexahedral cells of the computational grid and declares the respective quadrilateral cell faces according to the node indices. Figure 8 illustrates the node numbering of a single grid cell and the adjacency between the cells, defined by the cell faces. Three functions serve to declare the topology of the computational grid. One sets up the cell matrix �. Two others declare the inner and outer cell faces ��M� and ���� including adjacency relationships. Fig. 7 − Flow chart of automated meshing. Fig. 8 − Node numbering and topology of the grid cell. Fig. 9 − Face declaration of a general hexahedron. Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1043 - Algorithm 1: getNodes.m Input: �, �, �, n� Output: � /* Initialization */ i � 1 /* Setup of the node matrix */ for k � n�: ?1: 0 /* Loop over layers */ for n � n�: ?1: 1 /* Loop over y coordinates */ for m � n�: ?1: 1 /* Loop over x coordinates */ /* Node coordinates */ x � ��m, n� y � ��m, n� z � ��m, n� ∙ �1 ? 1/n�� ��i, : � � �x y z  i � i � 1 end for end for end for The function getCells.m links the nodes to hexahedral cells and thereby determines the respective cell nodes. The input parameters are N , N� and N�. Starting from the origin, Algorithm 2 assembles the cells first in � direction, then in � direction and finally in � direction in three nested loops. Output data is the cell matrix ��¡�T, where NK is the number of cells in the computational domain. Algorithm 2: getCells.m Input: n�, n�n� Output: ¢, nK /* Initialization */ i � 1 nK � �n� ? 1�£n� ? 1¤n� /* Setup of the cell matrix */ for k � 0: �n� ? 1� for n � 1: �n� ? 1� for m � 1: �n� ? 1� /* Cell nodes */ nd�,@ � n � �m ? 1� ⋅ n� � k ⋅ n� nd�,A � nd�,@ � 1 nd�,g � n � m ⋅ n� � k ⋅ n� nd�,m � nd�,g � 1 nd�,6 � nd�,@ � n� nd�,e � nd�,A � n� nd�,f � nd�,g � n� nd�,T � nd�,m � n� ¢�i, : � � ¦nd�,@ nd�,A nd�,g nd�,m nd�,6 nd�,e nd�,f nd�,T§ i � i � 1 end for end for end for As shown in Figure 8, each quadrilateral cell face has at least one adjacent cell. The adjacency definition is based on the Cartesian law system. The right adjacent cell ¨© is relatively located in the respective positive axis direction and the left ¨ª in the negative one. A distinction is made between inner and outer cell faces. While the inner cell faces are located within the fluid domain and have one right and one left adjacent hexahedron each, the outer cell faces have only one Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1044 - left or one right adjacent cell, depending on their location. Furthermore, the outer cell surfaces serve to define the necessary boundary conditions for the simulation. Normal vectors describe the orientation of the cell faces in local coordinate systems. For this purpose, Figure 9 introduces the face declaration. The maximum component of the normal vector on the respective plane defines the face designation. This means that the face with maximum � component is denoted by «. Analogously, the normal vectors with maximum � and � components result in the faces ¬ and ­. The function getIntFaces.m aims to determine the inner cell faces of the computational grid with the two respective adjacent cells. The input is the cell matrix �, the number of cells NK and the number of nodes N , N� and N� in �, � and � direction. Algorithm 3 shows the setup of the inner cell faces « . The inner cell faces ¬ and ­ can be determined analogously. For this purpose, the loop structure shown is repeated for each of the other two spatial directions. In each run of the inner loop, the corresponding face nodes are extracted from the cell matrix �. As shown in Figure 8, the cell nodes 2, 4, 6, and 8 constitute the cell face «® of cell ¯. The right adjacent cell ¨�,© corresponds to the current cell ¯ and the left adjacent cell ¨�,ª to the previous cell ¯ ? 1. The result is the matrix b�°±²�e, containing the N�M� inner cell faces with their respective adjacent cells ¨�,© and ¨�,ª. Algorithm 3: getIntFaces.m Input: ¢, nK, n�, n� n� Output: ³�M�, n�M� /* Initialization */ i � 1 j � 1 n�M� � nK/n��n� ? 1� µ�n� ? 2�£n� ? 1¤ � £n� ? 2¤�n� ? 1�¶ n� /* Setup of the inner cell faces */ for k � 1: n� for n � 1: nK/n� for m � 1: nK/n� ? 1 /* Face declaration */ F � C�j, �2 4 8 6  cª � j /* Left neighbor */ c© � j � 1 /* Right neighbor */ ³�M��i, : � � �F c© cª  i � i � 1 j � j � 1 end for j � j � 1 end for j � n� ⋅ k � 1 end for The function getExtFaces.m serves to specify the outer cell faces of the hexahedrons at the borders of the computational domain. The main script transfers the cell matrix �, the total number of cells NK, the number of cell layers N� and the number of nodes N , N� and N� in �, � and � direction to the subroutine shown in Algorithm 4. The function performs a case distinction to determine the position of the outer boundary faces. The orientation of the normal vector ¹ is passed for this purpose. Algorithm 4 considers the outer cell faces with normal vector �¹º. The outer cell faces at the remaining boundaries of the computational domain are determined analogously. Only the initialization of the indexing of the cells and the cell faces differs. Algorithm 4 determines the face nodes of cell face «® and the two adjacent cells ¨© and ¨ª. As the normal vector �¹º is oriented positively and is located on a cell face at the border of the Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1045 - computational grid, there is no right adjacent cell. Therefore, the right adjacent ¨© is assigned a 0. Output parameters is the outer cell face matrix ����,<». Algorithm 4: getExtFaces.m Input: C, nK, n�, ±¹ Output: ³��� switch(±¹� case(�¹º� /* Initialization i � 1 j � n� ? 1 for k � 1: n� for m � 1: n� ? 1 F � ¢�j, �2 4 8 6 � cª � j /* Left neighbor */ c© � 0 ./* Right neighbor */ ³����i, : � � �F c© cª  i � i � 1 j � j � �n� ? 1� end for j � nK/n� ⋅ k � �n� ? 1� end for end case case(?¹º� … end case … end switch After the subroutines set up the matrices describing the grid topology of the computational domain, the main script writes the matrices to a FLUENT grid file (*msh). This file format has the advantage that it can be imported into many common CFD applications. A detailed description of the structure of the file format can be found in (ANSYS, 2021). Figure 10 shows the discretized computational domain of the lubricant film in the sealing gap of a rotary shaft seal. Boundary conditions The pre-processing and solving of the computational domain is performed with ANSYS CFX 19.1. For this purpose, the generated FLUENT grid file is imported into CFX. The material definition depends on the temperature and consequently on the circumferential speed. The lubricant data for the operating conditions considered are listed in Table 1. The boundary cell faces on the air and fluid sides are defined as boundary conditions "openings". This boundary condition type is appropriate when the complete description at this position is not known in advance. For example, the pressure at the corresponding position is known, but the direction of the flow is not. The top boundary cell faces correspond to the interface to the solid sealing edge surface and the bottom to the interface to the solid shaft surface. Hence, they are defined as "walls". This boundary condition type represents an impermeable boundary for the fluid flow. The bottom wall (shaft surface) is subjected to the circumferential velocity �[ . The top wall (sealing edge surface) is stationary. The considered computational domain represents a segment of the lubricant film, analogous to the structural simulations in Figure 3. Consequently, the boundary cell faces are coupled in circumferential direction via a translational periodic interface. Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1046 - Fig. 10 – Computational domain. RESULTS The plausibility of the simulation results is checked by comparison with experimental results. For this purpose, the two sealing characteristics pumping rate ½[ and friction torque ¾ have become established (Bauer, 2021a. In the numerical analyses, the sealing edge surface is rough and the shaft surface is ideally smooth. In order to satisfy this assumption in the test rig experiments as accurately as possible and to be able to achieve comparable results, polished shafts provide the sealing counterfaces. The shafts employed have an arithmetic mean deviation of the roughness profile d¿ � 0.00649 … 0.0163 -m, a maximum height of the roughness profile d� � 0.0876 … 0.248 -m and a maximum peak-to-valley height of the sampling lengths on the roughness profile dG¿� � 0.142 … 0.322 -m. The pumping rate is measured experimentally as described in (Kawahara and Hirabayashi, 1979). The sealing ring is mounted inversely so that fluid is constantly available on the actual air side and can be pumped through the sealing gap. The difference between the mass flow over the cell faces on the fluid side and the mass flow over the cell faces on the air side results in the pumping rate in the simulated lubricant film segment. The computed total pumping rate of the rotary shaft seal results from the extrapolation to the circumference of the shaft. Figure 11 compares the computed pumping rates with the measured pumping rates. The presented pumping rates of each experiment result from the mean value over five measured values. The pumping rates for the different lubricant film thickness equations differ only slightly. The computated pumping rates are lower than the measured pumping rates. The calculated as well as the measured pumping rates show a linear dependency over the circumferential velocity. Thus, the presented model qualitatively approximates the correlation between circumferential speed and pumping rate, but quantitatively shows differences. One reason for the differences between measurement results and numerical results is that the polished shaft surface is quite smooth but does not correspond to the model idea of an ideally smooth surface. This leads to different hydrodynamics in the lubricant film and can affect the pumping rate. The friction torque is determined in additional experiments according to (Schulz et al., 2018). The friction torque in the lubricant film segment is obtained by multiplying the simulated tangential force in the segment under consideration and the shaft radius. The calculated friction torque of the rotary shaft seal results from the extrapolation to the shaft circumference. Figure 12 shows the measured and calculated friction torques versus circumferential speed. Similar to the pumping rates, there are only marginal differences in the friction torques for the different lubricant film thicknesses. The friction torque from the experimental tests is higher than the calculated torque. The friction torque curves are fitted based on rational models. The computed Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1047 - friction torque converges to a constant value after a rapid increase. The rational model used has a numerator and a denominator polynomial of the first degree. The friction torque measurements also show an initial rapid increase followed by a slight decrease in the friction torque over the circumferential velocity. The curve fitting employs a rational model with a first- degree numerator polynomialand a second-degree denominator polynomial. This allows a better approximation of the decrease in the friction torque at higher circumferential speeds. Despite the quantitative differences, the numerical model qualitatively shows the same physical behavior. The lubrication modes provide an explanation for the quantitative differences. While the CFD model applied here only considers the case of full-film lubrication, mixed and boundary lubrication can also occur in the experimental tests. This increases the friction. Another reason for the differences is that the present model does not consider the churning torque in the fluid chamber. As described in (Feldmeth et al., 2013), this can have a significant influence on the measured friction torque. Fig. 11 – Pumping rate vs. circumferential velocity. Fig. 12 − Friction torque vs. circumferential velocity. In summary, although the model provides quantitatively different values, it can qualitatively describe the operating behavior of rotary shaft seals. The main benefit of the computed results is that they provide an insight into the microscopic flow processes in the sealing gap of rotary shaft seals. Figure 13 shows the pressure distribution in the lubricant film in the sealing gap. Here, the sealing gap height is ℎ1 ≅ 30 nm based on the film thickness equation given in (17) at �[ ≅ 12.6 m s:@. The hydrodynamic pressure has an asymmetric pressure distribution in axial � direction. The maximum pressure is oriented to the fluid side. Pressure peaks arise in the areas in front of the compressed asperities of the sealing edge surface, where the minimum gap height is located. In the valleys of the roughness, the pressure decreases significantly. These pressure differences cause vortices in the flow in the lubricant film, as shown in Figure 14. In the areas where the sealing gap height is comparatively high, the streamlines are oriented in the circumferential direction and the velocity corresponds to the relative velocity between the sealing edge and shaft surface. At the points with the minimum sealing gap heights and the pressure peaks, the fluid is retarded, vortexed and thus axially deflected. This effect causes the active back-pumping mechanism of rotary shaft seals in operation, according to (Kammüller, 1986; Müller, 1987). Proceedings of M2D2022 conference, Funchal/Portugal 26-30 June 2022 Symp-9: Advances in Tribological Systems - 1048 - Fig. 13 – Pressure distribution in the sealing gap. Fig. 14 – Lubricant flow in the sealing gap. CONCLUSIONS This paper deals with Computational Fluid Dynamics (CFD) of the fluid domain respectively the lubricant film in the sealing gap of rotary shaft seals. The deformed sealing edge surface and the pressure distribution after mounting obtained from Finite Element Analysis (FEA) serve as input for the modeling of the lubricant film. A script is presented which automatically discretizes lubricant films with different geometries based on an initial surface and the minimum gap height. The minimum gap height is estimated via established lubricant film thickness equations. The lubricant data required as input is estimated for the different circumferential speeds. The discretization and the generation of the computational grid are described in detail. Experimental results are compared with the numerical results and discussed on the basis of the sealing parameters pumping rate and friction torque. The numerical model qualitatively shows the same physical behavior as the real sealing system, but underestimates the pumping rate and the friction torque. 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