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Shaft roughness effect on elasto-hydrodynamic lubrication of rotary lip seals: Experimentation and numerical simulation
Tribology International 88 (2015) 218–227
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Shaft roughness effect on elasto-hydrodynamic lubrication of rotary lip seals: Experimentation and numerical simulation M’hammed El Gadari a,n, Aurelian Fatu b,1, Mohamed Hajjam b,1 a b
Université Moulay Ismail, ENSAM Meknès, Morocco Institute Pprime, Department GMSC, University of Poitiers, France
art ic l e i nf o
a b s t r a c t
Article history: Received 21 November 2014 Received in revised form 16 February 2015 Accepted 8 March 2015 Available online 28 March 2015
Numerical analyses of the isothermal elasto-hydrodynamic lubrication (EHL) have made considerable advances in order to identify the most important features in the successful operation of rotary lip seal, and the results have shown a good agreement with experiments. Most of the models previously published are capable of predicting the combined effects of thin film through deformed lip and rotating shaft, but they assume a smooth surface of the shaft. Although this assumption is only verified for shaft roughness much smaller than that of the seal lip, it is the best solution to avoid a transient model. First, the present study describes an experimental work that provides a basis upon which a numerical EHL model of rotary lip seal is constructed by taking into account both the shaft and lip roughness. After confirming the validity of the current model by comparing experimental with numerical results, simulations have been performed and have underlined the effect of shaft roughness amplitude and profile on the rotary lip seal performance. It is shown that for shaft roughness beyond half of the lip roughness, the seal may leak. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Elasto-hydrodynamic Roughness Leakage Friction torque
1. Introduction The rotary lip seal is the most common type of rotary shaft seals (Fig. 1). It is used to withstand differences in pressure, to contain lubricant and to exclude contaminants such as air, water and dust particles. The functioning of radial lip seal is based on the formation of an oil film between the lip and the shaft. As described by Kammüller [1], elastomer lip seals operate by adopting a viscous inverse pumping regime, generated by the micro-geometry of the lip. It is shown that the pumping flow strongly depends on lip roughness [2] or microundulations [3–5]. In installed conditions, the lip geometry of a successful radial lip seal leads to a contact approach angle at the oil side significantly larger than the contact approach angle at the air side. Therefore, the resulting contact pressure profile has a maximum near the oil side of the contact. The frictional shear stress generated by the shaft rotation leads to a maximum tangential displacement that coincides with the maximum radial contact pressure. Therefore, the skewed asperities act like micro pumps that could pump oil into the contact from either side, but with a higher potential pumping capacity
n
Corresponding author. Tel.: þ212 666909018; fax: þ212 535467163. E-mail addresses: [email protected] (M. El Gadari), [email protected] (A. Fatu), [email protected] (M. Hajjam). 1 Tel.: þ33 545251979; fax: þ33 545670237. http://dx.doi.org/10.1016/j.triboint.2015.03.013 0301-679X/& 2015 Elsevier Ltd. All rights reserved.
at the air side. In steady state non-leaking conditions, a hydrodynamic equilibrium is established and a meniscus separates the liquid from the air side of the seal [6]. Starting with Jagger’s work [7,8], the shaft is assumed smooth compared to lip roughness. Typically, the arithmetic average roughness Ra of the shaft is 10 times less than the lip surface. However, practical experiences reflected in the Rubber manufactures Association (RMA) standard on upper and lower limits on the shaft surface roughness indicate that the shaft roughness influences significantly the rotary lip seal performances. Referring to Horve book [9]: “if the shaft is too rough or too smooth, the seal will leak”. Recently, experimentations have shown that not only the shaft roughness amplitude (measured by Ra) affects the lip seal performance, but also that the profile is important [10–12]. The first numerical study that takes into account both lip and shaft roughness has been presented by Salant and Shen [8]. The authors first presented a hydrodynamic model to determine the effects of the shaft surface micro geometry without considering material deformation on the lip surface. Later, they incorporated the elastic deformations of the lip into the model [13] and then they added an asperity contact model [4]. The model has been successfully used to predict the performance characteristics of a lip seal, such as load support, contact area ratio, cavitation area ratio, reverse pumping and average film thickness. In the present study, a numerical isothermal full film lubrication model is developed and validated by experimentations so as to investigate the shaft roughness effect on rotary lip seal behavior.
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Nomenclature shaft roughness amplitude [μm] lip roughness amplitude [μm] width contact [mm] compliance matrix (radial and circumferential respectively), [mm3/N] Ca instantaneous friction torque [N m] Ca friction torque averaged [N m] C10, C01 Mooney Rivlin coefficients [MPa] D Universal function, describing the pressure “p” in active zone, and the difference between replenishment and film thickness “rf–h” in non active zone D vector of nodal values of the universal variable Da shaft diameter [mm] F cavitation index (F¼1 when p 40) else (F¼0) h film thickness [mm] h0(¼havg) average film thickness [mm] h1 shaft sinusoidal roughness [mm] h2 lip sinusoidal roughness [mm] Nx peak number according to circumferential direction Ny peak number according to leakage direction NEx finite element number according to circumferential direction NEy finite element number according to leakage direction nep number of gauss points A1 A2 b C1, C2
Lip Metal insert
Garter spring Fig. 1. Rotary lip seal.
The model is an alternative of the one proposed by Salant and Shen. The differences come from a different numerical treatment of the Reynolds equation and also from the solutions used to obtain the elastic deformations generated by the hydrodynamic pressure. Salant and Shen used finite volumes method to discretize the Reynolds equation, while the present model is based on the finite elements method. The influence coefficient (or compliance) matrix is obtained here by considering a tridimensional behavior of the lip close to the contact, while Salant and Shen considered a two-dimensional axisymmetric model. It must be mentioned that the present model does not take into account the asperity contact, but the studied functioning conditions lead to full film lubrication conditions, and consequently, there is no contact observed between the lip and shaft roughness.
2. Experimental part 2.1. Test ring description Fig. 2 shows the experimental device used to test the seals. A two-phase asynchronous motor of 1.1 kW is used to control the rotational speed of the spindle. A Plexiglas tank contains the liquid
Nnk NN p ps R R Ra Rsk Sku S t T U x, y, z λ( ¼λx) λy μ ρ ρ0 τ δ δx δz
219
interpolation function relative to k node total nodes number film pressure [MPa] contact static pressure [MPa] shaft radius [mm] residual vector of the discretized modified Reynolds equation arithmetic roughness [mm] skewness measurement kurtosis measurement right hand side (RHS) vector member of modified Reynolds equation time [s] calculation cycle (where T¼ λ/U) [s] shaft velocity [mm/s] cartesian coordinates [mm] wavelength according to circumferential direction [μm] wavelength according to leakage direction [μm] lubricant dynamic viscosity [Pa s] lubricant density [Kg/m3] air density [Kg/m3] shearing according to circumferential direction [MPa] interference between shaft and lip [mm] lip circumferential displacement [mm] lip radial displacement [mm]
to seal. The tank is mounted on a needle bearing and is blocked in rotation by a torque-meter. This way, the friction force generated in the sealing zone is directly measured by the torque-meter. The Plexiglas tank is installed on a micrometric mobile support in order to obtain a very precise adjustment of the seal. The calibration of the torque meter is carried out by using different suspended masses and varying their attachment position according to the torque meter axis. From measurements, the accuracy of torque meter, which is 0.036 N m, can be deduced. The temperature measurements are carried out by type K thermocouples. A first thermocouple is placed inside the oil tank. Two additional thermocouples are used to measure the temperature of the seal near the contact zone. Sub-millimeter holes are made on the seal and the thermocouple is positioned as near as possible to the contact surface. The mean value measured by these two thermocouples will be referred to as the film temperature. The oil used is TOTAL ACTIVA 5000 15 petrol. Fig. 3 shows the variation of the oil dynamic viscosity with the temperature. The oil density is 816 kg/m3 at the ambient temperature (20 1C).
2.2. Seal description Experiments are performed on classical lip seals (material: fluorocarbon elastomer, Paulstra-Hutchinson, reference 702619), designed to be mounted on an 85 mm shaft. The radial force is measured for two different seals with and without the elastic spring. The measurements are shown in Table 1 and have been carried out on a laboratory test device according to DIN 3761-9. Shaft and lip roughness have been measured by an optical interferometry device. The results presented in Fig. 4 show that the shaft and lip topography do not have a Gaussian distribution (Sku43), with a profile that has more cavities for both surfaces (Rsko0). It must be noted that the shaft roughness amplitude (measured by Ra) is here 10 times smaller than the lip roughness amplitude, which is in accordance with the seal manufacture specifications.
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Testshaft sleeve
Acquisition unit measurements
spindle torquemeter Electric Motor Lip seal test Adjustabletable
Drilling to incorporate thermocouples Plexiglas tanks Thermocouples
Spindle
Torque meter
tested
Driving belt
Adjustable table
Motor Frame
Fig. 2. (a) Photograph of the test bench and (b) layout of the test bench.
Table 1 Values of the static load for two lip seals.
Fig. 3. Experimental measurement of the oil dynamic viscosity as a function of the temperature.
2.3. Experimental results The friction torque and film temperature measurements were taken for each rotational shaft speed from 500 rpm to 4000 rpm, increasing the velocity with step of 500 rpm. For each speed level,
Part
Radial load without spring [N]
Radial load with spring [N]
Rotary lip seal
10.1 10.2
17 16
the stabilization of the seal temperature is waited before moving to the next level. It must be mentioned that, for all operating conditions, the seal functions without any leakages. Fig. 5 shows an example of friction torque and temperature variation at 500 rpm over a period corresponding to the time needed for the film temperature stabilization. Fig. 6 contains plots of friction torque and film temperature in thermal steady state condition versus rotating speed of shaft. As expected, the film temperature increases with the shaft rotating speed. The friction torque slightly increases between 500 and 1500 rpm. Between 1500 and 2500 rpm it is almost constant and it slightly decreases for shaft speeds higher than 2500 rpm. However, the friction torque variation is low when the shaft rotating speed varies. The results presented in this paragraph will be used later to compare with the isothermal EHL numerical model described in the next section.
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Fig. 4. (a) Lip roughness: Ra ¼ 1.01 μm, λavg_x ¼ 18.88 μm, λavg_y ¼ 31.0 μm, Rq ¼ 1.12 μm; Sku ¼5.26; Rsk ¼–1.21. (b) Shaft roughness: Ra ¼ 0.09 μm, λavg_x ¼ 21.04 μm, λavg_y ¼ 42.25 μm, Rq ¼ 0.14 μm; Sku ¼ 9 ; Rsk ¼ 1.8.
seal and the rod and therefore solving Reynolds equation in order to find pressure and film thickness distribution into the contact. Fig. 7 shows a schematic diagram of a typical lip seal and the sealing zone. It is assumed that x is the circumferential direction and y is the axial direction – the shaft curvature is neglected. The upper stationary surface represents the lip surface, while the lower moving surface represents the shaft surface. The seal operates at a constant rotational speed U, the viscosity of the lubricant m is constant and the asperity contact is not considered in this full film lubrication model. The Reynolds equation in a Cartesian coordinate system is ∂ ∂ ∂h ∂h 3 ∂p 3 ∂p h þ h ¼ 6μU þ 12μ ð1Þ ∂x ∂x ∂y ∂y ∂x ∂t where p is the film pressure and h represents film thickness such as hðx; y; t Þ ¼ h2 ðx δx ; yÞ h1 ðx; y; t Þ þ h0 þ δz ðx; y; t Þ Fig. 5. Variation of the film temperature and the friction torque versus time.
with h1 the shaft roughness, h2 the lip roughness, δx the circumferential lip displacement, δz the radial lip displacement and h0 the average film thickness. Eq. (1) can be solved only for the full film zones. In order to take into account both active and non-active film zones, the modified Reynolds equation proposed by Hajjam and Bonneau [13] is used: ∂ ∂ ∂h ∂h ∂D ∂D 3 ∂D 3 ∂D F h þF h ¼ 6μU þ 12μ þ 6μð1 F Þ U þ 2 ∂x ∂x ∂y ∂y ∂x ∂t ∂x ∂t ð3Þ where D is a universal variable and F a cavitation index: D ¼ p; D Z 0 in the full film zone : F ¼1 in the cavitated ðnon activeÞ film zone :
Fig. 6. Variation of the film temperature and friction torque according to the shaft velocity.
3. Numerical model 3.1. Flow equations The evaluation of sealing performances referring to leakage and friction is carried out supposing a lubricated contact between the
ð2Þ
D ¼ r h; D o 0 F ¼0
ð4Þ
ð5Þ
where r ¼ ρh=ρ0 is the so-called effective film thickness, ρ represents the density of the lubricant–gas mixture and ρ0 is the density of the lubricant. As shown in References [15,16], the boundary conditions used for the solution of Eq. (3) are reduced to outside boundaries of the domain: at y¼b, p is equal to the sealed fluid pressure and at y¼0, p is equal to the air pressure. In the circumferential direction x, periodicity boundary conditions are imposed. Once the pressure and film thickness distribution in the contact are obtained, the hydrodynamic friction torque is computed from: Z λZ b 1 ∂p U Ca ¼ R h þ μ dx dy ð6Þ 2 ∂x h 0 0
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Fig. 7. Rotary lip seal mounted on shaft.
Fig. 8. The FE model of the seal.
The instantaneous leakage is calculated on the air side contact boundary from: Z λ 3 h ∂p dx ð7Þ Q¼ ∂y 12 0
3.2. Elastic deformations As it can be easily understood from the previous section, the evaluation of the pressure in the contact leads to an elastohydrodynamic lubrication problem: there is a strong link between the pressure, the lip elastic deformation and the film thickness. The first difficulty in modeling elastomeric seals is attributed to the non-linear stress–strain elastomer elastic behavior. However, in rotary lip seals, the radial strain imposed by the fluid film in contact is much smaller in comparison with the normal strain imposed by the seal/shaft interference. Therefore, it can be assumed that the radial load that will be supported by the lubricated contact is equal to the static radial load computed by the integration of the contact static pressure imposed by the seal/ shaft interference. It is also supposed that the width of the contact will not be modified by the radial strain imposed by the presence of the fluid film in functioning conditions. Consequently, a structural non-linear mechanic analysis must be carried out in order to predict the contact static pressure field and the contact width due to the mounting of the lip on the shaft. Fig. 8 shows the Finite Elements (FE) model of the seal. The lip is meshed with axisymmetric stress elements. The computations are made in large displacement and deformation hypotheses. The shaft is usually made in more rigid material (typically steel) than the elastomeric seal. Consequently, it is reasonable to consider the shaft as rigid. The seal material is considered as an incompressible,
hyperelastic material and approached with the Mooney–Rivlin model (C01 ¼2.49 MPa and C10 ¼0.5 MPa).2 Fig. 8 also shows the predicted contact static pressure and the contact width. The numerical integration of the contact static pressure gives 17 N – in concordance with the measurements shown in Table 1. The axial contact width b defines the study domain length in the axial direction. The second length λ is chosen equal to the lip surface roughness periodicity in the circumferential direction which, as shown in Fig. 4, is quite similar to the shaft wavelength. As the radial strain imposed by the fluid film in contact is small in comparison with the radial strain imposed by the seal/shaft interference, it makes it possible to use a linear elastic perturbation of the mounted seal in order to compute that the additional elastic deformations generated by the fluid film presence. Therefore, once the non-linear FE computation described above is made, the deformed geometry of the seal is kept and used to compute two (radial and tangential) compliance matrixes that can only be computed if the seal elastic behavior is considered as linear (Hookean model) and the computations are made in small displacement and deformation hypotheses. The method used to compute the compliance matrix has been previously presented in [15]. The lip is considered to have, along a height d, a tridimensional (3D) behavior. The elastic deformation of the lip is treated by the FE method using elements with 20 nodes
2
The Mooney–Rivlin model expresses the strain energy potential as 2 W ¼ C 10 ðI 1 3Þ þ C 01 ðI 2 3Þ þ ð J el 1 =DÞ where J el is the elastic volume ratio, C10, C01 and D are material parameters. I1 and I2 are the first and second deviatoric strain invariants, respectively. By analogy with the classical Hookean model, D is related to the material bulk modulus K¼2/D. If the material is considered incompressible (Poisson ratio ν ¼ 0.5) D is nil.
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variable. One term of the [M] matrix can be written as M jk ¼
npg ne X X n¼1m¼1
2
Fig. 9. Junction between the two-dimensional and three-dimensional meshes.
for the 3D part and eight nodes axisymmetric 2D elements for the rest of the seal structure (Fig. 9). In order to take into account the global periodicity hypothesis, rigid beams connect the two faces of the 3D domain, giving the same displacement for the connected nodes. Two compliance matrixes [C1] and [C2] are calculated. [C1] is used to compute the radial displacement and [C2] is used to compute the circumferentially tangential displacement. Thus, the radial and circumferential lip nodal displacements can be computed with the following equations: ðδ z Þi ¼
NN X
ðC 1 Þi;j pj psj
ð8Þ
ðC 2 Þi;j τj
ð9Þ
j¼1
ðδ x Þi ¼
NN X j¼1
where pj is the nodal film pressure, psj is the nodal contact static pressure, τj is the nodal shear stress and NN is the number of nodes. The reader must keep in mind that the radial compliance matrix [C1] can only be used to predict the lip deformation generated by the difference between the hydrodynamic film pressure (pj) and the contact static pressure (psj). 3.3. Numerical treatment In this paper, the FE formulation proposed by Hajjam and Bonneau [15,16] is used to deal with the EHL lubrication presented problem. The numerical algorithm is illustrated in Fig. 10. Two different problems are formulated: Problem 1. The film thickness is known and the active/inactive film zone separation coordinates are searched. Problem 2. The active and the inactive film zones are known; the film pressure and thickness, which verify the Reynolds equation, and the elastic equations, are searched. Both problems must be solved at each step. To solve Problem 1 we must solve the modified Reynolds equation (Eq. (3)). Problem 2 is solved using the classic Reynolds equation (Eq. (1)). Both equations are processed using the Galerkin approach. Previous studies show that, for EHL problems, the best domain discretization uses quadratic eight node elements. The particular form of the modified Reynolds equation, when it is applied to the cavitated film zones needs, for its discretization, the use of four node linear elements. Depending on whether Problem 1 or 2 is considered, the discretization uses four or eight nodes, the value of the central node parameters is obtained by interpolation. Finally, in the case of Problem 1 (Problem 2 is similar), a linear system of algebraic equations in D, is obtained: R ¼ ½M D þ S
ð10Þ
where R is the residual vector of the discretized modified Reynolds equation and D is the vector of nodal values of the universal
3 nne nne X ∂Nmj ∂N mk ∂Nmj ∂N mk ∂N mj hm X þ Fk N ð1 F k Þ 6μ k ¼ 1 ∂x ∂x ∂y ∂y ∂x mk k¼1 !
nne 1 X N mj N mk ð1 F k ðtÞÞ ΔΩm Δt k ¼ 1
ð11Þ
where m represents one of the ngp Gauss points on n elements and nne the number of nodes per element. Nmj is the weight function relative to the j node, while Nmk is the interpolation function relative to k node. Fk represents the status of k node and takes the value 1 if it is in the active zone and 0 in the opposite case. The vector S is the RHS member of modified Reynolds equation. The j term of S can be written as npg ne X X ∂hm hm ðtÞ hm ðt ΔtÞ Sj ¼ þ2 N mj U Δt ∂x n¼1m¼1 þ2
nne 1 X N N ðð1 F k ðt ΔtÞÞDk ðt ΔtÞÞΔΩm Δt k ¼ 1 mj mk
ð12Þ
The system of equations describing Problem 1, non-linear in p and h is solved through the Newton–Raphson method. The convergence criterion is the equality between the applied load and the hydrodynamic computed load at 1%. The structural mechanical analysis simulates the rotary lip seal assembly in dry condition, by taking into account the shaft radius R¼42.5 mm and the interference δ¼1.1 mm. The width of the contact is b¼0.088 mm. Fig. 11 shows the static contact pressure distribution over the hydrodynamic study domain. The maximum pressure is 2.4 MPa. For all the computational results presented afterward, the sealed pressure is 0.016 MPa when the air side pressure is nil. In the cavitation zones, the pressure is 0.1 MPa. The wavelength, according to circumferential direction, is λx ¼0.02 mm. A mesh sensibility parametric study leads us to choose 12 quadratic elements along the circumferential direction and 20 along the direction of the leakage. This choice is a compromise between the accuracy of the solution and the computing time. Using more accurate meshes leads to the same results concerning the maximum pressure and the minimum film thickness. Due to the consideration of the roughness on both lip and shaft surfaces, the problem is time dependent. However, due to the periodicity boundary conditions, the results present a time periodicity, depending on U and λ.
4. Numerical versus experimental comparison As described in the previous paragraph, the numerical model is isothermal. In order to compare with the experimental results, the oil viscosity used by the numerical model is approximated by using the temperature results presented in Fig. 6 and referring to the viscosity temperature shown in Fig. 3. Table 2 shows the oil dynamic viscosity corresponding to each shaft rotational speed, in thermal steady state conditions. The confrontation between the numerical simulations and the experimental measurements was made for several rotational speeds of the shaft: from 500 rpm to 4000 rpm with a 500 rpm step between each level of speed. As a first approximation already made in previous studies [3,4,8,14–16], it is chosen to define the roughness with simple analytical functions (Eq. (13) and (14)). The asperities on the lip and shaft are represented by a two-dimensional cosine function. However, we have respected measurements of the arithmetic roughness and also the wavelengths in leakage (λy ¼ 42.25 μm for
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shaft roughness and λy ¼31.0 μm for lip roughness) and circumferential direction (λx ¼20 μm for both surfaces):
Lip roughness: h2 ðx; y; t Þ ¼
A2 2π 2π N x2 ðx δx ðx; y; t ÞÞ N y2 y cos 1 cos λ b 2
ð13Þ
Shaft roughness:
A1 2π 2π N x1 ðx tU Þ N y1 y cos h1 ðx; y; t Þ ¼ 1 cos λ b 2
ð14Þ
where Nx1 ¼1 is the number of peaks of the shaft roughness according to x, Nx2 ¼ 1 is the number of peaks of the lip roughness according to x, Ny1 ¼ 2 is the number of shaft peaks along the y direction, Ny2 ¼3 is the number of lip peaks along the y direction, Read data: shaft and lip roughness, viscosity, shaft velocity, compliances matrix Initialization: computation h(x,y,0) and p(x,y,0) Loop 1: Do for each step “t” Loop 2: Do until stability of domain h(x,y,t) and p(x,y,t) Loop 3: Do until stability of cavitation domain Compute “D” (Modified Reynolds equation) Update the cavitation boundaries End loop 3 Loop 4: While residues (h, p)>ε (Newton-Raphson method) Compute h(x,y,t) and p(x,y,t) Update elastic displacement: δx and δz End loop 4 End loop 2 Write: pressure, film thickness, leakage, power loss, friction torque End loop 1 End of algorithm
A1 ¼0.1 μm is the shaft roughness amplitude and A2 ¼1 μm is the lip roughness amplitude. Fig. 12 shows the lip and the shaft profiles that correspond to Eqs. (13) and (14). Fig. 13 shows the variation of the friction torque versus the shaft rotational speed. The friction torque values are calculated by integrating the instantaneous friction torque over the time period T: Ca ¼
1 T
Z
T 0
C a ðt Þdt;
where : T ¼
λx U
ð15Þ
According to Fig. 13, it can be concluded that the numerically predicted friction torque gives a correct concordance with experimental measurements and both considered cases (smooth or rough shaft) are within the gap of experiments validity. This result gives a global validation of the current numerical model and also shows the low impact of the shaft roughness on the friction torque when the roughness amplitude is 10 times smaller than the lip one. To highlight the effect of the shaft roughness on friction torque, Fig. 14 shows, at 2000 rpm, the contribution of the Couette and Poiseuille terms. It can be noted that, for both cases (rough and smooth shaft), the Couette contribution is more important than the Poiseuille contribution. Fig. 15 shows that, the predicted average film thickness slightly increases for the rough shaft case. This E 3% increase of the film thickness does not have an important effect on the Couette term (proportional to the ratio between U and h), and consequently, the global friction torque is not really influenced by the considered shaft roughness amplitude.
Fig. 10. EHL algorithm.
5. Effect of shaft roughness on seal performance This section numerically explores the effect of the shaft roughness on the seal performances. By varying both the roughness amplitude and profile, friction torque and reverse pumping are investigated. The functioning characteristics are the same with the above presented lip seal at 2000 rpm. Table 2 Values of the oil dynamic viscosity depending on the shaft rotational speed. Shaft speed (rpm)
Dynamic viscosity (Pa s)
500 1000 1500 2000 2500 3000 3500 4000
0.0625 0.0375 0.025 0.02 0.0175 0.0175 0.01 0.01
Fig. 11. Adopted size mesh (contact pressure illustrated).
Fig. 12. (a) Lip roughness profile and (b) shaft roughness profile.
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Fig. 16. Shaft roughness profile defined as SH#1.
Fig. 13. Numerical and experimental friction torque variation with the shaft rotational speed.
The used lip profile is given by Eq. (9) (represented in Fig. 12 a) with A2 ¼ 1 mm, Nx2 ¼1 and Ny2 ¼3. Fig. 17 plots the torque friction and leakage versus time corresponding to three time periods T ¼λx/U. It can be noted that the shaft roughness amplitude influences the dynamic behavior of the global parameters of the lip seal. The increase of the roughness amplitude leads to significant fluctuations of the flow. Moreover, it can be observed that, for amplitudes higher than 0.4 mm, the seal will leak – the leakage rate became greater than zero. Concerning the friction torque, the increase of the roughness amplitude induces an increase of the film thickness that results in a decrease of the shear stress. However, for a shaft amplitude A1 ¼0.4 mm a sudden increase of the friction is predicted. In this case, the film thickness is locally very thin, which leads to a severe increase of the friction and of the pressure gradient. This result can be observed in Fig. 18 that compares the film thickness and the pressure fields for the smooth and two rough shaft configurations. 5.2. Effect of the shaft roughness profile
Fig. 14. Comparison between “Poiseuille” and “Couette” torque for two cases: rough and smooth shaft.
In order to underline the shaft roughness profile effect, 4 sinusoidal forms are studied: SH#1 (represented in Fig. 16), SH#2 given by Eq. (16) and represented in Fig. 19a), SH#3 given by Eq. (14) with Ny1 ¼4 (Fig. 19b)) and finally SH#4 also given by Eq. (14) with Ny1 ¼6 (Fig. 19c)). 2π ðx tU Þ ð16Þ h1 ðx; y; t Þ ¼ A1 sin λ For all these shaft roughness profiles, A1 ¼0.1 mm. Fig. 20 represents the friction torque and the leakage versus time corresponding to three time periods. It can be noted that, even if the shaft roughness amplitude is very small compared with the lip roughness amplitude, it generates important variations of the instantaneous friction torque and leakage. However, as already observed in Section 4, the mean values of both friction and leakage are not very different compared with the smooth shaft case.
6. Conclusions
Fig. 15. Comparison between averaged film thicknesses for two cases: rough and smooth shaft.
5.1. Effect of the shaft roughness amplitude To explore the effect of shaft roughness amplitude, we consider four shaft micro-undulations defined by Eq. (14) with identical numbers of peaks Nx1 ¼1, Ny1 ¼2 but different amplitudes: A1 ¼0.1 μm, 0.2 μm, 0.3 μm and 0.4 μm. Fig. 16 shows the used shaft roughness profile.
This work presents a numerical tool capable of predicting the combined effects of lip deformation, lubricating film through the seal and micro-undulation of shaft and rotary lip seal. The numerical results show a correct concordance with experiments. To explore the effect of shaft roughness on rotary lip seal performance, simulations were performed by changing the roughness amplitude and then, by changing the roughness profile. The results showed that when the shaft roughness amplitude increases, the friction torque decreases. However, if the shaft is too rough, an important increase of the friction is observed. This result seems to indicate that an optimal value for the shaft roughness amplitude can be found. The research of the optimum value must
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Fig. 17. Effect of shaft roughness amplitudes on the seal performances: (a) friction torque and (b) leakage.
Fig. 18. Predicted pressure (a, b & c) and film thickness (d, e & f) for different shaft roughness amplitudes.
also take into account the flow rate. It is shown that an increase of the shaft roughness amplitude induces important flow variations with time and can even lead to leakage.
A second investigation focuses on the shaft roughness profile impact on the friction torque and leakage. It is shown that changing the profile can induce important variations of the
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Fig. 19. Shaft roughness profiles. a) SH#2, b)SH#3, and c)SH#4.
Fig. 20. Effect of shaft profile on the seal performances: (a) friction torque and (b) leakage.
instantaneous friction torque and leakage but has only a small influence on the mean values of these parameters. It must be reminded that, all the results presented in this study define the lip and shaft roughness with simple analytical functions. Consequently, a direct perspective is to incorporate into the model the real surface roughness (like those represented in Fig. 4). Acknowledgments The authors are grateful to the Technical Centre for the Mechanical Industry (CETIM-Lerded) – Nantes that financially supported this work. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.triboint.2015.03. 013. References [1] Kammüller M. Zur abdichtwirkung von radial-wellendichtringen [Dr. Ing. thesis]. Germany: University of Stuttgart; 1986.
[2] Tonder K, Salant RF. Non-leaking lip seals: a roughness effect study. ASME J Tribol 1992;114:595–9. [3] Salant RF. Numerical analysis of the flow field within lip seals containing microundulations. ASME J Tribol 1992;114:485–91. [4] Shen D, Salant RF. A transient mixed lubrication model of rotary lip seal with a rough shaft. STLE Tribol Trans 2006;49:621–34. [5] Jia X, Jug S, Haas W, Salant RF. Numerical simulation and experimental study of shaft pumping by laser structured shafts with rotary lip seals. Tribol Int 2011;44:651–9. [6] Stakenborg MJL. On the sealing mechanism of radial lip seals. Tribol Int 1988;21:335–40. [7] Jagger ET, Walker PS. Further studies on the lubrication of synthetic rubber rotary shaft seals. Proc Inst Mech Eng 1966;9:191–204. [8] Salant RF, Shen D. Hydrodynamic effects of shaft surface finish on lip seal operation. STLE Tribol Trans 2002;45:404–10. [9] Horve LA. Shaft seals for dynamic applications. NewYork: Marcel Dekker; 1996. [10] Qu J. Non-Ra roughness parameters of shaft surfaces for radial lip seal applications. SAE paper 951195; 1995. [11] Qu J. The application of discriminant analysis in shaft surface classification for radial lip seals. SAE paper 960214; 1996. [12] Jackowski RA, Lavoie R. Recommendations for shaft surface finish parameters for radial lip seals. SAE paper 01-0886; 1999. [13] Shen D, Salant RF. Elastohydrodynamic analysis of the effect of shaft surface finish on rotary lip seal behavior. STLE Tribol Trans 2003;46:391–6. [14] Maoui A, Hajjam M, Bonneau D. Effect of 3D lip deformations on elastohydrodynamic lip seals behavior. Tribol Int 2008;4:901–7. [15] Hajjam M, Bonneau D. Influence of the roughness model on the thermoelastohydrodynamic performances of lip seals. Tribol Int 2006;39:198–205. [16] Hajjam M, Bonneau D. A transient finite element cavitation algorithm with application to radial lip seals. Tribol Int 2007;40:1258–69.
Contents lists available at ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Shaft roughness effect on elasto-hydrodynamic lubrication of rotary lip seals: Experimentation and numerical simulation M’hammed El Gadari a,n, Aurelian Fatu b,1, Mohamed Hajjam b,1 a b
Université Moulay Ismail, ENSAM Meknès, Morocco Institute Pprime, Department GMSC, University of Poitiers, France
art ic l e i nf o
a b s t r a c t
Article history: Received 21 November 2014 Received in revised form 16 February 2015 Accepted 8 March 2015 Available online 28 March 2015
Numerical analyses of the isothermal elasto-hydrodynamic lubrication (EHL) have made considerable advances in order to identify the most important features in the successful operation of rotary lip seal, and the results have shown a good agreement with experiments. Most of the models previously published are capable of predicting the combined effects of thin film through deformed lip and rotating shaft, but they assume a smooth surface of the shaft. Although this assumption is only verified for shaft roughness much smaller than that of the seal lip, it is the best solution to avoid a transient model. First, the present study describes an experimental work that provides a basis upon which a numerical EHL model of rotary lip seal is constructed by taking into account both the shaft and lip roughness. After confirming the validity of the current model by comparing experimental with numerical results, simulations have been performed and have underlined the effect of shaft roughness amplitude and profile on the rotary lip seal performance. It is shown that for shaft roughness beyond half of the lip roughness, the seal may leak. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Elasto-hydrodynamic Roughness Leakage Friction torque
1. Introduction The rotary lip seal is the most common type of rotary shaft seals (Fig. 1). It is used to withstand differences in pressure, to contain lubricant and to exclude contaminants such as air, water and dust particles. The functioning of radial lip seal is based on the formation of an oil film between the lip and the shaft. As described by Kammüller [1], elastomer lip seals operate by adopting a viscous inverse pumping regime, generated by the micro-geometry of the lip. It is shown that the pumping flow strongly depends on lip roughness [2] or microundulations [3–5]. In installed conditions, the lip geometry of a successful radial lip seal leads to a contact approach angle at the oil side significantly larger than the contact approach angle at the air side. Therefore, the resulting contact pressure profile has a maximum near the oil side of the contact. The frictional shear stress generated by the shaft rotation leads to a maximum tangential displacement that coincides with the maximum radial contact pressure. Therefore, the skewed asperities act like micro pumps that could pump oil into the contact from either side, but with a higher potential pumping capacity
n
Corresponding author. Tel.: þ212 666909018; fax: þ212 535467163. E-mail addresses: [email protected] (M. El Gadari), [email protected] (A. Fatu), [email protected] (M. Hajjam). 1 Tel.: þ33 545251979; fax: þ33 545670237. http://dx.doi.org/10.1016/j.triboint.2015.03.013 0301-679X/& 2015 Elsevier Ltd. All rights reserved.
at the air side. In steady state non-leaking conditions, a hydrodynamic equilibrium is established and a meniscus separates the liquid from the air side of the seal [6]. Starting with Jagger’s work [7,8], the shaft is assumed smooth compared to lip roughness. Typically, the arithmetic average roughness Ra of the shaft is 10 times less than the lip surface. However, practical experiences reflected in the Rubber manufactures Association (RMA) standard on upper and lower limits on the shaft surface roughness indicate that the shaft roughness influences significantly the rotary lip seal performances. Referring to Horve book [9]: “if the shaft is too rough or too smooth, the seal will leak”. Recently, experimentations have shown that not only the shaft roughness amplitude (measured by Ra) affects the lip seal performance, but also that the profile is important [10–12]. The first numerical study that takes into account both lip and shaft roughness has been presented by Salant and Shen [8]. The authors first presented a hydrodynamic model to determine the effects of the shaft surface micro geometry without considering material deformation on the lip surface. Later, they incorporated the elastic deformations of the lip into the model [13] and then they added an asperity contact model [4]. The model has been successfully used to predict the performance characteristics of a lip seal, such as load support, contact area ratio, cavitation area ratio, reverse pumping and average film thickness. In the present study, a numerical isothermal full film lubrication model is developed and validated by experimentations so as to investigate the shaft roughness effect on rotary lip seal behavior.
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Nomenclature shaft roughness amplitude [μm] lip roughness amplitude [μm] width contact [mm] compliance matrix (radial and circumferential respectively), [mm3/N] Ca instantaneous friction torque [N m] Ca friction torque averaged [N m] C10, C01 Mooney Rivlin coefficients [MPa] D Universal function, describing the pressure “p” in active zone, and the difference between replenishment and film thickness “rf–h” in non active zone D vector of nodal values of the universal variable Da shaft diameter [mm] F cavitation index (F¼1 when p 40) else (F¼0) h film thickness [mm] h0(¼havg) average film thickness [mm] h1 shaft sinusoidal roughness [mm] h2 lip sinusoidal roughness [mm] Nx peak number according to circumferential direction Ny peak number according to leakage direction NEx finite element number according to circumferential direction NEy finite element number according to leakage direction nep number of gauss points A1 A2 b C1, C2
Lip Metal insert
Garter spring Fig. 1. Rotary lip seal.
The model is an alternative of the one proposed by Salant and Shen. The differences come from a different numerical treatment of the Reynolds equation and also from the solutions used to obtain the elastic deformations generated by the hydrodynamic pressure. Salant and Shen used finite volumes method to discretize the Reynolds equation, while the present model is based on the finite elements method. The influence coefficient (or compliance) matrix is obtained here by considering a tridimensional behavior of the lip close to the contact, while Salant and Shen considered a two-dimensional axisymmetric model. It must be mentioned that the present model does not take into account the asperity contact, but the studied functioning conditions lead to full film lubrication conditions, and consequently, there is no contact observed between the lip and shaft roughness.
2. Experimental part 2.1. Test ring description Fig. 2 shows the experimental device used to test the seals. A two-phase asynchronous motor of 1.1 kW is used to control the rotational speed of the spindle. A Plexiglas tank contains the liquid
Nnk NN p ps R R Ra Rsk Sku S t T U x, y, z λ( ¼λx) λy μ ρ ρ0 τ δ δx δz
219
interpolation function relative to k node total nodes number film pressure [MPa] contact static pressure [MPa] shaft radius [mm] residual vector of the discretized modified Reynolds equation arithmetic roughness [mm] skewness measurement kurtosis measurement right hand side (RHS) vector member of modified Reynolds equation time [s] calculation cycle (where T¼ λ/U) [s] shaft velocity [mm/s] cartesian coordinates [mm] wavelength according to circumferential direction [μm] wavelength according to leakage direction [μm] lubricant dynamic viscosity [Pa s] lubricant density [Kg/m3] air density [Kg/m3] shearing according to circumferential direction [MPa] interference between shaft and lip [mm] lip circumferential displacement [mm] lip radial displacement [mm]
to seal. The tank is mounted on a needle bearing and is blocked in rotation by a torque-meter. This way, the friction force generated in the sealing zone is directly measured by the torque-meter. The Plexiglas tank is installed on a micrometric mobile support in order to obtain a very precise adjustment of the seal. The calibration of the torque meter is carried out by using different suspended masses and varying their attachment position according to the torque meter axis. From measurements, the accuracy of torque meter, which is 0.036 N m, can be deduced. The temperature measurements are carried out by type K thermocouples. A first thermocouple is placed inside the oil tank. Two additional thermocouples are used to measure the temperature of the seal near the contact zone. Sub-millimeter holes are made on the seal and the thermocouple is positioned as near as possible to the contact surface. The mean value measured by these two thermocouples will be referred to as the film temperature. The oil used is TOTAL ACTIVA 5000 15 petrol. Fig. 3 shows the variation of the oil dynamic viscosity with the temperature. The oil density is 816 kg/m3 at the ambient temperature (20 1C).
2.2. Seal description Experiments are performed on classical lip seals (material: fluorocarbon elastomer, Paulstra-Hutchinson, reference 702619), designed to be mounted on an 85 mm shaft. The radial force is measured for two different seals with and without the elastic spring. The measurements are shown in Table 1 and have been carried out on a laboratory test device according to DIN 3761-9. Shaft and lip roughness have been measured by an optical interferometry device. The results presented in Fig. 4 show that the shaft and lip topography do not have a Gaussian distribution (Sku43), with a profile that has more cavities for both surfaces (Rsko0). It must be noted that the shaft roughness amplitude (measured by Ra) is here 10 times smaller than the lip roughness amplitude, which is in accordance with the seal manufacture specifications.
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Testshaft sleeve
Acquisition unit measurements
spindle torquemeter Electric Motor Lip seal test Adjustabletable
Drilling to incorporate thermocouples Plexiglas tanks Thermocouples
Spindle
Torque meter
tested
Driving belt
Adjustable table
Motor Frame
Fig. 2. (a) Photograph of the test bench and (b) layout of the test bench.
Table 1 Values of the static load for two lip seals.
Fig. 3. Experimental measurement of the oil dynamic viscosity as a function of the temperature.
2.3. Experimental results The friction torque and film temperature measurements were taken for each rotational shaft speed from 500 rpm to 4000 rpm, increasing the velocity with step of 500 rpm. For each speed level,
Part
Radial load without spring [N]
Radial load with spring [N]
Rotary lip seal
10.1 10.2
17 16
the stabilization of the seal temperature is waited before moving to the next level. It must be mentioned that, for all operating conditions, the seal functions without any leakages. Fig. 5 shows an example of friction torque and temperature variation at 500 rpm over a period corresponding to the time needed for the film temperature stabilization. Fig. 6 contains plots of friction torque and film temperature in thermal steady state condition versus rotating speed of shaft. As expected, the film temperature increases with the shaft rotating speed. The friction torque slightly increases between 500 and 1500 rpm. Between 1500 and 2500 rpm it is almost constant and it slightly decreases for shaft speeds higher than 2500 rpm. However, the friction torque variation is low when the shaft rotating speed varies. The results presented in this paragraph will be used later to compare with the isothermal EHL numerical model described in the next section.
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Fig. 4. (a) Lip roughness: Ra ¼ 1.01 μm, λavg_x ¼ 18.88 μm, λavg_y ¼ 31.0 μm, Rq ¼ 1.12 μm; Sku ¼5.26; Rsk ¼–1.21. (b) Shaft roughness: Ra ¼ 0.09 μm, λavg_x ¼ 21.04 μm, λavg_y ¼ 42.25 μm, Rq ¼ 0.14 μm; Sku ¼ 9 ; Rsk ¼ 1.8.
seal and the rod and therefore solving Reynolds equation in order to find pressure and film thickness distribution into the contact. Fig. 7 shows a schematic diagram of a typical lip seal and the sealing zone. It is assumed that x is the circumferential direction and y is the axial direction – the shaft curvature is neglected. The upper stationary surface represents the lip surface, while the lower moving surface represents the shaft surface. The seal operates at a constant rotational speed U, the viscosity of the lubricant m is constant and the asperity contact is not considered in this full film lubrication model. The Reynolds equation in a Cartesian coordinate system is ∂ ∂ ∂h ∂h 3 ∂p 3 ∂p h þ h ¼ 6μU þ 12μ ð1Þ ∂x ∂x ∂y ∂y ∂x ∂t where p is the film pressure and h represents film thickness such as hðx; y; t Þ ¼ h2 ðx δx ; yÞ h1 ðx; y; t Þ þ h0 þ δz ðx; y; t Þ Fig. 5. Variation of the film temperature and the friction torque versus time.
with h1 the shaft roughness, h2 the lip roughness, δx the circumferential lip displacement, δz the radial lip displacement and h0 the average film thickness. Eq. (1) can be solved only for the full film zones. In order to take into account both active and non-active film zones, the modified Reynolds equation proposed by Hajjam and Bonneau [13] is used: ∂ ∂ ∂h ∂h ∂D ∂D 3 ∂D 3 ∂D F h þF h ¼ 6μU þ 12μ þ 6μð1 F Þ U þ 2 ∂x ∂x ∂y ∂y ∂x ∂t ∂x ∂t ð3Þ where D is a universal variable and F a cavitation index: D ¼ p; D Z 0 in the full film zone : F ¼1 in the cavitated ðnon activeÞ film zone :
Fig. 6. Variation of the film temperature and friction torque according to the shaft velocity.
3. Numerical model 3.1. Flow equations The evaluation of sealing performances referring to leakage and friction is carried out supposing a lubricated contact between the
ð2Þ
D ¼ r h; D o 0 F ¼0
ð4Þ
ð5Þ
where r ¼ ρh=ρ0 is the so-called effective film thickness, ρ represents the density of the lubricant–gas mixture and ρ0 is the density of the lubricant. As shown in References [15,16], the boundary conditions used for the solution of Eq. (3) are reduced to outside boundaries of the domain: at y¼b, p is equal to the sealed fluid pressure and at y¼0, p is equal to the air pressure. In the circumferential direction x, periodicity boundary conditions are imposed. Once the pressure and film thickness distribution in the contact are obtained, the hydrodynamic friction torque is computed from: Z λZ b 1 ∂p U Ca ¼ R h þ μ dx dy ð6Þ 2 ∂x h 0 0
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Fig. 7. Rotary lip seal mounted on shaft.
Fig. 8. The FE model of the seal.
The instantaneous leakage is calculated on the air side contact boundary from: Z λ 3 h ∂p dx ð7Þ Q¼ ∂y 12 0
3.2. Elastic deformations As it can be easily understood from the previous section, the evaluation of the pressure in the contact leads to an elastohydrodynamic lubrication problem: there is a strong link between the pressure, the lip elastic deformation and the film thickness. The first difficulty in modeling elastomeric seals is attributed to the non-linear stress–strain elastomer elastic behavior. However, in rotary lip seals, the radial strain imposed by the fluid film in contact is much smaller in comparison with the normal strain imposed by the seal/shaft interference. Therefore, it can be assumed that the radial load that will be supported by the lubricated contact is equal to the static radial load computed by the integration of the contact static pressure imposed by the seal/ shaft interference. It is also supposed that the width of the contact will not be modified by the radial strain imposed by the presence of the fluid film in functioning conditions. Consequently, a structural non-linear mechanic analysis must be carried out in order to predict the contact static pressure field and the contact width due to the mounting of the lip on the shaft. Fig. 8 shows the Finite Elements (FE) model of the seal. The lip is meshed with axisymmetric stress elements. The computations are made in large displacement and deformation hypotheses. The shaft is usually made in more rigid material (typically steel) than the elastomeric seal. Consequently, it is reasonable to consider the shaft as rigid. The seal material is considered as an incompressible,
hyperelastic material and approached with the Mooney–Rivlin model (C01 ¼2.49 MPa and C10 ¼0.5 MPa).2 Fig. 8 also shows the predicted contact static pressure and the contact width. The numerical integration of the contact static pressure gives 17 N – in concordance with the measurements shown in Table 1. The axial contact width b defines the study domain length in the axial direction. The second length λ is chosen equal to the lip surface roughness periodicity in the circumferential direction which, as shown in Fig. 4, is quite similar to the shaft wavelength. As the radial strain imposed by the fluid film in contact is small in comparison with the radial strain imposed by the seal/shaft interference, it makes it possible to use a linear elastic perturbation of the mounted seal in order to compute that the additional elastic deformations generated by the fluid film presence. Therefore, once the non-linear FE computation described above is made, the deformed geometry of the seal is kept and used to compute two (radial and tangential) compliance matrixes that can only be computed if the seal elastic behavior is considered as linear (Hookean model) and the computations are made in small displacement and deformation hypotheses. The method used to compute the compliance matrix has been previously presented in [15]. The lip is considered to have, along a height d, a tridimensional (3D) behavior. The elastic deformation of the lip is treated by the FE method using elements with 20 nodes
2
The Mooney–Rivlin model expresses the strain energy potential as 2 W ¼ C 10 ðI 1 3Þ þ C 01 ðI 2 3Þ þ ð J el 1 =DÞ where J el is the elastic volume ratio, C10, C01 and D are material parameters. I1 and I2 are the first and second deviatoric strain invariants, respectively. By analogy with the classical Hookean model, D is related to the material bulk modulus K¼2/D. If the material is considered incompressible (Poisson ratio ν ¼ 0.5) D is nil.
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variable. One term of the [M] matrix can be written as M jk ¼
npg ne X X n¼1m¼1
2
Fig. 9. Junction between the two-dimensional and three-dimensional meshes.
for the 3D part and eight nodes axisymmetric 2D elements for the rest of the seal structure (Fig. 9). In order to take into account the global periodicity hypothesis, rigid beams connect the two faces of the 3D domain, giving the same displacement for the connected nodes. Two compliance matrixes [C1] and [C2] are calculated. [C1] is used to compute the radial displacement and [C2] is used to compute the circumferentially tangential displacement. Thus, the radial and circumferential lip nodal displacements can be computed with the following equations: ðδ z Þi ¼
NN X
ðC 1 Þi;j pj psj
ð8Þ
ðC 2 Þi;j τj
ð9Þ
j¼1
ðδ x Þi ¼
NN X j¼1
where pj is the nodal film pressure, psj is the nodal contact static pressure, τj is the nodal shear stress and NN is the number of nodes. The reader must keep in mind that the radial compliance matrix [C1] can only be used to predict the lip deformation generated by the difference between the hydrodynamic film pressure (pj) and the contact static pressure (psj). 3.3. Numerical treatment In this paper, the FE formulation proposed by Hajjam and Bonneau [15,16] is used to deal with the EHL lubrication presented problem. The numerical algorithm is illustrated in Fig. 10. Two different problems are formulated: Problem 1. The film thickness is known and the active/inactive film zone separation coordinates are searched. Problem 2. The active and the inactive film zones are known; the film pressure and thickness, which verify the Reynolds equation, and the elastic equations, are searched. Both problems must be solved at each step. To solve Problem 1 we must solve the modified Reynolds equation (Eq. (3)). Problem 2 is solved using the classic Reynolds equation (Eq. (1)). Both equations are processed using the Galerkin approach. Previous studies show that, for EHL problems, the best domain discretization uses quadratic eight node elements. The particular form of the modified Reynolds equation, when it is applied to the cavitated film zones needs, for its discretization, the use of four node linear elements. Depending on whether Problem 1 or 2 is considered, the discretization uses four or eight nodes, the value of the central node parameters is obtained by interpolation. Finally, in the case of Problem 1 (Problem 2 is similar), a linear system of algebraic equations in D, is obtained: R ¼ ½M D þ S
ð10Þ
where R is the residual vector of the discretized modified Reynolds equation and D is the vector of nodal values of the universal
3 nne nne X ∂Nmj ∂N mk ∂Nmj ∂N mk ∂N mj hm X þ Fk N ð1 F k Þ 6μ k ¼ 1 ∂x ∂x ∂y ∂y ∂x mk k¼1 !
nne 1 X N mj N mk ð1 F k ðtÞÞ ΔΩm Δt k ¼ 1
ð11Þ
where m represents one of the ngp Gauss points on n elements and nne the number of nodes per element. Nmj is the weight function relative to the j node, while Nmk is the interpolation function relative to k node. Fk represents the status of k node and takes the value 1 if it is in the active zone and 0 in the opposite case. The vector S is the RHS member of modified Reynolds equation. The j term of S can be written as npg ne X X ∂hm hm ðtÞ hm ðt ΔtÞ Sj ¼ þ2 N mj U Δt ∂x n¼1m¼1 þ2
nne 1 X N N ðð1 F k ðt ΔtÞÞDk ðt ΔtÞÞΔΩm Δt k ¼ 1 mj mk
ð12Þ
The system of equations describing Problem 1, non-linear in p and h is solved through the Newton–Raphson method. The convergence criterion is the equality between the applied load and the hydrodynamic computed load at 1%. The structural mechanical analysis simulates the rotary lip seal assembly in dry condition, by taking into account the shaft radius R¼42.5 mm and the interference δ¼1.1 mm. The width of the contact is b¼0.088 mm. Fig. 11 shows the static contact pressure distribution over the hydrodynamic study domain. The maximum pressure is 2.4 MPa. For all the computational results presented afterward, the sealed pressure is 0.016 MPa when the air side pressure is nil. In the cavitation zones, the pressure is 0.1 MPa. The wavelength, according to circumferential direction, is λx ¼0.02 mm. A mesh sensibility parametric study leads us to choose 12 quadratic elements along the circumferential direction and 20 along the direction of the leakage. This choice is a compromise between the accuracy of the solution and the computing time. Using more accurate meshes leads to the same results concerning the maximum pressure and the minimum film thickness. Due to the consideration of the roughness on both lip and shaft surfaces, the problem is time dependent. However, due to the periodicity boundary conditions, the results present a time periodicity, depending on U and λ.
4. Numerical versus experimental comparison As described in the previous paragraph, the numerical model is isothermal. In order to compare with the experimental results, the oil viscosity used by the numerical model is approximated by using the temperature results presented in Fig. 6 and referring to the viscosity temperature shown in Fig. 3. Table 2 shows the oil dynamic viscosity corresponding to each shaft rotational speed, in thermal steady state conditions. The confrontation between the numerical simulations and the experimental measurements was made for several rotational speeds of the shaft: from 500 rpm to 4000 rpm with a 500 rpm step between each level of speed. As a first approximation already made in previous studies [3,4,8,14–16], it is chosen to define the roughness with simple analytical functions (Eq. (13) and (14)). The asperities on the lip and shaft are represented by a two-dimensional cosine function. However, we have respected measurements of the arithmetic roughness and also the wavelengths in leakage (λy ¼ 42.25 μm for
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shaft roughness and λy ¼31.0 μm for lip roughness) and circumferential direction (λx ¼20 μm for both surfaces):
Lip roughness: h2 ðx; y; t Þ ¼
A2 2π 2π N x2 ðx δx ðx; y; t ÞÞ N y2 y cos 1 cos λ b 2
ð13Þ
Shaft roughness:
A1 2π 2π N x1 ðx tU Þ N y1 y cos h1 ðx; y; t Þ ¼ 1 cos λ b 2
ð14Þ
where Nx1 ¼1 is the number of peaks of the shaft roughness according to x, Nx2 ¼ 1 is the number of peaks of the lip roughness according to x, Ny1 ¼ 2 is the number of shaft peaks along the y direction, Ny2 ¼3 is the number of lip peaks along the y direction, Read data: shaft and lip roughness, viscosity, shaft velocity, compliances matrix Initialization: computation h(x,y,0) and p(x,y,0) Loop 1: Do for each step “t” Loop 2: Do until stability of domain h(x,y,t) and p(x,y,t) Loop 3: Do until stability of cavitation domain Compute “D” (Modified Reynolds equation) Update the cavitation boundaries End loop 3 Loop 4: While residues (h, p)>ε (Newton-Raphson method) Compute h(x,y,t) and p(x,y,t) Update elastic displacement: δx and δz End loop 4 End loop 2 Write: pressure, film thickness, leakage, power loss, friction torque End loop 1 End of algorithm
A1 ¼0.1 μm is the shaft roughness amplitude and A2 ¼1 μm is the lip roughness amplitude. Fig. 12 shows the lip and the shaft profiles that correspond to Eqs. (13) and (14). Fig. 13 shows the variation of the friction torque versus the shaft rotational speed. The friction torque values are calculated by integrating the instantaneous friction torque over the time period T: Ca ¼
1 T
Z
T 0
C a ðt Þdt;
where : T ¼
λx U
ð15Þ
According to Fig. 13, it can be concluded that the numerically predicted friction torque gives a correct concordance with experimental measurements and both considered cases (smooth or rough shaft) are within the gap of experiments validity. This result gives a global validation of the current numerical model and also shows the low impact of the shaft roughness on the friction torque when the roughness amplitude is 10 times smaller than the lip one. To highlight the effect of the shaft roughness on friction torque, Fig. 14 shows, at 2000 rpm, the contribution of the Couette and Poiseuille terms. It can be noted that, for both cases (rough and smooth shaft), the Couette contribution is more important than the Poiseuille contribution. Fig. 15 shows that, the predicted average film thickness slightly increases for the rough shaft case. This E 3% increase of the film thickness does not have an important effect on the Couette term (proportional to the ratio between U and h), and consequently, the global friction torque is not really influenced by the considered shaft roughness amplitude.
Fig. 10. EHL algorithm.
5. Effect of shaft roughness on seal performance This section numerically explores the effect of the shaft roughness on the seal performances. By varying both the roughness amplitude and profile, friction torque and reverse pumping are investigated. The functioning characteristics are the same with the above presented lip seal at 2000 rpm. Table 2 Values of the oil dynamic viscosity depending on the shaft rotational speed. Shaft speed (rpm)
Dynamic viscosity (Pa s)
500 1000 1500 2000 2500 3000 3500 4000
0.0625 0.0375 0.025 0.02 0.0175 0.0175 0.01 0.01
Fig. 11. Adopted size mesh (contact pressure illustrated).
Fig. 12. (a) Lip roughness profile and (b) shaft roughness profile.
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Fig. 16. Shaft roughness profile defined as SH#1.
Fig. 13. Numerical and experimental friction torque variation with the shaft rotational speed.
The used lip profile is given by Eq. (9) (represented in Fig. 12 a) with A2 ¼ 1 mm, Nx2 ¼1 and Ny2 ¼3. Fig. 17 plots the torque friction and leakage versus time corresponding to three time periods T ¼λx/U. It can be noted that the shaft roughness amplitude influences the dynamic behavior of the global parameters of the lip seal. The increase of the roughness amplitude leads to significant fluctuations of the flow. Moreover, it can be observed that, for amplitudes higher than 0.4 mm, the seal will leak – the leakage rate became greater than zero. Concerning the friction torque, the increase of the roughness amplitude induces an increase of the film thickness that results in a decrease of the shear stress. However, for a shaft amplitude A1 ¼0.4 mm a sudden increase of the friction is predicted. In this case, the film thickness is locally very thin, which leads to a severe increase of the friction and of the pressure gradient. This result can be observed in Fig. 18 that compares the film thickness and the pressure fields for the smooth and two rough shaft configurations. 5.2. Effect of the shaft roughness profile
Fig. 14. Comparison between “Poiseuille” and “Couette” torque for two cases: rough and smooth shaft.
In order to underline the shaft roughness profile effect, 4 sinusoidal forms are studied: SH#1 (represented in Fig. 16), SH#2 given by Eq. (16) and represented in Fig. 19a), SH#3 given by Eq. (14) with Ny1 ¼4 (Fig. 19b)) and finally SH#4 also given by Eq. (14) with Ny1 ¼6 (Fig. 19c)). 2π ðx tU Þ ð16Þ h1 ðx; y; t Þ ¼ A1 sin λ For all these shaft roughness profiles, A1 ¼0.1 mm. Fig. 20 represents the friction torque and the leakage versus time corresponding to three time periods. It can be noted that, even if the shaft roughness amplitude is very small compared with the lip roughness amplitude, it generates important variations of the instantaneous friction torque and leakage. However, as already observed in Section 4, the mean values of both friction and leakage are not very different compared with the smooth shaft case.
6. Conclusions
Fig. 15. Comparison between averaged film thicknesses for two cases: rough and smooth shaft.
5.1. Effect of the shaft roughness amplitude To explore the effect of shaft roughness amplitude, we consider four shaft micro-undulations defined by Eq. (14) with identical numbers of peaks Nx1 ¼1, Ny1 ¼2 but different amplitudes: A1 ¼0.1 μm, 0.2 μm, 0.3 μm and 0.4 μm. Fig. 16 shows the used shaft roughness profile.
This work presents a numerical tool capable of predicting the combined effects of lip deformation, lubricating film through the seal and micro-undulation of shaft and rotary lip seal. The numerical results show a correct concordance with experiments. To explore the effect of shaft roughness on rotary lip seal performance, simulations were performed by changing the roughness amplitude and then, by changing the roughness profile. The results showed that when the shaft roughness amplitude increases, the friction torque decreases. However, if the shaft is too rough, an important increase of the friction is observed. This result seems to indicate that an optimal value for the shaft roughness amplitude can be found. The research of the optimum value must
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Fig. 17. Effect of shaft roughness amplitudes on the seal performances: (a) friction torque and (b) leakage.
Fig. 18. Predicted pressure (a, b & c) and film thickness (d, e & f) for different shaft roughness amplitudes.
also take into account the flow rate. It is shown that an increase of the shaft roughness amplitude induces important flow variations with time and can even lead to leakage.
A second investigation focuses on the shaft roughness profile impact on the friction torque and leakage. It is shown that changing the profile can induce important variations of the
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Fig. 19. Shaft roughness profiles. a) SH#2, b)SH#3, and c)SH#4.
Fig. 20. Effect of shaft profile on the seal performances: (a) friction torque and (b) leakage.
instantaneous friction torque and leakage but has only a small influence on the mean values of these parameters. It must be reminded that, all the results presented in this study define the lip and shaft roughness with simple analytical functions. Consequently, a direct perspective is to incorporate into the model the real surface roughness (like those represented in Fig. 4). Acknowledgments The authors are grateful to the Technical Centre for the Mechanical Industry (CETIM-Lerded) – Nantes that financially supported this work. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.triboint.2015.03. 013. References [1] Kammüller M. Zur abdichtwirkung von radial-wellendichtringen [Dr. Ing. thesis]. Germany: University of Stuttgart; 1986.
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