Fractal Characterization of Sealing Surface Topography and Leakage Model of Metallic Gaskets Xiu FENG a and Boqin GU b

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1 Key Engineering Materials Online: ISSN: , Vols , pp doi:10.408/ 007 Trans Tech Publications, Switzerland Fractal Characterization of Sealing Surface Topography and Leakage Model of Metallic Gaskets Xiu FENG a and Boqin GU b College of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing 10009, China a fxsue@163.com, b bqgu@njut.edu.cn Keywords: Metallic gasket; Sealing; Surface topography; Fractal; Leakage model. Abstract. In this paper, the fractal characterization of the sealing surface topography of metallic gaskets was studied. The influence of the compressive stress on the fractal parameters was also investigated. It s found that the sealing surface of metallic gaskets is fractal, and its topography can be characterized by the fractal dimension and the scale coefficient. The leakage model of metallic gaskets was established. The research results indicate that the larger the fractal dimension is, and the less the scale coefficient and the non-contact area are, the better the sealing performance of metallic gaskets is. Introduction Metallic gaskets seals are widely used in the process equipment operating at high-temperature, under high-pressure and in corrosion medium, where other types of gaskets are difficult to meet the sealing requirement. The metallic gasket seal is a type of contact seal. Experimental and theoretical researches have shown that the topography of contact surfaces has significant influence on the sealing performance of connections [1, ]. Fractal theory provides a new approach for characterizing the complicated surface topography. The fractal characterization of sealing surface profiles of metallic gaskets was studied in this paper. The influence of the compressive stress on the fractal parameters, such as the fractal dimension, the scale coefficient and the characteristic length scale, was also analyzed. Based on the research results of the surface topography of metallic gaskets and the laminar flow theory of incompressible viscous fluids, the leakage model of metallic gaskets was established. Experimental investigation on sealing surface topography of metallic gaskets Experimental principle. The topography of a machined surface appears random, irregular and multi-scale. Under different magnifications, a surface profile shows repetitive details and any thin part of the profile will be similar to the whole. So the topography of machined surfaces has self-similarity or self-affinity. Sealing surfaces of metallic gaskets have similar characteristics, which can be expressed by Weierstrass-Mandelbrot function (abbreviated to W-M function) [3] Z n ( 1) cos(πγ x) ( x) = G ( ) n= n l γ, (1) where Z(x) is the deviation of the surface profile from its mean line, parameter is the fractal dimension of Z(x) (1<<), G is a characteristic length scale of the surface, γ n is the discrete frequency spectrum of rough surfaces and n l is the low frequency cut-off determined by a low boundary of a non-scale range. The structure function of Z(x) is defined as [4] Γ( 3) sin ( 3) π ( 1) 4 4 S( τ) = [ Z( x+ τ) Z( x) ] = G τ = Cτ, () 4 lnγ ( ) All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (I: , Pennsylvania State University, University Park, USA-09/04/16,07:08:38)

2 978 Progresses in Fracture and Strength of Materials and Structures where τ is the displacement along the x direction, implies temporal average and C is a scale coefficient. Based on equation (), different τ and the corresponding S(τ) on lg-lg plot is a fitting line. Then, the fractal dimension can be related to the slope k of a fitting line on lg-lg plot = ( 4 k) /. (3) The value of parameter C can be obtained from the intercept B of the line on the lgs(τ) axis. B C = 10. (4) According to equations () and (4), equation (5) holds G = Γ 1 4 C( ) lnγ ( 3) sin[ ( 3) π ] ( 1). (5) The slope k and the intercept B can be determined from the structure function plot on lg-lg coordinates. Thus fractal dimension, scale coefficient C and characteristic length scale G can be obtained from the equations (3), (4) and (5). Samples and experimental conditions. Samples are three flat metallic gaskets and made of 10 steel. Their outer and inner diameters are m and m respectively, and their thickness is m. The traditional roughness parameters of one sample are Ra = m, Rz = m and Sm = m. The test temperature is the room temperature. The nominal contact stresses of metallic gaskets SG are 0, 15, 187, 18, 50, 31, 374, 41 MPa respectively. Experimental results and discussion Fractal character of the sealing surface of the metallic gasket. Fig.1 is the profiles of the sample under three compressive stresses. Fig. shows the structure function plot corresponding Fig.1. It can be found from Fig.1 and Fig. that the sealing surface of the metallic gasket under different compressive stresses has a strong random property. There is one straight line along a major portion of the lg-lg plots of structure functions. All the slopes of the straight lines satisfy 0<k<, which means, the profile of the sealing surface under different compressive stresses is fractal. Influence of compressive stress on fractal parameters. The relationship between fractal parameters and the compressive stress is shown in Fig.3~Fig.5. It can be found that the larger the compressive stress SG is, the larger the fractal dimension is, and the less the scale coefficient C is. The characteristic length scale G changes irregularly. The sealing surface becomes more and more smooth with the increase of the compressive stress. The fractal dimension and scale coefficient can be used for characterizing the surface topography of metallic gaskets under various compressive stresses. (a) S G =0 MPa (c) S G =31 MPa Fig.1 Profiles of the sample under three compressive stresses lgs(τ) [μm ] (b) S G =18 MPa 0.0 corresponding Fig. (a) corresponding Fig. (b) -0.5 corresponding Fig. (c) lgτ [μm] Fig. Structure function plot

3 Key Engineering Materials Vols Leakage model of metallic gaskets According to the above-mentioned research results, the contact between the flange and metallic gasket sealing surfaces can be modeled as the contact of a smooth rigid flat surface with a rough fractal surface. It can be assumed that the passage longitudinal section may be approximated to a cosine wave trough (Fig.6). According to equation (1), the passage longitudinal section can be expressed by the following function. 1 π l l z( x) = G l cos x, < x <. (6) l The gas flow through the leakage passage of the metallic gasket seal can be approximated to incompressible laminar flow. For the length element shown in Fig.6, the volumetric leakage is given by Eq.(7) [5]. 3 dxz ( ) ( p ) d l V =, (7) 1η B where p 1 and p are the pressure inside and outside the rig, respectively; B is the length of the Assuming l = a, Eq. (8) becomes p 3( 1) ( 7 3 ) lv = G a, (9) where a is the cross-section area of the passage. The distribution of the passage cross-section area is [3] lv 1+ C a n ( a) =, (10) a where a lv is the cross-section area of the largest passage. According to equations (9) and (10), the max volumetric leakage rate of all passages of the sealing surface is given by G [μm] Fig.3 Relationship between Fig.4 Relationship between Fig.5 Relationship between G and S G C and SG and S G passage, and η the dynamic viscosity. Substituting Eq. (6) into Eq. (7), and integrating Eq. (7) from x=-l/ to l/, Eq. (8) is obtained p 3( 1) 7 3 lv = G l. (8) l/ 0 dx l/ z( x) Fig.6 Passage longitudinal section x p 3( 1) LV = G alv. (11) 18πηB 7 4 According to Eq. (10), the true non-contact area A lv is given by alv Alv = n( a) ada = alv. (1) 0

4 980 Progresses in Fracture and Strength of Materials and Structures According to equations (11) and (1), Eq. (13) holds L V Summary p p = G 7 3 ( 1) 3 A lv. (13) The sealing surfaces of metallic gaskets are fractal. The larger the compressive stress is, the larger the fractal dimension is, and the less the scale coefficient is. The fractal dimension and scale coefficient can be used for characterizing the sealing surface topography of metallic gaskets under different compressive stresses. The leakage model for metallic gaskets was established, and the formula for calculating the leakage rate derived. The research results indicate that the bigger fractal dimension of the sealing surface and the compressive stress are, the less the leakage rate of metallic gaskets is; the bigger the scale coefficient and non-contact area are, the larger the leakage rate is. References [1] N. Patir, H. S. Cheng: ASME, J. Lub. Tech. Vol. 100 (1978), p.1~17 [] P. T. Manning: Wear, Vol. 57 (1979), p.365~376 [3] A. Majumdar, C. L. Tien: Wear, Vol.136 (1990), p.313~37 [4] L. He, J. Zhu: Wear, No. 08 (1997), p.17~4 [5] H. K. Müller, B. S. Nau: Fluid Sealing Technology (Marcel ekker Inc. Publications, New York, 1998)

5 Progresses in Fracture and Strength of Materials and Structures / Fractal Characterization of Sealing Surface Topography and Leakage Model of Metallic Gaskets /