Application of Fractal Contact Model in Dynamic Performance Analysis of Gas Face Seals
© The Author(s) 2018
Received: 8 May 2016
Accepted: 16 March 2018
Published: 20 April 2018
Fractal theory provides scale-independent asperity contact loads and assumes variable curvature radii in the contact analyses of rough surfaces, the current research for which mainly focuses on the mechanism study. The present study introduces the fractal theory into the dynamic research of gas face seals under face-contacting conditions. Structure-Function method is adopted to handle the surface profiles of typical carbon-graphite rings, proving the fractal contact model can be used in the field of gas face seals. Using a numerical model established for the dynamic analyses of a spiral groove gas face seal with a flexibly mounted stator, a comparison of dynamic performance between the Majumdar-Bhushan (MB) fractal model and the Chang-Etsion-Bogy (CEB) statistical model is performed. The result shows that the two approaches induce differences in terms of the occurrence and the level of face contact. Although the approach distinctions in film thickness and leakage rate can be tiny, the distinctions in contact mechanism and end face damage are obvious. An investigation of fractal parameters D and G shows that a proper D (nearly 1.5) and a small G are helpful in raising the proportion of elastic deformation to weaken the adhesive wear in the sealing dynamic performance. The proposed research provides a fractal approach to design gas face seals.
However, Sayles and Thomas  revealed that many engineered surfaces have the multi-scale characteristic. Bhushan et al.  found that statistical parameters depend strongly on the resolution of measuring instruments, and are not unique for a surface because of the multi-scale characteristic. It leads to the result that measurements with different resolutions and scanning lengths wouldn’t yield unique statistical parameters for a surface. Moreover, statistical contact models overlook the fact that the curvature radius of an asperity is a function of asperity size, and surely assume a constant curvature radius for all asperities. Majumdar and Bhushan  used the Weierstrass-Mandelbrot (WM) function to develop the first fractal contact model for real rough surfaces where the assumption of variable curvature radius was adopted. This fractal contact model has been of interest to many researchers, and has been applied to various applications. Wang and Komvopoulos [15, 16] researched the interfacial temperature factor in the fractal contact. Komvopoulos and Yan  generated a three-dimensional fractal surface by the WM function and introduced it into the contact model. Sahoo and Chowdhury [18, 19] analyzed the friction and the wear of fractal surfaces. Ciavarella et al.  investigated the elastic contact stiffness and the contact resistance of fractal surfaces. Kogut and Jackson  used both statistical and fractal approaches to characterize simulated surfaces, and obtained substantial differences between the two. Morag and Etsion (ME model)  argued that a single asperity transferring from plastic to elastic when the load increases and the contact area becomes larger in the MB model is in contrast with classical contact mechanics. They suggested the real deformation is an independent parameter ranging from zero to full interference, and thus developed a revised elastic-plastic contact model with respect to a single fractal asperity.
Face contact can affect sealing performance, such as dynamic behavior, thermodynamics, friction, wear. The dynamics of face seals has been an active area since the works by Etsion [23–25]. In the aspect of gas face seals, direct numerical method [26–30] is detailed but computationally intensive, whilst semi-analytical method including perturbation method [31–34] and step jump method , on the other hand, is a more practical and efficient alternative but intensively depends on small displacement precondition. Before the introduction of asperity contact, stability and tracking have been used to illustrate the potential menace of face contact. Variables such as minimum film thickness, axial relative displacement and angular transmissibility are regarded as the indirect evaluation indexes of face contact. With the development of simulation, asperity contact models have been involved to directly gauge their influence on sealing dynamic performance. Harp and Salant  developed an axial dynamic model including the Abbott-Firestone plastic asperity contact model. Green  researched the transient performance of coned gas face seals during the startup operation by using the CEB model. Ruan  also used the CEB model to investigate the transient performance of spiral groove gas face seals during the startup and shutdown operations.
The above dynamic works all used the statistical contact models, while the fractal contact theory has not been applied. The present study is an attempt to involve the fractal theory in the dynamic research of gas face seals. And a comparison of dynamic performance between statistical and fractal approaches is performed. The widely used, original contact models (i.e., the MB and the CEB models) are selected, respectively, as the representatives of fractal and statistical asperity contact models accounting for excluding extra factors (such as the impact of work hardening, the influence of neighboring asperities, the introduction of new bi-Gaussian stratified surface parameters [39–45], the revision of contact mechanic in ME model) involved in advanced statistical or fractal models.
2.1 MB Model
2.2 Application of MB Model in Sealing Dynamics
Then, the resulting Pc can be used in the sealing dynamics to take the place of Fc in Eq. (8).
3 Results and Discussion
3.1 Test of Fractal Characteristic in Gas Face Seals
3.2 Comparison of Steady-state and Transient Sealing Performance
Pressure at outer radius po (MPa)
Shaft speed Ω (rad/s)
Pressure at inner radius pi (MPa)
Fundamental parameters of a gas face seal
Outer radius ro (m)
Inner radius ri (m)
Balance radius rb (m)
Spiral angle α (°)
Number of grooves N
Groove to land width ratio λ
Groove to dam length ratio β
Groove depth δg (μm)
Stator mass m (kg)
Support axial stiffness k sz (kN/m)
Support axial damping c sz (Ns/m)
Hertz elastic modulus E (GPa)
Softer material hardness H (GPa)
Maximum contact pressure factor K
Parameters of the CEB and the MB models for three profiles
Asperity summit density η (μm−2)
Asperity summit radius R (μm)
Standard deviation of asperity height σ (μm)
Fractal dimension D
Fractal roughness parameter G (10−13 m)
Profile 3 is selected to gauge the differences of sealing performance between the MB and the CEB approaches. A whole-ring numerical model is established for the dynamic analyses of a spiral groove gas face seal with a flexibly mounted stator using the finite element method. In regards to the accuracy and stability of a complex dynamic system, a very fine mesh and a very small time step would be ideal. Yet, a balance should be struck between accuracy and computing time. After experiments, it is determined that a time step of 1 × 10−6 s with a 3160-finite element mesh is adopted. The simulation is carried out on a 2.3 GHz workstation.
To sum up, the MB and the CEB approaches induce different steady-state and transient performance in the dynamic simulation. In particular, under current sealing parameters, even if the distinctions of the two approaches in film thickness and leakage rate are tiny, the two approaches will induce apparent differences in the occurrence and the level of face contact, which affect contact mechanism and end face damage. In addition, since not all surfaces own fractal characteristic, it should be emphasized that the use of the MB model in other types of seals or other fields has an extra limitation.
3.3 Investigation of Fractal Characteristic
Adhesive wear is a significant physical phenomenon in the dynamic analyses of gas face seals for its effect on the end face damage and service life. As adhesive wear is more easily caused by plastic deformation, it is better to raise the proportion of elastic deformation in the face contact. In the MB model, a feasible approach to raise the proportion of elastic deformation is to control the fractal parameters D and G. Therefore, a parametric investigation of D and G is performed. Note that the present study focuses on the influence of D and G on the proportion of elastic deformation. The detailed approach to realize the control of D and G should be researched in the manufacture-process field.
Fractal asperity contact model can provide an asperity contact load out of the influence of measuring instruments or sample length for its scale-independence. It also uses an assumption of variable curvature radius. The MB fractal contact model is selected as the representative of fractal contact models in the attempt of incorporating the fractal theory into the dynamic research of gas face seals.
Structure-Function method is adopted to process the surface profiles of a typical carbon-graphite ring which is an industrial product, proving the MB model can be utilized in the typical gas face seals.
The CEB statistical model is selected to compare with the MB model to gauge the differences of the two approaches in dynamic performance. The MB and the CEB approaches will induce differences in terms of the occurrence and the level of face contact. Although the distinctions in film thickness and leakage rate may be tiny, the distinctions in contact mechanism and end face damage are apparent. The CEB approach offers a more optimistic consideration of face contact.
An investigation of fractal parameters D and G is performed to explore a feasible approach to raise the proportion of elastic deformation to weaken adhesive wear in the sealing dynamic performance.
S-TH was in charge of the whole trial; S-TH wrote the manuscript; W-FH, X-FL and Y-MW assisted with sampling and laboratory analyses. All authors have read and approved the final manuscript.
Song-Tao Hu, born in 1989, is currently a postdoctoral fellow at State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, China. He received his doctorate degree from State Key Laboratory of Tribology, Tsinghua University, China, in 2017, and received his bachelor degree from Northwestern Polytechnical University, China, in 2012. His research interests include mechanical face seals. Tel: +86-18221229658; E-mail: email@example.com.
Wei-Feng Huang, born in 1978, is currently an associate professor at State Key Laboratory of Tribology, Tsinghua University, China. His research interests include mechanical face seals. Tel: +86-10-62795124; E-mail: firstname.lastname@example.org.
Xiang-Feng Liu, born in 1961, is currently a professor at State Key Laboratory of Tribology, Tsinghua University, China. His research interests are machine design and mechanical face seals. Tel: +86-10-62795122; E-mail: email@example.com.
Yu-Ming Wang, born in 1941, is currently a professor at State Key Laboratory of Tribology, Tsinghua University, China, and an academician of Chinese Academy of Engineering. His research interest is fluid sealing technology. Tel: +86-10-62771865; E-mail: firstname.lastname@example.org.
The authors declare no competing financial interests.
Ethics Approval and Consent to Participate
Supported by China Postdoctoral Science Foundation (Grant No. 2017M621458), National Science and Technology Support Plan Projects (Grant No. 2015BAA08B02), National Natural Science Foundation of China (Grant No. 11632011), and National Natural Science Foundation of China (Grant No. 11372183).
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