液体の温度－粘度特性を利用したフランジ・ガスケ
ットの漏れ制御メタデータ

言語: English
出版者:
公開日: 2022-06-30
キーワード (Ja): シール、ガスケット, 液体, 潤滑油, 粘度,
温度, 漏れ, 振動, 平坦度
キーワード (En): Seal, Gasket, Liquid, Lubricant,Viscosity, Temperature, Leakage, Vibration, Flatness
作成者: ガオ, ソン
メールアドレス:所属:

URL

https://doi.org/10.15118/00010859

Doctoral Dissertation

LEAKAGE CONTROL FOR FLANGE GASKET USINGTEMPERATURE–VISCOSITY CHARACTERISTICS
OF LIQUIDSMURORAN INSTITUTE OF TECHNOLOGY
Doctor Degree of Division of Engineering
Course of Advanced Production Systems Engineering
Mechanical Systems Design Laboratory
Admission in 2019Student ID Number 19096005

GAO SONG
Submission in 2022

LEAKAGE CONTROL FOR FLANGE GASKET USINGTEMPERATURE–VISCOSITY CHARACTERISTICS OF
LIQUIDSbyGAO SONG(19096005)

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor ofEngineering in the subject of Division of Engineering
Full Supervisor:
Associate Supervisor:

Prof. Toshiharu Kazama
Prof. Naohiko Hanajima & Prof. Koji Teramoto

Examination Committee:

Prof. Toshiharu Kazama
Prof. Naohiko Hanajima
Prof. Koji Teramoto

Nationality:

The People’s Republic of China

Previous Degree:

Bachelor of Mechanical Designing
Henan Polytechnic University, China
Master of Mechanical Engineering
Henan Polytechnic University, China

Scholarship Donor:

Japan Student Services Organization (JASSO)
Hokkaido Foreign Student Grant
Muroran Institute of Technology (MuroranIT)
Scholarship

Muroran Institute of Technology
Muroran, Hokkaido, Japan
March 2022

- i -

Acknowledgement
The author wishes to express the profound gratitude to his full supervisor, Prof. Toshiharu
Kazama for his supporting study, helping with living in Japan, friendly discussions, reducing
study stress, and invaluable advice and suggestions for accomplishment of his research. All the
research work and thesis writing involved in this dissertation are completely under the careful
guidance of Prof. Toshiharu Kazama. If the author could make some progress and achievementsscientifically, it is exactly the result of the appropriate guidance by Prof. Toshiharu Kazama.
Prof. Toshiharu Kazama’s academic rigorous, realistic attitude, responsible dedication and
excellent personal morality make author deeply admire. His profound knowledge, broad vision,brisk academic insight, active academic thinking and rigorous research style have greatly
benefited the author academically and ideologically. From the topic selection to completion ofthis dissertation, every step is operated under the guidance of Prof. Toshiharu Kazama withplenty of efforts. On the occasion of completion of this dissertation, the author would like toexpress heartfelt thanks and highest respect to his supervisor, Prof. Toshiharu Kazama.
The author also thanks to his two associate supervisor, Prof. Naohiko Hanajima and Prof.
Koji Teramoto, and Assoc. prof. Yukihito Narita for kindly academic advice and supporting
study. Thanks to gasket research group members, Mr. Kazuki Aizawa, Mr. Wataru Tomiyama,and Mr. Taiga Sato for kindly assistance with the experiment. Thanks to the precious friends
Mr. Hiroto Kurata, Mr. Hayato Okanishi, and Mr. Wataru Mita for taking care all the time, theauthor’s doctoral life has become rich and colorful with their company. Recollecting every
moment with them and the years of friendship, it is the most beautiful picture during author’s
life in Japan. Thanks to Mr. Toshiki Haga, Mr. Takumi Furuya, Mr. Yudai Takahashi, Mr.
Izuru Takeda, Mr. Toma Shiraki, Mr. Yuma Sasaki and all members in the Mechanical Systems
Design Laboratory B320 for excluding difficulty, relieving stress, and encouragement.
The author also would like to express the special thanks to Ms. Hatsuki Noda, Ms. Kozue
Takekawa, Mr. Jinro Endo and all the staffs in the Center for International Relations of Muroran
IT for making his life comfortable and enjoyable in Japan. Thanks to Prof. Naoko Yamaji and
Assoc. prof. Masatsugu Ono for providing the authentic knowledge of Japanese language.
Last but not least, sincerely thanks to Muroran Institute of Technology for giving the
precious opportunity to entrance and dense academic atmosphere which make the author’s
worthwhile graduate life has become much meaningful.
This study is intended for the self-development of the author. Giving knowledge, ability,
and living with others, patience, diligence. The author is extremely cheerful, and very gratefulto these precious experiences in all over the life.
GAO SONG
March 2022
Muroran Institute of Technology- ii -

博士論文題目：
液体の温度－粘度特性を利用したフランジ・ガスケットの漏れ制御
氏名：
GAO SONG

論文の要旨
ガスケットは、静止固体面間における隙間の漏れを防ぐために、機械装置から生活用品の静的シールとして広く使用されている。液体の漏れは、隙間の高さの 3 乗に比例することから、基本的には隙間を狭める方策が採られる。通常、材料の弾性および
塑性変形を用いることで防止される。その場合、しばしばクランプ力を増加させるが、過度のクランプ力は、表面の損傷を引き起こす可能性が高い。さらに、固体表面に存在する粗さやうねりのため、隙間を完全に埋めることは不可能である。一方で、液体の物性値に着目すると、温度が低いほど粘度が高くなる。漏れは液体の粘度に反比例する。よって、漏れの流路の温度を制御することで、漏れを抑え、ガスケットのシー
ル性能を一段と高める可能性が考えられる。本研究では、その可能性を熱流体潤滑理論に基づく数値モデルとふたつの金属円板で構成させたフランジ・ガスケットのモデルによる実験により検証した。本論文は、全体で 8 章で構成される。
第 1 章は序論である。静的および動的シールに関する概要をまとめ、数学モデルの基礎となる流体潤滑理論ならびに液体の粘度特性を概観し、ガスケットを中心に従来の研究のレビューならびに本研究の位置づけを述べている。
第 2 章は理論モデルである。本数学モデルの基礎となる、熱流体潤滑理論に基づき、レイノルズ方程式、エネルギー方程式、熱伝導方程式などの基礎方程式ならびに圧力と温度の境界条件を導出している。また、等粘度理論との比較、数値計算方法、圧力や温度分布を示している。
第 3 章は実験装置と方法である。2 枚の金属製円板によるフランジ・ガスケットのモデルを述べている。ペルチェモジュールにより、その円板を冷却・加熱することにより、隙間を流れる液体の粘度を制御する装置の構成を示し、加圧や加振なども含めて、実験および計測の方法について述べている。
第 4 章は定常状態下における実験結果である。隙間、供給圧力、シール面寸法、潤滑油の粘度グレード、隙間の傾斜の影響などについて述べている。
第 5 章は加振条件下における実験結果である。加振条件の実験や計測の方法、加振の周波数や振幅に関する実験結果に加えて、振動条件下における傾斜隙間の計算結果などを述べている。

- iii -

第 6 章は湾曲したシール面の場合の数値結果である。理論計算により、凹凸面を呈する、温度制御されたシール面が漏れに及ぼす影響について述べている。
第 7 章は実際のフランジガスケットを模擬した実験結果である。JIS 規格のフランジにペルチェモジュールを装着し、シール面を垂直方向に設置し、加圧した潤滑油を供給する条件において、ガスケットを冷却することで漏れの効果を検証している。
第 8 章は本論文の結論である。定常状態下はもとより、振動、寸法、潤滑油種、隙間形状、設置方法等によらず、フランジガスケットのシール面を冷却・加熱すること
により、漏れを制御することができることを理論と実験の両面により検証することができたこと、ならびにその有効性や発展性を述べている。

キーワード ：シール、ガスケット、液体、潤滑油、粘度、温度、漏れ、振動、平坦度- iv -

Abstract
Gaskets are widely used as static seals in industry, machinery, and living ware forpreventing leakage between the static solid surfaces. Since leakage of fluid is proportional tothe cube of gap height, the basic method for preventing leakage is taken to narrow the gap.
Generally, leakage is decreased by using elastic and plastic deformation of material, whichcaused by the increased clamping force. However, excessive clamping force may lead to surface
damage. Moreover, it is impossible to fill the gap completely because of the rough nature andwaviness of the solid surfaces of gasket. On the other hand, according to the liquid physicalcharacteristics in viscosity, leakage is inversely proportional to the lubricant viscosity, as lower
temperatures correspond to higher viscosities. Therefore, it is possible to control the leakage bychanging the temperature of sealing flanges to enhance the sealing performance of gasket
further. In this dissertation, the possibility was confirmed by using the mathematical model
based on the thermo-hydrodynamic lubrication theory (THL) and iso-viscous (IV) theory, andexperiments with a flange-type gasket which composed of two metal disks. The dissertationconsists of eight chapters in total.
Chapter 1 is the introduction. It summarizes the outline of static and dynamic sealing, theoverview of fluid lubrication theory and liquid viscosity characteristics that form the basics ofmathematical models were shown. And it reviewed plenty of conventional studies focusing on
gaskets, and described the position of this research.
Chapter 2 is the theoretical model. The mathematical model is based on the thermohydrodynamic lubrication theory, the basic differential equations such as the Reynolds equation,
the energy equation, and the heat conduction equation, as well as the boundary conditions ofpressure and temperature are derived. The theoretical solution by using the THL theory wascompared with the IV theory, the numerical calculation procedure, and the pressure andtemperature distribution were shown.
Chapter 3 is the experimental apparatus and method. The flange-type gasket model
consists of two metal disks was depicted. The configuration of the experimental devices that
controls the viscosity of the liquid flowing through the gap by cooling and heating the disk fromthe Peltier module was shown, and the experimental and measurement methods includingpressurization and vibration were also described.
Chapter 4 is the experimental results under steady-state condition. The effect of the gap
height, supply pressure, sealing land size, oil viscosity grade, and gap parallelism on leakage offlow were described.
Chapter 5 is the experimental results under the vibration condition. The experimental andmeasurement methods of the vibration conditions and the experimental results on the vibration
frequency and amplitude, including the theoretical solutions of the inclined gap with vibrations
were described.

- v -Chapter 6 is the numerical results for the curved sealing surface of flange. The effect oftemperature-controlled sealing flange with uneven surfaces on leakage was solved by iteration
calculation based on the THL theory.
Chapter 7 is the experimental results simulating an actual flange-type gasket. The Peltier
module was mounted on the JIS standard flange and the sealing surfaces were installedvertically. The leakage–restrain phenomenon was verified by cooling the gasket flanges under
the condition of supplying bolt preload torque and pressurized lubricating oil.
Chapter 8 is the conclusion of this dissertation. The leakage can be controlled by coolingand heating the sealing surface of the flange-type gasket regardless of vibration, sealing landsize, oil viscosity grade, gap height, installation direction, etc., as well as under steady-state
conditions. It verified that the leakage from gasket clearance can be effective controlled in bothof theoretically and experimentally, which has effectiveness and development potential insociety.
Keywords : Seal, Gasket, Liquid, Lubricant, Viscosity, Temperature, Leakage, Vibration,
Flatness- vi -

List of Abbreviations
SymbolName of Measure𝐴𝐴

output vibration amplitude

𝑐𝑐𝑝𝑝

specific heat conductivity of lubricant

𝑓𝑓

vibration frequencyℎ𝑚𝑚𝑚𝑚𝑚𝑚

minimum clearance𝑃𝑃𝑒𝑒atmospheric pressure

𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜

instant leakage flow at a specific time point

𝑇𝑇𝑎𝑎

ambient temperature

𝑇𝑇𝑤𝑤

solid wall temperature∗

one period of vibration

𝛽𝛽viscosity–temperature index

𝜇𝜇

oil viscosity𝜌𝜌

oil density

𝐴𝐴0

specified amplitude

𝑑𝑑1

inner diameter of sealing ringℎgap clearanceℎ𝑡𝑡

heat transfer coefficient𝑃𝑃𝑟𝑟center recess pressure

𝑅𝑅1

inner radius

𝑇𝑇𝑖𝑖𝑖𝑖

inlet oil temperature𝐵𝐵

disk thickness

𝑑𝑑2

outer diameter of sealing ringℎ𝑐𝑐

center clearanceℎ0

mean clearance

𝑄𝑄𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

mean leakage flow

𝑅𝑅2

outer radius𝑇𝑇𝑟𝑟

recess temperature of the disk𝛼𝛼

gap inclined angle

𝜆𝜆′disk thermal conductivity𝜈𝜈

kinematic viscosity𝑇𝑇𝜆𝜆

oil thermal conductivity

𝜇𝜇𝑚𝑚

mean viscosity𝜌𝜌𝑚𝑚mean density

Unit Symbolmmm

J⁄(kg ∙ K)
m
m

Hz
m
m
m

W⁄(m2 ∙ K)
m

Pa
Pa

m 3 ⁄sm 3 ⁄s
m
m

℃
℃
℃
℃
s
°

1⁄ ℃

W⁄(K ∙ m)
W⁄(K ∙ m)
Pa ∙ sPa ∙ sm 2 ⁄s

kg⁄m3
kg⁄m3

- vii -

CONTENTS
Acknowledgement .............................................................................................................. ii
論文の要旨 ....................................................................................................................... iii
Abstract .............................................................................................................................. v
List of Abbreviations ........................................................................................................ vii
CONTENTS .................................................................................................................... viii
List of Figures and Tables ................................................................................................. xi
1 Introduction .................................................................................................................. 1
1.1 General Background .......................................................................................... 1
1.1.1 Static Seal ............................................................................................... 1
1.1.2 Dynamic Seal ......................................................................................... 1
1.1.3 Selection of Seal Type............................................................................ 2
1.2 Lubrication ........................................................................................................ 3
1.2.1 Hydrostatic Lubrication.......................................................................... 3
1.2.2 Hydrodynamic Lubrication .................................................................... 4
1.2.3 Elasto-hydrodynamic Lubrication .......................................................... 5
1.2.4 Boundary Lubrication ............................................................................ 5
1.2.5 Solid Lubrication .................................................................................... 5
1.3 Characteristic of Viscosity ................................................................................ 6
1.3.1 Dynamic Viscosity and Kinematic Viscosity......................................... 6
1.3.2 Viscosity-temperature Relationship ....................................................... 7
1.4 Current Research ............................................................................................... 8
1.5 Objective of this Study ...................................................................................... 9
1.6 Chapter Summary............................................................................................ 10
References ........................................................................................................................ 10
2 Theoretical Model ...................................................................................................... 13
2.1 The THL Theory ............................................................................................. 13
2.1.1 Establishment of the THL Model ......................................................... 13
2.1.2 Generalized Reynolds Equation ........................................................... 15
2.1.3 Energy Equation of Oil Film ................................................................ 16
2.1.4 Heat Conduction Equation of Upper Disk ........................................... 18
2.1.5 Boundary Conditions of Reynolds Equation........................................ 18
2.1.6 Boundary Conditions of Temperature Field......................................... 18
2.1.7 Gap Clearance, Flow Velocity and Leakage Flow ............................... 19
2.1.8 Viscosity-temperature Relationship of Lubricant ................................ 19
2.2 The IV Theory ................................................................................................. 19
2.3 Analysis of Pressure Distribution.................................................................... 20- viii -2.3.1 Principle and Difference Format of FDM ............................................ 21
2.3.2 Division of Discrete Grid in Oil Film .................................................. 22
2.3.3 Difference Format in Reynolds Equation............................................. 23
2.3.4 Boundary Conditions of Pressure ......................................................... 24
2.4 Analysis of Temperature Distribution ............................................................. 24
2.4.1 Discrete Grid and Computing Scheme ................................................. 24
2.4.2 Iteration Format of Energy Equation ................................................... 26
2.4.3 Iteration Format of Heat Conduction Equation .................................... 27
2.4.4 Iteration Form of Temperature Boundary Conditions.......................... 28
2.4.5 Preview Iterative Program of Temperature .......................................... 29
2.5 Construction of Program and Calculation Results .......................................... 31
2.5.1 Initial Parameters and Calculation Process .......................................... 32
2.5.2 Iteration Result of Pressure Distribution under Parallel Gap ............... 34
2.5.3 Iteration Result of Pressure Distribution under Inclined Gap .............. 36
2.5.4 Iteration Result of Temperature Distribution under Parallel Gap ........ 40
2.5.5 Iteration Result of Temperature Distribution under Inclined Gap ....... 44
2.6 Chapter Summary............................................................................................ 47
References ........................................................................................................................ 47
3 Experimental Apparatus and Method......................................................................... 51
3.1 Experimental Principle and Objective............................................................. 51
3.2 Experimental Apparatus .................................................................................. 51
3.3 Photographs of Apparatuses and their Parameters .......................................... 54
3.4 Experimental Method ...................................................................................... 56
3.5 Chapter Summary............................................................................................ 56
References ........................................................................................................................ 56
4 Experimental Results under Steady-state Condition .................................................. 59
4.1 Effects of Gap Height on Leakage .................................................................. 59
4.2 Effects of Supply Pressure on Leakage ........................................................... 60
4.3 Effects of Sealing Land Size on Leakage........................................................ 61
4.4 Effects of Oil Type Viscosity Grade on Leakage ........................................... 62
4.5 Effects of Inclined Angle on Leakage ............................................................. 63
4.6 Chapter Summary............................................................................................ 64
References ........................................................................................................................ 65
5 Experimental Results under Vibration Condition ...................................................... 67
5.1 Explanation of Experimental Row Data.......................................................... 68
5.2 Comparison Result of Vibrations with Parallel Disks .................................... 68
5.2.1 Effects of Vibration Frequency on Leakage ........................................ 68
5.2.2 Effects of Vibration Amplitude on Leakage ........................................ 69- ix -

5.3 Calculation Result of Vibrations with Inclination........................................... 70
5.3.1 Effects of Vibration Frequency on Leakage in Nonparallel ................ 71
5.3.2 Effects of Vibration Amplitude on Leakage in Nonparallel ................ 72
5.4 Chapter Summary............................................................................................ 72
References ........................................................................................................................ 73
6 Simulation of Effect of Sealing Surface Flatness ...................................................... 75
6.1 Theoretical Model and Grid Nodes with Surface Flatness ............................. 75
6.2 Theoretical Solutions of Effect of 𝛿𝛿 on Temperature Distribution ............... 77
6.2.1 Temperature Solutions under a Parallel Gap Condition....................... 77
6.2.2 Temperature Solutions in Gap-disk Interface under Inclination .......... 79
6.3 Theoretical Solutions of Effect of 𝛿𝛿 on Flow Velocity and Leakage............ 80
6.3.1 Flow Velocity under a Parallel Gap Condition .................................... 80
6.3.2 Leakage Flow under Static Conditions of Parallel and Inclination ...... 82
6.3.3 Leakage Flow under Vibrations of Parallel and Inclination ................ 83
6.4 Simulations in Parallel Conditions of Keeping Same Outlet Gap .................. 84
6.5 Chapter Summary............................................................................................ 85
References ........................................................................................................................ 86
7 Vertical Flange-type Gasket Model ........................................................................... 87
7.1 Experimental Apparatus .................................................................................. 87
7.2 Experimental Parameters................................................................................. 88
7.3 Experimental Method and Conversion of Leakage ......................................... 88
7.3.1 Experimental Method ........................................................................... 88
7.3.2 Conversion of Leakage......................................................................... 89
7.4 Experimental Results under Steady-state Condition ....................................... 90
7.4.1 Effects of Bolt Preload Torque on Leakage ......................................... 90
7.4.2 Effects of Recess Pressure on Leakage ................................................ 91
7.5 Chapter Summary............................................................................................ 92
References ........................................................................................................................ 93
8 Conclusion and Prospect ............................................................................................ 95
8.1 Conclusion....................................................................................................... 95
8.2 Prospect ........................................................................................................... 96
8.3 Chapter Summary............................................................................................ 97
References ........................................................................................................................ 97
Curriculum Vitae .............................................................................................................. 99
Muroran Institute of Technology Doctoral Dissertation Copyright Use Authorization 101- x -List of Figures and Tables
Fig. 1-1 Typical flange-type gasket.................................................................................... 1
Fig. 1-2 Section of hydraulic cylinder ................................................................................ 2
Fig. 1-3 Velocity distribution in Newton viscous flow ...................................................... 7
Fig. 1-4 Viscosity–temperature relationship of various lubricating oils ............................ 8
Fig. 2-1 Schematic diagram of the THL lubrication model ............................................. 14
Fig. 2-2 Schematic diagram of oil film ............................................................................ 16
Fig. 2-3 Schematic diagram of pressure grid nodes in oil film ........................................ 22
Fig. 2-4 Schematic diagram of temperature grid nodes in oil film .................................. 25
Fig. 2-5 Schematic diagram of temperature grid nodes in upper disk ............................. 25
Fig. 2-6 Flowchart for THL analysis ................................................................................ 33
Fig. 2-7 Relation between central gap ℎ𝑐𝑐 and time 𝑡𝑡 under vibration .......................... 34
Fig. 2-8 Pressure distribution in parallel gap (static; 𝑡𝑡 = 0s) ......................................... 35
Fig. 2-9 Pressure distribution in parallel gap (𝑡𝑡 = 0.025s; 𝑡𝑡 = 0.05s) .......................... 35
Fig. 2-10 Pressure distribution in parallel gap (𝑡𝑡 = 0.075s; 𝑡𝑡 = 0.1s) .......................... 35
Fig. 2-11 Pressure distribution in parallel gap (𝑡𝑡 = 0.125s; 𝑡𝑡 = 0.15s) ........................ 36
Fig. 2-12 Pressure distribution in parallel gap (𝑡𝑡 = 0.175s; 𝑡𝑡 = 0.2s) .......................... 36
Fig. 2-13 Pressure distribution in inclined gap (static; 𝑡𝑡 = 0s) ...................................... 37
Fig. 2-14 Pressure distribution in inclined gap (𝑡𝑡 = 0.025s; 𝑡𝑡 = 0.05s) ....................... 37
Fig. 2-15 Pressure distribution in inclined gap (𝑡𝑡 = 0.075s; 𝑡𝑡 = 0.1s).......................... 38
Fig. 2-16 Pressure distribution in inclined gap (𝑡𝑡 = 0.125s; 𝑡𝑡 = 0.15s) ....................... 38
Fig. 2-17 Pressure distribution in inclined gap (𝑡𝑡 = 0175s; 𝑡𝑡 = 0.2s) .......................... 38
Fig. 2-18 Pressure contour line in inclined gap (𝑡𝑡 = 0s; 𝑡𝑡 = 0.025s) ............................ 39
Fig. 2-19 Pressure contour line in inclined gap (𝑡𝑡 = 0.075s; 𝑡𝑡 = 0.1s) ......................... 39
Fig. 2-20 Theoretical solution of 𝑇𝑇′ in upper disk (ℎ𝑐𝑐 = 0.01 mm) ............................. 40
Fig. 2-21 Theoretical solution of 𝑇𝑇 in oil film (ℎ𝑐𝑐 = 0.01 mm) ................................... 41
Fig. 2-22 Theoretical solution of 𝑇𝑇′ in upper disk (ℎ𝑐𝑐 = 0.1 mm) ................................ 41
Fig. 2-23 Theoretical solution of 𝑇𝑇 in oil film (ℎ𝑐𝑐 = 0.1 mm) ..................................... 41
Fig. 2-24 Temperature contour in disk and oil film (ℎ𝑐𝑐 = 0.01 mm)............................. 42
Fig. 2-25 Temperature contour in disk and oil film (ℎ𝑐𝑐 = 0.1 mm) ............................... 42
Fig. 2-26 Temperature contour in disk and oil film (𝑇𝑇𝑤𝑤 = 15℃) .................................... 43
Fig. 2-27 Temperature contour in disk and oil film (𝑇𝑇𝑤𝑤 = 25℃) .................................... 43
Fig. 2-28 Temperature contour in disk and oil film (𝑇𝑇𝑤𝑤 = 35℃) .................................... 43
Fig. 2-29 Temperature contour in disk and oil film (𝑇𝑇𝑤𝑤 = 45℃) .................................... 44
Fig. 2-30 Contour of 𝑇𝑇 in gap–disk interface (𝛼𝛼 = 0.057°) .......................................... 44
Fig. 2-31 Contour of 𝑇𝑇 in gap–disk interface (𝛼𝛼 = 0.086°) .......................................... 45- xi -

Fig. 2-32 Contour of 𝑇𝑇 in gap–disk interface (𝑇𝑇𝑤𝑤 = 15℃) ........................................... 45
Fig. 2-33 Contour of 𝑇𝑇 in gap–disk interface (𝑇𝑇𝑤𝑤 = 25℃) ........................................... 46
Fig. 2-34 Contour of 𝑇𝑇 in gap–disk interface (𝑇𝑇𝑤𝑤 = 35℃) ........................................... 46
Fig. 2-35 Contour of 𝑇𝑇 in gap–disk interface (𝑇𝑇𝑤𝑤 = 45℃) ........................................... 46
Fig. 3-1 Schematic diagram of the experimental rig ........................................................ 52
Fig. 3-2 Internal section surface of gasket tester .............................................................. 52
Fig. 3-3 Sealing land between two disks .......................................................................... 53
Fig. 3-4 Schematic of all the experimental apparatuses ................................................... 53
Fig. 3-5 Photograph of gasket facility .............................................................................. 54
Fig. 3-6 Photograph of function parts .............................................................................. 54
Fig. 3-7 Photograph of sealing land ................................................................................. 55
Fig. 3-8 Inner structure of gasket facility ......................................................................... 55
Fig. 4-1 Effect of gap height on leakage (𝑃𝑃𝑟𝑟 : 1 kPa) ........................................................ 59
Fig. 4-2 Effect of gap height on leakage (𝑃𝑃𝑟𝑟 : 0.6 kPa) ..................................................... 60
Fig. 4-3 Effect of recess pressure on leakage (ℎ𝑐𝑐 : 0.15 mm) ........................................... 60
Fig. 4-4 Effect of recess pressure on leakage (ℎ𝑐𝑐 : 0.2 mm) ............................................. 61
Fig. 4-5 Effect of sealing size on leakage (ℎ𝑐𝑐 : 0.15 mm, 𝑃𝑃𝑟𝑟 : 1.4 kPa) ............................ 61
Fig. 4-6 Effect of sealing size on leakage (ℎ𝑐𝑐 : 0.20 mm, 𝑃𝑃𝑟𝑟 : 1 kPa) ............................... 62
Fig. 4-7 Effect of oil type on leakage (ℎ𝑐𝑐 : 0.2 mm, 𝑃𝑃𝑟𝑟 : 1 kPa) ....................................... 62
Fig. 4-8 Effect of oil type on leakage (ℎ𝑐𝑐 : 0.3 mm, 𝑃𝑃𝑟𝑟 : 0.6 kPa) .................................... 63
Fig. 4-9 Effect of inclined angle on leakage (ℎ𝑐𝑐 : 0.15 mm, 𝑃𝑃𝑟𝑟 : 1.4 kPa) ........................ 63
Fig. 4-10 Effect of inclined angle on leakage (ℎ𝑐𝑐 : 0.2 mm, 𝑃𝑃𝑟𝑟 : 1 kPa) ........................... 64
Fig. 5-1 Experimental row data of vibration parameters of two disks ............................. 68
Fig. 5-2 Effect of frequency on leakage (ℎ𝑐𝑐 : 0.15 mm, 𝑃𝑃𝑟𝑟 : 1.4 kPa) .............................. 69
Fig. 5-3 Effect of frequency on leakage (ℎ𝑐𝑐 : 0.2 mm, 𝑃𝑃𝑟𝑟 : 1 kPa) ................................... 69
Fig. 5-4 Effect of amplitude on leakage (ℎ𝑐𝑐 : 0.15 mm, 𝑃𝑃𝑟𝑟 : 1.4 kPa) .............................. 70
Fig. 5-5 Effect of amplitude on leakage (ℎ𝑐𝑐 : 0.2 mm, 𝑃𝑃𝑟𝑟 : 1 kPa) ................................... 70
Fig. 5-6 Calculation for effect of frequency with inclination (𝛼𝛼 = 0.057°) .................... 71
Fig. 5-7 Calculation for effect of frequency with inclination (𝛼𝛼 = 0.086°) .................... 71
Fig. 5-8 Calculation for effect of amplitude with inclination (𝛼𝛼 = 0.057°) .................... 72
Fig. 5-9 Calculation for effect of amplitude with inclination (𝛼𝛼 = 0.086°) .................... 72
Fig. 6-1 Schematic diagram of gasket model under convex condition ............................ 76
Fig. 6-2 Variation in gap ℎ with radius 𝑟𝑟 under a parallel gap condition .................... 76
Fig. 6-3 3-D schematic diagram under convex condition ................................................ 77
Fig. 6-4 Main view of schematic diagram under convex condition ................................. 77
Fig. 6-5 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with flat flange ................................................ 78
Fig. 6-6 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with convex flange ......................................... 78
Fig. 6-7 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with concave flange ........................................ 78- xii -

Fig. 6-8 Temperature map of gap–disk interface in inclined with flat disk ..................... 79
Fig. 6-9 Temperature map of gap–disk interface in inclined with convex disk ............... 80
Fig. 6-10 Temperature map of gap–disk interface in inclined with concave disk ........... 80
Fig. 6-11 Distribution of Velocity 𝑢𝑢 for flat sealing surface ......................................... 81
Fig. 6-12 Distribution of Velocity 𝑢𝑢 for convex sealing surface ................................... 81
Fig. 6-13 Distribution of Velocity 𝑢𝑢 for concave sealing surface .................................. 81
Fig. 6-14 Effect of 𝛿𝛿 on leakage under static parallel gap conditions............................ 82
Fig. 6-15 Effect of 𝛿𝛿 on leakage under static inclined conditions ................................. 82
Fig. 6-16 Effect of 𝛿𝛿 on leakage in parallel gap under vibrations.................................. 83
Fig. 6-17 Effect of 𝛿𝛿 on leakage in inclined gap under vibrations ................................. 83
Fig. 6-18 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with convex flange with same ℎ𝑜𝑜𝑜𝑜𝑜𝑜 ............. 84
Fig. 6-19 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with concave flange with same ℎ𝑜𝑜𝑜𝑜𝑜𝑜 ........... 84
Fig. 6-20 Effect of 𝛿𝛿 on leakage under parallel gap with same ℎ𝑜𝑜𝑜𝑜𝑜𝑜 ............................ 85
Fig. 7-1 Schematic diagram of vertical gasket rig ........................................................... 88
Fig. 7-2 Layout of all the vertical flange-type gasket devices ......................................... 89
Fig. 7-3 Relationship between the oil mass of one drop and temperature ....................... 90
Fig. 7-4 Effects of bolt preload torque on leakage (𝑃𝑃𝑟𝑟 : 7.0 kPa) ...................................... 91
Fig. 7-5 Effects of bolt preload torque on leakage (𝑃𝑃𝑟𝑟 : 20.0 kPa) .................................... 91
Fig. 7-6 Effect of recess pressure on leakage (Torque: 5.0 N ∙ m) .................................. 92
Fig. 7-7 Effect of recess pressure on leakage (Torque: 10.0 N ∙ m) ................................ 92
Table 2-1 Gasket parameters and lubricant properties ..................................................... 34
Table 3-1 Parameters of three types of sealing land ........................................................ 55- xiii -- xiv -1 Introduction

1

Introduction

1.1 General Background
Gaskets and gasketed joints [1-3] are widely used as static seals in industry, machinery,and living ware. Sealing refers to suppress the leakage of fluid or solid particles from theadjacent joint surface, and to prevent the external impurities such as dust and moisture frominvading the internal parts or measures of machinery and equipment, it is also used to avoid theliquid from flowing out of mechanical components. Seal can be mainly divided into static seal
and dynamic seal.

1.1.1

Static Seal

Static seal is also called fixed seal, which refers to the seal between two stationary surfaces,
and the sealed components are in the condition of static [4-6]. In the working environment ofstatic seal, there is no relative motion between the two coupling seal parts.
Static seal mainly includes gasket seal, sealant seal and direct contact seal. According to
the working pressure, static seal can also be divided into medium-pressure or low-pressure static
seal and high-pressure static seal. The high-pressure static seals are usually made of hard gasket
with narrow contact width. Figure 1-1 shows the flange-type gasket, which is a typical static
seal element. Dynamic seal can be divided into rotary seal and reciprocating seal. Other typesof seal are including contact seal and non-contact seal according to whether they are in contact
with the relative moving parts, they can also be divided into circumferential seal and edge sealaccording to the contact position, and edge seal is also called mechanical seal.

Fig. 1-1 Typical flange-type gasket

1.1.2

Dynamic Seal

Dynamic seal refers to the seal between relative moving elements in machine or equipment[7] such as the hydraulic cylinder in Fig. 1-2. Reciprocating dynamic seal has formed packing- 1 -PhD Dissertation of Muroran Institute of Technology
seal, stuffing box seal and expansion ring seal. Rotary dynamic seals include: contact dynamic
seals [8] with moving parts in direct contact with stationary parts; non-contact dynamic seals
without direct contact; semi fluid dynamic seals [9] with special properties of fluid added in thesealing gap. The semi fluid dynamic seal also includes magnetic fluid seal. Several of machines
and equipment must be absolutely sealed, some allow a small amount of leakage, so the sealing
structure should be selected and designed according to the nature of the working medium,
temperature, pressure, relative speed and other operating conditions as well as the requirements
for sealing performance.

Fig. 1-2 Section of hydraulic cylinder(Huahui hydraulic machinery Co., Ltd, 2019. http://www.gdhuahui.com/news/1071.html)

1.1.3

Selection of Seal TypeFor mechanical elements, the sealing methods of parts in different working environments
are also different. The selection of sealing type has a crucial impact on its performance. The
main factors affecting the sealing performance of gasket and packing are as follows:(1) The condition of sealing surface. The shape and roughness of the sealing surface havea certain influence on the sealing performance, and the smooth surface is conducive tosealing. Soft gasket is not sensitive to surface condition because it is easy to deform,
while hard gasket has great influence on surface condition.
(2) The contact width of sealing surface. The larger the contact width between the sealing
surface of gasket or packing, the longer the path of fluid leakage and the greater theloss of flow resistance, which is conducive to sealing.
(3) The properties of fluid. The viscosity of fluid has a great influence on the sealing
performance of packing and gasket. The fluid with high viscosity is easy to seal
because of its poor fluidity. The larger the molecular volume of fluid is, the easier it is
to be blocked by narrow sealing gap, so it is easier to seal; The wettability of liquid tosealing material also has a certain impact on sealing.

- 2 -1 Introduction(4) The temperature of fluid. The temperature affects the viscosity of the fluid, thus
affecting the sealing performance. With the increase of temperature, the viscosity ofliquid decreases. On the other hand, the change of temperature often results in the
deformation of the sealing components, which is easy to cause leakage.
(5) Material of gasket and packing. Soft materials are easy to produce elastic or plastic
deformation under the action of jacking force, thus blocking the leakage channel of
fluid, which is conducive to sealing.
(6) The external conditions. The vibration of the pipeline system, the deformation of theconnecting components, the offset of the installation position and other reasons willproduce additional force on the seals, which will cause adverse effects.

1.2 Lubrication
The measure to reduce the friction coefficient and wear rate by adding substance with low
shear resistance between the interacting surfaces with relative motion is called lubrication. The
added substance can be liquid, gas or solid. Effective lubrication method could reduce friction
coefficient to about 0.0001 and the wear rate to zero.
Lubrication can be divided into five basic types in mechanics: hydrostatic lubrication,hydrodynamic lubrication, elasto-hydrodynamic lubrication, boundary lubrication and solid
lubrication.

1.2.1

Hydrostatic Lubrication

Hydrostatic lubrication refers to the fluid lubricant with a certain pressure is transported
to the supported oil chamber, forming a lubrication film with enough static pressure to bear theload, and separating the surface, which is also called external pressure lubrication. With the
combination of hydraulic and pneumatic technology, hydrostatic lubrication has developed intoa relatively independent discipline. Its achievements, such as hydrostatic bearing, hydrostatic
guide rail, hydrostatic lead screw, have been widely used in machine tools, power machinery,
instrumentation and other industrial sectors, and achieved good economic results.
There are two types of lubricants for hydrostatic lubrication: oil and gas. The oil film isformed with oil flowing from outside to lubricating surface by external pressure. Hydrostatic
lubrication has incomparable advantages over other lubrication forms. It could completely
avoid wear with properly designed. The main characteristic of hydrostatic lubrication is that theoil film could bears external force in a wide speed range and static state without wear. The
advantages of hydrostatic lubrication are as follows:(1) Small starting friction, energy conservation and wide application range.
(2) High precision, anti-vibration performance, and adapted to various requirements.
(3) It is unnecessary to form convergence gap and enough relative velocity between two
relative moving surfaces. Moreover, the capacity and film thickness are independentof sliding speed.
- 3 -PhD Dissertation of Muroran Institute of Technology(4) Excellent oil film stiffness. By using special oil supply system and throttling device,
the stiffness of oil film could be infinite, and even when the load increases, the oil film
thickness adversely increase through the appropriate design of film thickness, so as tocompensate the deformation of other parts.
(5) Low friction coefficient. For large bearings, it could be started under the conditions offull film lubrication to eliminate the influence of static friction and reduce torque.

1.2.2

Hydrodynamic Lubrication

Hydrodynamic lubrication is to rely on the relative motion between a pair of solid friction
surfaces to produce pressure in the lubricating fluid film between the solids, so as to bear theexternal load, avoid solid contact, reduce friction resistance, and then protect the solid surface.
There are many researches study hydrodynamic lubrication. Wang and Zhu [10] conducted thenumerical analysis of micropolar fluids lubricating in journal bearing with considering thermaland cavitation effects, they discussed that the effect of different values of fluid parameter in the
bearing on operation such as pressure, temperature friction coefficient, and so on. Jeng andZhou [11] researched the thermohydrodynamic solution of pivoted thrust pads theoretically,
they solved the temperature and pressure distribution of pads with complicated iteration method.
Hydrodynamic lubrication theory was discovered in 1883 by B. tower, a German engineer. In
1886, a British scholar whose name is O. Reynolds applied the knowledge of hydromechanics
to explain tower’s experimental results. Then the partial differential equation describing thehydrodynamic pressure distribution was derived and the analytical solution of the equation was
obtained. The work of Tower and Reynolds announced the birth of the fluid lubrication theory.
Hydrodynamic lubrication is also operated with the oil film formed by lubricant to separate
two solid surfaces completely [12]. Different from hydrostatic lubrication, the pressure in fluid
film is produced by the relative motion of the surface to carry the viscous fluid into theconvergent gap. Therefore, dynamic pressure lubrication is an economical way for lubrication,and it is also the most widely used. The friction resistance of this kind of lubrication is mainly
produced by the internal friction of the liquid. The friction and wear characteristics mainly
depend on the viscosity of liquid, and no relationship with the material characteristics andmorphology of two solid surfaces. The necessary conditions for the formation of hydrodynamic
lubrication are as follows:(1) The wedge-shaped is exited between two solid surfaces.
(2) The clearance is filled with viscous fluid, which can be attached on the solid surfaces.
(3) The relative motion of the two solid surfaces drives the lubricating fluid to move along
the direction of large clearance to small clearance.
In addition, because that the lubricating oil film is established by the relative movementof two surfaces in hydrodynamic lubrication system, there is a stage when the machine starts
and brakes, which is not belong to full film lubrication, so it can not completely avoid wear,- 4 -1 Introduction
and the equipment generally could not start with load.

1.2.3

Elasto-hydrodynamic Lubrication

In the elasto-hydrodynamic lubrication, the friction between the two surfaces and thethickness of the lubricant oil film depend on its elastic deformation and the lubrication of the
lubricant rheological properties. Its object is the high contact stress parts, such as gear tooth
surface, rolling element and raceway of rolling bearing, cam and tappet, etc. Lubricants areusually mineral oils. Shinkarenko et al. [13] theoretically examined the surface texturing effectin soft elasto-hydrodynamic lubrication, the optimum range of parameters of surface texting
and operation conditions were studied. Xiang et al. [14] developed an elasto-hydrodynamic
lubrication simulation of a reciprocating rod seal with a texted rod, the effect of different texture
types and parameters on the sealing performance was analyzed by using stiffness matrix method.
The theory of elasto-hydrodynamic lubrication is the properties of hydrodynamic lubrication
film between two elastic objects under the condition of rolling or sliding with rolling. The
elastic equation for calculating the deformation of friction surface under oil film pressure, theviscous pressure equation for describing the relationship between lubricant viscosity andpressure, and the main equations for hydrodynamic lubrication are combined to solve theproblems of oil film pressure distribution and lubrication film thickness distribution.
For some cases, surface contact lubrication pairs with low stress will produce large elastic
deformation due to the small stiffness of the parts, which may cause a significant impact on the
lubricant. This lubrication state is often called the generalized elasto-hydrodynamic lubrication,
which is different from the classical elasto-hydrodynamic lubrication in high stress contact area.

1.2.4

Boundary Lubrication

In the condition of the lubricant can be added to the surface of the friction pair and form avery thin film when the hydrodynamic pressure film and elasto-hydrodynamic lubrication film
could not be achieved. This film can also reduce friction and reduce wear. This is called theboundary lubrication.
The boundary lubrication is the critical state before the transition from liquid friction to
dry friction. It is the lubrication condition of partial surface contact between unsmooth surfaces.
The viscosity characteristics of the lubricating oil is meaningless in this case. The tribology
properties of friction surfaces are determined by the interaction between lubricant and surface
and the properties of boundary film.

1.2.5

Solid Lubrication

Solid lubrication refers to the use of solid powder, film or integral material to reduce thefriction and wear of two surfaces in relative motion and protect the surface from damage. The
definition of solid lubrication is that solid powder or film used to protect the relative moving
surface from damage and reduce its friction and wear. Solid lubrication is usually existed when

- 5 -

PhD Dissertation of Muroran Institute of Technology
other lubrication methods cannot be used. Solid lubricants are generally layered solids, such asgraphite and molybdenum dioxide, or other inorganic solids, such as lead oxide and calcium
fluoride. PTFE and other plastics can also be used as solid lubricants.
The method of coating a layer of soft metal film on the surface of friction pair to reduce
friction also belongs to solid lubrication. Soft metals can be the cadmium, silver, etc. Solidlubrication is often used for the hinge of spacecraft. Under normal conditions, the clean metal
surface will be oxidized and polluted quickly, so that its surface is always covered with a layerof solid film. This kind of natural solid film also has the ability to reduce friction.
It is worth mentioning that various lubrication states are not divided strictly. Under certain
working conditions, there may be two or more kinds of lubrication types at the same time, whichis called mixed lubrication.

1.3 Characteristic of Viscosity
In lubrication theory, the most important physical property of lubricant is the viscosity. It
is the main factor to determine the oil film thickness. For example, in hydrodynamic lubrication,
the oil film thickness is directly proportional to viscosity, for elasto-hydrodynamic lubrication,
the oil film thickness is approximately proportional to the 0.7 power of viscosity.
On the other hand, viscosity is also an important factor affecting friction. High viscosity
grade of lubricant could not only cause great friction loss and heating, but also difficult toconvective heat dissipation. In this way, the increase of friction temperature may result in therupture of oil film and surface wear. Therefore, for any working condition, there is a reasonableviscosity range.
The viscosity and density of lubricant is varied with temperature, pressure and otherparameters. In hydrodynamic lubrication, the relationship between viscosity and temperature ismuch important.

1.3.1

Dynamic Viscosity and Kinematic Viscosity

When the fluid flows, the shear deformation of fluid particles can be existed due to theadhesion between fluid and solid surface, the interaction between molecules and the momentum
exchange between fluid particles. However, shear deformation can be resisted by the viscosityof fluid, which is used to describe the internal friction during flow.
Newton proposed the flow model of viscous fluid firstly. He proposed that the fluid flowis the sliding between many extremely thin layers. Due to the viscosity of the fluid, shear force
will be generated between the sliding layers, that is, the fluid internal friction, which can transfer
the motion to the adjacent fluid layers, so that the layer with faster flow will decelerate, while
the layer with slower flow will accelerate, and then forming the velocity distribution according
to a certain law. The viscosity is numerically equal to the shear stress in unit velocity gradient.
The velocity gradient also represents the angular deformation rate in fluid motion, so theviscosity can also be expressed by the ratio between shear stress and angular deformation rate.
- 6 -1 Introduction
Figure 1-3 shows that there is a flat plate with 𝐴𝐴 acreage on the fluid surface with a thicknessof ℎ, which moves at the velocity of 𝑢𝑢 under the force of 𝐹𝐹. The velocity 𝑢𝑢 of each layer islinearly distributed in this case because of the internal friction of viscous fluid, the motion istransferred to each layer of fluid in turn.

Fig. 1-3 Velocity distribution in Newton viscous flow
Newton also proposed the assumption that the viscous shear stress is proportional to theshear strain rate, which is called Newton's viscosity law now.
(1-1)

𝜏𝜏 = 𝜇𝜇𝛾𝛾̇

wherein, 𝜏𝜏 is the shear stress, i.e., the friction force per unit area, 𝜏𝜏 = 𝐹𝐹 ⁄𝐴𝐴, and its positive
direction is tangent to the plane. 𝛾𝛾̇ is the shear strain rate, i.e., the change rate of shear strain
with time 𝑡𝑡.
d𝛾𝛾

d d𝑥𝑥

d d𝑥𝑥

d𝑢𝑢

𝛾𝛾̇ = d𝑡𝑡 = d𝑡𝑡 d𝑧𝑧 = d𝑧𝑧 d𝑡𝑡 = d𝑧𝑧(1-2)

It can be seen that the shear strain rate is equal to the change rate of flow velocity along
the direction of fluid thickness. In this way, Newton's viscosity law can be written as𝜏𝜏 = 𝜇𝜇 d𝑧𝑧

d𝑢𝑢

(1-3)

𝜇𝜇

(1-4)

wherein, the coefficient 𝜇𝜇 is defined as the dynamic viscosity or absolute viscosity.
In the international system of units, the unit of dynamic viscosity is Pa ∙ s, or N ∙ s⁄m2 .
The instrument for measuring the fluid viscosity is called viscometer. The dynamic viscosity
could not be measured directly by many viscometers, but they can measure the ratio of dynamic
viscosity 𝜇𝜇 and density 𝜌𝜌. This ratio is called kinematic viscosity and is usually represented
by the symbol ν, its unit is m2 ⁄s.

1.3.2

ν = 𝜌𝜌

Viscosity-temperature Relationship

When the temperature increases, the distance between the molecules in the lubricating oil
increases and the intermolecular force decreases, so the viscosity decreases sharply and the- 7 -PhD Dissertation of Muroran Institute of Technology
lubrication situation becomes worse. Thermal analysis and temperature calculation are the maintasks of lubrication theory. At present, various forms of empirical formulas are widely used inlubrication analysis. For example, the following empirical formulas are often used in thermal
analysis of sliding bearings.
𝜇𝜇 = 𝜇𝜇0 exp[−𝛽𝛽(𝑇𝑇 − 𝑇𝑇0 )](1-5)

wherein, 𝜇𝜇0 is called the ambient viscosity, which is corresponding to oil temperature 𝑇𝑇0 ; 𝛽𝛽
is the viscosity-temperature index. Figure 1-4 shows the relationship between viscosity andtemperature of various types of lubricating oil.

Fig. 1-4 Viscosity–temperature relationship of various lubricating oils
(Jiaxingshun Electronic Technology Co., Ltd, 2020. http://www.sunonfan.cn/a/rhpghyzcfsdkkx.html)

1.4 Current Research
Existing studies have theoretically and experimentally proposed numerous feasible
methods for forecasting leakage and improving the gasket sealing performance. Jenco and Hunt[15] researched the generic issues effecting spiral-wound gasket performance, they discussed
the actions proven to mitigate leakage incidents, and offers suggestions for plant operators toenhance the sealing performance of gasket generally. Abid and Nash [16] studied two types of
conventional gasketed and non-gasketed joints under different conditions, the compact nongasketed joints were shown preferable alternative to the conventional gasket joints based on
tress results. Saeed et al. [17] introduced an all-metal material gasket which was combined- 8 -1 Introduction
circumferential annular lips as seal lines. They study verified that not only can effective gaskets
be developed using material methods but that optimization approaches based on the Taguchi
and finite difference method (FDM) can also be applied in this context. Jaszak [18] presented
the research of a bolted joint with an elastic serrated gasket numerically and experimentally.
They proved that making the core elastic by its side notch enhances tightness even of an orderof magnitude compared to the standard solution.
In recent years, with the development of electronic computer technology and scientificcomputational abilities, the influence of impact of parameters such as elastic, temperature andthe numerical methods have been received increasing attention. Krishna et al. [19] studied thesealing performance of bolted flange gasketed joint by using finite element analysis. They
presented that the contact stress was an important parameter effect on the sealing performance
than the limit on flange rotation specified. Zhang and Wang [20] applied the fluctuation entropy
and financial percolation model with random jump on gasket fractal lattice. Zhang et al. [21]presented the intense wear behavior of spiral wood gasket in the circumferential direction ofsmall displacement and the radial friction process of large displacement. The materials of gasket,
load conditions [22] and radial arrangement [23] also affect the performance of the generalgasket that provide critical seals between pipe segments and ensure the satisfactory operationof a board range in industry.

1.5 Objective of this StudyIn this dissertation, a prototype gasket [24] with a gap is modeled between flat flanges witha central recess. The prototype was developed using the inverse relationship between fluid
viscosity and the temperature dependence of hydraulic oils, i.e., the lower the temperature, thehigher the viscosity. Accordingly, the leakage is reduced by cooling the gasket sealing parts.
Additionally, the sealing land size, oil viscosity grade, inclined angle, and vibration parameters
also affect gasket sealing performance. The phenomena involved in gap leakage, controlledtemperature, and the relationship between the cooling of sealing parts and reducing leakage isresearched theoretically and confirmed experimentally. The experiment is conducted undervarious conditions and includes aspects such as the solid wall temperature, sealing land size,
oil viscosity grade, inclined angle, and vibration parameters such as frequency and amplitude.
The THL and IV theories are both applied to gap flow under conditions of static and harmonic
vibrations. The influence of parameters on leakage is numerically solved by finite difference
method (FDM). Additionally, the THL theory is used for estimating leakage in nonparallel
vibration conditions. The flatness of sealing flange surface was simulated theoretically in flat,
convex, and concave. The effect of flatness on temperature distribution in disk and oil film,
flow velocity, and leakage were compared in parallel and inclination. Finally, a vertical flangetype gasket model was established, that is closer to the practical application in realistic society.
The experiment focused on factors of bolt preload torque and recess pressure.

- 9 -PhD Dissertation of Muroran Institute of Technology

1.6 Chapter Summary
This chapter introduces the several types of sealing elements and the classification andbasic knowledge of fluid lubrication theory, leads to the research motivation and content of thisdissertation subsequently.

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- 10 -

1 Introduction[13] Shinkarenko, A., Kligerman, Y. and Etsion, I., The effect of surface texturing in soft elastohydrodynamic lubrication, Tribology International, Vol.42 (2009), pp.284-292.
[14] Xiang, C., Tan, L., Guo, F., Huang, X. and Wang, Y. M., Elastohydrodynamic lubrication
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International Journal of Pressure Vessels and Piping, Vol.77 (2000), pp.825-830.
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Mechanics and Materials Engineering, Vol.2, No.1 (2008), pp.105-114.
[18] Jaszak, P., The elastic serrated gasket of the flange bolted joints, International Journal ofPressure Vessels and Piping, Vol.176 (2019), pp.103-115.
[19] Krishna, M. M., Shunmugam, M. S. and Prasad, N. S., A study on the sealing performanceof bolted flange joints with gaskets using finite element analysis, International journal ofPressure Vessels and Piping, Vol.84 (2007), pp.349-357.
[20] Zhang, Y. L. and Wang, J., Fluctuation entropy and complexity of financial percolation
model with random jump on gasket fractal lattice, Physica A, Vol.531 (2019), 121754.
[21] Zhang, Z., Wang, D. G. and Guo, Y. B., Fretting friction and wear behaviors of spiralwound gasket (SWG) sealing surface, Tribology International, Vol.133 (2019), pp.236-245.
[22] Elsevier Ltd, Material differences and gasket efficiency in low load conditions, World
Pumps, Vol.2, No.62 (2008), pp.25-28.
[23] Kim, Y. and Gostick, J. T., Measuring effective diffusivity in porous media with a gasketfree, radial arrangement, International Journal of Heat and Mass Transfer, Vol.129 (2019),
pp. 1023-1030.
[24] Gao, S. and Kazama, T., Leakage characteristics of prototype gasket using viscositytemperature relation under vibration (effects of sealing land size and viscosity grade), Proc.
ICFP Conf (2021), S2304.

- 11 -

PhD Dissertation of Muroran Institute of Technology- 12 -

2 Theoretical Model

2

Theoretical ModelIn the mechanical engineering, the mathematical analytical solution of lubrication projects
is very limited. A large number of practical lubricating projects can only be solved by numericalmethod. In this chapter, a prototype gasket with a gap is modeled between flat flanges with acentral recess. The theoretical model was developed using the inverse relationship between
fluid viscosity and the temperature dependence of hydraulic oils, i.e., the lower the temperature,
the higher the viscosity. The thermal effect is taken as the main analysis object under the bothconditions of steady-state and vibration. The thermal effect and the squeezing effect between
the two disks are selected as the main factors to be considered in analysis.
Thermo-hydrodynamic lubrication (THL) has always been a hot topic for researchers [13], and there are lots of valuable literatures about thermo-hydrodynamic lubrication domestic
and abroad. Among them, the process of discretizing the energy equation and heat conduction
equation of temperature control by numerical method is the main object of the study [4-7], andthe selection of high-precision and effective numerical method is also the core content ofresearchers. Chun [8] examined the effects of variable density and specific heat on maximum
pressure, temperature, bearing load, frictional loss and side leakage in high-speed journalbearing operation, the results calculated under constant density and specific heat were compared.
Anandan et al. [9] numerically investigated the starvation effects in lubricated high rolling
speed line contacts by using the second-order of Legendre polynomial. Chauhan et al. [10]found that the temperature of the lubricant film and thermal pressure in central elliptical bearing
were proportional to the sliding speed and eccentricity ratio. Gu [11] developed a model of gas
leakage through nonmetallic gaskets in order to predict leakage of sealing performance, themodel was applied to a bolted flanged connection in service in order to assess the tightness ofthe connection. Chun [12] examined the influence of aerated oil on high-speed journal bearingsby the THL theory coupled with analytical models for viscosity and density of air-oil mixture
in fluid-film bearing.
The characteristics of lubrication state are modeled by using the thermo-hydrodynamic
THL and IV theories respectively, and the differences between two theoretical calculation
results are compared. Although the shear heat generation is small in this gasket model, thecalculation based on both the THL and IV theories is conducted to examine the temperature
distributions in the gap, to discuss the differences in solutions with both theories, and confirm
them with the experimental data in the Chapter 4 and 5.

2.1 The THL Theory
2.1.1

Establishment of the THL Model

The steps to set up mathematical model are generally divided into four parts:- 13 -

PhD Dissertation of Muroran Institute of Technology(1) Simplify the research object reasonably and draw the necessary schematic diagram.
(2) Establish the coordinate system which is easy to analyze.
(3) List each basic control equations to ensure that the number of equations is the same as
the number of unknowns connected to demand.
(4) Put forward the necessary boundary conditions or initial conditions for the differential
equations in order to form a unique solution.
Leakage in the case of a flange-type gasket is modeled with a gap between two stationary
disks, as shown in Fig. 2-1. The THL theory is applied for calculating leakage under the parallel
and inclined conditions [13], wherein, the thermal distortion and elastic deformation are both
ignored because these values are relatively low.
In the cylindrical coordinate systems in Fig. 2-1, for the oil film, the three coordinate
components are 𝑟𝑟 , 𝜃𝜃 , and 𝑧𝑧 ; for the upper disk, they are 𝑟𝑟 ′ , 𝜃𝜃 ′ and 𝑧𝑧 ′ . It is worth
mentioning that the coordinate directions of 𝑟𝑟 and 𝑟𝑟 ′ , 𝜃𝜃 and 𝜃𝜃 ′ , and 𝑧𝑧 and 𝑧𝑧 ′ can be
regarded as indistinguishable. In addition, the original points of 𝜃𝜃 and 𝜃𝜃 ′ can be regarded tothe same locations. Therefore, 𝑟𝑟 ′ cannot be replaced by 𝑟𝑟, but 𝜃𝜃 ′ can be replaced by 𝜃𝜃. In
Fig. 2-1, 𝑧𝑧 ′ cannot be replaced by 𝑧𝑧, this is because that direction 𝑧𝑧 ′ starts from the uppersurface of the oil film, but 𝑧𝑧 starts from its lower surface.

Fig. 2-1 Schematic diagram of the THL lubrication model- 14 -

2 Theoretical Model

2.1.2

Generalized Reynolds Equation

The fundamental coupled equations in the THL theory include the generalized Reynolds
equation, the energy equation, the heat conduction equation, and other equations for physicallubricant properties.
The diagram of oil film is shown in Fig. 2-2, the pressure gradients along three coordinate
axes are expressed as below.
∂𝑝𝑝

∂

∂𝑢𝑢

= ∂𝑧𝑧 �𝜇𝜇 ∂𝑧𝑧 �⎫
⎪
∂
∂𝑣𝑣
=
�𝜇𝜇
�
∂𝜃𝜃
∂𝑧𝑧
∂𝑧𝑧 ⎬
∂𝑝𝑝
⎪
=0
⎭
∂𝑧𝑧
∂𝑟𝑟
∂𝑝𝑝

(2-1)

The boundary conditions of velocity are given as follows:𝑢𝑢(𝑟𝑟, 𝜃𝜃, 0) = 0, 𝑢𝑢(𝑟𝑟, 𝜃𝜃, ℎ) = 0
𝑣𝑣(𝑟𝑟, 𝜃𝜃, 0) = 0, 𝑣𝑣(𝑟𝑟, 𝜃𝜃, ℎ) = 0 �

(2-2)

dℎ

𝑤𝑤(𝑟𝑟, 𝜃𝜃, 0) = 0, 𝑤𝑤(𝑟𝑟, 𝜃𝜃, ℎ) = d𝑡𝑡

Integrating the above partial differential equation and using the boundary conditions, thevelocity component 𝑢𝑢 can be written as follows:
𝑧𝑧 𝑧𝑧

∂𝑝𝑝

𝑧𝑧 d𝑧𝑧𝐹𝐹(2-3)

𝑢𝑢 = ∂𝑟𝑟 �∫0 𝜇𝜇 d𝑧𝑧 − 𝐹𝐹1 ∫0 𝜇𝜇 �
0The continuity equation of fluid is expressed in the cylindrical coordinate system as below.
∂𝜌𝜌∂𝑡𝑡

1 ∂(𝜌𝜌𝜌𝜌𝜌𝜌)

+ 𝑟𝑟

∂𝑟𝑟

1 ∂

𝑧𝑧

1 ∂(𝜌𝜌𝜌𝜌)

+ 𝑟𝑟

∂𝜃𝜃

+

∂(𝜌𝜌𝜌𝜌)
∂𝑧𝑧

∂ℎ

(2-4)

+ 𝜌𝜌 ∂𝑡𝑡 = 0

Integrating continuity equation toward 𝑧𝑧 and the velocity component 𝑤𝑤 can be written.
1 ∂

𝑧𝑧

1

𝑧𝑧 ∂𝜌𝜌

(2-5)

𝑤𝑤 = − 𝜌𝜌𝜌𝜌 ∂𝑟𝑟 ∫0 𝜌𝜌𝜌𝜌𝜌𝜌 d𝑧𝑧 − 𝜌𝜌𝜌𝜌 ∂𝜃𝜃 ∫0 𝜌𝜌𝑣𝑣 d𝑧𝑧 − 𝜌𝜌 ∫0 ∂𝑡𝑡 d𝑧𝑧

Substituting velocity 𝑢𝑢 in Eq. (2-3) to Eq. (2-4) and arrangement.
1 ∂𝑟𝑟 ∂𝑟𝑟

∂𝑝𝑝

𝑧𝑧 𝑧𝑧

𝐹𝐹

𝑧𝑧 d𝑧𝑧1 ∂(𝜌𝜌𝜌𝜌)�𝜌𝜌𝜌𝜌 ∂𝑟𝑟 �∫0 𝜇𝜇 d𝑧𝑧 − 𝐹𝐹1 ∫0 𝜇𝜇 �� + 𝑟𝑟
0

∂𝜃𝜃

+

∂(𝜌𝜌𝜌𝜌)
∂𝑧𝑧

∂𝜌𝜌

∂ℎ

+ ∂𝑡𝑡 + 𝜌𝜌 ∂𝑡𝑡 = 0(2-6)

Integrating above equation toward 𝑧𝑧 from 0 to ℎ and lubricant density 𝜌𝜌 is regarded
as constant, the final form of generalized Reynolds equation in the THL model can be given.
wherein,1 ∂𝑟𝑟 ∂𝑟𝑟

∂𝑝𝑝

1 ∂

∂𝑝𝑝

∂ℎ

�𝐹𝐹2 𝑟𝑟 ∂𝑟𝑟 � + 𝑟𝑟 2 ∂𝜃𝜃 �𝐹𝐹2 ∂𝜃𝜃� + 𝜌𝜌 ∂𝑡𝑡 = 0ℎ d𝑧𝑧

𝐹𝐹0 = ∫0 𝜇𝜇

⎫

⎪
ℎ 𝑧𝑧
𝐹𝐹1 = ∫0 𝜇𝜇 dz
⎬
ℎ
𝑧𝑧 𝑧𝑧
𝐹𝐹 𝑧𝑧 d𝑧𝑧
𝐹𝐹2 = ∫0 𝜌𝜌 �∫0 𝜇𝜇 d𝑧𝑧 − 𝐹𝐹1 ∫0 𝜇𝜇 � d𝑧𝑧⎪⎭
0
- 15 -

(2-7)

(2-8)

PhD Dissertation of Muroran Institute of Technology

Fig. 2-2 Schematic diagram of oil film

2.1.3

Energy Equation of Oil Film

In isothermal lubrication theory, the influence of thermal effect is ignored. The isothermal
lubrication theory is acceptable for low-speed or light-loaded sliding bearings in mechanicalsystems. However, in the condition of high-speed or heavy-loaded sliding bearings, temperatureof the oil film can be reached more than 100℃, so the thermal effect and temperature field in
lubrication must be considered in this situation.
During the lubrication process, the oil film is heated by viscous shearing and squeezing,
which resulting in a large increase of temperature. At the same time, the generated heat istransferred by heat convection and conduction of ambient. The heat generation and dissipation
form a stable temperature field after the mechanical system reaches thermal equilibrium, which
is determined by the energy equation and its boundary conditions.
If the effects of volume force and heat radiation are not considered, the generalized energy
equation of viscous fluid has the following general form.
𝜌𝜌D�𝑐𝑐𝑝𝑝 𝑇𝑇�
D𝑡𝑡𝑇𝑇∂𝜌𝜌 D𝑝𝑝= ∇ ∙ (𝜆𝜆∇𝑇𝑇) − 𝜌𝜌 ∙ ∂𝑇𝑇 ∙ D𝑡𝑡 + 𝜙𝜙

(2-9)

wherein, 𝑐𝑐𝑝𝑝 is the isobaric specific heat of the fluid. For lubricating oil, the isobaric specificheat 𝑐𝑐𝑝𝑝 is almost same with the specific heat conductivity 𝑐𝑐. The 𝜆𝜆 is thermal conductivityof fluid, and the value of 𝑐𝑐𝑝𝑝 and 𝜆𝜆 can be considered as constants for lubricating oil.

In the Eq. (2-9), the item D⁄D𝑡𝑡 is a differential operator which is commonly used in fluid
mechanics, and its differential operation is shown below.
D

D𝑡𝑡

∂

∂

∂

∂

= ∂𝑡𝑡 + 𝑢𝑢 ∂𝑥𝑥 + 𝑣𝑣 ∂𝑦𝑦 + 𝑤𝑤 ∂𝑧𝑧

The energy equation consists of four parts, and their physical meanings are as follows:𝜌𝜌D�𝑐𝑐𝑝𝑝 𝑇𝑇�
D𝑡𝑡— Convection heat transfer∇ ∙ (𝜆𝜆∇𝑇𝑇) — Heat conduction- 16 -(2-10)

2 Theoretical Model𝑇𝑇∂𝜌𝜌 D𝑝𝑝∙ ∙— Pressure work𝜌𝜌 ∂𝑇𝑇 D𝑡𝑡

𝜙𝜙 — Dissipation function
For the Newton fluid, the expression of the dissipation function 𝜙𝜙 is defined,
∂𝑢𝑢 2

∂𝑣𝑣 2

∂𝑤𝑤 2

∂𝑣𝑣 2

∂𝑢𝑢

∂𝑣𝑣 2

∂𝑣𝑣

∂𝑢𝑢 2

∂𝑤𝑤

𝜙𝜙 = 𝜇𝜇 �2 �∂𝑥𝑥� + 2 �∂𝑦𝑦� + 2 � ∂𝑧𝑧 � + �∂𝑦𝑦 + ∂𝑥𝑥� + �∂𝑧𝑧 + ∂𝑦𝑦� + � ∂𝑥𝑥 + ∂𝑧𝑧 � �(2-11)

When the above generalized energy equation is applied to the gasket oil film in this THL
theory, the following simplifications can be done as follows:(1) The oil film is in the condition of thermally stable, so each physical quantity does notchange with time 𝑡𝑡.
D

D𝑡𝑡

∂

∂

∂

∂2 𝑇𝑇

∂2 𝑇𝑇

∂2 𝑇𝑇

(2-12)= 𝑢𝑢 ∂𝑥𝑥 + 𝑣𝑣 ∂𝑦𝑦 + 𝑤𝑤 ∂𝑧𝑧(2) The specific heat conductivity 𝑐𝑐𝑝𝑝 and the thermal conductivity 𝜆𝜆 of lubricating oil
are also constant.
∂𝑇𝑇

∂𝑇𝑇

∂𝑇𝑇

𝑇𝑇

∂𝜌𝜌

∂𝑝𝑝

∂𝑝𝑝

∂𝑝𝑝

𝑐𝑐𝑝𝑝 𝜌𝜌 �𝑢𝑢 ∂𝑥𝑥 + 𝑣𝑣 ∂𝑦𝑦 + 𝑤𝑤 ∂𝑧𝑧 � = 𝜆𝜆 �∂𝑥𝑥 2 + ∂𝑦𝑦 2 + ∂𝑧𝑧 2 � − 𝜌𝜌 ∙ ∂𝑇𝑇 �𝑢𝑢 ∂𝑥𝑥 + 𝑣𝑣 ∂𝑦𝑦 + 𝑤𝑤 ∂𝑧𝑧 � + 𝜇𝜇𝜇𝜇

(2-13)(3) The velocity of lubricating oil along 𝑧𝑧 direction is much smaller than its along 𝑥𝑥 and𝑦𝑦 directions. Therefore, except for ∂𝑢𝑢⁄∂𝑧𝑧 and ∂𝑣𝑣⁄∂𝑧𝑧, the other velocity gradientscan be ignored for simplification. The dissipation function 𝜙𝜙 can be expressed.
∂𝑢𝑢 2

∂𝑣𝑣 2

𝑇𝑇

∂𝜌𝜌

∂𝑝𝑝

∂

∂𝑇𝑇

∂𝑢𝑢 2

1

𝑧𝑧 ∂(𝜌𝜌𝜌𝜌)d𝑧𝑧 − 𝜌𝜌 ∫0 ∂𝑡𝑡 d𝑧𝑧(2-14)

𝜙𝜙 = 𝜇𝜇 �� ∂𝑧𝑧 � + �∂𝑧𝑧 � �(4) The scale of oil film is much smaller than that of gasket upper disk along 𝑥𝑥 and 𝑦𝑦
directions, it can be assumed that ∂𝑝𝑝⁄∂𝑧𝑧 = 0. And the heat conduction along 𝑥𝑥 and𝑦𝑦 directions can be ignored as well. The energy equation can be simplified further.
∂𝑇𝑇

∂𝑇𝑇

∂2 𝑇𝑇

∂𝑇𝑇

∂𝑢𝑢 2

∂𝑝𝑝

∂𝑣𝑣 2

𝑐𝑐𝑝𝑝 𝜌𝜌 �𝑢𝑢 ∂𝑥𝑥 + 𝑣𝑣 ∂𝑦𝑦 + 𝑤𝑤 ∂𝑧𝑧 � = 𝜆𝜆 ∂𝑧𝑧 2 − 𝜌𝜌 ∙ ∂𝑇𝑇 �𝑢𝑢 ∂𝑥𝑥 + 𝑣𝑣 ∂𝑦𝑦� + 𝜇𝜇 �� ∂𝑧𝑧 � + �∂𝑧𝑧 � �(2-15)(5) In this gasket model, the change value of temperature range is small, and the oil density𝜌𝜌 is regarded as constant. Moreover, the mechanical elements of this model are two
disks. Therefore, the energy equation should be expressed in the cylindrical coordinate
system as shown below.
∂𝑇𝑇

𝑣𝑣 ∂𝑇𝑇∂𝑇𝑇∂𝑣𝑣 2𝜌𝜌𝑐𝑐𝑝𝑝 �𝑢𝑢 ∂𝑟𝑟 + 𝑟𝑟 ∂𝜃𝜃 + 𝑤𝑤 ∂𝑧𝑧 � = ∂𝑧𝑧 �𝜆𝜆 ∂𝑧𝑧 � + 𝜇𝜇 �� ∂𝑧𝑧 � + �∂𝑧𝑧� �(2-16)The item of flow velocity 𝑤𝑤 in Eq. (2-16) could cause difficulties in theoretical analysis,but it can be replaced by integrating the continuity equation (Eq. (2-4)) toward 𝑧𝑧, and taking
the boundary condition into consideration, the flow velocity 𝑤𝑤 can be given.
1 ∂𝑧𝑧

𝑤𝑤 = − 𝜌𝜌𝜌𝜌 ∂𝑟𝑟 ∫0 𝜌𝜌𝜌𝜌𝜌𝜌 d𝑧𝑧 − 𝑟𝑟 ∫0- 17 -∂𝜃𝜃1𝑧𝑧 ∂𝜌𝜌

(2-17)

PhD Dissertation of Muroran Institute of Technology
The complete form of energy equation in this THL model can be expressed as below after
taking Eq. (2-17) to Eq. (2-16).
∂𝑇𝑇 𝑣𝑣 ∂𝑇𝑇
1 𝑧𝑧 ∂(𝜌𝜌𝜌𝜌)∂𝑇𝑇
1 ∂ 𝑧𝑧1 𝑧𝑧 ∂𝜌𝜌𝜌𝜌𝑐𝑐𝑝𝑝 �𝑢𝑢++ �−� 𝜌𝜌𝜌𝜌𝜌𝜌 d𝑧𝑧 − �d𝑧𝑧 − �d𝑧𝑧� �
∂𝑟𝑟 𝑟𝑟 ∂𝜃𝜃
𝑟𝑟 0 ∂𝜃𝜃∂𝑧𝑧𝜌𝜌𝜌𝜌 ∂𝑟𝑟 0
𝜌𝜌 0 ∂𝑡𝑡
∂

∂𝑢𝑢 2

∂𝑇𝑇

∂𝑣𝑣 2

(2-18)= ∂𝑧𝑧 �𝜆𝜆 ∂𝑧𝑧 � + 𝜇𝜇 �� ∂𝑧𝑧 � + �∂𝑧𝑧� �

The energy equation (Eq. (2-18)) is a parabolic second-order nonlinear partial differential
equation, its form looks like quite complicated. But when it is solved with the numerical method,
the integral item in bracket before ∂𝑇𝑇⁄∂𝑧𝑧, the two items after ∂𝜌𝜌⁄∂𝑇𝑇 and dissipation function
𝜙𝜙 can be treated as the term which are independent of temperature. So, the complete form of
the energy equation (Eq. (2-18)) is actually not very complicated.

2.1.4

Heat Conduction Equation of Upper Disk

According to the heat transfer theory, the heat conduction equation of the upper disc is athree-dimensional Laplace equation as follow:∂2 𝑇𝑇 ′
∂𝑥𝑥 2

∂2 𝑇𝑇 ′

∂2 𝑇𝑇 ′

(2-19)+ ∂𝑦𝑦 2 + ∂𝑧𝑧 2 = 0

The Eq. (2-19) is an elliptic partial differential equation, it can be expressed in cylindricalcoordinate system as shown below.
∂𝑇𝑇 ′

1 ∂

2.1.5

1 ∂2 𝑇𝑇 ′∂2 𝑇𝑇 ′

�𝑟𝑟 ′ ∂𝑟𝑟 ′ � + 𝑟𝑟′2 ∂𝜃𝜃2 + ∂𝑧𝑧′2 = 0
𝑟𝑟 ′ ∂𝑟𝑟 ′(2-20)

Boundary Conditions of Reynolds Equation

As shown in Fig. 2-2, the pressure boundary conditions of Reynolds equation are:

2.1.6

𝑝𝑝(𝑅𝑅1 , 𝜃𝜃) = 𝑃𝑃𝑟𝑟𝑝𝑝(𝑅𝑅2 , 𝜃𝜃) = 𝑃𝑃𝑒𝑒 �𝑝𝑝(𝑟𝑟, 𝜃𝜃) ≥ 0(2-21)

Boundary Conditions of Temperature Field

Although there are two partial differential equations control the temperature, i.e., Eqs. (218) and (2-20), there is only one actual temperature field. Thus, all the temperature boundary
conditions should be given collectively as follows:𝑇𝑇(𝑟𝑟, 𝜃𝜃, 0) = 𝑇𝑇𝑤𝑤
𝑇𝑇(𝑅𝑅1 , 𝜃𝜃, 𝑧𝑧) = 𝑇𝑇𝑖𝑖𝑖𝑖𝑇𝑇′(𝑅𝑅1 , 𝜃𝜃, 𝑧𝑧′) = 𝑇𝑇′𝑟𝑟−𝜆𝜆′𝜕𝜕𝑇𝑇 ′ �𝑅𝑅2 ,𝜃𝜃,𝑧𝑧 ′ �𝜕𝜕𝜕𝜕′= ℎ𝑡𝑡 [𝑇𝑇′(𝑅𝑅2 , 𝜃𝜃, 𝑧𝑧′) − 𝑇𝑇𝑎𝑎 ]

⎫
⎪
⎪
⎪

⎬
′
−𝜆𝜆
= ℎ𝑡𝑡 [𝑇𝑇′(𝑟𝑟′, 𝜃𝜃, 𝐵𝐵) − 𝑇𝑇𝑎𝑎 ] ⎪
𝜕𝜕𝜕𝜕′
⎪
⎪
′
𝜕𝜕𝜕𝜕(𝑟𝑟,𝜃𝜃,ℎ)′ 𝜕𝜕𝑇𝑇 (𝑟𝑟′,𝜃𝜃,0)
−𝜆𝜆
= −𝜆𝜆
⎭
𝜕𝜕𝜕𝜕′
𝜕𝜕𝜕𝜕
𝜕𝜕𝑇𝑇 ′ (𝑟𝑟′,𝜃𝜃,𝐵𝐵)- 18 -(2-22)

2 Theoretical Model
The heat level at the interfaces between the upper and side surfaces in the disk and theenvironment is transmitted by natural convection of air into the ambient operation surroundings.

2.1.7

Gap Clearance, Flow Velocity and Leakage Flow

The gap height ℎ, the velocity of flow 𝑢𝑢 during the radial direction, and the leakage flow
𝑄𝑄 from gap clearance are derived as follows:ℎ = ℎ𝑐𝑐 + 𝛼𝛼𝛼𝛼cos(𝜃𝜃 − 𝜑𝜑)(2-23)
∂𝑝𝑝

𝑧𝑧 𝑧𝑧

𝐹𝐹

𝑧𝑧 d𝑧𝑧𝑢𝑢 = ∂𝑟𝑟 �∫0 𝜇𝜇 d𝑧𝑧 − 𝐹𝐹1 ∫0 𝜇𝜇 �

2.1.8

2π

ℎ

0

𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜 = ∫0 ∫0 𝑢𝑢𝑅𝑅2 d𝑧𝑧 d𝜃𝜃(2-24)(2-25)

Viscosity-temperature Relationship of Lubricant

The viscosity is changed with temperature, and the viscosity-temperature relationship ofthe lubricant is given as follow:𝜈𝜈 = 𝜈𝜈0 exp[−𝛽𝛽(𝑇𝑇 − 𝑇𝑇0 )](2-26)

where 𝛽𝛽 is the constant associated with temperature 𝑇𝑇0 . In this study, the lubricant density isconstant and the viscosity is only a function of temperature. The pressure effect on the viscosity
is neglected because that the supply pressure is relatively low. The physical properties of fluid,
such as specific heat apparatus, density, thermal conductivity, and expansibility is influenced
by the parameters of temperature and pressure as well, whereas they are taken for constantsbecause those changes are too small.

2.2 The IV Theory
The THL theory can precisely compute temperature and pressure distributions of oil film,
as well as other lubricant properties such as load-carrying capacity, friction force, leakage flow.
The fundamental differential equations comprise the Reynolds equation, energy equation, andheat conduction equation, as well as other equations related to the physical characteristics oflubricants, which must be solved collectively. As such, the THL model is relativity complicated,
and the calculation program presents a momentous and time-consuming scheme. Kazama andGao [14] proposed applying the THL theory to gap flow using a temperature-controlled wall.
They computed the temperature distribution of oil film and disk by FDM and proposed that theoil temperature in the gap could be controlled by the wall temperature. They also proposed theIV theory for estimating the gap flow. The temperature of oil film can be directly replaced bysolid wall temperature when the gap height is relatively small, which is also a feasible approachfor effectively estimating leakage. Accordingly, the IV theory is also applied in the present
paper for calculating leakage flow from flange-type gasket.
For the IV theory, the Reynolds equation is as follow:- 19 -

PhD Dissertation of Muroran Institute of Technology

1 ∂−ℎ3 𝜌𝜌

∂ℎ�𝑟𝑟 12 𝜇𝜇𝑚𝑚 � = −𝜌𝜌𝑚𝑚 ∂𝑡𝑡
𝑟𝑟 ∂𝑟𝑟

(2-27)ℎ = ℎ0 + 𝐴𝐴sin(2π𝑓𝑓𝑓𝑓)(2-28)

𝑚𝑚

The gap clearance ℎ is changed with time 𝑡𝑡 under the harmonic vibration condition.

The velocity 𝑢𝑢 during direction 𝑟𝑟, and instant flow rate 𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜 are derived as follows:𝑢𝑢 =−𝑧𝑧(ℎ−𝑧𝑧) 𝜕𝜕𝜕𝜕
2𝜇𝜇𝑚𝑚

ℎ

𝜕𝜕𝜕𝜕

𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜 = 2π ∫0 𝑢𝑢𝑅𝑅2 d𝑧𝑧
1

𝑇𝑇 ∗𝑄𝑄𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑇𝑇 ∗ ∫0 𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜 d𝑡𝑡(2-29)(2-30)(2-31)The Reynolds equation, i.e., Eq. (2-27) is discretized by FDM, and other lubricant property
equations are numerically solved simultaneously. Additionally, under condition of harmonic
vibration, the negative pressure could not be generated in the cross-section of oil film; thus,
once the instant pressure became a negative value, it was replaced by zero. Mean flow rate
𝑄𝑄𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 during one complete vibration period can be calculated by Eq. (2-31).

2.3 Analysis of Pressure Distribution

The analysis of pressure and temperature distribution is the central content in lubrication
project. In the THL theory of Section 2.1, the generalized Reynolds equation which is suitable
for this prototype gasket model has been derived. In this section, the numerical solution thatsatisfies with pressure boundary conditions will be discussed, the partial differential equation
will be transformed into the algebraic equation by finite different method and on this basis, thepressure of any node in the calculation field as a linear combination of the pressures of adjacentnodes will be expressed.
For one-dimensional lubrication projects with simple geometric shapes and a few of twodimensional lubrication projects, if the influence of temperature and surface deformation are
not considered, the specific equation of the pressure distribution can be obtained analyticallysuch as the classic lubrication theory about infinite sliders and cylinders. However, for moreslightly complicated problem, the analytical method is powerless. The numerical methods haveto be used in this case, such as finite difference method (FDM) [15-16], finite element method(FEM), multigrid method and so on. Kyuho and Tae [17] proposed the thermal model that
accounts for heat transfers within the rotor in gas foil bearings, they incorporated analyticalmodels for bump thermal contact and inlet flow mixing to improve the prediction accuracy.
Liang et al. [18] modeled lubrication performance of rubber-supported water-lubricated thrustbearings with the coated pads and optimized the onset ratio of rubber cushions. The authors
designed a durability test rig and verified the rubber cushion offset ratio under different loadings
and speeds. Lin et al. [19] modeled a thermohydrodynamic cavitation flow lubrication including
cavitation effect, inertia and turbulence for high speed water-lubricated spiral groove thrustbearing, the methods for temperature rise due to cavitation were researched. Wang et al. [20]
- 20 -

2 Theoretical Modelextended the thermohydrodynamic analysis of journal bearings including couple stress blended
with high polymer additives, they modified the energy equation in more general and discussed
that couple stress produce a higher load capacity and friction force.
In current prototype gasket lubrication model, the FDM is an ideal algorithm for pressure
analysis. The basic idea of this algorithm is to mesh the solution domain firstly, and then, thederivative items are replaced with the difference quotient on the grid points according to theappropriate differential formula, the original equation is discretized into the difference format
at this moment. Finally, the specific numerical solution can be obtained with combining theboundary conditions. In addition, it is necessary to confirm the existence and uniqueness of thenumerical solution from the difference scheme and the numerical stability, the error estimation
between the solution of difference scheme and real situation of the original problem. The
characteristics of FDM are simplicity, flexibility, and versatility, it can be easily implemented
by electronic computer.

2.3.1

Principle and Difference Format of FDM

In numerical analysis, the slope of tangent between a certain interval can be used to replace
the derivative value of a point in this section approximately, which is the basic mathematicalidea of the FDM. It is based on approximating the difference item by the difference equation
between the function values on two or more adjacent nodes. The one-dimensional first-order
central difference equation of ∂𝑓𝑓⁄∂𝑥𝑥 is given by:∂𝑓𝑓�∂𝑥𝑥 � =𝑖𝑖

𝑓𝑓𝑖𝑖+1 −𝑓𝑓𝑖𝑖−12∆𝑥𝑥(2-32)

wherein, the subscripts 𝑖𝑖 − 1 and 𝑖𝑖 + 1 represent the interpolation points before and after thecenter position 𝑖𝑖, and ∆𝑥𝑥 is the step length between the two adjacent nodes.
Similarly, the second-order difference ∂2 𝑓𝑓⁄∂𝑥𝑥 2 is expressed by subtracting the function
difference value of node position 𝑖𝑖 − 0.5 and 𝑖𝑖 + 0.5. The second-order difference equation
can be given by using the first-order ∂𝑓𝑓⁄∂𝑥𝑥 dividing by ∆𝑥𝑥.
wherein,∂2 𝑓𝑓

�∂𝑥𝑥 2 � =
∂𝑓𝑓

𝑖𝑖

�

∂𝑓𝑓
∂𝑓𝑓
�
−� �
∂𝑥𝑥 𝑖𝑖+0.5∂𝑥𝑥 𝑖𝑖−0.5�∂𝑥𝑥 �
∂𝑓𝑓

𝑖𝑖+0.5�∂𝑥𝑥 �

𝑖𝑖−0.5∆𝑥𝑥=

𝑓𝑓𝑖𝑖+1 −𝑓𝑓𝑖𝑖−1∆𝑥𝑥(2-33)(2-34)

𝑓𝑓𝑖𝑖+1 −𝑓𝑓𝑖𝑖−1(2-35)

𝑓𝑓𝑖𝑖+1 −2𝑓𝑓𝑖𝑖 +𝑓𝑓𝑖𝑖−1(2-36)=∆𝑥𝑥

The final form of one-dimensional second-order difference equation can be obtained.
∂2 𝑓𝑓�∂𝑥𝑥 2 � =
𝑖𝑖

∆𝑥𝑥 2

The formula of two-dimensional difference equation in FDM can be derived by adding thedifference function value along 𝑦𝑦 coordinate axe, i.e., by introducing second node parameter
𝑗𝑗. The two-dimensional difference equation can be expressed as shown below.
- 21 -

PhD Dissertation of Muroran Institute of Technology∂𝑓𝑓�∂𝑥𝑥 �

𝑖𝑖,𝑗𝑗

∂2 𝑓𝑓

𝑖𝑖,𝑗𝑗

=

(2-37)

𝑓𝑓𝑖𝑖+1,𝑗𝑗 −2𝑓𝑓𝑖𝑖,𝑗𝑗 +𝑓𝑓𝑖𝑖−1,𝑗𝑗(2-39)

𝑓𝑓𝑖𝑖,𝑗𝑗+1 −2𝑓𝑓𝑖𝑖,𝑗𝑗 +𝑓𝑓𝑖𝑖,𝑗𝑗−1(2-40)

𝑓𝑓𝑖𝑖+1,𝑗𝑗 +𝑓𝑓𝑖𝑖−1,𝑗𝑗 +𝑓𝑓𝑖𝑖,𝑗𝑗+1 +𝑓𝑓𝑖𝑖,𝑗𝑗−1 −4𝑓𝑓𝑖𝑖,𝑗𝑗(2-41)

𝑖𝑖,𝑗𝑗

𝑖𝑖,𝑗𝑗

�∂𝑦𝑦 2 ��∂𝑥𝑥 ∂𝑦𝑦�2∆𝑥𝑥(2-38)

�∂𝑥𝑥 2 �∂2 𝑓𝑓

𝑓𝑓𝑖𝑖+1,𝑗𝑗 −𝑓𝑓𝑖𝑖−1,𝑗𝑗
𝑓𝑓𝑖𝑖,𝑗𝑗+1 −𝑓𝑓𝑖𝑖,𝑗𝑗−1

∂𝑓𝑓

�∂𝑦𝑦�

∂2 𝑓𝑓

=

𝑖𝑖,𝑗𝑗

=

=

=

2∆𝑦𝑦

∆𝑥𝑥 2

∆𝑦𝑦 2

∆𝑥𝑥∆𝑦𝑦The FDM is not only easily calculate, but also does not cause difficulties to the boundary
conditions. When finite difference operator is located at boundary of solution domain, special
processing is needed for virtual nodes outside the boundary. For interpolation points that require
higher accuracy, more complex operators can be used. However, it is necessary to get specificvalue of the node outside solution domain, the solution method is much more complicated, so
the higher accuracy difference operator is rarely used even if the accuracy is high.

2.3.2

Division of Discrete Grid in Oil Film

The diagram division of discrete grid of pressure analysis in the oil film region is as shownin Fig. 2-3. It can be seen that along 𝑟𝑟 direction, there are 𝑚𝑚1 equidistant nodes arranged
between the inner radius 𝑅𝑅1 and the outer radius 𝑅𝑅2 . The subscript starts from 𝑖𝑖 = 1 to 𝑖𝑖 =𝑚𝑚1 , and the grid spacing ∆𝑟𝑟 = (𝑅𝑅2 − 𝑅𝑅1 )⁄(𝑚𝑚1 − 1); Along 𝜃𝜃 direction, the number of 𝑛𝑛𝑛𝑛ℎ𝑖𝑖
nodes are set in the range of a complete circumference from 0 to 2π. The subscript starts from𝑗𝑗 = 1 to 𝑗𝑗 = 𝑛𝑛𝑛𝑛ℎ𝑖𝑖, and the grid spacing ∆𝜃𝜃 = 2π⁄(𝑛𝑛𝑛𝑛ℎ𝑖𝑖 − 1).
In Fig. 2-3, in order to plot grid nodes intuitively and visible, the number of 𝑚𝑚1 and 𝑛𝑛𝑛𝑛ℎ𝑖𝑖
was choose to 6 and 36. In the process of programing, under the conditions that calculation
accuracy is guaranteed and computing capacity of the program is considered, the actual numberof discrete grids of 𝑚𝑚1 and 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 was choose to 30 and 120, respectively.

Fig. 2-3 Schematic diagram of pressure grid nodes in oil film (𝑚𝑚1 = 6, 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 = 36)- 22 -

2 Theoretical Model

2.3.3

Difference Format in Reynolds Equation

The generalized Reynolds equation in this prototype gasket model is given in Eqs. (2-7)
and (2-8).
1 ∂𝑟𝑟 ∂𝑟𝑟wherein,

∂𝑝𝑝

∂𝑝𝑝

1 ∂

𝜕𝜕ℎ

(2-7)

�𝐹𝐹2 𝑟𝑟 ∂𝑟𝑟 � + 𝑟𝑟 2 ∂𝜃𝜃 �𝐹𝐹2 ∂𝜃𝜃� = −𝜌𝜌𝑚𝑚 𝜕𝜕𝜕𝜕ℎ d𝑧𝑧

𝐹𝐹0 = ∫0 𝜇𝜇

⎫

⎪
ℎ 𝑧𝑧
𝐹𝐹1 = ∫0 𝜇𝜇 dz
⎬
ℎ
𝑧𝑧 𝑧𝑧
𝐹𝐹 𝑧𝑧 d𝑧𝑧
𝐹𝐹2 = ∫0 𝜌𝜌 �∫0 𝜇𝜇 d𝑧𝑧 − 𝐹𝐹1 ∫0 𝜇𝜇 � d𝑧𝑧⎪
⎭
0

(2-8)

It can be simplified to the following form by chain derivation rule.
∂𝐹𝐹

𝐹𝐹

∂2 𝑝𝑝

∂𝑝𝑝

1 ∂𝐹𝐹

∂𝑝𝑝

𝐹𝐹 ∂2 𝑝𝑝𝜕𝜕ℎ

� ∂𝑟𝑟2 + 𝑟𝑟2 � ∂𝑟𝑟 + 𝐹𝐹2 ∂𝑟𝑟2 + �𝑟𝑟 2 ∂𝜃𝜃2 � ∂𝜃𝜃 + 𝑟𝑟22 ∂𝜃𝜃2 = −𝜌𝜌𝑚𝑚 𝜕𝜕𝜕𝜕(2-42)

𝑝𝑝𝑖𝑖+1,𝑗𝑗 −𝑝𝑝𝑖𝑖−1,𝑗𝑗(2-43)

The adopted difference format is the center difference which with higher accuracy in the
FDM, and the specific formula along the 𝑟𝑟 and 𝜃𝜃 directions are respectively as follows:∂𝑝𝑝� ∂𝑟𝑟 �
∂𝑝𝑝

�∂𝜃𝜃�

∂2 𝑝𝑝

�∂𝑟𝑟 2 �∂2 𝑝𝑝�∂𝜃𝜃2 �

𝑖𝑖,𝑗𝑗

𝑖𝑖,𝑗𝑗

𝑖𝑖,𝑗𝑗

𝑖𝑖,𝑗𝑗

=

=

=

=

2∆𝑟𝑟

𝑝𝑝𝑖𝑖,𝑗𝑗+1 −𝑝𝑝𝑖𝑖,𝑗𝑗−12∆𝜃𝜃(2-44)

𝑝𝑝𝑖𝑖+1,𝑗𝑗 −2𝑝𝑝𝑖𝑖,𝑗𝑗 +𝑝𝑝𝑖𝑖−1,𝑗𝑗(2-45)

𝑝𝑝𝑖𝑖,𝑗𝑗+1 −2𝑝𝑝𝑖𝑖,𝑗𝑗 +𝑝𝑝𝑖𝑖,𝑗𝑗−1(2-46)

∆𝑟𝑟 2

∆𝜃𝜃2

Substituting above difference formulas to Eq. (2-42),

𝑝𝑝𝑖𝑖+1,𝑗𝑗 − 2𝑝𝑝𝑖𝑖,𝑗𝑗 + 𝑝𝑝𝑖𝑖−1,𝑗𝑗∂𝐹𝐹2 𝐹𝐹2 𝑝𝑝𝑖𝑖+1,𝑗𝑗 − 𝑝𝑝𝑖𝑖−1,𝑗𝑗�+ ��� + 𝐹𝐹2 ��(∆𝑟𝑟)2
∂𝑟𝑟
𝑟𝑟
2∆𝑟𝑟
𝑝𝑝
−𝑝𝑝𝑖𝑖,𝑗𝑗−1𝑝𝑝−2𝑝𝑝𝑖𝑖,𝑗𝑗 +𝑝𝑝𝑖𝑖,𝑗𝑗−1
𝜕𝜕ℎ
1 ∂𝐹𝐹
+ �2𝑟𝑟 ∂𝜃𝜃2 � � 𝑖𝑖,𝑗𝑗+1
� + 𝐹𝐹2 � 𝑖𝑖,𝑗𝑗+1 (𝑟𝑟∆𝜃𝜃)� = −𝜌𝜌𝑚𝑚 𝜕𝜕𝜕𝜕2(𝑟𝑟∆𝜃𝜃)(2-47)

After arrangement, put the items which only contain 𝑝𝑝𝑖𝑖,𝑗𝑗 to the left side of the equal sign

and items which contain 𝑝𝑝𝑖𝑖+1,𝑗𝑗 , 𝑝𝑝𝑖𝑖−1,𝑗𝑗 , 𝑝𝑝𝑖𝑖,𝑗𝑗+1 and 𝑝𝑝𝑖𝑖,𝑗𝑗−1 to the right side. The generalized
difference equation of pressure distribution of the lubricant film was obtained as below.
1

∂𝐹𝐹

𝐹𝐹

2
2
⎡ 2∆𝑟𝑟 � ∂𝑟𝑟 + 𝑟𝑟 � �𝑝𝑝𝑖𝑖+1,𝑗𝑗 − 𝑝𝑝𝑖𝑖−1,𝑗𝑗 � ⎤
𝐹𝐹2
⎢
⎥
+
�𝑝𝑝𝑖𝑖+1,𝑗𝑗 + 𝑝𝑝𝑖𝑖−1,𝑗𝑗 �(∆𝑟𝑟)2
⎢
⎥
2𝐹𝐹2
2𝐹𝐹2
𝑝𝑝𝑖𝑖,𝑗𝑗 = ⎢1∂𝐹𝐹2⎥��(∆𝑟𝑟)2 + (𝑟𝑟∆𝜃𝜃)2 �⎢ + 2𝑟𝑟(𝑟𝑟∆𝜃𝜃) � ∂𝜃𝜃 � �𝑝𝑝𝑖𝑖,𝑗𝑗+1 − 𝑝𝑝𝑖𝑖,𝑗𝑗−1 � ⎥
⎢
𝜕𝜕ℎ⎥
𝐹𝐹2
⎣+ (𝑟𝑟∆𝜃𝜃)2 �𝑝𝑝𝑖𝑖,𝑗𝑗+1 + 𝑝𝑝𝑖𝑖,𝑗𝑗−1 � + 𝜌𝜌𝑚𝑚 𝜕𝜕𝜕𝜕 ⎦wherein, the difference item ∂ℎ⁄∂𝑡𝑡 in Eq. (2-48) was given.
- 23 -(2-48)

PhD Dissertation of Muroran Institute of Technology∂ℎ∂𝑡𝑡= 𝐴𝐴cos(2π𝑓𝑓𝑓𝑓) ∙ 2π𝑓𝑓

(2-49)

2.3.4 Boundary Conditions of Pressure

In steady-state, the boundary conditions of pressure are determined by the recess oil supply
pressure 𝑃𝑃𝑟𝑟 and the ambient pressure 𝑃𝑃𝑒𝑒 at the edge of disk; under the condition of harmonic
vibration, the negative pressure would be existed by the cavitation effect when two disks wereseparated with each other. However, the negative pressure could not be generated in the crosssection of oil film; thus, once the instant pressure became a negative value, it was replaced byzero. All the boundary conditions of pressure are given as below.
𝑝𝑝(𝑅𝑅1 , 𝜃𝜃) = 𝑃𝑃𝑟𝑟𝑝𝑝(𝑅𝑅2 , 𝜃𝜃) = 𝑃𝑃𝑒𝑒 �𝑝𝑝(𝑟𝑟, 𝜃𝜃) ≥ 0(2-50)

2.4 Analysis of Temperature Distribution

It is relatively difficult to analyze the temperature distribution of mechanical components,
and the main reasons are as follows:(1) The number of grid nodes used in three-dimensional analysis is pretty large, so theamount of calculation is a heavy project.
(2) The equation form of the energy equation of lubricant film is different from the heat
conduction equation of disk. Generally, it is necessary to solve the two differential
equations separately by repeated iteration for matching the continue conditions of heatflow and temperature during interface, which is further increased calculation amount.
(3) In this prototype gasket model, the shape of the oil film is cylinder. In more generalcases, the geometry of the oil film may become irregular and uncertain in advance, so
it is difficult to choose an appropriate numerical method.
(4) In the bearing and slider model, the counter flow will occur in the inlet due to the fast
convergence speed along clearance. This phenomenon may affect dominant diagonalin liner systems of equations, and destroying the stability of iteration process further.

2.4.1

Discrete Grid and Computing Scheme

The entire definition domain of temperature analysis is the oil film and upper disk together.
Kim et al. [21] analyzed the thermohydrodynamic performance of bearing after taking accountthe three-dimensional various of lubricant viscosity and density, the effect of pivot position andoperating and environment conditions on performance was studied. Zhang ta al. [22] analyzed
temperature distribution and heat transfer mechanism of circular bearing, the numerical model
was verified by experiment with considering thermal effect. Figures 2-4 and 5 show the threedimensional schematic diagram of temperature field in oil film and upper disk respectively. The
section surface becomes a rectangle when disk inclined angle 𝛼𝛼 is zero. The height represents
gap ℎ and disk thickness 𝐵𝐵 in two figures. Equidistant grids were used in coordinate direction.
- 24 -

2 Theoretical ModelIn the domain of lubricating film, the discrete grids selected of temperature is exactly samewith pressure analysis in Section 2.3, i.e., along 𝑟𝑟 direction, there are 𝑚𝑚1 equidistant nodes
arranged between the inner radius 𝑅𝑅1 and the outer radius 𝑅𝑅2 . And the subscript starts from𝑖𝑖 = 1 to 𝑖𝑖 = 𝑚𝑚1 , and the grid spacing ∆𝑟𝑟 = (𝑅𝑅2 − 𝑅𝑅1 )⁄(𝑚𝑚1 − 1) ; along 𝜃𝜃 direction, thenumber of 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 nodes are set in the range of a complete circumference from 0 to 2π. The
subscript starts from 𝑗𝑗 = 1 to 𝑗𝑗 = 𝑛𝑛𝑛𝑛ℎ𝑖𝑖, and the grid spacing ∆𝜃𝜃 = 2π⁄(𝑛𝑛𝑛𝑛ℎ𝑖𝑖 − 1); along 𝑧𝑧
direction, the number of 𝑛𝑛1 nodes are selected from 0 to h. The subscript starts from 𝑘𝑘 = 1
to 𝑘𝑘 = ℎ, the grid spacing ∆𝑧𝑧 = ℎ⁄(𝑛𝑛1 − 1).
In the domain of upper disk, 𝑚𝑚2 nodes are selected along 𝑟𝑟′ direction, and ∆𝑟𝑟′ =(𝑅𝑅2 − 𝑅𝑅1 )⁄(𝑚𝑚2 − 1); 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 nodes are set along 𝜃𝜃 direction, which is the same with pressure
analysis and ∆𝜃𝜃 = 2π⁄(𝑛𝑛𝑛𝑛ℎ𝑖𝑖 − 1) ; 𝑛𝑛2 nodes are selected along 𝑧𝑧′ direction, and ∆𝑧𝑧′ =𝐵𝐵⁄(𝑛𝑛2 − 1). It is worth to mentioning that the gap-disk interface is a share flat which subscript
is 𝑘𝑘 = 𝑛𝑛1 and 𝑘𝑘 = 1 along 𝑧𝑧 and 𝑧𝑧′ direction in the domain of oil and disk respectively.

Fig. 2-4 Schematic diagram of temperature grid nodes in oil film(ℎ𝑐𝑐 = 0.2 mm, 𝛼𝛼 = 0°, 𝑚𝑚1 = 6, 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 = 36, 𝑛𝑛1 = 6)

Fig. 2-5 Schematic diagram of temperature grid nodes in upper disk
(𝐵𝐵 = 10 mm, 𝛼𝛼 = 0°, 𝑚𝑚2 = 6, 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 = 36, 𝑛𝑛2 = 6)
- 25 -

PhD Dissertation of Muroran Institute of Technology
In the actual program, it is because that the length of oil film and disk area in 𝑟𝑟 directionis the exactly same, the number of grid points are also set same value, that is 𝑚𝑚1 = 𝑚𝑚2 . Thus,
there are 𝑚𝑚1 × 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 × (𝑛𝑛1 + 𝑛𝑛2 − 1) nodes altogether during temperature analysis domainshown in Figs. 2-4 and 5. In the oil film, the temperature value of 𝑚𝑚1 nodes which located in(𝑖𝑖 = 1,2, … , 𝑚𝑚1 ; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 1) are set by the boundary condition 𝑇𝑇(𝑟𝑟, 𝜃𝜃, 0) = 𝑇𝑇𝑤𝑤 ;the 𝑛𝑛1 nodes which located in (𝑖𝑖 = 1; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 1,2, … , 𝑛𝑛1 ) are determined byboundary condition 𝑇𝑇(𝑅𝑅1 , 𝜃𝜃, 𝑧𝑧) = 𝑇𝑇𝑖𝑖𝑖𝑖 . In the upper disk, the temperature value of 𝑛𝑛2 nodes
which located in (𝑖𝑖 = 1; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 1,2, … , 𝑛𝑛2 ) are determined by the temperatureboundary condition 𝑇𝑇′(𝑅𝑅 ′1 , 𝜃𝜃, 𝑧𝑧′) = 𝑇𝑇′𝑟𝑟 . Therefore, the number of actual differential equationof temperature is only (𝑚𝑚1 𝑛𝑛1 + 𝑚𝑚2 𝑛𝑛2 − 2𝑚𝑚1 − 𝑛𝑛1 − 𝑛𝑛2 ) × 𝑛𝑛𝑛𝑛ℎ𝑖𝑖.
The temperature distribution of oil film and upper disk can be obtained by solving thesedifference equations altogether with iteration method. Besides, the number of grid nodes are set
as 𝑚𝑚1 = 𝑚𝑚1 = 30, 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 = 120, 𝑛𝑛1 = 10, and 𝑛𝑛2 = 20 in the process of coding program.

2.4.2

Iteration Format of Energy Equation

The energy equation in this prototype gasket model is given in Eq. (2-16).
∂𝑇𝑇

𝑣𝑣 ∂𝑇𝑇∂𝑇𝑇∂𝑇𝑇

𝑣𝑣 ∂𝑇𝑇

∂𝑇𝑇

∂

∂𝑢𝑢 2

∂𝑇𝑇

∂𝑣𝑣 2

𝜌𝜌𝑐𝑐𝑝𝑝 �𝑢𝑢 ∂𝑟𝑟 + 𝑟𝑟 ∂𝜃𝜃 + 𝑤𝑤 ∂𝑧𝑧 � = ∂𝑧𝑧 �𝜆𝜆 ∂𝑧𝑧 � + 𝜇𝜇 �� ∂𝑧𝑧 � + �∂𝑧𝑧� �(2-16)

After assuming 𝜆𝜆 is constant, it can be simplified to following form by chain derivation.
∂2 𝑇𝑇

∂𝑢𝑢 2

∂𝑣𝑣 2

𝜌𝜌𝑐𝑐𝑝𝑝 �𝑢𝑢 ∂𝑟𝑟 + 𝑟𝑟 ∂𝜃𝜃 + 𝑤𝑤 ∂𝑧𝑧 � = 𝜆𝜆 � ∂z2 � + 𝜇𝜇 �� ∂𝑧𝑧 � + �∂𝑧𝑧 � �(2-51)

When using the FDM for numerical analysis, the key for scheme success is to choose thereasonable difference format. The mathematical essence of the energy equation is the parabolic
equation, the direction 𝑟𝑟 represents the flow of the lubricating oil, and there is no temperatureboundary condition at outlet. The backward difference formula was selected as the difference
format of ∂𝑇𝑇⁄∂𝑟𝑟 in the 𝑟𝑟 direction.
∂𝑇𝑇� ∂𝑟𝑟 �=𝑖𝑖,𝑗𝑗,𝑘𝑘𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 −𝑇𝑇𝑖𝑖−1,𝑗𝑗,𝑘𝑘∆𝑟𝑟(2-52)

The central difference format with higher accuracy was used in 𝜃𝜃 and 𝑧𝑧 direction.
∂𝑇𝑇

�∂𝜃𝜃�
∂𝑇𝑇

� ∂𝑧𝑧 �∂2 𝑇𝑇

�∂𝑧𝑧 2 �𝑖𝑖,𝑗𝑗,𝑘𝑘𝑖𝑖,𝑗𝑗,𝑘𝑘𝑖𝑖,𝑗𝑗,𝑘𝑘===𝑇𝑇𝑖𝑖,𝑗𝑗+1,𝑘𝑘 −𝑇𝑇𝑖𝑖,𝑗𝑗−1,𝑘𝑘2∆𝜃𝜃𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+1 −𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−12∆𝑧𝑧
𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+1 −2𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 +𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1∆𝑧𝑧 2(2-53)(2-54)(2-55)The difference formula of velocity component in energy equation is given.
∂𝑢𝑢� ∂𝑧𝑧 �𝑖𝑖,𝑗𝑗,𝑘𝑘=

𝑢𝑢𝑖𝑖,𝑗𝑗,𝑘𝑘+1 −𝑢𝑢𝑖𝑖,𝑗𝑗,𝑘𝑘−12∆𝑧𝑧- 26 -(2-56)

2 Theoretical Model

∂𝑣𝑣

�∂𝑧𝑧�

𝑖𝑖,𝑗𝑗,𝑘𝑘𝑣𝑣−𝑣𝑣= 𝑖𝑖,𝑗𝑗,𝑘𝑘+12∆𝑧𝑧 𝑖𝑖,𝑗𝑗,𝑘𝑘−1(2-57)In the temperature analysis domain of oil film in Fig. 2-4, except for the temperature ofsolid wall, the entrance of oil film inlet, and gap-disk interface. The values of other grid nodes
should be expressed by energy equation by using above formulas. A most common grid node
was selected as an example to derive the difference form of energy equation.
For these nodes which located at (𝑖𝑖 = 2,3, … , 𝑚𝑚1 ; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 2,3, … , 𝑛𝑛1 − 1),
the above difference formulas can be substituted to energy equation (Eq. (2-51)) directly.
𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 − 𝑇𝑇𝑖𝑖−1,𝑗𝑗,𝑘𝑘𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+1 − 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1
𝑣𝑣 𝑇𝑇𝑖𝑖,𝑗𝑗+1,𝑘𝑘 − 𝑇𝑇𝑖𝑖,𝑗𝑗−1,𝑘𝑘𝜌𝜌𝑐𝑐𝑝𝑝 �𝑢𝑢 �
�+ �
� + 𝑤𝑤 �
��
∆𝑟𝑟
𝑟𝑟
2∆𝜃𝜃
2∆𝑧𝑧
𝑇𝑇

= 𝜆𝜆 � 𝑖𝑖,𝑗𝑗,𝑘𝑘+1−2𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 +𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1
∆𝑧𝑧 2

𝑢𝑢

2

−𝑢𝑢

𝑣𝑣

−𝑣𝑣

2

� + 𝜇𝜇 �� 𝑖𝑖,𝑗𝑗,𝑘𝑘+12∆𝑧𝑧 𝑖𝑖,𝑗𝑗,𝑘𝑘−1 � + � 𝑖𝑖,𝑗𝑗,𝑘𝑘+12∆𝑧𝑧 𝑖𝑖,𝑗𝑗,𝑘𝑘−1 � �(2-58)

After arrangement, put the items which only contain 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 to the left side of the equal sign

and items which contain 𝑇𝑇𝑖𝑖−1,𝑗𝑗,𝑘𝑘 , 𝑇𝑇𝑖𝑖,𝑗𝑗−1,𝑘𝑘 , 𝑇𝑇𝑖𝑖,𝑗𝑗+1,𝑘𝑘 , 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1 and 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+1 to the right side. The
generalized difference form of energy equation in the lubricant film was obtained as below.

𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 =2.4.3

2𝜆𝜆∆𝑟𝑟∆𝜃𝜃�𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+1 + 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1 �
⎧
⎫
2
2
1
⎪
⎪
𝜇𝜇∆𝑟𝑟∆𝜃𝜃
��𝑢𝑢
−
𝑢𝑢
�
+
�𝑣𝑣
−
𝑣𝑣
�
�
+
𝑖𝑖,𝑗𝑗,𝑘𝑘+1
𝑖𝑖,𝑗𝑗,𝑘𝑘−1
𝑖𝑖,𝑗𝑗,𝑘𝑘+1
𝑖𝑖,𝑗𝑗,𝑘𝑘−1
⎪ 2
⎪
⎨
⎪
⎪
⎩

1

+2𝜌𝜌𝑐𝑐𝑝𝑝 𝑢𝑢𝑖𝑖,𝑗𝑗,𝑘𝑘 ∆𝜃𝜃∆𝑧𝑧 2 𝑇𝑇𝑖𝑖−1,𝑗𝑗,𝑘𝑘− 𝑟𝑟 𝜌𝜌𝑐𝑐𝑝𝑝 𝑣𝑣𝑖𝑖,𝑗𝑗,𝑘𝑘 ∆𝑟𝑟∆𝑧𝑧 2 �𝑇𝑇𝑖𝑖,𝑗𝑗+1,𝑘𝑘 − 𝑇𝑇𝑖𝑖,𝑗𝑗−1,𝑘𝑘 �−𝜌𝜌𝑐𝑐𝑝𝑝 𝑤𝑤𝑖𝑖,𝑗𝑗,𝑘𝑘 ∆𝑟𝑟∆𝜃𝜃∆𝑧𝑧�𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+1 − 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1 �Iteration Format of Heat Conduction Equation2𝜌𝜌𝑐𝑐𝑝𝑝 𝑢𝑢𝑖𝑖,𝑗𝑗,𝑘𝑘 ∆𝜃𝜃∆𝑧𝑧 2���
+4𝜆𝜆∆𝑟𝑟∆𝜃𝜃⎬⎪⎪⎭(2-59)The heat conduction equation in this prototype gasket model is given in Eq. (2-20).
1 ∂𝑟𝑟 ′ ∂𝑟𝑟 ′∂𝑇𝑇 ′1 ∂2 𝑇𝑇 ′∂2 𝑇𝑇 ′

(2-20)∂2 𝑇𝑇 ′

1 ∂2 𝑇𝑇 ′∂2 𝑇𝑇 ′

(2-60)

�𝑟𝑟 ′ ∂𝑟𝑟 ′ � + 𝑟𝑟′2 ∂𝜃𝜃2 + ∂𝑧𝑧′2 = 0

It can also be simplified to the following form by chain derivation rule.
1 ∂𝑇𝑇 ′

𝑟𝑟 ′ ∂𝑟𝑟 ′+ ∂𝑟𝑟′2 + 𝑟𝑟′2 ∂𝜃𝜃2 + ∂𝑧𝑧′2 = 0

The heat conduction equation is Laplace equation mathematically. The central difference
scheme in FDM was applied to heat conduction equation. The difference formula along 𝑟𝑟 ′ , 𝜃𝜃and 𝑧𝑧′ directions are as shown below.
∂𝑇𝑇 ′�∂𝑟𝑟 ′ �∂2 𝑇𝑇 ′

� ∂𝑟𝑟′2 �∂𝑇𝑇 ′�∂𝜃𝜃2 �𝑖𝑖,𝑗𝑗,𝑘𝑘𝑖𝑖,𝑗𝑗,𝑘𝑘𝑖𝑖,𝑗𝑗,𝑘𝑘

𝑇𝑇 ′

−𝑇𝑇 ′

= 𝑖𝑖+1,𝑗𝑗,𝑘𝑘2∆𝑟𝑟 ′𝑖𝑖−1,𝑗𝑗,𝑘𝑘𝑇𝑇 ′= 𝑖𝑖+1,𝑗𝑗,𝑘𝑘𝑇𝑇 ′= 𝑖𝑖,𝑗𝑗+1,𝑘𝑘′′−2𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+𝑇𝑇𝑖𝑖−1,𝑗𝑗,𝑘𝑘

∆𝑟𝑟′2

′
′
−2𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+𝑇𝑇𝑖𝑖,𝑗𝑗−1,𝑘𝑘- 27 -∆𝜃𝜃2(2-61)(2-62)(2-63)

PhD Dissertation of Muroran Institute of Technology∂2 𝑇𝑇 ′

� ∂𝑧𝑧′2 �𝑖𝑖,𝑗𝑗,𝑘𝑘𝑇𝑇 ′= 𝑖𝑖,𝑗𝑗,𝑘𝑘+1′′−2𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1(2-64)∆𝑧𝑧′2

Similarly, except for the temperature of gap-disk interface, inner boundary, and outerboundary. The values of other grid nodes should be expressed by heat conduction by using
above formulas. A most common grid node was also selected to derive the common difference
form of heat conduction equation.
For these nodes which located at (𝑖𝑖 = 2,3, … , 𝑚𝑚2 ; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 2,3, … , 𝑛𝑛2 − 1),
the above formulas can be substituted to heat conduction equation (Eq. (2-60)) directly.
′
′
′
′
′
− 𝑇𝑇𝑖𝑖−1,𝑗𝑗,𝑘𝑘
𝑇𝑇𝑖𝑖+1,𝑗𝑗,𝑘𝑘− 2𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+ 𝑇𝑇𝑖𝑖−1,𝑗𝑗,𝑘𝑘
1 𝑇𝑇𝑖𝑖+1,𝑗𝑗,𝑘𝑘
�
�+�
�
𝑟𝑟 ′
2∆𝑟𝑟 ′∆𝑟𝑟 ′ 2
1

𝑇𝑇 ′+ 𝑟𝑟′2 � 𝑖𝑖,𝑗𝑗+1,𝑘𝑘′′−2𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+𝑇𝑇𝑖𝑖,𝑗𝑗−1,𝑘𝑘

∆𝜃𝜃2

𝑇𝑇 ′

� + � 𝑖𝑖,𝑗𝑗,𝑘𝑘+1′′−2𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1

∆𝑧𝑧′2

�=0

(2-65)′The items which only contain 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘
to the left side of the equal sign and other items which

′
′
′
′
′
′
contain 𝑇𝑇𝑖𝑖−1,𝑗𝑗,𝑘𝑘, 𝑇𝑇𝑖𝑖+1,𝑗𝑗,𝑘𝑘, 𝑇𝑇𝑖𝑖,𝑗𝑗−1,𝑘𝑘, 𝑇𝑇𝑖𝑖,𝑗𝑗+1,𝑘𝑘, 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1
and 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘+1
to right side. The generalized

difference form of heat conduction equation in the upper disk was obtained as below.
1

2.4.4

′
′
⎡ 2𝑟𝑟 ′ ∆𝑟𝑟 ′ �𝑇𝑇 𝑖𝑖+1,𝑗𝑗,𝑘𝑘 − 𝑇𝑇 𝑖𝑖−1,𝑗𝑗,𝑘𝑘 � ⎤⎢ + 1 �𝑇𝑇 ′⎥′𝑖𝑖+1,𝑗𝑗,𝑘𝑘 + 𝑇𝑇 𝑖𝑖−1,𝑗𝑗,𝑘𝑘 � ⎥
′ )2
(∆𝑟𝑟
⎢
2
2
1
′
𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘=⎢1⎥��(∆𝑟𝑟 ′ )2 + (𝑟𝑟 ′ ∆𝜃𝜃)2 + (∆𝑧𝑧 ′ )2 �′′⎢+ (𝑟𝑟 ′ ∆𝜃𝜃)2 �𝑇𝑇 𝑖𝑖,𝑗𝑗+1,𝑘𝑘 + 𝑇𝑇 𝑖𝑖,𝑗𝑗−1,𝑘𝑘 �⎥
⎢
⎥
1
′
′
⎣ + (∆𝑧𝑧 ′ )2 �𝑇𝑇 𝑖𝑖,𝑗𝑗,𝑘𝑘+1 + 𝑇𝑇 𝑖𝑖,𝑗𝑗,𝑘𝑘−1 � ⎦(2-66)

Iteration Form of Temperature Boundary Conditions

At the gap-disk interface, the difference form of the grid nodes can be derived according
to the boundary conditions of heat continuity. For the convenience of analysis, all the boundary
conditions of temperature and their difference forms were given as follows:(1) For the nodes located at (𝑖𝑖 = 1,2, … , 𝑚𝑚1 ; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 1) in oil film, thetemperature of this surface 𝑇𝑇𝑤𝑤 controlled by solid wall, which can be input from
Peltier module in experiment.
(2) For the nodes located at (𝑖𝑖 = 1; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 2,3, … , 𝑛𝑛1 − 1) in oil film, thissurface is the inlet of oil film, the temperature 𝑇𝑇𝑖𝑖𝑖𝑖 is oil temperature.
(3) For the nodes located at (𝑖𝑖 = 1; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 2,3, … , 𝑛𝑛2 ) in upper disk, thisis the inner surface of disk. Its temperature 𝑇𝑇 ′ 𝑟𝑟 is the original temperature of innerparts. Generally, it can be regarded as the same value with oil temperature 𝑇𝑇𝑖𝑖𝑖𝑖 .
(4) For the nodes located at (𝑖𝑖 = 𝑚𝑚2 ; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 1,2, … , 𝑛𝑛2 ) in upper disk, itis the lateral interface of disk-ambient. The forward difference form can be adopted.
∂𝑇𝑇 ′� ∂𝑟𝑟 ′ �𝑖𝑖,𝑗𝑗,𝑘𝑘

𝑇𝑇 ′

−𝑇𝑇 ′

= 𝑖𝑖,𝑗𝑗,𝑘𝑘 ∆𝑟𝑟𝑖𝑖−1,𝑗𝑗,𝑘𝑘′- 28 -(2-67)

2 Theoretical Model
Substituting it into the forth temperature boundary condition in Eq. (2-22).
1′′′−𝜆𝜆′ ∆𝑟𝑟 ′ �𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘− 𝑇𝑇𝑖𝑖−1,𝑗𝑗,𝑘𝑘
� = ℎ𝑡𝑡 �𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘− 𝑇𝑇𝑎𝑎 �(2-68)

The final iteration format at the lateral interface of disk-ambient can be derived.
′
𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘� ′𝑟𝑟 =𝑅𝑅2=∆𝑟𝑟′ℎ𝑡𝑡 𝑇𝑇𝑎𝑎 +𝜆𝜆′𝑇𝑇′𝑖𝑖−1,𝑗𝑗,𝑘𝑘∆𝑟𝑟′ℎ𝑡𝑡 +𝜆𝜆′(2-69)(5) For the nodes located at (𝑖𝑖 = 1,2, … , 𝑚𝑚2 ; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 𝑛𝑛2 ) in upper disk, itis the roof interface of disk-ambient. The forward difference format can be adopted.
∂𝑇𝑇 ′

𝑇𝑇 ′

−𝑇𝑇 ′

= 𝑖𝑖,𝑗𝑗,𝑘𝑘 ∆𝑧𝑧𝑖𝑖,𝑗𝑗,𝑘𝑘−1′

� ∂𝑧𝑧 ′ �𝑖𝑖,𝑗𝑗,𝑘𝑘(2-70)

Substituting it into the fifth temperature boundary condition in Eq. (2-22).
1′′′−𝜆𝜆′ ∆𝑧𝑧 ′ �𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘− 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1
� = ℎ𝑡𝑡 �𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘− 𝑇𝑇𝑎𝑎 �(2-71)

The final iteration format at the roof interface of disk-ambient can be derived.
′
𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘� ′𝑧𝑧 =𝐵𝐵=∆𝑧𝑧′ℎ𝑡𝑡 𝑇𝑇𝑎𝑎 +𝜆𝜆′𝑇𝑇′𝑖𝑖,𝑗𝑗,𝑘𝑘−1(2-72)

𝑇𝑇′𝑖𝑖,𝑗𝑗,𝑘𝑘+1 −𝑇𝑇′𝑖𝑖,𝑗𝑗,𝑘𝑘(2-73)∆𝑧𝑧′ℎ𝑡𝑡 +𝜆𝜆′(6) For nodes located at (𝑖𝑖 = 1,2, … , 𝑚𝑚1 ; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 𝑛𝑛1 ) in oil film. Besides,
they can be expressed as (𝑖𝑖 = 1,2, … , 𝑚𝑚2 ; 𝑗𝑗 = 1,2, … , 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; 𝑘𝑘 = 1) in coordinates ofupper disk as well. This is the gap-disk interface. The difference format can be derived
by using the heat continuity condition at this special surface.
𝜕𝜕𝑇𝑇 ′

�𝜕𝜕𝑧𝑧 ′ �𝜕𝜕𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘

� 𝜕𝜕𝜕𝜕 �𝑖𝑖,𝑗𝑗,𝑘𝑘

=

=

∆𝑧𝑧 ′
𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 −𝑇𝑇′𝑖𝑖,𝑗𝑗,𝑘𝑘−1∆𝑧𝑧(2-74)

Substituting it into the sixth temperature boundary condition in Eq. (2-22). Besides, inthis surface, 𝑇𝑇′𝑖𝑖,𝑗𝑗,𝑘𝑘 = 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 .
𝜆𝜆′

∆𝑧𝑧 ′

𝜆𝜆

�𝑇𝑇′𝑖𝑖,𝑗𝑗,𝑘𝑘+1 − 𝑇𝑇′𝑖𝑖,𝑗𝑗,𝑘𝑘 � = ∆𝑧𝑧 �𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 − 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1 �(2-75)

The final iteration format at the gap-disk interface can be derived.
𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 |𝑧𝑧=ℎ = 𝑇𝑇′𝑖𝑖,𝑗𝑗,𝑘𝑘 |𝑧𝑧′=0 =∆𝑧𝑧𝑧𝑧′𝑇𝑇′𝑖𝑖,𝑗𝑗,𝑘𝑘+1 +∆𝑧𝑧′𝜆𝜆𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘−1

2.4.5 Preview Iterative Program of Temperature∆𝑧𝑧′𝜆𝜆+∆𝑧𝑧𝑧𝑧′(2-76)

As can be seen from the previous four sections, although there are only two governing
equations for controlling the temperature [23-24], i.e., energy equation (Eq. (2-16)) and heat
conduction equation (Eq. (2-20)). However, there are eight difference equations with different
expressions in all. That is, one equation is about the temperature of the inner region in oil film,
one equation is about the inner region in upper disk, and other six equations are about the special
interface. Among these eight difference equations, the temperature value of grid nodes before- 29 -

PhD Dissertation of Muroran Institute of Technology
the equal sign is unknown, while the temperature value of nodes after the equal sign is always
treated as the known number. For this gasket model, the total number of unknown grid nodes
is (𝑚𝑚1 𝑛𝑛1 + 𝑚𝑚2 𝑛𝑛2 − 2𝑚𝑚1 − 𝑛𝑛1 − 𝑛𝑛2 ) × 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 in the temperature iteration field. That is to say
the number of (𝑚𝑚1 𝑛𝑛1 + 𝑚𝑚2 𝑛𝑛2 − 2𝑚𝑚1 − 𝑛𝑛1 − 𝑛𝑛2 ) × 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 difference equations about the nodetemperature is needed to list. The first subscript 𝑖𝑖 of these nodes ranges from 1 to 𝑚𝑚1 (𝑚𝑚2 );the second subscript 𝑗𝑗 ranges from 1 to 𝑛𝑛𝑛𝑛ℎ𝑖𝑖; and the third subscript 𝑘𝑘 ranges from 1 to𝑛𝑛1 (𝑛𝑛2 ). According to the subscript value of each node, the unique difference equation which iscorresponded to this node can always be selected from the above eight difference equations.
In the iteration process in temperature domain, the pressure 𝑝𝑝(𝑟𝑟, 𝜃𝜃) and film thicknessℎ(𝑟𝑟, 𝜃𝜃, 𝑡𝑡) are always regarded as the known function. The values of these nodes in each groupremain unchanged in one iteration time of the temperature calculation. Before the iteration, theinitial temperature value of all nodes must be given. If the temperature is iterated for the firsttime in the whole program, the initial temperature of all nodes in the oil film and upper disk
region can be equal to the target temperature 𝑇𝑇𝑤𝑤 as shown below.
𝑇𝑇 (0) = 𝑇𝑇𝑤𝑤 (𝑖𝑖 = 1~𝑚𝑚1 , 𝑗𝑗 = 1~𝑛𝑛𝑛𝑛ℎ𝑖𝑖, 𝑘𝑘 = 1~𝑛𝑛1 )𝑇𝑇 ′(0) = 𝑇𝑇𝑤𝑤 (𝑖𝑖 = 1~𝑚𝑚2 , 𝑗𝑗 = 1~𝑛𝑛𝑛𝑛ℎ𝑖𝑖, 𝑘𝑘 = 1~𝑛𝑛2 )(2-77)(2-78)

Otherwise, the temperature value of the previous iteration is used as its initial value of thisiteration time. Then, the values of functions 𝜇𝜇, 𝑢𝑢, 𝑣𝑣, ∂𝑢𝑢⁄∂𝑧𝑧 and ∂𝑣𝑣⁄∂𝑧𝑧 at all nodes of theoil film can be calculated from the 𝑝𝑝, ℎ, 𝑇𝑇 and 𝑇𝑇′. After the above preparation, the stepscanning solution process of temperature field can be started. For the convenience of coding
program, the scanning could start from the first node located in (𝑖𝑖 = 1, 𝑗𝑗 = 1, 𝑘𝑘 = 1) in theoil film domain. Then increasing the coordinate 𝑖𝑖 and 𝑘𝑘 respectively under the condition of
𝑗𝑗 is fixed until reach the nodes at (𝑖𝑖 = 𝑚𝑚1 , 𝑗𝑗 = 1, 𝑘𝑘 = 𝑛𝑛1 ). The next section surface is scannedby increasing the 𝑗𝑗 coordinate from 1 to 𝑛𝑛𝑛𝑛ℎ𝑖𝑖. This cross section scanning method is also
adopted to calculate the temperature in the upper disk domain.
The iteration loop can be stopped until the difference of the temperature value between
two iterations before and after satisfied the below convergence accuracy.
(𝑚𝑚+1)

�𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘′(𝑚𝑚+1)

�𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘(𝑚𝑚)− 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 � ≤ 1 × 10−3′(𝑚𝑚)

− 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 � ≤ 1 × 10−3

(2-79)(2-80)

wherein, 𝑚𝑚 and 𝑚𝑚 + 1 is the iteration time.
Then, it is time to provide the new initial values for the next time iteration. In this prototype
gasket lubrication model, the result of the last iteration can be directly used as the new initialvalue of this iteration and then iterate again and again until the convergence accuracy is satisfied.
(𝑚𝑚)

In this step, the final numerical solution at this group nodes can be selected from either 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘(𝑚𝑚+1)or 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘. In more complex mathematical models, factor 𝜔𝜔 can be used to provide new initial- 30 -

2 Theoretical Modelvalues for the next iteration.
(𝑚𝑚+1)

𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘(𝑚𝑚−1)= 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘(𝑚𝑚)(𝑚𝑚−1)+ 𝜔𝜔�𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘 − 𝑇𝑇𝑖𝑖,𝑗𝑗,𝑘𝑘�

(2-81)

wherein, 𝜔𝜔 is called the relaxation factor, it is necessary to solve the nonlinear equations inhydromechanics by controlling the change of variables, which is realized by relaxation factor.
Its function is to control variable change in each iteration, that is, the new value of the variable
is the original value plus the relaxation factor multiplied by the change. 𝜔𝜔 > 1 is called theover-relaxation factor to accelerate the convergence speed, and 𝜔𝜔 < 1 is called the underrelaxation factor. And the smaller the relaxation factor is, the smaller the change between thetwo iterations is, the more stable iteration system is, but the slower the convergence speed is.
In order to consider the efficiency and accuracy of the program, the upper and lower limitsof the number of iterations can be set in the specific program. If the number of iterations has
exceeded the upper limit, the temperature analysis will be terminated immediately even if theconvergence accuracy has not been reached; otherwise, even if the convergence accuracy has
been met, if the number of iterations has not reached the lower limit, the iteration will still
continue. In general, the upper limit of iterations can be 30 or more, and the lower limit can be
10 or less. In the process of solving the difference equations, temperature analysis and pressure
analysis are carried out alternately. The initial value of temperature may become better in later
iterations, and the convergence could be guaranteed only after a few iterations.

2.5 Construction of Program and Calculation Results
From Section 2.2 to 2.4, it was discussed the thermal hydrodynamic lubrication system of
the prototype gasket models. In Section 2.2, the mathematical model was established; In Section
2.3, the iterative formula of pressure distribution is obtained; In Section 2.4, the energy equationand heat conduction equation of the temperature field of the lubricant film and upper disk werediscretized by the FDM. The application of numerical iterative method in analysis of pressure
and temperature was also discussed. There are many formulas are derived among these chapter,
it is necessary to combine these formulas organically and then compile them into calculation
programs. The reliable numerical solutions can be obtained under given input parameters after
debugging in the end.
The analytical method can only solve the very limited simple fluid mechanics problems,
therefore, the specific thermo-hydrodynamic lubrication projects [25-28] are realized bynumerical methods in general. Durany et al. [29] analyzed a coupled model for pressure andtemperature computations in a plain journal bearing which includes shaft and bushing thermal
exchange with the external environment, which was solved by numerically for both steady andtime-dependent conditions. The temperature and pressure distribution of journal bearing wereobtained by using FDM. Wang and Zhu [30] studied the non-Newtonian behavior for a finite
journal bearing lubricated with micro polar fluids is undertaken considering both thermal and- 31 -

PhD Dissertation of Muroran Institute of Technology
cavitation effects, they modified the Reynolds equation and energy equation by using theElord’s cavitation algorithm. In gasket research, the key point is to find out the way to improvesealing performance and the main factors affecting the leakage by numerical analysis. Fessler
et al. [31] examined the leakage between flange joints without using gaskets. They noted thatthe gap space can be decreased by larger loading and that a divergent gap will cause significant
leaking. Guo et al. [32] presented a study of the effects of viscosity and sliding speed onboundary slippage in thin film hydrodynamic lubrication using a slider-on-disc tester.
There is no doubt that the analytic solution is the result expressed by formula, while thenumerical solution is the result expressed by numerical value. Therefore, when the analyticalsolution is discussed, the input data is unnecessary to mention, but it's almost meaningless to
conduct the numerical solution without input data. In this chapter, it can be seen that numericalsolution of the current model is related to more than 10 geometric and physical quantities. The
reasonable numerical solution can be obtained only when these quantities have definite specificvalues. In order to explore overall perspective of one solution, it is necessary to calculate plentyof numerical solutions of all these quantities under all possible combinations. It is applicable toimagine that all possible combinations of more than ten physical quantities are the huge number,so it is unrealistic to get all of the numerical solutions. For this reason, the common method in
research is to fix most of the relevant quantities, and allow several of the important quantities
to change in a reasonable range. Through a number of typical numerical solutions, the influenceof important quantities on the solutions and the regular things can be investigated. This is anefficiency way and approach to access to one complex phenomena in academic research.

2.5.1

Initial Parameters and Calculation Process

The gasket parameters and lubricant properties and initial values of physical quantities are
as shown in Table 2-1. Wherein, the lubricant is selected to ISO VG32 oil. In this prototype
gasket lubrication model, there are (𝑚𝑚1 𝑛𝑛1 + 𝑚𝑚2 𝑛𝑛2 − 2𝑚𝑚1 − 𝑛𝑛1 − 𝑛𝑛2 ) × 𝑛𝑛𝑛𝑛ℎ𝑖𝑖 number of
unknown grid nodes in the temperature controlling area. The unique difference equation whichis corresponded to this node can always be selected from the difference equations listed in
Section 2.4 according to the subscript value of each node. The calculation flowchart is shownin Fig. 2-6 includes two-layer loops, which nested between pressure and temperature analysis.
The governing equation is solved by FDM, and program is implemented by following steps:(1) Input specific parameters of lubricant characteristics, initial conditions, the initialize
temperature and viscosity.
(2) Calculate the viscosity-temperature equation and compute lubricant viscosity.
(3) Solve generalized Reynolds equation and compute the pressure distribution by FDM.
(4) Repeat step 3 until pressure convergence in oil film is achieved.
(5) Compute velocity of lubricant along 𝑟𝑟, 𝜃𝜃, and 𝑧𝑧 direction.
(6) Solve energy equation and compute the temperature of oil film domain.

- 32 -

2 Theoretical Model(7) Solve heat conduction equation and compute temperature of upper disk domain.
(8) Repeat steps 2-7 until achieving the both convergence of pressure and temperature.
(9) Modify the thermal properties of lubricant and compute the coefficients.
(10) Judge whether the final revolution it is. If yes, all procedure is completed; otherwise,
calculate the parameters of next grid nodes and go back to step 2.

Fig. 2-6 Flowchart for THL analysis- 33 -

PhD Dissertation of Muroran Institute of Technology
Table 2-1 Gasket parameters and lubricant properties
Parameters

ValuesDisk thickness 𝐵𝐵 (mm)102Heat transfer coefficient ℎ𝑡𝑡 (W⁄m ℃)
Inner radius 𝑅𝑅1 (mm)

Inner radius 𝑅𝑅2 (mm)3

Lubricant density 𝜌𝜌 (kg⁄m )Oil thermal conductivity 𝜆𝜆 (W⁄m℃)

Disk thermal conductivity 𝜆𝜆′ (W⁄m℃)
Oil specific heat capacity 𝑐𝑐𝑝𝑝 (J⁄kg℃)
Oil viscosity 𝜈𝜈0 (m2 ⁄s)

Viscosity-temperature index 𝛽𝛽 [1⁄℃]

2.5.2

50
60
100
869
0.13
16.3
1860
32.6 × 10−6 (40℃)
0.02969

Iteration Result of Pressure Distribution under Parallel Gap

Figure 2-7 shows the central gap–time relationship in vibration conditions during one
complete period, the central gap space changes as a sinusoidal function over time in this model.
And the vibration parameters were set as 𝑓𝑓 = 5 Hz, 𝐴𝐴 = 0.00205 mm, and ∆𝑡𝑡 = 0.025s.

Fig. 2-7 Relation between central gap ℎ𝑐𝑐 and time 𝑡𝑡 under vibration condition

Figures 2-8 – 12 show the iteration results of pressure distribution under parallel condition.
Wherein, the left side of Fig. 2-8 depicts the steady-state. The right side of Fig. 2-8, and Figs.
2-9 – 12 show the instant vibrated condition corresponding to the instant time point during thewhole complete vibration period, and time interval ∆𝑡𝑡 was selected as 0.025s. The input
parameters are as follow: ℎ𝑐𝑐 = 0.2 mm, 𝑃𝑃𝑟𝑟 = 1 kPa, sealing land size: size 1, oil viscosity
grade: ISO VG32, 𝑇𝑇𝑤𝑤 = 5℃, 𝑇𝑇𝑖𝑖𝑖𝑖 = 20℃, 𝛼𝛼 = 0°; And vibration parameters as: 𝑓𝑓 = 5 Hz
(𝑇𝑇 ∗ = 0.2s), 𝐴𝐴 = 0.00205 mm. The negative pressure was replaced by zero in the cavitation
effect, which is represented by red color in these figures.
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2 Theoretical Model

Fig. 2-8 Pressure distribution in parallel gap (left: static, right: 𝑡𝑡 = 0s, vibrated)

Fig. 2-9 Pressure distribution in parallel gap (left: 𝑡𝑡 = 0.025s, right: 𝑡𝑡 = 0.05s; vibrated)

Fig. 2-10 Pressure distribution in parallel gap (left: 𝑡𝑡 = 0.075s, right: 𝑡𝑡 = 0.1s; vibrated)- 35 -

PhD Dissertation of Muroran Institute of Technology

Fig. 2-11 Pressure distribution in parallel gap (left: 𝑡𝑡 = 0.125s, right: 𝑡𝑡 = 0.15s; vibrated)

Fig. 2-12 Pressure distribution in parallel gap (left: 𝑡𝑡 = 0.175s, right: 𝑡𝑡 = 0.2s; vibrated)

It can be seen that the pressure is distributed linearly along radius direction for both parallel
conditions of static and vibrated. In the steady-state, the pressure only contains the hydrostatic
effect. In the vibration condition, the pressure distribution is the result of hydrostatic effect andsqueeze effect. Besides, the cavitation effect is strongest at the points of 𝑡𝑡 = 0s and 𝑡𝑡 = 0.2s;
the squeeze effect is strongest at the points of 𝑡𝑡 = 0.1s. This is because at these points in time,
the pressure changes most with time 𝑡𝑡, that is, the absolute value of the pressure gradient is thelargest, so the squeeze effect is also the most severe. At the points of 𝑡𝑡 = 0s and 0.2s, thepressure gradient is the maximum of negative value, so the cavitation effect is the most obvious
when the hydrostatic effect is impacted together; At the point of 𝑡𝑡 = 0.1s, pressure gradient is
the maximum of positive value, and the squeeze effect is the most obvious after impacting thehydrostatic effect. In addition, the squeeze effect during the oil film is very sensitive with thegap height, especially under the condition with small gap.

2.5.3

Iteration Result of Pressure Distribution under Inclined Gap

The important part in the numerical analysis is under the condition of small clearance. The- 36 -

2 Theoretical Modelreasons are as follows: first, when gap clearance is small, the squeeze effect is obvious, which
results in greater pressure. From the perspective of numerical method, a solution can converge
under the condition of heavy load is an effective way to judge whether calculation is applicableor not; secondly, from the perspective of engineering practice, the sealing element working inheavy load is likely to break down, the sealing characteristics have to be paid much attention.
Therefore, we focus on the pressure distribution under the conditions of inclined gap.
Figures 2-13 – 17 show the iteration result of pressure under the inclined conditions.
Similar with the parallel conditions, the left side of Fig. 2-13 depicts the steady-state, the rightside of Fig. 2-13 and Figs. 2-14 – 17 show the instant inclined vibrated condition corresponding
to the instant time point during one complete vibration period, and the time interval ∆𝑡𝑡 wasalso selected as 0.025s. The input parameters under the inclined conditions are almost samewith parallel condition except for the inclined angle. Wherein, ℎ𝑐𝑐 = 0.2 mm, 𝑃𝑃𝑟𝑟 = 1 kPa,
sealing land size: size 1, oil viscosity grade: ISO VG32, 𝑇𝑇𝑤𝑤 = 5℃, 𝑇𝑇𝑖𝑖𝑖𝑖 = 20℃, 𝛼𝛼 = 0.057°;
the vibration parameters as: 𝑓𝑓 = 5 Hz (𝑇𝑇 ∗ = 0.2s), 𝐴𝐴 = 0.00205 mm. The negative pressure
was also represented by red color in Figs. 2-13 – 17.

Fig. 2-13 Pressure distribution in inclined gap (left: static, right: 𝑡𝑡 = 0s, vibrated)

Fig. 2-14 Pressure distribution in inclined gap (left: 𝑡𝑡 = 0.025s, right: 𝑡𝑡 = 0.05s; vibrated)- 37 -

PhD Dissertation of Muroran Institute of Technology

Fig. 2-15 Pressure distribution in inclined gap (left: 𝑡𝑡 = 0.075s, right: 𝑡𝑡 = 0.1s; vibrated)

Fig. 2-16 Pressure distribution in inclined gap (left: 𝑡𝑡 = 0.125s, right: 𝑡𝑡 = 0.15s; vibrated)

Fig. 2-17 Pressure distribution in inclined gap (left: 𝑡𝑡 = 0175s, right: 𝑡𝑡 = 0.2s; vibrated)- 38 -

2 Theoretical Model
The pressure distribution under the static inclined condition shows in the left side of Fig.
2-13, it seems that the pressure distribution is also symmetrical along 𝜃𝜃, and it is distributed
linearly along 𝑟𝑟, which is same with parallel cases. However, after checking the specific value
in pressure matrix of program, it was found that under inclined conditions, the pressure along𝜃𝜃 direction is not completely symmetric although the pressure difference is very little, it could
become smaller as closer to the ℎ𝑚𝑚𝑚𝑚𝑚𝑚 point. The inclined angle was selected as 𝛼𝛼 = 0.057°
in above figures, which is a small value. Thus, the pressure changes along 𝜃𝜃 direction is not
obvious. When the inclined angle is larger, the pressure changes along 𝜃𝜃 will become more
obvious, and it has been verified in program. It is worth mentioning that in static, the pressure
distribution is only determined by the hydrostatic effect. While in the vibration condition, it is
the collective result of hydrostatic effect and squeeze effect together.

Fig. 2-18 Pressure contour line in inclined gap (left: 𝑡𝑡 = 0s, right: 𝑡𝑡 = 0.025s; vibrated)

Fig. 2-19 Pressure contour line in inclined gap (left: 𝑡𝑡 = 0.075s, right: 𝑡𝑡 = 0.1s; vibrated)

Figures 2-18 and 19 depict the instant pressure contour of the cavitation and squeeze effectrespectively. Figure 2-18 is corresponding to the time points of 𝑡𝑡 = 0s and 0.025s, which is
impacted by cavitation effect. Negative pressure was not replaced by zero temporarily in order- 39 -

PhD Dissertation of Muroran Institute of Technology
to observe the influence of cavitation effect on pressure more conspicuous. Figure 2-19 iscorresponding to the time points of 𝑡𝑡 = 0.075s and 0.1s, which is impacted by squeeze effect.
It can be seen that both effects are indeed the most intense near the ℎ𝑚𝑚𝑚𝑚𝑚𝑚 area.
From above figures, it can be seen that under both conditions of parallel and inclined, thecavitation effect is strongest at the points of 𝑡𝑡 = 0s and 0.2s; the squeeze effect is strongest
at the point of 𝑡𝑡 = 0.1s. This is because at these time points, the pressure 𝑝𝑝 changes most withtime 𝑡𝑡, that is, the absolute value of pressure gradient ∂𝑝𝑝⁄∂𝑡𝑡 is the largest, so squeeze effectis also the most severe. At the points of 𝑡𝑡 = 0s and 0.2s, pressure gradient is the maximumof negative value, so the pressure was reached to minimum value when the hydrostatic effect is
impacted together; at the point of 𝑡𝑡 = 0.1s, pressure gradient is the maximum of positive value,
and the pressure was reached to maximum after impacting the hydrostatic effect.

2.5.4

Iteration Result of Temperature Distribution under Parallel Gap

Figures 2-20 – 23 show three-dimensional diagram of theoretical solutions for temperature
distribution with two types of gap space in oil film and upper disk with a parallel gap derived
by the THL theory. Figures 2-20 – 21 and 22 – 23 show the temperature maps of disk and oilfilm in terms of gap space as ℎ𝑐𝑐 = 0.1 and 0.01 mm, respectively. The other parameters are
as follows: 𝑃𝑃𝑟𝑟 = 1 kPa, 𝑇𝑇𝑤𝑤 = 5℃ , 𝑇𝑇𝑖𝑖𝑖𝑖 = 𝑇𝑇𝑎𝑎 = 20℃ , 𝑅𝑅1 = 60 mm, 𝑅𝑅2 = 100 mm, oil
viscosity grade: ISO VG32, 𝛼𝛼 = 0°. Among Figs. 2-20 – 23, the inner surfaces of disk and oilfilm shows the oil temperature 𝑇𝑇𝑖𝑖𝑖𝑖 , which is represented in yellow; the lower surface of oil film
is the solid wall temperature 𝑇𝑇𝑤𝑤 , it is shown in dark blue; the lower surface of disk and theupper surface of oil film is exactly the same surface, which is the gap–disk interface. Thus, all
the temperature values at the grad nodes on this interface should be correspondingly equal; theupper and outter surface of disk are both the disk–ambient interface, whose temperature value
is controlled by the boundary conditions. It can be seen that the temperature of oil film area iscloser to the target value in the condition with a smaller gap.

Fig. 2-20 Theoretical solution of 𝑇𝑇′ by THL theory in upper disk (ℎ𝑐𝑐 = 0.01 mm)- 40 -

2 Theoretical Model

Fig. 2-21 Theoretical solution of 𝑇𝑇 by THL theory in oil film (ℎ𝑐𝑐 = 0.01 mm)

Fig. 2-22 Theoretical solution of 𝑇𝑇′ by THL theory in upper disk (ℎ𝑐𝑐 = 0.1 mm)

Fig. 2-23 Theoretical solution of 𝑇𝑇 by THL theory in oil film (ℎ𝑐𝑐 = 0.1 mm)- 41 -

PhD Dissertation of Muroran Institute of Technology
Figures 2-24 and 25 show the temperature contour map in the cross-section of the disk andoil film, which is corresponding to Figs. 2-20 – 21 and 22 – 23 respectively. The upper andlower half sections represent the disk and oil film areas. The basic line (𝑧𝑧 = 0) is the target
temperature of the solid wall (𝑇𝑇𝑤𝑤 = 5℃). It is shown that, in the program solutions of the THL
model, the temperature values at different space points in the oil film show a small discrepancy;
nevertheless, they are close to the set temperature values of the solid wall, i.e., the temperatureof the oil film can feasibly be controlled and it more closely matches the solid wall temperature
𝑇𝑇𝑤𝑤 in the condition with a thinner gap. Based on the IV theory, the temperature value of all
space points in the oil film is approximately taken the temperature value of the solid wall.
During one group of specific experiment, the target temperature 𝑇𝑇𝑤𝑤 of solid wall was selected
as 5℃, 15℃, 25℃, 35℃ and 45℃, respectively.

Fig. 2-24 Temperature contour in the cross-section of disk and oil film (ℎ𝑐𝑐 = 0.01 mm)

Fig. 2-25 Temperature contour in the cross-section of disk and oil film (ℎ𝑐𝑐 = 0.1 mm)

Figure 2-25 shows temperature distribution in parallel gap of ℎ𝑐𝑐 = 0.1 mm, 𝑃𝑃𝑟𝑟 = 1 kPa,
𝑇𝑇𝑖𝑖𝑖𝑖 = 20℃, 𝛼𝛼 = 0°, and 𝑇𝑇𝑤𝑤 = 5℃. The temperature contour line in a cross section of oil film
- 42 -

2 Theoretical Modeland upper disk under other temperatures of 𝑇𝑇𝑤𝑤 = 15℃, 25℃, 35℃, and 45℃ are shown in
Figs. 2-26 – 29 respectively. These are the temperature distribution in a cross section of oil film
and upper disk under all the solid wall temperatures during one group of experiment.

Fig. 2-26 Temperature contour in the cross-section of disk and oil film (𝑇𝑇𝑤𝑤 = 15℃)

Fig. 2-27 Temperature contour in the cross-section of disk and oil film (𝑇𝑇𝑤𝑤 = 25℃)

Fig. 2-28 Temperature contour in the cross-section of disk and oil film (𝑇𝑇𝑤𝑤 = 35℃)
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PhD Dissertation of Muroran Institute of Technology

Fig. 2-29 Temperature contour in the cross-section of disk and oil film (𝑇𝑇𝑤𝑤 = 45℃)

2.5.5

Iteration Result of Temperature Distribution under Inclined Gap

Figures 2-30 and 31 show the THL solutions of temperature counter in the gap–diskinterface, i.e., 𝑇𝑇𝑧𝑧 = ℎ (𝑇𝑇′𝑧𝑧 ′ = 0 ), under the inclined disk condition. The input parameters are as

follows: ℎ𝑐𝑐 = 0.15 mm, 𝑃𝑃𝑟𝑟 = 1.4 kPa, and 𝑇𝑇𝑤𝑤 = 5℃. It shows the temperature contours fordisk inclined angle 𝛼𝛼 = 0.057° and 𝛼𝛼 = 0.086°, respectively. The temperature of inlet oiland solid wall was determined as 𝑇𝑇𝑖𝑖𝑖𝑖 = 20℃, 𝑇𝑇𝑤𝑤 = 5℃. The center gap ℎ𝑐𝑐 was set as 0.15mm, and supply pressure 𝑃𝑃𝑟𝑟 was 1.4 kPa. There were temperature variations during the gap–
disk interface, which affected by the specific value of the inclined angle. The temperature
distributions coincided with solid wall temperature 𝑇𝑇𝑤𝑤 , the temperature point near the locationof the minimum gap (ℎ𝑚𝑚𝑚𝑚𝑚𝑚 ) most closely matched the wall temperature for both cases; thetemperatures had the distributions.

Fig. 2-30 Contour map of 𝑇𝑇 in gap–disk interface with inclined angle (𝛼𝛼 = 0.057°)- 44 -

2 Theoretical Model

Fig. 2-31 Contour map of 𝑇𝑇 in gap–disk interface with inclined angle (𝛼𝛼 = 0.086°)

Figure 2-31 shows the temperature distribution under inclined condition of ℎ𝑐𝑐 = 0.15mm, 𝑃𝑃𝑟𝑟 = 1.4 kPa, 𝑇𝑇𝑖𝑖𝑖𝑖 = 20℃, 𝛼𝛼 = 0.086°, and 𝑇𝑇𝑤𝑤 = 5℃. The temperature contour line ofgap–disk interface in other target temperature conditions of 𝑇𝑇𝑤𝑤 = 15℃, 25℃, 35℃, and
45℃ are shown in Figs. 2-32 – 35 respectively. These are diagrams of temperature distribution
in a cross section of oil film and upper disk under all the solid wall temperatures during acomplete set of experiments in detail as well. In all cases of these solid wall temperature 𝑇𝑇𝑤𝑤 ,the thermal effect can be transmitted to the disk through lubricating oil, that is, the temperatureof gap–disk interface can be controlled by the solid wall effectively. Besides, the temperature
difference between the interface from Figs 2-31 – 35 is larger under the extreme temperature
conditions of 𝑇𝑇𝑤𝑤 differs greatly from 𝑇𝑇𝑎𝑎 , such as 𝑇𝑇𝑤𝑤 = 5℃ and 45℃.

Fig. 2-32 Contour map of 𝑇𝑇 in gap–disk interface with inclined angle (𝑇𝑇𝑤𝑤 = 15℃)

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PhD Dissertation of Muroran Institute of Technology

Fig. 2-33 Contour map of 𝑇𝑇 in gap–disk interface with inclined angle (𝑇𝑇𝑤𝑤 = 25℃)

Fig. 2-34 Contour map of 𝑇𝑇 in gap–disk interface with inclined angle (𝑇𝑇𝑤𝑤 = 35℃)

Fig. 2-35 Contour map of 𝑇𝑇 in gap–disk interface with inclined angle (𝑇𝑇𝑤𝑤 = 45℃)
- 46 -

2 Theoretical Model

2.6 Chapter Summary
In this chapter, the THL and IV theories are both applied to the gap flow under conditions
that include static and harmonic vibration. The basic equations of fluid mechanics applicableto this prototype gasket model are derived, i.e., Reynolds equation, energy equation and heat
conduction equation. And the mathematical analytical equations of flow velocity, gap clearance
and leakage flow along the radius direction are derived, which is the theoretical basis foranalyzing gasket sealing performance.
The solution domain of the pressure is meshed, and the central difference format was used
to replace the differential quotient in the Reynolds equation, so as to discretize it into difference
scheme, and then the numerical solution can be obtained. Then, the forward difference andcentral difference formulas of FDM are used to discretize the difference equation in the domainof oil film and upper disk respectively. The specific difference forms of energy equation andheat conduction equation are derived, the discretization methods of two temperature control
equations and five boundary conditions were discussed in detail. On this basis, the temperatureof any node in the calculation region can be expressed as a linear combination of adjacent nodes.
The end condition in iteration program of the temperature control equation were discussed.
Finally, it is combined methodically from Section 2.2 to 2.4 and constructed the loop
structure and termination conditions of pressure and temperature of iteration respectively in thespecific program. The Reynolds equation, the energy equation and the heat conduction equation
were nested by the iterative method with two-layer loops. The difference equations are solved
simultaneously and debugged, and the reliable numerical solutions reached the accuracy
conditions under the given input parameters are obtained. The contours of pressure distribution
in lubricant film in several typical cases were drawn, and the temperature distribution of oil
film and upper disk were obtained collectively. The THL application of iterative method inpressure and temperature analysis was discussed.

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lubrication simulation of reciprocating rod seals under transient condition, Tribology
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[3] Dousti, S. and Allaire, P., A thermohydrodynamic approach for single-film and double-film
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PhD Dissertation of Muroran Institute of Technology[4] Shyu, S. H., Jeng, T. R. and Li, F., A Legendre collocation method for thermohydrodynamic
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[5] Tang, H. S., Ren, Y. and Xiang, J. W., A novel model for predicting thermo-elastohydrodynamic lubrication characteristics of slipper pair in axial piston pump, International
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[6] Wang, X. L., Zhu, K. Q. and Wen, S. Z., Thermohydrodynamic analysis of journal bearings
lubricated with couple stress fluids, Tribology International, Vol.34 (2001), pp.335-343.
[7] Maraiy, S. Y., Crosby, W. A. and El-gamal, H. A., Thermohydrodynamic analysis of airfoil
bearing based on bump foil structure, Alexandria Engineering Journal, Vol.55 (2016),
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[8] Chun, S. M., Thermohydrodynamic lubrication analysis of high-speed journal bearing
considering variable density and variable specific heat, Tribology International, Vol.37(2004), pp.405-413.
[9] Anandan, N., Pandey, R. K. and Jagga, C. R., An efficient numerical analysis of starved
thermohydrodynamically lubricated rolling line contacts, Tribology International, Vol.41(2008), pp.940-946.
[10] Amit, C., Rakesh, S. and Rajesh, K. S., Thermohydrodynamic analysis of elliptical journalbearing with different grade oils, Tribology International, Vol.43 (2010), pp. 1970-1977.
[11] Gu, B. Q., Chen, Y. and Zhu, D. S., Prediction of leakage rates through sealing connections
with nonmetallic gaskets, Journal of Nanjing University, Vol.15, No.6 (2007), pp.837-841.
[12] Chun, S. M., A parametric study on bubbly lubrication of high-speed journal bearings,
Tribology International, Vol.35 (2002), pp.1-13.
[13] Gao, S. and Kazama, T., Leakage control for a flat flange model with a gap based on
sealing liquid viscosity, Journal of Advanced Mechanical Design, Systems, andManufacturing, Vol.15, No.3 (2021), 21-00065.
[14] Kazama, T. and Gao, S., Gap flow between two circular plates with temperature-controlled
wall: application of thermohydrodynamic lubrication theory and comparison with an isoviscous model, Mechanical Engineering Letters, Vol.6 (2020), 20-00292.
[15] Feng, H. H., Jiang, S. Y. and Ji, A. M., Investigations of the static and dynamic
characteristics of water-lubricated hydrodynamic journal nearing considering turbulent,
thermo-hydrodynamic and misaligned effects, Tribology International, Vol.130 (2019),
pp.245-260.
[16] Li, B., Sun, J., Zhu, S. Y., Fu, Y. Y., Zhao, X. Y., Wang, H., Teng, Q., Ren, Y. P., Li, Y.
Q. and Zhu, G, X., Thermohydrodynamic lubrication analysis of misaligned journal bearing
considering the axial movement of journal, Tribology International, Vol. 135 (2019),
pp.397-407.

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2 Theoretical Model[17] Kyuho, S. and Tea, H. K., Thermohydrodynamic analysis of bump-types gas foil bearings
using bump thermal contact and inlet flow mixing models, Tribology International, Vol.48(2012), pp.137-148.
[18] Liang, X. X., Yan, X. Q., Ouyang, W., Wood, R. J. K. and Liu, Z. L., Thermo-elastohydrodynamic analysis and optimization of rubber-supported water-lubricated thrustbearings with polymer coated pads, Tribology International, Vol.138 (2019), pp.365-379.
[19] Lin, X. H., Jiang, S. Y., Zhang, C. B. and Liu, X., Thermohydrodynamic analysis of highspeed water-lubricated spiral groove thrust bearing considering effects of cavitation, inertia
and turbulence, Tribology International, Vol.119 (2018), pp.645-658.
[20] Wang, X. L., Zhu, K. Q. and Wen, S. Z., Thermohydrodynamic analysis of journal bearing
lubricated with couple stress fluids, Tribology International, Vol.34 (2001), pp.335-343.
[21] Kim, K. W., Tanaka, M. and Hori, Y., A three-dimensional analysis ofthermohydrodynamic performance of sector-shaped, tilting-pad thrust bearings,
Transactions of the ASME, Vol.105 (1983), pp.406-412.
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[23] Kazama, T., Thermohydrodynamic lubrication model applicable to a slipper of swashplate
type axial piston pumps and motors (effects of operating conditions), Tribology Online,
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[24] Kazama, T., Suzuki, M. and Suzuki, K., Relation between sliding-part temperature andclearance shape of a slipper in swashplate axial piston motors, Proc. JFPS Conf (2015),
pp.10-17.
[25] Zhang, K., Zhan, X. Y., Feng, K. and Zhao, Z. L., Thermohydrodynamic analysis andthermal management of hybrid bump-metal mesh foil bearings: experimental tests andtheoretical predictions, International Journal of Thermal Sciences, Vol.127 (2018), pp.91104.
[26] Kyuho, S. and Tae, H. K., Thermohydrodynamic analysis of bump-type gas foil bearings
using bump thermal contact and inlet flow mixing models, Tribology International, Vol.48(2012), pp.137-148.
[27] Meng, F. M., Zhang, Y. Y., Hu, Y. Z. and Wang, H., Thermo-elasto-hydrodynamic
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[28] Wang, W. M., Liu, B. B., Zhang, Y., Shao, X. and Allaire, P. E., Theoretical andexperimental study on the static and dynamic characteristics of tilting-pad thrust bearing,
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[29] Durany, J., Pereira, J. and Varas, F., Numerical solution to steady and transient problems
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[30] Wang, X. L. and Zhu, K. Q., Numerical analysis of journal bearings lubricated withmicropolar fluids including thermal and cavitating effects, Tribology International, Vol.39(2006), pp.227-237.
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pp.85-93.

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3 Experimental Apparatus and Method

3

Experimental Apparatus and Method

Gaskets and its joints are sealing the devices and static seals in industry and machinery.
Gaskets prevent the leaking of fluid from various pipe connections and casings onto stationary
surfaces of elements. Since a machined surface cannot be absolutely flat, irregularities can be
filled with gaskets between connecting surfaces to ensure that two work-pieces are joined
closely together. The gasket can fill irregularities, thereby helping to overcome defects.
There are many kinds of gaskets according to their applications, such as size, sealing object,service environment and working conditions. In order to prevent leakage, the pressure of solid
surface is basically increased to the narrow gap. Generally, leakage is reduced by increasing
the preload of sealing components and by blocking flow. However, the overcharged preload
force may cause damage to the gasket surface because of excessive extrusion. Vibration may
also occur, which can cause the loosening of screws, leading to an increase in existing gaps.
Due to the vibration and deformation, it could be difficult to intercept leakage owing toirregularities and undulation on the clamped surface. Thus, to improve the sealing performanceof gaskets, it is necessary to develop methods for reducing leakage between uneven surfaces in
the field of mechanical manufacturing under both conditions of steady-state and vibrations.

3.1 Experimental Principle and Objective
The viscosity of fluid is the lubricant character about resistance to flow. It is one of theimportant factors affecting the properties of fluids. When the fluid flows, the viscosity makes
the fluid produce resistance along the flow direction. Leakage is directly proportional to the
cube of gap height and inversely proportional to viscosity. Moreover, the viscosity of fluid,particularly oil, strongly depends on temperature, as lower temperatures correspond to higher
viscosities. In other words, oil leakage can be reduced by decreasing its temperature. Therefore,
it is possible to control viscosity by changing the temperature of sealing parts of gasket, andthen to reduce leakage further. Through experiments, the approach of reducing leakage can beresearched even in the case of constant clearance. In addition, the leakage was also examined
under the vibration conditions. The experimental results of static and vibrations were compared.
The main idea of experiment is to research the applicable way to reduce the leakage of gasket
and increase gasket sealing performance.

3.2 Experimental Apparatus
Figure 3-1 provides a schematic diagram of the experimental rig, Fig. 3-2 shows its section
surface and Fig. 3-3 depicts the sealing land. The experimental device included a gasket tester,which comprised a set of stainless-steel disks, a Peltier module, three piezo-activators, a flow
meter, a pressure gauge, two laser detectors, a radiation thermometer, and three types of sealing
land with different widths. In this gasket model, the upper and lower flange was selected to the- 51 -

PhD Dissertation of Muroran Institute of Technology
A5052 in JIS and SUS303, respectively. The diameter of the upper and lower disks was 200
and 220 mm, the thickness of each disk and sealing ring was 10 and 5 mm. For ISO VG22 oil,𝜈𝜈0 = 23 × 10−6 m2 ⁄s at 𝑇𝑇0 = 40℃ and the other oil properties of 𝛽𝛽 = 0.0298 [1⁄℃],𝜌𝜌0 = 866 kg⁄m3 , 𝑐𝑐𝑝𝑝 = 1968 J⁄(kg ∙ K), and 𝜆𝜆 = 0.134 W⁄(K ∙ m) were assumed to beconstant. For ISO VG32 oil, 𝜈𝜈0 = 32.6 × 10−6 m2 ⁄s (at 40℃ ), 𝛽𝛽 = 0.02969 [1⁄℃] ,𝜌𝜌0 = 869 kg⁄m3 , 𝑐𝑐𝑝𝑝 = 1860 J⁄(kg ∙ K) , 𝜆𝜆 = 0.130 W⁄(K ∙ m) ; 𝜌𝜌𝑚𝑚 = 𝜌𝜌0 , 𝜇𝜇𝑚𝑚 = 𝜌𝜌𝑚𝑚 𝜈𝜈 ,and 𝜆𝜆′ = 16.3 W⁄(K ∙ m) , ℎ𝑡𝑡 = 50 W⁄(m2 ∙ K) . The experimental operating conditions
were as follows: ℎ𝑐𝑐 = 0.05 − 0.3 mm, 𝑃𝑃𝑟𝑟 = 0.5 − 2 kPa, and 𝑇𝑇𝑤𝑤 = 5 − 45℃. The value ofclearance ℎ𝑐𝑐 selected in experiment was relatively larger than the actual working conditionsof gasket. One of the reasons is that we tried to confirm the effects of temperature changes onleakage control of a flange type gasket and also under surface contacting conditions the oilleakage was infinitesimal so that it could not be measured accurately in experiment.

Fig. 3-1 Schematic diagram of the experimental rig

Fig. 3-2 Internal section surface of gasket tester- 52 -

3 Experimental Apparatus and Method

Fig. 3-3 Sealing land between two disks
Figure 3-4 shows the schematic diagram of all the experimental apparatuses. The Peltier
module was mounted above the upper disk to control the temperature of the sealing parts. The
gasket test section mainly consists of upper disk, lower disk, and sealing land as shown above.
The gap height from the sealing land to upper disk could be set by rotating hexagon bolts, which
supported tester facility. The temperature sensors were mounted on the apparatus. Oil supply
pressure was controlled by adjusting the surface height of the reservoir, and the accuratepressure value can be measured using the pressure gauge located near the oil inlet. Leakage wasmeasured by a flowmeter. The lubricant flowed into the gasket tester after passing through theflowmeter and pressure gauge. The actual temperature distribution of each disk was measured
and confirmed by using a radiation thermometer. The effective sealing size could be adjusted
using metal sealing land with different widths, which were inserted between two disks. The
vibration mode, input frequency, and supply voltage of the piezo-activators could be inputted
directly using a function generator. The output frequency and amplitude of the two disks were
measured by laser detectors, respectively, under vibration conditions.

Fig. 3-4 Schematic of all the experimental apparatuses- 53 -

PhD Dissertation of Muroran Institute of Technology

3.3 Photographs of Apparatuses and their Parameters
Figures 3-5 and 6 is the photograph of gasket tester section and function parts, respectively.
The test section of gasket facility mainly consists of upper disk, lower disk, and sealing land.
Wherein, the sealing land was inserted between two disks. The gap refers to the tiny distancefrom the sealing land to upper disk. The function of pump was to return the lubricating oil fromreservoir to tank, so as to ensure the oil surface in tank could kept the same level, so that the oilpressure of gasket recess was consistent during one group of experiment.

Fig. 3-5 Photograph of gasket facility

Fig. 3-6 Photograph of function parts
Figure 3-7 is the photograph of sealing land. The size of sealing land can be adjusted byusing different widths of rings inserted between two disks. There are three types of sealing landin experiment as shown in Table 3-1. wherein, size 1 is the widest, size 3 is the narrowest.
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3 Experimental Apparatus and Method
Figure 3-8 shows the internal structure of the gasket test section. Wherein, the function of shimring was to keep the minimum clearance. If the clearance is too small, the leakage will also bea small amount, so that it could not be measured accurately by flowmeter.
Table 3-1 Parameters of three types of sealing landType
Size 1Size 2Size 3

Inner diameters 𝑑𝑑1 (mm)Outer diameters 𝑑𝑑2 (mm)

140

180

120
130

Fig. 3-7 Photograph of sealing land

Fig. 3-8 Inner structure of gasket facility- 55 -

200
190

PhD Dissertation of Muroran Institute of Technology

3.4 Experimental Method
The specific experimental method is as follow: After setting the initial gap height value
and opening the tank valve, the lubricant flowed into the gasket tester by tube from central holeat the bottom of lower disk under the action of supply pressure, and flowed out circumferential
at the edge of disks from the gap between sealing land and upper disk. Subsequently, theflowmeter began to work. The supply pressure was always set to relatively small value during
the whole process, thus, the lubricant flowing can be regarded as the slow laminar flow. Under
the steady-state, the three piezo-activators were non-operational. Thus, it was unnecessary tomeasure the vibration meters (e.g., frequency and amplitude) because that gasket is operated
between two stationary disks in these circumstances. Under vibration conditions, the vibration
mode of three piezo-activators must be set to the simple harmonic vibration, and the specificfrequency and supply voltage from the function generator must be inputted. When the piezoactivators were functioning, the lower disk and sealing land were vibrated in the same way withthe piezo-activators, while upper disk remained relatively at rest. Output frequency and output
amplitude of two disks were measured by laser detectors. Single-side amplitude 𝐴𝐴 in simple
harmonic equation (Eq. (2-28)) was calculated by the amplitude difference between two disks.
The target temperature was changed in the order of 25°C, 15°C, 5°C, 25°C, 35°C, 45°C, and 25°C
from the Peltier module. Once the goal temperature became stable, the actual temperature oftwo disks were measured by radiation thermometer. Then, leakage was recorded by flowmeter
during the specific period of time. The gasket sealing acreage was adjusted by inserting sealing
land with different widths and fluid properties was changed by providing different viscosity oil
grades; other conditions remained the same. In this way, the factors that affected leakage wereresearched.

3.5 Chapter Summary
In this chapter, the theoretical basis is introduced, that is, trying to control the viscosity of
fluid by changing the temperature of sealing parts of gasket from Peltier module, and then to
reduce leakage further. The method of applications and the functions of each apparatus whichused in experiment such as gasket facility, Peltier module, laser detector, piezo-activator, pump,
and amplification driver were described. The specific steps and the precautions in the processof the experiment were introduced in detail.

References[1] Gao, S. and Kazama, T., Effect of vibration condition on leakage of characteristics ofprototype gasket using oil viscosity-Temperature relations, Proc. JSME Annual Conf
(2020), S11304.

- 56 -

3 Experimental Apparatus and Method[2] Gao, S. and Kazama, T., Leakage characteristics of prototype gasket using viscositytemperature relation under vibration condition (effects of sealing land size and viscosity
grade), Proc. ICFP Conf (2021), S2304.
[3] Gao, S. and Kazama, T., Leakage control for a flat flange model with a gap based on sealingliquid viscosity, Journal of Advanced Mechanical Design, Systems, and Manufacturing,
Vol.15, No.3 (2021), 21-00065.

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PhD Dissertation of Muroran Institute of Technology- 58 -

4 Experimental Results under Steady-state Condition

4

Experimental Results under Steady-state Condition
Under the conditions of the amplitude of two disks 𝐴𝐴𝑢𝑢𝑢𝑢 = 𝐴𝐴𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 0, the gap height does

not change with time 𝑡𝑡, that is, the gasket is operated under steady-state in this case. Under thecondition that the vibration is very small, it can be roughly regarded as the static-state. In theexisting research on gasket, many researches [1-3] are focused on the factors affecting gasket
leakage and how to reduce the leakage under static conditions. Fischer et al. [4] analyzed thefluid leakage in metallic seals, they investigated the role of plastic deformation in the leak-rateof metallic seals and the plastic deformation increases the area of real contact and reduce
interfacial separation at critical constriction was found. Roos et al. [5] researched the gasket
characteristics for the design of bolted flange connections of the metal-to-metal contact type.
Haruyama et al. [5] presented the influence of surface roughness on leakage of new metal gasket,
they investigated the helium leakage quantity for a flange with different surface roughness
values through a simulation by using finite element method and a leakage test.
The experiment was conducted under steady-state first, and then qualitatively analyze andforecast experiment situation under vibrations, and carry out by experimental verification. In
this paper, in order to show the contrast between the experimental data and theoretical results
more intuitively, theoretical results were compared with experimental data as represented in all
figures by lines and points, respectively. In the theoretical results, the dashed and solid curves
represent the program solutions derived by the THL and IV theories, respectively.

4.1 Effects of Gap Height on Leakage
The gap height from the sealing land to upper disk can be adjusted by turning the hexagon
bolt which supporting the tester facility shown in Fig. 3-5. In order to research the effects ofgasket height on the leakage flow under steady-state conditions, the supply pressure of the twogroups of experiment was set as 𝑃𝑃𝑟𝑟 = 0.6 and 1.0 kPa, respectively. The effects of different
clearance values on the leakage rate of lubricating oil was compared.

Fig. 4-1 Effect of gap height on leakage (𝑃𝑃𝑟𝑟 : 1 kPa, size 1, ISO VG32, 𝛼𝛼 = 0°)- 59 -

PhD Dissertation of Muroran Institute of Technology

Fig. 4-2 Effect of gap height on leakage (𝑃𝑃𝑟𝑟 : 0.6 kPa, size 1, ISO VG32, 𝛼𝛼 = 0°)

The experimental results are shown in Figs. 4-1 and 2. It can be seen that when the oiltemperature was kept same, the larger gap was corresponding to the greater leakage, which is
obviously consistent with common sense. In addition, as the temperature was lower, the leakage
decreased for both conditions of small and large gap.

4.2 Effects of Supply Pressure on Leakage
Nelson and Prasad [7] studied a gasketed joint by using three dimensional finite different
analyze with single and twin gaskets having nonlinear material properties, the flange joint with
twin gasket can withstand higher internal pressure was observed. In this experiment, the supply
pressure can be changed by adjusting the height of the reservoir. The specific value of pressure
was measured by a pressure gage, which was installed on the tube nearby oil inlet. Lubricant
flows into the gasket facility after passing through the flowmeter and pressure gauge, theleakage could be measured accurately in this way. In these groups of experiment, the gasket
clearance ℎ𝑐𝑐 was 0.15 and 0.20 mm, respectively. Other conditions were kept same, the effects
of recess supply pressure on leakage was researched in this case.

Fig. 4-3 Effect of recess pressure on leakage (ℎ𝑐𝑐 : 0.15 mm; size 1, ISO VG32, 𝛼𝛼 = 0°)- 60 -

4 Experimental Results under Steady-state Condition

Fig. 4-4 Effect of recess pressure on leakage (ℎ𝑐𝑐 : 0.2 mm; size 1, ISO VG32, 𝛼𝛼 = 0°)

The experimental results are shown in Figs. 4-3 and 4. When the oil temperature is thesame, the larger the supply pressure is, the greater the leakage is. It is also consistent with
common sense. Besides, as the temperature became lower, the leakage flow rate was decreased
independent of supply pressure.

4.3 Effects of Sealing Land Size on Leakage
The previous experimental results confirmed that the sealing performance of gasket canbe improved by cooling the sealing parts. Besides, sealing acreage is also one of the significant
parameters which affect sealing performance. Abid et al. [8] presented results of detailed nonlinear finite element analysis of an optimized bolt tightening strategy of different sizes forachieving proper preload close to target stress value. Wang et al. [9] investigated the influenceof effect of rod surface roughness on lubrication characteristic of a hydraulic O-ring seal. The
net leakage of O-ring seal was calculated and analyzed according to experimental results. The
effective sealing acreage was determined by inner 𝑑𝑑1 and outer diameter 𝑑𝑑2 of sealing land
which in inserted between two disks as shown in Fig. 3-7. The parameters of these sealing lands
used in experiment as shown in Table 3-1.

Fig. 4-5 Effect of sealing size on leakage (ℎ𝑐𝑐 : 0.15 mm, 𝑃𝑃𝑟𝑟 : 1.4 kPa, ISO VG32, 𝛼𝛼 = 0°)- 61 -

PhD Dissertation of Muroran Institute of Technology

Fig. 4-6 Effect of sealing size on leakage (ℎ𝑐𝑐 : 0.20 mm, 𝑃𝑃𝑟𝑟 : 1 kPa; ISO VG32, 𝛼𝛼 = 0°)

Figures 4-5 and 6 show the effect of the size of sealing land on leakage for different gaps
and supply pressures under parallel conditions. The leakage was decreased using a wider sealing
land independent of temperature. Decreased leakage was the most obvious at the widest sealing
land in both cases. The leakage ratio of the narrowest (size 3) and widest (size 1) sealing land
𝑄𝑄𝑛𝑛 ⁄𝑄𝑄𝑤𝑤 was 2.003; the amount of reduced leakage was proportional to the effective sealingacreage.

4.4 Effects of Oil Type Viscosity Grade on Leakage

As the common sense, different types of lubricating oil correspond to different viscosity
grade at the same temperature, and the gradient of viscosity with temperature is also different.
Some types of lubricating oil are very sensitive to temperature, and the tens of degree Celsius
temperature difference may cause their viscosity to differ by more than ten times; while some
types of lubricating oil are inert to temperature, their viscosity changes little even if there is alarge temperature difference [10-11]. Generally, the lubricant used in this experiment is ISO
VG22. In order to explore the effects of oil type viscosity grade on leakage flow, the oil of ISO
VG32 was also selected as the control groups.

Fig. 4-7 Effect of oil type on leakage (ℎ𝑐𝑐 : 0.2 mm, 𝑃𝑃𝑟𝑟 : 1 kPa, size 1, 𝛼𝛼 = 0°)- 62 -

4 Experimental Results under Steady-state Condition

Fig. 4-8 Effect of oil type on leakage (ℎ𝑐𝑐 : 0.3 mm, 𝑃𝑃𝑟𝑟 : 0.6 kPa, size 1, 𝛼𝛼 = 0°)

Figures 4-7 and 8 present comparison results for the oil type viscosity grade under parallel
conditions. For ISO VG22, the lubricant density and kinematic viscosity were 866 kg⁄m3 and23 mm2 ⁄s at 40 ℃ ; for ISO VG32, these results were 869 kg⁄m3 and 32.6 mm2 ⁄s ,respectively. The leakage was reduced by 57.6% and 60.6% as a result of cooling sealing
parts from 45℃ to 5℃ for lower (VG 22) and higher (VG 32) viscosity grade oils, respectively.
This indicates that cooling gasket sealing parts is an effective approach for leakage suppression.

4.5 Effects of Inclined Angle on Leakage
The parallelism structure of gasket is also one of the impart factors affecting leakage. The
gaskets cannot be kept completely parallel in some special cases in application. Therefore, it isnecessary to research the effect of inclined angle on leakage. In experiment, the inclination of
upper disk can be realized by setting the different gap height at the four equal angle points at
the edge of disk. In extreme cases, in order to maximize the inclined angle, the two disks must
contact with each other at left bisection point, that is, ℎ𝑚𝑚𝑚𝑚𝑚𝑚 = 0. The gap height is zero at thiscontacted point in this case, and Reynolds equation is not applicable for calculation in thisspecial point. Fortunately, this singular point could be regarded negligible comparing to theentire region of oil film. Thus, the Reynolds equation still can be used in numerical iteration.

Fig. 4-9 Effect of inclined angle on leakage (ℎ𝑐𝑐 : 0.15 mm, 𝑃𝑃𝑟𝑟 : 1.4 kPa, size 1, ISO VG32)- 63 -

PhD Dissertation of Muroran Institute of Technology

Fig. 4-10 Effect of inclined angle on leakage (ℎ𝑐𝑐 : 0.2 mm, 𝑃𝑃𝑟𝑟 : 1 kPa, size 1, ISO VG32)

Figures 4-9 and 10 show the effect of the inclined angle on leakage with different gaps andpressures. The leakage increased with a larger inclined angle when the center clearance andsupply pressure kept constant for both cases. The leakage–restrain phenomenon was the mostobvious in the parallel condition (𝛼𝛼 = 0°) in both cases. Moreover, although the leakage flow
increased as the inclined angle increased, the leakage could still be restrained by cooling thegasket sealing parts between the stationary disks for both parallel and inclined conditions.

4.6 Chapter Summary
In this chapter, the several typical cases of experiment under the steady-state conditions
were conducted. The leakage flow and oil temperature were measured when the gap height,
recess pressure, oil viscosity grade, sealing land size, and inclined angle were taken as the main
parameters. When the gasket sealing parts were cooled, the amount of leakage was reduced
based on a wide range of all the above cases. In addition, the THL and IV theories using solid
wall temperature were applied to gap flow for verification.
The leakage flow varies with the temperature of sealing parts under each cases above. The
lower the temperature, the higher the viscosity of fluids, the lower the leakage further, and viceversa. It can be seen that the increase and decrease of the leakage rate caused by heating andcooling by the solid wall was reversible and reproducible. And experimental data are comparedwith the THL and IV theories, the conclusions are as follows:(1) The theoretical solutions that applied by THL theory were more closely resembled thesituations of experiment. The IV theory can also provide a feasible way for analyzing
qualitatively the tendency of leakage of flow.
(2) The leakage of fluid was suppressed successfully by decreasing the gap space or supply
pressure, increasing the effective sealing acreage, adopting higher viscosity grade typeof oil, and reducing the inclined angle of disks.
(3) Regardless of the sealing land sizes and the oil viscosity grades, the leakage could be
restrained by more than 50% vis cooling sealing parts of gasket from 45℃ to 5℃.
- 64 -

4 Experimental Results under Steady-state Condition(4) Within the allowable margin of error, the experimental data show good agreement withthe analytical iteration solutions of each conditions.

References[1] Varela, A. C., Nielsen, B. B. and Santos, I. F., Steady state characteristics of a tilting pad
journal bearing with controllable lubrication: comparison between theoretical andexperimental results, Tribology International, Vol.58 (2013), pp.85-97.
[2] Zhang, X., Jiang, J. B., Peng, X. D., Zhao, W. J. and Li, J. Y., Leakage reduction by
optimization of hole-pattern damping seal with inclined hole cavity, International Journalof Heat and Mass Transfer, Vol.169 (2021), 120924.
[3] Gao, F., Naini, S. S., Wagner, J. and Miller, R., An experimental and numerical study ofrefrigerator heat leakage at the gasket region, International Journal of Refrigeration, Vol.73(2017), pp.99-110.
[4] Fischer, F. J., Schmitz, K., Tiwari, A. and Persson, B. N. J., Fluid leakage in metallic seals,
Tribology Letters, Vol.125, No.68 (2020), pp.1-13.
[5] Roos, E., Kockelmann, H. and Rolf, H., Gasket characteristics for the design of bolted flange
connections of metal-to-metal contact type, International Journal of Pressure Vessels andPiping, Vol.79 (2002), pp.45-52.
[6] Haruyama, S., Nurhadiyanto, D., Choiron, M. A. and Kaminishi, L., Influence of surface
roughness on leakage of new metal gasket, International Journal of Pressure Vessels andPiping, Vol.111-112 (2013), pp.146-154.
[7] Nelson, N. R. and Prasad, N. S., Sealing behavior of twin gasketed flange joints,
International Journal of Pressure Vessels and Piping, Vol.138 (2016), pp.45-50.
[8] Abid, M., Khan, A., Nash, D. H., Hussain, M. and Wajid, H. A., Optimized bolt tightening
strategies for gasketed flanged pipe joints of different sizes, International Journal ofPressure Vessels and Piping, Vol.139-140 (2016), pp.22-27.
[9] Wang, B. Q., Meng, X. K., Peng, X. D. and Chen, Y., Experimental investigations on theeffect of rod surface roughness on lubrication characteristics of a hydraulic O-ring seal,
Tribology International, Vol.156 (2021), 106791.
[10] Kererat, C., Effect of oil-contamination and water saturation on the bearing capacity andshear strength parameters of silty sandy soil, Engineering Geology, Vol.257 (2019), 105138.
[11] Pradhan, D. and Mishra, A. K., Analysis of ISO VG 68 bearing oil for condition monitoringcollected from an externally pressurized ball bearing system, Materials Today: Proceedings,
Vol.10, No.16 (2020), pp.2214-2219.

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PhD Dissertation of Muroran Institute of Technology- 66 -

5 Experimental Results under Vibration Condition

5

Experimental Results under Vibration Condition

In this chapter, all the experimental results were obtained under the conditions of vibration
with two parallel disks. In addition, it is difficult to contact the experiment under the vibration
condition with an inclined angle, thus, the leakage flow under inclined vibration conditions
were obtained by the THL theoretical calculation. In the practical application of gasket, theoperation ambient is important [1-3]. It is hard to reach the absolutely stationary in the working
environment. The vibration is inevitable when the machine is in operation, so it is necessary toresearch the leakage of gasket in the vibration conditions. Awad and Jorge [4] researched thehydrodynamic self-excited vibrations in leaking spherical valves with annular seal, the results
showed that the system ability depends on the relation between main line inertia and leakage
kinetic energy, leakage reduced velocity, seal surface ratio, pilot line geometry and hydraulicloss. Boubendir et al. [5] analyzed the hydrodynamic self-lubricating journal bearings by using
Rabinowitsch lubricant. It was shown that the use of pseudo-plastic fluids reduced the bearing
performance but increased with porous properties.
Compared with steady-state [6], the experimental operating parameters under the vibration
conditions could be more complicated. This is because that excepting for operating the Peltier
module and pump, it was also necessary to control the parameters such as the vibration mode,
input frequency and voltage of the piezo-activators and laser detectors. The range of rated input
frequency of piezo-activators is 0~12 Hz, and the voltage is 0~7 Vpp. In the actual
experiment of vibrations, the vibration modes were set to simple harmonic, and the input
frequency was 4~7 Hz, and the input voltage was 2~5 Vpp. The specific vibration of piezoactivators was realized by inputting the sinusoidal vibration mode, rated frequency and voltage
from the function generator. In this case, the lower disk and the sealing land was vibrated withthe piezo-activators in the same way, and the upper disk was relatively rest. Setting the vibration
mode was the simple harmonic vibration, the central gap ℎ𝑐𝑐 was changed with time 𝑡𝑡 insinusoidal, and the output vibration frequency 𝑓𝑓 and amplitude 𝐴𝐴 of the two disks can bemeasured by laser detectors which are shown in Fig. 3-5. After data analysis, it was found that
output frequency is almost equal to the input value through function generator. And the output
amplitude of the relative motion between the two disks were calculated.
From the perspective of gasket structure in Fig. 3-5, the lower disk and sealing land was
supported by three piezo-activators, and the upper disk was supported by pillars. Theoretically,
the amplitude of the lower disk should be greater than the amplitude of the upper disk. However,it was found that the amplitude of the upper disc was larger in experiments. This is due to the
influence of oil viscosity and the squeezing effect, the upper disk was pushed by the lower disk
to vibrate via the lubricating oil. In addition, the smaller the gap space, the more obvious thesqueeze effect. The squeeze effect can be reduced by increasing the gap space between the two- 67 -

PhD Dissertation of Muroran Institute of Technology
disks while it could also weaken the thermal effect of Peltier module, and cause the temperatureof sealing parts could not be controlled effectively further. Comprehensive consideration, it was
adopted the absolute value of amplitude differences between two discs 𝐴𝐴 = �𝐴𝐴𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝐴𝐴𝑢𝑢𝑢𝑢 � in

Eq. (2-28). In addition, we focused on achieving the approximate same value of the amplitude
value at each target temperature point during one group of experiment. The output amplitude
was defined as 𝐴𝐴 = 𝐴𝐴0 ± ∆𝐴𝐴, where, ∆𝐴𝐴 is the error range, which could be regulated within
the range of 0.3 × 10−3 mm in the experiment.

5.1 Explanation of Experimental Row Data

Figure 5-1 is the row data of the output frequency and amplitude of two disks which were
measured by laser detectors. The mean clearance was ℎ0 = 0.2 mm, and the input frequency
and voltage in function generator were set to 4 Hz and 5 Vpp, respectively. The red and blue
curves represent the actual vibration phases of upper and lower disk respectively. It can be seenfrom the row data that the phases of two disks were exactly same, and the vibration period wereboth 0.25 s, so the output frequency was 4 Hz. That is, the output frequency measured by laser
detectors is almost equal to its input value in function generator. The amplitude of the upperdisc 2𝐴𝐴𝑢𝑢𝑢𝑢 and lower disc 2𝐴𝐴𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 was 0.0849 and 0.0212 mm, respectively. Therefore, theexactly amplitude 𝐴𝐴 in Eq. (2-28) can be calculated as 𝐴𝐴 = �𝐴𝐴𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝐴𝐴𝑢𝑢𝑢𝑢 � = 0.03185 mm

in this case. The corresponding motion of simple harmonic vibration was depicted as below.
ℎ𝑐𝑐 = 0.2 + 0.03185 ∙ sin(2π ∙ 4𝑡𝑡) (mm)(5-1)

Fig. 5-1 Experimental row data of vibration parameters of two disks

5.2 Comparison Result of Vibrations with Parallel Disks
5.2.1

Effects of Vibration Frequency on Leakage

The input frequency can be directly input from the function generator, the frequency was
set to 5 Hz and 7 Hz respectively. In order to depict the distinguish the phenomenon of leakage- 68 -

5 Experimental Results under Vibration Condition
suppression between static and vibration conditions more intuitively, the results of the static
were also shown altogether.

Fig. 5-2 Effect of frequency on leakage (ℎ𝑐𝑐 : 0.15 mm, 𝑃𝑃𝑟𝑟 : 1.4 kPa, size 2, ISO VG32)

Fig. 5-3 Effect of frequency on leakage (ℎ𝑐𝑐 : 0.2 mm, 𝑃𝑃𝑟𝑟 : 1 kPa, size 2, ISO VG32)

Figures 5-2 and 3 show the effect of vibration frequency on leakage flow. It represents thetwo cases with different gap height ℎ0 , and central pressure 𝑃𝑃𝑟𝑟 , respectively. The leakage flowfrom the flat flange gasketed joint under the steady-state was the smallest compared with thevibration conditions. Furthermore, when the amplitude remained constant, as the frequency
increased, the leakage increased. Although the increase in frequency will increase fluid leakage,cooling the sealing parts remains a feasible approach for suppressing leakage.

5.2.2

Effects of Vibration Amplitude on Leakage

The magnitude of the amplitude can be adjusted by changing the input voltage value fromfunction generator. The input frequency is controlled at 5 Hz and remains unchanged, and the
input voltage was set from 3 to 5 Vpp appropriately. The greater the input voltage, the greater
the amplitude. The results under the static was also shown in this case. Figures 5-4 and 5 show
the effect of output vibration amplitude value 𝐴𝐴 on leakage flow at each temperature point for- 69 -

PhD Dissertation of Muroran Institute of Technology
two cases, respectively. It shows effect of two types of amplitude on leakage with considering
the temperature. The results indicate that, under the harmonic vibrations, the leakage increased
observably compared with the steady-state. Additionally, a higher amplitude corresponded tomore significant leakage at each of the targeted temperature points.

Fig. 5-4 Effect of amplitude on leakage (ℎ𝑐𝑐 : 0.15 mm, 𝑃𝑃𝑟𝑟 : 1.4 kPa, size 2, ISO VG32)

Fig. 5-5 Effect of amplitude on leakage (ℎ𝑐𝑐 : 0.2 mm, 𝑃𝑃𝑟𝑟 : 1 kPa; size 2, ISO VG32)

5.3 Calculation Result of Vibrations with Inclination

In the above comparison between the experimental data and theoretical solutions, the twotheoretical models both show good agreement with experimental data. Among the theoreticalresults, the iteration solutions derived using the THL theory more closely match experimental
data. The calculation program requires a significant amount of time to compute results by using
the THL theory in iterations, particularly in the case of nonparallel condition with an inclined
angle. In the steady-state, several hours are required to calculate the leakage flow at each target
temperature point from the gap flow with an inclined angle. Furthermore, under the vibration
condition with the inclination, it is much more complicated to calculate the flow rate by using
the THL theory.

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5 Experimental Results under Vibration Condition
The comparison of above experimental data shows that the calculation results of the IVtheory were not as accurate compared with the THL theory. Therefore, the calculation results
by using the THL theory are more feasible for presenting the leakage tendency for accurately
and effectively estimating the specific leakage value. Additionally, it was relatively difficult to
conduct the simple harmonic vibration condition with inclined disk in real specific experiment.
It is able to estimate the leakage at each target temperature point in the nonparallel harmonic
vibration condition with an inclined angle by using the THL theory.

5.3.1

Effects of Vibration Frequency on Leakage in Nonparallel

Figures 5-6 and 7 show the theoretical solution of leakage under condition of nonparallel
vibrations with different frequencies. It can be seen that the relationship between frequency andleakage was same with the parallel vibrations, that is, the lower the frequency, the smaller the
leakage. And the leaked oil was also suppressed by reducing the temperature during gasket
sealing parts.

Fig. 5-6 Calculation results for effect of frequency on leakage with inclination(ℎ𝑐𝑐 : 0.15 mm, 𝑃𝑃𝑟𝑟 : 1.4 kPa, size 1, ISO VG32, 𝛼𝛼 = 0.057°, 𝐴𝐴0 = 0.0018 mm)

Fig. 5-7 Calculation results for effect of frequency on leakage with inclination(ℎ𝑐𝑐 : 0.2 mm, 𝑃𝑃𝑟𝑟 : 1 kPa, size 1, ISO VG32, 𝛼𝛼 = 0.086°, 𝐴𝐴0 = 0.0026 mm)- 71 -

PhD Dissertation of Muroran Institute of Technology

5.3.2

Effects of Vibration Amplitude on Leakage in Nonparallel

Figures 5-8 and 9 depict the theoretical solution of leakage with different amplitudes under
the nonparallel vibrations. It depicts that the lower amplitude corresponding the smaller leakage.
And it was still applicable to reduce the leakage by cooling sealing area in gasket.

Fig. 5-8 Calculation results for effect of amplitude on leakage with inclination(ℎ𝑐𝑐 : 0.15 mm, 𝑃𝑃𝑟𝑟 : 1.4 kPa, size 1, ISO VG32, 𝛼𝛼 = 0.057°, 𝑓𝑓 = 5 Hz)

Fig. 5-9 Calculation results for effect of amplitude on leakage with inclination(ℎ𝑐𝑐 : 0.2 mm, 𝑃𝑃𝑟𝑟 : 1 kPa, size 1, ISO VG32, 𝛼𝛼 = 0.086°, 𝑓𝑓 = 5 Hz)

5.4 Chapter Summary

In this chapter, the theoretical solution and experimental data shown that whether understeady-state or harmonic vibration conditions, the leakage flow can be suppressed by reducing
the temperature during the gasket sealing parts. That is, restraining leakage by cooling thesealing parts is applicable for both conditions of static and vibrations. Furthermore, the leakage
flow was strongly affected by the vibration parameters such as frequency and amplitude. The
conclusions are as follows:- 72 -

5 Experimental Results under Vibration Condition(1) The phenomenon of leakage flow reduced with the decrease of temperature can berealized for both static and vibration conditions independent of gap height and recesspressure.
(2) Whether under the conditions of parallel or inclined vibrations, the leakage flow was
reduced with the decrease of vibration parameters such as frequency and amplitude.
(3) The theoretical solution by using the THL theory is more closely to the experimental
data, while the IV solution was also a feasible way for presenting and estimating
effectively the tendency toward leaked flow.

References[1] Abid, M. and Nash, D. H., Structural strength: gasketed vs non-gasketed flange joint underbolt up operating condition, International Journal of Solids and Structures, Vol.43 (2006),
pp.4616-4629.
[2] Ding, X. X., Chen, D. L., Zhang, W. Z. and Yu, S. R., Experiment of frictional vibration
performance of the micro-texture of DLC thin with dry gas seal rings, Tribology
International, Vol.147 (2020), 106267.
[3] Fukuta, M., Sotani, T. and Motozawa, M., Leakage and friction characteristics at sliding
surface of tip seal in scroll compressors, International Journal of Refrigeration, Vol.125(2021), pp.104-112.
[4] Awad, H. and Parrondo, J., Hydrodynamic self-excited vibrations in leaking sphericalvalves with annular seal, Alexandria Engineering Journal, Vol.59 (2020), pp.1515-1524.
[5] Boubendir, S., Larbi, S., Malki, M. and Bennacer, R., Hydrodynamic self-lubricating journalbearings analysis using Rabinowitsch fluid lubricant, Tribology International, Vol.140(2019), 105856.
[6] Kazama, T., Saito, K., Narita, Y. and Hanajima, N., Lubrication characteristics of atemperature-viscosity-controlled hydrodynamic journal bearing: prototype and experiment,
Tribology Online, Vol.11, No.2 (2016), pp.75-80.

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PhD Dissertation of Muroran Institute of Technology- 74 -

6 Simulation of Effect of Sealing Surface Flatness

6

Simulation of Effect of Sealing Surface Flatness

In the practical application of gasket, especially in the case of high oil supply pressure, thesealing flange of gasket is barely maintain absolute flat, the elastic deformation occur in general.
In the vibration conditions, the sealing flange is more prone to elastic deformation due to thesqueeze effect and cavitation effect of lubricating oil film. The previous theoretical calculation
is based on the assumption that the disk is absolutely flat. If the elastic deformation formula in
the project of elasto-hydrodynamic lubrication (EHL) is introduced into theoretical calculation,
the calculation will become very complicated, the program coding will be more difficult, andthe iteration time of the program will be greatly increased. Therefore, in order to simulate thedeformation of the upper disk as simply as possible, the sealing surface of upper disk can be
regarded as different shapes in calculation, the gap space can be calculated by the convex
surface formula, and the deformation can be simulated by the gap changes along the radiusdirection. We can qualitatively analyze the task by observing the different degree of gap spacechange along the radius direction by comparing with the flat situation.
There are many researchers work on the project of EHL [1-3]. Gadari et al. [4] describedthe 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. It is shown that theseal may leak in shaft roughness beyond half of the lip roughness. Shinkarenko et al. [5]developed a theoretical model to study the potential use of laser surface texting (LST) in theform of spherical micro-dimples for soft EHL, they discussed the optimum parameters of thesurface texting and shows that LST effectively increased load capacity and reduced friction.
Ciuperca and Tello [6] analyzed the problem of a deformable surface moving over a flat plane,
the general results for nonlocal coefficients with applications to particular problems in EHL
were presented. Lehn et al. [7] researched a 3D thermo-elasto-hydrodynamic model for air foil
thrust bearings (AFTBs), the relative magnitude of the different heat fluxes occurring in AFTBs
was analysis in detail.
In this chapter, the lower surface of upper flange is replaced with convex and concave
shape, and the gap space changed along radius direction. The numerical solution of temperature
distribution, flow velocity and leakage is calculated under the several types of sealing surface
shape in flat, convex, and concave by using the THL theory. The effect of sealing surface
flatness on temperature distribution, flow velocity, and leakage were mainly discussed for bothparallel and inclined conditions.

6.1 Theoretical Model and Grid Nodes with Surface Flatness
The leakage of a flange-type gasket is modeled as a gap between two circular disks asshown in Fig. 6-1 [8]. The THL theory is applied to the gap flow with considering the sealingsurface flatness under static conditions of parallel and inclination.

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PhD Dissertation of Muroran Institute of Technology

Fig. 6-1 Schematic diagram of gasket model under convex condition
The gap height ℎ changed along 𝑟𝑟 direction is given as follow:
𝑟𝑟−𝑅𝑅

2

ℎ = ℎ𝑐𝑐 + 𝛼𝛼𝛼𝛼cos(𝜃𝜃 − 𝜑𝜑) + 𝛿𝛿 �𝑅𝑅 −𝑅𝑅1 �
2

1

(6-1)

where 𝛼𝛼 is the inclination angle and 𝛿𝛿 is surface-flatness coefficient. 𝛿𝛿 < 0 indicates flange
surface is concave. 𝛿𝛿 = 0 indicates a flat surface. 𝛿𝛿 > 0 indicates convex flange surface.

Fig. 6-2 Variation in gap ℎ with radius 𝑟𝑟 under a parallel gap condition (ℎ𝑐𝑐 = 0.15 mm)- 76 -

6 Simulation of Effect of Sealing Surface Flatness
Figure 6-2 shows the gap ℎ changes with flatness coefficient 𝛿𝛿 along 𝑟𝑟 in parallel witha central gap of ℎ𝑐𝑐 = 0.15 mm, and the recess pressure set to 𝑝𝑝𝑟𝑟 = 1.4 kPa in calculations.
Figures 6-3 and 4 present the schematic diagram of oil film and upper disk under convex
condition from two perspectives. The value of gap space increases along the radial direction
and reaches the maximum at the edge of disk, that is, 𝑟𝑟 = 𝑅𝑅2 . The larger the coefficient 𝛿𝛿corresponding to the more fiercely degree of the gap change. In these schematic diagrams, inorder to better observe the change in gap space, the central clearance ℎ𝑐𝑐 and deformation
coefficient 𝛿𝛿 was set to 10 mm and 0.1 respectively. However, these two values are both
selected smaller in the actual program.

Fig. 6-3 3-D schematic diagram under convex condition

Fig. 6-4 Main view of schematic diagram under convex condition

6.2 Theoretical Solutions of Effect of 𝛿𝛿 on Temperature Distribution
6.2.1

Temperature Solutions under a Parallel Gap Condition

Figures 6-5 – 7 show the theoretical results of the temperature contour map in the sectionof upper flange and oil film for parallel flange with flat (𝛿𝛿 = 0), convex (𝛿𝛿 = 0.05), and- 77 -

PhD Dissertation of Muroran Institute of Technology
concave (𝛿𝛿 = −0.05) sealing surfaces, respectively. The upper and lower parts indicate theflange and oil domains. The temperature in the abscissa (𝑧𝑧 = 0) represents the solid wall
temperature (𝑇𝑇𝑤𝑤 = 5℃). Although the temperature values at each grid node in the lubricant
film are slightly different, they can be feasibly controlled by 𝑇𝑇𝑤𝑤 in all three cases.

Fig. 6-5 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with flat flange by THL theory (𝛿𝛿 = 0)

Fig. 6-6 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with convex flange by THL theory (𝛿𝛿 = 0.05)

Fig. 6-7 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with concave flange by THL theory (𝛿𝛿 = −0.05)- 78 -

6 Simulation of Effect of Sealing Surface Flatness
Figure 6-5 shows the temperature distribution for a flat flange (𝛿𝛿 = 0). Figure 6-5 shows
the convex flange condition (𝛿𝛿 = 0.05). The gap space increases to ℎ = 0.2 mm at the edgeof disk at 𝑟𝑟 = 𝑅𝑅2 . It decreases to ℎ = 0.1 mm in the concave flange (𝛿𝛿 = −0.05), as shownin Fig. 6-5. We need to mention that in order to intuitively observe the temperature contour inthe cross section of flange and oil film, disk thickness 𝐵𝐵 and gap space ℎ are plotted at
equidistant positions in Figs. 6-5 – 7. However, the gap space changes along the radial directionat 𝛿𝛿 ≠ 0. Therefore, we show the specific values of the gap space at the oil inlet (𝑟𝑟 = 𝑅𝑅1 ) andoutlet (𝑟𝑟 = 𝑅𝑅2 ) in Figs. 6-5 – 7.
In Figs. 6-5 – 7, the contour line of 𝑇𝑇 = 7.50℃ and 7.92℃, which is marked in red, toindicate the positions of the same contour in three cases. By comparing the result with the flat
flange case, the locations of the marked contours point downward in the convex flange (𝛿𝛿 =0.05), and the overall temperature in the oil film is relatively deviated from target value 𝑇𝑇𝑤𝑤 .
These values shift upward in the concave flange case (𝛿𝛿 = −0.05), and the oil film temperatureapproaches 𝑇𝑇𝑤𝑤 . In other words, the temperature of oil film area can be easily controlled by thesolid wall using a flange with a concave sealing surface.

6.2.2

Temperature Solutions in Gap-disk Interface under Inclination

Figures 6-8 – 10 show the temperature contour in the interface between the gap and theupper flange, i.e., 𝑇𝑇𝑧𝑧=ℎ (𝑇𝑇′𝑧𝑧 ′ =0 ) under an inclined flange condition (𝛼𝛼 = 0.057°) with flat(𝛿𝛿 = 0), convex (𝛿𝛿 = 0.05), and concave (𝛿𝛿 = −0.05) sealing surfaces, respectively. The solid
wall temperature is set at 𝑇𝑇𝑤𝑤 = 5℃. Although the gap–disk interface is not a flat circle becauseof the axial deformation at 𝛿𝛿 ≠ 0, as shown in Figs. 6-9 and 10, they are still expressed in acircular form for comparison with the flat case shown in Fig. 6-8.

Fig. 6-8 Temperature map of gap–disk interface in inclined with flat disk (𝛿𝛿 = 0)

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PhD Dissertation of Muroran Institute of Technology

Fig. 6-9 Temperature map of gap–disk interface in inclined with convex disk (𝛿𝛿 = 0.05)

Fig. 6-10 Temperature map of gap–disk interface in inclined with concave disk (𝛿𝛿 = −0.05)

We can observe that the small variations in temperature exist in the gap–disk interface,which is affected by the different values of flatness coefficient 𝛿𝛿. The temperature at theinterface agrees with that of the solid wall, and the temperature point near the minimum gap
position (ℎ𝑚𝑚𝑚𝑚𝑚𝑚 ) the most coincides with 𝑇𝑇𝑤𝑤 in all the three cases. In comparison with the flat
flange, the overall temperature at the interface is generally deviated from the solid wall
temperature 𝑇𝑇𝑤𝑤 in the gap with the convex flange. However, it is closer to 𝑇𝑇𝑤𝑤 in concave casein Fig. 6-10. The overall temperature values in the oil film was much more theoretically agrees
with the target value 𝑇𝑇𝑤𝑤 in the smaller gap space condition.

6.3 Theoretical Solutions of Effect of 𝛿𝛿 on Flow Velocity and Leakage
6.3.1

Flow Velocity under a Parallel Gap Condition

Figures 6-11 – 13 show the distribution of flow velocity 𝑢𝑢 along radius in terms of the 𝑧𝑧
value in parallel using flange with flat (𝛿𝛿 = 0), convex (𝛿𝛿 = 0.05), and concave (𝛿𝛿 = −0.05)- 80 -

6 Simulation of Effect of Sealing Surface Flatness
sealing surfaces, respectively. The solid wall temperature is 𝑇𝑇𝑤𝑤 = 5℃. The specific values of
the gap space at the oil inlet and outlet are also shown. Compared with the flat flange case in
Fig. 6-11, the flow velocity of with convex sealing surface decreases near oil outlet in Fig. 612, and the closer the surface is to the outlet, the larger is the decrease in the gradient. The flow
velocity of with concave sealing surface increases in Fig. 6-13. The velocity difference in threecases can be mainly attributed to the difference in the rate of change in gap along 𝑟𝑟 direction.

Fig. 6-11 Distribution of Velocity 𝑢𝑢 for flat sealing surface (𝛿𝛿 = 0)

Fig. 6-12 Distribution of Velocity 𝑢𝑢 for convex sealing surface (𝛿𝛿 = 0.05)

Fig. 6-13 Distribution of Velocity 𝑢𝑢 for concave sealing surface (𝛿𝛿 = −0.05)- 81 -

PhD Dissertation of Muroran Institute of Technology

6.3.2

Leakage Flow under Static Conditions of Parallel and Inclination

Figure 6-14 shows the effects of flatness coefficient 𝛿𝛿 of the flange surface on the leakage
flow under parallel gap condition. Figure 6-15 shows the effects of 𝛿𝛿 on the leakage flow
under the inclined gap with 𝛼𝛼 = 0.057°. Solid wall temperature 𝑇𝑇𝑤𝑤 is varied from 5℃ to
45℃, and the temperature interval is 5℃.
Comparison of the parallel and inclined gap cases reveals that the leakage increases under
the inclined case at the same values of ℎ𝑐𝑐 and 𝑃𝑃𝑟𝑟 . The leakage also increases with larger
flatness coefficient 𝛿𝛿, and the leakage–restraint phenomenon is most obvious in the flange withconcave surface in both cases of parallel and inclination, as shown in Figs. 6-14 and 15. The
main reason is that the a larger flatness coefficient corresponds to a larger gradient in gap spacealong the 𝑟𝑟 direction, i.e., the gap space is larger at the edge of disk (𝑟𝑟 = 𝑅𝑅2 ) while the centergap remains the same. Another reason is that when the gap is larger, controlling the oil film
temperature by solid wall is more difficult. However, although the leakage increases with alarger flatness coefficient, it still can be suppressed by lowering the temperature in the flanges
with different types of sealing surfaces under parallel and inclined gaps.

Fig. 6-14 Effect of 𝛿𝛿 on leakage under static parallel gap conditions (𝛼𝛼 = 0°)

Fig. 6-15 Effect of 𝛿𝛿 on leakage under static inclined conditions (𝛼𝛼 = 0.057°)- 82 -

6 Simulation of Effect of Sealing Surface Flatness

6.3.3

Leakage Flow under Vibrations of Parallel and Inclination

Figure 6-16 shows the theoretical solution in effects of flatness coefficient 𝛿𝛿 of the flange
surface on the leakage in parallel gap under simple harmonic vibrations. Figure 6-17 shows theeffects of 𝛿𝛿 on the leakage under vibrations with inclined gap 𝛼𝛼 = 0.057° . The input
vibration parameters are both selected as frequency 𝑓𝑓 =5 Hz, amplitude 𝐴𝐴 = 0.0018 mm.
Comparing with steady state condition, leakage from the gap was largely increased undervibration for both parallel and inclined gap. In addition, a gasket with an inclination angle has
a much larger leakage than parallel case, which is the coincided in Chapter 5. The leakage flow
also increases with the increase of the flatness coefficient 𝛿𝛿, and the gasket with concave flange
has the strongest ability to restrain leakage as well for both conditions of parallel and inclination.
In this section, it was discussed the effects of 𝛿𝛿 on leakage under vibrations. There is only one
set of vibration parameters were selected. Regarding the specific effects of vibration frequency
and amplitude on leakage, please refer to Chapter 5.

Fig. 6-16 Effect of 𝛿𝛿 on leakage in parallel gap under vibrations(ℎ𝑐𝑐 :0.15 mm, 𝑃𝑃𝑟𝑟 :1.4 kPa, 𝛼𝛼 = 0°; 𝑓𝑓 =5 Hz, 𝐴𝐴 = 0.0018 mm)

Fig. 6-17 Effect of 𝛿𝛿 on leakage in inclined gap under vibrations(ℎ𝑐𝑐 :0.15 mm, 𝑃𝑃𝑟𝑟 :1.4 kPa, 𝛼𝛼 = 0.057°; 𝑓𝑓 =5 Hz, 𝐴𝐴 = 0.0018 mm)- 83 -

PhD Dissertation of Muroran Institute of Technology

6.4 Simulations in Parallel Conditions of Keeping Same Outlet Gap
In the Section 6.1 – 6.3, the central gap ℎ𝑐𝑐 was kept same and gap space ℎ was changed
along radius, which was given in Eq. (6-1). The changes of leakage are mainly caused by thechanges in the gap space. In this section, we focus to keep the same value of outlet gap ℎ𝑜𝑜𝑜𝑜𝑜𝑜
for analyzing the temperature distribution and leakage under the conditions parallel gap.

Fig. 6-5 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with flat flange by THL theory (𝛿𝛿 = 0)

Fig. 6-18 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with convex flange with same ℎ𝑜𝑜𝑜𝑜𝑜𝑜 (𝛿𝛿 = 0.05)

Fig. 6-19 Theoretical result of 𝑇𝑇 and 𝑇𝑇′ with concave flange with same ℎ𝑜𝑜𝑜𝑜𝑜𝑜 (𝛿𝛿 = −0.05)- 84 -

6 Simulation of Effect of Sealing Surface Flatness
Figures 6-18 and 19 show temperature contour in the cross section of upper flange and oilfilm in parallel with convex, and concave sealing surfaces, while the outlet gap at the edge offlange was kept same and set as ℎ𝑜𝑜𝑜𝑜𝑜𝑜 = 0.15 mm. Figure 6-5 was referenced for comparison,and the specific values of the gap space at the oil inlet and outlet were also shown.
The location of marked contour lines of 𝑇𝑇 = 7.50℃ and 7.92℃ were shifted upwardand downward as well due to the gap changed along radius. And the temperature value of oil
film more followed with solid wall temperature 𝑇𝑇𝑤𝑤 under the smaller gap condition.

Fig. 6-20 Effect of 𝛿𝛿 on leakage under parallel gap with same ℎ𝑜𝑜𝑜𝑜𝑜𝑜

Figure 6-20 shows the effects of 𝛿𝛿 on leakage under parallel gap with same ℎ𝑜𝑜𝑜𝑜𝑜𝑜 . It
depicts that leakage increases under flange with concave sealing surface, which showed the
opposite results with Section 6.3. It is because in this section, the gap space at the outlet ℎ𝑜𝑜𝑜𝑜𝑜𝑜
was remained same, and in the case of concave surface, the gap increases radially toward ℎ𝑐𝑐 .
The size of gap space is the main influencing factor and is proportional to the leakage. Moreover,leakage can be reduced by cooling the sealing flange for all the above conditions.

6.5 Chapter Summary
In this chapter, the theoretical model for analyzing the temperature, flow velocity, andleakage of a flange-type gasket was developed using the viscosity–temperature relationship.
The THL theory was applied to the gap flow under the sealing surface of installed flat, convex,
and concave flanges. The leakage and oil film temperature were theoretically calculated using
the surface flatness coefficient and inclination angle were considered as the main parameters.
When the temperature of the sealing flange was controlled, the leakage varied under the rangeof the flatness coefficient and inclination angle. The conclusions are presented as follows:(1) The oil film temperature could theoretically agree with that of solid wall under various
conditions independent of the shape of sealing surface and parallel configuration of the gap.
(2) Comparing to flat flange, the flow velocity in the concave flange theoretically increased
along radius for both conditions of parallel and inclined gap. The closer the outlet was, thelarger was the gradient increase.
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PhD Dissertation of Muroran Institute of Technology(3) Irrespective of the inclination angle, the leakage could be most obviously restrained bylowering temperature of the sealing parts with small average gap space.
(4) Under vibrations, the leakage was largely increased compared with steady state, the leakage
was most restrained in the gasket with concave flange in both parallel and inclination.

References[1] Xiao, Z. L., Zhou, C. J., Li, Z. D. and Zheng, M., Thermo-mechanical characteristics ofhigh-speed and heavy-load modified gears with elasto-hydrodynamic contacts, Tribology
International, Vol.131 (2019), pp.406-414.
[2] Ruggiero, A., Sicilia, A. and Affatato, S., In silico total hip replacement wear testing in theframework of ISO 14242-3 accounting for mixed elasto-hydrodynamic lubrication effects,
Wear, Vol. 460-461 (2020), 203420.
[3] Bobzin, K., Brogelmann, T., Stahl, K., Michaelis, K., Mayer, J. and Hinterstoiber, M.,
Friction reduction of highly-loaded rolling-sliding contacts by surface modifications underelasto-hydrodynamic lubrication, Wear, Vol.328-329 (2015), pp.217-228.
[4] Gadari, M. E., Fatu, A. and Hajjam, M., Shaft roughness effect on elasto-hydrodynamic
lubrication of rotary lip seals: experimentation and numerical simulation, Tribology
International, Vol.88 (2015), pp.218-227.
[5] Shinkarenko, A., Kligerman, Y., and Stsion, I., The effect of surface texturing in soft elastohydrodynamic lubrication, Tribology International, Vol.42 (2009), pp.284-292.
[6] Ciuperca, I. S. and Tello, J. I., On a variational inequality on elasto-hydrodynamic
lubrication, Journal of Mathematical Analysis and Applications, Vol.383 (2011), pp.597607.
[7] Lehn, A., Mahner, M. and Schweizer, B., A thermal-elasto-hydrodynamic model for air foil
thrust bearings including self-induced convective cooling of the rotor disk and thermal
runaway, Tribology International, Vol.119 (2018), pp.281-298.
[8] Gao, S. and Kazama, T., Effect of sealing surface flatness on leakage characteristics offlange-type gasket model using oil viscosity-temperature relations, Proc. JFPS Hakodate
2020 Conf (2021), General Section, Oil hydraulics, GS4-01.

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7 Vertical Flange-type Gasket Model

7

Vertical Flange-type Gasket ModelIn the previous chapters, the operating parameters of gasket model we discussed were in
conditions of relatively large gap space and low pressure theoretically and experimentally, andgasket facility was placed lying in horizontally [1]. The phenomena between gap flow andtemperature of lubricant and the relationship between cooling the sealing flanges and reducing
leakage was confirmed and analyzed qualitatively. However, the practical application of gasket
in society was generally operated under conditions of tiny clearance and high recess pressure.
There are many studies proposed numerous feasible methods for estimating leakage andenhance sealing performance of gaskets theoretically and experimentally. Jaszak [2] presented
the results of experimental tests regarding assessment of tightness of a gasketed-flange-bolted
joint with an elastomer-fiber gasket as sealing node in industrial pipeline system. The cyclicallyalternating load bending of a gasketed-bolted-joint case a significant change of leakage in
relation to the assembly state was proved. Reducing the extrusion between bolts and the outerring through controlling gasket sizes and improving installation method was proposed foravoiding failure of gasket. Nelson [3] theoretically analyzed in spiral wound gasket by using
micromechanical approach, the volume element is chosen such that the individual constituentsin the gasket are included and the uniaxial behavior was confirmed experimentally. To improve
the sealing performance and efficiency in various environments, the development of machineelement under vibration conditions is also necessary. Aldakheel et al. [4] investigated the effects
of sealing gaskets on mass transport properties of the gas diffusion layer. They focused on theway to improve gas permeability for estimating compared with gas diffusivity changing before
and after compression and with or without sealing gaskets.
In this chapter, a vertical flange-type gasket with two circular disks, whose condition iscloser to the actual working environment was also established experimentally for connecting
the qualitative analyzation in the previous discussions to gasket practical applications. The
experiment focused on factors of bolt preload torque, and recess pressure.

7.1 Experimental Apparatus
The schematic diagram of experimental rig is shown in Fig. 7-1. The experimental device
is mainly composed a gasket tester, a vibrator, an air-dryer, a Peltier module, a pump, DC power
suppliers of air-dryer and vibrator, a pressure gauge, and thermometer. The gasket tester wasplaced vertically and consisted of left- and right-side sealing flanges clamped with four shimrings. The representative experimental parameters were as follows: the diameter of left-side
flange (SUS F304 10K 15 SH200052) and right-side flange (SUS F304 10K 15 CJ190098) was
both 95 mm. the thickness of each flange was 12 mm. The viscosity grade of lubricant used inexperiment was also selected as ISO VG 32. The experimental operating conditions were as
follows: 𝑀𝑀𝑒𝑒 = 5 − 15 N ∙ m, 𝑃𝑃𝑟𝑟 = 4.0 − 20.0 kPa, and 𝑇𝑇𝑤𝑤 = 0 − 20℃.
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Fig. 7-1 Schematic diagram of vertical gasket rig

7.2 Experimental Parameters
In experiment, the Peltier module was mounted connecting to sealing flange on the rightsurface to control the temperature of seal parts; the sensors of temperature were mounted on theapparatus; oil supply pressure can be changed through pump, and the accurate pressure value
and oil temperature was measured by the electronic manometer and thermometer nearby the oil
inlet; the vibration mode, input frequency and supply voltage of vibrator can be input from the
DC power supplier directly in vibrations. In the experiment, the gasket tester is placed vertically,
the lubricating oil flows in from the center hole of left-side flange and drips from the bottom of
the gasket flanges.
There are four shim rings inserted between sealing flanges, the function of shim rings is
to keep the minimum clearance in steady-state. The actual clearance between sealing flanges is
too small to be measured directly in this case, but it can be adjusted by the bolt preload torque.
In addition, one of the main measured parameters is the dropping seconds period during acertain number of oil droplets. When the target temperature value of Peltier module is lower
than room temperature, the water in the air could liquefy gradually onto the surface of gasket
flanges and drip with the leaked oil collectively, which may intervene the accuracy ofexperimental results. Therefore, the air-dryer was placed nearby gasket flanges and operated
for dehumidification during the whole process of experiment.

7.3 Experimental Method and Conversion of Leakage
7.3.1

Experimental Method

Figure 7-2 is the layout of vertical flange-type gasket devices, the specific experimental
method is as follow: After setting the initial value of the gasket bolt preload torque, opening the- 88 -

7 Vertical Flange-type Gasket Modeltank valve and starting the pump, the lubricating oil flows to the gasket facility through thecentral round hole of left-side flange after raising the supply pressure by pump. The values ofoil supply pressure and oil temperature can be measured by the electronic manometer andthermometer nearby oil inlet. Subsequently, the air-dryer was started to operate with the air
inlet facing towards the gasket flanges. The temperature target value of Peltier module wasadjusted in the order of 20℃, 15℃, 10℃, 5℃ and 0℃ by the sensor controller, and thecooling effect was realized by the coolant circulation system of refrigerator. After the aim
temperature become stable, the radiation thermometer was used to detect the actual temperatureof flanges. Then, the period of seconds during certain number drops of oil droplet from gasket
clearance was recorded. The electronic scale was used to measure the mass of one drop of oil,and then the leakage rate of flow per unit time can be calculated effectively.
The vibrator was mounted to connecting the sealing flange on the left surface for fixation.
when the experiment was operated in vibration, the vibration mode, frequency and amplitude
parameters of the vibrator can be set by adjusting the input signal of the DC amplifier. In thiscase, the left- and right-side flanges were both vibrated with the vibrator. However, thevibration parameters (frequency and amplitude) of two flanges are different due to the influenceof elastic deformation by O-rings. The output value of frequency and amplitude were measured
by laser detectors. Adjusting the gasket bolt preload and recess pressure by torque spanner andpump respectively, the factors that evident effect the leakage were researched.

Fig. 7-2 Layout of all the vertical flange-type gasket devices

7.3.2

Conversion of Leakage

As discussed in Section 7.2, one of the main measured parameters in this vertical flangetype gasket model is the dropping seconds 𝑡𝑡 during 𝑛𝑛 number of drops. In order to intuitively
express the relationship between the fluid leakage and many various experimental parameters,
it is necessary to convert the unit from measured seconds of oil dropping period to leakage.

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PhD Dissertation of Muroran Institute of Technology

𝑚𝑚 (g)∙𝑛𝑛𝑠𝑠
𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜 (ml/min) = 𝜌𝜌(g/cm
3 )∙𝑡𝑡(s) ∙ 60

(7-1)

Figure 7-3 depicts the way to measure oil droplets mass with electronic scale [5]. It can beobtained by calculating the difference mass of the empty tinfoil case and it with oil droplets.
The influence of temperature on oil mass was also considered. The relationship between the oilmass of one drop and temperature is shown in Fig. 7-3. In unit conversion, the mass of oil drop
was corresponded to temperature by using the approximate data.

Fig. 7-3 Relationship between the oil mass of one drop and temperature
It is feasible to convert mass into volume by using the parameter of oil density, and thenfluid leakage can be calculated by Eq. (7-1). Wherein, the parentheses after physical quantity
in the equation indicate the unit. For ISO32 oil, the density 𝜌𝜌 = 0.869 g⁄cm3 , and since theleaked oil is relatively small, the unit of fluid leakage is selected as milliliter per minute.

7.4 Experimental Results under Steady-state Condition
7.4.1

Effects of Bolt Preload Torque on Leakage

When experiment was operated under steady-state, the bolt preload torque between flanges
can be controlled by torque spanner, the actual value can be read from torque dial. In order toresearch the effects of preload torque on leakage flow, the recess pressure of the two groups ofexperiment was controlled as 𝑃𝑃𝑟𝑟 = 3.0 and 20.0 kPa by pump, respectively. The effects of
preload torque on leakage rate was compared and researched.
When the experiment was operated under steady-state condition, it is necessary to keep
the gap space between flanges unchanged during one group of experiment. Therefore, the shimrings were inserted between sealing flanges. The experimental results are shown in Figs. 7-4
and 5. When flange controlled temperature was same, the smaller preload torque corresponded
to the greater leakage. And as temperature was lower, the leakage can be reduced for bothconditions of small and large value of torque.

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7 Vertical Flange-type Gasket Model

Fig. 7-4 Effects of bolt preload torque on leakage (𝑃𝑃𝑟𝑟 : 7.0 kPa)

Fig. 7-5 Effects of bolt preload torque on leakage (𝑃𝑃𝑟𝑟 : 20.0 kPa)

7.4.2

Effects of Recess Pressure on Leakage

The recess pressure is proportional to the revolving speed of rollers inside pump. Recesspressure can be controlled by adjusting the input parameter of pump. The pressure value wasmeasured accurately by electronic manometer, which was installed on the tube nearby oil inlet.
Lubricant flows into the gasket facility after passing through the electronic manometer andthermometer, and then flows out from the gap space between vertical sealing flanges. The bolt
preload torque was set to 5.0 and 10.0 N ∙ m, respectively. The effects of recess supply pressure
on leakage can be researched.
The experimental results are shown in Figs. 7-6 and 7. It shows that when temperature is
the same, the larger the recess pressure is, the greater the leakage is, which is consistent with
common sense. And as the temperature became lower, the leakage was decreased independentof recess pressure.

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PhD Dissertation of Muroran Institute of Technology

Fig. 7-6 Effect of recess pressure on leakage (Torque: 5.0 N ∙ m)

Fig. 7-7 Effect of recess pressure on leakage (Torque: 10.0 N ∙ m)

7.5 Chapter Summary

In this chapter, a vertical flange-type gasket which is more close to practical application
was established and modeled. The experimental investigation for reducing leakage on thecharacteristics of a vertical flange-type gasket under static have been proposed. The leakage
flow and temperature were measured. The effects of cooling the sealing flanges of gasket interms of temperature, bolt preload torque, and recess pressure were taken as main parameters.
The relationship between leakage flow and temperature were mainly discussed, the conclusions
are as follows:(1) Under steady state, as the temperature became lower, the leakage was decreased. The
tendency was independent of the bolt preload torque and recess pressure.
(2) The way to suppress leakage flow by cooling the sealing flanges is always applicable
regardless of gasket placement in parallel and vertical for a wide range of all conditions
under steady-state.
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7 Vertical Flange-type Gasket Model

References[1] Gao, S. and Kazama, T., Leakage control for a flat flange model with a gap based on sealingliquid viscosity, Journal of Advanced Mechanical Design, Systems, and Manufacturing,
Vol.15, No.3 (2021), 21-00065.
[2] Jaszak, P., Prediction of the durability of a gasket operating in a bolted-flange-joint
subjected to cyclic bending, Engineering Failure analysis, Vol. 120 (2021), 105027.
[3] Nelson, N. R., Effective modeling of spiral wound gasket with graphite filler in gasketedflange joint subjected to bending loads, Materials Today: Proceedings, Vol. 44 (2021),
pp.2199-2204.
[4] Aldakheel, F., Ismail, M. S., Hughes, K. J., Ingham, D. B., Ma, L. and Pourkashanian, M.,
Effects of compression on mechanical integrity, gas permeability and thermal stability ofgas diffusion layers with/without sealing gaskets, International Journal of Hydrogen Energy,Vol. 46 (2021), pp.22907-22919.
[5] Kazama, T., A comparative Newtonian and thermal EHL analysis using physical lubricant
properties, Boundary and Mixed Lubrication: Science and Applications, Elsevier Science
B.V. (2002), pp.435-446.

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8 Conclusion and Prospect

8

Conclusion and Prospect

8.1 Conclusion
In this dissertation, a method for gasket leakage suppression by cooling the sealing flange
was researched and introduced. It can be applied in many occasions to reduce fluid leakage insociety. One of the potential applications is the underwater tank. When leakage occurs, water
will enter the tank and cause damage of precision instruments and property loss. It will also
endanger the life safety of operators inside the tank. Besides, the road is not absolutely flat, andvibration is also unavoidable when the tank walks underwater. Therefore, the experiments wereoperated under vibrations also has a great research significance. The temperature of sealing
parts can be changed and controlled by using the thermo-electric conversion elements and heatpumps in applications.
Using the changes in viscosity as a strong function of temperature, methods for reducing
leakage in characteristics of a flat flange gasket were proposed and experimental confirmation
was achieved [1-2]. Based on the knowledge of thermo-hydrodynamic lubrication andnumerical calculation method, the mathematical model of prototype gasket based on the THL
theory was established, the three-dimensional discrete forms of the Reynolds equation, energy
equation, and heat conduction equation as well as the boundary conditions of pressure andtemperature domain were also derived by using FDM. The complete program was coded withthe coupling of mentioned difference equations above with iteration method. The numericalsolutions of leakage flow and oil temperature were obtained by THL theory when gap space,recess pressure, sealing size, oil viscosity grade, inclined angle, vibration frequency andamplitude were taken as typical parameters.
The IV theory using solid wall temperature was also applied to the gap flow under staticand vibration conditions. When the gasket sealing parts were cooled, the leaked oil was reduced
based on a wide range of various parameters for both static and harmonic vibration conditions.
The instruction of each experimental apparatuses were discussed followed [3]. Almost all of
the theoretical solutions were verified by experiment for several repeated times. The effects oftemperature, gap space, recess pressure, sealing size, oil type, vibration frequency, amplitude
and other parameters on the leakage rate of lubricating oil were obtained from the specificexperimental data under the steady-state and harmonic vibration conditions. The main research
work was summarized as follows:(1) On the basis of theoretical analysis and thermo-hydrodynamic lubrication theory, themathematical model of prototype flange gasket was established, and the Reynolds
equation, energy equation and heat conduction equation were derived. And the solution
domain of pressure and temperature were meshed by using the FDM respectively, andthe appropriate difference formulas were applied to partial differential equations, and- 95 -

PhD Dissertation of Muroran Institute of Technology
boundary conditions were also discretized. The deformation of all nodes in calculation
domain can be expressed as a linear combination of the pressure and temperature ofeach nodes coupled with their boundary conditions. The two-layer nested iteration wasused to connect all the formulas. The difference equations were solved and debugged
altogether. The pressure and temperature distribution of oil film in several typical cases
were calculated, the temperature distribution during oil film and upper disk was solved
and obtained.
(2) Through the specific experimental apparatuses, the rated parameters of gasket facility,
Peltier module, piezo-activator, laser detector, function generator, flowmeter, pumpand other devices and their operation methods were described. And the specific stepsof the experiment and the announcements in the process were introduced in detail aswell. Finally, several groups of experiments were carried out under static and harmonic
vibration conditions, and the experimental data were discussed to research the effectof temperature, gap space, recess pressure, sealing land size, oil viscosity grade,
inclined angle, vibration frequency, amplitude and other parameters on leakage. The
experimental data were compared with solutions based on the THL and the IV theories.
(3) Theoretical simulating the different types of flange sealing surface by THL method,
and the temperature distribution, velocity along radius direction, and leakage flow arecompared in flat, concave and convex surface conditions. The influence of sealing
surface flatness on gasket sealing performance was analyzed qualitatively [4]. Finally,
it is concluded that the temperature of oil film are more matched with solid wall andthe sealing performance of gasket is also better in the gasket with concave surface.
(4) In the end, a vertical flange-type gasket model was established to confirm relationshipbetween temperature of sealing parts and fluid leakage, the condition is more matchedwith the actual gasket application in our society. The experiment was conducted understeady-state. The bolt preload torque and recess pressure were taken as the main
parameters. It was confirmed that the fluid leakage was indeed suppressed by coolingsealing flanges independent of above parameters for under steady-state.

8.2 Prospect
Although this paper studies and analyzes the factors that affect the fluid leakage in the
prototype gasket model by using the THL and IV theories and specific experiment, and has
achieved some academic results. However, due to the limitation of time and the academic levelof the author, there are still many aspects need to be researched and improved further.
Under vibrations, the new form of energy equation in the THL theory for iteration and themore complex methods such as analyzing the shape of the cavity bubble for calculating theinfluence of the cavity effect on leakage flow could be introduced; the impulse equation andelastic deformation could be considered to make the theoretical results more accurate.

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8 Conclusion and Prospect

8.3 Chapter Summary
In this chapter, the specific contents of the whole dissertation were summarized, and put
forward the research areas to be continued study and future prospects.

References[1] Kazama, T. and Gao, S., Gap flow between two circular plates with temperature-controlled
wall: application of thermohydrodynamic lubrication theory and comparison with an isoviscous model, Mechanical Engineering Letters, Vol.6 (2020), 20-00292.
[2] Gao, S. and Kazama, T., Leakage control for a flat flange model with a gap based on sealingliquid viscosity, Journal of Advanced Mechanical Design, Systems, and Manufacturing,
Vol.15, No.3 (2021), 21-00065.
[3] Gao, S. and Kazama, T., Leakage characteristics of prototype gasket using viscositytemperature relation under vibration condition (effects of sealing land size and viscosity
grade), Proc. ICFP Conf (2021), S2304.
[4] Gao, S. and Kazama, T., Effect of sealing surface flatness on leakage characteristics offlange-type gasket model using oil viscosity-temperature relations, Proc. JFPS Hakodate
2020 Conf (2021), General Section, Oil hydraulics, GS4-01.

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Curriculum Vitae

Curriculum Vitae
Basic Information
Name: GAO SONG
Gender: Male
Date of Birth: 1992.11.16
Nationality: The People’s Republic of China
Research Area: Advanced Production Systems Engineering
Educational Experience:
2011.9-2015.6 Bachelor of Mechanical Designing, Henan Polytechnic University, China
2015.9-2018.12 Master of Mechanical Engineering, Henan Polytechnic University, China
2017.4-2017.10 Short Term Exchange Study, Muroran Institute of Technology, Japan
Internship Experience:
2021.9.1-2021.9.29 Long-term Internship, Development Division, Dynax Co., Ltd, Japan.
Journals
1. Toshiharu, KAZAMA. and Song, GAO., Gap flow between two circular plates withtemperature-controlled wall: application of thermohydrodynamic lubrication theory
and comparison with an iso-viscous model, Mechanical Engineering Letters, Vol.6(2020), DOI: 10.1299/mel.20-00292.
2. Song, GAO. and Toshiharu KAZAMA., Leakage control for a flat flange model witha gap based on sealing liquid viscosity, Advanced Mechanical Design, Systems, andManufacturing, Vol. 15, No. 3 (2021), DOI: 10.1299/jamdsm.2021jamdsm0031.
Proceedings of Conferences
1. Gao, Song. and Kazama, Toshiharu., Effect of vibration condition on leakage
characteristics of prototype gasket using oil viscosity-temperature, Annual conferenceof Japan Mechanical Society (JSME), Room 23, S11304, pp.14-18, Nagoya, Japan(online), September 14-16, 2020.
2. Song, GAO. and Toshiharu, KAZAMA., Leakage characteristics of prototype gasket
using viscosity-temperature relation under vibration condition (effect of sealing landand viscosity grade), The 10th International conference on fluid power transmission
and control (ICFP 2021), Room 23, S2304, pp.685-690, Zhejiang university, China(online), April 11-13, 2021.
3. Song, GAO. and Toshiharu, KAZAMA., Effect of sealing surface flatness on leakage
characteristics of flange-type gasket model using oil viscosity-temperature relations,
The 11th JFPS International Symposium on Fluid Power, General Section, Oil
hydraulics, GS4-01, pp.399-405, Hakodate, Japan (online), October 12-13, 2021.
Society Member
1. Japan Society of Mechanical Engineers (JSME), Membership Number: 2014404.
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Curriculum Vitae
Rewards
1. 2017.4-2017.10 Japan Student Services Organization (JASSO) Scholarship for shortterm exchange student.
2. 2019.4-2020.3 Japan Student Services Organization (JASSO) Scholarship for
privately-financed international student.
3. 2020.8 Hokkaido International Exchange and Cooperation Center (HIECC)
Scholarship for privately-financed international student.
4. 2021.2 Japan Student Services Organization (JASSO) Scholarship (special additional
adoption) for privately-financed international student.
5. 2021.4-2022.3 Muroran Institute of Technology (MuroranIT) Scholarship for
privately-financed international student.

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Copyright Use Authorization

Muroran Institute of Technology Doctoral Dissertation
Copyright Use Authorization
I fully understand the relevant regulations regarding university dissertation copyright.
Copyright of theses at university during PhD period belongs to Muroran Institute of Technology,
allowing theses to be consulted and borrowed. University has right to retain theses and submit
copies and electronic editions to national departments and institutions concerned. University
can index relevant database for retrieval using part or the whole of this dissertation. University
can photocopy, print in reduced format or scan to keep and compile this dissertation.

Dissertation Title:
Leakage control for flange gasket using temperature–viscosity characteristics of liquids
Author’s Name and Date:GAO SONG, December 24, 2021

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