CHAPTER 1 Properties of gases.1 

1.1  Gas equations 1 

1.1.1  Ideal gas equations.. 2 

1.1.2  Gas index equation.. 3 

1.1.3  Actual gas equation. 5 

1.1.4  Degree of gas molecular freedom .. 5 

1.1.5  Specific heat capacity  7 

1.2  Viscosity ..7 

1.3  Property of wet gas .11 
1.3.1 Pressure .. 12 
1.3.2 Humidity. 13 
1.3.3 Dew point temperature 14 
References 14 

CHAPTER 2 Gas lubrication equations. 15 

2.1  Reynolds equation 15 

2.1.1  Derivation of Reynolds equation.. 16 

2.1.2  Reynolds equation in the polar coordinate system . 19 

2.1.3  Reynolds equation in the cylindrical coordinate system.. 19 
2.1.4  Lubrication parameters .. 20 

2.2  Energy equation.22 

2.2.1  Chang of gas inner energy 23 

2.2.2  External work on gas and energy loss 24 

2.3  Solid heat conduction equation and the interface equation ..26 

2.4  Numerical analysis method.27 

2.4.1  Finite difference method 27 

2.4.2  Flow conservation. 28 

2.4.3 Friction force balance . 34 
References 35 

CHAPTER 3 Isothermal gas lubrication .. 37 

3.1  Sliders .37 

3.1.1  Lubrication equation  38 

3.1.2  Pressure boundary condition 40 

3.1.3  Lubrication performance parameters.. 40 

VI Contents 
3.2 
3.3 
CHAPTER 4 
4.1 
4.2 
4.3 
4.4 
3.1.4  Hydrodynamic lubrication characteristics of sliders.. 41 

3.1.5  Hydrodynamic lubrication characteristics of divergent sliders .. 43 
3.1.6  Lubrication characteristics of the magnetic head slider . 45 Journal bearing and radial seal .51 
3.2.1 Lubrication equations.. 52 
3.2.2 Boundary conditions  52 
3.2.3 Lubrication parameters .. 53 
3.2.4 Lubrication characteristics 55 
Spiral groove thrust bearing56 
3.3.1 Gas lubrication equations.. 59 
3.3.2 Pressure boundary conditions . 60 
3.3.3 Lubrication parameters .. 60 
3.3.4 Lubrication characteristics 61 
3.3.5 Spiral groove face seal 64 
3.3.6 Lubrication equations.. 65 

3.3.7  Pressure boundary conditions . 65 

3.3.8  Seal performance parameters.. 66 

3.3.9 Lubrication regularity . 66 
References 71 

Gas thermohydrodynamic lubrication of rigid surfaces.. 73 Sliders .73 
4.1.1  Gas lubrication equations.. 73 

4.1.2  Boundary conditions  75 

4.1.3 Thermal lubrication characteristics . 75 
Journal bearing and radial seal .79 

4.2.1  Lubrication equations.. 79 

4.2.2  Thermal boundary condition 80 

4.2.3 Lubrication property  80 
Spiral groove thrust bearing83 

4.3.1  Lubrication equations.. 83 

4.3.2  Boundary conditions  85 

4.3.3 Lubrication property  86 
Spiral groove face seal..86 

4.4.1  Temperature characteristics of gas film  91 

4.4.2 Lubrication property  94 
References 99 

Contents VII 
CHAPTER 5  Gas thermoelastohydrodynamic lubrication of face seals . 101 
5.1  Fundamental equations101 

5.1.1  Lubrication equations 101 

5.1.2  Boundary conditions . 104 

5.2  Choked fluid effect107 

5.2.1  Model validation . 107 

5.2.2  Pressure distribution characteristics . 109 

5.3  Characteristics of thermoelastic distortions of seal faces..109 
5.4  Characteristics of gas thermoelastohydrodynamic lubrication..111 
5.4.1  Mechanical distortions . 111 

5.4.2  Thermal distortions  115 

5.4.3 Thermoelastic distortions 117 
References. 120 

CHAPTER 6  Transient thermoelastohydrodynamic gas lubrication of face seals  121 
6.1  Fundamental equations121 

6.1.1  Dynamic equations. 122 

6.1.2  Lubrication equations 122 

6.1.3  Boundary conditions . 123 

6.1.4  Dynamic characteristic parameter . 123 

6.2  Dynamic characteristics of isothermal gas lubrication .124 

6.2.1  Axial stiffness and damping . 124 

6.2.2  Angular stiffness and damping  126 

6.2.3  Amplitude-frequency characteristics of gas film .. 129 

6.3  Dynamic characteristics of thermal gas lubrication of rigid surfaces 130 
6.3.1  Axial stiffness and damping . 131 

6.3.2  Angular stiffness and damping  134 

6.3.3  Amplitude-frequency characteristics of gas film .. 137 

6.4  Dynamic characteristics of gas thermoelastohydrodynamic lubrication..137 
6.4.1  Axial stiffness and damping . 137 

6.4.2  Angular stiffness and damping of gas film .. 139 

6.4.3  Amplitude-frequency characteristics of gas film. 140 References. 141 
VIII Contents 
CHAPTER 7  Vapor-condensed gas lubrication of face seals .. 143 
7.1  Fundamental equations143 

7.1.1  Wet gas equations .. 143 

7.1.2  Vapor-condensed parameter . 145 

7.2  Characteristics of vapor condensation in gas lubrication film 145 
7.3  Laws of vapor condensation in gas lubrication film..146 

7.3.1  Humidity.. 146 

7.3.2  Seal clearance 147 

7.3.3  Seal pressure . 148 

7.3.4  Rotational speed.. 148 

7.4  Movement of liquid drops on gas lubrication surfaces.150 

7.4.1  Surface wetting model . 150 

7.4.2  Surface-texture geometry parameters and wettability 152 
7.4.3 Droplet motion . 158 
7.4.4 Drop adsorption on seal surface . 164 
References. 164 

CHAPTER 8  Cryogenic gas lubrication of face seals 167 

8.1  Fundamental equations167 

8.1.1  Lubrication equations 168 

8.1.2  Boundary conditions . 171 

8.2  Phase change in gas lubrication film..172 

8.3  Characteristics of thermoelastic distortions of seal faces 173 

8.4  Characteristics of cryogenic gas lubrication173 

8.4.1  Ambient temperature  174 

8.4.2 Rotation speed.. 174 
References. 177 

CHAPTER 9  Surface grooves of gas face seals and testing technology.. 179 
9.1  Surface grooves of gas face seals .179 

9.2  Testing technology of gas face seals ..185 

9.2.1  Experimental setup. 185 

9.2.2  Face groove machining 185 

9.2.3  Face morphology test 187 

9.3  Experimental characteristics of gas face seals ..188 

9.3.1  Seal opening characteristic  188 

9.3.2  Hydrodynamic characteristics.. 190 

Contents IX 
9.3.3 Surface wear.. 192 
References. 195 

CHAPTER 10 Design of gas face seals  197 

10.1 Force analysis of gas seals197 

10.1.1 Opening force. 198 

10.1.2 Closing force.. 198 

10.1.3 O-ring friction force 199 

10.2 Geometric parameters of gas face seals 199 

10.2.1 Seal clearance. 199 

10.2.2 Seal face width.. 199 

10.2.3 Balance diameter.. 200 

10.2.4 Seal face and shaft (shaft sleeve) clearance.. 200 

10.3 Performance parameters of gas face seals201 

10.3.1 Leakage rate  201 

10.3.2 Gas film stiffness . 201 

10.4 Materials of the seal couple .202 

10.5 Dimension design of seal rings..202 

10.5.1 Design of rotor dimension .. 202 

10.5.2 Design of stator dimension . 203 

10.5.3 Design of the face groove 204 

10.6 Design of the secondary seal ..205 

10.7 Process of seal design and illustration205 

10.7.1 Process of seal design  205 

10.7.2 Design conditions. 206 

10.7.3 Design steps  206 
References. 218 

Index .219 

Gas lubrication is a form of fluid lubrication, which uses air or working medium gas as a lubricant to separate two friction pairs moving in relation to each other. It has the advantages of low friction resistance, high working accuracy, and a wide range of applicable temperatures, and is widely used in the design of gas-eous static pressure bearings, high-speed thrust bearings, foil bearings, mechanical seals, and other mechanical parts and equipment under extreme conditions such as extremely high or low temperatures, ultra-high or -low speeds, and ultra-precision. 
In 1854, Hirm came up with the idea of using gas as a lubricant. In 1886, Reynolds deduced the Reynold’s equation describing the pressure distribution of fluid lubrication film, which raised people’s understanding of the principle of fluid lubrication to a theoretical height. In the 1950s, research on gas lubrication theory based on bearing design requirements developed rapidly. In 1959, Elrod and Burgdorfer theoretically explained that the temperature rise effect inside the lubrication gas film could be ignored under general working conditions, and the isothermal hypothesis was generally accepted in gas lubrication analysis. After the 1980s, the development of magnetic storage technology pushed the film thick-ness of gas bearings from microns to nanometers, and microscopic effects such as the gas thinning effect and surface roughness were widely studied, which pro-moted the development of gas thin film lubrication theory. 
With an increase of rotation speed in mechanical equipment and the appear-ance of new bearing structures, the problem of gas thermodynamic lubrication became increasingly prominent. For example, in a gas hydrostatic bearing under the conditions of a film thickness of 20 μm, pressure of 0.7 MPa, and rotation speed of 20,000 rpm, shear heat can cause the gas film lubrication temperature to rise above 30°C. The increase of rotor temperature rise and gas viscosity can improve the stiffness and damping coefficient of the bearing, but in the absence of sufficient cooling, the bearing is prone to thermal instability, which is particu-larly prominent in a high-speed bearing design. 
Compared with bearing gas lubrication dominated by shear flow, there is also pressure flow caused by seal pressure in the gas seal lubrication area. In the pro-cess of gas leakage flow from the high-pressure side to the low-pressure side, gas film temperature decreases because of rapid volume expansion, resulting in ther-mal distortion and other gas thermohydrodynamic lubrication problems. 
In 1968, the John Crane Company first developed the circular arc surface spiral groove gas-lubricated seal and introduced plane spiral groove gas-lubricated seal products. With the development of gas sealing technology to high-temperature, high-pressure, high-speed, and other high parameters; the diversification of sealing media; and the continuous expansion of application fields, the surface thermal dis-tortion, supersonic flow, media phase change, and other gas thermohydrodynamic lubrication problems are increasing. 
Preface 
Based on the research methods and results of elastohydrodynamic lubrication theory of Prof. Wen Shizhu, this book summarizes the research results of the authors in recent years. Grounded in the application background of lubrication design of high-speed gas bearings and high-pressure gas seals, theory and analysis methods of gas thermohydrodynamic lubrication are systematically expounded. In this book, theoretical model and lubrication characteristics of gas lubrication are discussed for typical bearings and seals. 
The book is divided into 10 chapters. 
In Chapter 1, Properties of gases, properties of gases are introduced. Based on the principle of energy equipartition, the ideal gas state equation is decomposed into two independent gas equations. 
In Chapter 2, Gas lubrication equations, the derivation of the Reynolds equa-tion, energy equation, heat conduction equation, interface equation, and other basic lubrication equations are explained. The lubrication analysis of force bal-ance and flow conservation problems are also discussed. 
In Chapter 3, Isothermal gas lubrication, we introduce the modeling method and basic lubrication characteristics of isothermal gas lubrication of typical struc-tures such as the slider bearing, radial bearing, thrust bearing, and face seal. 
In Chapter 4, Gas thermohydrodynamic lubrication of rigid surfaces, the gas thermodynamic lubrication law of gas-lubricated bearings and low-pressure gas seals under high-speed working conditions are analyzed without consideration of surface distortions. 
Chapter 5, Gas thermoelastichydrodynamic lubrication of face seals, discusses the high-pressure gas face seal, the modeling method of TEHL, and fundamental characteristics of choked flow effect, thermoelastic distortion of seal faces, and the temperature distribution of gas film. 
In Chapter 6, Transient thermoelastichydrodynamic gas lubrication of face seals, targets the gas face seal, dynamic load characteristics of the seal gas film under the conditions of isothermal lubrication, rigid surface thermal lubrication, and TEHL, exploring these in detail. 
Chapter 7, Vapor-condensed gas lubrication of face seals, explores the face seal of high-pressure gas, the law of condensation and precipitation of water vapor in the seal gas film, and the movement of liquid drops in the seal gap under TEHL conditions. 
Chapter 8, Cryogenic gas lubrication of face seals, uses the inclined ellipse dimpled gas face seal as an example and natural gas as the sealing medium to introduce the modeling method of cryogenic gas lubrication and the fundamental characteristics of phase change in seal film and thermoelastic distortion of seal faces. 
In Chapter 9, Surface grooves of gas face seals and testing technology, typical seal grooves and their laser processing methods and experimental measurement methods are presented. Opening characteristics, hydrodynamic characteristics, and friction characteristics of gas face seals are discussed by taking the micro-dimpled face seal as an example. 
Preface III 
In Chapter 10, Design of gas face seals, an internal flow single-face spiral groove gas seal is used as an example to introduce the basic method and design process of gas face seals. 
In the process of compiling this book, Prof. Huang Ping has given great sup-port and help. Here I would like to express my sincere gratitude. At the same time, I would like to express my heartfelt thanks to my colleagues and graduate students who have given me warm support and help in the preparation of this book. 
Bai Shaoxian 
Hangzhou