Non-equilibrium Evaporation and Condensation Processes: Analytical Solutions (Mathematical Engineering)

✍ Scribed by Yuri B. Zudin

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 573
• Category: Library

✦ Synopsis

This present book is concerned with analytical approaches to statement and solution of problems of non-equilibrium evaporation and condensation. From analytical solutions, one is capable to understand and represent in a transparent form the principal laws, especially in the study of a new phenomenon or a process. This is why analytical methods are always employed on the first stage of mathematical modeling. Analytical solutions are also used as test models for validation of results numerical solutions. Non-equilibrium evaporation and condensation processes play an important role in a number of fundamental and applied problems: laser methods for processing of materials, depressurization of the protection cover of nuclear propulsion units, solar radiation on a comet surface, explosive boiling of superheated liquid, thermodynamic principles of superfluid helium. Analytical relations provide an adequate description of the essence of a physical phenomenon.

✦ Table of Contents

Preface
Contents
1 Introduction to the Problem
1.1 Kinetic Molecular Theory
1.2 Discussing the Boltzmann Equation
1.3 Precise Solution to the Boltzmann Equation
1.4 Intensive Phase Change
References
2 Non-equilibrium Effects on the Phase Interface
2.1 Conservation Equations of Molecular Flows
2.2 Evaporation into Vacuum
2.3 Extrapolated Boundary Conditions
2.4 Accommodation Coefficients
2.5 Linear Kinetic Theory
2.6 Introduction to the Problem of Strong Evaporation
2.6.1 Conservation Equations
2.6.2 The Model of Crout
2.6.3 The Model of Anisimov
2.6.4 The Model of Rose
2.6.5 The Mixing Model
References
3 Approximate Kinetic Analysis of Strong Evaporation
3.1 Introduction
3.2 Conservation Equations
3.3 Mixing Surface
3.4 Limiting Mass Flux
3.5 Reflection of Molecules from the Surface
3.5.1 Condensation Coefficient
3.5.2 Diffusion Scheme for Reflection of Molecules
3.6 Thermodynamic State of Vapor
3.7 Laser Irradiation of Surface
3.8 Integral Heat Balance Method
3.8.1 Analytical Solutions
3.8.2 Heat Perturbation Front
3.9 Heat Conduction Equation in the Target
3.10 The “Thermal Conductivity–Evaporation” Conjugate Problem
3.11 Linear Evaporation Problem
3.12 Nonlinear Evaporation Problem
3.13 Irradiation of Saturated Fluid
3.14 Conjugate Problem for Hyperbolic Heat Conduction Equation
3.14.1 Generalized Form for the Solution
3.14.2 Fourier’s Heat Conduction Hypothesis
3.14.3 Cattaneo–Vernotte’s Heat Conduction Hypothesis
3.14.4 Spatially Inhomogeneous Structures
3.14.5 Estimate of the Relaxation Time
3.15 General Analytical Solution of the Conjugate Problem
3.15.1 Integral Laplace Transform
3.15.2 Relative Heat Transfer Coefficient
3.15.3 Dimensionless Heat Transfer Coefficients
3.15.4 Two-Zone Approximation of the Solution
3.15.5 Algorithm of Solution of the Conjugate Problem
3.16 Conclusions
References
4 Semi-Empirical Model of Strong Evaporation
4.1 Strong Evaporation
4.2 Approximate Analytical Models
4.3 Analysis of the Available Approaches
4.4 The Semi-Empirical Model
4.5 Validation of the Semi-Empirical Model
4.5.1 Monatomic Gas
4.5.2 Sonic Evaporation
4.5.3 Polyatomic Gas
4.5.4 Maximum Mass Flow
4.6 Hydrodynamic Instability
4.6.1 Small Disturbance Method
4.6.2 Kelvin–Helmholtz Instability
4.6.3 Kinetic Shock Wave
4.6.4 Thermodynamic System
4.6.5 Onsager Reciprocal Relation
4.6.6 Prigogine Theorem
4.6.7 Change of the Entropy
4.7 Inverse and Ill-Posed Problems
4.7.1 Conditions for Well-Posedness
4.7.2 Numerical Methods
4.8 Direct and Inverse Problems of Strong Evaporation
4.8.1 Strong Evaporation
4.8.2 Macroscopic Models of Strong Evaporation
4.8.3 Solution of Inverse Problem
4.9 Conclusions
References
5 Approximate Kinetic Analysis of Strong Condensation
5.1 Introduction
5.2 Macroscopic Models
5.3 Strong Evaporation
5.4 Strong Condensation
5.5 The Mixing Model
5.6 Solution Results
5.7 Sonic Condensation
5.8 Approximate Solution
5.9 Supersonic Condensation Regimes
5.9.1 Introduction
5.9.2 Calculation Results
5.10 Diffusion Scheme of Reflection of Molecules
5.11 The General Case of Boundary Conditions
5.12 The Effect of β on the Condensation Process
5.13 Conclusions
References
6 Linear Kinetic Analysis of Evaporation and Condensation
6.1 Introduction
6.2 Conservation Equations
6.3 Equilibrium Coupling Conditions
6.4 Linear Kinetic Analysis
6.4.1 Linearized System of Equations
6.4.2 Symmetric and Asymmetric Cases
6.4.3 Kinetic Jumps
6.4.4 Effect of Condensation Coefficient
6.4.5 Short Description
6.5 Conclusions
References
7 Binary Schemes of Vapor Bubble Growth
7.1 Introduction
7.2 Limiting Schemes of Growth
7.3 The Energetic Thermal Scheme
7.4 Binary Schemes of Growth
7.4.1 The Viscous–Inertial Scheme
7.4.2 The Inertial–Thermal Scheme
7.4.3 The Region of High Superheatings
7.4.4 The Non-equilibrium-Thermal Scheme
7.5 Conclusions
References
8 Pressure Blocking Effect in a Growing Vapor Bubble
8.1 Introduction
8.2 The Inertial-Thermal Scheme
8.3 The Pressure Blocking Effect
8.4 The Stefan Number in the Metastable Region
8.5 Effervescence of the Butane Drop
8.6 Seeking an Analytical Solution
8.6.1 Qualitative Analysis
8.6.2 Asymptotically Analytical Solution
8.7 Conclusions
References
9 Evaporating Meniscus on the Interface of Three Phases
9.1 Introduction
9.2 Evaporating Meniscus
9.3 Approximate Analytical Solution
9.4 Nanoscale Film
9.5 The Averaged Heat Transfer Coefficient
9.6 The Kinetic Molecular Effects
9.7 Conclusions
References
10 Kinetic Molecular Effects with Spheroidal State
10.1 Introduction
10.2 Assumptions in the Problem Description
10.3 Hydrodynamics of Flow
10.4 Equilibrium of Drop
10.5 Conclusions
References
11 Solution of Special Problems of Film Condensation
11.1 Condensation of Vapor Flowing around a Cylinder
11.1.1 Introduction
11.1.2 Limiting Heat Transfer Laws
11.1.3 Asymptotics of Immobile Vapor
11.1.4 Pressure Asymptotics
11.1.5 Tangential Stresses at the Interface Boundary
11.1.6 Analysis of the Results
11.2 Condensation from a Vapor-Gas Mixture
11.2.1 Survey of the Literature
11.2.2 The Mechanistic Model
11.2.3 The Chilton–Colburn Analogy
11.2.4 The Principal Equation of the Mechanistic Model
11.2.5 Iteration-Free Solution Procedure
11.2.6 Evaluation by the Mechanistic Model
11.2.7 The Modified Mechanistic Model
11.2.8 Asymptotic Analysis of Mechanistic Model
11.3 Conclusions
References
12 Nucleate Pool Boiling
12.1 Metastable Liquid
12.2 Conditions for the Onset of Boiling
12.3 Nucleation Sites
12.4 Boiling Regimes
12.4.1 Boiling Curve
12.4.2 Nucleate Boiling
12.4.3 Film Boiling
12.4.4 Transition Boiling
12.5 Vapor Bubble Growth Laws
12.5.1 Bubble Growth in a Bulk Liquid
12.5.2 Bubble Growth on a Rigid Surface
12.6 Mechanisms of Nucleate Boiling Heat Transfer
12.6.1 Applied Significance of Nucleate Boiling
12.6.2 Classification of Utilized Liquids
12.6.3 Heat Transfer Modeling (Dynamics of Bubbles)
12.6.4 Heat Transfer Modeling (Integral Characteristics)
12.6.5 Difficulties of Theoretical Descriptions
12.7 Periodic Model of Nucleate Boiling
12.7.1 Oscillations of the Thickness of a Liquid Film
12.7.2 Nucleation Site Density
12.8 Effect of Thermophysical Characteristics of Wall Materials
12.9 Effect of Oscillating Interface on Heat Transfer
12.9.1 Oscillation Structure of Convective Heat Transfer
12.9.2 Correct Averaging of the Heat Transfer Coefficient
12.9.3 Model of Periodical Contacts
12.10 Conjugate Heat Transfer Problem in Boiling
12.10.1 Nucleate Boiling
12.10.2 Transition Boiling
12.11 Conclusions
References
13 Heat Transfer in Superfluid Helium
13.1 Introduction
13.2 Practical Applications of Superfluidity
13.3 Two-Fluid Model of Superfluid Helium
13.4 Peculiarities of “Boiling” of Superfluid Helium
13.4.1 Some Properties of Superfluid Helium
13.4.2 Heat Transfer in Superfluid Heilum
13.4.3 Kapitza Resistance
13.5 Theory of Laminar Film Boiling of Superfluid Helium
13.5.1 Laminar Film Boiling
13.5.2 Cylinder
13.5.3 Plate
13.6 Thermodynamic Principles of Superfluid Helium
13.6.1 Two-Fluid Model
13.6.2 Microscopic Analysis
References
14 Pseudoboiling
14.1 Area of Supercritical Pressures
14.1.1 The Relevance of the Problem
14.1.2 Theoretical Studies of Heat Transfer
14.2 Surface Renewal Model
14.2.1 Periodic Structure of Near-Wall Turbulent Flow
14.2.2 Relative Correspondence Method
14.3 Mathematical Description
14.3.1 Conservation Equations
14.3.2 Boundary Conditions
14.3.3 Dimensionless Variables
14.4 Solution of the Main Equation
14.4.1 Exact Solution
14.4.2 Approximate Solution
14.5 Heat Transfer and Friction in the Turbulent Boundary Layer
14.5.1 Mathematical Description
14.5.2 Effect of Variable Thermophysical Properties
14.5.3 Effect of Thermal Expansion/Contraction
14.6 Wall Blowing/Suction
14.7 Heat Transfer Evaluation
14.7.1 Relative Law of Heat Transfer
14.7.2 Deteriorated and Improved Regimes
14.7.3 Comparison with Experimental Data
14.8 Conclusions
References
15 Bubble Rising in a Liquid
15.1 Bubble Rising in Liquid Column
15.1.1 Spherical Bubbles
15.1.2 Force Balance
15.1.3 Energy Balance
15.1.4 Semiempirical Models
15.1.5 Empirical Models
15.1.6 Instability of Buoyancy Trajectory for a Bubble
15.1.7 Base Underpressure Model
15.1.8 Gravitational Asymptotics
15.1.9 Capillary Asymptotics
15.2 Taylor Bubble Rising in Vertical Tube
15.2.1 Theoretical Solutions
15.2.2 Correct Statement of the Problem
15.2.3 Stagnation Point
15.2.4 Method of Collocations
15.2.5 Asymptotical Solution
15.2.6 Plane Taylor Bubble
15.2.7 Non-uniqueness of the Solution
15.3 Conclusions
References
16 Bubbles Dynamics in Liquid
16.1 Bubble Dynamics in a Tube
16.1.1 The Generalized Rayleigh Equation
16.1.2 The General Case
16.1.3 Collapse of a Bubble in a Tube
16.2 Homogeneous Nucleation
16.2.1 Introduction
16.2.2 Classical Theory
16.2.3 Quantum Mechanical Model
16.3 Bubble Size in a Turbulent Fluid Flow
16.3.1 Structure of Turbulent Vortices
16.3.2 Richardson–Kolmogorov Cascade
16.3.3 Models of Turbulence
16.3.4 Bubbles Breakup
16.3.5 Local Isotropic Turbulence
16.3.6 Empirical Formulas and Experimental Data
16.3.7 Calculation of Turbulence Energy Dissipation
16.3.8 Modified Kolmogorov–Hinze Model
16.4 Conclusions
References
17 Heat Transfer to a Disperse Two-Phase Flow
17.1 Droplet Size in Dispersed Two-Phase Flow
17.1.1 Free Oscillations of Droplet
17.1.2 Analysis of the Solution
17.1.3 Theory of Locally Isotropic Turbulence
17.1.4 Resonance Model of Droplets Breakup
17.2 Effect of Droplets on Heat Transfer to a Disperse Two-Phase Flow
17.2.1 Analytical Solutions
17.2.2 Energy Equation for Dispersed Two-Phase Flow
17.2.3 Analytical Solution
17.2.4 Relative Law of Heat Transfer
17.3 Conclusions
References
18 Thermal–Hydraulic Stability Analysis of Supercritical Fluid
18.1 Thermal–Hydraulic Instability
18.2 Small Variations Method
18.3 Density Wave Instability
18.3.1 Physical Analysis
18.3.2 Mathematical Description
18.4 Problem 1. Pressure Losses in Throttles
18.4.1 The Process of Solution
18.4.2 Approximation of the Solution
18.5 Problem 2. Pressure Losses Over the Channel Length
18.5.1 Construction of the General Solution
18.5.2 Analytical Solution
18.5.3 Instability Region
18.5.4 Stability Region
18.6 Conclusions
References
19 Heat Transfer in a Pebble Bed
19.1 Introduction
19.2 Experimental Facility
19.3 Measurement Results
19.4 Processing of the Results
19.5 Conclusions
References
Appendix A Film Boiling Heat Transfer
Appendix B Critical Heat Flux and Landau Instability
Landau Instability
Problems of Hydrodynamic Instability
Problem Statement
Consistency Conditions
Analysis of Stability
Critical Heat Flux
Kutateladze–Zuber Model
Boiling in a Tube with Twisted Tape
Conclusions
References