Multicomponent Flow Modeling

ISBN-13: 9781461272021 / Angielski / Miękka / 2012 / 321 str.

Vincent Giovangigli

Opis

The goal of this is book to give a detailed presentation of multicomponent flow models and to investigate the mathematical structure and properties of the resulting system of partial differential equations. These developments are also illustrated by simulating numerically a typical laminar flame. Our aim in the chapters is to treat the general situation of multicomponent flows, taking into account complex chemistry and detailed transport phe- nomena. In this book, we have adopted an interdisciplinary approach that en- compasses a physical, mathematical, and numerical point of view. In par- ticular, the links between molecular models, macroscopic models, mathe- matical structure, and mathematical properties are emphasized. We also often mention flame models since combustion is an excellent prototype of multicomponent flow. This book still does not pretend to be a complete survey of existing models and related mathematical results. In particular, many subjects like multi phase-flows, turbulence modeling, specific applications, porous me- dia, biological models, or magneto-hydrodynamics are not covered. We rather emphasize the fundamental modeling of multicomponent gaseous flows and the qualitative properties of the resulting systems of partial dif- ferential equations. Part of this book was taught at the post-graduate level at the Uni- versity of Paris, the University of Versailles, and at Ecole Poly technique in 1998-1999 to students of applied mathematics.

Recenzje

"This book (written with originality by a recognized expert actively working in the title area) can be considered as an interdisciplinary presentation of multicomponent flow models, their mathematical properties, and selected numerical simulations...The book is well organized, clearly written, and can serve as a very useful reference book for specialists (circa 300 contemporary references) and as a good introduction to the subject for senior and graduate-level students in applied mathematics."

-Zentralblatt Math

Spis treści

1. Introduction.- 2. Fundamental Equations.- 2.1. Introduction.- 2.2. Conservation equations.- 2.2.1. Species, momentum, and energy.- 2.2.2. Total mass conservation.- 2.2.3. Species mass fractions.- 2.2.4. Kinetic and internal energy.- 2.2.5. Species independent specific forces.- 2.3. Thermodynamics.- 2.3.1. Density and internal energy.- 2.3.2. Enthalpy.- 2.3.3. Mole fractions and molar concentrations.- 2.3.4. Enthalpy and temperature equations.- 2.3.5. Entropy and Gibbs function.- 2.3.6. Alternative formulations.- 2.3.7. Thermodynamic data.- 2.4. Chemistry.- 2.4.1. Elementary reactions.- 2.4.2. Maxwellian production rates.- 2.4.3. Total mass conservation.- 2.4.4. Notation for three-body reactions.- 2.4.5. Chemistry data.- 2.5. Transport fluxes.- 2.5.1. Viscous tensor, species mass fluxes, and heat flux.- 2.5.2. Diffusion velocities and diffusion matrix.- 2.5.3. Alternative formulations.- 2.5.4. Transport coefficients.- 2.6. Entropy.- 2.6.1. Entropy differential.- 2.6.2. Entropy equation.- 2.6.3. Entropy production.- 2.7. Boundary conditions.- 2.7.1. Dirichlet and Neumann boundary conditions.- 2.7.2. Porous walls.- 2.7.3. Catalytic plates.- 2.8. Notes.- 2.9. References.- 3. Approximate and Simplified Models.- 3.1. Introduction.- 3.2. One-reaction chemistry.- 3.2.1. One-reaction kinetics.- 3.2.2. Approximations for one-reaction kinetics.- 3.2.3. Simplified equations.- 3.2.4. Deficient reactants.- 3.3. Small Mach number flows.- 3.3.1. Orders of magnitude.- 3.3.2. Momentum equation and pressure splitting.- 3.3.3. Energy equation.- 3.3.4. Isobaric equations.- 3.3.5. Vorticity-velocity formulation.- 3.3.6. Strained flows.- 3.3.7. Shvab-Zeldovitch formulation.- 3.4. Coupling.- 3.4.1. Coupling of partial differential equations.- 3.4.2. Dilution approximation.- 3.4.3. Constant density approximation.- 3.5. Notes.- 3.6. References.- 4. Derivation from the Kinetic Theory.- 4.1. Introduction.- 4.2. Kinetic framework.- 4.2.1. Distribution functions.- 4.2.2. Macroscopic properties.- 4.2.3. Boltzmann equations.- 4.2.4. Scattering source terms.- 4.2.5. Reactive source terms.- 4.2.6. Examples.- 4.3. Kinetic entropy.- 4.3.1. Definition of the kinetic entropy.- 4.3.2. Kinetic entropy equation.- 4.3.3. Positivity of entropy production.- 4.4. Enskog expansion.- 4.4.1. Asymptotic orders.- 4.4.2. Collisional invariants of the fast operator.- 4.4.3. Macroscopic equations.- 4.5. Zero-order approximation.- 4.5.1. Maxwellian distributions.- 4.5.2. Zero-order macroscopic equations.- 4.5.3. Zero-order time derivatives.- 4.6. First-order approximation.- 4.6.1. Linearized Boltzmann operator.- 4.6.2. Linearized Boltzmann equations.- 4.6.3. Expansion of perturbed distributions.- 4.6.4. Macroscopic equations and transport fluxes.- 4.6.5. Transport coefficients.- 4.6.6. Chemistry source terms.- 4.6.7. Thermodynamics.- 4.7. Transport linear systems.- 4.7.1. Galerkin method.- 4.7.2. Basis functions.- 4.7.3. Structure of transport linear systems.- 4.7.4. Sparse transport matrix.- 4.7.5. Vanishing mass fractions.- 4.8. Notes.- 4.9. References.- 5. Transport Coefficients.- 5.1. Introduction.- 5.2. Transport algorithms.- 5.2.1. Transport linear systems.- 5.2.2. Mathematical structure.- 5.2.3. Direct inversion.- 5.2.4. Iterative methods.- 5.2.5. Empirical expressions.- 5.2.6. Operational count.- 5.2.7. Stability for vanishing mass fractions.- 5.3. Molecular parameters.- 5.3.1. Interaction potentials.- 5.3.2. Collision integrals.- 5.3.3. Viscosity of pure gases and binary diffusion.- 5.3.4. Relaxation and diffusion of internal energy.- 5.3.5. Transport data.- 5.4. Shear viscosity.- 5.5. Volume viscosity.- 5.6. Diffusion matrix.- 5.7. Thermal conductivity.- 5.8. Thermal diffusion ratios.- 5.9. Partial thermal conductivity.- 5.10. Thermal diffusion coefficients.- 5.11. Notes.- 5.12. References.- 6. Mathematics of Thermochemistry.- 6.1. Introduction.- 6.2. Thermodynamics with volume densities.- 6.2.1. State variables (T, ?1, …, ?n).- 6.2.2. Energy and enthalpy per unit volume.- 6.2.3. Entropy and Gibbs function per unit volume.- 6.2.4. Assumptions.- 6.2.5. Differentials and convexity.- 6.3. Thermodynamics with mass densities.- 6.3.1. State variables (T, p, Y1, …, Yn).- 6.3.2. Energy and enthalpy per unit mass.- 6.3.3. Entropy and Gibbs function per unit mass.- 6.3.4. Assumptions.- 6.3.5. Differentials and convexity.- 6.3.6. Miscellaneous.- 6.4. Chemistry sources.- 6.4.1. Chemical reactions.- 6.4.2. Maxwellian production rates.- 6.4.3. Assumptions.- 6.4.4. Mass weights.- 6.4.5. Mass conservation.- 6.4.6. Creation and destruction rates.- 6.4.7. Symmetric formulation for the rates of progress.- 6.5. Positive equilibrium points.- 6.5.1. Definition of equilibrium points.- 6.5.2. Equilibrium points with T and ? fixed.- 6.5.3. Equilibrium points with h and ? fixed.- 6.5.4. Smoothness of equilibrium points.- 6.6. Boundary equilibrium points.- 6.6.1. Definition of boundary equilibrium points.- 6.6.2. Decomposition chain property.- 6.7. Inequalities near equilibrium.- 6.7.1. Production rates and chemical dissipation.- 6.7.2. Entropy difference and chemical dissipation.- 6.8. A global stability inequality.- 6.9. Notes.- 6.10. References.- 7. Mathematics of Transport Coefficients.- 7.1. Introduction.- 7.1.1. Definition of transport fluxes.- 7.1.2. Diffusion velocities.- 7.1.3. Alternative formulations.- 7.2. Assumptions on transport coefficients.- 7.3. Properties of diffusion matrices.- 7.3.1. First properties of the diffusion matrix D.- 7.3.2. First properties of the flux diffusion matrix C.- 7.3.3. Flux splitting.- 7.3.4. Generalized inverses of C and D.- 7.3.5. Modified diffusion coefficients.- 7.4. Properties of other coefficients.- 7.4.1. Alternative coefficients.- 7.4.2. Waldmann coefficients.- 7.5. Diagonal diffusion.- 7.5.1. Irreducibility of C and D.- 7.5.2. Matrix E and mass fraction gradients.- 7.5.3. Irreducibility of CE and DE.- 7.5.4. Diagonal diffusion of C and D over U?.- 7.5.5. Diagonal diffusion of CE and DE over U?.- 7.5.6. Diagonal diffusion of C and D for n — 1 species.- 7.5.7. Diagonal diffusion of CE and DE for n — 1 species.- 7.6. Diffusion inequalities.- 7.6.1. Fundamental diffusion inequality.- 7.6.2. Positivity properties of C.- 7.7. Stefan-Maxwell equations.- 7.7.1. Matrices ? and D.- 7.7.2. Matrices ? and C.- 7.7.3. Diagonal first-order diffusion.- 7.7.4. Asymptotic expansions of D and C.- 7.8. Notes.- 7.9. References.- 8. Symmetrization.- 8.1. Introduction.- 8.2. Vector notation.- 8.2.1. Conservative and natural variables.- 8.2.2. Vector equations.- 8.3. Quasilinear form.- 8.3.1. The map Z ? U.- 8.3.2. Dissipation matrices and quasilinear form.- 8.4. Symmetrization and entropic variables.- 8.4.1. Symmetric conservative forms.- 8.4.2. Entropic variables.- 8.4.3. The equivalence theorem.- 8.5. Normal forms.- 8.5.1. Definition of normal forms.- 8.5.2. Nullspace invariance property.- 8.5.3. Description of normal variables.- 8.6. Symmetrization for multicomponent flows.- 8.6.1. Entropy and symmetric conservative form.- 8.7. Normal forms for multicomponent flows.- 8.7.1. Nullspace of dissipation matrices.- 8.7.2. First normal form.- 8.7.3. Natural normal form.- 8.7.4. Intermediate normal form.- 8.8. Notes.- 8.9. References.- 9. Asymptotic Stability.- 9.1. Introduction.- 9.2. Governing equations.- 9.2.1. Abstract system.- 9.2.2. Equilibrium points.- 9.2.3. Entropy equation.- 9.2.4. Functional spaces.- 9.3. Local dissipative structure.- 9.3.1. Linearized equations.- 9.3.2. Locally stable source terms.- 9.3.2. Global dissipative structure.- 9.4. Global existence theorem.- 9.4.1. Main result.- 9.4.2. Local existence.- 9.4.3. A priori estimates.- 9.4.4. More a priori estimates.- 9.4.5. Global existence proof.- 9.5. Decay estimates.- 9.6. Local dissipativity for multicomponent flows.- 9.6.1. Chemical sources.- 9.6.2. Local dissipative structure.- 9.6.3. Linearized source term.- 9.7. Global existence for multicomponent flows.- 9.7.1. Linearized normal form.- 9.7.2. Global existence and asymptotic stability.- 9.8. Notes.- 9.9. References.- 10. Chemical Equilibrium Flows.- 10.1. Introduction.- 10.2. Governing equations.- 10.2.1. Notation associated with equilibrium.- 10.2.2. Atomic species and formation reactions.- 10.2.3. Equations at chemical equilibrium.- 10.2.4. Conservative and natural variables.- 10.2.5. Fluxes at chemical equilibrium.- 10.2.6. Quasilinear form at chemical equilibrium.- 10.3. Entropy and symmetrization.- 10.3.1. Entropy at chemical equilibrium.- 10.3.2. Symmetrized equations.- 10.4. Normal forms.- 10.4.1. Nullspace invariance property.- 10.4.2. Intermediate normal form.- 10.5. Global existence.- 10.5.1. Local dissipativity.- 10.5.2. Global existence.- 10.6. Notes.- 10.7. References.- 11. Anchored Waves.- 11.1. Introduction.- 11.2. Governing equations.- 11.2.1. Conservation equations.- 11.2.2. Thermodynamic properties.- 11.2.3. Maxwellian chemistry.- 11.2.4. Transport fluxes.- 11.2.5. The temperature equation.- 11.2.6. Boundary conditions.- 11.2.7. Equilibrium limit.- 11.2.8. The matrix L.- 11.2.9. Entropy conservation equation.- 11.3. First properties.- 11.3.1. Preliminaries.- 11.3.2. Reduction to a problem on [0, ?).- 11.3.3. Extension to (—?, 0).- 11.4. Existence on a bounded domain.- 11.4.1. Preliminaries.- 11.4.2. Fixed point formulation.- 11.4.3. Existence of the degree.- 11.4.4. Calculation of the degree.- 11.5. Existence of solutions.- 11.5.1. Uniform estimates for c.- 11.5.2. Convergence towards equilibrium.- 11.5.3. Passage to the limit a ? ?.- 11.6. Notes.- 11.7. References.- 12. Numerical Simulations.- 12.1. Introduction.- 12.2. Laminar flame model.- 12.2.1. Governing equations.- 12.2.2. Boundary conditions.- 12.2.3. Chemical mechanism.- 12.3. Computational considerations.- 12.3.1. Discretized equations.- 12.3.2. Multiple time scales.- 12.3.3. Multiple space scales.- 12.3.4. Nonlinear solvers.- 12.3.5. Pseudo-unsteady iterations.- 12.3.6. Thermochemistry and transport software.- 12.4. Hydrogen-Air Bunsen flame.- 12.4.1. Burner geometry.- 12.4.2. Numerical results.- 12.5. References.