Multicomponent Flow Modeling

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.

Contenu

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. Conservat…