Showing papers on "Non-equilibrium thermodynamics published in 1989"

A review of reaction rates and thermodynamic and transport properties for the 11-species air model for chemical and thermal nonequilibrium calculations to 30000 K

Abstract: Reaction rate coefficients and thermodynamic and transport properties are provided for the 11-species air model which can be used for analyzing flows in chemical and thermal nonequilibrium. Such flows will likely occur around currently planned and future hypersonic vehicles. Guidelines for determining the state of the surrounding environment are provided. Approximate and more exact formulas are provided for computing the properties of partially ionized air mixtures in such environments.

Conservation equations and physical models for hypersonic air flows in thermal and chemical nonequilibrium

Abstract: The conservation equations for simulating hypersonic flows in thermal and chemical nonequilibrium and details of the associated physical models are presented These details include the curve fits used for defining thermodynamic properties of the 11 species air model, curve fits for collision cross sections, expressions for transport properties, the chemical kinetics models, and the vibrational and electronic energy relaxation models The expressions are formulated in the context of either a two or three temperature model Greater emphasis is placed on the two temperature model in which it is assumed that the translational and rotational energy models are in equilibrium at the translational temperature, T, and the vibrational, electronic, and electron translational energy modes are in equilibrium at the vibrational temperature, T sub v The eigenvalues and eigenvectors associated with the Jacobian of the flux vector are also presented in order to accommodate the upwind based numerical solutions of the complete equation set

The dynamic origin of increasing entropy

Abstract: Thermodynamic states are assumed to be characterized by densities. Recent ergodic-theory results on the evolution of densities are used to give a unified treatment of the origin of classical nonequilibrium thermodynamic behavior. Asymptotic periodicity is sufficient for the existence of at least one state of (metastable) thermodynamic equilibrium and for the evolution of the entropy to a relative maximum that depends on the way the system is prepared. Ergodicity is necessary and sufficient for a unique state of thermodynamic equilibrium to exist. Exactness, a property of chaotic semidynamical (irreversible) systems, is necessary and sufficient for the global evolution of the entropy to its unique maximum for all initial states. Since all of the laws of physics are formulated as (reversible) dynamical systems, it is unclear why entropy is observed to approach a maximum. Setting aside the possibility that all of the laws of physics are incorrectly formulated, it is demonstrated that either observation of a subset of the complete dynamics (trivial coarse graining) or interactions with an external heat bath (addition of noise) may induce exactness with a consequent evolution of entropy to a maximal state.

Galilean invariance and entropy principle for systems of balance laws

Abstract: The Galilean invariance of a generic system of balance laws dictates a specific dependence of the densities and fluxes on velocity. Thus these quantities decompose in a unique manner into convective and non-convetive parts. Such a decomposition permits the elimination of velocity dependencies in the entropy principle, which becomes a constraint on the constitutive functions only. These results clarify the mathematical structure of extended thermodynamics. They also provide a connection between the equations of continuum thermodynamics and the Boltzmann equation.

The thermodynamics of solutions

Abstract: This paper explores the relationship between thermodynamics and two other types of investigation. Structure determinations by neutron diffraction and computer simulation give a clear picture of first-shell solvation for a number of cations in water. Replacement of a water molecule in the complex M(H20): by an organic ligand S in a mixed aqueous solvegt S-H20 may result in distortion which raises the free energy of transfer(AtG ) of the cation M+ from water above that expected from an unhindered base-line. The sequence of these deviations is that predictable from the known geometries, i.2. Li+>Na'>Cs+>Ag+, H' . relative viscosities of solutions are examined by Transition-State Tneory. Electrolytes like CsCl lower the free energy, and markedly, the enthalpy and entropy of activation for viscous flow of water, but do not necessarily break down solvent structure as in the classical view. Enchanced co-ordination of solvent to the ions could occur in a transition-state solvent more weakly structured and bonded than the ground-state solvent. Enthalpies of transfer of the "hydrophobic" solute E-butanol in methanoluater mixtures suggest that TBA makes strong solute-solvent bonds, but breaks solvent-solvent bonds. The large positive activation parameters for viscous flow in highly aqueous mixtures suggest that water encages this type of solute in the ground state, thereby enhancing the solute-solvent interaction. The INTRODUCTION Our basic thermodynamic process is the transfer of a solute between standard states in two different solvents; it will be accompanied by changes in the free energy,&G enthalpy, Aty , and entropy, AtS , of the system, which reflect differences in the solvation of the solute in the two solvents (ref. I). Most of our solutes will be electrolytes. Most of the transfers will involve binary aqueous mixtures. We shall first show how recent structural studies (refs 2 & 3) help us to understand steric influences on the free energies of transfer, AtG , of some simple electrolytes from water to mixed aqueous solvents. We shall then discuss briefly a simple theoretical model (ref. 4 ) for the enthalpy of transfer, AtH*, in binary solvent systems. Finally, we shall consider viscous flow (ref. 5). This process played a crucial part in the development of the established models (refs. 6 & 7 ) for ions in solution. It can be treated quasi-thermodynamically, by Transition-State theory (ref. 8). Our transfer quantities now involve something we call the transitionstate solvent. This access to an unusual type of solvent helps us to a clearer understanding of the solvation process. 8 8 6

Equilibrium and nonequilibrium Lyapunov spectra for dense fluids and solids

Abstract: The Lyapunov exponents describe the time-averaged rates of expansion and contraction of a Lagrangian hypersphere made up of comoving phase-space points. The principal axes of such a hypersphere grow, or shrink, exponentially fast with time. The corresponding set of phase-space growth and decay rates is called the ''Lyapunov spectrum.'' Lyapunov spectra are determined here for a variety of two- and three-dimensional fluids and solids, both at equilibrium and in nonequilibrium steady states. The nonequilibrium states are all boundary-driven shear flows, in which a single boundary degree of freedom is maintained at a constant temperature, using a Nose-Hoover thermostat. Even far-from-equilibrium Lyapunov spectra deviate logarithmically from equilibrium ones. Our nonequilibrium spectra, corresponding to planar-Couette-flow Reynolds numbers ranging from 13 to 84, resemble some recent approximate model calculations based on Navier-Stokes hydrodynamics. We calculate the Kaplan-Yorke fractal dimensionality for the nonequilibrium phase-space flows associated with our strange attractors. The far-from-equilibrium dimensionality may exceed the number of additional phase-space dimensions required to describe the time dependence of the shear-flow boundary.

Covariant fluid mechanics and thermodynamics: An introduction

Abstract: These lectures are intended as a pedagogical introduction to the covariant formulation of nonequilibrium and statistical thermodynamics and kinetic theory. The various formulations of nonequilibrium thermodynamics and their individual difficulties — with causality, stability, shock structure — are critically reviewed. As an illustration of the general formalism, a fairly detailed treatment is presented of the covariant statistical thermodynamics of superfluids.

Nonequilibrium statistical mechanics of concentrated colloidal dispersions: Hard spheres in weak flows with many-body thermodynamic interactions

Abstract: A theory is presented which relates the colloidal interactions to the microstructure of a Brownian suspension under weak shear and then to the bulk stresses via a new technique for renormalizing the thermodynamic contribution. Further derivations of the interparticle stress provide an independent test of the accuracy of requisite closures. The results are very sensitive to the coupling between equilibrium and nonequilibrium distribution functions in the three-body closures; a closure in the spirit of the Percus-Yevick equation provides the most consistent results while superposition predicts aphysical results. Comparison with the available measurements on hard-sphere systems indicates that the Brownian stresses, renormalized into a hydrodynamic function, are responsible for the divergence in the low shear limiting viscosity in dense suspensions. However, pairwise additive hydrodynamics adequately predict neither the high frequency limiting complex viscosity nor the steady shear viscosity in dense suspensions.

Agreement of models of chemical reactions in gases with the second law of thermodynamics

Abstract: In the specified caloric equation of state, the rate constants of the direct and inverse chemical reactions must agree, taking account of the second law of thermodynamics. In particular, the free energy of the chemical-equilibrium gas mixture with specified density and temperature must be minimal. If this condition is not met, the phase portraits of the dynamic systems may be distorted, which is especially significant in analyzing the existence, uniqueness, and stability of solutions close to the bifurcation points. In the present work, corrections to the kinetic (equilibrium) models [1-4] are proposed for accurate agreement with the second law of thermodynamics. In these models, the composition of the chemically reacting gas is determined by a variable molecular mass ~ and the molecular masses in the limiting dissociated and recombined states, the constants ~min and BmaxThe total internal energy of the mixture per unit mass in [1-4] is represented as a function of two variables

Conservation laws from Hamilton's principle for nonlocal thermodynamic equilibrium fluids with heat flow.

Abstract: Extended thermodynamics of heat-conducting fluids is used to give explicit formulas for non- equilibrium energy density of ideal gas expressed as functions of classical variables and the diffusive entropy flux (a nonequilibrium variable). A Lagrangian density associated with the energy density is used to obtain the components of energy-momentum tensor and corresponding conservation laws on the basis of Hamilton's principle of stationary action and Noether's theorem. The heat flux appears naturally as a consequence of a free entropy transfer (independent of mass transfer) and a momentum transport is associated with tangential stresses resulting from this entropy transfer. The compatibility of the present description with the kinetic theory is shown. Hamilton's principle is extended so that the flux of entropy as well as the fluxes and densities of mass are varied independently. The concept of thermal momentum as the derivative of the kinetic potential with respect to the entropy flux is introduced; this quality plays a fundamental role in the extension of Gibbs's equation to describe a nonequilibrium fluid with heat flux.

Nonequilibrium lattice models: Series analysis of steady states

Abstract: A perturbation theory for steady states of interacting particle systems is developed and applied to several lattice models with nonequilibrium critical points near an absorbing state. The expansion is expressed directly in terms of the kinetic parameter (creation rate), rather than in powers of the interaction. An algorithm for generating series expansions for local properties is described. Order parameter series (16 terms) and precise estimates of critical properties are presented for the one-dimensional contact process and several related models.

One-dimensional kinetic Ising model with competing dynamics: Steady-state correlations and relaxation times

Abstract: A one-dimensional kinetic Ising model with dynamics characterized by a combination of spins flips at temperature T and spin exchanges at T=\ensuremath{\infty} is studied. The two-spin correlations in the steady state are calculated exactly and the decay times describing the relaxation of both the magnetization and the two-spin correlations are also given. We find that neither the steady-state nor the dynamic quantities show any sign of a phase transition that could exist in this one-dimensional, nonequilibrium system. Two remarkable features of the solution are that (i) the correlation length in the steady state with random spin exchanges is larger than the correlation length in the corresponding equilibrium state without spin exchanges, and (ii) a fluctuation-dissipation theorem is satisfied in the nonequilibrium steady state.

Biophysical Chemistry: Molecules to Membranes

Abstract: 1 Molecules, Membranes, and Modeling.- I Review of Thermodynamics.- 2 Thermodynamics: An Introductory Glance.- 2.1. Overview.- 2.1.1. The First Law: "The Energy of the Universe Is Conserved".- 2.1.2. The Second Law: "The Entropy of the Universe Increases".- 2.2. Defining Thermodynamic Terms.- 2.2.1. Systems, Surroundings, and Boundaries.- 2.2.2. Properties of a System.- 2.2.3. State Functions and the State of a System.- 2.2.4. Changes in State.- 2.3. Work.- 2.3.1. Electrical Work.- 2.3.2. Pressure-Volume Work.- 2.3.3. Mechanical Work.- 2.3.4. A Return to the Laws.- 3 The First Law.- 3.1. Understanding the First Law.- 3.1.1. Specialized Boundaries as Tools.- 3.1.2. Evaluating the Energy of a System.- 3.2. Derivation of the Heat Capacity.- 3.3. A System Constrained by Pressure: Defining Enthalpy.- 4 The Second Law.- 4.1. Understanding the Second Law of Thermodynamics.- 4.2. A Thought Problem: Designing a Perfect Heat Engine.- 4.2.1. Reversible Versus Irreversible Path.- 4.2.2. A Carnot Cycle.- 4.3. Statistical Derivation of Entropy.- 4.3.1. Limits of the Second Law.- 4.3.2. Statistical Distributions.- 4.3.3. The Boltzmann Distribution.- 4.3.4. A Statistical Mechanical Problem in Entropy.- 4.4. The Third Law and Entropy.- 5 Free Energy.- 5.1. The Gibbs Free Energy.- 5.2. A Moment of Retrospection Before Pushing On.- 5.3. The Properties of the Gibbs Free Energy.- 5.4. Introduction of , the Free Energy per Mole.- 5.5. Transforming the General Ideal Equation to a General Real Equation.- Appendix 5.1. Derivation of the Statement, qrev > qirrev.- 6 Multiple-Component Systems.- 6.1. New Systems, More Components.- 6.2. Chemical Potential and Chemical Systems.- 6.2.1. Characteristics of ?.- 6.2.2. An Immediate Biological Relevance of the Chemical Potential.- 6.3. The Entropy and Enthalpy and Free Energy of Mixing.- 6.4. Free Energy When Components Change Concentration.- 6.4.1. A Side Trip: Derivation of a General Term, the Activity.- 6.4.2. Activity of the Standard State.- 6.4.3. Returning to the Problem at Hand.- 6.5. The Thermodynamics of Galvanic Cells.- 7 Phase Equilibria.- 7.1. Principles of Phase Equilibria.- 7.1.1. Thermodynamics of Transfer Between Phases.- 7.1.2. The Phase Rule.- 7.2. Pure Substances and Colligative Properties.- 7.2.1. Colligative Properties and the Ideal Solution.- 7.2.2. Measurements of the Activity Coefficient Using Colligative Properties.- 7.3. Surface Phenomena.- Appendix 7.1. Equilibrium Dialysis and Scatchard Plots.- Appendix 7.2. Derivation of the Clausius-Clapeyron Equation.- Appendix 7.3. Derivation of the van't Hoff Equation for Osmotic Pressure.- 8 Engineering the Cell: A Modeling Approach to Biological Problem Solving.- II The Nature of Aqueous Solutions.- 9 Water: A Unique Structure, A Unique Solvent.- 9.1. Introduction.- 9.2. Hydrogen Bonds in Water.- 9.3. The Structure of Crystalline Water.- 9.4. Theories of the Structure of Liquid Water.- 10 Introduction to Electrolytic Solutions.- 10.1. Introduction to Ions and Solutions.- 10.1.1. The Nature of Electricity.- 10.2. Intermolecular Forces and the Energies of Interaction.- 10.3. The Nature of Ionic Species.- Appendix 10.1. Derivation of the Energy of Interaction Between Two Ions.- 11 Ion-Solvent Interactions.- 11.1. Understanding the Nature of Ion-Solvent Interactions Through Modeling.- 11.1.1. Overview.- 11.1.2. The Born Model.- 11.2. Adding Water Structure to the Continuum.- 11.3. The Energy of Ion-Dipole Interactions.- 11.4. Dipoles in an Electric Field: A Molecular Picture of Dielectric Constants.- 11.5. What Happens When the Dielectric Is Liquid Water?.- 11.6. Extending the Ion-Solvent Model Beyond Born.- 11.7. Recalculating the Born Model.- 11.7.1. Ion-Solvent Interactions in Biological Systems.- Appendix 11.1. Derivation of the Work to Charge and Discharge a Rigid Sphere.- Appendix 11.2. Derivation of Xext = 4? (q - qdipole) by Gauss's Law.- 12 Ion-Ion Interactions.- 12.1. Ion-Ion Interactions.- 12.2. Testing the Debye-Huckel Model.- 12.3. A More Rigorous Treatment of the Debye-Hu qirrev.- 6 Multiple-Component Systems.- 6.1. New Systems, More Components.- 6.2. Chemical Potential and Chemical Systems.- 6.2.1. Characteristics of ?.- 6.2.2. An Immediate Biological Relevance of the Chemical Potential.- 6.3. The Entropy and Enthalpy and Free Energy of Mixing.- 6.4. Free Energy When Components Change Concentration.- 6.4.1. A Side Trip: Derivation of a General Term, the Activity.- 6.4.2. Activity of the Standard State.- 6.4.3. Returning to the Problem at Hand.- 6.5. The Thermodynamics of Galvanic Cells.- 7 Phase Equilibria.- 7.1. Principles of Phase Equilibria.- 7.1.1. Thermodynamics of Transfer Between Phases.- 7.1.2. The Phase Rule.- 7.2. Pure Substances and Colligative Properties.- 7.2.1. Colligative Properties and the Ideal Solution.- 7.2.2. Measurements of the Activity Coefficient Using Colligative Properties.- 7.3. Surface Phenomena.- Appendix 7.1. Equilibrium Dialysis and Scatchard Plots.- Appendix 7.2. Derivation of the Clausius-Clapeyron Equation.- Appendix 7.3. Derivation of the van't Hoff Equation for Osmotic Pressure.- 8 Engineering the Cell: A Modeling Approach to Biological Problem Solving.- II The Nature of Aqueous Solutions.- 9 Water: A Unique Structure, A Unique Solvent.- 9.1. Introduction.- 9.2. Hydrogen Bonds in Water.- 9.3. The Structure of Crystalline Water.- 9.4. Theories of the Structure of Liquid Water.- 10 Introduction to Electrolytic Solutions.- 10.1. Introduction to Ions and Solutions.- 10.1.1. The Nature of Electricity.- 10.2. Intermolecular Forces and the Energies of Interaction.- 10.3. The Nature of Ionic Species.- Appendix 10.1. Derivation of the Energy of Interaction Between Two Ions.- 11 Ion-Solvent Interactions.- 11.1. Understanding the Nature of Ion-Solvent Interactions Through Modeling.- 11.1.1. Overview.- 11.1.2. The Born Model.- 11.2. Adding Water Structure to the Continuum.- 11.3. The Energy of Ion-Dipole Interactions.- 11.4. Dipoles in an Electric Field: A Molecular Picture of Dielectric Constants.- 11.5. What Happens When the Dielectric Is Liquid Water?.- 11.6. Extending the Ion-Solvent Model Beyond Born.- 11.7. Recalculating the Born Model.- 11.7.1. Ion-Solvent Interactions in Biological Systems.- Appendix 11.1. Derivation of the Work to Charge and Discharge a Rigid Sphere.- Appendix 11.2. Derivation of Xext = 4? (q - qdipole) by Gauss's Law.- 12 Ion-Ion Interactions.- 12.1. Ion-Ion Interactions.- 12.2. Testing the Debye-Huckel Model.- 12.3. A More Rigorous Treatment of the Debye-Huckel Model.- 12.4. Consideration of Other Interactions.- 12.4.1. Bjerrum and Ion Pairs.- 12.5. Perspective.- 13 Molecules in Solution.- 13.1. Solutions of Inorganic Ions.- 13.2. Solutions of Small Nonpolar Molecules.- 13.3. Solutions of Organic Ions.- 13.3.1. Solutions of Small Organic Ions.- 13.3.2. Solutions of Large Organic Ions.- 14 Macromolecules in Solution.- 14.1. Solutions of Macromolecules.- 14.1.1. Nonpolar Polypeptides in Solution.- 14.1.2. Polar Polypeptides in Solution.- 14.2. Transitions of State.- III Membranes and Surfaces in Biological Systems.- 15 Lipids in Aqueous Solution: The Formation of the Cell Membrane.- 15.1 The Form and Function of Biological Membranes.- 15.2. Lipid Structure: Components of the Cell Membrane.- 15.3. Aqueous and Lipid Phases in Contact.- 15.4. The Physical Properties of Lipid Membranes.- 15.4.1. Phase Transitions in Lipid Membranes.- 15.4.2. Motion and Mobility in Membranes.- 15.5. Biological Membranes: The Complete Picture.- 16 Irreversible Thermodynamics.- 16.1. Transport: An Irreversible Process.- 16.2. Principles of Nonequilibrium Thermodynamics.- 17 Flow in a Chemical Potential Field: Diffusion.- 17.1. Transport Down a Chemical Potential Gradient.- 17.2. The Random Walk: A Molecular Picture of Movement.- 18 Flow in an Electric Field: Conduction.- 18.1. Transport in an Electric Field.- 18.2. A Picture of Ionic Conduction.- 18.3. The Empirical Observations Concerning Conduction.- 18.4. A Second Look at Ionic Conduction.- 18.5. How Do Interionic Forces Affect Conductivity?.- 18.6. The Special Case of Proton Conduction.- 19 The Electrified Interface.- 19.1. When Phases Meet: The Interphase.- 19.2. A More Detailed Examination of the Interphase Region.- 19.3. The Simplest Picture: The Helmholtz-Perrin Model.- 19.4. A Diffuse Layer Versus a Double Layer.- 19.5. Combining the Capacitor and the Diffuse Layers: The Stern Model.- 19.6. The Complete Picture of the Double Layer.- 20 Electrokinetic Phenomena.- 20.1. The Cell and Interphase Phenomena.- 20.2. Electrokinetic Phenomena.- 21 Colloidal Properties.- 21.1. Colloidal Systems and the Electrified Interface.- 21.2. Salting Out Revisited.- 22 Forces Across Membranes.- 22.1. Energetics, Kinetics, and Force Equations in Membranes.- 22.1.1. The Donnan Equilibrium.- 22.1.2. Electric Fields Across Membranes.- 22.1.3. Diffusion Potentials and the Transmembrane Potential.- 22.1.4. Goldman Constant Field Equation.- 22.1.5. Electrostatic Profiles of the Membrane.- 22.1.6. The Electrochemical Potential.- 22.2. Molecules Through Membranes: Permeation of the Lipid Bilayer.- 22.2.1. The Next Step: The Need for Some New Tools.- Appendices.- Appendix I Further Reading List.- Appendix II Study Questions.- Appendix III Symbols Used.- Appendix IV Glossary.

Extended thermodynamics of molecular ideal gases

Abstract: The objective of extended thermodynamics of molecular ideal gases is the determination of the 17 fields ofmass density, velocity, energy density, pressure deviator, heat flux, intrinsic energy density and intrinsic heat flux.

Hypervelocity atmospheric flight: Real gas flow fields

Abstract: Flight in the atmosphere is examined from the viewpoint of including real gas phenomena in the flow field about a vehicle flying at hypervelocity. That is to say, the flow field is subject not only to compressible phenomena, but is dominated by energetic phenomena. There are several significant features of such a flow field. Spatially, its composition can vary by both chemical and elemental species. The equations which describe the flow field include equations of state and mass, species, elemental, and electric charge continuity; momentum; and energy equations. These are nonlinear, coupled, partial differential equations that have been reduced to a relatively compact set of equations in a self-consistent manner (which allows mass addition at the surface at a rate comparable to the free-stream mass flux). The equations and their inputs allow for transport of these quantities relative to the mass-average behavior of the flow field. Thus transport of mass by chemical, thermal, pressure, and forced diffusion; transport of momentum by viscosity; and transport of energy by conduction, chemical considerations, viscosity, and radiative transfer are included. The last of these complicate the set of equations by making the energy equations a partial integrodifferential equation. Each phenomenon is considered and represented mathematically by one or more developments. The coefficients which pertain are both thermodynamically and chemically dependent. Solutions of the equations are presented and discussed in considerable detail, with emphasis on severe energetic flow fields. Hypervelocity flight in low-density environments where gaseous reactions proceed at finite rates chemical nonequilibrium is considered, and some illustrations are presented. Finally, flight where the flow field may be out of equilibrium, both chemically and thermodynamically, is presented briefly.

Thermodynamics of chemical systems far from equilibrium

Abstract: A critique is presented of some recent work in this and other journals on the relation of thermodynamics to the mass action law of kinetics. For most chemical reactions, the thermodynamic variables change on the same time scale as the progress variable and there is no need for an extended thermodynamics.

On electromagnetic waves in isotropic media with dielectric relaxation

Abstract: In some previous papers one of us discussed dielectric relaxation phenomena from the point of view of nonequilibrium thermodynamics. If the theory is linearized one may derive a dynamical constitutive equation (relaxation equation) which has the form of a linear relation among the electric field E, the polarization P, the first derivatives with respect to time of E and P and the second derivative with respect to time of P. The Debye equation for dielectric relaxation in polar liquids and the De Groot-Mazur equation (obtained by these authors with the aid of methods which are also based on nonequilibrium thermodynamics) are special cases of the more general equation of which the structure has been described above. It is the purpose of the present paper to investigate the propagation and damping of electromagnetic waves. We consider the case in which the dielctric relaxation may be described by the above mentioned relaxation equation derived by one of us, the case in which the Debye equation may be used and the case in which one may apply the De Groot-Mazur equation. We derive solutions of the relaxation equations which also satisfy Maxwell's equations. We limit ourselves to plane waves of a single frequency in isotropic homogeneous linear media with vanishing electric conductivity. It is also assumed that the media are at rest. From thermodynamic arguments several inequalities are derived for the coefficients which occur in the relaxation equations. Explicit expressions are given for the complex wave vector, the complex dielectric permeability and for the phase velocity of the waves. All these quantities are functions of the frequency ω of the waves. For ω→0 the complex permeability e(compl) → e(eq) where e(eq) is the equilibrium value of the permeability for static fields. If ω→∞ we find e(compl) → 1 except for the case of the Debye equation. This is due to the fact that a part of the polarization changes in a reversible way in media for which the Debye equation holds.

From classical irreversible thermodynamics to extended thermodynamics

Abstract: A brief outline of classical and extended irreversible thermodynamics is presented. Classical irreversible thermodynamics (CIT) is known as an active and fast developing field with numerous applications in continuum mechanics, chemistry and statistical mechanics. Its foundations and principal results are shortly commented. Extended irreversible thermodynamics (EIT) has fuelled an increasing interest during the last decade. Its main objective is to extend the domain of validity of classical non-equilibrium thermodynamics. Here, a short analysis of Lebon-Jou-Casas' version of EIT is presented, it is compared with CIT and the Gyarmati wave approach.

Aspects of Non-Equilibrium Thermodynamics: Lectures on Fundamentals and Methods

Abstract: In six lectures aspects of modern non-equilibrium thermodynamics of discrete systems as well as continuum theoretical concepts are represented. Starting out with survey and introduction, state spaces are defined, the existence of internal energy is investigated, and Clausius inequality including negative absolute temperature is derived by diagram technique. Non-equilibrium contact quantities, such as contact temperature - the dynamic analogue of thermostatic temperature - and chemical potentials are phenomenologically defined and quantumstatistically founded. Using Clausius inequality the existence of non-negative entropy production is proved which allows to formulate a dissipation inequality in continuum thermodynamics. The transition between thermodynamics of discrete systems and continuum thermodynamics with respect to contact quantities is considered. Different possibilities of exploiting the dissipation inequality for getting constraints for constitutive equations are discussed. Finally hyperbolic heat conduction in non-extended thermodynamics is treated.

The computation of thermo-chemical nonequilibrium hypersonic flows

Abstract: Several conceptual designs for vehicles that would fly in the atmosphere at hypersonic speeds have been developed recently. For the proposed flight conditions the air in the shock layer that envelops the body is at a sufficiently high temperature to cause chemical reaction, vibrational excitation, and ionization. However, these processes occur at finite rates which, when coupled with large convection speeds, cause the gas to be removed from thermo-chemical equilibrium. This non-ideal behavior affects the aerothermal loading on the vehicle and has ramifications in its design. A numerical method to solve the equations that describe these types of flows in 2-D was developed. The state of the gas is represented with seven chemical species, a separate vibrational temperature for each diatomic species, an electron translational temperature, and a mass-average translational-rotational temperature for the heavy particles. The equations for this gas model are solved numerically in a fully coupled fashion using an implicit finite volume time-marching technique. Gauss-Seidel line-relaxation is used to reduce the cost of the solution and flux-dependent differencing is employed to maintain stability. The numerical method was tested against several experiments. The calculated bow shock wave detachment on a sphere and two cones was compared to those measured in ground testing facilities. The computed peak electron number density on a sphere-cone was compared to that measured in a flight test. In each case the results from the numerical method were in excellent agreement with experiment. The technique was used to predict the aerothermal loads on an Aeroassisted Orbital Transfer Vehicle including radiative heating. These results indicate that the current physical model of high temperature air is appropriate and that the numerical algorithm is capable of treating this class of flows.

Comments ``On the Relationship Between Extended Thermodynamics and the Wave Approach to Thermodynamics''

Abstract: In this short paper we analyse the discussion of Garcia-Colin and Rodriguez on the relationship between two modern thermodynamic theories: the Wave Appro­ ach to Thermodynamics and the Extended Irreversible Thermodynamics. We demonstrate that Gyarmati's wave theory is , indeed, no more and no less than the adequately generalized form - for case of continua - oft he Machlup-Onsager theory of adiabatically isolated nonequilibrium systems with kinetic energy. Fur­ ther, we emphasize, that up to this time there are not enough mathematical arguments at hand to decide which theory is more general. lntroduction This short paper is a reflection to the article of L. S. Garcia-Colin and R. F. Rodriguez [1], for we cannot accept some parts of their analysis of the relationship between extended irreversible thermodynamics (E[T) and the wave approach to thermodynamics (W AT). First of all we emphasize that it was Fekete [2] and Lengyel [3] who first analyzed this relationship. [n the abstract of the paper [1] we may read: "A comparative analysis of two formulations of non­ linear irreversible processes is performed. We first disclose the difference between the well known Onsager-Machlup linear theory and the wave approach to ther­ modynamics (W A T) introduced by Gyarmati. We then discuss the essential con­ tent of the goal and postulates of extended irreversible thermodynamics (EIT) and compare them with those of W A T. From this comparison the difference , scopes and limitations of each theory follow and precise special conditions under which both theories yield similar results are identified." We have to clarify two problems: 1. The relationship between the Onsager-Machlup theory and Gyarmati's wave

Nonequilibrium equations of state and thermal waves

Abstract: When dissipative fluxes are added as independent variables into the entropy, the equations of state become dependent on the fluxes and are accordingly modified. Such a modification may have observable consequences on thermal waves propagating in a nonequilibrium reference state. Here, the equation of state for a generalized absolute temperature under a heat flux and some of its consequences are examined from several points of view.

Extended thermodynamics and statistical mechanics of a polyatomic ideal gas

Abstract: A statistical theory for an ideal gas with vibrating molecules is developed. The basic fields in this theory are the seventeen scalar fields of density, velocity, pressure tensor, translational heat flux, vibrational energy density and vibrational heat flux. Agreement has been found between the results of statistical and phenomenological theories. Introduction The molecules of polyatomic gases are considered to have internal energies, which are related to the rotational and vibrational modes. In this sense, we could say that there are internal variables which contribute to the energy density of a polyatomic gas. A phenomenological theory of an ideal gas with one internal variable was formulated by Kremer [l], [2] within the framework of extended thermodynamics [3]-[6]. In this theory, it has been considered that the basic fields were the seventeen scalar fields of density, velocity, pressure tensor, translational heat flux, energy density of the internal variable and its flux. The seventeen balance equations for these fields were supplemented by constitutive equations for the fluxes and production terms of the pressure tensor, translational heat flux and of the flux of the internal variable. Restrictions imposed by the principle of material frame indifference and by the entropy principle were exploited. The coefficients in the representations of the constitutive functions were related to the thermal and caloric equations of state and to measurable quantities, such as the attenuation coefficient and the coefficients of shear viscosity and self-diffusion. J. Non-Equilib. Thermodyn., Vol. 14, 1989, No. 4 Copyright © 1989 Walter de Gruyter Berlin · New York