High Temperature Phenomena in Shock Waves

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Shock Wave Science and Technology Reference Library

The new Springer collection, Shock Wave Science and Technology Reference Library, conceived in the style of the famous Handbuch der Physik has as its principal motivation to assemble authoritative, state-of-the-art, archival reference articles by leading scientists and engineers in the field of shock wave research and its applications. A numbered and bounded collection, this reference library will consist of specifically commissioned volumes with internationally renowned experts as editors and contributing authors. Each volume consists of a small collection of extensive, topical and independent surveys and reviews. Typical articles start at an elementary level that is accessible to non-specialists and beginners. The main part of the articles deals with the most recent advances in the field with focus on experiment, instrumentation, theory, and modeling. Finally, prospects and opportunities for new developments are examined. Last but not least, the authors offer expert advice and cautions that are valuable for both the novice and the well-seasoned specialist.

Shock Wave Science and Technology Reference Library

Collection Editors Hans Grönig Hans Grönig is Professor emeritus at the Shock Wave Laboratory of RWTH Aachen University, Germany. He obtained his Dr. rer. nat. degree in Mechanical Engineering and then worked as postdoctoral fellow at GALCIT, Pasadena, for one year. For more than 50 years he has been engaged in many aspects of mainly experimental shock wave research including hypersonics, gaseous and dust detonations. For about 10 years he was Editorin-Chief of the journal Shock Waves.

Yasuyuki Horie Professor Yasuyuki (Yuki) Horie is internationally recognized for his contributions in high-pressure shock compression of solids and energetic materials modeling. He is a co-chief editor of the Springer series on Shock Wave and High Pressure Phenomena and the Shock Wave Science and Technology Reference Library, and a Liaison editor of the journal Shock Waves. He is a Fellow of the American Physical Society, and Secretary of the International Institute of Shock Wave Research. His current interests include fundamental understanding of (a) the impact sensitivity of energetic solids and its relation to microstructure attributes such as particle size distribution and interface morphology, and (b) heterogeneous and nonequilibrium effects in shock compression of solids at the mesoscale.

Kazuyoshi Takayama Professor Kazuyoshi Takayama obtained his doctoral degree from Tohoku University in 1970 and was then appointed lecturer at the Institute of High Speed Mechanics, Tohoku University, promoted to associate professor in 1975 and to professor in 1986. He was appointed director of the Shock Wave Research Center at the Institute of High Speed Mechanics in 1988. The Institute of High Speed Mechanics was restructured as the Institute of Fluid Science in 1989. He retired in 2004 and became emeritus professor of Tohoku University. In 1990 he launched Shock Waves, an international journal, taking on the role of managing editor and in 2002 became editorinchief. He was elected president of the Japan Society for Aeronautical and Space Sciences for one year in 2000 and was chairman of the Japanese Society of Shock Wave Research in 2000. He was appointed president of the International Shock Wave Institute in 2005. His research interests range from fundamental shock wave studies to the interdisciplinary application of shock wave research.

R. Brun (Ed.)

High Temperature Phenomena in Shock Waves With 179 Figures, 48 in Color, and 41 Tables

ABC

Prof. Raymond Brun Université de Provence 42 Chemin des Petits Cadeneaux 13170 Les Pennes Mirabeau France E-mail: [email protected]

ISBN: 978-3-642-25118-4

e-ISBN: 978-3-642-25119-1

DOI 10.1007/978-3-642-25119-1 Library of Congress Control Number: 2011940964 © 2012 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com

Contents

General Introduction.................................................................................................. 1 R. Brun References .................................................................................................................... 9 Chapter 1: Thermodynamic Properties of Gases behind Shock Waves .............. 11 M. Capitelli, D. Bruno, G. Colonna, G. d’Ammando, A. d’Angola, D. Giordano, C. Gorse, A. Laricchiuta, S. Longo 1 Introduction ...................................................................................................... 11 2 Partition Functions for Atomic Levels. General and Few-Level Approaches ........................................................................................................ 13 3 Partition Functions for Diatomic Molecules..................................................... 15 4 Transport Cross Sections and Collision Integrals............................................. 18 4.1 Phenomenological Approaches for Unknown Colliding Systems ........... 19 4.2 Resonant Charge Transfer........................................................................ 20 4.3 Neutral-Neutral Interactions.................................................................... 22 4.4 Neutral-Ion Interactions ........................................................................... 25 4.5 Charge-Charge Interactions ..................................................................... 26 4.6 Electron-Neutral Interactions ................................................................... 27 5 Thermodynamic Properties of Thermal Plasmas: The Cut-Off Issue............... 28 5.1 Cut-Off Criteria........................................................................................ 29 5.1.1 The Ground State Method ............................................................. 29 5.1.2 Debye-Hückel Based Criteria........................................................ 29 5.1.3 The Fermi Criterion....................................................................... 30 5.2 The Cut-Off Criteria Based on the Solution of Schrödinger Equation..... 30 5.3 Case Study: Air Plasma............................................................................ 34 6 Transport of Two-Temperature Plasmas .......................................................... 41 7 Transport Properties of Plasmas: The Role of Electronically Excited States ... 43 8 Conclusions ...................................................................................................... 47 Acknowledgments.. ............................................................................................... 47 Appendix: Thermodynamic and Transport Properties of Air Plasmas .................. 48 References .................................................................................................................. 54 Chapter 2: Non-equilibrium Kinetics and Transport Properties behind Shock Waves .......................................................................................... 59 E.V. Kustova, E.A. Nagnibeda 1 Introduction ...................................................................................................... 59 2 State-to-State Approach.................................................................................... 60

VI

Contents

2.1 Distribution Functions and Macroscopic Parameters............................... 60 2.2 Governing Equations................................................................................ 61 2.3 First-Order Approximation. ..................................................................... 63 3 Quasi-stationary Approaches............................................................................ 71 3.1 Vibrational Distributions: Governing Equations...................................... 71 3.2 Transport Terms....................................................................................... 75 3.3 Production Terms..................................................................................... 77 4 Non-equilibrium Processes behind Shock Waves in Air Components and CO2 Mixtures ............................................................................................. 79 4.1 Non-equilibrium Kinetics and Transport Properties in Diatomic Gas Mixtures............................................................................. 79 4.1.1 Governing Equations and Flow Parameters ................................... 79 4.1.2 Transport Properties ....................................................................... 83 4.1.3 Electronic Excitation and Radiation............................................... 84 4.2 Non-equilibrium Kinetics and Transport Processes in Air Mixture......... 87 4.3 Mixtures Containing CO2 Molecules ....................................................... 90 References .................................................................................................................. 96 Chapter 3: Non-equilibrium Kinetics behind Shock Waves Experimental Aspects .................................................................................................... 99 L. Ibraguimova, O. Shatalov 1 Introduction ...................................................................................................... 99 2 Vibrational Relaxation of Diatomic Molecules .............................................. 100 2.1 Vibrational Relaxation of O2, N2, CO, NO ............................................ 100 2.2 Vibrational Relaxation of Diatomic Molecules in Collisions with Potentially Reactive Atoms .................................................................... 104 2.3 Vibrational Relaxation of H2 and D2 ...................................................... 106 2.4 Vibrational Relaxation of Halides and Hydrogen Halides ..................... 107 3 V-T Relaxation of Three- and Multi-atomic Molecules ................................. 111 3.1 CO2 V-T Relaxation............................................................................... 113 3.2 N2O V-T Relaxation............................................................................... 115 3.3 H2O V-T Relaxation............................................................................... 117 3.4 NO2 V-T Relaxation............................................................................... 118 4 Chemical Reactions ........................................................................................ 119 4.1 Chemical Reactions in the System C-O ................................................. 120 4.2 Chemical Reactions in the System N-C-O ............................................. 128 5 Non-equilibrium Radiation............................................................................. 135 References ................................................................................................................ 141 Chapter 4: Ionization Phenomena behind Shock Waves .................................... 149 W.M. Huo, M. Panesi, T.E. Magin 1 Introduction .................................................................................................... 149 2 Production, Reaction and Removal of Charged Species................................. 151 2.1 Electrons ................................................................................................ 151 2.1.1 Production of Electrons............................................................... 153

Contents

VII

2.1.2 Reactions Involving Electrons .................................................... 159 2.1.3 Electron Recombination.............................................................. 164 2.1.4 Interparticle Interactions and Free Electron Number Density..... 164 2.2 Ions......................................................................................................... 165 3 Modeling Collisional and Radiative Processes in a Weakly Ionized Plasma ............................................................................................................ 165 3.1 The Collisional-Radiative Model ........................................................... 165 3.1.1 Transport Equations .................................................................... 166 3.1.2 Reaction Source Terms ............................................................... 168 3.1.3 Radiative Processes and Radiative Transport.............................. 173 3.2 Results.................................................................................................... 179 3.2.1 Fire II Flight Experiment ............................................................ 179 3.2.2 Chemistry and Flow Field Energy Distribution .......................... 180 3.2.3 Radiative Transport and Interaction between Radiation and Matter................................................................................... 182 3.2.4 Quasi-steady State Distribution................................................... 184 3.2.5 Comparison with Experimental Data .......................................... 186 4 Conclusions .................................................................................................... 188 Acknowledgments. .............................................................................................. 188 References ................................................................................................................ 189 Chapter 5: Radiation Phenomena behind Shock Waves..................................... 193 M.Y. Perrin, Ph. Rivière, A. Soufiani 1 Introduction .................................................................................................... 193 2 Radiative Mechanisms and Radiative Properties............................................ 195 2.1 Bound-Bound Transitions....................................................................... 195 2.1.1 General Formulation .................................................................... 195 2.1.2 Atomic Line Spectra .................................................................... 198 2.1.3 Diatomic Line Spectra.................................................................. 200 2.2 Bound-Free Transitions.......................................................................... 204 2.2.1 General Formulation ................................................................... 204 2.2.2 Atomic Photoionization .............................................................. 205 2.2.3 Molecular Photodissociation ....................................................... 206 2.2.4 Molecular Photoionization .......................................................... 206 2.2.5 Photodetachment ......................................................................... 207 2.3 Free-Free Transitions ............................................................................. 207 3 Example of Application.................................................................................. 208 4 Radiative Transfer Modeling.......................................................................... 212 4.1 The Escape Factor Approach ................................................................. 213 4.2 Spectral Models...................................................................................... 214 4.2.1 Statistical Narrow Band Models ................................................. 214 4.2.2 Global Models............................................................................. 218 4.3 Geometrical Treatment of Radiative Transfer........................................ 219 4.3.1 The Discrete Ordinate Method (DOM) ....................................... 220 4.3.2 Spherical Harmonics (PN) and Related Methods........................ 220

VIII

Contents

4.3.3 The Tangent Slab Approximation ............................................... 222 4.4 The Monte Carlo Method....................................................................... 223 5 Radiation and Flow-Field Coupling ............................................................... 225 6 Conclusion and Perspectives .......................................................................... 227 References ................................................................................................................ 227 Chapter 6: Structure of Shock Waves .................................................................. 231 A.A. Raines, F.G. Tcheremissine 1 Introduction .................................................................................................... 231 2 Methodology of Computations ....................................................................... 233 2.1 Solution of the Boltzmann Equation for a Pure Monatomic Gas ........... 233 2.2 Solution of the Generalized Boltzmann Equation .................................. 237 2.3 Two Levels Kinetic Model of RT Relaxation ........................................ 239 2.4 Solution of the Boltzmann Equation for a Gas Mixture......................... 241 2.5 Statement of the Boundary Problem and Presentation of the Computed Data ....................................................................................... 244 3 Shock Wave Structure in a Pure Monatomic Gas........................................... 245 3.1 Gas of Hard Sphere Molecules .............................................................. 245 3.2 The Lennard-Jones Gas.......................................................................... 250 4 Shock Wave Structure in a Polyatomic Gas ................................................... 253 4.1 Shock Wave Structure with Frozen Vibrational Levels ......................... 253 4.2 Shock Wave Structure with Excited Rotational and Vibrational Levels .................................................................................. 259 5 Shock Wave Structure in a Mixture of Monatomic Gases.............................. 261 6 Conclusion...................................................................................................... 266 References ................................................................................................................ 266 Chapter 7: Shock Waves in Hypersonic Rarefied Flows .................................... 271 V. Lago, A. Chpoun, B. Chanetz 1 Introduction .................................................................................................... 271 2 General Phenomena in Rarefied Flows .......................................................... 274 2.1 Flow Regime Classification ................................................................... 274 2.2 Shock Waves Thickness and Stand-Off Distance .................................. 275 2.2.1 Mixing Reynolds Number ........................................................... 275 2.2.2 Shock-Wave Stand-Off Distance Ahead of Blunt Bodies........... 276 2.2.3 Shock Wave Thickness Ahead of Blunt Bodies.......................... 276 2.2.4 Impact of Flat-Faced Leading-Edge Effects on Shock Stand-Off Distance and Shock Wave Thickness ......................... 277 2.2.5 Characterisation of Shock Waves in Rarefied Regime over a Flat Plate .......................................................................... 278 2.3 Heat Flux in Rarefied Conditions........................................................... 279 2.4 Leading Edge Flow and Viscous Interaction in Supersonic Rarefied Flow ......................................................................................... 280 2.5 Wall Pressure in Free Molecular Flow Regime ..................................... 281

Contents

IX

3 Experimental Approach.................................................................................. 281 3.1 Hypersonic Rarefied Wind Tunnel ........................................................ 281 3.2 Shock Wave-Boundary Layer Interactions in Low Density Flow ......... 283 3.3 Shock-Shock Interferences in Low Density........................................... 286 3.4 Pressure Measurements in Rarefied Flow Regimes ............................... 288 3.4.1 Pressure Measurements and Orifice Diameter Effects ................ 288 3.4.2 Pitot Pressure Measurement ........................................................ 288 3.4.3 Static Pressure Measurement ..................................................... 290 3.4.4 Pressure Transducers................................................................... 291 3.5 Heat Flux Measurements........................................................................ 291 3.6 Shock Wave Control .............................................................................. 293 References ................................................................................................................ 297 Chapter 8: High Enthalpy Non-equilibrium Shock Layer Flows: Selected Practical Applications ......................................................................... 299 S. Karl, J. Martinez Schramm, K. Hannemann 1 Introduction .................................................................................................... 299 2 Chemical Relaxation in High Enthalpy Cylinder Shock Layer Flow ............. 300 2.1 High Enthalpy Shock Tunnel Göttingen (HEG) .................................... 300 2.2 Phase Step Holographic Interferometry ................................................. 305 2.3 CFD Code .............................................................................................. 307 2.4 Experimental Setup and Results............................................................. 308 2.5 Summary and Conclusions..................................................................... 313 3 CFD Modeling of Radiation Phenomena in Shock Layers............................. 314 3.1 Introduction, Definitions and Nomenclature.......................................... 314 3.2 The Radiative Transfer Equation in Participating Media....................... 316 3.3 One-Dimendional Approximations for the Solution of the Radiative Transfer Equation ................................................................. 317 3.3.1 The Infinite Slab Model .............................................................. 317 3.3.2 Infinite Cylinder .......................................................................... 319 3.4 Approximate Solution Methods of the Radiative Transfer Equation in Three Dimensions ............................................................... 320 3.4.1 The Discrete Transfer Model ...................................................... 320 3.4.2 Solution of the Radiative Transfer Equation Using a Monte Carlo Method ...................................................... 321 3.4.3 Isothermal Cylinder..................................................................... 324 3.5 Huygens Entry Peak Heating Prediction ................................................ 327 4 Summary and Conclusions ............................................................................. 334 References ................................................................................................................ 335 Author Index........................................................................................................... 337

General Introduction R. Brun Université de Provence, Marseille, France

The production of high temperatures in gases constitutes one major feature of shock waves which represent one of the best means to transform kinetic energy into thermal energy. One of the most important consequences lies in the physical and chemical phenomena which may arise from these high temperatures such as rotational and vibrational excitation of molecules, dissociation, ionization and various chemical reactions, as well as associated radiation. Another fundamental feature proceeds from the fact that, in collisional regime (continuum), the shock wave may be considered as a discontinuity, so that the temperature rise is quasi-instantaneous. Thus, as the characteristic time required for the development of the physical and chemical phenomena is non-negligible, since it is related to collisions between elementary particles, the gaseous medium behind shock waves is in a non-equilibrium thermodynamic and chemical state. These essential points have thus contributed to the fundamental knowledge of the gaseous reactive flows at high temperature and, in particular, to the kinetics of the above phenomena from an experimental and theoretical point of view. Moreover, many applications of these properties, related to the development of aero-spatial flights, thermonuclear fusion or combustion, may thus be found in hypersonic flow, plasma generation or propulsion. After crossing a shock wave, the fluid particles, in absence of other perturbations, tend to a physical and chemical equilibrium state corresponding to the pressure and temperature conditions determined by flow conservation equations (Euler or NavierStokes) and boundary conditions, using adequate expressions for internal energy (or enthalpy). This variable, in equilibrium conditions, may be calculated a priori as a function of pressure and temperature by methods of statistical mechanics involving the computation of partition functions and transport cross-sections, keys of the determination of thermodynamic and transport properties in high temperature gases. Details on these computations are given in Chapter 1, with a particular emphasis on air plasmas obtained behind very strong shock waves. A simple introductory example is given by a straight shock wave for which the downstream equilibrium flow quantities result from the Rankine-Hugoniot relations: thus, in Fig. 1, equilibrium temperature and density ratios across a straight shock wave propagating in pure oxygen [1], i.e. T2 T1 and ρ 2 ρ1 are represented as functions of shock Mach number Ms . In the same way, as an example of chemical reactions taking place at high temperatures, the equilibrium composition of air as a function of temperature [2,3] is represented in Fig. 2.

2

High Temperature Phenomena in Shock Waves

Fig. 1. Equilibrium temperature and density ratios across a straight shock wave : Temperature, : Density (Oxygen, T1 = 300K , p1 = 10 3 Pa )

Fig. 2. Equilibrium air composition (Mass concentrations, p = 10 7 Pa )

Before reaching equilibrium, the gas “just” behind the shock wave may be considered in a “frozen” state corresponding to its chemical state in front of the shock. Then, various reactions start with, generally, different characteristic times. Detailed kinetics of chemical reactions and vibrational populations is given in Chapter 2, in which the “state to state” approach as well as more global relaxation models are presented, including the computation of transport properties in non-equilibrium conditions. Examples of the kinetics of air and carbon dioxide are also given.

General Introduction

3

Many experimental aspects of vibrational relaxation and chemical kinetics behind shock waves are proposed in Chapter 3, owing to the numerous measurements of relaxation times and chemical rate constants carried out in shock tube [1,4,5] by means of various optical and spectroscopic diagnostic methods. As introductory examples, Fig. 3 represents the experimental evolution of vibrational populations of carbon monoxide behind a shock wave [6] and Fig. 4 show experimental values of dissociation rate constants of oxygen as functions of temperature [4].

Fig. 3. Evolution of the relative population of the 3rd and the 6th vibrational level behind a straight shock wave (CO ; M s = 5, 60; T1 = 293 K ; p1 = 196 Pa )

Fig. 4. Measured values of dissociation rate constant (Oxygen)

4


The interaction of these various processes with the flow parameters behind shock waves results in a variation of all macroscopic quantities in the non-equilibrium region. Thus in Fig. 5, an example of the variation of translation-rotation temperature, vibrational temperature and density behind a straight shock wave in nitrogen is presented [6].

Fig. 5. Evolution of temperatures and density ratio behind a straight shock wave (Nitrogen, M s = 6,12, pi = 3947 Pa , Ti = 295 K )

In the same way, the variation of temperatures and species concentrations behind a strong straight shock wave in air [6] is represented in Figs. 6 and 7 respectively.

Fig. 6. Spatial variation of temperatures behind a straight shock wave in air ( M s = 25, p1 = 8, 5 Pa, T1 = 205 K )


5

Fig. 7. Spatial variation of concentrations behind a straight shock wave in air (Conditions of Fig.6)

The coupling between the chemical processes themselves also may be important, as illustrated by Fig. 8, in which the evolution of the dissociation rate constant of nitrogen is represented, with (curve A) and without (curve B) neglecting the influence of the vibrational relaxation [6]. It may be also noted that the interaction between dissipative processes and non-equilibrium phenomena may lead to complex, indeed “anomalous” situations [7].

Fig. 8. Dissociation rate constants of nitrogen behind a straight shock wave in air ( M s = 25, p1 = 8, 5 Pa, T1 = 205 K ) A:

kD

Arrhenius, B:

kD

with vibrational interaction

6


When the Mach number becomes sufficiently high, the collisions between elementary particles are more intense and ionization phenomena can become important. Thus, the aim of Chapter 4 is to describe the various collision processes contributing to create ions and electrons as well as the reactions involving charged species. Applications to air plasmas in shock tube are given including radiative processes. An introductory example of the spatial variation of ionized species behind a straight shock wave in air is presented in Fig. 9.

Fig. 9. Spatial variations of ionized species concentrations behind a straight shock wave in air (Conditions of Fig.6)

It is of course impossible to dissociate the physical and chemical processes described in Chapters 1-4 from the radiation inherently connected to these processes. It is the aim of Chapter 5 to describe various radiative mechanisms in hot gases, particularly in non-equilibrium conditions. Examples are given, showing the importance of radiative fluxes in hypersonic flight and emphasizing the coupling between radiation and aerothermodynamics. An introductory example is given in Fig.10 which represents the experimental evolution of the spontaneous emission of the Δv = 0 band of the electronic transition B2Σ+ ↔ X2Σ+ of CN behind a strong shock wave in the Titan simulated atmosphere [8] ( 92%N 2 , 3%CH 4 , 5%Ar ). It is thus possible to deduce from Fig.10 the intensity profiles of the lines as functions of the wavelength (Fig. 11) at different instants. In the same way, the time evolution of the intensity of different lines may be determined, as represented in Fig. 12 where a strong overshoot of non-equilibrium radiation is clearly visible. From this type of results, vibrational populations, species concentrations and (or) temperatures can be deduced.


7

Fig. 10. Example of streak image behind a straight shock wave ( Δv = 0 , CN band of the Titan mixture CH 4 / N 2 / Ar ) U s = 5560m / s, p1 = 220 Pa

Fig. 11. Experimental spectra of the (Conditions of Fig.10)

Δv = 0 band of CN at two instants behind the shock

8


Fig. 12. Time evolution of the 0-0, 1-1, 2-2 vibrational bands (Conditions of Fig.10)

As stated above, in continuum regime, a shock wave can be represented as a discontinuity, but, in reality, the passage from an upstream state to a downstream one (frozen state) through a shock wave requires a few collisions between elementary particles, so that the “thickness” of the shock wave is of the order of several mean free paths: in this zone, the gas can be considered in strong translational and rotational non-equilibrium. Thus, Chapter 6 deals with methods used for the study of the shock wave itself, in which Euler or Navier-Stokes equations are invalid. These methods essentially consist in directly solving the Boltzmann equation. Distribution functions and profiles of macroscopic quantities can then be obtained inside the shock wave; solutions are presented for pure monatomic and diatomic gases, as well as for gas mixtures, and non-monotonous temperature profiles can be found. An example of temperature profiles obtained for a mixture He/Ar with the Direct Simulation Monte-Carlo Method [9] (DSMC) is presented in Fig.13. When the gaseous medium becomes “rarefied”, the mean free path is lengthening, so that the shock wave can no longer be considered as a discontinuity and presents a non-negligible thickness; then, non-equilibrium phenomena take particular aspects which are examined in Chapter 7: thus, different flow regimes may be defined between the continuum regime and the “free molecular” one, which are likely to be analysed by modified Navier-Stokes equations (slip flow) or purely numerical methods (DSMC); in the same way, specific diagnostic experimental methods are presented in this chapter. In the last chapter (Chapter 8), concrete examples of specific non-equilibrium flows are presented: thus, results of measurements of static pressure, heat flux, standoff distance, phase shift in the dissociated air flow around a cylinder placed in the test section of a free piston shock tunnel (HEG Göttingen) are presented. Moreover, various models for the computation of radiative heat flux in non-equilibrium conditions are presented, completed by a calculation of the peak heat flux undergone by the Huygens probe during its entry into the Titan atmosphere.


(

9

)

Fig. 13. Temperature profiles across a shock wave ⎡ M s = 8, nAr = 0.1⎤ , nHe a ⎣⎢ ⎦⎥ : TAr , a: upstream, b: downstream :T , : THe ,

Finally, we might consider that combustion phenomena induced by shock waves should be included in the present book; the particular specificity of this topics however, as well as the important developments required for its treatment, would be beyond the framework of one single chapter and should be the subject of one another book. To sum up, the main aspects of phenomena related to high temperatures prevailing behind shock waves are presented in the present book, at least the actual knowledge on the matter, which of course remains an active research field.

References 1. Gaydon, A.G., Hurle, I.R.: The Shock Tube in High Temperature Research. Chapman and Hall, London (1963) 2. Park, C.: Nonequilibrium Hypersonic Aerothermodynamics. J.Wiley, New-York (1990) 3. Vincenti, W.G., Krüger, C.H.: Introduction to Physical Gas Dynamics. R.G.Krieger, Florida (1965) 4. Stupochenko, Y.V., Losev, S.A., Osipov, A.I.: Relaxation in Shock Waves. Springer, Berlin (1967) 5. Oertel, H.: Stossrohre. Springer, Wien (1966) 6. Brun, R.: Introduction to Reactive Gas Dynamics. Oxford Univ. Press, Oxford (2009) 7. Belouaggadia, N., Armenise, I., Capitelli, M., Esposito, F., Brun, R.: J. Therm. Heat Transf. 24(4), 684 (2010) 8. Ramjaun, D., Dumitrescu, M.P., Brun, R.: J. Therm. Heat Transf. 13(2), 219 (1999) 9. Bird, G.A.: Rarefied Gas Dynamics, vol. 175. Tokyo Univ. Press, Tokyo (1984)

Chapter 1

Thermodynamic Properties of Gases behind Shock Waves M. Capitelli1,2, D. Bruno2, G. Colonna2, G. D’Ammando1, A. D’Angola3, D. Giordano4, C. Gorse1,2, A. Laricchiuta2, and S. Longo1,2 1

Department of Chemistry, University of Bari (Italy) CNR Institute of Inorganic Methodologies and Plasmas (IMIP) Bari (Italy) 3 Università della Basilicata, Potenza (Italy) 4 ESA ESTEC Aerothermodynamics Section Noordwijk (The Netherlands)

2

1 Introduction The research on high-energy shock wave is a field of large interest including nuclear explosion, hypersonic flows as well as laser forming plasmas. During the relevant interaction a high-temperature, high-pressure plasma is formed, which in some cases can be ascribed to the family of thermal plasmas, characterized by equilibrium between the different degrees of freedom, including chemical and internal ones. Thermal plasmas can be described by equilibrium chemical thermodynamics, in particular statistical thermodynamics is used in this field to get information about the input data (entropy, enthalpy and specific heat of single species). Thermal plasmas are usually characterized by a single temperature for all species, including the vibrationally, rotationally and electronically distributions among the excited states, while dissociation and ionization (Saha) equilibria characterize them. On the other hand thermal plasmas with different temperatures are still accepted in this kind of literature, the different temperatures characterizing the corresponding reservoirs of energy. The internal distributions are still Boltzmann at a given (different) temperature; chemical equilibrium thermodynamics again characterizes the plasma properties even though caution must be exercised in using it. Typical conditions for thermal plasmas are temperatures in the range of 5 000-50 000 K, pressure in the 10-2103 atm. range and ionization degree larger than 10-5. Characterization of thermal plasma flows is obtained by using NS (Navier-Stokes) fluid-dynamic codes; thermodynamics in this case provides to the CFD (computational fluid dynamics) community important input data for the different species as well as the properties of the mixture if the hypothesis of local equilibrium holds in the flow. Moreover NS equations need the transport properties (thermal conductivity, diffusion coefficients, viscosity and electrical conductivity) of the plasma components as well as of the mixture, these quantities determining the heat flux from plasma to solid samples that can be heated during the plasma-material interaction.

12


The experimental determination of both thermodynamic and transport properties of ionized gases is very difficult to be achieved so that one demands to the theory the calculation of these quantities. Statistical thermodynamics is used, as anticipated, to get information about the thermodynamic properties of the high temperature components, while statistical mechanics is used for getting information of transport properties through the Chapman-Enskog solution of the Boltzmann equation[1,2]. The key point in this characterization is the calculation of partition functions of atomic and molecular species as well as of transport cross sections (collision integrals) for the relevant interactions. The knowledge of the partition function, in fact, is the basis for calculating the thermodynamic properties of single species as well as of the mixture, while the transport cross sections are essential ingredients to calculate the transport properties of the system. In both cases we present in this chapter simplified and accurate methods to calculate these two quantities. Concerning the partition function of atomic species we will present essentially two methods, the first one based on the inclusion in the partition function of a complete set of energy levels, subjected to an appropriate cut-off criterion to avoid the divergence. This approach involves the insertion of thousands of electronic energy levels, representing a computational problem when these partition functions must be calculated in the mathematical grid used by CFD. An alternative approach is based on particular grouping of levels such to reproduce the thermodynamic behavior of the multilevel system. As an example the nitrogen atom is reduced to a three-level system composed by the ground state (4S), a second level, which coalesces, with appropriate energy and multiplicity, the two low-lying excited states 2P and 2D, and a third level, which accounts for the huge number of electronically excited states. Two levels of accuracy are also used to characterize the viscosity-type collision integrals of the different atom-atom, atom-ion interactions. In this case in fact accurate transport cross sections can be obtained by averaging the different contributions coming from the numerous potentials arising in the interaction. As an example two nitrogen atoms in the ground state (4S) can interact along four potentials 1,3,5,7 Σ and the transport cross sections are obtained by averaging the different contributions with suitable statistical weights. This procedure becomes prohibitive when the interaction occurs through unknown potentials, a situation met in the interaction between electronically excited states. An alternative is the use of an average potential and in this direction the phenomenological potential, which is actually an improvement of the Lennard-Jones, is a good candidate to get accurate values for transport cross sections. The situation is more complicated when dealing with atom-parent-ion interactions, the diffusion-type transport cross section being governed by the resonant chargeexchange process. In turn these cross sections can be obtained by using the huge number of gerade-ungerade (g-u) potential pairs arising in the interaction (e.g. for N(4S)-N+(3P) the g-u electronic pairs 2,4,6Σgu, 2,4,6Πgu of the molecular ion should be considered). The asymptotic theory could also be alternatively used for getting these data, avoiding the quantum mechanical derivation of the relevant potentials. Partition functions and transport cross sections are the ingredients for the calculation of plasma properties. However problems are met in the derivation of thermodynamic and transport properties of plasmas. In particular we refer to the dependence of thermodynamic properties of thermal plasmas on the cut-off criterion

Thermodynamic Properties of Gases behind Shock Waves

13

used in the calculation of self-consistent partition function. A similar problem arises in the calculation of transport coefficients i.e. the role of electronically excited states in affecting the transport properties of the plasmas. All these concepts will be analyzed in this chapter, which is divided in different sections. Section 2 is devoted to the calculation of partition function of atomic species, either using the complete set of levels or by using a three-level approach. Section 3 deals with the calculation of partition function of diatomic species. Section 4 focuses on the calculation of transport cross sections for interactions involving atomic species by using both multi-potential and phenomenological approaches, also discussing the estimation of inelastic corrections to diffusion-type collision integrals in ion-parent-atom collisions. Section 5 reports examples of thermodynamic properties of thermal plasmas, emphasizing the role of cut-off criteria in affecting the results. Section 6 deals with the treatment of thermodynamics and transport for two-temperature plasmas. Finally Section 7 discusses the role of electronically excited states in affecting transport properties of thermal plasmas.

2 Partition Functions for Atomic Levels. General and Few-Level Approaches The partition function of an atomic system is the product of translational and internal contributions

Qa = Q tr ⋅ Q int

(2.1)

The translational partition function is given by the closed form ⎡ 2π mkT ⎤ , Qtr = ⎢ 2 ⎥⎦ V ⎣ h 32

(2.2)

whereas the internal partition function is expressed as Qint = ∑ g n e − En

kT

,

(2.3)

n

where gn and En represent, in the order, the degeneracy and the energy of the nth level. In the case of atomic hydrogen we sum over the principal quantum number, keeping in mind that 1⎤ ⎡ En = I H ⎢1 − 2 ⎥ ⎣ n ⎦

and

g n = 2n 2

(2.4)

These equations lead to the so-called divergence of partition function, in fact, once the exponential factor converge to e − I H kT , the factor gn diverges as n2. A suitable cutoff criterion is then necessary for the truncation of electronic partition function of atoms. This problem will be discussed in details in Section 5.

14


In general the sum in equation (2.3) includes thousands of levels so that, in practical calculations, few-level approaches have been developed, based on grouping criteria for electronic energy levels. These approaches will be demonstrated in the following for benchmark atomic systems. For atomic hydrogen, in the frame of the two-level approach, the ground state is characterized by EH,0=0 and gH,0=2, while the large number of electronically excited states is reduced to one lumped level, having the degeneracy equal to the sum of degeneracies and the energy equal to the mean value in the excited manifold nHm

nHm

n =2

n=2

EH ,1 = ( g H ,1 ) ∑ g H , n EH , n and g H ,1 = ∑ g H ,n

(2.5) m

The summation is performed up to a maximum number of levels nN . It is straightforward that both the value of energy and its degeneracy factor depend on the m choice of nN . Atomic nitrogen is the case study for the three-level system. The ground state configuration is 4S3/2 having a statistical weight gn = 4 . There are other two low-lying levels, having the same 2s22p3 electronic configuration of the ground state, i.e. 2D5/2,3/2 (EN=2.3839 eV, gN = 10 ) and 2P3/2,1/2 (EN=3.5756 eV, g N = 6 ), grouped to form the first excited level in the three-level model, with energy at EN,1=2.8308 eV and statistical weight g N ,1 = 16 . All other levels are grouped to form the third level, whose energy and degeneracy have been calculated applying equations (2.5), starting the summation from n = 3 to the selected maximum number n Nm . Excited state energies can be calculated in the hydrogen-like approximation or extending available data following the Ritz-Rydberg series. In Fig.1, the internal specific heat of atomic nitrogen considered as a three-level system, for different values of the maximum number of levels actually included, is reported. Note that the curve labeled with nNm = 2 corresponds to neglect the third level and exhibits a well-defined maximum at T~15 000 K. Including an increasing number of excited states in the definition of the higher lumped level, transient bimodal behaviors are found leading, for very high degeneracy of the third level, to the disappearance of the maximum due to the first levels. Note that the results reported in Fig.1 have been obtained by using the following expressions for the energy and degeneracy of electronically excited states En ≅ I N −

Ry n2

and

g n = g core ⋅ 2n 2 = 9 ⋅ 2n 2 ,

(2.6)


15

The accuracy of two- and three-level models has been validated by comparison with partition functions including thousands of energy levels, as shown in Refs[3,4].

Fig. 1. Internal specific heat of atomic nitrogen as a function of temperature for different number of levels included in the highest lumped level.

3 Partition Functions for Diatomic Molecules Closed form for the vibrational and rotational partition functions can be obtained separating the different degrees of freedom. In doing so we should be aware that, summing over the vibrational and rotational ladders, we are considering levels with energy exceeding the dissociation limit. In the most general case, each electronic state is described by its own potential energy curve that can be approximated by an harmonic oscillator only in a small region close to the minimum. As a consequence, the momentum of inertia depends on vibrational state. Moreover, the potential energy curve should be corrected by adding the contribution of the centrifugal force, which depends on the rotational state. In this way the vibrational and rotational states are strictly related and we cannot consider them separately. For this reason the relative motion of molecular nuclei is described by ro–vibrational levels. In this picture, the energy of internal levels of the sth diatomic molecule depend on the electronic state n, vibrational v and rotational j quantum numbers, writing the partition function as

Q

int

=

1

m nsm vsm js ( nv )

∑ σ ∑∑ n = 0 v = 0 j =0

g s ,n ( 2 j + 1) e

−

ε s ,nvj kT

,

(3.1)

16


where we can understand that the vibrational energy depends not only on the vibrational quantum number but also on the electronic state, as well as the rotational energy depends on the electronic and vibrational state. The relevant sums are extended on the available electronic states and the limiting values vsm , jsm represent the maximum vibrational and rotational quantum numbers such that the maximum total energy is below the dissociation limit of the corresponding electronic state. Using this approach, it is not possible to obtain closed form for the partition function and thermodynamic quantities and the sum over the energy levels must be calculated directly, as for the atomic species. Energy of ro–vibrational levels is calculated by semi–empirical formula, with coefficients determined from molecular spectra [5]. The treatment for diatomic molecules follows the method developed by Drellishak et al. [6,7] and by Stupochenko et al.[8] In this method the energy of a molecular state is split into three contributions: the electronic excitation, the vibrational and the rotational energy rot ε s,nvj = ε sel,n + ε svib , nv + ε s , nvj

(3.2)

The vibrational energy associated with the vth vibrational level of the nth electronic state of a diatomic molecule, referred to the ground electronic state, is expressed in analytical form as

ε svib,v hc

= ω0 v − ω0 x0 v 2 + ω0 y0 v 3 + ω0 z0 v 4 + ω0 k0 v 5

(3.3)

The rotational energy for a non–rigid rotor, associated to each vibrational level in a given electronic state, is given by

ε srot, nvj hc

= Bs , nv j ( j + 1) − Ds , nv j 2 ( j + 1)

2

(3.4)

The spectroscopic constants entering equations (3.2)-(3.4) can be found in Refs[9,10,11,12]. Equation (3.3) can be used for calculating the maximum vibrational quantum number of each electronic state, while a more elaborate approach can be used to calculate the maximum rotational quantum number for each vibrational level[9,10,11,12]. The method has been extensively applied to many diatomic molecules existing in planetary atmospheres[9,10,11,12]. A sample of results for N2, N2+ and O2, O2+ has been presented in Fig.2 (a) and (b). In each figure we report the temperature dependence of the internal partition function, Qint, of the reduced internal energy, Eint/RT, and of the reduced specific heat C int p / R . The last two quantities can be written as a function of the logarithmic derivatives of the internal partition functions as follows Eint d ln Qint = RT d ln T

(3.5)


2 int ⎡ d ln Q int ⎤ ⎤ 2 ⎡ d ln Q = 2⎢ ⎥ +T ⎢ ⎥ 2 R ⎣ d ln T ⎦ ⎣ dT ⎦

C int p

17

(3.6)

very low temperature (activation of rotational degree of freedom), reaching asymptotically the value 2 (activation of rotational and vibrational degrees of freedom), that is kept constant from a given high temperature. Quantitatively the same numbers occur for the reduced internal specific heat.

Fig. 2. Partition function, first logarithmic derivative and specific heat as a function of temperature for diatomic species at 1 atm pressure. (left) nitrogen (right) oxygen.

We qualitatively recover these numbers by inspection of the results obtained by the more complicated approach. Looking at the trend of Eint/RT for N2 reported in Fig.2 (a) we can observe that the reduced internal energy presents a value of 1 at very low temperature asymptotically reaching a value of 2 at high temperature. For T>10 000 K the internal energy starts increasing reaching a maximum, after which it starts decreasing up to zero (at very high temperature not reported in the figure). The same trend is observed for the reduced internal specific heat, rapidly converging to zero after a strong maximum. We therefore note that the behavior of Eint/RT and for low and intermediate temperatures (i.e. in the temperature range important C int p / R for the activation of rotational and vibrational degrees of freedom) can be explained by the well-known ho-rr approximation. On the other hand for T>10 000 K strong

18


the internal partition function, neglected in the ho-rr approximation as well as the insertion of finite number of rotational, vibrational and electronic states in our partition function, differently from that obtained in the frame of ho-rr approximation, including an infinite number of rotational and vibrational states. The results for N2+ and O2, O2+ systems, also reported in Fig.2 (a) and (b), present a similar qualitative trend, strong differences appearing in the maximum of internal energy and specific heat due to the different number of electronic states inserted in the relevant partition functions [9,10,11,12]. In the given references the quality of the present results can be appreciated, looking at their comparison with existing accurate calculations as well as similar results for diatomic and polyatomic molecules for high temperature planetary atmospheres (Earth, Mars and Jupiter).

4 Transport Cross Sections and Collision Integrals The heart of the Chapman-Enskog theory lies on some hypotheses about the dynamics of collisions at a microscopic level, that are assumed to be binary, elastic, with isotropic interaction potentials. The dynamical information is contained in the socalled collision integrals which can be classically obtained by performing a threefold integral [2], i.e. integration over inter-particle distance r, leading to the deflection angle ϑ, over the impact parameter b for transport cross sections QA and, finally, over reduced energy γ2=E/kT ∞

⎡

ϑ (b, E ) = π − 2b ∫ ⎢1 − rc

⎣

b2 ϕ ( r ) ⎤ − ⎥ r2 E ⎦

−1/ 2

r −2 dr

(4.1)

∞

Q A ( E ) = 2π ∫ [1 − cos A (ϑ )] b db

(4.2)

0

Ω( A , s ) (T ) =

kT 2πμ

∞

∫Q

A

e − γ γ 2 s +3 d γ 2

(4.3)

0

where rc is the distance of closest approach and ϕ ( r ) is the spherically symmetric interaction potential. (A,s) represents the collision integral order§, related to the momentum, A, of transport cross section. A number of model potentials (inverse power[13], polarization potential, exponential repulsive[14,15], Morse potential[16], Lennard-Jones[17], modified Buckingham[18], Hulburt-Hirschfelder[19], Tang&Toennies[20]) have been proposed in literature, whose parameters could be estimated theoretically or experimentally, including generally a

§

Traditionally orders (1,1) and (2,2) are associated to so-called diffusion- and viscosity-type collision integrals


19

short-range repulsive term and long-range attractive term, and an attractive well, whose depth is related to the strength of the chemical bond. In general for these model potentials dimensionless reduced collision integrals are reported, physically representing the deviation of actual values from the rigid-sphere case Ω( A , s )∗ =

∞

2 Ω( A , s ) kT = Q l e − γ γ 2 s +3 d γ ( A,s ) ∫ Ω rs 2πμ 0

(4.4)

Actually interactions between open-shell chemical species rarely can be described by means of a single potential, due to the large number of molecular states, bound and repulsive, arising in the approaching of colliding species in a defined quantum state, predicted through the rules of momentum addition in different coupling schemes. The collision integral results from a weighted average of the contributions of each state, the statistical weight being the product of spin multiplicity (2s+1) and of the multiplicity due to the axial projection of the orbital angular momentum which, in the case of electronic terms for diatomic molecules, is 1 for Σ states and 2 for all other symmetries (Π, Δ, Φ, …) Ω(avA ,s )∗ =

∑

ωn Ω(nl ,s )∗ ∑ n ωn n

(4.5)

4.1 Phenomenological Approaches for Unknown Colliding Systems Despite the incredible advancement of theoretical chemistry, accurate knowledge of interaction potentials is still a challenging problem. For this reason phenomenological potentials are being reconsidered as a tool to describe in a satisfactory way the average interaction. Large interest in this direction is devoted to the potential energy function developed by Pirani et al.[21,22], which can be considered as an improvement of the Lennard-Jones potential. This potential is able to predict intermolecular interactions in a variety of systems (neutral-neutral and neutral-ion). Fundamental interaction features, i.e. binding energy and equilibrium distance, enter in relevant equations as parameters and their values are determined on the base of correlation formulas of the physical properties of colliding partners (polarizability, charge, number of electrons effective in polarization)[23,24,25,26]. The proposed full-range phenomenological potential, simulating the average interaction, could allow direct evaluation of internally consistent complete sets of collision integrals for different atmospheres. The validity of this approach was demonstrated[27] by comparing, for some benchmark systems, results obtained using the model potential with those calculated with more accurate methods[28,29,30,31]. The interaction potential is modeled with the function m ⎡ m ⎛ re ⎞n n ⎛ re ⎞ ⎤ − ⎜ ⎟ ⎜ ⎟ ⎥ ⎣⎢ n − m ⎝ r ⎠ n − m ⎝ r ⎠ ⎦⎥

ϕ ( r ) = ϕ0 ⎢

(4.6)

20


where n = β + 4 ( r re ) . The parameter m assumes different values depending on the nature of the interaction, i.e. 4 for neutral-ion and 6 for neutral-neutral interactions. The value of β parameter, ranging from 6 to 10 depending on the hardness of interacting electronic distribution densities, could be estimated through the following empirical formula[32] 2

β =6+

5 s1 + s2

(4.7)

where the subscripts 1 and 2 identify the colliding partners. The softness s, entering in equation (4.7), is defined as the cubic root of the polarizability. For open-shell atoms and ions a multiplicative factor, which is the ground state spin multiplicity, should be also considered. Useful bi-dimensional fitting relations have been derived, depending on both temperature and β parameter, allowing the estimation of classical reduced collision integrals up to order (4,4) for any colliding pair[32]. Another approach has been proposed[33], modeling the average interaction with the Lennard-Jones potential, estimating the parameters( σ , ϕ0 ) for the asymmetrical interactions through simplified mixing rules involving well-known parameters for symmetric colliding pairs, i.e. arithmetic mean of collision diameter and a geometric mean of the potential well depth

σ ij =

1 (σ ii + σ jj ) 2

(ϕ0 )ij = ⎡⎣(ϕ0 )ii (ϕ0 ) jj ⎤⎦

1

2

(4.8)

4.2 Resonant Charge Transfer The above considerations completely neglect inelastic channels, in fact cross sections for inelastic processes (internal energy transfer, chemical processes, …) are usually too small to be effective, but in low temperature range[34,35]. However the assumption is not acceptable for resonant processes, i.e. charge-transfer in neutral— parent-ion interactions and excitation-transfer in interaction involving identical atoms in excited states, characterized by high-value cross sections. The simplest theoretical treatment of resonant charge transfer processes relies on the two-state approximation, the atom and its parent ion interacting along two possible molecular states of different symmetry with respect to the interchange of nuclei, i.e. gerade or ungerade parity. If a higher number of (g-u) electronic states arise in the interaction the cross section results from the weighted average of various pairs. A quantum mechanical treatment leads to expression for the charge transfer cross section in terms of phase shifts for gerade and ungerade electronic terms

σ ex ( E ) =

π ∑ ( 2n + 1) sin 2 (ηng −ηnu ) κ2 n

(4.9)


21

with diffusion cross section defined as Q( ) = 1

4π

κ2

∑ ( n + 1) sin (η even

2

n

g n +1

− ηnu ) +

4π

κ2

∑ ( n + 1) sin (η odd

2

n

u n +1

− η ng )

(4.10)

Alternatively a very powerful theoretical tool is represented by the asymptotic approach, formulated by Firsov[36] and developed by Nikitin & Smirnov[37]. In the frame of the semi-classical impact parameter method the cross section for resonant charge transfer process can be written as ∞

∞

+∞

0

0

−∞

σ ex = 2π ∫ Pex ( b ) bdb = 2π ∫ bdb sin 2

∫

Δ ( R) 2h

dt

(4.11)

where Pex represents the charge exchange probability and Δ(R) is the exchange interaction potential, that is the g-u energy splitting Δ ( R ) = ϕu − ϕ g

(4.12)

The exchange interaction potential, in the frame of the asymptotic approach, is expressed in terms of the parameter describing the asymptotic behavior of the radial wave function of the valence electron undergoing the transition between the two ionic cores R (r ) = Ar (1/γ −1) e −γ r

(4.13)

−γ 2 2 representing the electron binding energy and A is the normalization factor,

evaluated tailoring the long-range asymptotic representation of electron wavefunction with accurate results obtained by ab-initio Hartree-Fock approach (see Refs.[36,37] for details). The g-u splitting decreases exponentially at large inter-particle distances, as molecular states become degenerate correlating with the same dissociation limit, thus the integral in equation (4.12) can be divided in two parts, the region of impact parameters b less than a critical value b*, and of high values of the phase ξ, where the probability value rapidly oscillates between 0 and 1 and may be replaced by its average value 1/2, while in the second region, at large b values, the probability falls off to zero ∞

σ ex = π ( b∗ ) + 2π ∫ Pex ( b ) bdb ≈ π ( b∗ ) 1 2

2

b∗

1 2

2

(4.14)

The calculation of cross sections thus reduced to the estimation of the critical impact parameter. The transport cross section is affected by elastic and inelastic scattering, and it can be demonstrated that resonant charge transfer processes do not affect even orders[38]. It can be also shown[38] that, neglecting the elastic contribution, the simplified expression holds

22


Qin = 2σ ex ,

(4.15)

and assuming a form for the dependence of the charge-transfer cross section on the relative velocity the inelastic contribution to the odd-order collision integrals have a closed form[39]. Finally the effective diffusion-type collision integral[40] can be defined as Ω(eff1,1)∗ =

4.3

( Ω ( ) ) + ( Ω( ) ) 1,1 ∗ in

2

1,1 ∗ el

2

(4.16)

Neutral-Neutral Interactions

A wide literature does exist on the exact multi-potential treatment of ground-state interactions relevant to hydrogen and air plasmas. The O(3P)-O(3P) can be a suitable example. The momentum-coupling of atomic electronic terms originates 18 molecular states (2 1,5 Σ +g , 1,5 Σu− , 2 3 Σu+ ,3 Σ −g ,1,3,5 Π gu ,1,5 Δ g , 3 Δ u ) and the collision integral is defined as the weighted average of contributions from each term, equation (4.5). The results by Yun & Mason[41], dated 1962, were based on accurate force laws and the advancement in the theoretical study of electronic structure of O2 molecule has been followed by an increasing accuracy in the calculation of corresponding collision integrals. The commonly adopted approach is the fitting of ab-initio data using model potentials, though quantum mechanical approaches can be also found in literature[42]. It is possible to trace the improvements either moving from a rigid classification of Morse and exponential decaying potentials, for bound and repulsive states respectively[43,44] to functional forms able to accommodate peculiar potential features, as with Hulburt-Hirschfelder potential[33,45] or using re-evaluated ab-initio potential energy curves. Table 1. Diffusion and viscosity-type collision integrals [Å2] for O(3P)-O(3P) interaction. (results of Ref.[33] have been obtained by using the fitting formula given in the corresponding paper)

σ 2rs Ω (1,1)∗

T[K] 2 000 4 000 6 000 8 000 10 000 12 000 14 000 16 000 18 000 20 000

σ 2rs Ω (2,2)∗

41

43

42

33

44

43

42

33

44

5.27 4.39 3.90 3.58 3.34 3.15 3.00

4.69 3.98 3.58 3.30 3.09 2.93 2.79 2.68 2.58 2.49

4.84 4.00 3.57 3.27 3.05 2.87 2.72 2.59 2.48 2.38

4.81 4.07 3.63 3.33 3.11 2.94 2.79 2.67 2.56 2.47

6.01 4.88 4.26 3.84 3.53 3.28 3.08 2.91 2.77 2.64

5.45 4.66 4.21 3.90 3.67 3.49 3.34 3.21 3.10 3.00

5.58 4.67 4.20 3.88 3.64 3.44 3.28 3.14 3.02 2.91

5.57 4.74 4.26 3.94 3.69 3.50 3.34 3.20 3.08 2.98

6.97 5.74 5.06 4.59 4.24 3.97 3.74 3.55 3.38 3.24


23

In Table 1, theoretical diffusion and viscosity-type collision integrals for groundstate oxygen-oxygen interaction by different authors[33,41-44] are reported, considering the temperature interval relevant to the existence of atomic oxygen in an LTE plasma. An excellent agreement is found among the results of Refs.[33,43], both based on analytical fits and the ones by Levin[42] based on quantum mechanically derived potential energy surfaces, with deviations below 4% in the whole temperature range. Instead discrepancies within 20% are observed in the low-temperature region with collision integrals by Ref.[44] obtained following the same approach used in Ref.[43] with up-dated interaction potentials. The relative difference decreases with temperature, reaching about 10% at 20 000 K. It should be noted that different sets show quite similar values (10% at T=2 000 K) for the relative distance from the old results by Yun & Mason[41], this agreement being also due to compensation effects between the contribution coming from the 18 potential curves. The case of O(3P)O(3P) interaction clarifies the critical point of the traditional approach, i.e. the availability of reliable curves for the ensemble of electronic terms. Recently a phenomenological approach has been proposed, overcoming this difficulty by considering a modified Lennard-Jones potential, describing the average interaction. The investigation on the applicability of the proposed methodology can proceed through the analysis of benchmark systems, such as N(4S)-N(4S). According to the Withmer-Wigner rules the interaction occurs along four different potential curves corresponding to 1 Σ +g , 3 Σ u+ , 5 Σ +g , 7 Σ u+ electronic terms. Following the pair-valence theory we can rationalize the increase of the unbound character of the state with spin multiplicity. So while the singlet state is characterized by a strong chemical bond, the septet exhibits a repulsive potential, as can be appreciated in Fig.3 (a), where relevant potential energy curves are reported.

Fig. 3. The interaction potential energy in the N2 system. (left) Potential energy curves for the electronic states correlating with N(4S)-N(4S), (right) detail of averaged potential (full line) and of phenomenological potential (dotted line).

24


In the same figure the curve is shown, resulting from the statistical average of the four potentials. This kind of averaging emphasizes the role of the repulsive states in smoothing the attractive parts of chemical bonds. The curve for the N2 system is compared, in Fig.3 (b), with the one obtained with the phenomenological procedure. The comparison shows that the wells are quite similar, with a depth three orders of magnitude lower than that of the ground singlet state, and located approximately in the same inter-nuclear-distance range. Table 2. Diffusion-type collision integrals [Å2] for N(4S)-N(4S) interaction. 2

σ rs Ω

T[K] 27

500 1 000 2 000 4 000 5 000 6 000 8 000 10 000 15 000 20 000

7.34 6.30 5.42 4.64 4.40 4.21 3.93 3.72 3.36 3.13

5.54 4.82 4.25 3.74 3.58 3.45 3.26 3.11 2.84 2.66

(1,1) ∗

46

42

7.76 6.79 5.25 4.50 4.27 4.09 3.79 3.55 3.12 2.82

7.03 5.96 5.15 4.39 4.14 3.94 3.61 3.37 2.92 2.62

In Table 2, diffusion-type collision integrals, obtained integrating the classical deflection angle on the averaged and phenomenological potentials, are reported. In the same table a comparison with results from literature[42,46], obtained with the standard procedure, i.e. adiabatically averaging the contributions coming from the four different states, is also performed. In particular, collision integrals by Capitelli et al.[46] result from a Morse fitting of experimental potential curves for the bound states and an exponential-repulsive function reproducing an Heitler-London calculation of septet state, while in the low temperature region (T2 000 K). 4.4 Neutral-Ion Interactions The procedure for estimation of elastic collision integrals in the case of neutral-ion interactions is the same already outlined for neutral-neutral collisions, thus characterized by the same drawbacks. Additionally in atom—parent-ion collisions the contribution coming from the resonant charge-transfer channel to odd-order terms should be estimated. Also for this class of interactions the phenomenological approach has been validated for the derivation of viscosity-type and of elastic contribution to diffusion-type collision integrals, considering benchmark systems[27], for example the N(4S)-N+(3P) system interacting along the 12 related electronic states 2,4,6 Σ gu ,2,4,6 Π gu. Table 3. Viscosity-type collision integrals [Å2] for atom-parent ion interactions O(3P)-O+(4S)

(2,2) ∗

2 (2,2) ∗ σ rs Ω

2 σ rs

T[K] 47

500 1 000 2 000 4 000 5 000 6 000 8 000 10 000 15 000 20 000

N(4S)-N+(3P)

9.32 8.64 7.67 6.99 5.91 5.25

Ω

29

46

27

29

46

27

16.41 13.27 10.50 8.33 7.74 7.26 6.48 5.84 4.66 3.87

13.25 11.32 9.55 7.85 7.32 6.90 6.26 5.79 4.99 4.46

18.54 11.65 7.88 6.09 5.72 5.45 5.09 4.83 4.41 4.13

14.78 11.14 8.72 6.94 6.39 5.95 5.26 4.75 3.92 3.41

10.19 8.73 7.40 6.09 5.65 5.29 4.73 4.31 3.61 3.18

15.22 9.58 6.50 5.05 4.74 4.53 4.23 4.02 3.67 3.45

In Table 3, the viscosity-type collision integrals, not affected by the charge transfer process, calculated with the phenomenological potential27 are reported together with data in Refs.[29,46,47]. Collision integrals calculated according to the phenomenological potential show a reasonable agreement with Stallcop et al.[29] and Capitelli et al.[46] results, especially in the temperature range (5 000-20 000 K) in which N and N+ are the major species. The results in Ref.[47] are, on the contrary, higher with maximum relative difference of about 35%. The behavior, in the considered temperature range, of the absolute error of data obtained with the phenomenological approach with respect to the accurate calculations, based on ab-initio potentials for each interaction channel, is the same displayed in Figure 1 of Ref.[31] Levin used the effective potential in the Tang & Toennies form, which is actually a more complex function than the phenomenological potential. However, it should be noted that in Ref.[31] the binding energy and the equilibrium distance, the two basic potential parameters, have been obtained using the

26


methodology outlined above. Same considerations apply to the case of O(3P)-O+(4S) collision, viscosity-type collision integrals being also presented in Table 3. A comparison with data in literature, still gives a satisfactory agreement, especially when compared with data in Refs.[29,46] in the temperature range of interest (5 00020 000 K) and confirming that an effective potential, not directly connected with details of the interacting system in different electronic states, can be used for transport cross section prediction. The inelastic contribution dominates the effective odd-order collision integral, mainly in the high-temperature region, and the estimation proceeds through the knowledge of corresponding resonant charge transfer cross sections. Due to its relevance in affecting the transport properties resonant process has been investigated, both theoretically and experimentally, for a number of systems. The analysis here is limited to the case of N(4S)-N+(3P) interaction. Theoretical cross sections for resonant charge transfer have been obtained by Stallcop & Partridge[29] by a phase-shift approach, based on accurate ab-initio potential energy curves for the molecular ion N2+ calculated at CASSCF level. Two sets of data refer to calculations performed in the framework of the asymptotic theory, the paper by Eletskii et al.[48]also extending to the treatment of highly excited states, and the re-evaluation recently done by Kosarim et al.[49] that critically selects the proper scheme of coupling momenta on the basis of hierarchy of interactions, including also the low-lying excited states. It should be emphasized the satisfactory agreement among different approaches also confirmed by experimental measurements by Belyaev[50]. This reflects on the corresponding inelastic contribution to collision integrals reported in Table 4. Deviations from the accurate quantum results of Ref.[29] of collision integrals based on asymptotic cross sections of Ref.[48] are within 15% and a little larger 18% for results by Kosarim et al[51]. In the same table, collision integrals by Capitelli et al.[46,52,53] have been also reported. The agreement is ∗ were derived by satisfactory but in the case of the oldest Refs.[52,53] where σ rs2 Ω(1,1) in integration of cross sections obtained by using the g-u splitting of the relevant states of the molecular ion. The observed discrepancies could be rationalized considering that the g-u splitting for different pairs were estimated in a range of atom-ion internuclear separations too close to the equilibrium distance of the molecular ion, far from the region relevant for charge-exchange process, thus underestimating the cross sections. 4.5 Charge-Charge Interactions The interaction between charged particles, i.e. ion-ion and electron-ion collision pairs, are modeled with the screened Coulomb potential

ϕ (r) +

zi z j r

e − r λD

(4.17)

zi and zj being the charge of i and j ions, e the electron charge and λD the Debye length.


27

Collision integrals for this potential do exist either in analytical[54] or in tabular form[55], and recently accurate collision integrals by Mason[56] have been fitted by the following equation[57]

( )

6

ln ⎡⎣Ω( l , s )∗ ⎤⎦ = ∑ c j ln j T ∗ j =0

(4.18)

Table 4. Inelastic diffusion-type collision integrals [Å2] for N(4S)-N+(3P) interaction. 2

52

500 1 000 2 000 3 000 4 000 5 000 6 000 7 000 8 000 9 000 10 000 12 000 14 000 15 000 16 000 18 000 20 000 30 000 40 000 50 000

(1,1)∗

σ sr Ω in

T[K]

14.5 14.2 13.9 13.8 13.6 13.4

12.5

53

16.3 16.0 15.8 15.7 15.6 15.3 15.2

29 38.17 34.27 31.39 29.95 29.00 28.30 27.73 27.26 26.86 26.50 26.19 25.65 25.20 25.00 24.82 24.48 24.18 23.05 22.28 21.70

46

40.6 37.5 34.5 32.8 31.6 30.7 30.0 29.4 28.9 28.5 28.1 27.4 26.8 26.5 26.3 25.9 25.5 24.0 23.0 22.3

48

38.88 36.52 34.24 32.94 32.03 31.34 30.78 30.31 29.90 29.55 29.23 28.69 28.23 28.03 27.84 27.50 27.19 26.04 25.23 24.61

49

40.61 38.17 35.81 34.46 33.52 32.80 32.22 31.73 31.31 30.94 30.61 30.05 29.58 29.37 28.82 28.50 28.50 27.30 26.46 25.82

4.6 Electron-Neutral Interactions Collision integrals for electron-neutral interactions are usually calculated by integration of theoretical or experimental differential elastic electron-scattering cross sections, so as to include quantum effects. An illustrative example is represented by the electron elastic scattering by atomic argon, exhibiting a low-energy Ramsauer minimum. This system has been deeply studied and, combining theoretical results in the low-energy region[58] and measured elastic differential cross sections[59,60,61], a wide energy range can be explored, allowing accurate estimation of transport cross sections and high-order collision integrals, displayed in Fig.4 (a). However the critical point still remains the knowledge of the differential cross sections. In fact experimental data are usually available for few collision energy values, also missing extreme values of the scattering angle, and accurate theoretical results are also difficult to be retrieved for all the interactions. On the contrary the integral transport cross sections, elastic term Q(0) and momentum transfer Q(1) are found in literature for a number of systems, allowing the straightforward derivation of

28


diffusion-type collision integrals, but again for the higher order viscosity-type collision integrals the Q(2) is not always readily available. These difficulties could be overcome by using different techniques to estimate this last from basic models or additional information. In Ref.[62] the ratio Q(2)/Q(1) has been determined from the known Q(1)/Q(0) assuming a model angular dependence of the differential cross section. The approach is demonstrated for the e-CO2 interaction, taking the elastic and momentum transfer cross section from Ref.[63], giving a satisfactory agreement with recommended data in literature[64] as shown in Fig.4 (b).

Fig. 4. Diffusion (continuous line) and viscosity-type (dotted lines) collision integrals for electron-neutral interactions, compared with recommended data in Ref.64 (left) e-Ar; (right)e-CO2.

5 Thermodynamic Properties of Thermal Plasmas: The Cut-Off Issue In this chapter we will show the importance of electronic excitation in deriving partition functions and their first and second logarithmic derivatives as well as thermodynamic properties of single atomic species and of plasma mixture. Recent results obtained by using different cut-off criteria are discussed and compared with the so ground state method i.e. by inserting in the electronic partition function only the ground electronic state of the atomic species. A rich literature does exist on the subject, indicating the existence of compensation effects in the calculation of the thermodynamic properties of thermal plasmas. These compensations hide in some cases the role of electronic excitation of atomic species in affecting the thermodynamic properties of thermal plasmas. Results for a case study, i.e. air plasma in a wide range of temperature (500-100 000 K) and pressure (1-1000 bar§), are reported and can be considered representative of many other systems. §

1 atm=1.01325 bar.


29

5.1 Cut-Off Criteria It has been already pointed out the necessity of introducing a suitable cut-off criterion to prevent the divergence of electronic partition functions of atomic (neutral and ionic) species. This section will be focused on the following criteria (a) the groundstate method (GS) (b) the Debye Hückel criteria (c) the Fermi criterion (F). Strong differences are expected especially when a complete set (observed and missing) of electronic levels is used in the calculation. 5.1.1 The Ground State Method The partition function includes only the ground state i.e.

Qej = g e0 ,

(5.1)

involving that the first and second logarithmic derivatives are zero. (5.2) In this case the electronic excitation is completely disregarded. 5.1.2 Debye-Hückel Based Criteria In this case we have two types of approaches, one due to Griem (G)[65] and the other one to Margenau and Lewis (ML)[66]. According to Griem we write the electronic partition function of the jth species as Qej =

Enj max

∑g

nj

e

− Enj kT

,

(5.3)

0

where Enj and gnj represent in the order the energy and the statistical weight of the nth level of the jth species. The sum includes all levels up to a maximum value given by

Enj max = I j − ΔI j , j +1

(5.4)

In turn, the lowering of the ionization potential ΔIj,j+1 is given by e3 ⎛ π ⎞ πζ 3 2 ⎜⎝ kT ⎟⎠

12

ΔI j , j +1 =

(∑

n 2 i =1 i i

z n

)

1

2

(z

j

+ 1)

(5.5)

Note the nth level in this case does not refer to the principal quantum number. Following Margenau and Lewis the electronic partition function is written as nmax

Qej = ∑ g nj e n

− Enj kT

,

(5.6)

30


where nmax is the maximum principal quantum number to be inserted in the partition function. In turn nmax is obtained by assuming that the classical Bohr radius does not exceed the Debye length λD. 2 a0 nmax = λD , Z eff

(5.7)

where Z eff = z + 1 is the effective charge seen by the electronic excited state (z is the charge of the atom/ion, z=0 for neutral and so on) and a0 is the Bohr radius. The two formulations coincide when use is made of hydrogen-like levels, presenting however large differences when the dependence of energy levels on the angular and spin momenta and their coupling are considered. In this last case the partition functions and related properties calculated according to ML method exceed the corresponding G values (see Refs.[67,68]). This point should be taken into account when comparing the well-known Drellishak et al.[69] partition functions based on the ML theory and the corresponding values obtained by the G method. 5.1.3 The Fermi Criterion According to the Fermi criterion[70] an electronic state is considered still bound and therefore to be included in the partition function if his classical Bohr’s radius does not exceed the inter-particle distance. One can therefore write 2 Z eff a0 nmax 1 = 1 3 ⇒ nmax = Z eff n a0 n1 3

,

(5.8)

where n, not to be confused with the principal quantum number, is the particle density [cm-3] linked to the pressure by

p = NkT

(5.9)

Therefore the electronic partition function depends on pressure, in particular a decrease of nmax is to be expected with the increase of pressure. 5.2 The Cut-Off Criteria Based on the Solution of Schrödinger Equation The results reported in the previous sections can be rationalized by solving the Schrödinger equation for atomic systems in particular for hydrogen. The energy levels of the hydrogen atom as well as the degeneracy can be obtained by solving the radial Schrödinger equation −

d ⎛ 2 dR ⎞ ⎡ h 2 l ( l + 1) ⎤ ⎥ R = ER ⎜r ⎟ + ⎢V ( r ) + 2 8π μ r dr ⎝ dr ⎠ ⎣ 8π μ r 2 ⎦ h2 2

2

(5.10)


31

E is the energy, h is the Planck constant, A (0, 1, 2, 3 …) is the azimuthal quantum number, r is the radial coordinate, V(r) is the potential energy and µ is the reduced mass for the electron-proton system. Energy levels from Eq. (2.4) are eigenvalues of Eq. (5.11) for a Coulomb potential V (r ) = −

e2 r

(5.11)

In this section we re-examine the problem by considering not an isolated atom but an atom closed in a spherical box of radius δ, i.e. we numerically solve the radial part of the Schrödinger equation for atomic hydrogen by considering the following boundary conditions[71] R (r = δ ) = 0

(5.12)

This boundary condition is completely different from the one appearing in the analytical solution of the Schrödinger equation i.e. R(r=∞)=0. In Ref.[71] results are reported for δ a0 = 103 and δ a0 = 10 4 values by imposing A=0, i.e. ns levels. More in details Fig.5 reports the non-dimensional energy level values

α=

En IH

(5.13)

obtained with δ a0 = 103 as a function of the number of grid points. In the same figure we have also reported the analytical reduced energy levels i.e. α=En/IH=-1/n2 (called Bohr), which show the well-known asymptotic trend of energy levels to α=0. The numerical results present values which closely follow the analytical ones suddenly becoming positive from n=28 on, clearly showing the existence of two types of energy levels, the negative ones, assimilated to the bound states, and the positive ones, representing the discretized continuum. These last levels strongly increase their energy with n asymptotically going to the analytical energy levels obtained by the particle in cell model, described by the following equations En =

αn =

h2 n2 8meδ 2

En ⎛ π n ⎞ =⎜ ⎟ I H ⎝ δ a0 ⎠

(5.14) 2

,

(5.15)

32


where me is the electron mass (see Fig.5). Similar results have been obtained for δ a0 = 10 4 where the numerical results reproduce the analytical ones up to n=89, suddenly becoming positive for n>89. Again the positive levels asymptotically go toward the corresponding particle in the box values. Different interesting points can be derived from these calculations. The first one is linked to the fact that the partition function of atomic hydrogen, including bound and positive levels, converges since the positive levels present energies increasing with n2. Thus the solution of the Schrödinger equation for hydrogen confined in a box can be considered as a natural cut-off criterion for the partition function. Moreover the principal quantum number at which occurs the sudden onset of the positive energy levels is in satisfactory agreement with the corresponding value obtained by applying the Fermi cut-off, which gives for the conditions above studied (i.e. δ/a0=103 and δ/a0=104) the values of nmax 40 and 120 (see Ref.[71] for details). Finally we want to mention that the equilibrium between bound and continuum levels can be used in the so-called physical picture to recover the well-known Saha’s equation in the chemical picture. All these effects have been obtained by using ns levels. Going beyond this approximation, it can be done by calculating the energy levels with different A values. The energy levels as a function of the azimuthal quantum number for the δ/a0=103 case start to be affected by A only for n>15, the dependence on A becoming dramatic when we consider very small δ/a0 values.

Fig. 5. Reduced energy levels calculated according to numerical δ/a0=103 and analytical Bohr atom solution of the Schrödinger equation. In the same figure are also reported the particle in the box energy levels calculated numerically and analytically.

As far we have presented results of the Schrödinger equation in the box considering a Coulomb potential. Now we solve the same problem accounting for a Debye potential[72,73,74], i.e.


V (r ) = −

e − r λD e r2

33

(5.16)

Fig.6 reports the energy levels for two values of the Debye length, i.e. λD/a0=102 3 and λD/a0=108 for a box δ a0 = 10 . In the same figure, we also report the Bohr’s results as well as the particle in the box values. Inspection of the figure shows again the transition from bound to continuum levels occurring respectively at nmax=11 for λD/a0=102 and nmax=29 for λD/a0=108. The last value coincides with the corresponding value in the presence of Coulomb potential i.e. the Debye length is too high for affecting the results. In both cases the positive levels asymptotically go toward the particle in the box, while the bound levels closely follow the Bohr results. On the other hand the energy levels are strongly affected by the Debye length for λD/a02000 K are Q[O( 3 P)] = 9 Q[O + ( 4 S)] = 4 Q[O2+ (2 P)] = 6

Qint ′ [O( 3 P)] = Qint ′′ [O(3 P)] = 0 Qint ′ [O + ( 4 S)] = Qint ′′ [O + ( 4 S)] = 0 2+ 2 Qint ′ [O ( P)] = Qint ′′ [O2+ (2 P)] = 0

Values for ground state nitrogen species can be found in Ref.[75]. In Fig.7 and Fig.8 electronic partition functions, their first and second logarithmic derivatives and internal specific heats, calculated according to G and F cut-off criteria, are reported as a function of temperature for the three species of oxygen (O, O+ and O2+) and of nitrogen (N, N+ and N2+) at different pressures. In both cases the two cut-off criteria give different values of partition function, these differences propagating on the first and second logarithmic derivatives as well as in the specific heat. In particular the F criterion introduces more levels in the partition function as compared with the G criterion with the consequence of increasing the partition function. Note also that, due to the energy range of electronic levels, the partition function of the different species present the sudden increase in well-defined and not overlapping temperature ranges. This aspect is better evidenced in the first and second logarithmic derivatives, which present well distinct maxima. Note that in the second logarithmic derivative the values calculated according to the Fermi criterion overcome the corresponding Griem values up to the maximum, the opposite occurring in the decreasing region. This behavior is reflected on the specific heat of the single species which in any case presents the trend characteristic of a system containing a finite number of excited levels i.e. the internal specific heat after the maximum asymptotically reaches a zero value (see Section 1). Moreover the large influence of electronic excitation on the specific heat can be understood by reminding that the corresponding values for the ground state are zero independently of temperature and the reduced translational contribution to the specific heat is 5/2.

36


Fig. 7. Partition function, first and second logarithmic derivatives and specific heat as a function of temperature for oxygen species, (left) p=1 bar, (right) p=100 bar

Fig. 8. Partition function, first and second logarithmic derivatives and specific heat as a function of temperature for nitrogen species, (left) p=1 bar, (right) p=100 bar


37

Fig.9 reports the entropy of the oxygen (O and O+) species, as a function of temperature at different pressures, calculated according to the ground state method (translational contribution) and to G and F cut-off criteria (translational and electronic excitation contribution). The differences between the three methods reflect the corresponding trend of the electronic partition function and of the corresponding first logarithmic derivative. The contribution of the electronic states is well evident in the different plots when G and F values start deviating from the corresponding values calculated from the ground state method. In any case the trend of the entropy for the different species monotonically increase passing from ground state to Griem and Fermi methods following the corresponding increase of electronic contribution. Before examining the dependence of total thermodynamic properties on the cut-off criterion we report the corresponding dependence of the molar fractions of the major species of the air plasma. In Fig.10 in particular the temperature dependence of molar fractions of selected species at different pressures is displayed, showing a small dependence although not negligible of the molar fractions on the adopted cut-off criterion.

Fig. 9. Entropy for oxygen species as a function of temperature for different pressures, (left) oxygen atom, (right) oxygen ion.

38


Fig. 10. Molar fractions for air species as a function of temperature for different pressures and different cut-off criteria, (left) oxygen, (right) nitrogen.

Fig. 11. Total entropy for air plasma as a function of temperature for different pressures with ground-state cut-off. The right axis reports percentage relative difference of results obtained with different criteria with respect to the ground-state cut-off.


39

Fig. 12. Specific heat, (left) frozen, (right) total, for air plasma as a function of temperature for different pressures and different cut-off criteria.

Let us now examine the behavior of the thermodynamic properties of air plasma mixture, starting with the total entropy, reported in Fig.11 as a function of temperature at different pressures, calculated with the different methods. In general the Fermi criterion presents larger entropy values compared to G and GS methods, the differences not exceeding 6%. The behavior of the frozen and total specific heats with temperature, for the different pressures, is reported in Fig.12. In this case the differences between the three methods can reach, at high pressure, a factor larger than 2, the values calculated by using the Fermi criterion overcoming in any case the values obtained by G and GS methods. It should be again reminded that the ground state method includes only the translational degree of freedom. More complicated is the situation for the reactive contribution. The dissociation regime is not affected by the cut-off of electronic partition function as confirmed by the results of Fig.10. The ionization regimes are strongly affected by the chosen cut-off criterion. The compensation between F, G and GS methods occurs only in the dissociation and first ionization regimes, while large differences are observed for the second, third and fourth ionization reactions, these differences increasing with pressure. At 1000 bar we observe the largest deviations in the three methods. The differences between G and F values are reduced in the total specific heat due to the partial compensation between frozen and reactive specific heats. On the other hand this compensation tends to disappear when comparing these values with the corresponding ground state values. Only at 1 bar the ground state values are in good agreement with the other two methods, while the differences strongly increase at high pressure.

40


Finally Fig.13 reports the frozen and the total isentropic coefficients for the air plasmas calculated according to the three methods. Once more the effects of the electronic excitation are well evident on the frozen coefficient, being in any case appreciable for the total isentropic coefficient. Note that the frozen isentropic coefficient for the ground state method reaches the constant value of 1.67 in the ionization regime (see Ref.[76]). Results for hydrogen and argon-hydrogen plasmas, recently reported by Sing et al.[77], confirm the above observations.

Fig. 13. Isentropic coefficients, (left) frozen, (right) total, for air plasma as a function of temperature for different pressures and different cut-off criteria.

The results reported in this section do not exhaust the numerous methods used in the literature for the calculation of the electronic partition functions of atomic (neutral and ionized) species. It is worth noting that many researchers calculate the partition function inserting in it only the observed levels[78,79], avoiding in this case any cutoff criterion. Of course this method or other similar methods, which insert in the partition only few levels above the ground state, dramatically underestimate the electronic contribution to the thermodynamic properties of thermal plasmas being not so far from the corresponding values obtained with the ground state method. Note that the famous Gurvich’s tables[80] include in the partition function only the electronic levels coming from the rearrangement of valence electrons i.e. only the low-lying excited states. As an example the partition function of an oxygen atom is obtained by inserting the ground state 3P and the 1D and 1S metastable excited states. On the other hand the well-known JANAF tables, as well as the pioneering calculations of Gordon


41

& McBride[81], include those levels whose energy is lower than I-kT, where I is the ionization potential of the species. Also in this case the electronic partition function is strongly underestimated.

6 Transport of Two-Temperature Plasmas The general equations of multi-temperature plasmas have been discussed in different papers. Simplified formulations are nowadays used to calculate the relevant transport coefficients[82,83,84]. The results depend not only on the transport equations but also on the thermodynamic model used to calculate composition and thermodynamic properties of multi-temperature plasmas. We can obtain in fact a multiplicity of Saha's equations depending on the equilibrium criterion adopted (minimization of Gibbs potential, maximization of entropy) as well as the definition of the different temperature existing in the plasma[85,86]. Let us consider hydrogen plasmas characterized by the dissociation and ionization reactions H 2 R 2H

(6.1)

H R H+ +e

The ionization constants are derived by minimization of Gibbs potential and maximization of entropy subjected to different constraints. •

minimization of Gibbs potential with the constraint Th = Tel ≠ Te ϑ

⎡ nH + ⎤ ( 2π me kTe ) ⎢ ⎥ ne = h3 ⎣ nH ⎦

•

ϑ

32

kTe

(6.2)

ϑ

⎡ Q + (Te ) ⎤ − lH Qe ⎢ H ⎥ e ⎣⎢ QH (Te ) ⎦⎥

kTe

(6.3)

maximization of entropy with the constraint Th ≠ Tel = Te ⎡ nH + ⎤ ( 2π me kTe ) ⎢ ⎥ ne = h3 ⎣ nH ⎦

32

•

ϑ

⎡ Q + (Th ) ⎤ − lH Qe ⎢ H ⎥ e ⎢⎣ QH (Th ) ⎥⎦

minimization of Gibbs potential with the constraint Th ≠ Tel = Te ⎡ nH + ⎤ ( 2π me kTe ) ⎢ ⎥ ne = n h3 ⎣ H ⎦

•

32

ϑ

⎡ Q + (Te ) ⎤ −lH Qe ⎢ H ⎥ e ⎣⎢ QH (Te ) ⎦⎥

kTe

(6.4)

maximization of entropy with the constraint Th = Tel ≠ Te ϑ

⎡ nH + ⎤ ( 2π me kTe ) ⎢ ⎥ ne = h3 ⎣ nH ⎦

32

ϑ

⎡ Q + (Th ) ⎤ − lH Qe ⎢ H ⎥ e ⎣⎢ QH (Th ) ⎦⎥

kTh

(6.5)

42


These equations can be further simplified in the specific case by putting Qe = 2 and

QH + (Th ) = QH + (Te ) = 1 . Comparison of the different equilibrium equations for our case study (i.e. H2 plasmas) shows differences in both the exponential ϑ= Th/Te factor disappearing in the equations which derive from the maximization of entropy as well as on the different temperatures appearing in the partition function and in the exponential term. Again we note that all exponential terms contain the electron temperature with the exception of equation (6.5) which contains the heavy particle temperature. This difference should have strong consequences in the relevant results. It should be noted that equation (6.4) is nowadays the most used equation for the two temperature plasmas based on the kinetic idea that electrons are responsible of the ionization equilibrium as well as of the excitation one, this idea i.e. mixing thermodynamic and kinetic concepts being a little contradictory. The presentation of results taken from Ref.[87] is made either as a function of Th (in the range 2 50010 000 K) keeping constant Te=10 000 K or as a function of Th (in the range 8 00030 000 K) at constant Th=8 000 K, thus meaning that the different plots are made at different 1/ϑ values rather than fixing it. In the first case only the electron density is strongly affected by the choice of the different equations while atomic and molecular hydrogen densities scarcely depend on this choice. In any case electron and ions densities keep values well below the corresponding values for atomic and molecular hydrogen (at Th=Te=10 000 K the electron density is a factor 100 less than atom density). As a consequence only the transport coefficients which depend on the electron density (i.e. total thermal conductivity λ and electrical conductivity σ) will be affected by the choice of the equilibrium constants.

Fig. 14. (left) Electrical conductivity, as a function of gas translation temperature at Te=10 000 K, (right) total thermal conductivity, as a function of the electron temperature at Th=8 000 K, of H2 plasma, corresponding to different Saha equations. (solid line)- Eq. (6.2), (dashed line)-Eq. (6.3), (dashed-dotted line)-Eq. (6.4), (dotted line)-Eq. (6.5)


43

This is indeed the case, as can be appreciated by looking at Fig.14 (a) where we have reported t versus Th at Te=10 000 K for an atmospheric hydrogen plasma calculated inserting in the transport equations the compositions coming from equations (6.2)-(6.5). No appreciable change is observed by using equations (6.2) and (6.3), while the use of equation (6.4) strongly increases the electrical conductivity. On the other hand use of equation (6.5) is such to strongly underestimate the electrical conductivity as a consequence of the exponential factor calculated at Th (see equation (6.5)). All the curves converge to the same values for the one temperature case i.e. Th= Te=10 000 K. Let us consider now the results obtained as a function of Te for Th=8 000 K. In this case the influence of Te can play an important role only when Te ≥ 2Th i.e. from 15 000 K on. This is indeed the case as shown in Fig.14 (b), where we have reported the total thermal conductivity calculated by inserting in the transport equations the different compositions coming from equations (6.2)-(6.5). Inspection of the results shows that equations (6.2) and (6.3) give practically the same results, while an appreciable change is observed when using the maximization of entropy in the form of equation (6.4). Again we note that use of equation (6.5) does not allow the onset of the ionization reaction yielding a total thermal conductivity basically given by the atomic hydrogen contribution. To conclude this section we want to emphasize that transport coefficients of two temperature plasmas strongly depend on the used Saha's equation, a problem still open to discussion despite the numerous researchers contributing to the field[88,89,90,91].

7 Transport Properties of Plasmas: The Role of Electronically Excited States The excited state issue is still debated, being the dependence of collision integrals on the quantum state of chemical species largely unknown. However it has been shown[92,93] the role of electronically excited states in affecting transport properties of equilibrium atomic hydrogen plasmas. The plasma is composed by H(n), H+ and electrons, where n is the principal quantum number of excited atomic hydrogen, i.e. each electronic excited state is considered as an independent species with its own transport cross section. Electronically excited state cross sections dramatically increase with n. In Table 5 the approaches adopted in the derivation of collision integrals for the different kind of interactions occurring in the plasma and involving excited atoms, H(n), with n≤12, are summarized. The diffusion-type collision integrals for the H(n)-H+ interaction are dominated by the inelastic contribution, due to resonant charge-transfer process, that is characterized by a strong increase with n (between n3 and n4)[94], while for H(n)-H(n) and e-H(n) interactions the dependence on n is weaker. In considering collisions between excited atoms, in asymmetric collisional schemes (n≠m), the inelastic channel due to resonant excitation transfer should be properly account in the estimation of diffusion-type collision integrals[92].

44

High Temperature Phenomena in Shock Waves Table 5. Collision integrals for interaction involving excited states in hydrogen plasma.

interaction

H(n)-H+

collision integrals ( 1,1) ∗

σ Ω el 2

σ Ω 2

( 2 , 2 )∗ ( 1,1 ) ∗

σ Ω in 2

H(n)-e H(n)-H(n)

2

σ Ω

σ 2Ω (1,1)∗ σ 2Ω (2,2)∗ ( 1,1) ∗

σ Ω el 2

H(n)-H(m)

(1,1)∗

σ 2Ω (2,2)∗ σ

2

(1,1) ∗ Ω in

adopted approach polarizability model extrapolation formulas of accurate resonant chargeexchange collision integrals for n≤5[94] integration of momentum transfer cross sections[95] extrapolation formulas of accurate collision integrals for n≤5[96] averaging of symmetric interactions scaling of accurate resonant excitation-exchange collision integrals n≤3[92]

The effect on the different contributions of the total thermal conductivity is reported in Fig.15, where it is shown the ratio between the relevant contributions calculated with this kind of cross sections (called abnormal) and with the ground state approximation (called usual). In the last case excited state collision integrals are set equal to the ground state ones. Fig.15(a) reports the ratio between the translational thermal conductivity values calculated with the abnormal cross sections (λha) and the corresponding results calculated with the usual cross sections (λhu) as a function of temperature for different pressures. For T2)[48] have been calculated with the asymptotic approach and results show a n4-n5 dependence on the principal quantum number of the atomic valence shell. Results for transport coefficients of air plasmas, limiting the analysis to the role of low-lying electronically excited states[75], demonstrate a large effect on the internal thermal conductivity value in the whole temperature range explored, thus confirming


47

the importance of the inclusion of higher excited states in transport calculations due to the enormous increase of transport cross sections on n. Future work in this direction should develop a complete set of state-specific collision integrals for electronically excited states, analogously to the database built for hydrogen.

8 Conclusions Simulation of shock wave entails the reliable estimation of thermodynamics and transport properties characterizing the plasma. Fundamental quantities are represented by translational and internal partition functions for thermodynamic properties, including Saha equilibrium constants fully determining the equilibrium plasma composition, and transport cross sections for the description of microscopic collisional dynamics of chemical species. Accurate theoretical methods are presented in this chapter together with alternative approximate approaches, representing valuable tools when a rigorous treatment is hindered by a high computational load or by lack in the knowledge of the fundamental ab-initio structural information about chemical systems. This is the case of few-level approaches for internal partition function of atoms, reducing the huge number of levels to be included in the summation by adopting a suitable grouping criterion, or the use of phenomenological potential for the average interaction in the calculation of collision integrals of colliding systems, whose multi-surface interaction potential is unknown. The central point in the development of accurate theoretical thermodynamic methods is the definition of physically sounded criteria for truncation of the internal partition function for atomic species, affected by divergence. For thermal plasmas actually accurate values of thermodynamic and transport coefficients, obtained by using sophisticated approaches, are nowadays available and in Appendix A recommended values for air plasmas[104] are tabulated. These data for atmospheric air plasmas are in good in agreement with corresponding ones reported by Boulos et al.[105], even though this comparison suffers to some extent of the compensation effects already pointed out in this chapter. Improvements of transport properties of air plasmas can be achieved from a more complete treatment of transport cross sections of electronically excited states, as done for atomic hydrogen plasma. Insertion of electronically excited states represents an important issue also in the calculation of the transport properties of multi-temperature air plasmas[106]. Finally we want to stress that the bulk of present results has been obtained in the assumption of ideal plasmas, whose validity can be questioned at very high pressure and temperature regimes[107,108,109]. Acknowledgments. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement n° 242311.

48


Appendix: Thermodynamic and Transport Properties of Air Plasmas Table A.1. Air plasma density, ρ [kg m-3], as a function of temperature for different pressures T [K] 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

pressure [atm]

10-2

10-1

1

10

102

0.03465120 0.01732560 0.01155040 0.00866281 0.00693025 0.00577521 0.00495018 0.00433140 0.00385014 0.00346512 0.00231006 0.00173001 0.00134653 0.00101631 0.00083214 0.00071898 0.00062014 0.00051192 0.00031352 0.00024800 0.00021184 0.00017757 0.00014036 0.00010536 0.00008212 0.00007001 0.00006312 0.00005830 0.00005440 0.00005104 0.00004792 0.00004477 0.00004127 0.00003329 0.00002692 0.00002318 0.00002093 0.00001922 0.00001478 0.00001131 0.00000954 0.00000792

0.34651200 0.17325600 0.11550400 0.08662810 0.06930250 0.05775200 0.04950180 0.04331400 0.03850140 0.03465120 0.02310080 0.01731750 0.01372700 0.01086230 0.00859941 0.00729257 0.00637863 0.00555825 0.00380173 0.00260883 0.00216573 0.00187718 0.00160975 0.00133530 0.00106728 0.00084792 0.00070257 0.00061502 0.00055905 0.00051886 0.00048632 0.00045856 0.00043317 0.00038244 0.00032463 0.00026839 0.00022758 0.00020155 0.00016333 0.00012852 0.00010172 0.00008657

3.46512000 1.73256000 1.15504000 0.86628100 0.69302500 0.57752000 0.49501800 0.43314000 0.38501400 0.34651200 0.23100800 0.17323100 0.13817400 0.11295300 0.09180140 0.07569730 0.06527960 0.05761850 0.04419050 0.03105880 0.02295250 0.01933550 0.01696560 0.01489580 0.01288570 0.01091190 0.00908662 0.00757016 0.00644765 0.00567209 0.00513651 0.00474947 0.00444922 0.00397611 0.00358058 0.00317687 0.00274892 0.00235249 0.00173932 0.00143504 0.00117183 0.00094912

34.6512000 17.3256000 11.5504000 8.66281000 6.93025000 5.77520000 4.95018000 4.33140000 3.85014000 3.46512000 2.31008000 1.73248000 1.38468000 1.14655000 0.96185800 0.80757500 0.68592900 0.59777100 0.47461300 0.37068700 0.27559000 0.21114100 0.17719800 0.15616600 0.13942700 0.12427700 0.10986700 0.09613230 0.08339210 0.07220580 0.06293000 0.05560990 0.04999950 0.04238760 0.03750020 0.03387140 0.03078790 0.02779030 0.02052750 0.01575240 0.01306980 0.01103000

346.512000 173.256000 115.504000 86.6281000 69.3025000 57.7520000 49.5018000 43.3140000 38.5014000 34.6512000 23.1008000 17.3254000 13.8562000 11.5231000 9.80484000 8.43599000 7.29747000 6.35892000 5.01336000 4.08181000 3.29544000 2.59484000 2.04911000 1.69664000 1.47733000 1.32233000 1.19611000 1.08400000 0.98021800 0.88296600 0.79208000 0.70894000 0.63478700 0.51540300 0.43155300 0.37389700 0.33286300 0.30155000 0.24163100 0.18981200 0.15005200 0.12422300


49

Table A.2. Air plasma enthalpy, H [kJ kg-1], as a function of temperature for different pressures T [K]

pressure [atm]

10-2 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

100.958 201.916 303.240 404.808 508.248 612.579 720.163 829.321 941.415 1055.425 1654.368 2323.130 3498.021 6070.441 7660.881 8698.956 10577.950 15049.494 33236.326 40446.037 44739.726 53430.686 72082.461 103389.677 135571.933 154883.548 164365.813 169871.874 174122.732 178361.867 183959.618 193464.910 210984.575 279812.636 362905.544 419073.559 449704.188 473912.266 628974.590 849979.015 975976.574 1200753.306

10-1 100.958 201.916 303.240 404.808 508.248 612.579 720.163 829.321 941.415 1055.425 1654.259 2305.327 3152.698 4804.604 7023.943 8340.692 9510.274 11475.684 22160.691 36908.484 42579.756 47098.151 54698.867 68354.953 90261.128 117569.949 142041.377 158386.225 168220.181 174593.747 179537.520 184168.540 189574.536 209290.177 254714.026 324232.491 390543.029 436641.372 509759.647 666287.352 863540.060 994645.516

1 100.958 201.916 303.240 404.808 508.248 612.579 720.163 829.321 941.415 1055.425 1654.221 2299.679 3036.460 4094.762 5789.172 7620.688 8929.649 10175.724 14960.316 26338.112 38411.294 44274.573 48741.888 54759.995 64040.455 77801.211 96334.081 117805.248 138488.040 155176.206 167288.618 175923.474 182447.346 193424.717 207299.354 232796.547 276868.060 334847.704 456654.276 533201.540 662213.596 840160.969

10 100.958 201.916 303.240 404.808 508.248 612.579 720.163 829.321 941.415 1055.425 1654.213 2297.896 2998.996 3829.926 4945.197 6422.216 7983.307 9328.274 12156.206 17495.214 27273.108 38054.914 44957.858 49658.797 54531.395 60800.710 69155.888 80237.510 93874.755 109816.833 126631.859 142712.940 156853.857 178182.253 192937.792 205756.099 221105.745 243944.321 345876.572 456846.869 537790.273 634877.978

102 100.958 201.916 303.240 404.808 508.248 612.579 720.163 829.321 941.415 1055.425 1654.209 2297.331 2987.081 3742.003 4608.160 5646.193 6863.672 8174.183 10731.689 13731.197 18496.571 25983.993 35217.812 43303.235 49186.526 53985.855 58847.149 64370.794 71103.168 79256.961 88659.677 99586.806 111679.541 137192.225 161050.705 181025.090 197503.079 212326.863 257609.617 335320.393 431910.288 518237.917

50


Table A.3. Air plasma entropy, S [J K-1 kg-1], as a function of temperature for different pressures T [K]

pressure [atm]

10-2 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

7319.489 7819.896 8230.788 8522.692 8753.887 8943.382 9109.891 9254.838 9387.601 9506.925 9991.966 10374.514 10890.202 11823.314 12319.994 12596.448 13035.602 13970.891 17279.175 18411.453 18981.905 19997.245 21949.676 24924.062 27728.255 29280.176 29985.485 30366.400 30640.886 30897.830 31217.116 31729.496 32625.472 35888.245 39505.807 41761.739 42900.465 43734.894 48435.833 54368.947 57340.319 62046.283

10-1 6654.542 7154.948 7565.841 7857.744 8088.939 8278.434 8444.943 8589.890 8722.654 8841.977 9326.934 9700.062 10075.241 10670.855 11357.156 11710.856 11985.636 12397.366 14313.497 16602.881 17367.726 17897.942 18693.978 19988.411 21886.790 24066.857 25880.286 27009.256 27644.687 28032.175 28314.784 28565.615 28842.949 29775.056 31739.000 34514.011 36971.663 38564.824 40818.914 44955.346 49613.580 52378.961

1 5989.594 6490.001 6900.893 7192.797 7423.992 7613.486 7779.995 7924.943 8057.706 8177.030 8661.945 9032.100 9359.926 9742.725 10262.768 10753.000 11062.363 11324.373 12183.588 13923.622 15545.002 16240.095 16709.937 17280.850 18084.003 19178.810 20543.958 22017.125 23346.350 24354.951 25045.605 25512.308 25847.377 26369.722 26971.198 27984.194 29607.500 31599.371 35371.510 37411.486 40423.931 44166.135

10 5324.647 5825.053 6235.946 6527.849 6759.044 6948.539 7115.048 7259.995 7392.758 7512.082 7996.997 8366.231 8678.776 8980.393 9322.957 9716.390 10084.536 10368.275 10880.865 11696.161 12995.618 14268.489 14999.085 15447.199 15869.437 16367.960 16981.410 17738.425 18607.395 19562.656 20512.724 21372.442 22090.143 23099.790 23738.277 24248.271 24813.824 25594.955 28690.005 31642.243 33539.827 35565.736

102 4659.699 5160.105 5570.998 5862.902 6094.097 6283.591 6450.100 6595.047 6727.811 6847.135 7332.050 7700.949 8008.679 8283.207 8549.864 8826.319 9112.905 9388.899 9855.392 10315.689 10948.398 11827.040 12799.466 13571.512 14083.893 14466.776 14824.119 15199.842 15627.022 16111.393 16634.408 17209.590 17813.327 18995.512 20006.749 20788.006 21386.844 21889.511 23256.962 25282.578 27513.263 29303.288


51

Table A.4. Air plasma specific heat, cp [J K-1 kg-1], as a function of temperature for different pressures T [K] 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

pressure [atm]

10-2

10-1

1

10

102

1014.520 1024.986 1035.496 1046.007 1056.632 1067.303 1078.075 1089.129 1100.460 1112.283 1197.811 1576.336 3852.955 4780.056 2224.724 2521.479 5381.352 13324.183 14199.251 3986.872 5566.082 12436.343 25858.943 35022.360 25917.821 12890.191 6447.673 4484.200 4539.567 5836.254 8580.667 13520.980 21472.278 41635.856 38583.471 19648.378 11861.544 14464.231 49705.790 29666.227 30732.499 55626.641

1007.379 1019.193 1031.067 1042.975 1055.002 1067.108 1079.354 1091.868 1104.679 1117.918 1201.673 1414.532 2310.936 4396.216 3534.237 2304.186 2819.383 5163.011 16723.042 9129.795 4161.324 5550.852 9990.580 17849.586 26178.366 27101.316 19688.604 11760.530 7201.355 5381.051 5132.901 5895.724 7616.617 15119.992 28861.573 37528.527 28695.069 17113.209 18594.428 42697.574 31126.417 26315.412

999.692 1012.905 1026.177 1039.520 1052.987 1066.540 1080.253 1094.242 1108.478 1123.082 1207.825 1353.060 1732.113 2736.026 3782.774 3108.601 2493.950 2899.424 7450.140 14323.441 8416.463 4507.565 4871.495 7112.580 11059.886 16452.059 21274.323 22257.378 18702.803 13514.762 9384.078 6976.605 5932.350 6368.178 9512.442 16242.392 25924.269 31322.438 16157.240 18627.342 33526.227 33745.157

991.135 1005.712 1020.358 1035.057 1049.905 1064.828 1079.881 1095.180 1110.624 1126.371 1212.330 1326.512 1523.123 1926.976 2627.059 3128.177 2877.939 2605.873 3634.367 7355.384 11594.896 9193.429 5453.854 4465.663 5095.517 6677.107 9130.361 12337.200 15649.028 17795.560 17691.110 15504.099 12482.605 7927.448 6462.797 7012.757 9149.809 13188.905 25600.297 17641.543 16552.424 24246.007

980.921 996.771 1012.707 1028.671 1044.764 1060.917 1077.178 1093.577 1110.069 1126.802 1214.302 1314.418 1443.388 1633.423 1925.484 2293.618 2541.829 2543.750 2651.560 3674.722 5921.948 8779.046 9341.867 7153.383 5214.909 4555.560 4774.278 5541.464 6758.608 8396.067 10338.294 12268.423 13669.132 13373.046 10341.591 7963.109 7124.192 7395.310 11542.517 19040.416 18912.212 15715.859

52


Table A.5. Air plasma viscosity µ, 104 [kg m-1 s-1], as a function of temperature for different pressure T [K]

pressure [atm]

10-2 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

0.070478 0.130294 0.180187 0.224078 0.263933 0.300875 0.335592 0.368557 0.400092 0.430435 0.569453 0.695858 0.836173 1.037112 1.198476 1.330414 1.466503 1.624842 1.879189 2.062503 2.146783 1.849861 1.204350 0.606252 0.260733 0.118569 0.070457 0.056345 0.053812 0.056137 0.059585 0.060515 0.055158 0.033235 0.019890 0.014835 0.013904 0.014255 0.011272 0.007566 0.007388 0.006370

10-1 0.070478 0.130294 0.180187 0.224078 0.263933 0.300875 0.335592 0.368557 0.400092 0.430435 0.569453 0.695156 0.820403 0.981896 1.171143 1.319642 1.450967 1.587976 1.903264 2.106463 2.280353 2.346231 2.115625 1.591983 1.015144 0.574555 0.312060 0.182016 0.124785 0.103038 0.095507 0.096062 0.098504 0.095813 0.073339 0.048236 0.033162 0.026430 0.024102 0.018756 0.013457 0.012461

1 0.070478 0.130294 0.180187 0.224078 0.263933 0.300875 0.335592 0.368557 0.400092 0.430435 0.569447 0.694933 0.814916 0.946145 1.111411 1.285015 1.431124 1.563677 1.849121 2.153168 2.343793 2.508764 2.603900 2.508539 2.167383 1.675900 1.187453 0.794995 0.523175 0.355312 0.260472 0.211367 0.188537 0.174879 0.175703 0.159413 0.125436 0.091633 0.051352 0.044750 0.036925 0.027489

10 0.070478 0.130294 0.180187 0.224078 0.263933 0.300875 0.335592 0.368557 0.400092 0.430435 0.569447 0.694864 0.813126 0.931844 1.064314 1.218951 1.379692 1.526672 1.796698 2.092230 2.390949 2.595010 2.756634 2.891139 2.935977 2.828907 2.553361 2.161841 1.736515 1.349495 1.033251 0.794597 0.625603 0.438867 0.367527 0.345225 0.337274 0.313844 0.194299 0.121025 0.097066 0.083487

102 0.070478 0.130294 0.180187 0.224078 0.263933 0.300875 0.335592 0.368557 0.400092 0.430435 0.569447 0.694843 0.812549 0.926965 1.043886 1.170230 1.308917 1.454556 1.735612 2.007099 2.301107 2.604812 2.850717 3.029418 3.182606 3.309411 3.380286 3.363595 3.241351 3.018712 2.720730 2.390710 2.066136 1.519453 1.151920 0.936084 0.821281 0.763227 0.676374 0.503380 0.347650 0.264329


53

Table A.6. Air plasma total thermal conductivity, λ [W m-1 K-1], as a function of temperature for different pressures T [K]

pressure [atm]

10-2 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

0.010347 0.019126 0.026511 0.033244 0.039759 0.046247 0.052733 0.059184 0.065562 0.071851 0.101662 0.169482 0.621638 0.774534 0.347090 0.550411 1.475929 3.575126 3.343822 0.890515 1.040140 1.719839 2.775433 3.267816 2.388875 1.473512 1.119406 1.062241 1.116666 1.224217 1.357653 1.504522 1.654833 1.934224 2.165996 2.429127 2.774950 3.173694 4.115690 5.065981 6.350769 7.559955

10-1 0.010347 0.019126 0.026511 0.033244 0.039759 0.046247 0.052733 0.059184 0.065562 0.071851 0.101453 0.145464 0.334716 0.805333 0.594779 0.418192 0.694294 1.543614 4.654813 2.361559 1.039848 1.196090 1.717283 2.454675 3.107835 3.109148 2.517998 1.983151 1.742015 1.713820 1.788025 1.929433 2.104051 2.504460 2.914643 3.282035 3.610524 3.987431 5.267572 6.546212 7.859838 9.532709

1

10

102

0.010347 0.019126 0.026511 0.033244 0.039759 0.046247 0.052733 0.059184 0.065562 0.071851 0.101387 0.137798 0.222197 0.483524 0.750142 0.583697 0.504607 0.750090 2.467025 4.356947 2.423929 1.317651 1.364111 1.750623 2.289554 2.889072 3.360348 3.486108 3.283431 3.000136 2.829727 2.812017 2.910924 3.306078 3.859325 4.459614 5.037119 5.548666 6.849608 8.650775 10.488277 12.282686

0.010347 0.019126 0.026511 0.033244 0.039759 0.046247 0.052733 0.059184 0.065562 0.071851 0.101366 0.135367 0.184226 0.299898 0.511206 0.667127 0.618947 0.584469 1.141813 2.681341 4.022729 2.990622 1.847529 1.657301 1.927490 2.389902 2.943731 3.519161 4.029105 4.380553 4.549753 4.598192 4.628028 4.875871 5.430368 6.210712 7.137552 8.092531 9.994419 11.869134 14.389841 17.134915

0.010347 0.019126 0.026511 0.033244 0.039759 0.046247 0.052733 0.059184 0.065562 0.071851 0.101360 0.134598 0.171972 0.231259 0.332557 0.469773 0.581421 0.620108 0.723585 1.297034 2.424558 3.608618 3.620807 2.797966 2.313369 2.369820 2.771441 3.368757 4.064296 4.783103 5.466364 6.068133 6.582626 7.447319 8.318966 9.371899 10.663627 12.161004 15.865971 18.572724 21.436432 25.063154

54


Table A.7. Air plasma electrical conductivity, σ [S m-1], as a function of temperature for different pressures T [K]

pressure [atm]

10-2 500 1 000 2 000 3 000 4 000 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 21 000 22 000 23 000 24 000 25 000 26 000 27 000 28 000 29 000 30 000 32 000 34 000 36 000 38 000 40 000 42 000 44 000 46 000 48 000 50 000

10-1

1

102

10

0

0

0

0

1.0632·10-18 1.0016·10-5 0.17 8.43 68.54 350.45 1010.99 1631.16 2199.05 2741.77 3241.13 3676.99 4067.29 4441.00 4812.06 5182.68 5542.91 5860.13 6063.66 6076.90 5930.40 5772.09 5708.86 5753.60 5879.75 6061.33 6277.30 6510.88 6747.01 6969.80 7302.48 7418.15 7446.47 7573.16 7828.16 8162.27 8508.67 8798.67 8990.36 9113.16

2.9968⋅10-19 3.1574⋅10-6 0.064 4.75 41.46 175.07 681.92 1493.24 2260.52 2986.86 3685.75 4345.01 4940.96 5468.36 5949.25 6408.06 6857.69 7301.67 7729.61 8115.27 8408.03 8545.59 8507.50 8360.64 8215.79 8149.32 8179.25 8293.27 8471.19 8695.07 9210.38 9719.56 10104.20 10296.50 10370.90 10481.60 10704.70 11028.70 11405.00 11776.20

5.4226⋅10-20 9.7366⋅10-7 0.021 2.35 24.39 101.61 318.76 999.82 1992.66 2977.39 3934.26 4869.85 5772.02 6620.13 7394.90 8091.58 8725.14 9317.21 9883.90 10433.60 10966.20 11466.30 11905.10 12241.80 12437.00 12475.30 12391.80 12258.80 12148.30 12104.40 12255.60 12660.70 13211.00 13789.90 14301.60 14670.50 14880.00 15013.50 15176.30 15424.70

6.1694⋅10-21 2.5908⋅10-7 0.0063 0.88 12.66 59.44 171.88 443.79 1182.84 2360.71 3618.40 4896.25 6175.89 7441.91 8671.96 9846.78 10941.10 11945.90 12863.70 13707.20 14492.80 15235.00 15942.50 16616.10 17248.10 17821.60 18310.10 18686.40 18930.50 19044.60 18984.40 18836.10 18871.70 19155.90 19649.30 20245.30 20870.90 21448.80 21920.20 22265.60

0 6.2623⋅10-22 4.2388⋅10-8 0.0015 0.25 4.73 28.88 94.12 225.25 496.76 1134.63 2309.08 3821.74 5454.82 7155.50 8903.00 10668.50 12421.60 14141.20 15790.10 17347.00 18802.70 20153.50 21404.90 22566.60 23650.10 24665.60 25619.20 26514.50 27347.40 28108.60 29367.60 30181.30 30571.90 30683.90 30715.80 30838.00 31125.30 31575.30 32149.40 32795.80

References 1. Ferziger, J.H., Kaper, H.G.: Mathematical theory of transport processes in gases. NorthHolland, Amsterdam (1972) 2. Hirschfelder, J., Curtiss, C., Bird, R.: Molecular theory of gases and liquids, p. 525. Wiley, New York (1964) 3. Colonna, G., Capitelli, M.: Spectrochimica Acta B 64, 863 (2009)


55

4. D’Ammando, G., Colonna, G., Pietanza, L.D., Capitelli, M.: Spectrochimica Acta B 65, 603 (2010) 5. http://webbook.nist.gov/ 6. Drellishak, K.S., Aeschliman, D.P., Cambel, A.B.: Tables of thermodynamic properties of argon, nitrogen and oxygen plasmas. Aedc-tdr 64-12 (1964) 7. Drellishak, K.S., Aeschliman, D.P., Cambel, A.B.: Physics of Fluids 8, 1590 (1965) 8. Stupochenko, E.V., Stakhenov, I.P., Samuilov, E.V., Pleshanov, A.S., Rozhdestvenskii, I.B.: American Rocket Society Journal Supplement 30, 98 (1960) 9. Capitelli, M., Colonna, G., Giordano, D., Marraffa, L., Casavola, A., Minelli, P., Pagano, D., Pietanza, L.D., Taccogna, F.: Tables of internal partition functions and thermodynamic properties of high-temperature Mars atmosphere species from 50 to 50 000 K. In: Giordano, D., Warmbein, B. (eds.) ESA STR–246. ESA Publication Division (2005) 10. Capitelli, M., Colonna, G., Giordano, D., Marraffa, L., Casavola, A., Minelli, P., Pagano, D., Pietanza, L.D., Taccogna, F.: Journal of Spacecrafts and Rockets 42, 980 (2005) 11. Pagano, D., Casavola, A., Pietanza, L.D., Capitelli, M., Colonna, G., Giordano, D., Marraffa, L.: Internal partition functions and thermodynamic properties of hightemperature Jupiter-atmosphere species from 50 to 50 000 K. In: Giordano, D., Fletcher, K. (eds.) ESA STR–257. ESA Communication Production Office (2009) 12. Pagano, D., Casavola, A., Pietanza, L.D., Colonna, G., Giordano, D., Capitelli, M.: Journal of Thermophysics and Heat Transfer 22, 434 (2008) 13. Kihara, T., Taylor, M.H., Hirschfelder, J.O.: Physics of Fluids 3, 715 (1960) 14. Monchik, L.: Physics of Fluids 2, 695 (1959) 15. Kalinin, A.P., Dubrovitskii, D.Y.: High Temperature 38, 848 (2000) 16. Smith, F.J., Munn, R.J.: Journal of Chemical Physics 41, 3560 (1964) 17. Neufeld, P.D., Janzen, A.R., Aziz, R.A.: Journal of Chemical Physics 57, 1100 (1972) 18. Mason, E.A.: Journal of Chemical Physics 22, 169 (1954) 19. Rainwater, J., Holland, P., Biolsi, L.: Journal of Chemical Physics 77, 434 (1982) 20. Tang, K.T., Toennies, J.P.: Journal of Chemical Physics 118, 4976 (2003) 21. Pirani, F., Maciel, G., Cappelletti, D., Aquilanti, V.: International Reviews in Physical Chemistry 25, 165 (2006) 22. Pirani, F., Albertí, M., Castro, A., Teixidor, M.M., Cappelletti, D.: Chemical Physics Letters 394, 37 (2004) 23. Liuti, G., Pirani, F.: Chemical Physics Letters 122, 245 (1985) 24. Cambi, R., Cappelletti, D., Liuti, G., Pirani, F.: Journal Chemical Physics 95, 1852 (1991) 25. Cappelletti, D., Liuti, G., Pirani, F.: Chemical Physics Letters 183, 297 (1991) 26. Aquilanti, V., Cappelletti, D., Piran, F.: Chemical Physics 209, 299 (1996) 27. Capitelli, M., Cappelletti, D., Colonna, G., Gorse, C., Laricchiuta, A., Liuti, G., Longo, S., Pirani, F.: Chemical Physics 338, 62 (2007) 28. Stallcop, J., Partridge, H., Pradhan, A., Levin, E.: Journal of Thermophysics and Heat Transfer 14, 480 (2000) 29. Stallcop, J., Partridge, H., Levin, E.: Journal of Chemical Physics 95, 6429 (1991) 30. Stallcop, J., Partridge, H., Levin, E.: Physical Review A 64, 0427221 (2001) 31. Levin, E., Wright, M.: Journal of Thermophysics and Heat Transfer 18, 143 (2004) 32. Laricchiuta, A., Colonna, G., Bruno, D., Celiberto, R., Gorse, C., Pirani, F., Capitelli, M.: Chemical Physics Letters 445, 133 (2007) 33. André, P., Bussiére, W., Rochette, D.: Plasma Chemistry and Plasma Processing 27, 381 (2007)

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34. Viehland, L.A., Dickinson, A.S., Maclagan, R.G.A.R.: Chemical Physics 211, 1 (1996) 35. Maclagan, R.G.A.R., Viehland, L.A., Dickinson, A.S.: Journal of Physics B 32, 4947 (1999) 36. Firsov, O.B.: Journal of Experimental and Theoretical Physics 21, 1001 (1951) (in Russian) 37. Nikitin, E.E., Smirnov, B.M.: Soviet Physics Uspekhi 21, 95 (1978) 38. Mason, E.A., Vanderslice, J.T., Yos, J.M.: Physics of Fluids 2, 688 (1959) 39. Devoto, R.S.: Physics of Fluids 10, 354 (1967) 40. Murphy, A.B.: Plasma Chemistry and Plasma Processing 15, 279 (1995) 41. Yun, K.S., Mason, E.A.: Physics of Fluids 5, 380 (1962) 42. Levin, E., Partridge, H., Stallcop, J.R.: Journal of Thermophysics and Heat Transfer 4, 469 (1990) 43. Capitelli, M., Ficocelli, E.: Journal of Physics B 5, 2066 (1972) 44. Laricchiuta, A., Bruno, D., Capitelli, M., Celiberto, R., Gorse, G., Pintus, G.: Chemical Physics 344, 13 (2008) 45. Sourd, B., Aubreton, J., Elchinger, M.F., Labrot, M., Michon, U.: Journal of Physics D 39, 1105 (2006) 46. Capitelli, M., Gorse, C., Longo, S., Giordano, D.: Journal of Thermophysics and Heat Transfer 14, 259 (2000) 47. Gupta, R.N., Yos, J.M., Thompson, R.A., Lee, K.P.: NASA Report RP-1232 (1990) 48. Eletskii, A.V., Capitelli, M., Celiberto, R., Laricchiuta, A.: Physical Review A 69, 042718 (2004) 49. Kosarim, A., Smirnov, B., Capitelli, M., Celiberto, R., Laricchiuta, A.: Physical Review A 74, 0627071 (2006) 50. Belyaev, Y.N., Brezhnev, B.G., Erastov, E.M.: Soviet Physics JEPT 27, 924 (1968) 51. Kosarim, A., Smirnov, B.: Journal of Experimental and Theoretical Physics 101, 611 (2005) 52. Capitelli, M., Devoto, R.S.: Physics of Fluids 16, 1835 (1973) 53. Capitelli, M.: Journal of Plasma Physics 14, 365 (1975) 54. Liboff, R.L.: Physics of Fluids 2, 40 (1959) 55. Hahn, H.S., Mason, E.A., Smith, F.J.: Physics of Fluids 14, 278 (1971) 56. Mason, E.A., Munn, R.J., Smith, F.J.: Physics of Fluids 10, 1827 (1967) 57. D’Angola, A., Colonna, G., Gorse, C., Capitelli, M.: European Physical Journal D 46, 129 (2008) 58. Bell, K.L., Scott, N.S., Lennon, M.A.: Journal of Physics B 17, 4757 (1984) 59. Gibson, J.C., Gulley, R.J., Sullivan, J.P., Buckman, S.J., Chan, V., Burrow, P.D.: Journal of Physics B 29, 3177 (1996) 60. Panajotovic, R., Filipovic, D., Marinkovic, B., Pejcev, V., Kurepa, M., Vuskovic, L.: Journal of Physics B 30, 5877 (1997) 61. Nahar, S.N., Wadehra, J.M.: Physical Review A 35, 2051 (1987) 62. Bruno, D., Capitelli, M., Catalfamo, C., Celiberto, R., Colonna, G., Diomede, P., Gorse, C., Laricchiuta, A., Longo, S., Pagano, D., Pirani, F.: Transport Properties of HighTemperature Mars Atmosphere Components, ESA STR 256. In: Giordano, D., Fletcher, K. (eds.). ESA Communication Production Office (2008) 63. Itikawa, Y.: J. Phys. Chem. Ref. Data 31, 749 (2002) 64. Wright, M.J., Bose, D., Palmer, G.E., Levin, E.: AIAA Journal 43, 2558 (2005) 65. Griem, H.R.: Physical Review 128, 1280 (1962) 66. Margenau, H., Lewis, M.: Rev. Modern Physics 31, 594 (1959) 67. Capitelli, M., Ficocelli, E.V.: Zeitschrift für Naturforschung A 25, 977 (1970)


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68. Capitelli, M., Molinari, E.: Journal of Plasma Physics 4, 335 (1970) 69. Drellishak, K.S., Knopp, C.F., Cambel, A.B.: Physics of Fluids 6, 1280 (1963) 70. Drellishak, K.S., Knopp, C.F., Cambel, A.B.: Partition functions and thermodynamic properties of argon plasmas, Arnold Engineering Development Center, Tullahome, Tennessee, report TDR 63-146 (1963) 71. Fermi, E., für, Z.: Physik 26, 54 (1924) 72. Capitelli, M., Giordano, D.: Physical Review A 80, 32113 (2009) 73. Ecker, G., Weizel, W.: Annalen der Physik 17, 126 (1956) 74. Ecker, G., Kroll, W.: Zeitschrift für Naturforschung 21A, 2012 (1966) 75. Roussel, K., O’Connell, R.: Physical Review A 9, 52 (1974) 76. Giordano, D., Capitelli, M.: Unpublished results 77. Capitelli, M., Bruno, D., Colonna, G., Catalfamo, C., Laricchiuta, A.: Journal of Physics D 42, 194005 (2009) 78. Capitelli, M., Giordano, D., Colonna, G.: Physics of Plasmas 15, 082115 (2008) 79. Sing, K., Sing, G., Sharma, R.: Physics of Plasmas 17, 72309 (2010) 80. Moore, C.E.: Atomic Energy Levels NBS Circular N 467, 1949 (1958) 81. http://physics.nist.gov/PhysRefData/ASD/levels_form.html 82. Gurvich, L.V., Veyts, I.V., Alcock, C.B.: Thermodynamic Properties of Individual Substances. Hemisphere Publishing Corporation, New York (1989) 83. Gordon, S., McBride, B.J.: Thermodynamic data to 20 000 K for monatomic gases. NASA/TP-1999-208523 (1999) 84. Aubreton, J., Elchinger, M.F., Fauchais, P.: Plasma Chemistry and Plasma Processing 18, 1 (1998) 85. Rat, V., André, P., Aubreton, J., Elchinger, M.F., Fauchais, P., Lefort, A.: Physical Review E 64, 026409 (2004) 86. Rat, V., Murphy, A.B., Aubreton, J., Elchinger, M.F., Fauchais, P.: Journal of Physics D 41, 183001 (2008) 87. Giordano, D., Capitelli, M.: Physical Review E 65, 16401 (2001) 88. Capitelli, M., Giordano, D.: Journal of Thermophysics and Heat Transfer 16, 283–285 (2002) 89. Capitelli, M., Colonna, G., Gorse, C., Minelli, P., Pagano, D., Giordano, D.: AIAA paper 2001-3018 (2001) 90. Potapov, A.: High Temperature 4, 48 (1966) 91. Chen, X., Han, P.: Journal of Physics D 32, 1711 (1999) 92. Van de Sanden, M.C.M., Schram, P.P.J.M., Peeters, A.G., van der Mullen, J.A.M., Kroesen, G.M.W.: Physical Review A 40, 5273 (1989) 93. Morro, A., Romeo, M.: Journal of Non-Equilibrium Thermodynamics 13, 339 (1988) 94. Capitelli, M., Colonna, G., Gorse, C., Minelli, P., Pagano, D., Giordano, D.: Journal of Thermophysics and Heat Transfer 16, 469 (2002) 95. Capitelli, M., Celiberto, R., Gorse, C., Laricchiuta, A., Pagano, D., Traversa, P.: Physical Review E 69, 26412 (2004) 96. Capitelli, M., Lamanna, U.: Journal of Plasma Physics 12, 71 (1974) 97. Ignjatovic, L., Mihajlov, A.A.: Contributions to Plasma Physics 37, 309 (1997) 98. Celiberto, R., Lamanna, U.T., Capitelli, M.: Physical Review A 58, 2106 (1998) 99. Capitelli, M., Celiberto, R., Gorse, C., Laricchiuta, A., Minelli, P., Pagano, D.: Physical Review E 66, 16403 (2002) 100. Bruno, D., Capitelli, M., Catalfamo, C., Laricchiuta, A.: Physics of Plasmas 14, 072308 (2007)

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101. Bruno, D., Laricchiuta, A., Capitelli, M., Catalfamo, C.: Physics of Plasmas 14, 022303 (2007) 102. Capitelli, M., Lamanna, U.T., Guidotti, C., Arrighini, G.P.: Chemical Physics 19, 269 (1977) 103. Nyeland, C., Mason, E.A.: Physics of Fluids 10, 985 (1967) 104. Sourd, B., André, P., Aubreton, J., Elchinger, M.F.: Plasma Chemistry and Plasma Processing 27, 35 (2007); ibidem 27, 225 (2007) 105. Laricchiuta, A., Pirani, F., Colonna, G., Bruno, D., Gorse, C., Celiberto, R., Capitelli, M.: Journal of Physical Chemistry A 113, 15250 (2009) 106. D’Angola, A., Colonna, G., Gorse, C., Capitelli, M.: European Physical Journal D 46, 129 (2008) 107. Boulos, M.I., Fauchais, P., Pfender, E.: Thermal plasmas: fundamentals and applications. Plenum Press, New York (1994) 108. Ghorui, S., Heberlein, J.V.R., Pfender, E.: Plasma Chemistry and Plasma Processing 28, 553 (2008) 109. Kremp, D., Schlanges, M., Kraeft, W.: Quantum statistics of non-ideal plasmas. Atomic, Molecular and Plasma Physics Series, vol. 25. Springer, Heidelberg (2005) 110. Zivny, O.: European Physical Journal D 54, 349 (2009) 111. Zaghoul, M.R.: Physics of Plasmas 17, 062701 (2010)

Chapter 2

Non-equilibrium Kinetics and Transport Properties behind Shock Waves E.V. Kustova and E.A. Nagnibeda Saint Petersburg State University, Saint Petersburg, Russia

1 Introduction In high-temperature and hypersonic flows of gas mixtures, the energy exchange between translational and internal degrees of freedom, chemical reactions, ionization and radiation may result in noticeable violation of thermodynamic equilibrium when the characteristic times of kinetic and gas-dynamic processes are comparable. Therefore the non-equilibrium effects become important and for a correct prediction of gas flow parameters, non-equilibrium kinetics and gas dynamics should be considered jointly. In shock waves occurring in hypersonic flows, the rapid gas compression within a thin shock front results in a temperature jump which leads to excitation of internal degrees of freedom of molecular species and chemical reactions. Experimental data show the significant difference in relaxation times of various kinetic processes. The theoretical models adequately describing physical-chemical kinetics behind shock waves depend on relations between relaxation times of various kinetic processes. At the high temperature conditions which are typical just behind the shock front, the equilibrium between the translational and rotational degrees of freedom is established in a substantially shorter time than that of vibrational relaxation and chemical reactions, and therefore the following relation takes place[1]: ~

~ .

(1)

Here, , , , and are, respectively, the relaxation times for the translational, rotational and vibrational degrees of freedom, and the characteristic time for chemical reactions; is the mean time of macroscopic parameters variation. In this case it is usually supposed that translational and rotational relaxation occurs in a thin shock front (with a characteristic length of about several mean free paths of molecules) essentially without variation in the mixture composition and distributions over the vibrational energies. Then, in the relaxation zone (with a length of many tens or even hundreds mean free paths) behind the shock front, the excitation of vibrational degrees of freedom and chemical reactions take place, while equilibrium or weakly non-equilibrium distributions over translation and rotational energies established in the shock front are maintained. For the description of the non-equilibrium flow under the

60


condition (1) it is necessary to consider the equations of the state-to-state vibrational and chemical kinetics coupled to the gas dynamic equations. It is the most detailed description of the non-equilibrium flow. More simple models are based on quasi-stationary multi-temperature or onetemperature vibrational distributions. In the vibrationally excited gas at moderate temperatures, the near-resonant vibrational energy exchanges between molecules of the same chemical species occur much more frequently compared to the non-resonant transitions between different molecules as well as transfers of vibrational energy to the translational and rotational ones and chemical reactions[1,2]: ~

~ .

(2)

Here , , are, respectively, the mean times for the VV1 vibrational energy exchange between molecules of the same species, VV2 vibrational transitions between molecules of different species and TRV transitions of the vibrational energy into other , quasi-stationary (multimodes. Under the condition (2), during the time temperature) distributions over the vibrational levels establish, which then maintain in non-equilibrium chemical reactions (model of non-equilibrium multi-temperature kinetics). For tempered reaction regime, with the chemical reaction rate considerably lower than that for the internal energy relaxation, the following characteristic time relation takes place: ~ ,

(3)

where is the mean time for the internal energy relaxation. Under this condition, the non-equilibrium chemical kinetics can be considered on the basis of the maintaining thermal-equilibrium one-temperature Boltzmann distributions over the internal energy levels of molecular species (the model of one-temperature chemical kinetics [3], often used in applications). The most accurate theoretical description of non-equilibrium flows behind shock waves can be given using the kinetic theory methods. The kinetic theory makes it possible to develop mathematical models of a flow under different non-equilibrium conditions, i.e. to obtain closed systems of the non-equilibrium flow equations and to elaborate calculation procedures for transport and relaxation properties.

2 State-to-State Approach 2.1 Distribution Functions and Macroscopic Parameters In reacting mixtures with rapid and slow physical-chemical processes, the kinetic , , over chemical species c, vibrational equations for the distribution functions i and rotational j energy levels in the phase space of the velocity coordinates and time may be written in the form [4,5]: ·

.

(4)

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

Under the conditions (4), the integral operator of rapid processes

61

describes elastic

collisions and rotational energy exchange whereas the operator of slow processes describes the vibrational energy exchange and chemical reactions: ,

.

(5)

The parameter in Eqs.(4) represents the ratio of the characteristic times: ⁄ , ~ , ~ . The integral operators (5) are given in [4,6]. Modification of the Chapman–Enskog method for the solution of the kinetic equations (4), (5) [7,8], makes it possible to derive governing equations of the flow, expressions for the dissipative and relaxation terms in these equations and algorithms for the calculation of transport and reaction rate coefficients. The solution of the kinetic equations in the zero-order approximation 0

(6)

is specified by the independent collision invariants of the most frequent collisions. These invariants include the momentum and particle total energy which are conserved at any collision, and additional invariants for the most probable collisions which are given by any value independent of the velocity and rotational level j and depending arbitrarily on the vibrational level i and chemical species c. The additional invariants appear because vibrational energy exchange and chemical reactions are supposed to be frozen in rapid processes. Based on the above set of the collision invariants, the zero-order solution of Eqs.(4) takes the form exp

.

(7)

is the population of vibrational level i of species c, , is the Here, macroscopic velocity, the rotational energy of the molecule at jth rotational and ith vibrational levels, is the gas temperature, the molecular mass, k the Botzmann the rotational partition constant, the rotational statistical weight, functions. For the rigid rotator model, , , is the moment of inertia, h is the Planck constant, is the symmetry factor. The distribution functions (7) are specified by the macroscopic gas parameters , ( 1, , , 0,1, , , is the number of chemical species, is the number of excited vibrational levels in species c), , , and , which correspond to the set of the collision invariants of rapid processes. 2.2 Governing Equations The closed set of equations for the macroscopic quantities , , , , and , follows from the kinetic equations and includes the conservation equations of

62


momentum and total energy coupled to the equations of detailed state-to-state vibrational and chemical kinetics [4]: ·

·

,

1,

·

, ,

0,1,

0,

·

:

,

,

(8)

(9) 0.

(10)

diffusion velocities of Here is the pressure tensor, the total energy flux, molecules at different vibrational states and the total energy per unit mass ∑

∑

(11)

,

is the rotational energy per unit mass, the vibrational energy of a molecule of species con the i-th vibrational level and the energy of formation of the particle of species c. The source terms in the equations (8) are expressed via the integral operators of slow processes ∑

,

(12)

and characterize the variation of the vibrational level populations and atomic number densities caused by different vibrational energy exchanges and chemical reactions. For this approach, the vibrational level populations are included to the set of main macroscopic parameters, and the equations for their calculation are coupled to the equations of gas dynamics. Particles of various chemical species in different vibrational states represent the mixture components, and the corresponding equations contain the diffusion velocities of molecules at different vibrational states. In the zero-order approximation of Chapman-Enskog method ,

0,

0

, ,

(13)

and the set of governing equations takes the form ·

,

1,

, ,

0, ·

0.

0,1,

,

(14) (15) (16)


63

contain the microscopic rate coefficients for vibrational The expressions for energy exchanges and chemical reactions averaged with the Maxwell–Boltzmann distribution over the velocity and rotational energy levels and depending on the vibrational states and chemical species of interacting particles. The equations (14)–(16) describe detailed state-to-state vibrational and chemical kinetics in an inviscid non-conductive gas mixture flow in the Euler approximation. Taking into account the first-order approximation makes it possible to consider dissipative properties in a non-equilibrium viscous gas. 2.3 First-Order Approximation. The first-order distribution functions can be written in the following structural form [4]: · ln

∑

·

:

·

.

(17)

depend on the derivatives of all macroscopic The distribution functions parameters: temperature , velocity , and vibrational level populations via the diffusive driving forces ln .

(18)

, , , and depend on the peculiar velocity and the The functions flow parameters, and satisfy the linear integral equations with linearized operator for rapid processes. The transport kinetic theory in the state-to-state approximation was developed, for the first time, in [8] and also given in [4]. The expressions for the transport terms in the equations (8)–(10) in the first order approximation are derived on the basis of the distribution functions (17). The viscous stress tensor is described by the expression: 2

·

.

(19)

is the relaxation pressure, and are the coefficients of shear and bulk Here, viscosity. The additional terms connected to the bulk viscosity and relaxation pressure appear in the diagonal terms of the stress tensor in this case due to rapid inelastic TR exchange between the translational and rotational energies. The existence of the relaxation pressure is caused also by slow processes of vibrational and chemical 0. relaxation. If all slow relaxation processes in a system disappear, then The diffusion velocity of molecular components c at the vibrational level i is specified in the state-to-state approach by the expression [4,8]: ∑

ln ,

(20)

64


where and are the multi-component diffusion and thermal diffusion coefficients for each chemical and vibrational species. The total energy flux in the first-order approximation has the form: ∑

∑

,

(21)

where is the thermal conductivity coefficient, is the mean and are responsible for the energy transfer rotational energy. The coefficients associated with the most probable processes which, in the present case, are the elastic collisions and inelastic TR- and RR rotational energy exchanges. In the state-to-state approach, the transport of the vibrational energy is described by the diffusion of vibrationally excited molecules rather than by the thermal conductivity. In particular, the diffusion of the vibrational energy is simulated by introducing independent diffusion coefficients for each vibrational state. It should be noted that all transport coefficients are specified by the cross sections of rapid processes except the relaxation pressure depending also on the cross sections of slow processes of vibrational relaxation and chemical reactions. From the expressions (20), (21), and (18), it is seen that the energy flux and diffusion velocities include along with the gradients of temperature and atomic number densities also the gradients of all vibrational level populations. This constitutes the main feature of the heat transfer and diffusion in the state-to-state and and the diffusion approach and the fundamental difference between velocities and heat flux obtained on the basis of one-temperature, multi-temperature or weakly non-equilibrium approaches. The transport coefficients in the expressions (19)–(21) can be written in terms of , , , and : functions 10

,

,

, ,

,

,

, ,

, ,

,

.

(22)

Here , are the bracket integrals associated with the linearized operator of rapid processes. They were introduced in [4] for strongly non-equilibrium reacting mixtures similarly to those defined in [9] for a non-reacting gas mixture under the conditions for weak deviations from the equilibrium. For the transport coefficients calculation, the functions , , , and are expanded into the Sonine polynomials in the reduced peculiar velocity and those of Waldmann-Trübenbacher in the dimensionless rotational energy. For the coefficients of these expansions, the linear transport systems are derived, and the transport coefficients are expressed in terms of the solutions of these systems. in Eqs. (8) describe slow processes of vibrational relaxation The source terms and chemical reactions. These terms can be written as follows ,

,

,

,

,

(23)


(24)

, ,

,

,

and dissociation-recombination ,

,

(25)

,

(26)

They contain the rate coefficients for the energy transitions ,

,

,

,

,

,

65

, ,

, exchange reactions

respectively

,

. (27)

,

The expressions for the zero-order rate coefficients for binary reactions have the following form , ,

∑

exp

,

exp

,

,

.

(28)

is the reduced mass, the relative velocity, the integral cross section , of the collisions resulting in a binary reaction. The expressions for rate coefficients for the remaining processes have a similar form. In the first order approximation, the rate coefficient, contrarily to the zero-order coefficients, depend not only on the temperature but also on the vibrational level populations and includethe term proportional to the velocity divergence · . The procedure for calculation of the zero-order and first-order rate coefficients is given in [4]. It should be noted, that in the practical simulations of dynamics of viscous conducting gases, in the equations of non-equilibrium kinetics (in the state-to-state, multi-temperature or one-temperature approaches), the reaction rate coefficients are calculated using the zero-order distribution function. Up to now, no reliable calculations for the first order state-depending reaction rate coefficients are available. Such estimations were proposed only for the multi-temperature model of coupled dissociation and vibrational relaxation in [10], and for a one-temperature approach in [11]. In the literature, a number of theoretical and experimental estimates for the zeroorder rate coefficients for vibrational energy transitions in different temperature intervals are available (see for example [12]). Up to the recent time, the most commonly used are the formulas of the Schwartz, Slawsky and Herzfeld theory (known as the SSH-theory), developed for the harmonic oscillator model in [13] and later generalized for anharmonic oscillators in [2,14], as well as the Landau-Teller theory (for the VT exchange) with various semi-empirical expressions for the vibrational relaxation time [15,1]. In addition, semi-empirical formulas for the rate coefficients of vibrational energy transitions written in a form similar to the

66


expressions of the SSH-theory are often used; they provide a satisfactory consistency with the experimental results due to introduction of some additional empirical parameters. More accurate results are based on the quantum-mechanical and semiclassical techniques applied to the calculation of the cross sections for inelastic collision and probabilities for vibrational and rotational energy transitions in various gases [16,17], as well as on the trajectory calculations [18,19,20]. In particular, in [21] it is shown that at low temperatures, the SSH-theory does not provide a satisfactory accuracy for the evaluation of the atoms efficiency in VT energy transitions. Furthermore, at high temperatures, the SSH-theory overestimates the probabilities for VT transitions from high vibrational states (compared to those obtained in [16,17]). However, practical implementation of the quantum-mechanical methods and trajectory calculations is restricted by the computational costs of the calculation of the cross section for each specific transition. Among the up-to-date analytical models for the vibrational transition probabilities, we can recommend the semi-classical model of forced harmonic oscillator (FHO) [22,23] which makes it possible to obtain correct values for the rate coefficients of VV and VT transitions (including multi-quantum jumps which are particularly important if the partner in the collision is an atom [16] at high temperatures. A model proposed in [24] based on the information theory can also be appreciated. The analytical approximations useful for practical calculations of the probabilities of different vibrational energy transitions in air components are obtained in [25,26] by interpolation of accurate numerical results presented by G. Billing in [16,17]. These approximate formulas are valid for temperatures below 12000 K. Comparison of the results obtained using the SSH formulas and the expressions proposed in [25,26] , on the show [4] that, while the dependences of the coefficients , and , vibrational quantum number are similar to those given by the formulas of the generalized SSH theory, for the coefficient , , an essential difference can be noticed. Figure 1 presents the dependence of the rate coefficients for VT transitions 1 1 (b) on the (a), and vibrational level i at T = 6000 K. One can observe a qualitative agreement of the VT rate coefficients for the transition in a collision with a molecule. Quantitatively, the generalized SSH theory gives considerably higher rates of VT transitions for upper vibrational levels ( 20). On the other hand, for the transition proceeding through a collision with an atom, theoretical models and quasi-classical trajectory (QCT)calculations give a different dependence of the rate coefficient on the vibrational state. According to QCT results, it increases almost linearly with whereas analytical models provide non-linear rising of the rate coefficient. Therefore using the SSH and FHO models for the description of non-equilibrium vibrational kinetics in mixtures with high atomic concentrations can lead to a certain error. Nevertheless, for investigation of kinetics in strong shock waves, this transition does not play a crucial role because the vibrational distributions are established at a rather short distance close to the shock front, where atomic concentrations remain low.


67

Fig. 1. Rate coefficients for VT transitions in N2 in a collision with N2 molecule (a) and N atom (b) respectively. T=6000 K. 1: SSH model for anharmonic oscillators [2]; 2: SSH model for harmonic oscillators; 3: FHO model [22]; 4: formulas [25, 26].

Fig. 2. Vibrational distributions behind a shock wave. 1: SSH model for anharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHO model. Curves 1-3 in graph correspond to x=0.01cm; 1'-3’ to x=2cm. (The conditions in the free stream: T=293 K, p=100 Pa, Mach number M=15).

In Figs.2 and 3, vibrational distributions and gas temperature calculated in [27] behind a shock wave using different models for the VV and VT rate coefficients are presented. One can notice that the SSH model for anharmonic oscillators provides higher population of the upper vibrational levels and lower values of temperature. The SSH model for harmonic oscillators gives a slower excitation of high vibrational levels, and, as a result, lower rate of temperature decrease. The discrepancy between the results obtained on the basis of the FHO model and the SSH model for anharmonic oscillators is small: it does not exceed 2% for the temperature. Since the

68


FHO model can be considered as the most accurate one in the high temperature conditions, one can conclude that the SSH model for harmonic oscillators leads to a noticeable error in the predicted values of gas dynamic parameters whereas its generalization for anharmonic oscillators works rather well in shock heated gases.

Fig. 3. Temperature behind a shock wave. 1: SSH model for anharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHO model.

The important role of multi-quantum transitions in a collision of N2 molecules with atoms is discussed in [25,26]. Solution of equations for the vibrational level populations in N2-N mixture [28] shows that multi-quantum transitions in collisions with atoms influence significantly vibrational distributions and macroscopic flow parameters reducing the vibrational relaxation time. Multi-quantum transitions in molecule-molecule collisions play a weaker role and can be neglected. The rate coefficients for dissociation from different vibrational levels have been studied much less widely than those for vibrational energy transitions. Two models are commonly used in the literature: the ladder-climbing model assuming dissociation only from the last vibrational level (see, for instance [29,25,26]), and that of Treanor and Marrone [30] allowing for dissociation from any vibrational state. Originally this model was proposed for the two-temperature approximation; its modification for the state-to-state approach [4,31] makes it possible to present the rate coefficient for dissociation of a molecule on the vibrational level iin the form: ,

where

,

,

is the state-dependent non-equilibrium factor

(29)


,

exp

,

69

(30)

is the thermal equilibrium dissociation rate coefficient, the , equilibrium vibrational partition function, and the parameter of the model. For , the empirical Arrhenius law can be applied: , ,

exp

(31)

,

where is the dissociation energy; the coefficients and are generally obtained from a best fit of experimental data. The tables of the coefficients in the Arrhenius formula for various chemical reactions can be found in [1,3,12,32,33]. For a practical implementation of the Treanor–Marrone model, it is important to select the parameter so that a good consistency is reached for the dissociation rate coefficient with experimental data or the results of calculations based on more accurate models. The following approximations for are commonly used: ∞, ⁄6 , and 3 . For ∞, dissociation is assumed to be equi-probable for each vibrational level, whereas the other values of the parameter describe preferential dissociation from high vibrational states. In [34], the dissociation rate coefficients calculated within the framework of the Treanor–Marrone model are compared , with those obtained from trajectory calculations [20]. The Figs.4 present the for temperature dependence of the state-dependent dissociation rate coefficient , 0 (a) and 20 (b) obtained in [20] and using the Treanor-Marrone model with different U values.

Fig. 4. Temperature dependence of the dissociation rate coefficient 20 (b). Curve 1: results of Ref. [20], curves 2–4 correspond to ∞

,

for ⁄6 ,

0 (a) and 3 , and

70


It can be noticed that the choice ∞results in significant overestimation for at low vibrational levels. With the increase of the vibrational quantum number , a better agreement with the results of accurate trajectory calculations is found. It confirms the common assumption of the preferential dissociation from high ⁄6 and vibrational states. The values 3 provide good consistency for at intermediate levels. It is also shown that using the same value of the , parameter for any i and T can result in considerable error in the calculation of statedependent dissociation rate coefficients. The choice of the parameter should be specified by the conditions of a particular problem. In some studies [35,25,26], a possibility for dissociation from any vibrational state is suggested within the framework of the ladder-climbing model. To this end, it is supposed that a transition to the continuum occurs as a result of multi-quantum vibrational energy transfers. From this point of view, the authors of [35] conclude that dissociation from low vibrational levels is preferential for high gas temperatures. A similar effect is also mentioned in [36]. However, this conclusion has not been justified either by accurate trajectory calculations or by experiments. The influence of the dissociation model on the vibrational level populations behind the shock wave is shown in Figure 5, where the population of the tenth level versus the distance from the shock front is presented for different values for the parameters U, A, and n, as well as for the ladder-climbing model. Three values for U are ⁄6 , and considered: ∞, 3 , the parameters in the Arrhenius formula are taken from [37,32]. We can see that the vibrational distributions calculated with the ladder-climbing dissociation model are significantly different from those obtained using the Treanor–Marrone model. The rate coefficients for bimolecular exchange reactions depending on the vibrational states of reagents and products have been less thoroughly studied than those for dissociation processes. Theoretical and experimental studies for the influence of the vibrational excitation of reagents on reaction rates were started by J. Polanyi [38]; some experimental results were also obtained in [39]. The accurate theoretical approach to this problem primarily requires a calculation for the statedependent differential cross sections for collisions resulting in chemical reactions, and their subsequent averaging over the velocity distributions. In the recent years, the dynamics of atmospheric reactions has been studied, and quasi-classical trajectory calculations for the cross sections and state-dependent rate coefficients for the and have been carried out by reactions several authors. The reactions of NO formation are considered in [40]; the effect of translational, rotational, and vibrational energy of reagents on the reaction is discussed in [41,42]. For the application of the existing results to the problems of non-equilibrium fluid dynamics, the analytical expressions for the dependence of the reaction rate coefficients on the vibrational states of molecules participating in the reactions are needed. Two kinds of such expressions are available in the literature. The first kind includes analytical approximations for numerical results obtained for particular reactions (see [43,12,44]). These expressions are sufficiently accurate and convenient for practical use; however, their application is


71

restricted by the considered temperature range. Another approach is based on the generalizations of the Treanor–Marrone model to exchange reactions suggested in [45,46].

⁄6 (the parameters A, n are taken Fig .5. 10th level population as a function of x. 1: ⁄6 ;3: from Ref. [37]); 2: 3 ; 4: ∞ (for 2–4A, n are from Ref. [32]); 5: ladderclimbing model. (The conditions in the free stream: T=293 K, p=100 Pa, Mach number M=15).

These models can be used for more general cases, but the theoretical expressions for the rate coefficients contain additional parameters, which should be validated using experimental data. A lack of data for these parameters restricts the implementation of the above semi-empirical models. Therefore, the development of justified theoretical models for cross sections of reactive collisions and statedependent rate coefficients for exchange reactions remains a very important problem of the non-equilibrium kinetics.

3 Quasi-stationary Approaches 3.1 Vibrational Distributions: Governing Equations Practical implementation of the state-to-state kinetic model leads to serious difficulties. The important problem encountered in the realization of the state-to-state model is its computational cost. Indeed, the solution of the fluid dynamics equations coupled to the equations of the state-to-state vibrational and chemical kinetics in multi-component mixtures requires numerical simulation of a great number of equations for the vibrational level populations of all molecular species. Moreover, in

72


the viscous gas approximation, numerical simulations require the calculation of a large number of transport coefficients, particularly, diffusion coefficients in each space cell and at each time step, which significantly complicates the study of specific flows. Simplifications proposed in [47] make it possible to reduce the number of state-dependent transport coefficients, but even after that, the state-to-state model for multi-component reacting flows remains time consuming and numerically expensive. Therefore simpler models based on quasi-stationary vibrational distributions are rather attractive for practical applications. In quasi-stationary approaches, the vibrational level populations are expressed in terms of a few macroscopic parameters; consequently, non-equilibrium kinetics can be described by a considerably reduced set of governing equations. Commonly used models are based on the Boltzmann distribution with the vibrational temperature different from the gas temperature. However, such a distribution is valid solely for the harmonic oscillator model, which describes adequately only the low vibrational states. The more accurate quasistationary model is based on the Treanor two-temperature vibrational distribution for anharmonic oscillators. The state-to-state kinetic model of non-equilibrium flow may be reduced to multitemperature description under the condition (2). In this case, the integral operator of the most frequent collisions in the kinetic equations (4) includes the operator of VV1 vibrational energy transitions between molecules of the same species along with the operators of elastic collisions and collisions with rotational energy exchanges; the operator of slow processes consists of the operator of VV2 vibrational transitions between molecules of different species, as well as the operators describing the transfer of vibrational energy into rotational and translational modes and chemical reactions ,

.

(32)

The zero-order distribution functions are specified by the invariants of the most frequent collisions. In addition to the invariants which are conserved in any collision, under the condition (2) there are additional independent invariants of rapid processes: the number of the vibrational quanta in each molecular species c, and an arbitrary value independent of the velocity, vibrational i and rotational j quantum numbers and depending arbitrarily on the particle chemical species c. Conservation of vibrational quantum for VV transitions in a single-component gas was found for the first time in [48] and the non-equilibrium two-temperature distribution was derived, now it is called the Treanor distribution. In a gas mixture, conservation of vibrational quanta in each species during VV1 transitions takes place. The existence of the other additional invariants is explained by the fact that under the condition (2), slow chemical reactions remain frozen in the most rapid process. Taking into account the system of collision invariants one can obtain the zero-order distribution functions and the following expressions for vibrational level populations: ,

exp

,

(33)


73

, the non-equilibrium is the number density of cth species, Where partition function of vibrational degrees of freedom and the temperature the first vibrational level for each molecular species c. The distributions (33) generalize the Treanor distributions [48] for a multicomponent reacting mixture. Similarly to a single-component gas, the distribution (33) describes adequately only the populations of low vibrational levels , . This is explained by the where corresponds to the minimum of the function fact that the conservation of vibrational quantum takes place only at low levels . However, in the high temperature gas when , the level appears to be close 2 to the last vibrational level . Therefore the Treanor distribution may be used for all vibrational levels in the relaxation zone behind shock waves. On condition that the anharmonic effects can be neglected, the distribution (33) is reduced to the non-equilibrium Boltzmann distribution with the vibrational different from T. In the case of the temperature of molecular components local thermal equilibrium, the vibrational temperatures of all molecular species are equal to the gas temperature , and the Treanor distribution (33) is reduced to the one-temperature Boltzmann distribution. The zero-order distribution functions depend on the macroscopic parameters , , , , , , and , . In the present case, in contrast to the state-tostate model, the number of main macroscopic parameters is reduced, and instead of it includes the vibrational the level populations of all vibrational states temperatures and number densities of chemical species. The governing , , , , , , and , . are equations for the macroscopic quantities derived in [7,49]. A closed system of reacting multi-component mixture dynamics consists of the equations of the multi-temperature chemical kinetics for the species number densities, conservation equations for the momentum and the total energy, and additional relaxation equations for molecular species: ·

·

, · ·

·

,

1,

, ,

(34)

0, :

(35) 0, ·

(36) ,

1,

,

,

(37)

where are diffusion velocities of different chemical species, is a specific number of vibrational quanta in molecules of c species and , the flux of vibrational quanta of c molecular species:

74


,

∑

.

(38)

In the multi-temperature approach, the total energy is a function of , , , in contrast to the state-to-state model where it depends on all level populations and gas temperature. The source terms in Eqs. (34), (37) have the form: ∑

, ,

∑

(39)

,

,

(40)

.

The equations (34)–(37) form a closed system of equations for the macroscopic parameters of a reacting gas mixture flow in the multi-temperature approach. It is obvious that the system (34)–(37) is considerably simpler than the corresponding system (8)–(10) in the state-to-state approach, as it contains much fewer equations. Thus, instead of ∑ equations for the vibrational level populations, one should equations for the numbers of quanta and equations for the number solve densities of the chemical components ( is the number of vibrational levels in the is the number of the molecular species in a mixture). molecular species c, Consequently, for a two-component mixture containing nitrogen molecules and and should be solved instead of 46 equations for atoms, two equations for the level populations of N2 molecules. While studying the (important for practical applications) five-component air mixture N2, O2, NO, N, O in the state-to-state 114 equations for the vibrational approach, one should solve level populations. In the multi-temperature approach, they are reduced to six equations: three for the molecular number densities , , and three for , , and . vibrational temperatures In a system of harmonic oscillators, the relaxation equations (37) are transformed into those for the specific vibrational energy which is defined by non-equilibrium Boltzmann distribution with vibrational temperature : ,

·

,

,

,

·

,

1,

,

, (41)

with ,

,

,

(42)

.

In the zero-order approximation of the Chapman–Enskog method, the transport terms are as follows ,

,

0,

0

,

(43)


75

and the system (34)–(37) takes the form typical for inviscid non-conductive flows ·

,

1,

, ,

(44)

0,

·

(45)

0,

(46)

,

1,

,

(47)

.

The production terms in equations (44), (47) are given by the formulas ∑

∑

,

(48)

,

and contain the zero-order operators of VV2 and TRV vibrational energy exchanges and chemical reactions. 3.2 Transport Terms In the multi-temperature approach, the first-order distribution functions have the following form [7,4]: 1 ∑

1

· ln

·

:

· ln ·

.

(49)

The coefficients , , , , and are functions of the peculiar velocity and macroscopic parameters and satisfy the linear integral equations with linearized operator of rapid processes of VV2, VT vibrational transitions and chemical reactions. First-order distribution functions (49) define transport terms in the equations (34)– (37). The pressure tensor has the form (19) where the relaxation pressure and bulk viscosity coefficient are presented as sums of two terms: ,

,

(50)

where the first term is due to inelastic RT rotational energy exchange, whereas the second is connected to the VV1 transitions in each vibrational mode.

76


The diffusion velocity takes the form ∑

ln ,

(51)

and are the diffusion and thermal diffusion coefficients . The total energy flux and the fluxes of vibrational quanta depend on the gradients of the gas temperature T, temperatures of the first vibrational level , and molar fractions of chemical species ⁄ : ∑

∑

∑

,

.

∑

,

(52) (53)

In Eqs. (52), (53), , , , are thermal conductivity coefficients and is the specific enthalpy of c particles. Transport coefficients are defined by bracket integrals, depending, in the multitemperature approximation, on the cross sections of elastic collisions and collisions resulting from the RT and non-resonant VV1 energy exchanges. The expressions for shear and bulk viscosity and relaxation pressure via bracket integrals are the same as in the state-to-state approximation (see Eq. (22)). However, bracket integrals in these two approaches are different because they are defined with cross sections of various rapid processes [4]. In the expressions (51)–(53), the diffusion, thermal diffusion and heat conductivity coefficients have the forms ,

, ,

, ,

,

,

, ,

, ,

.

(54)

The coefficient describes the transport of the translational, rotational energy and a small part of the vibrational energy, which is transferred to the translational mode as a result of the non-resonant VV1 transitions between molecules simulated by anharmonic oscillators and is presented as a sum of three corresponding terms: . The coefficients are associated with the transport of vibrational quanta in each molecular species and thus describe the transport of the . The cross coefficients , are specified by main part of vibrational energy both the transport of vibrational quanta and the vibrational energy loss (or gain) as a ⁄ , the result of non-resonant VV1 transitions. For low values of the ratio , , and are much smaller than , and for the harmonic coefficients 0 since VV1 transitions appear to be strictly oscillator model and disappear in a system resonant. For the same reason, the coefficients of harmonic oscillators.


77

The number of independent diffusion coefficients in the multi-temperature model is considerably smaller than that in the approach accounting for the detailed vibrational kinetics. Therefore the use of the quasi-stationary vibrational distributions noticeably facilitates the heat fluxes calculation in a multi-component reacting gas mixture. The proposed kinetic theory was applied in [49] for the simulation of gasdynamic parameters, transport coefficients and heat fluxes in non-equilibrium reacting air flows behind strong shock waves. In the one-temperature approach based on the thermal equilibrium Boltzmann vibrational distributions, the closed set of governing equations includes equations for , , which have the form (34), (35), (36). However, one should keep in mind that transport and relaxation terms in these equations differ from those obtained in the multi-temperature approach because they are defined by different collision processes: in the one-temperature approximation rapid processes include along with elastic collisions all internal energy transitions while slow processes are specified by only chemical reactions. The total heat flux is described by the expression ∑

∑

,

(55)

where , is the vibrational thermal conductivity coefficient. In this approach the bulk viscosity and relaxation pressure in the stress , . tensor are Possibility of limit transition from the state-to-state heat transfer description to the quasi-stationary models is discussed in [4]. 3.3 Production Terms in the equations (34) describes the variation of The chemical production term the particle c number density due to chemical reactions, whereas the term in in Eq.(37) characterizes the variation of the specific number of vibrational quanta molecular species c due to both slow vibrational energy exchanges and chemical reactions. The term describes exchange reactions, dissociation and recombination and can be written in the form (56)

, where ∑ ∑

,

,

,

.

(57)

(58)

78


Here, is the multi-temperature rate coefficient of the exchange reaction (during a ). The coefficients , , collision of two molecules or a molecule and an atom are the rate coefficients of dissociation and recombination reactions. , In the quasi-stationary approaches the zero-order reaction rate coefficients are expressed in terms of state-dependent rate coefficients ,

∑

,

,

∑

,

(59)

,

,

(60)

, where denotes quasi-stationary distributions and , are state, , dependent rate coefficients of bimolecular reactions (28) and dissociation. For the generalized Treanor distribution, the rate coefficients of exchange reactions occurring in the collision of two molecules have the form

,

,

,

∑

, ,

exp ,

,

(61)

and depend on the gas temperature and vibrational temperatures of the first levels of reagents. Rate coefficients for dissociation depend on two temperatures

,

,

,

∑

exp

(62)

.

,

Neglecting the anharmonic effects, we can find the reaction rate coefficients averaged over the non-equilibrium Boltzmann distribution ,

∑

,

,

∑

,

,

exp

exp

,

,

(64)

.

,

(63)

In a thermal equilibrium gas mixture, the reaction rate coefficients depend only on the gas temperature and are specified by the expressions ∑

exp

, ,

,

(65)


∑

,

exp

and can be described by the Arrhenius law. The total recombination rate coefficient specific rate coefficients as follows ,

∑

,

,

(66)

is defined in terms of the state-

,

,

79

,

(67)

and depends on the gas temperature only. The superscript ”0” in the notations for the state-to-state rate coefficients indicates that they are calculated by averaging the corresponding inelastic collision cross sections with the Maxwell-Boltzmann distribution over the velocity and rotational energy. in the relaxation equations for can be expressed in The production term terms of macroscopic parameters substituting the zero-order or the first-order distribution functions into the formulas (40). In the zero-order approximation, includes the vibrational distributions (33) and the state-to-state rate coefficients of VV2, VT vibrational energy transitions and chemical reactions [4]. The expressions can be also simplified if non-equilibrium or thermal equilibrium Boltzmann for distributions are used instead of Treanor distributions.

4 Non-equilibrium Processes behind Shock Waves in Air Components and CO2 Mixtures 4.1 Non-equilibrium Kinetics and Transport Properties in Diatomic Gas Mixtures In this section the results of applications of the state-to-state, multi-temperature and one-temperature models for evaluation of non-equilibrium kinetics and transport properties behind shock waves in the air components are presented. 4.1.1 Governing Equations and Flow Parameters The gas state in the unperturbed flow before the shock front is supposed to be in equilibrium, and vibrational and chemical kinetics in the relaxation zone behind the shock front is studied in the Euler approximation for an inviscid non-conducting gas flow. If the flow is assumed to be one-dimensional and steady-state, the governing equations are substantially simplified. Thus, the flow of a binary mixture of molecules A2 and atoms A behind a plane shock wave with dissociation, recombination, TV and VV vibrational energy transitions is described by the following set of equations for , atomic number densities , macroscopic gas the vibrational level populations velocity , and temperature ,

1,

, ,

(68)

80


2∑

(69)

,

(70)

,

(71)

,

(72)

,

where is the distance from the shock front. The subscript ”0” denotes the parameters in the free stream and the specific enthalpy h is equal to (73)

, ,

are the mass fractions of molecules and atoms, ,

,

∑

.

(74)

The right-hand sides of Eqs.(68), (69) include the state-to-state coefficients for vibrational energy transitions, dissociation, and recombination. The distribution in the free stream is usually assumed to be the Boltzmann one with a given temperature T0. The results obtained from the numerical solution of the system (68)-(72) for the N2/N mixture under the following conditions in the free stream: T0 =293 K, p0 =100 Pa, M0 =15 are presented below. The vibrational energy is simulated by the Morse anharmonic oscillator, the rate coefficients for vibrational energy transitions are calculated on the basis of the SSH-theory generalized for anharmonic oscillators [2,14], dissociation is described using the generalized Treanor–Marrone model with different values for the parameter U and parameters A, n in the Arrhenius law (31). The vibrational level populations and macroscopic flow parameters are computed in the state-to-state, two-temperature, and one-temperature approaches. The quasistationary vibrational distributions for different distances from the shock front are calculated using the obtained values for the macroscopic parameters. In Figure 6, the dimensionless vibrational level populations / calculated with the above three models are given as functions of i for various distances from the shock front. We can see a substantial discrepancy between the vibrational distributions obtained with different models close to the shock front, where the quasistationary distributions have not been yet established. The two-temperature and onetemperature approaches overestimate the vibrational level populations in a thin layer immediately behind the shock front. This discrepancy decreases with the distance from the shock front.


81

Fig. 6. Vibrational level populations. 1, 2: state-to-state model; 1’, 2’: two-temperature approach; 1’’, 2’’: one-temperature model. Solid curves correspond to x = 0.03, dashed to x = 0.8 cm.

Fig. 7. Gas temperature T behind the shock as a function of x. The curves 1, 2, 3 represent the state-to-state, two-temperature, and one-temperature approaches, respectively.

Figure 7 presents the variation of the gas temperature, calculated with the three approaches, as a function of the distance from the shock front. With the onetemperature and two-temperature models, the temperature is underestimated, since these approaches assume the existence of quasi-stationary distributions immediately behind the shock front and do not take into account the process of vibrational excitation in the very beginning of the relaxation zone.

82


Fig. 8. Atomic molar fraction behind the shock front as a function of x. The curves 1, 2, 3 represent the state-to-state, two-temperature, and one-temperature approaches, respectively.

Fig. 9. Averaged dissociation rate coefficient as a function of x. The curves 1, 2, 3 represent the state-to-state, two-temperature, and one-temperature approaches, respectively.


83

In Fig.8, the number densities of nitrogen atoms calculated in the three approaches are compared. It can be noticed that the one-temperature model does not describe the dissociation delay immediately behind the shock front. Both quasi-stationary models overestimate the dissociation degree in the vicinity of the front. From the vibrational distributions found behind the shock wave for the state-tostate and two-temperature approaches, the averaged dissociation rate coefficient can be calculated. For this purpose, the vibrational level populations, molecular , and temperature T obtained in these approaches for various points number density behind the shock are substituted into the formula (60). In the one-temperature approach, the dissociation rate coefficient is calculated using the Arrhenius formula (31). The results are presented in Fig.9. It is seen that the one-temperature model describes the behavior of the dissociation rate coefficient inadequately, particularly close to the shock front. The two-temperature approach provides more realistic values in comparison to for the dissociation rate coefficient, overestimating however the state-to-state approximation at x 1.5·1017 cm-3 (Fig.13) demonstrating the appearance of some additive processes of effective electronic state quenching. The same is true for the effective rate constant of excitation of the term C2(d3Πg). The mechanism of electronic excitation of diatomic molecules from ground electronic state, taking NTs into account in relaxation processes, was formulated for a gas heated by shock wave[222]. The proposed mechanism was based on the following conception: Excitation of higher energy levels of a molecule immediately behind the shock front occurs primarily owing to rotational-vibrational relaxation in ground electronic state in collisions with other particles of the gas. The excited electronic states are populated mainly because of collisional NTs in regions of close approach to neighboring electronic states including the ground state. Through the agency of NTs, the energy redistribution over rotational-vibrational-electronic states continues during vibrational relaxation in the ground state. Keeping in view the CN, N2, C2, O2, CO molecules and gas temperatures not higher than 10,000K, the vibrational relaxation time was assumed to be much greater than the times of translational and rotational relaxation, and to be much less than the time of chemical reactions, especially, reactions of dissociation. This assumption was confirmed by a comparative analysis of the relaxation times[222]. Thus, near the front of a shock wave, the populations of excited states reach its equilibrium values during the time of vibrational relaxation. At next moments, when chemical reactions start to proceed and that the gas temperature changes, the distribution over all internal degrees of freedom is in equilibrium at the local gas temperature. The proposed mechanism of electronic state excitation is mainly collisional in nature, and, for this reason, is universal. At the same time, the participation of other processes of electronic excitation is not excluded, for example, any resonance energyexchange processes and exothermic chemical reactions. This mechanism was used in processing the experimental data[223,224] where radiation of CN, C2 and atoms C was studied behind the front of a shock wave in gas mixtures CO/N2/Ar at temperatures T=4000-9500K. The kinetics of the CN, C2, C radiation in heated gas region behind the shock front was simulated by nonequilibrium chemical reactions with the assumption of equilibrium distribution over all internal degrees of freedom of molecules at the local gas temperature. The experimental time histories of radiation obtained in experiments were satisfactorily described by this mechanism (Fig.14).

Non-equilibriu um Kinetics behind Shock Waves. Experimental Aspects

139

Fig. 14. Experimental (black points) and simulated (solid curves) profiles of CN, C2 annd C radiation intensity (in relative units) behind a shock front in gas mixture 10% CO/20% N2/770% Ar at V=3.07 km/s, P1=8.1 Torrr[224]

The processes selectively y populating the vibrational levels of the excited termss of molecules may complicate a total picture of relaxation dependent on selecting souurce power. Such an example waas presented in study[225], where the CN radiation spectrra in bands of Δv=0 sequence (traansition B2Σ→X2Σ) were investigated behind a shock fronnt in Titan atmosphere (92% N2-3 3% CH4-5% Ar). The experiments were carried out for two cases: with an initial pressurre P1=1.5 Torr and a shock wave velocity V=5.56 km/s (ccase A) and P1=8.2 Torr, and V= =5.13 km/s (case B). An imaging spectroscopy technique w was used to supply simultaneou us wavelength-intensity-time information behind the shhock front. The observation of tim me-resolved spectra of the band of CN showed evidencee of non-Boltzmann distribution ns in the vibrational populations. In both cases, an overpopulation of the level v=6 was found. With distance from the shock front, this ore rapidly in case A. At the higher initial pressure (case B), overpopulation decreased mo the population for higher leveels (v=6 and higher) deviated from the Boltzmann distribution and this took place during th he entire recording. The studies of the CN spectra in CO2-N2 mixture[226] showed that vib brational relaxation of CN reached the Boltzmann distribuution very rapidly. The results obtaained in N2-CH4 mixture may be explained by the availabiility of any resonance energy-ch hange processes or even exothermic chemical reactions w with production of CN molecules mainly in highly vibrational levels of the excited state. In this e for the higher initial pressure in case B (that is, forr the case, the more pronounced effect greater rate of chemical reacttion) becomes comprehensible. The conditions in heaated gases corresponding to the entry into planettary atmospheres with velocitiess higher than 10 km/s and pressures lower than 1 Torr w were modeled[227-229]. The stu udies were carried out in a free-piston double-diaphraagm shock tube. A spatial variattion of air radiation spectra behind strong shock waves w was obtained by spatially resolv ved imaging spectroscopy. Two peaks of radiation behhind the shock front were observ ved[227] when detecting the profiles of integral radiationn in the range 300-445 nm. Th he radiation of molecular species (N2, N2+) gave a m main contribution to the first peak immediately after the shock front, whereas the atom mic me intense shortly after it. Spatial profiles of temperatuures line spectra (O+, N) becam were obtained[228,229] forr a shock velocity 11.9 km/s and an ambient pressure off 0.3 Torr ahead of the shock wave, w using pure nitrogen as a test gas. In Fig.15 the tiime histories of rotational, vibraational and electronic temperatures behind the shock frront are shown and compared with the calculated ones obtained using Park’s model[14].

140


a

b

Fig. 15 (a, b). Distributions of temperature (rotational (a) and vibrational (b)). Shock velocity is 11.9 km/s, ambient pressure is 0.3 Torr[228]

The estimated electronic excitation temperature of N was found to be approximately 4000K at a 5-mm distance behind the shock front and to increase very gradually with the distance from the shock. The vibrational temperature of N2 was close to this temperature, whereas that of N2+ was two times higher and quickly increased with the distance from the shock front. The rotational temperatures of N2 and N2+ were turned out to be below 6500 and 23000K within 3-mm distance from the shock front, respectively, that is, much lower than the translational temperatures simulated by two-temperature Park’s model. Thus, these results indicated that the region immediately behind the shock front was in strong non-equilibrium, even for the rotational mode (Fig.15). The rotational and vibrational temperatures obtained[228] with a higher initial pressure and a lower velocity (P1=2.1 Torr, V=8.14 km/s) showed good agreement in equilibrium region with simulations using the two-temperature model. In many studies, the shock-heated gas in the conditions of vehicle entry into the atmospheres of Mars (CO2-N2) and Titan (N2-CH4) was investigated[230-235]. The authors of these studies were faced with unsatisfactory modeling description of results of their experiments in conditions of low pressure and high shock speed (Fig.16). In studies[230,231] the necessity of developing more sophisticated models adequately describing collisional-radiative processes and excited state formation in nonequilibrium zones was noticed. Many questions remain on chemical reaction modeling, especially on dissociation and, more especially, on carbon dioxide dissociation. This is not a simple problem because reliable data on the rate coefficients of dissociation at temperatures higher than 10,000 K are lacking and experimental studies are difficult because of the strongly non-equilibrium nature of dissociation.

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

141

Fig. 16. Comparison between experimental and numerical spectra[230]

With increase of the temperature, the chemical reaction rates strongly increase, and near the shock front all the processes take place simultaneously and proceed in the absence of equilibrium with respect to internal degrees of freedom. In addition, for molecules such as CN, C2, N2, N2+, rotational and vibrational relaxations are essentially non-adiabatic processes, especially in excited electronic states, and often, apparently, they are inseparable from relaxation over electronic states221. Thus, there exists a diversity of situations for non-equilibrium radiation in shock waves depending on molecule and atom properties as well as on the gas conditions which have an influence on population and deactivation of excited states, namely gas composition, pressure, temperature, and species concentrations, including electrons and chemically active species.

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204. Davidson, D.F., Dean, A.J., DiRosa, M.D., Hanson, K.: Int. J. Chem. Kin. 23, 1035 (1991) 205. Ibraguimova, L.B., Kuznetsova, L.A.: Chem. Phys. Reports 23, 82 (2004) 206. Dean, A.J., Hanson, R.K., Bowman, C.T.: J. Phys. Chem. 95, 3180 (1991) 207. Dean, A.J., Hanson, R.K., Bowman, C.T.: In: International Symposium on Combustion, vol. 23, p. 259 (1990) 208. Lindeckers, D., Burmeister, M., Roth, P.:In: International Symposium on Combustion, vol. 23, p. 251 (1990) 209. Natarajan, K., Woiki, D., Roth, P.: Int. J. of Chem. Kinet. 29, 35 (1997) 210. Natarayan, K., Roth, P.: International Symposium on Combustion, vol. 21, p. 729 (1988) 211. Sims, J.R., Queffelec, J.L., Defrance, A., et al.: J. Chem. Phys. 100, 4229 (1994) 212. Sims, J.R., Smith, W.M.: Chem. Phys, Lett. 151, 481 (1988) 213. Burmeister, M., Gulati, S.K., Natarayan, K., et al.: In: Int. Symp. on Combustion, vol. 22, p. 1083 (1989) 214. Patterson, W.L., Green, E.F.: J. Chem. Phys. 36, 1146 (1962) 215. Faibairn, A.R.: J. Chem. Phys. 51, 972 (1969) 216. Sommer, T., Kruse, T., Roth, P., Hippler, H.: J. Phys. Chem. A 101, 3720 (1997) 217. Zaloguin, G.N., Lunev, V.V., Plastinin, Y.A.: Fluid Dynamics 15, 85 (1980) 218. Ibraguimova, L.B.: Zhurnal Prikladnoi Spectroskopii. J. of Applied Spectrosc. 28, 612 (1978) (in Russian) 219. Ibraguimova, L.B.: Chem. Phys. Reports 15, 939 (1996) 220. Dvoraynkin, A.N., Ibraguimova, L.B., Kulagin, Y.A., Shelepin, L.A.: Review of Plasma Chemistry, Consultants Bureau, NY, p. 1 (1991) 221. Benesch, W., Fraedrich, D.: J. Chem. Phys. 81, 5367 (1984) 222. Ibraguimova, L.B.: Chem Phys. Reports 15, 959 (1996) 223. Dushin, V.K., Ibraguimova, L.B.: Fluid Dynamics 16, 253 (1981) 224. Ibraguimova, L.B., Losev, S.A.: Kinetika i Kataliz 24, 263 (1983) (in Russian) 225. Ramjaun, D.H., Dumitrescu, M.P., Brun, R.: In: Proc. 21th Int. Symp. Rarefied Gas Dynamics, vol. 2, p. 361 (1999) 226. Dumitrescu, M.P., Ramjaun, D.H., Chaix, A., et al.: In: Proc. 20th Int. Symp. Shock Waves (1997) 227. Morioka, T., Sakurai, N., Maeno, K., Honma, H.: In: Proc. 21th Int. Symp. Rarefied Gas Dynamics, vol. 2, p. 345 (1999) 228. Fujita, K., Sato, S., Ebinuma, Y., et al.: In: Proc. 21th Int. Symp. Rarefied Gas Dynamics, vol. 2, p. 353 (1999) 229. Fujita, K., Sato, S., Abe, T.: J. of Thermophysics, Heat Transfer 16, 77 (2002) 230. Rond, C., Boubert, P., Felio, J.-M., Chikhaoui, A.: Chemical Physics 340, 93 (2007) 231. Boubert, P., Rond, C.: J. of Thermophysics, Heat Transfer 24, 40 (2010) 232. Grinstead, J.H., Wright, M.J., Bogdanov, D.W., Alen, G.A.: J. Thermophysics, Heat Transfer, 23, 249 (2009) 233. Lee, E., Park, C., Chang, K.: J. Thermophysics, Heat Transfer 21, 50 (2007) 234. Lee, E., Park, C., Chang, K.: J. Thermophysics, Heat Transfer 23, 226 (2009) 235. Brandis, A.M., Morgan, R.G., McIntyre, T.J., Jacobs, P.A.: J. Thermophysics, Heat Transfer 24, 291 (2010)

Chapter 4 Ionization Phenomena behind Shock Waves W.M. Huo1,*, M. Panesi2, and T.E. Magin3 1

2

NASA Ames Research Center, Moffett Field, U.S.A. Institute for Computational Engineering and Sciences, University of Texas, Austin U.S.A. 3 Aeronautics and Aerospace Department, Von Karman Institute Rhode-S-Genèse, Belgium

1 Introduction During the hypersonic entry of a space vehicle into a planetary or lunar atmosphere, the flow field becomes partially ionized. The percentage of ionization depends on the entry speed and the vehicle size. The electrons and atomic/molecular ions produced by the ionization introduce new reaction mechanisms that significantly influence the radiative and convective heat loads. Thus a simulation of the flow field behind the shock wave needs to incorporate the production of ionic species, chemical reactions due to charge-neutral and charge-charge interactions, and the removal of the charged species by recombination. In this regime, electron collision provides an efficient means of producing electronic excited states of the atoms and molecules in the flow field. Thus it plays an important role in determining the internal energy and state distribution of the gaseous particles. The excited states in turn are the source of radiation observed during a hypersonic entry. Electron-atom/molecule collisions differ from heavy particle (atom-atom, atom-molecule and molecule-molecule) collisions in two aspects. First, the mass of an electron is more than four orders of magnitude smaller than the reduced mass of N2. Thus its average speed, and hence its average collision frequency, is more than 100 times larger. Even in the slightly ionized regime with only 1% electrons, the frequency of electronatom/molecule collisions is equal to or larger than that of heavy particle collisions, an important consideration in the low density part of the atmosphere where the reaction probability is frequently controlled by the collision frequency. Second, the interaction potential between a charged particle (electron) and a neutral particle is longer range than neutral-neutral interactions. Hence electron-atom/molecule collision cross sections tend to be larger. A characteristic of electron collisions is that it generally produces a variety of excited states whereas heavy particle collisions tend to produce specific excited states. Also, low-energy electron collisions can be effective in spin changing excitations. Recombination of electron and ions removes the charge particles from the flow field. Radiative recombination produces continuum radiation extending from the VUV to the far infrared, providing a radiation source for photoionization downstream. Due to their heavier mass, ions are not as effective in generating excited states of atoms and molecules as the free electron, but charge transfer between ions and neural *

Ames Associate.

150


atoms or molecules provides a means of generating new ionic species. Beside the radiation emitted upon recombination with electrons, excited states of ions also are a source of radiation from bound-bound transitions. Their spectra, however, are often imbedded under the more intense spectra of the neutral species. Modeling electron and ion collisions in nonequilibrium gas dynamics requires data to simulate their production and removal in the flow field. In the non-equilibrium regime their number densities, temperature, and reaction rate coefficients are part of the input data needed to determine the populations of the radiative species. Using a combination of experimental data and a variety of approximate formulas, a number of databases have been developed in the entry physics community. The data sets by Park[1,2], Losev[3] and Bird’s TCE[4,5] models are well-established examples of this approach. The NEQAIR package[6] that simulates nonequilibrium radiation in an entry flow employs Gryzinski’s classical formula [7] for electron-impact excitation of atoms whereas for molecules experimental data are used, sometimes by extrapolation or by analogy. More recent models have incorporated improved databases, based on new experimental data and/or theoretical calculations. The collisional radiative model by Bultel et al.[8] includes many updates. Similarly SPRADIAN07[9] incorporated new, improved data into the NEQAIR model. This chapter reviews the electron and ion collision processes relevant to hypersonic entry. Most of the examples given are for air. Since rare gases are also used in shock tube experiments, selected references to rare gas data are provided. In the ionized regime, molecules are mostly dissociated. Thus the discussions mainly concern the atomic species. Emphasis is on the collision data obtained either by using a quantum mechanical method or recent experimental data. A one-dimensional flow solver coupled to the gas kinetics based on a collisionalradiative (CR) model[8] and to the radiative transfer equation is used to illustrate how the ionization process influences the flow properties and its effect on the radiative and convective heat load. In the application considered here, where entry speeds exceed 9-10 km/s, the primary contributor to the radiative processes are atomic species (mainly nitrogen atoms), which account for about 90% of the overall radiation output. Note that a realistic representation of the ionization and radiative processes, occurring in shock heated air, can only be achieved through the explicit calculation of the population of the atomic electronic states using a state-to-state description of the gas kinetics, i.e., by treating the quantum states of atoms as separate pseudospecies[74,75]. Often in the literature the calculation of the radiation field is decoupled from the solution of the flow-field quantities (species densities, temperatures etc.) and escape factors are used in the flow equations to model the effects of the radiative processes on the population of the excited states[74,75,76,9]. The analysis of ionizing flow adopting this simplified treatment of radiation clearly shows the strong dependence of the result on the assumptions made when selecting the escape factors[74,75,77] (e.g. thin or thick assumptions). By substituting escape factors with the source terms resulting from the solution the radiative transfer equation, a fully consistent treatment of the radiation processes is employed in this chapter. The influence of the radiation processes on the population of the excited states as well as the cooling effects is thus correctly modeled. Modeling examples using this approach are presented using the conditions of FIRE II flight experiment and EAST shock tube facility.

Ionization Phenomena behind Shock Waves

151

2 Production, Reaction and Removal of Charged Species 2.1 Electrons Due to the light electron mass, in general e-atom/molecule collisions should be treated using quantum mechanics instead of classical mechanics. The Schrödinger equation for an e-atom/molecule system is given by ( H − E ) Ψ (τ 1

τ N +1 , R 1

RM ) = 0

(1)

Here H is the Hamiltonian of the e + target system, where the target is an atom or molecule, E the total energy, Ψ the corresponding wave function, τi the spatial (ri) and spin (si) coordinates of the ith electron and RK the spatial coordinate of the Kth nucleus. The indices i and j label the bound electrons, and N+1 labels the free electron. Due the large difference between electron and nuclear mass, it is common to use the center of mass of the target as the center of mass of the colliding system. The total Hamiltonian consists of the target Hamiltonian, HA, the kinetic energy operator of the free electron, Te, and the Coulomb potential V between the free electron and the target. H = H A +T e + V , HA = −

N 1 N 2 1 M 2 ∇i − ∑ ∇ K + ∑ ∑ 2 i =1 2 K =1 j >i

N

∑r i =1

M N M M 1 ZK ZK ZL − ∑∑ +∑∑ − r R R − ri K =1 i =1 i L > K K =1 K L − RK j

1 2

T e = − ∇ 2N +1 N

V =∑ i

,

M 1 ZK −∑ rN +1 − ri K =1 rN +1 − R K

(2)

,

(3)

(4)

(5)

Here ZK is the charge of the Kth nucleus. The origin of the coordinate system is chosen to be at the center of the mass of the target. Note that HA in Eq. (3) is the nonrelativistic Hamiltonian of the target. Even for light atoms such as N and O and molecules consisting of light atoms, relativistic corrections are generally required to obtain accurate energy levels. The solution to Eq. (1) are obtained by first solving the target Schrödinger equation ( H A − E A ) Φ (τ 1

τ N , R1

RM ) = 0

(6)

The solution of Eq. (6) is then used as a building block for solving Eq. (1). In the range of incident electron energy of interest to hypersonic flow modeling, from threshold to ≈100 eV, the close coupling method[10] provides the most reliable solution of Eq. (1)

152


Ψ(τ 1

τ N +1,R1

∞

R M ) = ∑ A {f m (τ N +1 )Φ m (τ1 m =1

τ N ,R1

R M )}.

(7)

The antisymmetrizer A permutes the free electron with the bound electrons to account for the fact that electrons are indistinguishable and must satisfy Fermi statistics. The summation is over all possible states of the target, including the continuum. The summation in Eq. (7) becomes integration in the continuum region. The excitation/ionization cross section is obtained by analyzing the asymptotic behavior of the function fm. Because there are an infinite number of discrete and continuum target states, for practical reasons the summation is necessarily truncated. The success of a close coupling calculation depends on the quality of the target wave functions used and the number of terms included in the summation. Two of the most successful closed coupling approaches used in e-atom collisions are the convergent close-coupling (CCC)[11-13] method and the R-matrix with pseudo states (RMPS)[14, 15]. The CCC method expands target states using squareintegrable functions. The convergence of such representation is tested by successively increasing the size of the basis. While the CCC method has demonstrated excellent results, so far its application is limited to atoms/ions with one or two electrons outside a closed shell core, e.g., the alkali and alkaline earth atoms. The current version is not applicable to N, O, and their ions that are important in the Earth entry environment. The RMPS method is an extension of the R-matrix method[16]. In the R-matrix approach the scattering problem is separated into two regions. Inside the R-matrix hypersphere of electronic radius r = ae, the full N+1-electron Schrödinger equation is used with the proper boundary condition. Outside ae, the collisional system is regarded as consisting of a well separated atom and the scattering electron. Analogous of the close coupling method, the quality of the calculation depends on the number of terms used in the expansion in Eq. (7). Insufficient number of terms in the expansion may lead to pseudo resonances, i.e., a false trapping of the electron. The RMPS method corrects this problem by constructing target functions using both physical wave functions determined from the solution of Eq. (6) and pseudo states. The latter approximately represent the high-lying bound states and continuum states. This method is applicable to atoms/ions of arbitrary structure. A more recent version of the R-matrix method, the B-spline RMPS method, employs nonorthogonal one-electron orbitals that lead to a more compact calculation. Simulation of nonequilibrium gas dynamics requires a complete set of cross sections involving all possible initial and final states of the atom/molecule. The NIST database[17] lists 381 levels for N atom and many high-lying Rydberg states are known to be missing in this database. Thus a very large set of cross-section data are needed. As discussed in the above, modern quantum mechanical calculations can provide reliable cross section data but the accurate calculations are limited to a small set of low-lying states for a given atom, or applied only to a particular class of atoms. Therefore it is necessary to employ more approximate methods. At present the best practice is to use experimental or more accurate quantal treatment when available, and supplement the data set using more approximate treatments.


153

2.1.1 Production of Electrons In the neutral atmosphere immediately behind the shock, molecules are first dissociated to form atoms. The initial production of electrons is achieved by associative ionization of two atoms. In the Earth’s atmosphere, these reactions are N + O → NO + + e,

(8)

N + N →N2+ + e,

(9)

O+ O →O2+ + e.

(10)

In Eqs. (8) – (10), the internal states of the reactants and products are not specified. This notation implies that a number of state-to-state reactions of the same type are possible. For a specific state-to-state reaction, the internal states are explicitly written out. See, for example, Eqs. (15) – (17). This convention is used throughout the chapter. Reaction (8) has the lowest threshold among the three reactions and it dominates the initialization of electron production in air. At high entry velocity, as the electron number density increases electron-impact ionization becomes dominant. N + e → N + + 2e,

(11)

O+ e →O+ + 2e.

(12)

The VUV radiation emitted upstream can also photoionize the neutral species downstream.

N + hν →N + + e.

(13)

O + hν →O+ + e.

(14)

Precursor photoionization has been reported previously[18]. (a) Associative Ionization Associative ionization (AI) is a resonant process involving both electronic and nuclear motions. In the simplest case (called the direct process), the potential energy curve for the relative motion of the two atoms crosses with the potential curve of the ion. Simultaneously at the crossing region the electronic wave functions of the two atoms are at resonance with a compound state of the diatomic ion and an electron. Autoionization can occur, producing a free electron. Thus Eq. (8) can be more explicitly written as

N + O ⇔ NO** → NO+ + e.

(8a)

Figure 1, taken from Vejby-Christensen et al.[19], illustrates this process. The ground states of the atoms, N (4S) and O (3P), can follow the A′ 2Σ+ curve and cross the NO+ (X1Σ+) ground state at the right limb of its potential curve. Autoionization at the proximity of the crossing point produces the ground state NO+ and an electron.

154

High Temperature Ph henomena in Shock Waves

The potential curves for ex xcited states of the two atoms can also cross the ion currve. For example, N (2D) and O (3P) can follow the B 2Π, B′ 2Δ and L 2Π curves and crross the NO+ ground state. An indirect i process, involving first crossing with the potenntial curve of a Rydberg state of molecule that converges to the ion state, can also occurr. mental and theoretical studies are devoted to the inveerse The majority of experim process, dissociative recom mbination (DR). The AI rate coefficient is then obtaiined using detailed balancing. In I the past two decades, storage ring experiments hhave greatly advanced the knowlledge on DR, including those for atmospheric ions. VejjbyChristensen et al.[19] and Hellberg H et al.[20] reported the DR cross sections and the branching ratios for the production of N (4So) + O (3P), N (4So) + O (1D) and N (2Do) + O (3P) from v = 0 of NO+. NO+ (X 1Σ+ ,vv = 0) + e →N( 4 S o ) + O( 3 P) + 2.7eV, NO+ (X 1Σ+ ,vv = 0) + e →N( 4 S o ) + O(1 D) + 0.80eV, +

1 +

((15)

NO (X Σ ,vv = 0) + e →N( D ) + O( P) + 0.38eV . 2

o

3

Hellberg et al. found 95% of the product to be N (2Do) + O (3P), 5% N (4So) + O ( P), and the production of N (4So) + O (1D) is negligible. Employing the Multichannel Quantum Deefect method[21] and potential curves determined ussing R-matrix[22] calculations adjusted to match experiment, Motapon et al.[[23] calculated the rate coefficieents for DR covering v = 0 – 14 of NO+ that can be usedd in plasma modeling. It should d be pointed out the application of detailed balancingg to determine AI rate coefficiients requires the reactants and products to be uniquuely defined. The DR rate coeffiicients tabulated in Ref.[23], on the other hand, correspoond to a mixed product of N (4So) + O (3P) and N (2Do) + O (3P). Their branching ratioo for v > 0 of NO+ have not beeen reported. Thus additional data is required for a uniique determination of the AI ratee coefficients. 3

Fig. 1. Reaction pathways for the associative ionization of N + O. The solid curves are forr the neutral N + O system whereas the dotted curve is for NO+. This figure is taken from Ref. [199].


155

As seen in Eq. (15), the electron production via AI is a competition between the reaction of the more abundant N (4So) with O (3P), but with a smaller cross section and higher threshold kinetic energy, versus the less abundant N (2Do) but with a larger cross section and lower threshold kinetic energy. Since N (4So) is expected to be the major dissociation product of N2 immediately behind the shock, the AI rate in this region is relatively small. For N2+ Peterson et al.[24] measured the DR rate coefficients for electron energy 10 meV – 30 eV. They found the DR rate coefficients to be weakly dependent on the N2+ vibrational level. At zero electron energy and v = 0 level of the ion, they determined the branching ratio of the dissociation products N 2 + ( X 2Σ g + , v = 0) + e → N ( 4 S o ) + N ( 4 S o ) + 5.82eV , N 2 + ( X 2Σ g + , v = 0) + e → N ( 4 S o ) + N (2 D o ) + 3.44eV , N 2 + ( X 2Σ g + , v = 0) + e → N ( 4 S o ) + N ( 2 P o ) + 2.25eV ,

(16)

N 2 + ( X 2Σ g + , v = 0) + e → N ( 2 D o ) + N ( 2 D o ) + 1.06eV .

to be 0:0.37:0.11:0.52. Even with the weak dependence of the DR rate coefficient on v, a unique determination of AI rate coefficients from the DR data is still not possible because the branching ratio is determined only at a single electron energy. The DR of O2+ has been measured by Peverall et al.[25] at electron energy 1meV – 3eV. O2+ (X 2Πg+ ,v = 0) + e →O( 3P) + O( 3 P) + 6.65eV, O2+ (X 2Πg+ ,v = 0) + e →O( 3P) + O(1 D) + 4.99eV, O2+ (X 2Πg+ ,v = 0) + e →O(1 D) + O(1 D) + 3.02eV, +

+

(17)

O2 (X Πg ,v = 0) + e →O( P) + O( S) + 2.77eV, 2

3

1

O2+ (X 2Πg+ ,v = 0) + e →O(1D) + O(1S) + 0.80eV,

The branching ratio for the formation of O (3P) + O (3P), O (3P) + O (1D) and O ( D) + O (1D) is 0.20:0.45:0.30 at zero electron energy. The branching ratio of O (3P) + O (1S) is negligible and for O (1D) + O (1S) it is less than 0.06. 1

(b) Electron-Impact Ionization There are two mechanisms in electron-impact ionization. The first is direct ionization where the colliding electron directly detaches a bound electron from the atom or molecule, as shown in Eqs. (11) and (12). The second mechanism is via autoionization. It is an indirect process where the atom is first excited to a metastable bound electronic state that lies in the continuum. As example consider the N atom,

N + e →N * + e →N + + 2e.

(18)

156


The lowest state of N+ is the 2s22p2 3P state. The next state of the ion is the 2s22p2 D state that is 15,316.2 cm-1 above the ground 3P state. Thus the series of neutral atomic states with the configuration 2s22p2(1D)nl converging to the 2s22p2 1D ionization limit include states that lie above the 2s22p2 3P ion state. One state of N atom that lies above the first ionization limit is the 2s2p4 2D state. It is 3974.8 cm-1 above the ground state of N+. Since these states lie above the first ionization limit they are metastable states. It can decay by radiation to a neutral state that is below the first ionization limit, or it can autoionize and emitted an electron. In Fig. 2 the initial state 2s22p3 2Do of N atom is first excited to the metastable state 2s2p4 2D by electron impact. The metastable state subsequently emitted an electron and produces the 2s22p3 3P state of N+. 1

Fig. 2. Schematic diagram of the indirect ionization process

The total electron-impact ionization cross section is the sum of the contributions from direct ionization and autoionization. σ I = σ DI + σ Auto

(19)

The cross term between direct and autoionization has been neglected. Generally experimental measurements of electron impact ionization are available for the ground state of the atom or low-lying metastable states, but not high-lying excited states. Quantum theory, on the other hand, has a demonstrated record of successful calculations of the total ionization cross sections by electron impact[26] and can be applied to excited states as well as the ground state. Different methods are used to calculate direct ionization and autoionization cross sections. The improved Binary-Encounter-Dipole (iBED) model[27] expresses the direct ionization cross section in two terms. σ DI = σ BinaryEncounter + σ BornDipole

(20)


157

The Binary-Encounter cross section σBinaryEncounter describes the close collision between the free electron and bound electron and the dipole Born cross section accounts for the long-range interaction between the free electron and the target. The autoionization cross section through a specific metastable state m is given by the product of the electron-impact excitation cross section σim from the initial state i to the metastable state m and the ionization probability of the metastable state, PmI.

σ Auto = PmI σ im

(21)

The ionization probability is given by PmI =

k I ,m k I ,m + k R ,m

,

(22)

with kI,m and kR,m the ionization and radiative rate coefficients of state m. Generally, autoionization can occur through several metastable states. In that case the total autoionization cross section is expressed as a sum of these processes. While direct ionization applies to any state of the atom, autoionization is possible only if electron impact can excite the initial state to an autoionizing state. Figure 3 presents σI of the 4So and 2Do states of N atom. For 2Do the two components of σI, σDI and σAuto, are also presented. σDI is calculated using the iBED method[28] whereas σAuto with 2s2p4 2D as the intermediate is from the calculation of Kim and Desclaux[29]. For the 4So state, the autoionization probability is expected to be small. Thus the σI of 4So includes only σDI. Figure 3 shows σDI of 2Do larger than that of 4So, mainly due to the lower ionization threshold. Note σDI and σAuto of 2Do have significantly different electron energy dependence. While σDI consistently increases with energy between threshold to 50 eV, σAuto reaches a plateau and then remains almost constant with energy. Also, σAuto is consistently smaller than σDI except near the threshold. However, under most entry conditions only the high-energy tail of the electron energy distribution reaches the ionization threshold. Thus the larger σAuto near the threshold makes it an important path for ionization. Figure 4 presents the ionization rate coefficients for 10 states of the N atom with the outermost electron at n=2 and 3[30,31]. The states are 2s22p3 4So, 2s22p3 2Do, 2s22p3 2Po, 2s22p2(3P)3s 4P, 2s22p2(3P)3p 4Do, 2s22p2(3P)3p 4Po, 2s22p2(3P)3d 4F, 2s22p2(3P)3d 4D, 2s22p2(3P)3s 2P, 2s22p2(1D)3s 2D. It is seen that the rate coefficients separate into two groups. The three lowest states with the outermost electron at n=2 are more tightly bound and have smaller ionization rate coefficients than the seven states that has one electron in the n=3 shell. The difference is particularly striking at low electron temperatures. The large ionization rates of the upper states lends to the possibility that the upper states will first reach Saha equilibrium with the free electrons and ions before they reach Boltzmann equilibrium with the lower states.

158


Fig. 3. Ionization cross sections for the ground 4So and first excited 2Do state of N atom.

Fig. 4. Ionization rate coefficients for 10 states of N atom as a function of electron temperature.

Electron-impact ionization cross sections for the ground states of O and C atoms have been calculated by Kim and Desclaux[29]. Straub et al.[32] measured the partial and total ionization cross sections of Ar.


159

(c) Photoionization In analogy to electron-impact ionization, photoionization consists of direct ionization and autoionization,

N + hν →N + + e,

(23)

N + hν → N * → N + + e.

(24)

The TOPBase data from the Opacity Project[33] tabulates the photoionization cross sections for a large collection of atoms as a function of their electronic states and photon frequency, including the data for N, O, and C of interest to Earth entry. It should be noted, however, that the TOPBase tabulation does not provide information on the final ion state. Detailed discussions on photoionization are given in Chapter 5. 2.1.2 Reactions Involving Electrons Electron collisions provide an efficient means for electronic excitation/de-excitation of atoms and molecules. In the ionized regime, this is a major source of radiative species. In addition, an electron can be attached to an atom or molecule to form a negative ion and molecules can be vibrationally excited/de-excited and dissociated by electron-impact. (a) Electronic Excitation by Electron-Impact Modeling the distribution of electronic excited states of atoms in a plasma requires a complete set of electron-impact excitation cross sections. However, both experimental and theoretical cross-section data cover only a small number of initial and final states. Experimental data on e - N, O collisions are sparse. The review of Laher and Gilmore[34] on O atom covers data prior to 1990. Landolt-Börnstein[35] covers the experimental database up to 1990’s. Also, the International Atomic Energy Agency (IAEA) Atomic Molecular Data Services[36] includes electron collisions in their database. For molecules a recent review is by Brunger and Buckman[37]. Recent theoretical calculations using improved quantal treatments produce cross section data with accuracy comparable to experiment. These include the B-spline RMPS calculations of N38, O39, and Ar40. Figure 5, taken from Tayal and Zatsarinny[38], presents the electron-impact excitation cross section for the 4So – 2Do transition in the N atom. It compares the experimental data by Yang and Doering[41] with the B-spline RMPS calculation by Tayal and Zatsarinny using 24 spectroscopic bound and autoionizing states together with 15 pseudo states in the close coupling calculation. The pseudo states are determined by the requirement that the polarizability for the ground 2s22p3 configuration is approximately accounted for. Earlier R-matrix calculations[38,42,43,44] are also included in the comparison. The oscillatory structures in the older calculations are due to pseudo resonances. These pseudo resonances have been removed in the latest calculation[38] by increasing the size of the calculation and the use of the pseudo states. The sharp structures between 10 – 13 eV in the RMPS curve are real resonance structures due to the transient formations of the N- ion [38]. Theoretical calculations are in general agreement with experiment[41]. Unlike the close coupling approach, perturbation theory can be used to treat excitation to high-lying electronic states without the corresponding problem of

160


increasing the size of the calculation. The Born approximation is a first order perburbation treatment.[45] Due to the simplicity in its calculation, it is frequently used in the database for plasma modeling. However, while the Born approximation describes long-range interactions such as dipole and quadrupole interactions well, it does not account for electron exchange, the polarization of the target electrons by the free electron, and the distortion of the free electron by the target. These features are described by higher order terms in the perturbation series. Thus the Born approximation is generally applicable only at high electron energies. The energy regime of interest in modeling hypersonic flow is too low for it to be valid. The BE scaling method by Kim[46] incorporates the high order effects approximately into the Born cross section by the use of energy scaling. This method has been applied successfully in treating electron collisions with neutral atoms and ions[46,47]. It should be noted, however, that this method does not include resonance effects, that is, the enhancement of the collision cross section due to the formation of a transient compound state. In this approach resonances should be treated separately. The distorted wave approximation is another perturbation approach. Here the incoming and outgoing electrons are described by elastically scattered waves and their coupling is treated by the first order Born approximation. This approximation is employed by the HULLAC[104] code in the calculation of electron-impact excitation cross sections.

Fig. 5. Electron-impact excitation cross section (10-22 m2) for the 4So – 2Do transition in N atom as a function of electron energy (eV). Solid curve: Tayal and Zatsarinny[38] 39-state B-spline RMPS result; long-dashed curve: Tayal and Zatsarinny[38] 21-state R-matrix result; shortdashed curve: 8-state R-matrix calculation of Berrington et al.[42]; dashed-dotted curve: 7-state R-matrix calculation of Ramsbottom and Bell[43]; dotted curve: 11-state R-matrix calculation of Tayal and Beatty[44]; diamonds: experimental cross section of Yang and Doering[41]. This figure is from Ref. [38].


161

Figure 6 presents the rate coefficients for the electron-impact excitation of 4So – 2s 2p2(3P)3s 4P of N atom. The three theoretical curves are calculated using the B-spline RMPS method[38], a 33-state R-matrix method[48] and the BE scaling method[31]. The experimental curve labeled as Stone and Zipf are calculated from their cross-section data[49] as recalibrated by Doering and Goembel[50]. The crosssection measurement of Doering and Goembel[50] does not cover the region below 30 eV electron energy and hence cannot be used to deduce rate coefficients. Figure 6 also include a single data point from the arc chamber measurement of Frost et al.[48] at 52,220 K. The B-spline RMPS and BE scaling curves are in good agreement whereas the R-matrix curve of Frost et al. is consistently larger. The experimental curve from Stone and Zipf are also higher than the B-spline RMPS and BE scaling curves, where the arc chamber measurement of Frost et al. is lower. The arc chamber data at higher electron temperatures, not shown in Fig. 6, are also lower than the B-spline RMPS/BE scaling curves. The above examples indicate that the optimal approach to build an electron-impact excitation data set is to use a combination of quantal methods with guidance from available experimental data. 2

(b) Vibrational Excitation by Electron-Impact Since the ionization potentials of air molecules are higher than their dissociation energies, electron production in the shock layer occurs after molecular dissociation has initiated. Thus e-molecule collisions do not play as important a role as e-atom collisions. Note, however, there is a significant resonance enhancement in the vibration excitation cross sections of N2 and O2 by electron impact[51,37]. For N2 the resonance enhancement extends to high vibrational levels[52,53]. As a result of this resonance, the e + N2 vibrational excitation rate coefficients are two orders of magnitude larger than the corresponding N + N2 vibrational excitation rates. Figures 7 and 8 compare the two sets of rate coefficients at initial v = 2 and 10, respectively, and both translational temperature (T) and electronic temperature (Te) at 10,000 K. The N + N2 vibrational excitation rate coefficient[54] is obtained from a weighed sum of rovibrational rate coefficients, K v →v ′ (T) = ∑ PvJ (T)K vJ →v ′J ′ (T), JJ ′

(25)

with PvJ(T) the statistical weight of the initial (v, J) level, KvJ→v’J’(T) the rovibration excitation rate coefficient and Kv→v’(T) the vibrational excitation rate coefficient. The rotational temperature is assumed to be the same as the translational temperature. The electron-impact vibrational excitation rate coefficient is calculated at J = 50[55] (26) The magnitude of this e + N2 vibrational excitation rate coefficient implies the e-N2 vibrational excitation/de-excitation may play a role in determining electron temperature in the nonequilibrium regime in the shock layer. The e-N2 vibrational rate coefficient have been tabulated by Huo et al.[55] up to v = 12 and J = 50.

162


Fig. 6. Electron-impact excitation rate coefficient for the 4So – 2s22p2(3P)3s 4P transition in N atom as a function of electron temperature. Theoretical curves are from the B-spline RMPS calculation of Tayal and Zatsarinny[38], 33-state R-matrix calculation of Frost et al.[48] and the BE-scaling method[31]. The experimental curve of Stone and Zipf[49] are calculated from their cross-section data as recalibrated by Doering and Goembel[50]. A single data point from the arc chamber measurement by Frost et al.[48] is also included.

1010 v = 2, T/T = 10,000K

108

3

Rate coefficient (m mole

-1

-1

s )

e

10

9

107 106 105

N+N 2 e+N 2

104

0

5

v'

10

15

Fig. 7. Comparison of e + N2 and N + N2 vibrational excitation rate coefficients at initial v = 2 and T = Te = 10,000 K. The e+ N2 data are from Ref. [55] and the N + N2 data are from Ref. [54].


163

Fig. 8. Comparison of e + N2 and N + N2 vibrational excitation rate coefficients at initial v = 10 and T = Te = 10,000 K. The e+ N2 data are from Ref. [55] and the N + N2 data are from Ref. [54].

(c) Electron-Impact Dissociation Electron-impact excitation of a molecule to a dissociative state leads to dissociation. The process is written as N 2 + e → N 2* + e → N + N + e.

(27)

N2*

Here denotes an electronic excited state of N2 that is dissociative. Cosby[56,57] measured the electron-impact dissociation cross sections of N2 and O2. His paper also reviewed older data for this process. (d) Electron Attachment For atom and molecules with positive electron affinity, electron collision can lead to the formation of a stable ion with the release of the excess energy. For atoms the energy is released by emitting a photon. The electron attachment of O atom is written as O + e →O− + hν .

(28)

The reverse process, photo detachment, has been studied both experimentally [58,59,60] and theoretically[61,62,63,64]. The cross section for electron attachment can be deduced using microscopic reversibility. For molecules the excess energy in the attachment can be transformed into vibrational energy. For example, the Bloch-Bradbury mechanism for O2¯ production

164


first forms a vibrational excited ion O2¯ *. The vibrational excitation is subsequently quenched by a second collision. O2 + e →O2−*, O2−* + M →O2− + M.

(29)

Electron attachment of O2 has been reviewed by Hatano and Shimamori[65]. The nitrogen species, N and N2, have negative electron affinity and no stable negative ions exist. Using a high-resolution electron beam experiment of the dissociative attachment of N2, Mazeau et al.[66] determined that the ground (3P) state of N¯ is located at 0.07 ± 0.02 eV above the ground (4So) state of N atom. Theoretical calculations of Thomas and Nesbet[67] give N¯ (3P) 0.1 eV above N (4So). Thus unlike the O atom, electron attachment of N and the reverse reaction, photo detachment of N¯ , are not possible. Instead, the presence of N¯ is seen as resonance structures in e - N collision cross sections. Similarly, the ground 2Πg state of N2¯ is seen as prominent resonance structures in the elastic and vibrational excitation cross sections in e - N2 collisions between 2 – 5 eV electron energy. See discussion in Sec. 2.1.2b. The e - N2 data have been reviewed by Brunger and Buckman[37]. 2.1.3 Electron Recombination Recombination of electrons and ion is the reverse of the ionization process discussed in Sec. 2.1.1. Thus radiative recombination is the reverse of direct photoionization in Eq.(23) and dielectronic recombination is the reverse of autoionization in Eq. (24). Both processes remove the charge species through the formation of a neutral atom and emit a photon. They can produce significant radiative heat load during the vehicle entry and detailed discussions are given in Chapter 5. It is worth noting that the TOPBase data[33] currently available do not distinguish transitions to different final states. This prevents the use of microscopic reversibility to deduce the recombination cross sections from the photoionization cross sections. The AMDPP (Atomic and Molecular Diagnostic Processes in Plasmas) database[68], on the other hand, provides state-to-state partial radiative recombination rate coefficients and partial dielectronic recombination rate coefficients that may be used for modeling purposes[28,30,31]. Dissociative recombination is an important recombination pathway for molecular ions. This process is discussed in Sec. 2.1.1. 2.1.4 Interparticle Interactions and Free Electron Number Density The atomic and molecular data discussed so far are either calculated for an isolated atom/molecule or from measurements extrapolated to zero pressure. The presence of neighboring atoms or molecules in a real plasma means the data must be modified to account for the effect of interparticle interactions. Griem[69] uses Debye shielding to describe the influence of all the charged species surrounding an atom and deduces an approximate expression for the lowering of ionization potential of the atom in a plasma. However, neutral-neutral interactions are ignored. A more general approach is the occupation probability formalism of Hummer and Mihalas[70] where the probability of finding a particle occupying states i is calculated directly from a physical description of interparticle interactions. This approach has been applied to astrophysical plasma with some success, but has not yet been applied to entry plasma.


165

2.2 Ions

The production of ions and electrons are in pairs. Thus the discussion in Sec. 2.1.1 for electron production also applies to ion production. Similarly the removal of ions by recombination is also paired with electrons and Sec. 2.1.3 also applies to ion removal. However, the reactions of ions with neutral particles are different from electrons. The heavier ion mass means the collision frequency between ions and neutral are two orders of magnitude smaller than electron-neutral collision frequency. The heavier mass of the ion also mean that ion-atom/molecule collisions may be amenable to semi-classical or quasi-classical calculations. Charge transfer between ion and neutral does not have an analogy in electron collisions. In nitrogen discharges with collision energy under 10 eV, fast N atoms are produced mainly by the charge exchange mechanism[71]. Freysinger et al.[72] reported measurements of the charge transfer reaction, N + + N 2 →N + N 2+ ,

(30)

from thermal to 100 eV. Excited states of ions can be produced by electron-impact and bound-bound transitions from these excited states provide another source of radiation. The R-matrix method, RMPS, and the BE-scaling method have all been applied to e-ion collisions.

3 Modeling Collisional and Radiative Processes in a Weakly Ionized Plasma 3.1 The Collisional-Radiative Model

The modeling of the kinetic and radiative processes and the analysis of available experimental and in-flight data put forward the importance of ionization processes in shocked heated air[76]. The radiative and convective heat loads generated during Earth entry are also shown to be strongly dependent on the degree of ionization in the gas. In ionizing air, the formation of the first electrons is due to the association of N and O atoms into NO+. This is the first step in a two-step ionization process. It is favored by the relatively low activation energy[73]. Also the reaction does not require the presence of charged particles. Thus it is well suited as the initialization step. In a second step, when the number of electrons is sufficiently large, the high-speed electrons ionize the neutral atoms, rapidly increasing the electron density. The accurate modeling of this two-step process requires the correct modeling of the thermo-chemical relaxation leading to the formation of the atomic species (e.g. dissociation), as well as a detailed modeling of the excitation and ionization of the atomic species, contributing to the production of electrons in the hypersonic regimes. One important aspect in the present discussion is the explicit coupling of the kinetic equations with radiation by solving of the radiative transfer equation, instead of employing an escape factor to approximately account for the effect of the radiative processes on the on the population of the excited states. This results in a consistent modeling of the radiative processes.

166


Two important issues for re-entry applications into the earth’s atmosphere are considered in this section: (1) the treatment of the time-dependent chemistry of an ionizing air plasma for different re-entry conditions; (2) the importance of a consistent treatment of kinetics and radiation, via the numerical solution of the radiative transport equation coupled with the kinetic set of equations. 3.1.1 Transport Equations In this section, consideration is given to the governing equations used to describe the dynamics of multi-component, multi-temperature, chemically reacting, relaxing and emitting flows. The detailed derivation of the equations used in this work is omitted, and instead the discussion is limited to a brief analysis of the fundamental assumptions underlying the mathematical construct. Further information concerning the derivation can be found in Refs.[78,79,80]. The air mixture used in this work comprises 95 chemical components, including the electronic energy levels of atomic nitrogen and oxygen. The populations of the vibrational energy levels of the molecules (N2, NO, O2, N2+, NO+, O2+) are assumed to follow Boltzmann distributions and share a common vibrational temperature TV. The rotational energy level populations are assumed to follow Boltzmann distributions at the translational temperature T of the gas. The CR model provides the electronic state populations of the N and O atoms. (a) Thermodynamics In this study, air is considered as a mixture of nitrogen and oxygen and their products. It is composed of neutral species (N2, O2, NO, N(1-46), and O(1-40)) and charged species (N2+ , O2+ , NO+, N+, O+, and e+). Forty-six electronic energy levels for N and 40 levels for O are employed[74,75]. The levels used are a combination of physically real states and lumped states, obtained by averaging energies and by summing the statistical weights of the states that are lumped together. The final reduced atomic model obtained allows one to accurately calculate: 1) ionization of the N and O atoms by electron impact and 2) the net population of the excited states resulting from the collisional and radiative processes. Furthermore, the coupling of the atom electronic energy levels through the different elementary processes considered in the following section allows for explicit determination of their excitation and consequently the radiative signature of the plasma without using any assumption on their populations a priori. The number of electronic levels used to compute the energy of the ions and molecules is tuned to yield the best matching agreement between values of the computed energies and the reference tables of Gurvich et al.[81]. Molecular energy is computed assuming the rigid rotor and harmonic oscillator approximations. Spectroscopic constants are taken from Ref.[81]. Electronic-specific data have been used for the vibrational and rotational constants of the molecules. In general, such simplified thermodynamic models for the rotation and vibration of molecules are not good approximations in high-temperature flow conditions. However, in this work the fraction of bound molecules is very low; thus our results are relatively insensitive to the molecular model chosen. Although the negative ions (e.g. O-2 and O-) can also be formed, their contribution to chemistry can be considered to be negligible, as a result of the high temperatures


167

reached behind the shock wave, and due to the high rate coefficient for detachment processes. On the other hand, these processes have to be accounted for, when considering radiation processes, due to the formation of background continuum radiation, which often characterizes the photo-detachment process. (b) Shock Tube Flow Solver: Mass, Momentum and Energy Equations We have developed a one-dimensional flow solver, SHOCKING to simulate air plasmas obtained in shock-tube facilities, based on the model presented in Ref.[82]. This model has been modified to simulate re-entries at speeds higher than 10 km/s. First, a radiative source term QRad has been added in the equation that expresses conservation of the total energy. The inclusion of this term is important as radiative transitions tend to deplete/replenish the flow energy for an optical thin/thick medium. Second, a separate source term in the species continuity equation has been added to account for the effect of radiation on the population of electronic states. Post-shock conditions are derived from the jump relations (Ranking-Hugoniot equations) assuming frozen-gas composition and vibrational and electronic energy modes, and the rotational mode is in equilibrium with the translational mode. It is important to stress that the Ranking-Hugoniot equations tend to overestimate the jump of the flow quantities across the shock as they do not account for the dissipative effects, which are not negligible within the shock, owing to the strong gradients. A better approximation of the physical phenomena can be obtained by using the shock slip conditions as suggested in Refs.[76,78]. The downstream flow-field is determined by solving a set of continuity equations for each chemical component, including its electronic structure in the case of the atoms, complemented by the Euler system of equations, namely mass, momentum and energy conservation, which allow one to retrieve the remaining characteristic flow quantities, such as pressure, temperature and flow speed. Also, the relaxation of the vibrational and free-electron energy is modeled using separate conservation equations accounting for the energy exchanges with the other modes (i.e. translation) and chemistry. Finally, the characterization of the radiative field is modeled solving the radiative transfer equation, discussed in details in Section 3.1.3. The physicalmathematical structure of the model is summarized as follows:

• Euler equations: conservation of mass for species i, momentum and total energy

⎛ ρiu ∂ ⎜ 2 ⎜ ρu + ∂x ⎜ ρ uH ⎝

⎛ ⎞ ⎞ ⎜ miωi + miωir ⎟ ⎟ ⎟ ⎜ p⎟ = ⎜ 0 ⎟ ⎟ ⎜ ⎟ ∂ ⎠ ⎜ − Qrad ⎟ ⎝ ⎠ ∂x

(31)

• MultiT models: additional energy conservation eqs., e.g. vibrational energy for the mth molecule in the mixture eq.

168


∂ ∂x

( ρ ue ) = Ω

m

(32)

m

• Radiation transport models:

μ ∂I λ κ λ ∂x

+ Iλ =

ηλ κλ

( x) = Sλ ( x) ,

(33)

where i stands for the set of indices of the mixture species (including species and pseudo-species), ρi the mass density of the ith species; mi the mass of the species;

ωi and ω ir the mass production source terms due to chemical and radiative processes, respectively. The velocity is indicated with u, while p stands for the static pressure and H stands for the total enthalpy and accounts for the kinetic and mixture enthalpy. The radiation losses resulting from the emission of radiation are indicated with Qrad In Eq.(32), em refers to the energy of the internal energy modes. In the particular case, the kinetic energy of the free electrons and all the internal energy modes with the exception of the rotational structure and the internal energy of the atomic species are included in em . The energy exchanged by the lumped energy modes em with chemistry and the translational energy modes are indicated with Ω m . In Eq.(33), I λ is the spectral intensity, μ is the cosine of the angle

ϑ between

ˆ and the axis x, κ indicates spectral absorption coefficient and η the direction Ω λ λ the spectral emission coefficients. The ratio between emission and absorption coefficients is indicated with Sλ (x) and it is referred to as source function.

(c) Internal Energy Modes of Gaseous Particles For atoms the only internal energy mode is the electronic energy. Molecules have three internal energy modes: electronic, vibrational and rotational energies. 3.1.2 Reaction Source Terms The electronic energy relaxation is accounted for by solving electronic master equations based on the kinetic processes previously discussed. In particular, the rate for the production of an atom s, at the electronic level i, via excitation or ionization by electron-impact or heavy-particle impact, can be written as ⎡

⎤

ωi = ∑ k eji N s j N e +∑ k lji N s j N l +N e Ni + ⎢ β i e ,b N e + ∑ β il ,b Nl + α iRRκ iRR + α iDRκ iDR ⎥ j∈A

j∈A l∈H

⎢ ⎣⎢

j∈A l∈H

⎡ ⎤ − N s i ⎢∑ kije N e + ∑ kijl N l + βi e , f N e + ∑ βi l , f Nl ⎥ ⎢ j∈A ⎥ j∈A j∈A l∈H l∈H ⎣⎢ ⎦⎥

⎥ ⎦⎥

(34)


169

where the symbol A stands for the set of indices for the electronic energy levels of the N and O atoms. The set of indices for companion electronic levels is denoted by Ai. This set is N for an electronic level of the nitrogen atom and O for oxygen, with A = N ∪ O . Symbol H stands for the set of indices for the heavy particles. Symbol Ni stands for the molar number density of the species or pseudo species “i”. The reaction rate coefficient for excitation “kei,j”, ionization “β fi”, and recombination “βbi” depends on the collision partner: electron-impact interactions are denoted by the superscript e, and heavy-particle impact interactions are denoted by the superscript l. For radiative and dielectronic recombinations, the recombination rate coefficients are given by κ iDR and κ iRR , respectively. For these two recombination processes, an escape factor α i is introduced in Eq. (34) because the corresponding photoionization processes are not included in our radiative transport model. Here α iDR is the escape factor for dielectronic recombination, and α iRR for radiative recombination. Note that escape factors are only used for these two processes. All other kinetic processes are properly coupled with radiation. (a) Kinetic Processes Included in the Model The closure of the complex non-linear set of equations previously discussed requires the knowledge of rate parameters governing the dynamics of the particles and photons. Recently, Bultel et al.[8] compiled an electronic specific kinetic mechanism for air, which was applied to the study of compressing and expanding flow situations using a 0D model. In Bultel’s model, also referred to as ABBA model, 13 species in their ground state and numerous electronic excited states were taken into account. Although similar models have already been proposed by Teulet et al.[83,84] and Sarrette et al.[85], none are valid for pressures between 1 kPa and the atmospheric pressure. Furthermore, numerous recent experiments and ab-initio calculations have been carried out to improve the accuracy of excitation cross sections by electron impact, dissociative recombination rates of NO+, O2+ and N2+ and their branching fractions as well as vibrational processes. The more recent data are incorporated in our database. Atomic Processes The inelastic collisions between the species lead to chemical changes. The N and O atoms are efficiently excited and ionized by electron-impact reactions. Due to their small mass and the long-range charge-neutral interaction potential, free electrons are effective in exciting the atomic electrons to a higher state and they provide the major source of excited populations of the atoms. A number of models exist for the related cross sections and rate coefficients. Whenever possible, our calculation makes use of recent data from ab initio calculations. For N atom, the excitation rate coefficients from the R-matrix calculation of Frost et al.[48] cover the transitions from the ground and first two metastable states to the first twenty levels, and have been included in the present model. Furthermore, the cross sections from the RMPS calculation by Tayal et al.[38] have been tabulated and included in the model as well. When the two sources provide different data for the same transitions, Tayal’s rate coefficients have been preferred to the data of Frost. Finally, all the data provided by Huo[30,31] have

170


been used to model the electron-induced ionization and excitation of atomic species. Additional calculations have been performed using the BE scaling method for transition among the following states of atomic nitrogen: 4S0(2s2.2p3)4Pe(2s2.2p2.3s); 4Pe(2s2.2p2.3s)-4D0(2s2.2p2.3p); 2Pe(2s2.2p2.3s)-2S0(2s2.2p2.3p); 4Pe(2s2.2p4)-4D0(2s2.2p2.3p). Drawin’s expressions [86, 87] of electron-impact excitation and ionization, based on a simplified model originally developed for atom-atom collisional excitations, provide an efficient method for this type of calculations and thus are adopted for excitation and ionization involving higher states. The corresponding rate coefficients have been expressed in analytical form obtained through the integration of Drawin’s cross-sections over a Maxwell-Boltzmann distribution at the electron Te. For an electronic transition from i to j level, where j > i, the rate coefficient kf is a function of the secondary quantum number l of each level involved. For an optically allowed transition (li ≠ lj): 2

⎛ E ⎞ k = 4π v a α ⎜ H ⎟ Σ1 ( ε ) ⎝ kBTe ⎠ e ij

2 e 0

(35)

where quantity ve = ⎡⎣8RTe / (π me ) ⎤⎦ is the electron thermal speed, R is the universal gas constant, me is the electron molar mass, a0 is the first Bohr radius, EH = 13.6 eV is the ionization energy of the hydrogen atom, α=0.05, and Σ1 ( ε ) = 0.63255ε −1.6454 exp(−ε ) with the reduced energy ε = ( E j − Ei ) / k BTe . 1/2

For an optically forbidden transition (li = lj): 2

⎛ E j − Ei ⎞ kijε = 4π ve a02α ⎜ ⎟ Σ 2 (ε ) ⎝ k BTe ⎠

(36)

with Σ2 ( ε ) = 0.23933ε −1.4933 exp( −ε ) . For ionization of an atom under electron impact, β ie , f , Eq.35 is used with α=1 and a reduced energy ⎛ E Ion − Ei ⎞ , a=⎜ i ⎟ ⎝ k BTe ⎠

(37)

where EiIon is the energy of the ground state of the ion related to that of the ground state of the atom.


171

The rate coefficients for the excitation of atoms from the ground state into the metastable states due to atom-atom collisions are taken from Capitelli et al.[88], while excitation of the remaining excited levels by heavy particle impact have been neglected. Molecular Processes Throughout this investigation, the population of the molecular excited states will be supposed to be in Maxwell-Boltzmann equilibrium at the kinetic temperature of the free-electrons, thus reducing the complexity of the model. In fact, based on the analyses presented in Ref.[76,89], the electronic states of the major molecular species are very likely to follow a Maxwell-Boltzmann distribution, making the state-specific treatment unnecessary. The kinetic mechanism comprises different types of forward and backward reactions involving molecular and atomic species: 1) dissociation of N2, O2, and NO by atomic or molecular impact/recombination; 2) dissociation of N2 by electron impact/recombination; 3) associative ionization/dissociative recombination; 4) radical reactions (including Zel’dovich reactions); and 5) charge exchange. Dissociation processes, as well as their coupling with vibration, are of key importance for hypersonic reentry applications, as they significantly affect aerodynamics, the radiative and convective heat fluxes, and the spectral signatures of vehicles flying at suborbital to super-orbital velocities in rarefied atmospheres. To determine the population of the internal energy states, it would be more accurate to consider the vibronic states as pseudo-species, then treat inelastic collisions as chemical reactions, and finally compute averaged quantities for the VT, VV, and VVT processes. This type of model requires a very large number of data in terms of transition rate coefficients. Theoretical calculations have become possible only recently [90, 91], but an exhaustive database for air is still lacking. In this Chapter, Park’s model is used to describe the influence of the vibration on the dissociation of the molecular species and to account for the influence of the chemistry on the vibrational energy. The T -Tv Park’s model[1] is the most widely spread and used model in the aerospace community, mainly due to its simplicity. The geometrical average temperature is used in the Arrhenius law for the rate coefficients. This model is purely heuristic and based on the analysis of experimental data coming from the study of the post shock radiative signature in shock-tube facilities. The dissociative recombination of molecular ions is known to play an important role in the case of recombining plasmas. The inverse process, associative ionization, allows for formation of the first electrons in many cases, such as in shock tubes and reentry problems. Consequently, it allows for many ionizing situations to be explained. Since N2, O2, and NO are present in the molecular air plasma described here, dissociative recombination has to be considered[8]. Zel’dovitch reactions are known to greatly influence the distribution of nitrogen and oxygen between atomic and molecular systems and to contribute to the destruction of O2 and N2 and the formation of NO. For these processes, we have used the rate coefficients estimated by Bose and Candler [92,93], which are based on quasi-classical trajectory study performed on ab-initio potential energy surfaces. Further details about the model are given in Ref. [8].

172


(b) Chemistry and Flow Field Energy Distribution Chemistry and energy exchange source terms are the key parameters in the set of equations governing the behavior of high enthalpy gases, allowing the changes in the mixture due to chemical reactions as well as accounting for the exchanges of energy among the internal energy modes. Also, chemistry and internal excitation are closely correlated and are known to influence each other. Non-equilibrium models are distinguished by the way they describe the interaction among the excitation of the internal energy modes and the macroscopic kinetic mechanisms employed. The model proposed is hybrid in the sense that it combines a state-to-state treatment of the atomic electronic levels with a multi-temperature approach for the remaining modes, thus requiring the modeling of the energy exchanges among the different energy modes. In the case of two temperature models the relaxation terms to be considered are: Ω = Ω VT + Ω CV + Ω ET + Ω E + Ω I m

(38)

Vibrational Translational exchanges ( ΩVT ): The rate of vibrational-translational energy transfer follows a Landau-Teller formula, where the species relaxation time is based on the Millikan–White formula, including Park’s high-temperature correction[48]. Exchanges between chemistry and vibration ( Ω CV ): The model employed was proposed by Candler [94]. Elastic exchanges among electrons and heavy species ( Ω ET ): The form of the source term is very similar to VT (Landau-Teller form). Note, however, due to the disparity of masses of an electron and a heavy particle, kinetic energy transfer between them is inefficient, resulting from the simultaneous requirement of energy and momentum conservation. Energy losses due to excitation/ionization ( Ω I / Ω E ): At high speeds, it is important to account for the energy lost by the free electrons during ionization and excitation of the atoms and molecules, as already stressed in Ref. [82]. If neglected, electron-impact ionization reactions (and in general, all the reactions involving free electrons) produce a large amount of free electrons without depleting their kinetic energy, thus enhancing their production. This phenomenon may lead to an “avalanche” ionization, which introduces related numerical problems, especially for high-speed conditions. The expression for the related source terms for electron-impact ionization and excitation reactions is: I

ΩI =

∑ω

e ,r

U

r

r

,

E

ΩE =

∑ω r

e,r

U

r

(39)


173

where Ur is the reaction enthalpy of the r reaction, ω e,r is the electron chemical production term of the r reaction, RI denotes the set of indices of the electron-impact ionization reactions, RE is the set of indices of electron-impact excitation reactions, Ω I accounts for the energy removed by electron-impact ionization reactions, and Ω E accounts for the energy removal by electron-impact excitation reactions. 3.1.3 Radiative Processes and Radiative Transport The system under study is composed of molecules, atoms, ions and electrons interacting with each other and with the radiation field. This system is modeled with the photons regarded simply as another species of particles. This simplifying assumption is appropriate for the description of the chaotic light characterizing the reentry environment. Under these assumptions, the kinetic theory of particles and photons is thus reduced to the set of kinetic equations for the material particles (e.g. Navier-Stokes system of equations, complemented by conservation equations for energy and the various chemical components) and the well-known Radiative Transfer Equation (RTE). The theory governing the dynamics of such systems in nonequilibrium is well established and is discussed in details in Ref.[95]. (a) Atomic and Molecular Spectral Properties When considering atomic systems, all the transitions characterized by absorption and emission of light is subdivided into three types: free-free transitions; bound-free transitions; and bound-bound transitions. The present analysis is carried out considering only the bound-bound radiation, which is the result of an electronic transition among bound atomic or bound molecular states. This type of radiation is also referred to as line radiation, owing to its discrete nature. When modeling atomic line radiation, three mechanisms have to be considered: spontaneous emission, absorption, and stimulated emission. These processes require the knowledge of one of the three transition probabilities also known as Einstein coefficients: Aji, Bij and Bji. The Einstein coefficients are not independent of each other and they must satisfy the Einstein relations:

B ji =

c2

8π hν 3 g j B ji = g i Bij

A ji

,

(40)

where c indicates the speed of light, h is the Planck’s constant and ν the frequency of the radiation. The upper states, in the electronic transition are indicated by j, while the lower states by i. The degeneracy of the states is indicated with g. Note that different definitions of B can be found in literature. A comprehensive list of atomic line probabilities, in terms of Einstein coefficients, for nitrogen and oxygen is provided by the NIST atomic line database[17], which has been used in this work. When the gas cannot be considered optically thin, an accurate modeling of the spectral line shape is also important. In this work we account for Doppler, natural and collisional broadening. Among the pressure broadening

174


mechanisms considered, the broadening due to collisions with charged particles (electron and ions), known as Stark broadening, has to be accounted for. Three different models have been included: Johnston's fits of experimental data[76], Cowley's and Arnold's curve fits[96] and experimental values taken from Griem[97]. (b) Energy Levels The atomic model used in radiation and the thermodynamic model used in the calculation of the flow properties differ significantly. In fact, the number of atomic levels considered on the radiation side is considerably larger than the corresponding number of pseudo-species, used in the solution of the flow field. This is due to the fact that while radiation is very sensitive to the accuracy of the atomic model, the kinetic model is relatively insensitive number of atomic levels employed. That is to say that the kinetics of the gas can be faithfully represented with a reduced number of levels, while a large number of levels is required in order to have a good representation of the radiative signature of the gas. The reduction in the number of the atomic levels in the flow case is obtained by lumping the electronic states into groups, assuming a uniform distribution within each bin [74, 75]. When computing the radiation, the population of each lumped level is distributed among the levels included in the considered group according to a Maxwell-Boltzmann distribution at a local temperature Te. Thus, the population of the ith state of the un-grouped system is obtained as: ⎛ ΔE ⎞ gi exp ⎜ − i ⎟ ⎝ kbTe ⎠ ni = nk Qˆ

(41)

k

where ni indicates the population of the ungrouped level, nk refers to the grouped level and the partition function of the grouped level is defined as: ⎛ ΔE ⎞ Qˆ k = ∑ gi exp ⎜ − i ⎟ i∈I k ⎝ kbTe ⎠

(42)

The definition of the partition function of each single bin is based upon the definition of ΔEi for each single level, which reads: ΔEi = Ek − Ei

where Ek is the energy of the grouped level Ek = ∑ i∈I k

gi Ei ∑ gj j∈I k

and I k is the set of indices of the states belonging to the group k.

(43)


175

A special treatment is reequired for the auto-ionizing states, which are assumedd to be in Saha equilibrium (cheemical equilibrium) with the free-state and their populattion is obtained as follows I ne ni = nIon

⎛ Ei − E Ion ⎞ gi 3 λ exp ⎜− ⎟ e kbTe ⎠ 2QIon ⎝

((44)

where λe = hP / (2π me kBTe ) is the thermal De Broglie wavelength of the free electrrons and Q Ion is the partition fun nction of the ion. In Fig.9 the distribution of the electronic states of the nitrogen atoms is shown. T The black part of the curve reprresents what is actually computed by the kinetic model and the red is the part, which is added in the calculation of the radiative properties. IIt is worth mentioning that any attempt to lump the auto-ionizing states in a unique grooup, as done for the bound stattes, resulted in a distortion of the tail of the distributtion function (red part in Fig.9), which is reflected in the spectrum. The inconsistent groupiing strategy adopted for kinetics, based on the unifoorm averaging, and radiation relying r on a Boltzmann distribution, is the main souurce inaccuracies in the spectru um and the calculation of the optical properties of the plasma. 1/2

Fig. 9. Electronic energy disttribution function for N. Equilibrium mixture at 10000 K and 1 atm

176


(c) Fluid Description of a Gas with Radiative Interaction This section considers the transport of radiation in a participating medium (a medium that absorbs and emits radiation). Thus, the output from the flow calculation in terms of flow quantities, i.e. pressure, temperature, electron density and the population of the electronic states of the atomic species is used to compute the spectral dependent emission and adsorption coefficients (see Eq.33). The radiative transfer problem behind the normal shock is modeled using the tangent slab approximation, thus assuming that the radiative properties, in terms of absorption and emission coefficients, of a plane layer of gas of finite thickness vary only in direction perpendicular to the surface (or the shock wave). This assumption is consistent with the assumption of one-dimensionality used for the flow-field calculations, and it is often successfully employed in literature for three-dimensional calculations to describe the radiation field in the shock-layer of blunt bodies, where the flow is almost one-dimensional. In the following, the formal solutions for the net radiative heat-flux, the divergence of the radiative heat-flux and the incident radiation for a plane-parallel medium are presented. (See Refs. [98,99] for the full derivation). The knowledge of the radiative intensity allows for the estimation of the energy source terms, thus accounting for the radiative power emitted or adsorbed by the gas. This term is simply obtained by computing the divergence of the radiative heat-flux. This can be obtained by: Incident intensity: ⎡ I + (τ b ) E (τ ) + I − (τ s ) E (τ s − τ ) + τ S (τ ) E (τ − τˆ ) dτˆ ⎤ 2 λ λ λ λ ∫τb λ λ 1 λ λ λ ⎥ ⎢ λ λ 2 λ Gλ (τ λ ) = 2π ⎢ ⎥ τs ⎢⎣ + ∫τ S λ (τ λ ) E1 (τˆλ − τ λ ) dτˆλ ⎥⎦

(45)

where En (τ λ ) is the exponential integral of nth kind. τ λ indicates thee optical depth, defined as τ λ = κ λ dx . The superscript “b” and “s” indicates the body and the shock

∫

location respectively, whereas the “+” and “-“ sign corresponds respectively to the positive or negative cosine of the angle ϑ . All the expressions above have been calculated by numerically integrating the product of the exponential integral and the source function. The solution of those integrals is easily obtained by assuming a piece-wise constant representation of the spatial evolution of the source function, given the existence of an analytical solution for the exponential functions. Divergence of Radiative Heating:

dqλ (τ λ ) = 4π Sλ (τ λ ) − G (τ λ ) dτ λ

(46)


177

Radiative Heating:

( )

( ) (

)

⎡ I + τ b E (τ ) − I − τ s E τ s −τ + τ S (τ ) E (τ −τˆ ) dτˆ ⎤ 3 λ λ λ λ ∫τb λ λ 2 λ λ λ ⎥ ⎢λ λ 3 λ qλR (τ λ ) = 2π ⎢ ⎥ τs ⎢⎣+∫τ Sλ (τλ ) E2 (τˆλ −τλ ) dτˆλ ⎥⎦

(47)

The other output from the radiation calculation are the rates of change of the number densities of the chemical component s in the quantum electronic state i. γ ⎡ ⎤ λmax ⎢ ⎥ i, j ⎢ Aj ,i ns , j − ( Bi , j ni − B j ,i n j ) ∫ Gλ ( x ) Φ λ d λ ⎥ λmin ⎢ ⎥ ⎣ ⎦ ωir = ∑ γ ⎡ ⎤ j >i λmax ⎢ ⎥ i, j −∑ ⎢ Ai , j ns ,i − ( B j ,i n j − Bi , j ni ) ∫ Gλ ( x ) Φ λ d λ ⎥ i> j ⎢ λmin ⎥ ⎣ ⎦

(48)

(a) Numerical Solution of Non-equilibrium Flows with Radiative Interactions The description of the non-equilibrium kinetic processes taking place behind a strong normal shock requires the solution of the system of stiff ordinary differential equations discussed, in Section 3.1.1. The reasons of the stiffness in the system of equations originate from differences in the time scales at which the different state variables relax towards their equilibrium value. In the case under analysis, the dynamics of the high lying excited states of the atomic species is extremely fast and tends to quickly equilibrate, reaching a state of quasi-equilibrium, whereas the lower states relax according to slower kinetics. The presence of different time scales suggested the development of simplified methods based on the solution of asymptotic analytical methods, also known as quasisteady state (QSS) approaches (or statistical equilibrium approaches (SE) in the astrophysical community). The use of QSS methods reduces the system of ODEs to a hybrid system of equations, since the population of the high lying states can be determined via the solution of a non-linear algebraic system of equations. Within the aerospace community, the application of the QSS methods is often used to determine the internal distribution of the excited states, when departures from the MaxwellBoltzmann distribution are expected to occur. In general however, the a priori use of these methods is highly discouraged, since the validity of the underlying assumptions strongly depends on the case to be investigated. An example of the inappropriate

178


application of the QSS method is discussed in Ref. [100], where Magin et al. demonstrate its inability to properly estimate the radiative signature of the hot gases surrounding the Huygens probe, by comparing QSS predictions and the results obtained using a time dependent model. Also, Ref. [82] clearly show that the low lying metastable states of nitrogen and oxygen do not reach steady-state conditions at the same time as the upper states, making any asymptotic approach inapplicable. For these reasons the direct numerical solution of the non-linear set of ODE is preferred to the asymptotic analytical methods. The introduction of the radiation coupling drastically changes the mathematical structure of the system of equations from ODE to integral-differential equations, causing the properties of very distant points in a medium to be non-locally coupled through the radiation field. In this work, the numerical solution of this problem is thus obtained through a semi-implicit iteration scheme, as follows:

∂y n = f ⎡⎣ y n , Γ( n −1,n ) Ω slx ⎤⎦ ∂x

( )

Γ where

( n −1, n )

(Ω ) sl x

∂ n −1 ⎤ ⎡ = ⎢ωir ( n−1, n ) , 0, 0, − Qrad ⎥ ∂x ⎣ ⎦

T

(49)

y n is the state vector, and it includes mass fractions for the chemical

components and the atomic excited states, velocity and temperatures. The Γ ( n −1,n ) is the vector of the coupling source terms, composed of the source terms in the species conservation equations and the divergence of the radiative heating for the total energy equation. The spatial domain is indicated using the greek letter Ω slx . It is written explicitly to stress the non-local nature of the coupling source term. The index n refers to the iterations needed to obtain convergence. The loosely coupled approach employed requires the independent solution of the kinetic and the radiative transfer problem. When solving the kinetic problem, the radiation source term used corresponds to the value at the previous step. Once the state vector y is updated the new source term can be computed and used in the following kinetic iteration. It is important to note that in order to improve the stability of the algorithm, part of G( Ω slx ) is treated implicitly.

ω r ±( n −1,n ) = ± nun Aul ∓ ( nln Blu − nun Bul ) ϒ( n −1) i

where

ϒ ( n −1) =

λmax

∫ λ

Gλ( n −1) ( x )Φ iλ, j d λ

(50) (51)

min

The only part of the source terms to be treated explicitly is ϒ {n−1} , which improves the stability and convergence of the numerical algorithm.


179

3.2 Results

In this section a hybrid CR model, modified from the ABBA model [8], is used to analyze the behavior of the electronically excited states of atomic species behind a strong shock. Sec. 3.2.1 is devoted to the analysis of the non-equilibrium ionization process occurring in the post-shock region of the FIRE II capsule, a well-known flight experiment from the 1960s. The dynamics of the electronic level populations of N and O is discussed. Comparison of the non-equilibrium distribution with the Boltzmann populations is made in order to quantify its departure from equilibrium. Furthermore, the effect of a self-consistent treatment of radiation coupling on the ionization process and on the internal electronic distribution is discussed. Sec. 3.2.2 studies the flow field energy distribution and Sec. 2.2.3 the interaction between radiation and matter. Sec. 3.2.4 is devoted to the analysis of the QSS regime of high-lying excited electronic states. In particular the limits of its applicability are discussed. Sec. 3.2.5 compares the modeling result with experimental measurement in the EAST shock tube facility. 3.2.1 Fire II Flight Experiment One of the primary objectives of the Fire project was to define the radiative heating environment associated with the re-entry of a large-scale Apollo vehicle at a velocity of 11.4 km/s. During this re-entry, a large portion of the overall wall heat flux was due to radiation. Most of the radiation (approximately 90%) came from atomic lines, and thus an accurate prediction of the populations of excited electronic states of the atoms is crucial. The aim of the present work is to test the CR model for different physicochemical conditions, from electronic energy level populations in strong nonequilibrium to populations following Boltzmann distributions. The shock-tube operating conditions corresponding to the trajectory point investigated here is presented in Table 1. Free stream characteristic quantities are denoted by the subscript 1, post-shock characteristic quantities by the subscript 2. Symbols U represents the shock velocity. The mole fractions of nitrogen and oxygen are assumed to be constant through the shock (xN2 = 0.79 and xO2 = 0.21). We recall that, after the shock, the rotational temperature is equal to the post-shock gas temperature T2, whereas the vibrational and electron/electronic temperatures are still equal to the free stream gas temperature T1. Table 1. Shock-tube operating conditions used in the simulation

Time P1 [Pa] T1 [K] U1 [m/s] P2 [Pa] T2 [K] U2 [m/s]

1634 s 2.0 195 11360 3827 62337 1899

180

High Temperature Ph henomena in Shock Waves

The numerical solution of o the governing equations requires iterations between the system of equation describiing the dynamics of the particles and the radiative transfer equation, solved within thee tangent slab approximation. Monitoring the relative errror in the coupling term betweeen the nth iteration and the previous (n-1)th iteration alloows one to assess the convergen nce of the numerical algorithm. Figure 10 shows the noorm of the error as a function off the iteration number. The relative error is steadily reduuced by ten orders of magnitud de in only 24 iterations. The stability of the algorithm m is ±{ n −1, n} ω achieved through the semi--implicit treatment of the term i . Every attemppt to treat this term explicitly, as done for the divergence of the radiative heating, has faiiled completely, due to the onseet of un-dumped oscillations in the coupling source terrms, which lead to the divergencce of the solution.

Fig. 10. Infinity no orm of the relative error on the coupling source terms

3.2.2 Chemistry and Flow w Field Energy Distribution The characterization of thee physical-chemical state of the plasma in the shock laayer requires the knowledge of its i chemical composition as well as the internal energies of the particles. To this aim m, Fig.11 shows the evolution of the rotational and translational temperature an nd the vibrational/free-electron temperature. After a juump in the translational temperaature across the shock (located on the left at x=0), the gas redistributes energy throug gh activation of the internal energy modes as well as the onset of chemical reactio ons, until the flow eventually reaches its post-shhock equilibrium state. The inteernal temperature profile, indicated by Tv, shows a raapid initial increase due to the excitation e of the vibrational energy states, but then flatttens out due to the eventual onseet of chemical reactions (see Fig.12). The extent of therm mal non-equilibrium (i.e. T ≠T TV) is dictated by the coupling of the internal energy and


181

chemistry. In the particular case considered here, the extent of thermal nonequilibrium strongly depends on the dissipation of free-electron energy by the excitation and ionization of atomic species. This relaxation process is modeled by Eqs.(39). The kinetic processes, acting as energy sinks, tend to slow down further reactions, thus reducing the rate at which the thermal energy stored in the form of translational energy is converted into chemical energy. As a result, a higher average kinetic temperature can be observed throughout the shock layer. The use of an electronic specific model allows for an accurate estimation of the energy losses to be obtained, without the use of ad-hoc parameters calibrated on the basis of experimental data. Most of the losses are due to excitation of the electronic structure of the atomic species (and, to a lesser extent, to the direct ionization). In this picture, radiative processes play a fundamental role, since they influence the dynamics of the thermal and chemical relaxation. The influence of the optical thickness of the gas on the chemistry is clear when observing the electron density profiles, shown in Fig. 12. When the medium is assumed optically thick, all of the radiation emitted is immediately self-absorbed and the electron density quickly reaches its equilibrium value. In the case of optically thin gas case, the ionization rate is considerably slower; this is due to the fact that radiative processes (when only emissive processes are considered) tend to deplete the excited states, resulting in a delay of the excitation as well as the ionization processes. In Fig. 12, the density plot exhibits a maximum and is monotonically decreasing due to the radiative energy losses, which work to consume energy, thereby preventing the gas from reaching the desired equilibrium electron density. The result of the coupled calculation is in close agreement with the optically thick case. This is the first time a rigorous modeling of the radiative processes and their interaction with the surrounding matter is used to test the escape factor approximation. 70000

T TV

Temperature [K]

60000 50000 40000 30000 20000 10000 0 0

0.02

0.04

0.06

0.08

Distance from the shock [m] Fig. 11. Post shock temperature profile

0.1

182


21

10 10 -3

21

a

Number Density [m ]

3

Electron Number Density [m ]

5.0×10

4.0×10

21

3.0×10

21

2.0×10

Coupled Model Optically Thick Optically Thin

21

1.0×10

0.0 0

0.02

0.04

0.06

0.08

Distance from the shock [m]

10 10

10 10 10 10

10

b

N

N2 O

21

N

+

O2

20 19

+

O

NO

18

+

NO

-

e

17

16

10 0.1

23

22

N2+

O+2

15

14

10

-4

10

-3

10

-2

10

-1

Distance from the shock [m]

Fig. 12. a) Electron density as a function of the optical thickness. b) density plots.

3.2.3 Radiative Transport and Interaction between Radiation and Matter At high shock speed, associative ionization reactions are responsible for the production of the prime electrons, thus enabling the ionization by electron collision with the atomic species. A comparison of the electron density profiles with the densities of NO+, N+ and O+ in Fig.12b shows this phenomenon very clearly: NO+ is responsible for the creation of free electrons only in the shock area (xi

k

(62)

Radiation Phenomena behind Shock Waves

213

where Aij designates the Einstein spontaneous emission coefficient from level i to level j, Afi the radiative recombination rate, Ne the electron number density and N+ the ion density. When the medium cannot be considered as optically thin, the absorption of radiation leads to non-local phenomena and several issues must be addressed to enable practical prediction of radiative quantities. On the one hand, spectrally resolved calculations of radiation transport require the use of a huge number of discrete wavenumbers. Approximate radiative properties are generally required. On the other hand, spatial and directional aspects of radiation transport involve also important computational times and simplified models are whished. Some simple models such as the escape factor approach can handle both aspects of spectral and geometrical complexities. This concept is discussed in the first part of this section. We discuss in the second part some approximate treatments of the spectral integration problem and, in the third part, some geometrical approximations and methods to solve the radiative transport equation. We conclude this section by a brief description of the Monte Carlo method, especially when applied to nonequilibrium radiation. 4.1 The Escape Factor Approach

The main idea in this approach consists in still using Eqs.(61) or (62) but with a correction to take into account absorption. For the total energy, equation (61) is transformed to [39] ∞

SR = −4π ∫ ησ e −κ σ R dσ , O

(63)

where R is a characteristic length of the medium. This expression is rigorous at the center of a uniform spherical medium of radius R if one neglects the radiation incoming on the sphere. It remains however very approximate for shock waves and the choice of the parameter R is not intuitive for 3D geometries. On the other hand, this approximation leads everywhere to a negative source term and cannot predict the net absorption in the cold regions. For the level number density equation, self-absorption is very important close to resonance atomic lines and it was early recognized that a specific treatment is required to account for the “imprisonment” of radiation in the medium [40,41]. Equation (62) is generally written with effective Einstein emission coefficients Ajkeff=Λjk Ajk and effective cross sections for radiative recombination where the escape factor Λjk is introduced to take into account the trapping of radiation in the medium [42]. Λjk =1 corresponds to an optically thin medium and Λjk =0 to the optically thick case where the effects of radiation vanish. The escape factor is often taken as an adjustable parameter and the results are presented in the limiting thin and thick cases (see e.g. Ref.43). The rigorous calculation of Λ requires to solve the full radiative transfer equation. Let us consider for instance the radiative de-excitation of an atom from level i to a lower level j. The effective rate due to the transition i-j can be written as N i Aijeff = N i Aij − ∫

∞

0

κ σ ij I dΩ dσ , hcσ ∫4 π σ

(64)

214


where κσij is the absorption coefficient due to transition i-j. Using the definition of the absorption coefficient and the relations between Einstein spontaneous emission, absorption and induced emission coefficients, the escape factor is given by Λ ij = 1− ∫

∞

0

Iσ f (σ )

1 ⎛ gi N j ⎞ − 1 dσ , 2hc 2σ 3 ⎜⎝ g j N i ⎟⎠

(65) 1

where Iσ is the directionally averaged incoming intensity, Iσ = ∫4π Iσ dΩ . Under 4π LTE conditions, and assuming that the wavenumber σ remains close to (Ei-Ej)/hc under the line profile, the last expression reduces to Λ ij = 1− ∫

∞

0

Iσ f (σ )dσ , Iσ0

(66)

which shows clearly that Λij vanishes for very thick lines since we have, locally, Iσ ≈ Iσ0 close to line center. Several other definitions of the escape factor, and other ways to use it either for radiative transfer or for population density calculations, exist in the literature. These definitions are summarized in Ref.44 where some approximate expressions may be found. Such approximate expressions are generally welcome since they avoid the solution of the radiation transport equation for several millions of wavenumbers, especially when radiation must be coupled to aero-thermal phenomena. However, the correct and self-consistent treatment of radiation transport in collisional-radiative models has been undertaken in some simple (small number of transitions) or with the Quasi-Steady-State (QSS) approximation [45-47]. 4.2 Spectral Models 4.2.1 Statistical Narrow Band Models Statistical Narrow-Band (SNB) models were first developed and widely used to study infrared molecular radiation in atmospheric and in combustion applications [48-49]. They are particularly suited to spectra dominated by a dense structure of molecular (or atomic) lines for which the statistical treatment inside a narrow band (but containing a great number of lines) is an efficient way to summarize the whole spectroscopic properties of the lines belonging to the band. We briefly recall here the principles of SNB models and discuss their possible use for nonequilibrium radiation. Let us consider first a uniform gaseous column of length l, at equilibrium at temperature T, and a spectral band Δσ containing a large number N of lines but sufficiently narrow to consider that the Planck function may be considered constant inside Δσ. Under the assumption that line centers may be considered as randomly located inside Δσ, the column transmissivity, averaged over Δσ may be written


τσ =

1 Δσ

⎛ W⎞ ⎟, ⎠

∫ σ exp ( −κσ l ) dσ = exp ⎜⎝ − δ Δ

215

(67)

where δ is the mean distance between two consecutive line centers δ = Δσ / N and is the mean value of the black equivalent line width given by W =

1 N

N

∞ ; Wi = ∫ (1 − exp( −κ σ ,i l ) ) d σ ,

∑W i =1

i

0

W

(68)

where κσ,i is the contribution of the ith line to the absorption coefficient. The expression of W depends in practice on the line broadening regime (Doppler, Lorentz or Voigt line shapes) and on the statistical distribution of line intensities characterized by a probability density function P(S). Here, the intensity S for a transition u-l is defined by S=

(N l Blu − N uBul ) hσ ul , p

(69)

where p is the partial pressure of the emitting/absorbing species. Figure 8 shows a typical behavior of the curve of growth − ln( τ) = W / δ as function of the column length l for CO2 lines at 2000K and 100 Pa. A linear growth is first observed in the optically thin limit, followed by a Doppler dominated growth, then a transition regime, and finally a Lorentz regime characterized by a root square power law l1/2 resulting from the Lorentz shape which dominates line wings at high optical thicknesses. There are no general analytical expressions for WV in the general case of Voigt line shapes. Several approximations exist in the literature and the following one, due to Ludwig et al [50], yields generally accurate results WV

δ

=

plS

δ

1 − Ω−1/ 2 , −2

(70) −2

⎡ ⎛ W / δ ⎞2 ⎤ ⎡ ⎛ W / δ ⎞2 ⎤ L D Ω = ⎢1 − ⎜ ⎟ ⎥ + ⎢1 − ⎜ ⎟ ⎥ −1 , ⎢⎣ ⎝ plS / δ ⎠ ⎥⎦ ⎢⎣ ⎝ plS / δ ⎠ ⎥⎦

where

S=

1 N ∑S N i=1 i

(71)

is the mean line intensity and W L and W D are the mean equivalent

line black widths in the pure Lorentz and Doppler regimes, respectively. Comparison between different line intensity distribution functions show that the closest agreement with line by line calculations is obtained with the inverseexponential tailed distribution for the Lorentz profile, and the exponential distribution for Doppler profile. These distributions lead to

216


WL

δ

WD

δ

=

=

⎞ 2γ L ⎛ ( S / δ ) pl ⎜ 1+ − 1⎟ , ⎜ ⎟ δ ⎝ γL /δ ⎠

γ D ⎛ ( S / δ ) pl ⎞ E⎜ ⎟ with δ ⎝ γD /δ ⎠

E ( y) =

1

π

(72)

+∞

y e −ξ

−∞

1 + y e −ξ

∫

2

2

dξ ,

(73)

where γ L and γ D are the mean Lorentz and Doppler line half-widths respectively, and

δ is an effective mean distance between two consecutive lines.

Fig. 8. Example of the curve of growth for CO2 transmission at 2000K

Equations (67, 70-73) show that SNB models lead to simple expression of the average transmissivity of uniform columns at equilibrium. Their extension to nonuniform columns, yet still at LTE, requires further approximations such as the Curtis-Godson approximation which are not detailed here (see Refs.51-54). The practical use of SNB models requires to rewrite the RTE in terms of transmissivities. It is easily shown for instance that for a medium at LTE, and if one neglects the incoming radiation at the boundaries, the RTE for the intensity averaged over Δσ may be written in the integral form


s

Iσ (s) = ∫ Iσ0 (s') 0

∂τσ (s',s) ds' . ∂s'

217

(74)

For a gaseous medium under NLTE, the source function should be replaced by the ratio ησ/κσ (s’) but this quantity may exhibit some strong variations with the wavenumber that could be spectrally correlated with the transmissivity. Figure 9 (left part) shows an example of the high resolution variations of ησ/κσ (s’) for the N2 Birge-Hopfield-2 system. These variations result from the mixing of rotational lines of different vibrational bands. The right part of this figure shows however that these variations are, to a good approximation, not correlated with the absorption coefficient or the transmissivity of the medium [53].

Fig. 9. Left part: Planck function at 10000K and 20000K, and the ratio ησ/κ σ in nonequilibrium conditions (Tv=Tel=10000K, T=Tr=20000K) for N2 Birge-Hopfield-2 system. Right part: net emission, averaged over spectral bands 1000 cm-1 wide, for the same N2 system and temperatures, and a uniform column of length 10 cm (from Ref.53).

The RTE, averaged over Δσ, may then be written in the form

Iσ (s) = ∫

ησ ∂τ (s') σ (s',s) ds' . κσ ∂s'

s

0

(75)

The above model has been implemented for the optically thick diatomic electronic systems of air plasmas in Ref.53 and was shown to predict accurately radiative transfer in Earth reentry flows. Model parameters

have been tabulated

versus two temperatures Ttr = Trot and Tv = Te, but the model can be applied to any nonequilibrium distribution of electronic levels in diatomic molecules. Note also that similar statistical models were applied to atomic lines in Ref. 8.

218


4.2.2 Global Models For media under LTE conditions, global radiation quantities, i.e. intensities or fluxes integrated over the whole spectrum, may be directly calculated from the distribution function of the absorption coefficient, weighted by the Planck function

F(k,Tp ,T) =

π σ SB Tp4

∫σ κ /

σ

(T )≤k

Iσ0 (Tp )dσ ,

(76)

where σSB is the Stefan-Boltzmann constant. This distribution function depends on the local medium temperature T and on another temperature Tp used to evaluate the Planck function. The integration of radiation is carried out over k values instead of σ in these models. where If the absorption spectrum can be assumed to be separable, i.e. designates the local state of the medium (temperature, pressure and concentrations) the set of elementary wavenumber intervals for which ξ(σ) belongs to [ξ, ξ +dξ] does not depend on the spatial position and the integration of the radiation intensity over k values, or more precisely over ξi values, is rigorous. This leads, after discretization of ξ space into N intervals [ξ j-, ξ j+], each one characterized by a representative value ξ j, to Itot =

∑I

j =1,...,N

j

∂ Ij ⎡ ⎤ σ T 4 (s) = k j ( χ ) ⎢a j ( χ ) SB − I j ⎥, ∂s π ⎣ ⎦

j = 1,..,N

(77)

where T is the temperature of the state χ kj(χ)=ξ jΦ(χ), and the weights aj(χ) are given by a j (χ ) =

π σ SBT 4

∫σ ξ /

− Φ ( χ ) ≤ κσ j

≤ ξ +j Φ ( χ )

Iσ0 (T ) d σ = F ( ξ +j Φ ( χ ), T , T ) − F ( ξ −j Φ ( χ ), T , T )

and satisfy the normalization relation

∑

N j =1

(78)

a j ( χ ) = 1.

In practice, and even at LTE conditions, the scaling approximation κσ (T)=ξ(σ)Φ(χ) is not satisfied in shock wave applications for two reasons. First, species concentrations strongly vary in the shock layer and the absorbing and emitting species are not the same everywhere. The second reason comes from the variations with temperature of line intensities. Even for a single species, the populations Nu of the excited emitting states vary strongly with T, which leads to a non-scaling behavior in the case of important temperature gradients. One remedy to this problem was proposed in Ref.55. It consists in splitting the actual spectra into fictitious spectra, each one gathering the radiation mechanisms and/or species which have similar variations with temperature. A correlation approximation remains however necessary to take into account the actual variations of the absorption coefficients. The α discretization of k space was first carried out in a reference condition denoted χ ref


219

α with a temperature Tref for a given fictitious species α, and the absorption coefficient in a current condition χ was deduced from the implicit relation

α α Fα (kαj ( χ ),Tref , χ )= Fα (kαj ( χαref ),Tref , χαref ) ,

(79)

where Fα is defined in a similar manner as in Eq.(76) for the spectrum of species α. α Fα (k,Tref , χ) =

π α4 σ SB Tref

∫σ κ /

α σ

( χ )≤k

Iσ0 (Tref )dσ

(80)

This led to the Mixture Absorption Distribution Function (MADF) model where the RTE has to be solved for instance 63=216 times (N=6 and 3 fictitious species)[56]. A similar approach was developed by Zhang and Modest [57] and called Multi-Scale Full-Spectrum Correlated-k Distribution. These authors introduced an overlap factor between the different spectra in order to reduce the number of RTEs to NM, where M is the number of fictitious species (or scales), instead of NM. This last approach was recently applied to CN radiation in nonequilibrium flows [58]. 4.3 Geometrical Treatment of Radiative Transfer

The Ray Tracing (RT) method, either in its deterministic or statistical form (given in the following subsection), is the most rigorous method for the geometrical integration of radiative transfer. It consists simply in discretizing the 4π total solid angle into M elementary solid angles ΔΩm, centered around the directions um, and performing angular integration of a given radiation quantity G according to M

∫ G(u)dΩ= ∑ω

4π sr

m=1

m

M

M

m=1

m=1

G(um ) = ∑ ΔΩm G(um ) = ∑ ∫

∫

Δϕ m Δθ m

sin(θ)dθ dϕ G(um ) ,

(81)

where θ and ϕ are the spherical angles. Rays are launched from each volume element and from each surface element of the boundaries, and the radiative transfer equation is solved along each ray according to

(

X

)

X

(

X

)

Iσ ( X , um ) = Iσ (0)exp −∫ κσ (s) ds + ∫ ησ exp −∫ κσ (s ') ds ' ds , 0

0

s

(82)

where s is the abscissa along the ray characterized by the unit vector um and s=0 designates the intersection point of this ray with the boundaries of the calculation domain. Such detailed geometrical calculations are however often prohibitive and approximate methods have been developed. A very brief overview of these methods is given below and the details may be found in classical textbooks dedicated to radiative transfer (see e.g., Refs.59-61).

220


4.3.1 The Discrete Ordinate Method (DOM) In this method, the discrete directions um and weights ωm in Eq.(81) are determined from a quadrature formula. In particular, ωm has no more the meaning of an elementary solid angle. Among the different available quadratures, the more popular is the SN quadrature [62] in which the directions are chosen from symmetry considerations. If a direction um, characterized by its direction cosines (μm, ηm, ξm) belongs to the set of quadrature directions, the 8 symmetric directions (±μm, ±ηm, ±ξm) must also belong to this set. The directions are thus defined on only one octant of the sphere. If n designates the number of possible latitudes of a direction cosine in an octant, the order of the quadrature is N=2n and the total number of directions is M=N(N+2). The weights are chosen in the SN quadrature in such a way that some moments are conserved. This leads for instance to

∫π

dΩ = 4π

∫π

μ 2 dΩ =

∫π

μ dΩ = π ⇒

4 sr

4 sr

2 sr

M

∑ω

⇒

m =1

4π 3

⇒

m

= 4π

M

∑ω m =1

m

M

∑ω

m m =1/ μ m >0

μ m2 =

4π 3

(83)

μ m = π , ....

These considerations are generally sufficient to determine the set of directions and associated weights. Another important difference with the ray tracing method is that the RTE, for a given direction, is solved by a finite volume type method instead of direct integration along a ray. This leads naturally to some numerical difficulties such as numerical diffusion or instabilities, depending on the used interpolation schemes. 4.3.2 Spherical Harmonics (PN) and Related Methods A decomposition of the radiation intensity on the spherical harmonics basis is used in these methods in order to simplify its directional dependence: N

Iσ (r,u) = ∑

l

∑

l = 0 m =− l

Al,mσ (r)Ylm (u) .

(84)

For a given wavenumber σ, the decomposition coefficients Alm only depend on the spatial position r and are governed by equations obtained by replacing this decomposition in the RTE and taking different moments of the result. The resulting equations are coupled partial differential equations that may be solved by classical finite volume or finite element methods. The most popular P1 method corresponds to a truncation at the first order N=1 in the decomposition. This yields the following equation for the zero order moment Gσ :


⎛1 ⎞ −∇⋅ ⎜ ∇Gσ (r)⎟ + 3κσ Gσ (r) =12π ησ (r), ⎝κσ ⎠

Gσ (r) = ∫

4π st

Iσ (r,u) dΩ.

221

(85)

The directional intensity, and then the radiative flux and its divergence, may be calculated from Iσ (r,u) =

1 4π

⎞ ⎛ 1 u ⋅ ∇Gσ (r)⎟ , ⎜ Gσ (r) − κσ ⎠ ⎝

(86)

once the above second order partial differential equation is solved for Gσ with appropriate boundary conditions. Let notice that Eqs.(84-86) are valid both for local equilibrium and nonequilibrium conditions. The source term ησ(r) can be replaced by κσ(r)Iσ0(r) under LTE conditions. The relative simplicity of the P1 approximation makes it very popular. The simple angular decomposition and truncation of the intensity to the first order leads generally to some discrepancies, in particular near the domain boundaries. Several studies show that the P1 approximation provides more accurate results for optically thick media than for optically thin ones (see e.g. Ref.56). The method can thus be used in combined manner with other methods to treat different spectral regions. It can be used for instance for wavenumbers close to the optically thick line centers while the ray tracing method, or Monte Carlo method, is used for the other non-thick parts of the spectrum. The use of higher order (P3) approximation leads of course to higher accuracy. Some attempts have been recently proposed to improve the efficiency of the PN approximation. In the SPN method developed by Larsen et al [63], the operator 1/ (1 + ε / κ σ u ⋅∇ ) , where ε = 1/(k ref Lref ) is a representative optical thickness, is developed for an optically thick wavenumber σ, in a power series. The resulting equations, for a truncation at the third order, are partial differential equations for two quantities (φ1σ and φ2σ) which can be written in non-dimensional form

−∇ ⋅

ε 2μ12 ∇φ1σ + κ σ φ1σ = 4πησ κσ

−∇ ⋅

ε 2μ 22 ∇φ 2σ + κ σ φ 2σ = 4πησ , κσ

(87)

with μi = (3 ± 6 /5) /7, i = 1,2 . This system of two partial differential equations is in fact coupled by the boundary conditions for φ1σ and φ2σ . These functions allow to calculate the radiative source term (in dimensional form) according to

222


∞ ⎛ 1 ⎞ PR = − ∫ ∇ ⋅ ⎜ ∇ ( a1φ1σ + a2φ2σ ) ⎟ dσ 0 κ ⎝ σ ⎠

(

(88)

)

where the constants ai are given by ai = 5 ∓ 5 6 30 , i = 1, 2 . The use of this method was shown to improve the P1 approximation for the treatment of photoionization in the problem of streamer propagation [64]. 4.3.3 The Tangent Slab Approximation When shock waves may be approximated as planar 1D media, the radiation transport can be simplified using analytical integrations over the two dimensions where the medium is assumed to be infinite. if μ denotes the direction cosine μ=cosθ, and θ is the angle between a propagation direction and the normal to the 1D layer, the radiative intensity at an abscissa z between the two boundaries (z=0 and z=E) is given by c − c'σ dc'σ ησ (c' )exp(− σ ) ,μ ≥ 0 κσ σ μ μ , c c −c c' −c dc' ησ σ Iσ (z, μ ) = Iσ ,2 exp(− σ ,E σ ) + ∫ (c' )exp(− σ σ ) σ , μ ≤ 0 cσ μ κσ σ μ μ

Iσ (z, μ ) = Iσ ,1 exp(−

cσ

μ

)+∫

cσ

0

(89)

,E

where Iσ,1 and Iσ,2 are the intensities leaving the boundaries 1 (z=0) and 2 (z=E), and z

cσ is the optical path given by cσ = ∫0κσ (z') dz' . Taking the moments of these intensities (multiplying by μ0 and μ1 and integrating over 4π steradians) leads respectively to the incoming intensity uσ,z and to the local radiative flux qσ,z along the z direction 1

uσ , z = 2π ∫ Iσ d μ = 2π ⎡⎣ Iσ ,1 E2 (cσ ) + Iσ ,2 E2 (cσ , E − cσ ) ⎤⎦ −1 +2π ∫

cσ

0

cσ η ησ σ (c 'σ ) E1 (cσ − c 'σ )dc 'σ + 2π ∫ (c 'σ ) E1 (c 'σ − cσ )dc 'σ cσ κσ κσ ,E

(90)

1

qσ , z = 2π ∫ Iσ μ d μ =2π ⎡⎣ Iσ ,1E3 (cσ ) − I σ ,2 E3 (cσ ,E − cσ ) ⎤⎦ −1

+2π ∫

cσ

0

cσ η ησ σ (c ' ) E2 (cσ − c 'σ ) dc 'σ − 2π ∫ (c ' ) E2 (c 'σ − cσ )dc 'σ cσ κσ σ κσ σ ,E

(91)

where the exponential-integral functions En are defined by 1 ⎛ c⎞ E n (c) = ∫ μ n −2 exp⎜ − ⎟ dμ 0 ⎝ μ⎠

(92)


223

It is worth noticing that Eq.(90) provides the incoming intensity that must be used to compute the absorption term in the balance equations of population densities of energy levels. The radiative source term may also be calculated from the intensity following

dqσ , z dz

= −2π κ σ ⎡⎣ Iσ ,1 E2 (cσ ) + Iσ ,2 E2 (cσ , E − cσ ) ⎤⎦ − 2πκσ ∫

cσ , E

0

ησ (c ' ) E ( c − c 'σ ) dc 'σ + 4πησ (cσ ) . κσ σ 1 σ

(93)

In practice the first and second order exponential-integral functions can be calculated from the third order function using the general formula dE n (c) = − E n −1 (c) n = 1, 2, 3, ... dc

(94)

Several studies have addressed the accuracy of the tangent slab approximation in the case of shock waves in front of atmospheric entry vehicles. The discrepancies are generally smaller in the region close to the stagnation line and higher in the detachment regions. The typical accuracy is about 10 to 30% in re-entry applications in the peak-heating region. 4.4 The Monte Carlo Method

The Monte Carlo method (MCM), applied to radiative transfer, can accommodate spatial, directional, and spectral aspects of the problem. Radiation transport is simulated stochastically in MCM by tracing energy bundles from emitting points until ending points. Bundle characteristics such as direction of emission, wavenumber of the bundle, etc, are chosen from random numbers and absorption is either considered as a local phenomenon with a randomly chosen point along the propagation direction, or calculated in a deterministic way from the length crossed by the bundle in a given cell. The random choice of each event follows of course probability densities in order to be physically consistent. The power emitted by a cell volume Vi is given by

Qie = ∫ dV ∫ Vi

π

θ =0

sinθ dθ ∫

2π

ϕ =0

dϕ ∫

∞

σ =0

ησ dσ ,

(95)

where θ and ϕ are the polar and azimuthal angles. The probability density P(M, θ, ϕ, σ) for a bundle to be emitted at point M, in a direction (θ, ϕ), with a wavenumber σ, is simply given by P( M,θ ,ϕ ,σ ) dV d θ d ϕ dσ =

dV sin θ dθ dϕ ησ dσ V i πsin θ d θ 2πdϕ ∞η d σ ∫ ∫ ∫ σ 0

0

0

(96)

224


and appears to be the product of independent probabilities related to each variable. As a general rule, the cumulated probability density function of any random variable may be seen as a random variable itself, with a uniform distribution in the range [0, 1]. Thus, the variables θ, ϕ and σ may be chosen from uniform random numbers Rθ , Rϕ , Rσ in this range according to

1 − cosθ ϕ Rθ = , Rϕ = , Rσ = 2 2π

∫ ∫

σ

0 ∞ 0

η (σ ') dσ ' η (σ ') dσ '

.

(97)

The choice of a point M inside Vi depends on the type of the used mesh. While the determination of θ and ϕ is performed analytically, the choice of σ is much more complex and may consume important CPU times if spectrally resolved calculations are performed with line structures. The inversion of the third relation in Eq.(97) can be drastically accelerated if one uses intermediate pre-tabulated values of Rσ . When absorption of the energy bundle is treated as a local phenomenon, advantage is taken from the fact that the transmissivity of the gaseous column between the emitting and absorbing points represents also the cumulated probability that the bundle escapes from the column. The transmissivity τσ is then considered as a uniform random variable in the range [0, 1] and the length labs traveled by the bundle is deduced from − ln(τσ ) = ∫

labs

0

κσ (s) ds ,

(98)

where the abscissa s = 0 designates the location of the emitting point. As a matter of fact, the bundle may be outside the calculation domain depending on the value of labs. In that case, miscellaneous surface events may occur such as absorption or reflection. It is found generally that the deterministic treatment of absorption is to be preferred to local deposition since it reduces significantly the statistical noise. When a bundle crosses a cell of volume Vj, entering it at length lin and leaving at length lout, its power is reduced by the fraction (τ σ (0, lin ) − τ σ (0, lout )) which is added to the power absorbed by cell Vj. The inspection of a given bundle is followed in this approach until the power carried by it becomes smaller than a cutoff criterion. In the most common MC methods, the number of bundles issued from each cell Vi or each surface element Sj is taken proportional to the power Qei or Qej emitted by this element and the number of bundles launched from Vi is calculated according to N i = NQie /(

Nv +Ns

∑ Q ) where N is the total number of bundles chosen for the simulation k =1

e k

and Nv and Ns stand respectively for the total number of volume and surface elements. The initial power carried by the different bundles is in this way almost the same. The net radiative power for a volume element is simply calculated at the end of the simulation as Pi = (Qia − Qie ) /Vi , where Qia is the total power absorbed by Vi, originating from all volume and surface elements.


225

When statistical narrow-band models are employed for the spectral behavior, the spectral correlation between emitted, transmitted and absorbed radiation must be treated carefully [65-66]. Taking the derivative of Eq. (74) or (75) with respect to s shows that absorption at abscissa s of radiation emitted at abscissa s’ requires the use of the second derivative ∂ 2τ σ / ∂s ∂s ' . In a discretized form, and assuming as discussed in Section 4.2.1 that η/κΔσ ≈ηΔσ /κΔσ in nonequilibrium conditions, the power of a bundle emitted at point si of the volume Vi and absorbed between the abscissa sjand sj+ of volume Vj is given by em,i Pabs ,j =

Qie Ni κ i

Δσ

)(

(

)

⎡ τ Δσ (s + δ s, s− ) − τ Δσ (s , s− ) − τ Δσ (s + δ s, s + ) − τ Δσ (s , s + ) ⎤ i j i j i j i j ⎥⎦ δ s ⎢⎣

(99)

In this expression, δs is the length of an elementary column in the source volume Vi. This length must be chosen sufficiently small so that the elementary column is optically thin. More details on the application of SNB models and MC methods in the framework of atmospheric entry applications may be found in Refs.65-66. Monte Carlo radiation simulations are known to possess a slow convergence rate scaling as 1/ N , where N is the total number of bundles. In order to control the convergence of MC simulations, N is generally subdivided into a number of M samples. Each sample m provides an estimation Pi,m of the radiative power at cell i and, at the limit of large numbers, an estimation of the variance σ 2 of the mean value Pi is given by

σ2 =

M 1 ⎡ Pi, m − Pi ⎤ ∑ ⎦ M ( M − 1) m=1 ⎣

2

(100)

5 Radiation and Flow-Field Coupling As stated above, radiative field strongly depend on the thermochemical state of the flow field, through the radiative properties κσ and ησ. Various models have been used to describe nonequilibrium thermochemical flows, from direct simulation Monte Carlo (DSMC) methods used to simulate high Mach numbers re-entry flow field at high altitude [67], to continuum fluid flow multi-temperaure [68] approaches, where the different energy modes (i.e. translation-rotation, vibration, electronic,…) are assumed to be characterized by Boltzmannian populations at distinct temperatures (Ttr, Tv, Tel, …). More detailed vibrational-specific or electronic-specific approaches [69] have been developed to account for non-Boltzmannian populations of levels of some specific modes. In such generally called collisional radiative description, these specific populations become state variables which can be determined in a full coupled manner with other aerothermal variables [70] or more usually as post-processing of the aerothermal fields (densities, velocities, temperatures) using additional approximations such as the QSS one [71], or the Lagrangian approach [72].

226


Once the populations of the energy levels of the radiating particle are determined from the thermochemical state of the fluid, emission and absorption coefficient can be evaluated. The radiative transfer equation may then be solved in order to obtain the field of radiative power PR and the distribution of the radiative flux qR on the boundaries of the medium. In return the radiative field affects the thermochemical state of the flow through source terms in energy balance and population balance equations. First, the radiative power - PR acts as a source term in the total energy balance equation. In multi-temperature approach, the radiative power - PR has moreover to be split into various relevant contributions which should appear as source terms in the energy balance equations of the various modes. In many approaches, where a unique temperature is used to describe vibrational and electronic energy modes [73-75] the total radiative power has been used as source term in both the total energy and the electro-vibrational energy balance equations. It has been generally observed that in near-equilibrium situations, radiation decreases the “equilibrium” temperature in the shock layer, increases the density, and thus reduces the shock standoff distance [73,74,76]. On the contrary, for strong non-equilibrium shock layer with significant ionization, radiation has been shown to reduce the electro-vibrational temperature and to increase the heavy particle translational one and thus the shock standoff distance [74]. The wall radiative fluxes are generally reduced in all situations since the electrovibrational temperatures are lowered. As pointed out in Ref.77, the coupling of radiative transfer to aerothermal fields through source terms in energy equations is usually achieved using an iteration procedure and a slight under-relaxation of the variations of the radiative source terms. In this last reference, wall ablation effects and their coupling to the flow simulations are investigated for various entry conditions (Fire II …). An additional coupling mechanism is thus considered, since the wall temperature in this model is strongly determined by the radiative flux. Here again, the coupling of the ablation model to the radiative transfer is obtained through a strongly under-relaxed iteration procedure. In more detailed approaches, radiative transfer should be accounted for as populating and depleting mechanisms of energy levels. Bound-free transitions such as photoionization or photodissociation even contribute as source terms in the balance equation of involved radiating species. In the frame of collisional radiative models, specific population equations such as Eqs.(62,64) have to be solved in a selfconsistent manner with the radiative transfer equation. Such studies remain quite sparse. In the case of calculations devoted to the Huygens probe entry into Titan, Johnston [75] evaluated escape factors related to the CN (B-X) violet system from a tangent slab Radiative Transfer approach iteratively coupled to a QSS approach. The B state populations obtained are slightly larger than those obtained by neglecting absorption (escape factor equal to 1) providing an increase of the radiative wall fluxes of about 16 %. More recently, in calculation cases representative of the Stardust Sample Return capsule, Sohn et al [47] have evaluated escape factors associated to the main resonant O and N transitions using a Monte Carlo RTE solver coupled to the NEQAIR QSS approach. Aerothermal fields were obtained from a DSMC simulation. Starting from escape factor equal to one, convergence of the relevant atomic populations was obtained at the first iteration; these populations was shown to be increased by a factor of two in comparison with the optically thin approach. In a similar way, the radiative wall fluxes also increased by a factor of two.


227

6 Conclusion and Perspectives This brief overview shows the complexity of radiation phenomena in strong shock waves, especially under non-equilibrium conditions. Several experimental, theoretical and numerical studies were devoted to this problem during the last two decades to support engineering aspects in the planned spatial missions. Modeling of nonequilibrium radiation remains however a challenging issue and further work is needed in several directions: - Uncertainties remain on the spectroscopic data in the UV and VUV regions where measurements are scarce or even lacking. - As the radiation is closely linked to the thermochemical state of the gases, a better understanding of the excitation processes in the flow is required to enable reliable radiation transport predictions. - The coupling between radiation and aerothermal fields, on one hand, and with energy level populations, on the other hand, is far to be fully understood. Selfconsistent modeling of these coupled phenomena is a formidable task and has not been fully undertaken yet. - The development of approximate and reliable spectral models should enable accurate prediction of radiation and aerothermal fields in complex 3D geometries. - The use of ablative thermal protection system, which produces pyrolysis gases and other ablative products, gives rise to additional difficulties. The radiative properties of products have to be determined. These products may absorb radiation from the shock layer and reduce the radiative heat flux to the surface through radiation blockage.

References 1. Zeldovich, Y.B., Raizer, Y.P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Academic Press, New York (1966) 2. Mihalas, D., Mihalas, B.W.: Foundations of Radiation Hydrodynamics. Dover Publications, Inc. (1999) 3. Park, C.: Nonequilibrium hypersonic aerothermodynamics. A Wiley-Interscience Publication, New York (1990) 4. Park, C.: Overview of Radiation Problems in Planetary Entry. Proceedings of the International Workshop on Radiation of High Temperature Gases in Atmospheric Entry, ESA-SP-583 (2005) 5. Park, C.: Nonequilibrium Air Radiation (NEQAIR) Program: User’s Manual. NASA TM 86707 (1995) 6. Hartung, L.C.: Predicting radiative heat transfer in thermo-chemical nonequilibrium flowfields: theory and user’s manual for the LORAN code. NASA TM 4564 (1994) 7. Fujita, K., Abe, T.: SPRADIAN. Structured Package for Radiation Analysis: theory and application, ISAS Report No. 669 (1997) 8. Surzhikov, S.: Radiation Modeling and Spectral Data. VKI Lecture Series 2002-2007 on Physico-Chemical Models for High Enthalpy and Plasma Flow, VKI (2002)

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9. Laux, C.: Radiation and nonequilibrium collisional-radiative models. In: VKI Lecture Series 2002-2007 on Physico-Chemical Models for High Enthalpy and Plasma Flows Modeling, VKI (2002) 10. Smith, A., Wood, A., Dubois, J., Fertig, M., Pfeiffer, N.: Technical Paper 3. ESTEC contract11148/94/NL/FG, FGE TR28/96 (2006) 11. Johnston, C., Hollis, B., Sutton, K.: Journal of Spacecraft and Rockets 45, 865 (2008) 12. Passarinho, P., Lino da Silva, M.: Journal of Molecular Spectroscopy 236, 148 (2006) 13. Perrin, M.Y., Rivière, P., Soufiani, A.: Radiation database for Earth and Mars entry. In: AVT-162 RTO AVT/VKI Lecture Series on Non-Equilibrium Gas Dynamics, from Physical Models to Hypersonic Flights, VKI (2008) 14. Chauveau, S., Perrin, M.Y., Rivière, P., Soufiani, A.: Journal of Quantitative Spectroscopy and Radiative Transfer 72, 503 (2002) 15. Chauveau, S., Deron, C., Perrin, M.Y., Rivière, P., Soufiani, A.: Journal of Quantitative Spectroscopy and Radiative Transfer 77, 113-130 (2003) 16. Babou, Y., Rivière, P., Perrin, M.Y., Soufiani, A.: Journal of Quantitative Spectroscopy and Radiative Transfer 110, 89 (2009) 17. Hollas, J.M.: High Resolution Spectroscopy. Butterworths (1982) 18. Cowan, R.D.: The Theory of Atomic Structure and Spectra. University of California Press, Berkeley (1981) 19. Herzberg, G.: Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules, 2nd edn. Van Nostrand Reinhold, New York (1950) 20. Lefebvre-Brion, H., Field, R.W.: Perturbations in the Spectra of Diatomic Molecules. Academic Press Inc. (1986) 21. Griem, H.R.: Principles of plasma spectroscopy. Cambridge University Press (1997) 22. Ralchenko, Y., Kramida, A.E., Reader, J.: NIST ASD Team, NIST Atomic Spectra Database (version 4.0), National Institute of Standards and Technology, Gaithersburg, MD (2010), http://physics.nist.gov/asd 23. The Opacity Project Team, The opacity Project, vol.1. Institute of Physics Publishing, Bristol and Philadelphia (1995), http://cdsweb.u-strasbg.fr/topbase/topbase.html 24. Traving, G.: Plasma Diagnostics. McGraw-Hill Book Company, New York (1964) 25. Rivière, P.: Journal of Quantitative Spectroscopy and Radiative Transfer 73, 91 (2002) 26. Babou, Y., Riviere, P., Perrin, M.Y., Soufani, A.: International Journal of Thermophysics 30, 416 (2009) 27. Zare, R.N., Schmeltejopf, A.L., Harrop, W.J., Albritton, D.L.: Journal of Molecular Spectroscopy 46, 37 (1973) 28. Kovacs, I.: Rotational structure in the spectra of diatomic molecules. American Elsevier Publishing company Inc., New York (1969) 29. Whiting, E.E., Schadee, A., Tatum, J.B., Hougen, J.T., Nicholls, R.W.: Journal of Molecular Spectroscopy 80, 249 (1980) 30. Hartmann, J.M., Boulet, C., Robert, D.: Collisional effects on molecular spectra. Elsevier (2008) 31. Breene, R.G.: Applied Optics 6, 141 (1967) 32. Lamet, J.M., Babou, Y., Rivière, P., Perrin, M.Y., Soufiani, A.: Journal of Quantitative Spectroscopy and Radiative Transfer 109, 235 (2008) 33. Morris, J.C., Key, R.U., Bach, G.R.: Physical Review 159, 113 (1967); Morris, J.C., Krey, R.U., Garrison, R.L.: Physical Review 180, 167 (1969) 34. Cruden, B.A., Martinez, R., Grinstead, J.H., Olejniczak, J.: AIAA Paper 2009-4240 (2009)


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35. Yamada, G., Takayanagi, H., Suzuki, T., Fujita, K.: AIAA paper 2009-4254 (2009) 36. Cauchon, D.L.: Radiative heating results from the FireII flight experiment at a reentry velocity of 11.4 kilometers per second, NASA TM X-1402 37. Mazoue, F., Marraffa, L.: Determination of the radiation emission during the FIRE II entry. In: Proceedings of the 2nd International Workshop on Radiation of High Temperature Gases in Atmospheric Entry, Rome, September 6-8 (2006) 38. Lamet, J.M.: Transferts radiatifs dans les écoulements hypersoniques de rentrée atmosphérique terrestre, Thèse de doctorat de l’Ecole Centrale, Paris (2009) 39. Lowke, J.J.: Journal of Quantitative Spectroscopy and Radiative Transfer 14, 111 (1974) 40. Holstein, H.: Physical Review 72, 1212 (1947) 41. Holstein, H.: Physical Review 83, 1159 (1951) 42. Irons, F.E.: Journal of Quantitative Spectroscopy and Radiative Transfer 22, 1 (1979) 43. Bourdon, A., Térésiak, Y., Vervisch, P.: Physical Review E 57, 4684 (1998) 44. Pestehe, S.J., Tallents, G.J.: Journal of Quantitative Spectroscopy and Radiative Transfer 72, 853 (2002) 45. Fisher, V.I., Fisher, D.V., Maron, Y.: High Energy Density Physics 3, 283 (2007) 46. Novikov, V.G., Ivanov, V.V., Koshelev, K.N., Krivtsun, V.M., Solomyannaya, A.D.: High Energy Density Physics 3, 198 (2007) 47. Sohn, I., Li, Z., Levin, D.A.: AIAA Paper 2011-533 (2011) 48. Goody, R., Yung, Y.: Atmospheric Radiation Oxford Univ. Press, New York (1989) 49. Taine, J., Soufiani, A.: Adv. Heat Transfer 33, 295 (1999) 50. Ludwig, C., Malkmus, W., Reardon, J., Thomson, J.: Handbook of infrared radiation from combustion gases, Technical Report NASA SP-3080, Washington DC (1973) 51. Young, S.: Journal of Quantitative Spectroscopy and Radiative Transfer 15, 483 (1975) 52. Young, S.: Journal of Quantitative Spectroscopy and Radiative Transfer 18, 1 (1977) 53. Lamet, J.-M., Rivière, P., Perrin, M.-Y., Soufiani, A.: Journal of Quantitative Spectroscopy and Radiative Transfer 111, 87 (2010) 54. Rivière, P., Soufiani, A.: Journal of Quantitative Spectroscopy and Radiative Transfer 112, 475–485 (2011) 55. Rivière, P., Soufiani, A., Perrin, M.-Y., Riad, H., Gleizes, A.: Journal of Quantitative Spectroscopy and Radiative Transfer 56, 29 (1996) 56. Kahhali, N., Rivière, P., Perrin, M.-Y., Gonnet, J.-P., Soufiani, A.: J. Phys. D: Appl. Phys. 43, 425204 (2010) 57. Zhang, H., Modest, M.F.: Journal of Quantitative Spectroscopy and Radiative Transfer 73, 349 (2002) 58. Bansal, A., Modest, M.F.: AIAA Paper 2011-247 (2011) 59. Chandrasekhar, S.: Radiative Transfer. Dover Publications Inc. (1960) 60. Siegel, R., Howell, J.R.: Thermal Radiation Heat Transfer. Taylor&Francis (2002) 61. Modest, M.F.: Radiative Heat Transfer. Elsevier (2003) 62. Carlson, B.G., Lathrop, K.D.: Discrete-ordinates angular quadrature of the neutron transport equation, Technical Information Series Report LASL-3186, Los Alamos Scientific Laboratory (1964) 63. Larsen, E.W., Thömmes, G., Klar, A., Seaïd, M., Götz, T.: J. Comput. Phys. 183, 652 (2002) 64. Ségur, P., Bourdon, A., Marode, E., Bessieres, D., Paillol, J.H.: Plasma Sources Sci. Technol. 15, 648 (2006) 65. Rouzeau, O., Tessé, L., Soubrié, T., Soufiani, A., Rivière, P., Zeitoun, D.: Journal of Thermophysics and Heat Transfer 22, 10 (2008)

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66. Lamet, J.M., Perrin, M.-Y., Soufiani, A., Rivière, P., Tessé, L.: In: Proc. Third Int. Workshop on Radiation of High Temperature Gases in Atmospheric Entry. ESA, Heraklion (2008) 67. Ozawa, T., Zhong, J., Levin, D.A.: Phys. Fluids 20, 046102 (2008) 68. Gnoffo, P.A., Gupta, R.N., Shinn, J.L.: Conservation Equations and Physical Models for Hypersonic Air Flows in Thermal and Chemical Nonequilibrium, NASA TP-2867, NASA Langley Research Center, Hampton, VA 23665-5225 (1989) 69. Capitelli, M. (ed.): Non-equilibrium vibrational kinetics, Topics in Current Physics, vol. 39. Springer, Heidelberg (1986) 70. Panesi, M., Magin, T., Bourdon, A., Bultel, A., Chazot, O.: Journal of Thermophysics and Heat Transfer 23, 236 (2009) 71. Park, C.: AIAA Paper 84-0306 (1984) 72. Magin, T.E., Caillault, L., Bourdon, A., Laux, C.O.: J. Geophys. Research 111, E07S12 (2006) 73. Gökçen, T., Park, C.: AIAA paper 91-0570 (1991) 74. Hartung, L.C., Mitcheltree, R.A., Gnoffo, P.A.: J. Thermophys. Heat Transfer 8(2), 244 (1994) 75. Johnston, C.: Nonequilibrium Shock-Layer Radiative Heating for Earth and Titan Entry. PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia (November 17, 2006) 76. Kay, R.D., Gogel, T.H.: AIAA Paper 94-2091 (1994) 77. Gnoffo, P.A., Johnston, C.O., Thompson, R.A.: AIAA Paper 2009-1399 (2009)

Chapter 6

Structure of Shock Waves A.A. Raines1 and F.G. Tcheremissine2 1

2

Feza Gursey Institute, Istanbul, Turkey Dorodnicyn Computing Center of RAS, Moscow, Russia

1 Introduction Shock wave that appears from the solution of gas dynamic Euler equations as a discontinuity of gas parameters related by Rankine-Hugoniot conditions, in reality presents a narrow layer inside which the dissipative processes form the transition from one thermodynamically equilibrium state to another one. The fastest dissipative process is due to the elastic collisions between gas molecules. It imposes the characteristic length scale of the molecular mean free path and makes the hydrodynamic equations inapplicable for rigorous analysis. Hence, the application of computational methods of the kinetic theory is required. For gas mixtures and polyatomic gases additional dissipative processes of lower rate should be considered as well. For inert mixtures it is a collisional momentum and energy exchange between species of different masses, and for polyatomic gases the energy transfer between translational and internal degrees of freedom. At the microscopic level, the high nonequilibrium shock wave layer is described by velocity distribution functions of the components of the gas, and by internal energy spectra. At the macroscopic level, the shock wave structure can be represented by the densities of gas components, their velocities, kinetic temperature tensors of the gas as a whole and of the gas components, by the temperatures of the internal energy modes, and by the different energy fluxes. Besides its physical interest as an example of high non-equilibrium layer, the problem of the shock wave structure presents an important benchmark for computational methods of the kinetic theory. Its advantage consists of its onedimensional structure in the physical space, simple boundary conditions, and property of conservation of mass, momentum, and energy fluxes that can be used for verification of computations. It is also important that there exists good experimental data [1,2]. Historically, the problem has been used for the development of a number of numerical methods and approximate kinetic theories. In the case of a pure gas, various methods were applied: moment methods [3], solution of relaxation equations [4-7], direct simulation Monte Carlo (DSMC) method (in [8,9], later in [10-13], and their references), and in [14-15]. The first numerical solutions of the Boltzmann equation [16] were obtained by Nordsieck, Hicks, and Yen [17-19] and then in [20]. Later, the problem was solved by discrete-ordinate methods for the Boltzmann equation with different techniques of evaluation of the

232


collision integral: use of high order polynomial approximation of the distribution function in the velocity space [21,22], with application of the polynomial correction in the course of computations for complying with the conservation laws [23], and use of the conservative projection method [24-27]. Analysis of the velocity distribution function in an infinitely strong shock wave was done in [28]. For polyatomic gases, experimental measurements were made in [1,29-33] and computations in [1,34,35] by DSMC, in [36] with model equation, and in [37-39] by solving the generalized Boltzmann equation. Review of DSMC phenomenological models for rotational relaxation is presented in [40,41]. For binary gas mixtures, the shock wave structure has been investigated experimentally [42,44] and theoretically by different approaches: with moment methods [45,46], with fluid-dynamics models [47,48], by solving model equations [49,50], by DSMC method [51,52], (see [12] and references therein.). The above cited method [22] was extended to binary gas mixtures [53], the method with polynomial correction [23] was also applied to binary gas mixtures [54,55], the method [25,26] was first extended to binary gas mixtures [56,59], then to 3-4-component mixtures [60], later the computations for 3-component mixtures were repeated by the same method [61]. Here, the shock wave structure is computed by the Conservative Projection Method (CPM) [24-26]. The method is based on a special projection technique for evaluation of the collision operator that provides its explicit discrete form which approximates the continuum operator at a set of nodes in the phase space. The computed discrete collision operator is conservative for density, momentum and energy, and is equal to zero when the solution has a form of Maxwellian distribution function. The latter feature strongly increases the efficiency of calculations, especially in near-equilibrium parts of the flow. The integration grid for evaluation of the collision integrals are given by the method [62] which provides more uniform and more efficient distribution of the grid nodes than obtained by random number generators. The differential part of the Boltzmann equation is approximated by the flux conservative finite-difference scheme of the 2nd order [63]. In this scheme the transport of mass, momentum and energy between the nodes of the configuration space is realized in a conservative way.The method was extended to gas mixtures [57] and to gases with internal degrees of freedom, where it can incorporate real physical parameters of molecular potential and internal energy spectrum [64]. It was also applied for joint solution of the Boltzmann equation and Navier-Stokes equations [65]. A simple model for the rotational relaxation in the framework of the CPM was proposed [66]. The discrete-ordinate method introduced in [22] for a single gas was extended in [53] to the case of a binary gas mixture. In this approach the collision integral is approximated by a decomposition of the distribution functions on piecewise quadratic functions by the longitudinal molecular velocity ξ x , and a system of Laguerre polynomials by the transversal velocity ξr . The conservation laws are not satisfied exactly but with an accuracy sufficient for stable calculations. In [53] the authors have analysed the accuracy of computations, defined a convergence criterion and obtained high precision results by using powerful computers.

Structure of Shock Waves

233

2 Methodology of Computations The SW structure is studied on the basis of the Boltzmann kinetic equation for a monatomic gas and a mixture of monatomic gases, and with the use of the Generalized Boltzmann Equation (GBE) for a polyatomic gas having internal degrees of freedom. The kinetic equation is solved by a deterministic finite-difference method on fixed grids in velocity and configuration spaces. For a long time the main deficiency of the methods of direct solution of the Boltzmann equation has consisted in non-conservative evaluation of the multidimensional collision integral that produced spurious sources of mass, impulse, and energy. The problem has been solved [24] by application of a special projection technique for conservative evaluation of the collision integral. The method was considerably improved in [26] by inclusion of “inverse collisions” that strongly increases the accuracy of computations in near equilibrium parts of the flow. Further development consisted of improvements of some numerical techniques for accelerating the calculations and of extension of the method to gas mixtures and molecular gases with internal degrees of freedom [27,57,64]. The detailed description of the method for a simple gas can be found in [27]. A short description of the method is given below. 2.1 Solution of the Boltzmann Equation for a Pure Monatomic Gas The Boltzmann kinetic equation may be written in the following form ∂f ∂f +ξ =I ∂t ∂x

(2.1.1)

The distribution function f (ξ, x, t ) and the collision integral I (ξ , x , t ) are defined in 6dimension phase space (ξ, x ) and depend on time t . The collision integral, omitting variables x and t , can be presented in the standard form I (ξ) =

2π

bm

0

0

∫ ∫ ∫

R3

( f ' f *' − ff * ) gbdbd ϕ dξ *

The functions f ' and f *' contain post-collision velocities

(2.1.2)

ξ ' , ξ*'

which are

determined for a given molecular potential by velocities before collision ξ , ξ* an impact parameter b , g =| ξ* − ξ | and an angular parameter of the collision ϕ . The equation (2.1.1) is solved in a domain Ω of a volume V of the velocity space on a grid of N 0 equidistant nodes ξ γ with a step h. In configuration space an arbitrary discrete grid x i is applied. In the basis of Dirac δ–functions the distribution function and the collision integral can be presented in the following form

234

High Temperature Phenomena in Shock Waves N0

f (ξ , x, t ) = ∑ fγ ( x, t )δ (ξ − ξ γ ) γ =1

,

N0

I (ξ, x, t ) = ∑ Iγ (x, t )δ (ξ − ξγ ) γ =1

After the evaluation of the collision integral, the problem is reduced to a system of linear equations ∂fγ ∂t

+ ξγ

∂fγ ∂x

= Iγ

(2.1.3)

For construction of the projection method the integral (2.1.2) at a point ξγ is written in the form I γ ≡ I ( ξγ ) =

2π

*

R3

Using

the

bm

∫ ∫ ∫ ∫ δ (ξγ − ξ )( f R3

0

'

f*' − ff* ) gbdbdϕ dξ*dξ

0

φ (ξγ ) = δ (ξ − ξγ ) + δ (ξ* − ξγ ) − δ (ξ' − ξγ ) − δ (ξ*' − ξγ )

notation

and

knowing the properties of the collision integral, one can write this integral in the symmetric form Iγ =

1 4 R∫3

2π

bm

∫ ∫ ∫ φ (ξγ )( f R

3

0

'

f*' − ff* ) gbdbdϕ dξ*dξ

(2.1.4)

0

To evaluate (2.1.4), we define the domain Ω × Ω × 2π × bm , in which a uniform integration grid ξ α , ξ β , bν , ϕν of Nν nodes is built, so that ξα and ξ β belong to the ν ν ν

ν

velocity grid, excluding the values of variables bν , ϕν , for which the post-collision

velocities ξ α' , ξ 'β are outside of Ω . As the points ξα' , ξ 'β in general, do not coincide ν ν ν

ν

with the velocity grid, a regularization of the sum (2.1.4) is needed. Let ξ λ and ξ μ ν ν be the nearest vertices of the cells, in which we find the points ξ α' , ξ 'β and let ξ λ + s , ν ν ν

ξ μν − s be a pair of other symmetrically situated vertices. Then, the last two

δ – functions in φ (ξ γ ) are replaced by the expressions

δ (ξα' ν − ξγ ) = (1 − rν )δ (ξ λν − ξγ ) + rν δ (ξ λν + s − ξγ )

(2.1.5)

δ (ξ'βν − ξγ ) = (1 − rν )δ (ξμν − ξγ ) + rνδ (ξμν −s − ξγ ) This means that the contributions to the points ξα' , ξ 'β are distributed among the ν ν nearest

nodes.

Defining

E0 = (ξα' ν )2 + (ξ 'βν )2 ,

E1 = (ξ λν )2 + (ξ μν )2 ,


235

E 2 = ( ξ λν + s ) 2 + ( ξ μν − s ) 2 , one of the following conditions is true: E1 ≤ E 0 < E 2 , or

E 2 < E0 ≤ E1 . The coefficient rν can be defined from the energy conservation law E 0 = (1 − rν ) E1 + rν E 2 , from which it results 0 ≤ rν < 1 . Using the obtained value of rν one finds the interpolation formula that is exact for the Maxwellian function f β = f M (ξ β )

fα'ν f β'ν = ( fλν f μν )1−rν ⋅ ( fλν +s f μν −s )rν

(2.1.6)

The integral sum (2.1.4) is calculated simultaneously in all nodes ξ γ of the velocity grid. Defining B = V π bm2 N 0 / (4 Nν ),

Δν = ( fαν f βν − fα'ν f β'ν ) gν , and introducing

Kroneker symbol δ γ ,β , one obtains an explicit discrete form of the Boltzmann collision integral in Cartesian velocity space, i.e. Nν

Iγ = B∑[−(δγ ,αν + δγ ,βν ) + (1 − rν )(δγ ,λν + δγ ,μν ) + rν (δγ ,λν +s + δγ ,μν −s )]Δν ν =1

The interpolation (2.1.6) annuls the collision integral from the Maxwellian distribution function

I ( fM , fM ) = 0

(2.1.7)

Assuming that the solution is close to the Maxwellian distribution function

f = f M + ε f (1) , ε 1 for all Mach numbers, when, for the above considered two cases, the curves cross the level Q = 1 at M ≈ 2.4 , in accordance with the experimental data.


253

nsity profile as a function of Mach number. Solid line - LJ Arrgon Fig. 12. Assymetry of the den gas, dashed line - HS gas, and d dash-dotted line- solution of the N.-S. equations; -- data [11].

4 Shock Wave Struccture in a Polyatomic Gas In general case, the shock k wave structure in a polyatomic gas is formed by thhree dissipative processes: elastic collisional relaxation, rotational-translational (R RT) energy transfers, and vibrattional-translational (VT) energy transfers. The first proccess has the highest rate, it is folllowed by the second one, and the rate of the VT processs is usually some order of magn nitude less compared to the RT transfers. For calculating the non-eequilibrium flow of molecular gases, the most widely uused technique is the DSMC meethod. Here, we apply the computational methodology for computing the shock wav ve structure in a polyatomic gas using the Generaliized Boltzmann Equation (GBE)). The whole problem that includes both (VT) and (RT) energy transfers is solvedd by applying a three-stage splittting procedure to the GBE. The three stages consist of ffree molecular transport, VT rellaxation, and RT relaxation. For the VT relaxation, GBE E is always solved. For the RT relaxation, r two approaches are theoretically possible. In the first approach, for the RT reelaxation GBE is solved. This approach is computationaally very intensive since the totaal number of excited levels is the product of rotational and vibrational levels. In the seecond approach, a two-level model of RT relaxation tthat equilibrates rotational and translational t energies is employed. The second approacch is computationally much less intensive than the first and therefore is much m more efficient. Using this approach the computations are performed for SW structuree at high Mach numbers accoun nting for both vibrational and rotational excitations. 4.1 Shock Wave Structurre with Frozen Vibrational Levels The vibrational and rotation nal quanta have very different magnitudes. As an examp mple, the rotational quantum of Nitrogen N is equal to 2.9K, when the vibrational quantum m is 3340K. For Oxygen the corrresponding values are 2.1K and 2230K [69]. The VT crross

254


sections for both gases are by some orders of magnitude less the RT cross sections. Because of such a difference, in many cases the VT process could be neglected and only RT relaxation should be taken into account. On the other hand, the small value of the rotational quantum and rather high value of the cross section of RT process, which may be of the order of 0.25 - 0.1 of the elastic cross section [1] requires the replacement for molecular gases of the classic Boltzmann equation by GBE at all temperatures. When the temperature in SW does not approach the value of the vibrational quantum or when the cross section of the VT transfer at that temperature is negligibly small, the SW structure can be computed by solving the GBE equation for RT process only. Numerical study of the shock wave structure in a polyatomic gas with rotational degrees of freedom presents interest for two main reasons. First, it provides additional data about the process of rotational-translational (R-T) energy exchange that cannot be obtained by a physical experiment. Second, it can serve as a test and verification of numerical methods by comparison with existing experimental data. The experimental data for Nitrogen shock structure are obtained by registration of electron beam induced fluorescence [29-31], absorption of an electron beam [1], and by Raman spectroscopy[32], with the use of jets, wind tunnels, and shock tubes. The most definite experimental conditions were realized for moving shock waves in shock-tube experiments [1], but the applied method of measurements provided only density profiles. In other experiments with steady shocks formed in expanding free jets and wind tunnels, the thermodynamic equilibrium between rotational and translational modes before the front may be distorted, influencing not only the shock wave structure, but also rotational spectrum and rotational temperature as well. Most computations of the shock wave structure by DSMC method were carried out with application of different phenomenological relaxation models for the internal energy that involve numerous assumptions, which are not always physically justified [40,41]. More rigorous Monte Carlo approach that uses classical trajectory calculations of the interactions of rotating molecules [34] requires enormous amount of calculations. A more economic DSMC relaxation model based on the trajectory calculations was recently proposed [35]. In the present computations the molecular collisions are described by the LennardJones interaction potential with parameters and rotational spectrum data taken from [68]. Thus, for molecular Nitrogen, the depth of the energy hole ε = 91K , the degeneration of rotational level qi = 2i + 1, i = 0,1,..∞ , and the rotational energy of the level eri = ε 0 i (i + 1), ε 0 = 2.9 K . The molecular interaction during the collision consists of two phases. In the first phase, the molecules interact in an elastic manner according to the molecular potential. This stage determines the deviation angle of the relative velocity. In the second stage, the modulus of the relative velocity changes according to the energy conservation equation. For the transition probabilities Pijkl we apply the formulas [36] that are obtained by fitting the experimental data of molecular dynamics simulations of interactions of rigid rotors that model N 2 molecules


P i jk l = P 0 ω

kl ij

[α

0

exp(− Δ

− Δ

1

− Δ

2

3

− Δ

4

) +

1

α

exp(− Δ

3

− Δ

255

4

)] ,

0

where

Δ1 =| Δe1 + Δe2 | / etr 0 ,

Δ 2 = 2 | Δe2 − Δe1 | / etot

Δ 3 = 4 | Δe1 | /(etr 0 + eri ) ,

Δ e1 = e r i − e r k , e tr 0 = m g

2

/4,

Δ 4 = 4 | Δe2 | /(etr 0 + erj )

Δ e 2 = e rj − e rl ,

α

0

= 0 . 4 e to t / e tr 0

etot = etr 0 + eri + erj .

The energy conservation law in a collision selects virtual collisions with non-zero probability. From the equation

m g ij2 / 4 + e ri + e rj = m g kl2 / 4 + e rk + e rl , it follows that Pijkl > 0 , if g kl2 ≥ 0 , otherwise Pijkl = 0 . The elastic collision is a particular case of the collision. The probabilities obey the normalization condition:

∑P

kl ij

k ,l

=1

that should be strictly satisfied in the

computations. The formula for transition probabilities is averaged over all interactions and does not depend on the impact parameter. In [37] the inelastic collisions were limited by some impact parameter common for all the interactions. In the presented computations, the deviation angle is limited to a value 0.13 below which the R-T transition is prohibited. The number of levels is selected according to the temperature range of the considered problem. For moderate Mach numbers, the SW structure in Nitrogen can be computed with real value of the spectral energy gap

ε 0 , but for the hypersonic

case the required number of levels becomes too high (up to 50-70 levels) and therefore the computations become very time consuming. To facilitate this problem it is possible to increase this energy gap and thereby reduce the number of levels, keeping the condition

ε 0*

High Temperature Phenomena in Shock Waves

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Magnetohydrodynamics waves and shock waves in curved space-time

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