The Measurement and Application of Electric E ects in ... · Appendix C {Electron Density...

The Measurement and Applicationof Electric Effects in Combustion

by

Daniel Corrigan Murphy

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Engineering – Mechanical Engineering

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor A. Carlos Fernandez-Pello, ChairProfessor Michael LustigProfessor Robert Dibble

Spring 2015

Abstract


by


Doctor of Philosophy in Engineering – Mechanical Engineering

University of California, Berkeley

Professor A. Carlos Fernandez-Pello, Chair

The existence of interactions between flames and electric fields has been known for quitesome time and has received experimental and theoretical study over the years but remainan active topic of research in the combustion community. The present work specificallyinvestigates the sensitivity of premixed flames electric fields. Prior work has demonstratedthat electric fields may be used to enhance and control combustion, but full potential andrange of applications for this effect have not been explored. Multiple theories have beenpresented to explain the process, but there is not yet a truly complete understanding of howelectric fields fundamentally change the process of combustion. This work serves to explorenew applications of electric-flame interactions and to provide experimental measurements tosupport the development of detailed theoretical models.

Thermo-diffusive and acoustic instabilities in freely propagating flames can trigger theformation of wrinkled flames and turbulence, which may or may not be desirable in differentscenarios. Electric fields present a means to interact with such instabilities either by gen-erating direct hydrodynamic forces (’ionic wind’) or by modifying the rate of combustion.Experiments performed on downward-propagating hydrocarbon flames have demonstratedthat electric fields can be applied to excite or suppress existing instabilities and control theonset of turbulence. Numerical models following from these experiments indicate that subtlespatial variations in chemical reactivity can achieve similar control of inherently unstableflames.

A practical limitation of combustion is the phenomenon of quenching, where heat losses tothe surroundings extinguish a flame. The enhancement achieved by electric fields is shown todramatically increase flame propagation speed and reduce quenching in methane-air flames,which may permit the miniaturization of combustion-based power generation systems. Inthe course of these experiments, flames were found to exhibit different behavior dependingon the direction of the applied field. This result is consistent with a proposed model of

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ion transport in laminar flames, although the experiments have shown a greater degree ofenhancement than predicted.

Ultimately, modeling the electrical aspects of combustion should be based on detailedaccounting of the ion species present in flames. To support the ongoing development ofdetailed ion-chemistry mechanisms, a non-intrusive microwave interferometer for use in shocktube studies of combustion kinetics was developed. This was used to measure the formationand consumption of free electrons in shock-induced combustion. The range of equivalenceratios tested and the variety of hydrocarbon fuels used provide a rich dataset. Comparisonof these results to complementary chemical kinetics simulations have been used to validateproposed improvements to existing mechanisms.

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Dedication

i

Table of Contents

Table of Contents ii

List of Figures iv

List of Tables vi

Nomenclature x

Acknowledgments xi

CHAPTER 1 – INTRODUCTION 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

CHAPTER 2 – FLAME STABILITY 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Results: Flame Acceleration . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Results: Flame Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

CHAPTER 3 – FLAMES IN NARROW CHANNELS 293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 V-Channel quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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3.2.2 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.3 Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.1 Quasi-2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.2 Quenching Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.3 SP as a surrogate for δq . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

CHAPTER 4 – MICROWAVE INTERFEROMETRY 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.1 Plasma Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.2 Thermal Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3 Experimental Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.1 Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.2 Test Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.3 Microwave Interferometer . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.4.1 Thermal Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.4.2 Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

CHAPTER 5 – CONCLUSIONS 885.1 Flame Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2 Quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3 Microwave Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Appendix A –Flame stability code 92

Appendix B –Image Analysis Code 96B.1 Flame Tracking with Background Subtraction . . . . . . . . . . . . . . . . . 96B.2 Video Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97B.3 V-channel Flame Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98B.4 Correlation Flame Velocimetry . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Appendix C –Electron Density Measurements 102

Appendix D –MWI-Related Codes 107

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D.1 Kinetics Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107D.2 Intrinsic Parameter Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 110D.3 I/Q data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113D.4 Electron Decay Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115D.5 Shock tube uncertainty estimation . . . . . . . . . . . . . . . . . . . . . . . . 116

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List of Figures

Figure 2.1: Diagram of the Flame Propagation Tube . . . . . . . . . . . . . . . . . . 9Figure 2.2: Electric field simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.3: Downward propagating flames in an electric field . . . . . . . . . . . . . 13Figure 2.4: Effect of partial electric fields on flame propagation speed . . . . . . . . . 14Figure 2.5: Suppression of turbulence by electric fields . . . . . . . . . . . . . . . . . 14Figure 2.6: Suppression of turbulence by electric fields . . . . . . . . . . . . . . . . . 21Figure 2.7: Flame stability with G(x, y) = µx enhancement . . . . . . . . . . . . . . 22Figure 2.8: Flame stability with G(x, y) = µy enhancement . . . . . . . . . . . . . . 23Figure 2.9: Flame stability with G(x, y) = µsin(0.179y) enhancement . . . . . . . . . 24Figure 2.10: Reaction rate profiles induced by enhancement effects . . . . . . . . . . . 25

Figure 3.1: Diagram of tapered channel quenching . . . . . . . . . . . . . . . . . . . 31Figure 3.2: Images of the V-Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 3.3: Schlieren Optics Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.4: Flame Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.5: Flame Velocimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.6: Computed flame structure under electric field . . . . . . . . . . . . . . . 40Figure 3.7: Quasi-2D quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.8: Quenching Distance Measurements . . . . . . . . . . . . . . . . . . . . . 42Figure 3.9: Laminar Flame Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 3.10: Quenching Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 3.11: Effects of electric field and stoichiometry on flame speed . . . . . . . . . 45Figure 3.12: Field strength and polarity effects . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.13: Effects field strength and polarity on flame speed . . . . . . . . . . . . . 48

Figure 4.1: Pre-computed reduced collision rates . . . . . . . . . . . . . . . . . . . . 56Figure 4.2: Shock reaction uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 4.3: Chemical kinetics simulation parameter sweep . . . . . . . . . . . . . . . 63Figure 4.4: Images of the MWI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.5: Simplified MWI Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.6: Phasor representation of I/Q signals . . . . . . . . . . . . . . . . . . . . 69Figure 4.7: MWI intrinsic parameter estimation . . . . . . . . . . . . . . . . . . . . 71

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Figure 4.8: Thermal Ionization of Argon . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 4.9: Argon Ionization Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 4.10: Example data from methane combustion . . . . . . . . . . . . . . . . . . 75Figure 4.11: Electron histories, experimental and numerical . . . . . . . . . . . . . . . 76Figure 4.12: Ignition Delay Time Comparison . . . . . . . . . . . . . . . . . . . . . . 77Figure 4.13: Evaluation of peak electron number density . . . . . . . . . . . . . . . . 78Figure 4.14: Comparison of chemical mechanisms . . . . . . . . . . . . . . . . . . . . 79Figure 4.15: Electron consumption measurements . . . . . . . . . . . . . . . . . . . . 81Figure 4.16: Electron decay rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Figure C.1: Electron histories mixture 1 . . . . . . . . . . . . . . . . . . . . . . . . . 102Figure C.2: Electron histories mixture 2 . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure C.3: Electron histories mixture 3 . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure C.4: Electron histories mixture 4 . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure C.5: Electron histories mixture 5 . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure C.6: Electron histories mixture 6 . . . . . . . . . . . . . . . . . . . . . . . . . 105Figure C.7: Electron histories mixture 7 . . . . . . . . . . . . . . . . . . . . . . . . . 105Figure C.8: Electron histories mixture 8 . . . . . . . . . . . . . . . . . . . . . . . . . 106Figure C.9: Electron histories mixture 9 . . . . . . . . . . . . . . . . . . . . . . . . . 106

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List of Tables

Table 2.1: Electrode Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Table 3.1: Ohmic Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Table 4.1: Shock tube test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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Nomenclature

Physical Constants

ε0 Free space permittivity

~ The Planck constant

µ0 Free space permeability

c Speed of light in a vacuum

kB The Boltzmann constant

R Ideal gas constant

Dimensionless Values

β Zel’dovich number, Ea(Tb − Tu)/(RT 2b )

γ Heat release parameter, (Tb − Tu)/Tb, ratio of specific heats (for shocks)

µ Flame speed eigenvalue

Ω Reaction rate

Φ Equivalence ratio, f/fs

φ Phase

θ Temperature

E Local electric field strength

F Fuel mass fraction

f Fuel-air ratio

G Reaction enhancement effect

Le Lewis number, Le = Dth/Dmol

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M Mach Number

Q Heat release (similar to ∆Hc)

U Speed (moving reference frame)

E0 External electric field strength

General Parameters

ν Mean collision rate

δφ MWI phase mismatch

∆Hc Heat of combustion

∆h0k Enthalpy of formation of species k

∆t Discrete time step

δf Flame thickness

δq Flame quenching distance

Q′′′C Volumetric heat release rate due to combustion

Q′′′E Volumetric Ohmic heating

r Reaction rate

η Index of refraction

V The space of trial functions

λ Thermal conductivity

J Current density

ω Electric field frequency (angular)

ω Reaction rate

ωp Plasma Frequency

ρ Density

σ Growth rate

τ Shifted electron recombination time scale

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τig Ignition delay time

εi Energy of the ith state of an atom or molecule

a(w, v) The bilinear functional in the weak formulation

A0 Arrhenius pre-exponential factor

AI,Q MWI sensitivity factors

cp Specific heat

D Diffusivity

D(w) A differential operator

E ′0 External electric field strength

Ea Activation energy

gi Degeneracy the ith state of an atom or molecule

hs Sensible enthalpy

I MWI in-phase signal

K Complex dielectric constant

k Kinetic rate constant

k Wavenumber

m Mass of a charged particle

ni Number density of species i

P Pressure

p Momentum of a particle

Q MWI quadrature signal

SL Laminar flame speed

SP Flame propagation speed

T Temperature

t Time

x

U Internal energy

ui Flow velocity in ith direction

V Volume

Vk,i Drift velocity of species k along direction i

Wk Molecular weight of species k

xi ith spatial coordinate

Yk Mass fraction of species k

Z Partition function

Image Processing

δx,u/ Image displacement

r Vector of pixels from a reference image

s Vector of pixels from an image to be aligned

Vn+1/2 Velocity vector of a flame between the n and n+ 1 images

In(x, y) Intensity map of the nth video frame

Abbreviations

MWI microwave interferometer

PMMA Polymethylmethacrylate

AC Alternating Current

DC Direct Current

RMS Root mean square

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Acknowledgments

To start, I would like to thank my parents for loving and supporting me all of my life. Fromthe start you’ve always inspired me to learn and be curious about the world. That has beenthe foundation of my education. You’ve helped me to find my path.

I’m very grateful to Professor Carlos Fernandez-Pello, for bringing me to Berkeley andfor helping me to pursue and enjoy my research. You’ve given me the freedom to learnand the guidance to succeed. It’s true Ph.D is about independent work, but we couldn’tsucceed alone, without your kindness and sincere advice. Also, thank you for indulging myside projects. I’m feel very lucky to have been able to pursue these interests and broaden byeducation.

Huge thanks to everyone in the lab, for all your friendship, your help and for makingmy time here amazing. The collaborative and supportive spirit of our group sets the toneof daily life and makes what we do something more than just ’work’. I’m very glad that Ihad the chance to work directly with Greg Noel. Having a true colleague and collaboratorwas a high point in the course of this work. I’d also like to like to recognize and thank twoformer members of the lab, Chris Lautenberger and David Rich. Chris for introducing me toCarlos, which was the invaluable first step towards my Ph.D. Dave for getting me started onmy research during my first semester and for his ongoing collaboration on our scale-modelingwork. I’m also very thankful to MaryAnne Peters whose patience, skill and kindness keepsus all on course.

I’ve learned a great deal from both the faculty and staff at Berkeley. A great manyconversations with Prof. Robert Dibble helped shape my understanding of a career in science.Thank you as well to Professors J.Y Chen, Michael Lustig and Omer Savas, for their excellentinstruction in classes and for participating in the committees needed for my degree.

The technical staff in Hesse and Etcheverry Halls do much more than support research,they are also valued teachers. I thank them for their commitment to developing the practicalskills to complement our coursework.

Thanks as well, to my colleagues at KAUST for the amazing opportunity to live andwork among you. My time there was a fascinating personal experience, that I’ll never forget.I also greatly value and appreciate the opportunity to develop the MWI and the chance toexpand my research horizons. I’d like to thank Professors Fabrizio Bisetti and Aamir Farooqfor their guidance and for inviting me to work with them. I’d especially like to thank AwadBin Saud Alquaity for working with me on this project. Your expertise and dedication wereessential to this work. I’m also grateful to KAUST as an institution for the AEA grant thatsupported me for the majority of my time at Berkeley.

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I consider myself lucky to have collaborated with Prof. Mario Sanchez-Sanz, whosetheoretical models have been very valuable in analyzing my experiments. Thank you fortaking an interest in my work and inviting me to assist you.

Lastly, thanks to the many undergraduate students who have helped so much with myresearch. You’ve made my work a fun process and have taught me a great deal as well.Special thanks to David Shulman and Joe Robbins for hanging in there during those earlydays, when I was still finding the course of my research. I’d also like to make special mentionof Henrik Hindborg, Harlan Kuo, Seth McFarland, Martin Thompson and Scott Sexton fortheir exceptional work on our PIV and scale-modeling experiments. That work is not coveredin this dissertation but was a terrific learning experience.

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CHAPTER 1

INTRODUCTION

1.1 Overview

The broad goal of this dissertation is to advance our understanding how the presence of elec-tric fields can influence the process of combustion and to recognize the practical implicationsof this interaction. The underlying mechanisms involve communication across a wide rangeof physical scales, from molecular interactions across nanometers to combustors that aremeters in length. For an experimentalist, it is both valuable and practical to identify testsand measurements that capture the macroscopic behavior that results from electrodynamicforces acting on ions and free electrons in flames. Such efforts relate well to practical appli-cations, while still exhibiting characteristics that can inform the development of theoreticalmodels, both inductively, suggesting new avenues of study, and deductively, by validatingpredictions. Large scale observations, however, are, substantially, global metrics that do notresolve the finer points of chemistry and transport. More direct methods, which can measureindividual physical quantities, provide a complementary perspective on the phenomena byisolating details, at the expense of the larger picture. The work that follows addresses bothapproaches by examining the dynamics of flames in electric fields, to observe the large scalemanifestations of electrical effects, and by directly measuring free electrons in combustion, tostudy the ionization reactions that are fundamentally responsible for the electrical propertiesof combustion.

1.2 Background

The scientific study of the electrical properties of flames is not new. Some reports of theseeffects date back as far as the year 1600 [1], although it was not until the mid-eighteenthcentury that notable interest began to develop among students of electricity[2], such asBacon, Boyle, Volta and Franklin [3]. Thereafter, the matter continued to receive attentionin scientific circles, with various activities reported throughout the nineteenth[4, 5] and early

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twentieth centuries [6]. Within this span, separate from the study of combustion, Chattock[7]developed the concept of momentum transfer between ions and neutrals (often called ’ionicwind’), which influenced later work in our field. The latter half of the twentieth centurysaw a sharp rise in interest among the combustion community [8], with colloquia on ’Ions inFlames’ and ’Chemi-Ionization’ becoming regular features at the international symposiumon combustion [9–11].

Some of the work has been very fundamental, attempting detailed characterization of theion species found in flames. Calcote[8, 12] developed the early evidence that ion concentra-tions found in flames greatly exceed that expected for thermal equilibrium and demonstratedthat chemical reactions are significant sources of ions. This has inspired experimental effortsto quantify ion populations by way of electrical measurements (e.g. Langmuir probes) andmass spectrometry[13–16]. Direct electrical probing is somewhat intrusive, requiring physicalcontact with the reacting mixture, but have provided useful data on the total concentrationsof charged species[17]. Mass spectroscopic methods are more useful for determining the rela-tive concentrations of specific ion species, which can greatly assist in identifying the chemicalreactions responsible for their formation. For example, it is now generally recognized thatthe reaction CH+O CHO+ +e− is key to producing the high levels of ionization found inhydrocarbon combustion, as compared to the burning of other fuels[18]. From these experi-ments and complementary data from other fields of study[19], chemical kinetics mechanismshave been formulated to model the chemi-ionization process [14, 20, 21], although addi-tional, detailed measurements of ions and free electrons would greatly assist in their furtherdevelopment.

Another topic of study has been the structure and dynamics of flames subject to electricfields. Here the goal has been to understand how ions, once present, are able to influencethe behavior flames as a whole, despite their relatively small concentrations. A commonlyreferenced explanation for the influence of electric fields on flame behavior is the ’ionic wind’proposed by Chattock, the exact interpretation of which may differ somewhat among authors.Some focus on the generation of a body force in the bulk gas flow as momentum imparted toions by the Lorenz force is transfered to neutral species [22, 23]. That force may simply shiftthe location of a flame or alter the burning rate by imparting hydrodynamic stretch [24].Another perspective on ’ionic wind’ is that charged species subject to electric fields developa biased diffusion velocity that redistributes them within the flame structure, influencingtheir availability for reaction. This redistribution could produce any number of effects, suchas influencing the growth of soot particles[25] or modifying the local rate of reaction[26].

Others, such Stockman et al.[27] and MacLatchy et al.[28], have presented alternatetheories and experiments to show that energy may be directly applied to flames by electricmeans. This has been particularly successful in examining the enhancement of flames byvery high frequency (MHz to GHz) electric fields. In such cases, energy is imparted directlyto electrons with negligible net momentum imparted by the rapidly oscillating fields. The

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resulting changes in flame behavior appear be the result of ohmic heating of the mixture[29] or the reaction kinetics of non-thermal electrons[30]. None of these theories are eitherdemonstrably false nor are they mutually exclusive. Each may be valid, but their relativeimportance could be situation-dependent.

A reasonable body of work on this topic has been rather more application-oriented.In service of other scientific interests, Strayer et al.[31] developed a technique by whichelectrodynamic forces on a flame could counteract buoyancy to approximate the effects ofmicrogravity. Kim et al.[32] tested changes in liftoff and blowoff of jet flames when exposedto electric fields. Their experiments were well controlled and defined (making them suitablefor the validation of theoretical models) but maintained practical relevance by demonstratingthat electric fields were means to expand the useful operation range of their burner. Workon microwave-assisted spark-ignition[33, 34] has successfully expanded the range of fuel-air mixtures over which gasoline engines can reliably operate, potentially improving fueleconomy and reducing pollutant formation. These endeavors, and many others of a similarspirit[35–39], show potential but most have not yet made the transition to real world use.

Although great strides have been made in both the theory and application of electricalphenomena in flames, it is hardly a solved problem. Increasingly sophisticated numericalmodels promise greater understanding but demand rich experimental data as inputs and asa means of validation. Future experimental work should address these requirements and,when possible, attempt to highlight practical uses.

1.3 Organization

This document is organized around chapters for three distinct classes of experiments whichexamine flame stability, flame quenching and chemi-ionization kinetics. Attendant theory,computations and comparison to literature are included within those chapters.

Chapter 2 is examines the stability of flames subject to electric fields. In the courseof the experiments, circular tubes were filled with methane-air mixtures and ignited. De-pending on the geometric factors (tube length and diameter) these flames were observed toremain laminar throughout the tube or transition to turbulence due to a thermo-acousticinstability. Electrodes placed selectively along the length of these tubes, establishing spa-tially non-uniform electric fields within, were then found to be effective at disturbing theflame front such that a normally stable system (not prone to turbulence) could be renderedunstable and that unstable systems could be made extremely stable. Both effects may beof practical interest, as a non-intrusive means of enhancing turbulence to increase burningrates or to suppress undesirable instabilities. Following from this result, a numerical modelwas developed to simulate the effects of spatially-varying enhancement effects on the thermo-diffusive wrinkling instability of flames. Concurrent with experimental results, the inherent

3

instabilities of the flames studied were found to be enhanced or suppressed depending on thedisposition and intensity of the enhancement effect.

Chapter 3 focuses on phenomenon of quenching, a practical imitation of combustion whereheat losses to the surroundings extinguish a flame. In the experiments flames were allowedto propagate into tapered channel millimeters wide, until quenching occurred. Analysisof high speed schlieren footage of the events recovered both flame propagation speed andquenching distance as measures of reactivity. To this apparatus, provisions were added toestablish electric fields both parallel and perpendicular to the flame surface. The effects ofDC polarity and oscillating fields were tested. An enhancement achieved by electric fields wasfound to dramatically increase flame propagation speed and reduce quenching in methane-air flames under some field orientations. In other configurations, electric effects negligible oreven detrimental effects. The prospect of reducing quenching effects is exciting in the contextof miniature, portable power generation, where the limitations imposed by quenching haveprecluded the use of combustion as an energy source. From a theoretical perspective, thedirection and polarity sensitivities valuable as clear and easily evaluated measures of by whichmodels may be judged. One such model, motivated by these experiments, was compared tothe present results to clarify the aspects potentially responsible for agreement and divergencebetween the two.

Chapter 4 addresses the more fundamental aspects of electrical phenomena by providingdirect measurements of free electrons produced by the combustion of hydrocarbon fuels.The core of this work was the development of a 94 GHz microwave interferometer (MWI),a tool able to non-intrusively detect the presence of free electrons due to the strong effectthat they have on the complex dielectric properties of a medium. The MWI was purpose-built to interface with a shock tube for gas phase chemical kinetics diagnostics. Thus,the MWI was able to record the production and decay of electrons in suddenly heated,homogeneous gas mixtures, providing datasets for which equivalent numerical predictionscan be readily computed. A set of validation tests, using the thermal ionization of inert argongas were used to calibrate the MWI in situ and permit comparison against reported studiesof that phenomenon. Electron histories from combustion with a variety of hydrocarbons andequivalence ratios were compared were compared to the predictions of two related chemicalmechanisms and electron-ion recombination rates reported in the literature.

4

1.4 References

[1] Howard Palmer. Combustion technology: some modern developments. Elsevier, 2012.

[2] Thomas Thomson. History of the Royal Society: From its institution to the end of theEighteenth Century. Cambridge University Press, 2011.

[3] I Bernard Cohen. “Benjamin Franklin’s experiments”. In: A new edition of Franklin’sex-periments and observations on electricity 159 (1941).

[4] John Cuthbertson. Practical electricity, and galvanism. 1807.

[5] William Thomas Brande. “The bakerian lecture: On some new electro-chemical phe-nomena”. In: Philosophical Transactions of the Royal Society of London (1814), pp. 51–61.

[6] WE Garner and SW Saunders. “Ionisation in gas explosions”. In: Trans. Faraday Soc.22 (1926), pp. 281–288.

[7] AP Chattock. “XLIV. On the velocity and mass of the ions in the electric wind inair”. In: The London, Edinburgh, and Dublin Philosophical Magazine and Journal ofScience 48.294 (1899), pp. 401–420.

[8] Hartwell F Calcote. “Ionization Flame Detectors”. In: Review of Scientific Instruments20.5 (1949), pp. 349–352.

[9] R Carabetta and RP Porter. “Twelfth Symposium (International) on Combustion”.In: The Combustion Institute, Pittsburgh (1969), p. 423.

[10] JC Bell, D Bradley, and LF Jesch. “Chemi-ionization, chemiluminescence, and elec-tron energy exchange in hydrocarbon-air flames”. In: Symposium (International) onCombustion. Vol. 13. 1. Elsevier. 1971, pp. 345–352.

[11] Itsuro Kimura, Nobuyuki Negishi, and Masaru Nakahara. “Effects of probe surfacecharacteristics on probe measurements in seeded combustion plasmas and arc plasmas”.In: Symposium (International) on Combustion. Vol. 17. 1. Elsevier. 1979, pp. 935–942.

[12] HF Calcote. “Mechanisms for the formation of ions in flames”. In: Combustion andFlame 1.4 (1957), pp. 385–403.

[13] L Delfau, P Michaud, and A Barassin. “Formation of small and large positive ions inrich and sooting low-pressure ethylene and acetylene premixed flames”. In: CombustionScience and Technology 20.5-6 (1979), pp. 165–177.

[14] JM Goodings, DK Bohme, and Chun-Wai Ng. “Detailed ion chemistry in methaneoxygen flames. I. Positive ions”. In: Combustion and Flame 36 (1979), pp. 27–43.

[15] Jozef Peeters and Christiaan Vinckier. “Production of chemi-ions and formation of CHand CH 2 radicals in methane-oxygen and ethylene-oxygen flames”. In: Symposium(International) on Combustion. Vol. 15. 1. Elsevier. 1975, pp. 969–977.

5

[16] Awad BS Alquaity et al. “Ion Measurements in Premixed Methane-Oxygen Flames”.In: 50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference (2014).

[17] TW Lester, DM Zallen, and SLK Wittig. “Chemi-ionization in shock-induced hydro-carbon combustion I. Ion concentrations in methane/air”. In: Combustion Science andTechnology 7.5 (1973), pp. 219–226.

[18] William J Miller. “Ions in flames: evaluation and prognosis”. In: Symposium (Interna-tional) on Combustion. Vol. 14. 1. Elsevier. 1973, pp. 307–320.

[19] TJ Millar, PRA Farquhar, and K Willacy. “The UMIST database for astrochemistry1995”. In: Astronomy and Astrophysics Supplement Series 121.1 (1997), pp. 139–185.

[20] Jens Prager. “Modeling and Simulation of Charged Species in Lean Methane-OxygenFlames”. In: (2005).

[21] J. Han et al. “A comprehensive ion chemistry in premixed, lean methane flames”. In:Poster presented at the European Combustion Meeting (2014).

[22] Bernard J Rezy and Robert J Heinsohn. “The Increase in the Maximum Heat-ReleaseRate and Apparent Flame Strength of Opposed-Jet Diffusion Flames Under ImpressedElectric Fields”. In: Journal of Engineering for Gas Turbines and Power 88.2 (1966),pp. 157–164.

[23] Matthew Rickard et al. “Maximizing ion-driven gas flows”. In: Journal of Electrostatics64.6 (2006), pp. 368–376.

[24] Chung K Law. Combustion physics. Cambridge University Press, 2006.

[25] Masahiro Saito, Toshihiro Arai, and Masataka Arai. “Control of soot emitted fromacetylene diffusion flames by applying an electric field”. In: Combustion and Flame119.3 (1999), pp. 356–366.

[26] Mario Sanchez-Sanz, Daniel C Murphy, and C Fernandez-Pello. “Effect of an externalelectric field on the propagation velocity of premixed flames”. In: Proceedings of theCombustion Institute 35.3 (2015), pp. 3463–3470.

[27] Emanuel S Stockman et al. “Measurements of combustion properties in a microwaveenhanced flame”. In: Combustion and Flame 156.7 (2009), pp. 1453–1461.

[28] CS MacLatchy, RM Clements, and PR Smy. “An experimental investigation of theeffect of microwave radiation on a propane-air flame”. In: Combustion and Flame 45(1982), pp. 161–169.

[29] D Bradley and Said MA Ibrahim. “The effects of electrical fields upon electron energyexchanges in flame gases”. In: Combustion and Flame 22.1 (1974), pp. 43–52.

[30] Anthony Cesar DeFilippo. “Microwave-Assisted Ignition for Improved Internal Com-bustion Engine Efficiency”. In: (2013).

6

[31] BA Strayer et al. “Simulating microgravity in small diffusion flames by using electricfields to counterbalance natural convection”. In: Proceedings of the Royal Society ofLondon. Series A: Mathematical, Physical and Engineering Sciences 458.2021 (2002),pp. 1151–1166.

[32] MK Kim et al. “Electric fields effect on liftoff and blowoff of nonpremixed laminar jetflames in a coflow”. In: Combustion and Flame 157.1 (2010), pp. 17–24.

[33] Vi H Rapp et al. “Extending lean operating limit and reducing emissions of methanespark-ignited engines using a microwave-assisted spark plug”. In: Journal of Combus-tion 2012 (2012).

[34] Anthony DeFilippo et al. Extending the lean stability limits of gasoline using a microwave-assisted spark plug. Tech. rep. SAE Technical Paper, 2011.

[35] Atsushi Nishiyama and Yuji Ikeda. Improvement of lean limit and fuel consumptionusing microwave plasma ignition technology. Tech. rep. SAE Technical Paper, 2012.

[36] Yoshinori Matsubara, Kenichi Takita, and Goro Masuya. “Combustion enhancement ina supersonic flow by simultaneous operation of DBD and plasma jet”. In: Proceedingsof the Combustion Institute 34.2 (2013), pp. 3287–3294.

[37] A Feng and ZA Munir. “The effect of an electric field on self-sustaining combustionsynthesis: Part I. Modeling studies”. In: Metallurgical and Materials Transactions B26.3 (1995), pp. 581–586.

[38] A Sakhrieh et al. “The influence of pressure on the control of premixed turbulent flamesusing an electric field”. In: Combustion and flame 143.3 (2005), pp. 313–322.

[39] Guillaume Pilla et al. “Stabilization of a turbulent premixed flame using a nanosecondrepetitively pulsed plasma”. In: Plasma Science, IEEE Transactions on 34.6 (2006),pp. 2471–2477.

7

CHAPTER 2

FLAME STABILITY

2.1 Introduction

There is a well-established body of work studying the behavior of flames in the presenceof electric fields [1, 2]. Such efforts have identified a number of flow altering and reactionenhancing mechanisms by which it is possible to exert influence over combustion processesand discern information regarding the structure of flames [3]. Due to their fundamentalnature, many of these experiments approximate a steady state and utilize fields with thehighest spatial homogeneity possible, seeking to isolate the electric interactions as funda-mental properties of flames [1, 4]. The implications of electric effects in the dynamics offlames have been examined by some [5], but the richness of combustion physics leaves muchroom for further study.

Although the existence of interactions between electric fields and charged species flameshas been known for some time, the exact physical mechanism by which flame speeds increaseremains an active area of research. Many have observed ionic wind phenomena in which bulkmotion of the fluid is affected by electrodynamics forces acting upon the charged species [6,7], positing that the transfer of momentum drives the observed variation in combustion.Others have investigated the case of alternating electric fields, with frequencies extendinginto the GHz range [8].

The present work examines the behavior of laminar premixed flames in the presenceof spatially non-uniform electric fields. We begin with experimental studies of downwardpropagating premixed flames subject to intense electric fields. In addition to a generalacceleration and enhancement effects we found a propensity for these fields to suppresswrinkling and to suppress the transition from laminar to turbulent propagation due to athermal-acoustic instability. Motivated by these observations we proceeded to a numericalstudy of spatially varying reaction enhancement effects on the wrinkling of flames with athermal-diffusive instability.

8

Figure 2.1: Diagram of the Flame Propagation Tube indicating the arrangement of the gassupply lines, high voltage electrodes and the combustion initiation chamber.

2.2 Experiments

An apparatus was constructed to observe the influence of alternating current (AC) electricfields on downward-propagating premixed flames. In all instances, the flame propagationenvironment was a vertically-oriented circular tube open at the top end and closed at thebottom. Fuel-air mixture was supplied using the system depicted in Figure 2.1.

2.2.1 Methodology

The gases used were dried ambient air, as well as chemically-pure methane and ethylene.In order to achieve flow control, fuel and air were separately released through sonic nozzles(O’Keefe Controls) with the upstream pressure monitored using dead-weight calibrated pres-sure gauges (Ashcroft and Omega). The downstream pressure of the fuel-air mixture wasmonitored to verify that the pressure drop across the nozzles was sufficient to ensure chokedconditions. Volumetric flow rates of fuel and air downstream of the nozzles were monitoredwith ball float rotameters (Omega and Matheson) to verify that the desired flow rates wereachieved.

Two sizes of clear acrylic tube were used to observe propagation behavior. They were

9

designated tube ’A’ and tube ’B’ with an inside diameter of 57 mm and lengths of 482mm and 787 mm, respectively. The tubes were mounted in in a purpose-built aluminumbase with ports for admitting and removing the gas mixture. The open upper ends werefitted with a 219 mm length, 115 mm ID clear acrylic combustion initiation chamber (CIC)containing two steel electrodes energized by an automotive spark ignition coil. The top ofthe CIC was closed with a loose fitting paper diaphragm to allow gas to escape during thefilling procedure and during combustion while isolating the contents from external disruptionand mixing with ambient air. The gas mixture was introduced to the tube for 20 s at a rateof no less than 40 liters per minute (lpm) and then allowed to equilibrate for an additional20 s prior to ignition by electric spark. These two periods ensured purging of initial contentsand subsequent settling of the flow. Appropriate duration of the two periods was verified byreducing them until repeatability suffered: 5 s for filling time and ∼10 s for settling time.

In order to obtain consistent and repeatable experimental results, it was imperative thataxisymmetric laminar flames were formed at the entrance of the tube. To do so, the tubeentrance was covered by a steel mesh flame arrestor ( 1 mm) with a 6 mm hole at thecenter. The practical benefit of the flame arrestor was that it would filter any asymmetric oroff-center flames into the desired laminar structure. The flame arrestor was not secured toany of the tubes and was ejected by the escaping products of combustion. Had the arrestorbeen fixed, it may have introduced a damping effect [9] that would have interfered with thethermo-acoustic instability of the flames.

These experiments were recorded using a high speed video camera (Kodak EKTAPROHG Model 2000) at 500 frames per second (fps) for stoichiometric methane-air and at 1000fps for stoichiometric ethylene and air. A script was written in Matlab to fully automatethe processing of the high speed video results. This program removed static image artifacts,enhanced the brightness, applied brightness thresholding and recorded the largest continuousbright region. The primary outputs of this script were measurements of the leading edge andgeometric center of flames as a function of time. These data were used to determine flamepropagation speeds and to observe the transition to turbulence.

Electric fields were generated by applying voltage to parallel flat conductive plates incontact with the outside of the combustion tube. The plates consisted of ridged acryliccoated with thin (.03 mm) aluminum with epoxy applied to edges and corners to minimizecorona discharge. All plates were made to match the diameter of their associated combustiontube. High voltage (HV) AC was produced by a Transco NT1512N3G transformer operatingat 15 kV r.m.s. at 60 Hz. Four pairs of plates were made for tube A, each of which extendedone quarter of the length of the tube. In experiments, these pairs were mounted at quarterlength intervals to produce nine distinct configurations (Cfgs.) of electric field as indicatedin Table 2.1. Flame propagation data were recorded with these plates energized to 15 kVr.m.s for all nine permutations with five replicates each. For tube B, a single plate extendingthe full length of the tube was used. This plate was energized by one of the active poles of

10

ConfigurationHeight 1 2 3 4 5 6 7 8 93/4-1 0 1 0 0 0 1 0 0 11/2-3/4 0 0 1 0 0 1 1 0 11/4-1/2 0 0 0 1 0 0 1 1 10-1/2 0 0 0 0 1 0 0 1 1

Table 2.1: Vertical Arrangement of electrodes for tube A. Segments with active electrodesnoted as 1

the transformer, producing 7.5 kVac relative to ground.

An essential aspect of the electric fields created for both tubes was that their magnitudeswere not spatially uniform across the cross-section of the gas volume. This is entirely tobe expected for tube B, due to the use of a single asymmetrically placed electrode. Fortube A, the inhomogeneity is the result of the finite size of the electrodes and the dielectricproperties of the acrylic walls of the tube. To quantify these effects, Maxwells equations forstatic, 2 dimensional electric fields were solved using COMSOL Multiphysics with dielectricproperties taken from [10]. In Figure 2.2 we see that the resulting electric field within thetube is greatest at the sides of the tube near the electrodes and decays as it approaches theplane of symmetry.

2.2.2 Results: Flame Acceleration

The extended regions of intense electric field used in tube A produced and sustained aforward-swept V-shaped flame front. The V flame exhibited a velocity approximately 4times that of the undisturbed laminar flame (average of 1.15 m/s). While the V-shapeformed quickly and persisted throughout the region of the electric field, it would dissipaterapidly to an asymmetric slanted flame upon leaving that region.

Figure 2.3(a-b) shows the evolution of flame shape with time for Cfgs. 1 and 3. Asexpected from their constant flame speeds, the curved and V-shaped flames maintain ahighly uniform shape throughout the tube. In Figure 2.3(b) one can observe the transitionfrom curved flame, to V-shaped flame and back again. Once beyond the electric field,the flame transitioned to a gently slanted (non-symmetric) shape traveling at the original(unenhanced) propagation speed.

Figure 2.4 graphs flame speed vs. height within the tube (averaged over five replicates) forCfgs. 1, 3, 7 and 9 as defined in Table 2.1, Cfgs. 1 and 9 are the two extreme cases of steadylaminar combustion without electric field and with continuous field throughout the length.

11

Figure 2.2: Finite element simulation of the electric field formed in the combustion tube bya potential difference between two electrodes

These two flames speeds provide the bounds of behavior for the remaining configurations.The plots for configurations 3 and 7 illustrate velocity change as flame travel into and outof the region of electric field influence. Note that the different duration of field exposure forthese different cases does not influence the subsequent propagation speed.

2.2.3 Results: Flame Dynamics

The critical distinction between the results obtained from tubes A and B was that the latterinherently supported the development of thermo-acoustic instabilities [9] as studied by Parket al.[11]. It was observed that application of the asymmetric electric field produced consis-tently higher propagation speeds than the unmodified laminar flames but entirely preventedthe excitation of the unstable mode and the resultant transition to turbulence. Figure 2.5plots the transit of flames under both conditions with five replicates.

In the absence of electric fields, flames propagating down tube B exhibited a steady flamespeed of 0.36 m/s and the usual axisymmetric, curved shape for the first 0.23 m of the tube.At this stage, flames began to oscillate intensely in the axial direction and underwent severedistortion and wrinkling until fully developing into turbulence. The transition to turbulence

12

Figure 2.3: Images of downward propagating flames. a) Tube A without electric field, b)tube A in Configuration 3, c) flattening of a flame in tube B without electric field, duringthe transition to turbulence, d) tube B with the electric field present. Color and contrasthave been augmented for visibility. Multiple frames have been overlaid for a stroboscopiceffect.

occurred over a length of approximately 0.05 m during which the flame speed increasedconsistently until reaching a steady value 0.96 m/s for the remainder of the tube.

The presence of the oscillating electric field significantly altered the flame shape and thedynamics of the combustion process within the tube. Consistent with the asymmetry of thefield, the flame accelerated dramatically at the face of the tube nearest the electrode, withthe acceleration effect decaying strongly with distance from the electrode. Figure 2.3(c,d)shows both the normal flame shape immediately before the transition to turbulence (c) andwith the electric field applied (d). Stroboscopic overlays taken at identical intervals of 20 mswere used to illustrate the change in flame shape. At that sampling rate, the neutral flameremains in the field of view for four full intervals, while the enhanced flame appears onlytwice, giving a basic sense of the relative speeds.

13

Figure 2.4: Propagation speed in tube A as flames pass through regions of electric fieldfor multiple electrode configurations. Dashed and dotted lines indicate regions of the tubewithout electric field.

Figure 2.5: Location of the leading edge of flames in tube B with and without electric field,showing suppression of the laminar-to-turbulent transition by the electric field

14

2.3 Numerical Study

The above experimental observation of enhanced combustion is consistent with prior workon this topic (van den Boom 2009). Of greater interest is the effect of inhomogeneousapplication of this enhancement to suppression of an inherent instability of the system. Thestudy of combustion instability has long been a theoretical tool used to develop a betterfundamental understanding of the essential phenomena governing the evolution of reactingflows. Further, combustion instabilities are a significant practical challenge as they caninterfere with the proper operation of engines [12] and can dramatically increase the hazardsposed by explosions [13]. Rather than directly modeling the very specific configuration ourexperiment, we have elected to proceed with more a fundamental study of flame stability.Our specific objective is to build an understanding of flame stability in the presence of aspatially inhomogeneous enhancement effect of any kind. Readers interested in the detailsof electrical aspects of flames are encouraged to consult the existing body of work such asthat by Lawton and Weinberg [1] or by Pedersen and Brown [3] as well as current ongoinginvestigations such as the work by Kim et al. [14] and Sanchez-Sanz et al. [15].

To construct a numerical tool for this investigation, we began with the well-studiedthermal-diffusive instability of low-Lewis number premixed flames. This phenomenon hasreceived thorough treatment both analytically and numerically. To this we added a relativelygentle modification of reaction rate as a function of space. We then observed the effect ofthese enhancements on the growth and decay of disturbances introduced into the flame sheet.

2.3.1 Formulation

The basis of this formulation and method for measurement of unstable modes comes fromDenet and Haldenwang [16] who present a numerical investigation of the thermal-diffusiveinstability in the context of spatial frequency. Under consideration is a lean premixed flamethat is initially propagating as a plane, but subsequently perturbed to introduce slight wrin-kling. The stability of the flame in this context is gauged by measuring the amplitude ofwrinkling (deviation from a planar front) in the reaction zone of the flame. For such a flame,equations for the conservation of mass, momentum, species and energy and may be writtenin general form[17]:

15

∂ρ

∂t+∂ρui∂xi

= 0 (2.3.1)

∂ρui∂t

+∂ρuiuj∂xj

= − ∂p

∂xi+∂τij∂xi

(2.3.2)

∂ρYk∂t

+∂

∂x(ρ(ui + Vk,i)Yk) = rk (2.3.3)

∂ρhs∂t

+∂ρuihs∂xj

=∂

∂xi

(λ∂T

∂xi

)−

N∑k=1

[∆h0

krk +∂

∂xi(ρhkYkVk,i)

](2.3.4)

In these, the influence of body forces and viscous heating have already been neglected andfurther assumptions follow. First, consideration is restricted to an incompressible mediumand, second, changes in density due to thermal expansion are assumed to be minimal. Thefirst of these is easily justified for low Mach number (M < 0.3) flow while the second ismore tenuous. For that condition, it is necessary to accept the premise of a very lean flame,such that temperature rise is minimal and the number of molecules remains nearly constant.For the infinite domain under consideration, these assumptions allow Equation 2.3.1 andEquation 2.3.2 to be satisfied by a single, constant velocity. The remaining transport phe-nomena of thermal conduction and mass diffusion are assumed to be constant and equal forall chemical species. Doing so reduces the terms for diffusion velocities (Vk,i) and thermalconduction to Laplacians of mass fraction and temperature, respectively. In similar fashion,constant and equal specific heats (Cp) eliminates heat transfer due to mass diffusion. Lastly,the chemical reactions are simplified to a single-step, irreversible Arrhenius reaction of fueland air to products. From these, the conservation of fuel is reduced to a more tractableform:

ρ∂YF∂t

+ ρSP∂YF∂x

= ρDmol

(∂2YF∂x2

+∂2YF∂y2

)− ρ2YF

YO2

WO2

A0 exp−(EaRT

)(2.3.5)

in which the molecular diffusivity (Dmol) has been introduced as has the molecular weightof oxygen, WO2 . Index notation for spatial dimensions has been replaced with x and y (theremaining z is not necessary for a 2-D formulation). The constant velocity, SP lies in x, whichis defined as the direction of flame propagation. These same methods may be applied to theenergy equation, in which the definition of sensible enthalpy simplifies to hs = Cp(T − T0),recasting the energy equation purely in terms of temperature. The expression can be made

16

more compact by dividing by the quantity ρCp and introducing both the thermal diffusivity,Dth = λ/ρCp.

∂T

∂t+ U ′

∂T

∂x= Dth

(∂2T

∂x2+∂2T

∂y2

)− ∆h

Cpρ2 YFWF

YO2

WO2

A exp−(EaRT

)(2.3.6)

As the question at hand is one of thermo-diffusive instability, it is expedient to expressthese equations in terms of the Lewis number, the ratio of thermal to molecular diffusivityLe = Dth/Dmol. Further, it is possible to non-dimensionalize a number of quantities in termsof the conditions in the unburnt fuel-air mixture and the fully reacted mixture (denoted bythe subscripts u and b, respectively). In this fashion we define dimensionless, normalizedtemperature and fuel concentration as θ = (T − Tu)/(Tb − Tu) and F = (YF − YF,u)/(YF,b −YF,u).

The reaction rate term is slightly more complicated, but is easily handled by the methodof Zel’dovich and Frank-Kamenetskii [18]. Due to its monotonicity, the exponential term canby normalized by its value for T = Tb. To express this term using θ, the activation energy(Ea) and gas constant (R) are combined into a single, reduced activation energy known asthe Zel’dovich number, β = Ea(Tb−Tu)/(RT 2

b ) [19]. A neat collection of terms also benefitsfrom the definition γ = (Tb − Tu)/Tb.

∂θ

∂t+ U

∂θ

∂x=∂2θ

∂x2+∂2θ

∂y2+ Ω(θ, F, x, y) (2.3.7)

∂F

∂t+ U

∂F

∂x=

1

Le

(∂2F

∂x2+∂2F

∂y2

)− Ω(θ, F, x, y) (2.3.8)

Ω(θ, F, x, y) = (1 +G(x, y))Fβ2

2Leexp

(β(θ − 1)

1 + γ(θ − 1)

)(2.3.9)

The term G(x, y) has been added to the reaction rate term Ω as a means to producea spatial dependence in the chemical reaction. The functional form of G is, in fact, thekey point of interest in the numerical aspects of this study. Subtle variations along thestreamwise and cross-stream directions may strongly influence the natural development ofhigh and low reactivity regions responsible for flame wrinkling.

2.3.2 Implementation

The governing equations were recast in variational form and solved using a finite element codewritten in FreeFEM++ [20]. FreeFEM++ is a substantially C-like programming language

17

with a streamlined syntax and collection of built-in functions suited to the formulation,meshing and solution of partial differential equations by the finite element method. Althoughit includes many tools to assist in this process, it is only fundamentally capable of solvinglinear problems in the weak (or variational) form[21], wherein an approximate solution (w)is determined to satisfy the following relation:

a(w, v) = L(v) ∀v ∈ V (2.3.10)

Here, a and L are linear functionals and V is a space of trial functions. It remains thetask of the user to recast their problem of interest in a form that can leverage the capabilitiesof FreeFEM++ to reach a solution. Commonly, one considers a differential operator, D, aforcing function, f , and a true solution w′ such that D(w′) = f . This differential operatoris adapted into a variational form by taking the inner product with v over the domain, S.

a(w, v) =

∫SD(w)v dx L(v) =

∫Sfv dx (2.3.11)

While this process is quite convenient for the advective and diffusive terms of Equa-tion 2.3.7 and Equation 3.3.2, it does not translate meaningfully to the time derivative.Additionally, as Ω is nonlinear in θ, and the resulting functional is non-linear as well. Thetime derivative is addressed by discretization. Specifically, by a Crank-Nicolson scheme. [22]

wn+1 − wn

∆t= Θ

∂wn+1

∂t+ (1−Θ)

∂wn

∂tΘ = 0.5 (2.3.12)

Which introduces the discrete time step, ∆t, to express the approximate relation betweenthe current (t = t0) value of the solution, wn, and its value a short time into the future(t = t0 +∆T ), wn+1. The time derivatives on the right hand side are simply a reorganizationof Equations 2.3.7 and 3.3.2. A simple explicit Euler method (using only ∂wn/∂t ) wouldbe significantly less computationally intensive, as wn+1 could be directly computed fromwn, but would provide only first order accuracy in time and would not guarantee stability.The addition of ∂wn+1/∂t necessitates the solving of a system of equations, but (with theconstraint that R=0.5) improves the accuracy of the computation to be second order in timeand provides unconditional stability. In the final formulation, the wn+1 appears in a, whilewn appears in L.

The non-linear component of Ω and the coupling between the equations for θ and F werehandled by iteration at each time step. In the solution of θ, Ω was computed using valuesfrom the previous iteration, θ∗ and F ∗, and placed in the linear functional Lθ. As Ω is linear

18

in F , it was only necessary to use θ∗ and Ω remained in the bilinear functional aF . The fullforms of these equations are:

aθ(θn+1, v) =

θn+1v

∆t+

Θ

[∂θn+1

∂x

∂v

∂x+∂θn+1

∂y

∂v

∂y+ Uv

∂θn+1

∂x

](2.3.13)

Lθ(v) =θnv

∆t−Θ vΩ(θ∗, F ∗, x, y)−

(1−Θ)

[∂θn

∂x

∂v

∂x+∂θn

∂y

∂v

∂y+ Uv

∂θn

∂x+ Ω(θn, F n, x, y)v

](2.3.14)

aF (F n+1, v) =F n+1v

∆t+

Θ

[1

Le

(∂F n+1

∂x

∂v

∂x+∂F n+1

∂y

∂v

∂y

)+ Uv

∂F n+1

∂x− Ω(θ∗, F n+1, x, y)v

](2.3.15)

LF (v) =F nv

∆t+

(1−Θ)

[∂F n

∂x

∂v

∂x+∂F n

∂y

∂v

∂y+ Uv

∂F n

∂x− Ω(θn, F n, x, y)v

](2.3.16)

Both θ and F were resolved with triangular second order polynomial elements. Thesolution was advanced in time using a Crank-Nicholson scheme where the implicit componentwas determined by iteratively solving with the exponential (θ dependent) term of with Ωfixed. In order to achieve high resolution at the flame sheet, a non-uniform mesh wasconstructed. This mesh mapped the verticies of a uniform rectangular grid on [0, 1] x [0, 1]onto the larger computational domain [-100, 100] x [-17.5, 17.5]. Mapping in the y directionwas uniform while mapping in x was governed by:

xfinal = A · tan(

2

[xinitial −

1

2

])+B ·

[xinitial −

1

2

]+ C ·

[xinitial −

1

2

]3

(2.3.17)

With the values A=1.29,B=-.044 and C=34.4. The imposed boundary conditions:

θ(−100, y) = 1− θ(100, y) = 1− F (100, y) = F (100, y) (2.3.18a)

19

θ(x,−17.5) = θ(x, 17.5) (2.3.18b)

F (x,−17.5) = F (x, 17.5) (2.3.18c)

These serve to approximate the ideal case of a flame propagating within an infinitemedium. The periodic conditions in Equation 2.3.18b and Equation 2.3.18c limit the lowestspatial frequency k (wavenumber) in the y direction to ∼ 0.18.

The cases examined reference a baseline condition where G(x, y) = 0, β = 10, Le = 0.6,and γ = 0.8 . Both the expansion theory shown by Clavin [19] and the numerical results ofDenet and Haldenwang [16] show that the flame so described will be unstable with respectto perturbations with low wave numbers between 0 and 0.5 (Clavin) or 0 and ∼ 0.6 (Denet).Due to the prescribed boundary conditions, only solutions periodic in y with wavenumbersthat are integer multiples of 0.179 are valid solution. Of those, only k = 0.179, 0.358 and0.537 wrinkles are unstable for the combustion parameters used.

In the analysis of one-dimensional laminar flames, it is common practice to formulatea time-invariant solution to the conservation equations [23, 24], resulting in an eigenvalueproblem that recovers both the structure and propagation speed of a flame when solved.The fluid velocity U is not a quantity of particular interest to the instabilities studied here,and the wrinkling phenomenon is inherently time-dependent. It is, however, possible toadopt a conceptually similar technique to constrain the flame sheet to a single region of thecomputational domain. Doing so can greatly reduce the computational cost of the solutionby using a non-uniform grid which is, globally, very coarse but highly refined at the flamesheet where strong gradients and strongly nonlinear effects are present. This method wasalso found to be preferable to adaptive meshing schemes which provide similar reduction inthe number of total grid points but introduce significant computational cost each time themesh is re-computed.

In the absence of thermal expansion and momentum effects, the advective term simplyprovides a moving reference frame that can balance the propagation rate of the flame. Inorder to evaluate the speed of the flame and select U it is useful to define a mean locationin the direction of propagation.

xave =

∫D2 xΩ dA∫D2 Ω dA

(2.3.19)

We have primarily examined the growth and decay of k = 0.358 wrinkles, being thefastest growing for these parameters. The intensity of wrinkling was measured by Fourier

20

Figure 2.6: Computed temperature distribution before (a) and after (b) disturbance by sinewave with amplitude 0.02 and k=0.358. Values have been remapped onto a uniform meshto make small features visible,

analysis of values of Ω along the line x = xave. Specifically, we had defined the amplitude ofwrinkling as

A0.358 =

∣∣∣∣∫ 17.5

−17.5

Ω(y)e0.385 i y dy

∣∣∣∣ (2.3.20)

Perturbations were introduced by distortion of the flame sheet. Beginning from a steadypropagating solution (uniform in y) both θ and F were translated in the x direction by asine function in y with amplitude 0.02 and k = 0.358. Following this somewhat non-physicaldisturbance, the flames would briefly settle before beginning to grow or decay. Figure 2.6shows the temperature distribution before and after perturbation. To make the relativelysmall disturbances visible, we have mapped the solution back onto the uniform [0, 1] x [0, 1]domain.

The results presented were computed using a 150 by 150 vertex grid and using a timestep of 0.033. Convergence was verified by separately reducing the spatial resolution to a100 by 100 grid and increasing the time step to 0.05. The growth rate for disturbances in theunenhanced case were also compared against and found to match those presented by Denetand Haldenwang [16].

21

0 20 40 60 80 100 120t

0.95

0.96

0.97

0.98

0.99

1.00

1.01

A0.3

58(Normalized) ∆σ=−3.3e−3

∆σ=−6.7e−3

∆σ=−3.2e−2

∆σ=−6.3e−2

G(x,y) =µx

µ=0

µ=1 ·10−5

µ=5 ·10−5

µ=1 ·10−4

µ=5 ·10−4

Figure 2.7: Development of induced flame wrinkles with G(x, y) = µx.

An example of the FreeFEM++ code implementation of the above is given in AppendixA.

2.3.3 Results

A natural starting point for inhomogeneous flame enhancement is one with the formG(x, y) =µx. With positive values of µ, a stabilizing effective is to be expected intuitively. Segmentsof the flame sheet lagging behind the mean will experience enhanced reaction rates and ac-celerate towards the mean location of the flame sheet, while segments ahead of the meanwill decelerate. Thus, wrinkles naturally decay.

This effect is present in Figure 2.7 which shows the impact of increasing values of µ ongrowth of the k = 0.358 mode. With each curve, the change in growth rate relative to theG(x, y) = 0 case is indicated as ∆σ. Here we have applied relatively modest values of µwhich do not overcome the intense natural instability generated by the selected values of β,γ, and Le. It is evident that the unstable mode grows more slowly as µ increases, confirmingthat this particular enhancement effect serves to stabilize the flame front against wrinkling.

From the previous example we proceed to the case G(x, y) = µy, where the enhancementeffect varies along rather than across the flame sheet. The implications of this effect aresomewhat less easily predicted by intuition or inspection. For the sake of clarity, we repeatthat the present results were computed with periodic conditions in the y direction. Thus,the functional form of G may be more accurately considered as a sawtooth function with a

22

0 20 40 60 80 100 120t

100A0.

358

(Normalized)

∆σ=−1.9e−2∆σ=−2.2e−2

∆σ=8.9e−1

∆σ=8.6e−1

G(x,y) =µyµ=0

µ=1 ·10−6

µ=5 ·10−6

µ=1 ·10−5

µ=5 ·10−5

Figure 2.8: Development of induced flame wrinkles with G(x, y) = µy.

period matching the height of the domain (i.e. k =0.179 ). Also recognize that this formcontains components at every wave number (with amplitudes decreasing as µk−1). Thus, weshould expect a direct (albeit non-linear) interaction between this excitation effect and theinduced perturbation at k =0.358.

We do find in Figure 2.7 that the effect of this enhancement effect is quite strong. Withthe sensitivity to µ being much higher when applied along the flame rather than across.Note also that this enhancement effect initially opposes growth of the unstable mode butultimately supports it as time increases, as seen for the chosen µ =1e-5 and 5e-5. The tem-porary decrease is potentially due to the large flame shape changes caused by this particularenhancement and the fact that the largest wavenumber of the sawtooth is 0.179, rather thanthe measured 0.358.

To simplify the infinitely many frequency components introduced by the sawtooth func-tion, we also investigated the influence of G(x, y) = µsin(0.179y). This is also a morephysically reasonable form than the previous case, which was discontinuous at the periodicboundaries. The effect is a strong and consistent enhancement of the induced mode, withoutany temporary reversal period.

Figure 2.10 examines, in more detail, how the enhancement effects presented caused theflame fronts to deviate from the baseline case. In blue, the first case (µx) presents a relativelysmall sinusoidal deviation and a constant offset from the base case. The offset occurs simplybecause this form of G shifts xave in the positive x direction, where all values of Ω are higher.Yet, the wave shape demands further examination. In the ideal case of reduced growth of the

23

0 20 40 60 80 100 120t

0.9

1.0

1.1

1.2

1.3

1.4

1.5

A0.3

58

(Normalized)

∆σ=1.9e−2∆σ=1.1e−1

∆σ=5.5e−1

∆σ=9.2e−1G(x,y) =µ sin(0.179 y)µ=0

µ=1 ·10−4

µ=2 ·10−4

µ=4 ·10−4

µ=5 ·10−4

Figure 2.9: Development of induced flame wrinkles with G(x, y) = µsin(0.179y).

perturbation, we would expect the deviation to be exactly a sine wave of k =0.358. Instead,the deviation is somewhat greater in the leading portions of the perturbation (y = [-17.5,-8.75], [0, 8.75]). The leading portions of the third case (µsin) are similarly more affectedthan the lagging regions of the flame sheet. The effect is quite pronounced in the secondcase (µy) where the leading portion for y<0 has expanded and substantially absorbed theadjacent lagging edge.

The sensitivity of the leading portions to local reaction rate is due to the mechanisms ofthermal-diffusivity. As is often discussed [25], low Lewis numbers create favorable conditionsin which chemical energy diffuses to the flame sheet more readily than heat is lost, andthis effect is enhanced by the local curvature of the flame sheet. This imbalance increasesflame temperature, increasing the local reaction rate and thus flame speed such that theproportionality U ∼

√Ω [19] holds well. The specific enhancements studied above are

multiplicative, rather than additive with reaction rate, Ω, such that their influence on flamespeed is greater for larger values of Ω. It is natural, therefore, that the leading componentsof a low Lewis number flame, having elevated temperature and reaction rates, would be themost influenced by the proposed enhancement mechanism.

24

15 10 5 0 5 10 15y

0.985

0.990

0.995

1.000

1.005

1.010

1.015

1.020

Ω(x

ave)

t=100

x10−1

x10

G(x,y) =1 ·10−4 x

G(x,y) =1 ·10−5 y

G(x,y) =1 ·10−4 sin(0.179 y)

Figure 2.10: Reaction rates along xave at t = 100 demonstrating the strong sensitivity ofwrinkle development to the functional form of G

2.4 Discussion

The experimental results presented demonstrate some of the effects achievable by applyingintense electric fields to transient combustion. The simple increase in flame speed is, as hasbeen noted elsewhere, potentially valuable in increasing the power of a combustion system,but it remains to be seen whether this can be accomplished in an energy efficient manner. Itis far more likely that there will be utility in the capacity to modify flame shape and providetemporary or local enhancements to control the combustion process. Either by design ofpassive electrode configurations or active controls, it could be feasible to excite and suppresscombustion instabilities as desired in engines and other power generation systems.

The first step in harnessing these observed effects is to understand what they are andhow they interact with existing combustion processes. To this end, we constructed a modelthat introduces external reaction rate modifications to a well-known and studied unstablecombustion process. We have found that, depending on disposition, these enhancements canserve to either quell or enhance the existing instability. From these results, we have alsobeen able to glean some insight into how the thermal-diffusive effects could produce greateror lower susceptibility to these effects.

Without limiting our view to electric-field effects, we recognize the broader study ofinhomogeneous combustion enhancement effects as fertile ground for further study. Exoticthermal, electrical, and chemical effects may each provide unique phenomena to be studied

25

and harnessed. Naturally, theoretical and numerical work should be expanded to includemore sophisticated models of both the combustion and enhancement, improve accuracy, andcapture more complex dynamics.

26

2.5 References

[1] J Lawton and FJ Weinberg. “Electrical Aspects of”. In: Combustion (1969).

[2] HC Jaggers and A Von Engel. “The effect of electric fields on the burning velocity ofvarious flames”. In: Combustion and Flame 16.3 (1971), pp. 275–285.

[3] Timothy Pedersen and Robert C Brown. “Simulation of electric field effects in premixedmethane flames”. In: Combustion and Flame 94.4 (1993), pp. 433–448.

[4] SD Marcum and BN Ganguly. “Electric-field-induced flame speed modification”. In:Combustion and Flame 143.1 (2005), pp. 27–36.

[5] DL Wisman, SD Marcum, and BN Ganguly. “Electrical control of the thermodiffusiveinstability in premixed propane–air flames”. In: Combustion and Flame 151.4 (2007),pp. 639–648.




[9] George H Markstein. Nonsteady Flame Propagation: AGARDograph. Vol. 75. Elsevier,2014.

[10] Mark A Heald and Charles B Wharton. “Plasma diagnostics with microwaves”. In:(1965).

[11] June Sung Park et al. “Phenomena in oscillating downward propagating flames inducedby external laser irradiation method”. In: Experimental Thermal and Fluid Science 34.8(2010), pp. 1290–1294.

[12] Michel Cazalens et al. “Combustion instability problems analysis for high-pressure jetengine cores”. In: Journal of Propulsion and Power 24.4 (2008), pp. 770–778.

[13] Sergey B Dorofeev. “Flame acceleration and explosion safety applications”. In: Pro-ceedings of the Combustion Institute 33.2 (2011), pp. 2161–2175.

[14] MK Kim, SH Chung, and HH Kim. “Effect of AC electric fields on the stabilizationof premixed bunsen flames”. In: Proceedings of the Combustion Institute 33.1 (2011),pp. 1137–1144.

27


[16] B Denet and P Haldenwang. “A numerical study of premixed flames Darrieus-Landauinstability”. In: Combustion science and technology 104.1-3 (1995), pp. 143–167.

[17] Thierry Poinsot and Denis Veynante. “Theoretical and numerical combustion”. In:(2005).

[18] Ya Be Zeldovich and DA Frank-Kamenetskii. “A theory of thermal propagation offlame”. In: Zh. Fiz. Khim 12.1 (1938), pp. 100–105.

[19] Paul Clavin. “Dynamic behavior of premixed flame fronts in laminar and turbulentflows”. In: Progress in Energy and Combustion Science 11.1 (1985), pp. 1–59.

[20] Frederic Hecht. “New development in freefem++”. In: Journal of Numerical Mathe-matics 20.3-4 (2012), pp. 251–266.

[21] Anders Logg, Kent-Andre Mardal, and Garth Wells. Automated solution of differentialequations by the finite element method: The FEniCS book. Vol. 84. Springer Science &Business Media, 2012.

[22] Daniel R Lynch. Numerical partial differential equations for environmental scientistsand engineers: a first practical course. Springer Science & Business Media, 2005.

[23] Vadim N Kurdyumov and Daniel Fernandez-Galisteo. “Asymptotic structure of pre-mixed flames for a simple chain-branching chemistry model with finite activation en-ergy near the flammability limit”. In: Combustion and Flame 159.10 (2012), pp. 3110–3118.

[24] JW Dold et al. “From one-step to chain-branching premixed flame asymptotics”. In:Proceedings of the Combustion Institute 29.2 (2002), pp. 1519–1526.


28

CHAPTER 3

FLAMES IN NARROW CHANNELS

3.1 Introduction

The study of combustion is ultimately motivated by the demands for power generation andpractical concerns. As such, many investigations of electric field effects on flames have beenaimed at improving power, efficiency and other engineering parameters. Fowler et al.[1] andJaggers et al.[2] demonstrated the increase of burning velocity by electricity, indicating thatelectric effects could increase the power of a combustor. Others have experimented withelectric fields as a means to control the formation of soot[3, 4], a significant pollutant. Verycurrent work involving microwave assisted spark ignition has found means to extend theoperating limits of automotive engines [5].

The utility of electric field effects on miniaturized combustion systems seems to be a par-ticular focus area that is missing from the literature. It may have been that the quenchingdistance was assumed to be inversely proportional to the flame speed, which would reducethe necessity for such a study. Flame phenomena on the millimeter scale are of great con-cern in the field of miniature combustion and power generation systems. Epstein et al.[6], Fuet al.[7], indicate that the high energy density of hydrocarbon fuels make them an attractivealternative to batteries for portable power but development of such systems is greatly com-plicated by heat losses leading to quenching, as evidenced by the work presented by Spragueet al.[8], Fernandez-Pello[9] and Ronney[10]. In addition to problematic heat losses, someminiature combustor designs exhibit inherent instabilities shown by Zamashchikov[11] andFateev et al.[12]. The ability to rapidly modulate the combustion process by electric meansmay provide novel means by which to mitigate oscillations and regain control of otherwiseunstable processes. If the observed electric enhancement of combustion processes can trans-late to miniature reactors it may facilitate the design of lightweight power supplies to replaceconventional batteries.

The goal of the present work was to study, through experimentation, the effect of electric

29

fields on the combustion phenomena at the millimeter scale. Premixed flames at the limitsof quenching were observed under multiple electric field scenarios. Changes in quenchingbehavior and propagation speed were observed. The dependence of the electrical effects onfuel air-mixture and field orientation were compared to the results of a theoretical model,to better understand the mechanisms responsible for the observed phenomena. The resultsserve to further clarify the physics of electric-flame integrations and demonstrate potentialmeans to improve miniature combustion processes.

3.2 Experimental Methods

Multiple experimental devices were constructed with the intent of observing the effect ofstrong electrical fields on the behavior and stability of flames in highly confined spaces.Despite their geometric differences, all of them functioned by providing an electrically insu-lating combustion chamber with optical access and a means of establishing external electricfields. Specific attention was paid to ensure that there was no direct conductive path betweenthe flame and any electrodes. This differs fundamentally from much of the prior work onelectric-flame interactions, where the flames were anchored to or impinging upon electrodes,making it possible for current to flow between the flame and the source of excitation [13–17].

The core concept of these devices is to observe a flame propagating through a channelthat becomes progressively more narrow in the direction of flame propagation. In such asystem, a the flame would continue to propagate until the width of the channel reachesthe quenching distance, δq, where heat losses to the solid walls exceed the chemical energyrelease from combustion. The taper of the channels was very gradual, generally less than 2.For the purposes, of most analysis, the walls may, therefore, be well approximated as beingparallel. From the know geometry of the channel it was simple to relate the easily observableparameter of position along the channel to the quenching distance, as shown in ??

For all devices, the gas supply systems, electrical excitation, optics and experimentalprocedure were identical:

• Methane (Praxair 99.9% pure research grade) and air (dried house compressed air) wereadmitted and controlled through the sonic nozzle system described in subsection 2.2.1.The fuel-air mixture entered through a sealed chamber at the base of the channel. Thechamber at the entrance of the channel contained electrodes for spark ignition of thefuel air mixture and was loosely sealed with a plastic diaphragm to prevent mixing withambient air prior to ignition but readily release the products of combustion during theexperiment.

• Electrical spark ignition of the gas mixture was performed at the top of the test volume.The flame propagated downward, with the direction of gravity, while exhaust gases wereexpelled from the top of the test volume.

30

Figure 3.1: Conceptual diagram of tapered channel quenching. Spark ignition occurs atthe open end (far left) and the flame propagates inward until quenching occurs. Due tothe known taper, the quenching distance is easily determined from the location l along thechannel.

• High voltage (HV) AC was produced by a Transco NT1512N3G transformer operatingat maximum of 15 kV r.m.s. at 60 Hz, with lower voltages being realized by reducingthe input voltage with a Powerstat variable autotransformer. DC high voltage wassupplied by a Spectra Physics 256 4.5 kV exciter.

• The high-speed schlieren video system is described below in subsection 3.2.2

• Procedure:

1. Open fuel-air mixture valve for 5 seconds to fully purge the combustion chamber.Close rapidly.

2. Start active buffering of the high speed camera

3. Activate high voltage supply, if appropriate

4. Trigger spark ignition and high speed camera

5. Deactivate high-voltage supply

3.2.0.1 Planar Combustion Channel

The initial intent of these experiments was to observe flame quenching in quasi-2D con-figuration, with fuel-air mixture confined between two approximately parallel plates. Thismethod would have been preferable to quasi-1D quenching experiments in that it would haveprovided a larger total area of flame to be observed in each experiment, providing a larger,more robust dataset.

The internal walls of the planar combustion chamber were sheets of clear polymethyl-methacrylate (PMMA) 3 mm thick, held in a structure of 12 mm thick black PMMA. One

31

plane was fixed while the other was secured with adjustment screws and a linear slide mech-anism. This design allowed fine control of the average gap width and taper through thechannel. In order to establish electric fields while permitting optical access, sheets of 127 µmthick PET coated with 130 nm electrically conducting indium tin oxide (ITO) were placedon the exterior faces of the PMMA walls and secured with circular aluminum frames. Intotal, these two electrodes reduced the total transitivity of the combustion chamber to 62%, which reduced the recorded image intensity but did not pose any practical problem inthe observation of flames.

3.2.1 V-Channel quenching

In response to the undesirable Taylor-Saffman instability encountered in the planar combus-tion channel, a new, significantly less wide, quenching test was designed. Reducing the widthof the channel below the scale of the previously observed viscous fingering eliminated thateffect. The new design was also amenable to viewing through, rather than across, the tapergap (comparable to the view in Figure 3.2), eliminating the need for transparent electrodes,which were prone to damage in the event of electrical arcing.

The apparatus, referred to here as the V-channel channel and shown schematically inFigure 3.2, consisted primarily of two opaque PMMA blocks which formed the walls of theV-Channel. Oriented vertically, these walls created a small gap which gradually decreasedfrom 5 mm at the entrance, to 2 mm at the bottom. This 3 mm change occurred over 100mm in the vertical direction. Above the linear test section was a curved section that allowedthe flame to enter the channel smoothly. The opaque PMMA channel walls were 12 mmthick and held between sheets of clear PMMA to enclose the channel while providing opticalaccess. The channel was operated in two configurations (Figure 3.2 left and center). Thefirst to establish an electric field perpendicular to the direction of flame propagation andchannel walls, the second to establish fields parallel to the direction of propagation. For theperpendicular field configuration, the electrodes were flat aluminum plates, 107 mm square,held symmetrically around the channel walls at a distance of 25 mm from one another. Forthe parallel configuration, the electrodes were 90 mm square plates with centrally located 15mm square holes to permit the V-channel to pass through. The parallel plates were alwaysspaced between 25 mm and 75 mm apart, depending on the electric field strength required.For very high electric field, with closely spaced plates, it was necessary to reposition theelectrodes along the length of the channel so that the electric field would be present at thelocation where quenching would normally occur in the absence of the electric field. Bothconfigurations were used to some extent, although parallel fields received more attention.Parallel and perpendicular configurations were tested with the electrodes energized usinga 20 kV 60Hz alternating current (AC) transformer, producing a nominal electric field of800kV/m. Due to atmospheric breakdown and arcing, it was not possible to use higher fieldstrengths. The parallel field configuration was also tested at 266 kV/m AC, ±60 kV/m DCand ±180 kV/m DC.

32

Figure 3.2: (left to right) Diagrams of the V-Channel in perpendicular and parallel fieldconfigurations. A photograph of the perpendicular configuration (gas supply detached)

To introduce variation in the chemical processes involved, experiments were conductedover a range of fuel-air ratios, f . For methane and air, a stoichiometric mixture has fs =0.105. Mixtures were recorded in terms of the normalized quantity, equivalence ratio, Φ =f/fs. For a mixture deficient in fuel (lean) Φ < 1 and one with excess fuel (rich) has Φ > 1.Both lean and rich conditions are non-optimal, as not all of the reactants are able to be fullyreacted. Such mixtures are less reactive and energetic, resulting in lower flame temperature,lower flame speed and larger quenching distances [18, 19]

In the perpendicular field configuration, experiments were performed at 5 equivalenceratios ,Φ, ranging from 0.85 to 1.2. In the parallel field configuration with AC, four valuesof Φ from .95 to 1.2 were tested. To minimize the impact of environmental factors andvariability in mixture, each nominal Φ and configuration was tested multiple times, alterna-tively applying electric field and allowing the flame to propagate freely, providing a referenceto which the electrically altered flames could be compared. In all cases, 6 tests were runwithout electric field and 6 were run with electric field. For Φ = 1.0 with perpendicular fieldan additional 6 tests without field and 6 tests with field were conducted. All DC tests wereconducted at Φ = 1.0. Each set of DC experiments included its own set of reference (E ′0 = 0)experiments to account for any drift or environmental changes.

33

3.2.2 Optics

Transient flame quenching phenomena are inherently difficult to observe due to the factthat they occur at very small scales in both time and space. The total duration of mostexperiments was generally less than 0.25 s, with the quenching event itself occurring overonly a fraction of that time. To provide satisfactory temporal resolution, a high speed videocamera (Kodak EktaPro HG Imager Model 2000) was used to record these experiments. Forall tests in which quantitative data were desired, video was recorded at the maximum framerate of the camera, 1000 or 2000 frames per second (FPS), depending on the aspect ratioof the images recorded. Lower frame rates were occasionally used to produce higher qualityimages for presentation as the longer exposure time reduced image noise.

The flames under study were not bright enough to observe directly at the frame ratesdesired. Methane burnt near stoichiometric conditions produces very little soot [19], whichgenerally contributes greatly to the visible light emission from flames [20]. This lack of lu-minosity, combined with the fact that the optical path through the flames was always lessthan ∼ 10 mm, simply did not yield enough light to excite the camera’s sensor. To com-pensate for this complication, a schlieren photography system was constructed to visualizethe density gradients produced by thermal expansion of the reacting gases. This methodutilizes external illumination which may be as powerful as desired to provide satisfactoryimage exposure.

The schlieren optics system used is diagrammed in Figure 3.3. This is, substantially,a common Z-type configuration as described by Settles [21] with the addition of significant”folding” of the optical paths by flat mirrors to limit the overall footprint to a few square me-ters. Rather than the traditional knife-edge cutoff, which produces a monochromatic imageof density gradients in one direction, the present system uses an transparent photographicslide with concentric circular bands of different colors. As a result, density gradients in anydirection (parallel to the image plane) appear as variations in color from blue to red.

While it is possible to recover quantitative information from the intensity of schlierenimages, the processes of doing is not always practical. In a chemically homogeneous flow,[Reasons Here]

3.2.3 Image Processing

The raw data collected from all of these experiments was in the form of high speed schlierenvideo. Given the number of total tests and high frame rate used, any sort of manual dataextraction would have been impractical. Human intervention could have also introduced thepossibility of operator error and bias. Fully automated image analysis methods were thereforedeveloped to provide rapid and consistent datasets by detecting and tracking the flames.These programs were used to recover flame location, propagation velocity and quenchingdistance. The code used can be found in Chapter B.

34

Figure 3.3: Diagram of the schlieren apparatus. A) LED light source, B) 3 m diameter, 3.94m focal length first surface mirror, C) Flat second surface mirror, D) 114 mm diameter, 914mm focal length first surface mirror, E) Bullseye schlieren slide F) 110 mm diameter, 240mm focal length achromatic lens, G)Kodak EktaPro HG Imager Model 2000 high speedcamera

3.2.3.1 Image Registration

The first step towards recovering accurate measurements from the video files was to correctlydetermine the location and orientation of the experiment within the image frame. Thanksto the collimated illumination used for schlieren imaging and firm fixturing of optics, thescale of the experiment was found not to change across the experimental campaign. Eachpixel in the recorded images was approximately 200 µm. Even small perturbations to theexperiment during maintenance and normal operation resulted in sufficient motion acrossthe field of view that key locations (e.g. the centerline of the experimental apparatus), couldnot be relied upon to coincide with the same pixels across experiments.

For the V-channel device described in subsection 3.2.1, only angular and horizontal reg-istration proved to be problematic. While vertical misalignment was a possibility, absoluteposition was only necessary for determination of quenching distance. For the very grad-ual taper used, a single pixel of misalignment would have introduced a 6 µm error in theestimate of the quenching distance ( ∼0.25% in most cases). It is also worth noting thatalthough the angular alignment was sufficient to complicate flame tracking, the actual anglesinvolved were very small (never more than 0.5 degrees from vertical), so that the small angle

35

assumptions (sin(θ) ≈ θ, cos(θ) ≈ 1, etc.) were quite accurate.

The distinctive patterns of light and dark created by the opaque walls of the V-channelprovided a strong reference pattern for measuring the location of the experiment within thefield of view. A single reference image was created, in which the V-channel was manuallycentered and vertically aligned. The first frame of each test was aligned relative to thatreference. From each, the 100th and 400th rows (20% and 80% of the vertical scale) wereextracted as 1D vectors (si and ri for the subject and reference). The horizontal displacementof upper and lower rows, δx, u and δx, l, in the subject image were determined from thelocation of the peak value of the cross correlation between the target and reference rows

δx,u/l = arg maxj

(∑m

s∗mrm+j

)(3.2.1)

From these, the angular error, δθ = (δx,u− δx,l)/300 and mean displacement error, δx,m =(δx,u + δx,l)/2 were readily computed. To complete the registration, each test video wasrotated by −δθ and translated by −δx,m using image transformation functions from theimage processing toolbox in MATLAB [22].

3.2.3.2 Flame Location Measurement

Determining the flame location along the V-channel was primarily a peak finding exercise.The shape of flames in the V-channel exhibited an extremely consistent appearance acrossall tests. The leading edge of the flame was always centered within the channel and formeda strong signal in the green channel of the image.

The column of pixels from the center of the channel were extracted across all frames ofthe video to create a 2D map of image intensity as a function of time, as shown in Figure 3.4b). These reduced datasets contained both a strong, constant background signal and time-varying pixel noise. The first was addressed by applying a normalized thresholding, settingpoints below 40% of the maximum intensity to 0. Pixel noise was eliminated by applying a3-by-3 median filter. Finally, the map was reduced to a binary from, as shown in Figure 3.4c).

Quenching location was taken to be the farthest point at which signal remained in themap. Strictly, that method introduced a small bias towards underestimating quenching dis-tance but the bias was constant across all experiments and did not interfere with comparisonwithin the present work.

In some early experiments, the flame position data was also used to determine the flamepropagation speed, SP . To do this, the portion of the binary map from 5 to 85 millisecondsprior to quenching was isolated. The value of SP was taken to be the slope of a linearleast-squares fit to the t,y coordinates of the non-zero pixels.

36

Figure 3.4: Sketch of the flame detection process. a) input image, b) unfiltered map of pixelbrightness along the image centerline, c) binary filtered brightness image showing quenchinglocation (yellow) and propagation speed prior to quenching (red)

3.2.3.3 Flame Velocimetry

Although it was possible to compute flame propagation speeds by detecting the flame, asdescribed above, that method was not particularly robust against disruptions introducedby the electrodes used in some experiments. Instead, an image cross-correlation techniquewas applied in order to detect relative motion without needing to determine any absolutelocations.

The premise was fundamentally equivalent to the 1D correlation method described insubsubsection 3.2.3.1, which was used to measure the displacement between to sets of data. Inthis case, the method was extended to two dimensions and the comparison was made betweensequential images within the video (In(x, y),In+1(x, y)). The resulting cross-correlation peakmeasured the displacement of the flame during a known interval, ∆t, which approximatedthe true velocity well.

Vn+1/2 =1

∆targ maxδx,δy

∫I∗n(x, y)In+1(x+ δx, y + δy)dA (3.2.2)

As with Equation 3.2.1 the ’*’, to denote complex conjugation, is retained from thegeneral definition of the operation, but is not actually necessary for the real-valued imagemaps used here. It is also most accurate to treat this as the approximate velocity halfwaybetween the n and n+1 frames [23].

37

Figure 3.5: Process flow of the flame velocimetry showing: a) raw image, b) cropped image,c) background-subtracted image, d) single image pair cross-correlation, e) ensemble cross-correlation (peak displacement indicated in red)

The actual implementation involved a few practicalities not specified in the general for-mulation:

• For this method, the field of view was reduced to capture only the width of the channeland the length over which electric fields were present. This area was large enough thatminor misalignment posed no problems and no registration process was needed.

• The static, or background, component of the image was estimated by taking the pixel-wise minimum value from five frames spaced evenly in time throughout the video.This background value was subtracted from the video prior to cross-correlation. Hadthe static component not been removed, the cross correlation would have contained astrong zero-displacement peak.

• As in the flame location method, median filtering (spatially) was fairly effective atremoving pixel noise.

• Due to the color pattern selected for the rainbow schlieren filter, little to no usefulsignals were found on the red channel. Rather than introduce what was essentiallypure noise, I(x, y) values were taken as the sum of the blue and green channels only.

• Under the assumption that propagation speed varied slowly, a technique for improvingthe robustness of displacement tracking was borrowed from ensemble methods in parti-cle image velocimetry [24]. Multiple, sequential cross-correlations were computed and

38

summed prior to locating their peak values. This process was immensely beneficial inimproving the signal to noise ratio in the cross-correlations. (N.B. this is not equivalentto averaging peak values. In the event that one or more cross-correlations contains aspurious peak, simple averaging would include that value while the ensemble methodwould reject the false information.)

• The actual cross-correlations of image pairs were not computed by summation of pixels,as implied by the integral in Equation 3.2.2, as this process was found to be compu-tationally inefficient for images of this size. Rather, a fastFourier transform basedcomputation was applied [25] to achieve equivalent results.

An example of the full process is shown in Figure 3.8.

3.3 Theoretical Formulation

An extension of the present work was a collaboration with Prof. Sanchez-Sanz of UniversidadCarlos III de Madrid, assisting in the development and analysis of a reduced-chemistry, one-dimensional laminar flame model aimed at clarifying the physical processes underlying theflame enhancement observed in the V-channel experiments. The computed solutions ofthis model will be of use in understanding the following experimental results and provideconceptual insight into the effect of an electric field on the internal structure of a flame.

The full derivation of this model is given by Sanchez-Sanz et al.[26]. For the most part,it begins with the conservation of mass, energy and species for lean, planar, premixed flamewith a two-step chemical reaction, similar to that proposed by Dold et al.[27]. The inter-mediate, radical species in that reaction is then given an additional consumption reaction,by which it can decompose into a positively charged radical and an electron, both of whichmust subsequently recombine to complete the reaction. Gauss’s Law is added to determinethe electric field E, which depends on the distribution of the two charged species and theexternally applied field, E0, which enters as a boundary condition. When reduced to a

39

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

x

YF

YZ

θ/ Q

a)

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

x

Ye-b)

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

x

YZ+

0 100−0.02

−0.01

0

0.01

E-E0c)

Figure 3.6: a) Profiles of fuel YF , radical YZ and temperature θ/Q, b,c)Electron and protonmass fraction profiles, for E0 = 0 (blue, solid line), E0 = 0.65 (black, dashed line) andE = −0.65 (red dot-dashed lines) with β = 10,∆β = 1,A = 0.1,B = 100, ε = 100,m+/m− = 100 and Q = 5. The inset of figure b) represents the auto-induced electric fieldcreated by the charge displacement near the flame. Adapted from Sanchez-Sanz et al.[26]

non-dimensional form, the model appears as:

dθ

dx=d2θ

dx2+ µQ

(rII + rIV

qIVqII

)(3.3.1)

dYFdx

=1

LeF

d2YFdx2

− µrI (3.3.2)

dYZdx

=1

LeZ

d2YZdx2

+ µ [rI − rII − rIII ] (3.3.3)

dYZ+

dx= −µ1/2d (EYZ+)

dx+

1

LeZ+

d2YZ+

dx2+ µ [rIII − rIV ] (3.3.4)

dYe−

dx= µ1/2

(m+

m−

)1/2d(EYe−)

dx+

(m+

m−

)1/21

LeZ+

d2Ye−

dx2+ µ [rIII − rIV ] (3.3.5)

Where the conserved quantities are temperature, θ, (energy) and the mass fractions of fuel,neutral radicals, positively charged radicals, and electrons, YF , YZ , YZ+ , Ye− , respectively.Two parameters of interest are the heat release Q, which indicates how energetic the mixtureis and the eigenvalue of the problem, µ, which is proportional to the inverse square of thelaminar flame speed, S−2

L . As µ−1/2 is the most comparable to experimental observations,it will receive the most attention. The reaction rate terms, rI−IV , also include terms tocontrol the relative importance of neutral ion-chemistry, although these are not necessary inthe present work. The structure of the resulting flames for negative, neutral, and positivevalues of E0 are shown in Figure 3.7, to highlight the changes in flame shape as electric fieldsare applied. The auto-induced electric field is important to maintaining the structure of theneutral flame. It balances the very high mobility of the electrons by producing an opposing

40

force as they diffuse away from the positive radicals due to the locally non-zero net charge.In this model, a positive E0 drives electrons forward into the unburnt mixture, allowing theexothermic reaction with the positive radicals to occur earlier. This subtle shift slightlysteepens the temperature gradient at the flame front, enhancing forward heat conductionwhich drives the initial decomposition reaction or, equivalently, flame propagation. Thereverse is true for E0 < 0, where the redistribution of electrons reduces flame propagation.

3.4 Results and Discussion

3.4.1 Quasi-2D

As had been alluded to earlier in this chapter, attempts at gleaning useful, quantitativedata from the planar combustion chamber were ultimately fruitless. A representative setof sequential images taken from it are shown in Figure 3.7. Across all test conditions at-tempted (various combinations of channel width, equivalence ratio, electric field strength)the same wrinkled flame patterns formed. This flame structure was quite irregular andproved very problematic to measure in an automated fashion, making it of little value forthe measurement of δq and SP .

Upon initial observation, a thermo-diffusive instability of the kind discussed in subsec-tion 2.3.1 was suspected. When tests with both lean and rich mixtures showed similar results(indicating no Lewis number dependence), this theory was discounted. Further review of theliterature [28, 29] found that flames in narrow channels are susceptible to a form of theSaffman-Taylor instability[30], which produces a ”finger” pattern as relatively inviscid fluidis forced into a viscous one. In essence, friction forces in the rapidly moving burnt gasesestablish favorable pressure gradients behind the flame, reinforcing deformation of the flamesheet. This effect is quite pronounced and could not be overcome within the limits of theexperiments at hand.

3.4.2 Quenching Distance

At all tested equivalence ratios, it was found that the perpendicular field had a detrimentaleffect on the flame, increasing quenching distance, while parallel fields enhanced the flamegreatly reducing quenching distance. In all perpendicular field tests the flame never quenchedwithin the region of applied field. Quenching would, however, occur immediately as the flamepassed beyond the electrodes. An inherent limitation of this apparatus is the limited span ofthe applied parallel field which could at most prevent quenching for 25mm along the lengthof the channel, reducing quenching distance by as much as 0.75 mm.

The addition of perpendicular field also increased the variance of the quenching distancemeasurement. Figure 3.8 displays the data graphically. Each point indicates the meanresult of a field/equivalence ratio normalized by the mean value observed in the absence of

41

Figure 3.7: Schlieren images of methane-air flames from the planar combustion chamber,showing the curve-cusp pattern characteristic of the hydrodynamic instability of flames in aHele-Shaw cell. Lines in red added to mark the flame front.

Figure 3.8: Variation in quenching distance, δq, with 800 kV/m AC electric field applied inboth the perpendicular and parallel orientations.

42

applied electric field. The patches indicate one standard deviation about the mean value.For the parallel case, no uncertainty is indicated because the observed quenching locationwas identical across all tests, where the flame exited the region of applied electric field. Thesevalues are not indicative of a true δq under the nominal E0. Rather, they mark an upperlimit on δq as it could be significantly smaller.

Practical interest in the flammability reduction caused perpendicular electric fields issomewhat limited. The magnitude of the effect is quite small, given the intensity of theelectric fields used. There may be some conceivable application in which this method couldbe useful as a means to control or limit an unstable combustion process. Should one wishto achieve control, it might be preferable to design around a modest E0 in a parallel field,to which flames are much more sensitive. A relative reduction could then be achieved byreducing E0.

The enhancement caused by a parallel electric field is significantly more interesting, bothbecause it appears to be quite strong and because it suggests a means by which millimeter-and micro-scale combustion systems could be improved. Determination of the true potentialof this effect was only limited by the specific geometry of the test apparatus and the practi-calities of observing quenching at very small scales. Rather than redesigning the experimentfor significantly smaller experiments, it was found to be much more practical to use flamepropagation speed, SP , as a measure of reactivity, through which quenching distance maybe estimated.

3.4.3 SP as a surrogate for δq

In a premixed flame, quenching distance (or diameter as appropriate) and laminar flamespeeds SL are closely related values. Inherently, both are governed by interactions betweenthe heat generation from combustion and conduction of that heat. Consider the conservationof energy as applied to the preheat and reaction zones from Figure 3.9 in a moving referenceframe. The value of SL is taken to be that at which the reacting volume of the flame sheetgenerates sufficient heat such that thermal conduction forward into the unburnt mixture issufficient to bring the reactants to some auto-ignition temperature, Tig.

This level of analysis simplifies the chemistry greatly by approximating the normalizedreaction rate as the ratio of initial fuel concentration, [Fuel], to a temperature-averagedreaction rate, r. The results are only approximate, but capture the parametric sensitivitywell. Extending this analysis to include conductive losses to a cold boundary allows asimilarly compact expression for quenching distance.

The value of the Equations 3.4.1 and 3.4.2 is that they can be recombined to reveal theapproximate relationship: δq ∝ S−1

L which we extend to δq ∝ S−1P . Although the present

experiments were not able to directly measure δq for flames excited by electric fields, theydo permit the measurement of SP of the enhanced flames where quenching would normally

43

SL =

√λ(Tb − Tig)r/[Fuel]

ρcp(Tig − Tu)(3.4.1)

Figure 3.9: The simplified structure and speed of a laminar, premixed flame. Figure adaptedfrom [18]

δq =

√2λ[Fuel](Tb − Tw)

r · ρucp(Tb − Ty)(3.4.2)

Figure 3.10: The simplified model of flame quenching. Figure minimally adapted from [18]

have occurred. We have, therefore proposed the use of S−1P as a surrogate for δq. This is

imprecise, but serves to indicate the magnitude of enhancement possible.

One should note that SP and SL are similar but not identical. SL refers to a pure,effectively theoretical speed of propagation of a flame through an infinite one-dimensionaldomain. SP is simply the observed propagation speed in the experiment at hand and maybe strongly influenced by thermal and hydrodynamic interactions with the walls of the V-channel.

3.4.3.1 Measured SP

We begin with a comparison between the parallel and perpendicular field configurations atthe highest field strength (800 kV/m). The results are presented in Figure 3.11 for bothelectric field conditions, and are supplemented by data for a parallel field at 266 kV/m AC.They show that the perpendicular field moderately lowered the propagation speed while theparallel field greatly increased it. The effect was more pronounced at the stoichiometricmixture fraction and when slightly lean, but the effect was consistently observed over theentire range of mixtures. Similar to the δq results, the addition of the electric field increasedthe scatter of the data at all equivalence ratios. Another similarity is that the propagationspeeds are clearly separated for all but the lean mixture, and even that mixture maintains ahigher average propagation speed with no electric field.

The clear distinction is that perpendicular fields are disadvantageous to the electric en-

44

Figure 3.11: Variation in observed SP due to 800 kV/m AC electric fields applied paralleland perpendicular to flame propagation. Patches mark ±standarddeviation

hancement of combustion. At present, understanding of the electrical sensitivity of flamesin only one dimension remains an active subject of research. Future extension to the two-dimensional analysis may serve to clarify the influence of perpendicular fields. One mayconsider that the fields serve to enhance microscopic transport (conduction or diffusion)to the channel walls thereby increasing heat losses or reducing the availability of radicalspecies through radical quenching [31, 32], although that theory remains unconfirmed byanalysis. Given that the suppression effect was quite weak and of less practical value tothe micro-combustion community, no further studies of the perpendicular configuration weremade.

Having probed the effects of AC fields, it was natural to extend study of the parallelfield enhancement effect to DC fields. The natural expectation is that there should besome polarity sensitivity, owing to the tremendous difference in mass between positive heavyions and primary negative species, electrons. The importance of field polarity has beenwell documented in earlier work in this field[citations here]. The transient nature of ACexcitation is also problematic for the purposes of theoretical analysis. Steady propagationmodels such as that referenced in section 3.3 are incompatible with periodic excitation, whichmay become increasingly problematic for models involving detailed transport and chemistry.

Flame acceleration for parallel DC fields is shown in Figure 3.12, with parallel AC resultsfor Φ = 1 included for reference. No change in propagation speed was evident for the E0 < 0,while flame acceleration greater than a factor of two was found for E0 > 0. It is interestingto note that the increase in flame speed with E0 in DC appears to coincide with the trendobserved for the higher voltage AC results. A naıve time-averaging of the observed DCeffects would suggest that AC excitation should be roughly half as effective as DC, on a

45

200 0 200 400 600 800

Electric Field (kV)

2

4

6

8

NormalizedFlameSpeed AC

DC

Figure 3.12: Variation in propagation speed with E0 for stoichiometric methane-air mixtures.For AC, E0 indicates the magnitude of the sinusoidal field.

peak-voltage basis, given that half of the cycle time is spent in a DC mode. This resultis even more surprising given that any structural changes to the flame (redistribution ofcharged species) caused during the E0 > 0 phase would be expected to be opposed in somefashion.

The model anticipates an upper limit on the enhancement effect possible (above a cer-tain value µ−1/2 decreases as E0), which we have not observed experimentally. The limitdoes, however increase as the energy content of the mixture, Q, increases. The analogousmeasurement in the experiments is proximity to a stoichiometric mixture, |Φ − 1|. In thisregard, we see a level of global agreement between the experiments and theory. At a moredetailed level, Q, as modeled, is substantially decoupled from the ionization rate. In real-ity, the concentration of ions has been shown to drop significantly as |Φ − 1| increases[33],which would amplify this sensitivity. An approximate theoretical complement to the ionconcentrations is the ionization rate constant, A (a coefficient of rIII found in Equations3.3.3,3.3.4,3.3.5). Quite unsurprisingly, a higher relative concentration of ions increases thepredicted sensitivity to electric fields. Thus, both theoretical parameters related to |Φ − 1|produce flame enhancement consistent with experimental observations.

The flame speed enhancement effects for E0 > 0 is certainly in line with the predictionsof the theoretical model, as shown in Figure 3.13a,3.13b, although the two diverge for E0,where the model predicts a reduction in flame speed. This partial agreement is encouraging,but the discrepancy for negative E0 and the lack of apparent limit on enhancement remainproblematic. One physical mechanism absent from the model which could improve agreementis Ohmic heating, energy deposited in the flame by the applied electric field. This addedenergy could counteract the detrimental effects of certain charge redistributions.

46

E0 (kV) Reduced Frequency (ω/N) Q′′′E/Q′′′C

800 6.16 · 10−23 0.790266 6.16 · 10−23 0.112180 - 0.10460 - 0.015

Table 3.1: Relative magnitude of Ohmic heating, Q′′′E and chemical heat release rate,Q′′′C

To estimate the relevance of Ohmic heating, it must be compared to the heat generationalready present in the flame. To arrive at the latter quantity, we compute the rate at whichchemical energy is released within the flame sheet. Given the heat of combustion of methane,∆Hc = 50MJ/kg[18], mass fraction of fuel YCH4 = 0.055, mixture density ρ = 1.114, flamethickness δf ≈ 3mm and SP = 0.3m/s (a representative value from the present experiments)the volumetric heat release rate from combustion, Q′′′C , is readily computed:

Q′′′C =ρYCH4∆Hc SP

δf(3.4.3)

Which gives t value of ∼ 255MJ/kg as a reasonable approximation. The equivalent Ohmicheating requires slightly more analysis. To begin, one must determine the concentration ofelectrons. Pedersen[34] reports positive ion concentrations in the reaction zone of stoichio-metric CH4/O2 flames as being ∼ 1017m−3. Under the assumptions of net charge neutrality,electrons as the dominant anion species and that dilution by nitrogen does not alter therelative concentrations of the reacting species indicates that the number density of electrons,ne, in the flame sheet is ∼ 2 · 1016. Computation of Ohmic heating from the present infor-mation (ne,ntotal,E0 etc.) requires solving the Boltzmann transport equation to determinethe thermal electron energy distribution function (EEDF) and the resultant rate of energytransfer from electrons to molecules. To do so we have used Bolsig+[35], which solves for theEEDF using a two-term expansion of the steady-state Boltzmann equation (with provisionfor oscillating electric fields). For these computations we have used 25 elastic and inelastice− + N2 collision cross sections from the work of Phelps et al.[36] and Boeuf et al. [37],recorded in the LXCAT database [38] and have assumed an intermediate temperature of1200 K.

The resulting reduced power dissipation values need only be multiplied by the numberdensity of electrons and molecules to recover the total volumetric energy deposition rate, Q′′′E .The results of these computations are given in Table 3.1. Although Q′′′E is consistently smallerthan Q′′′C , it is generally large enough to not be negligible, particularly for the highest electricfield tested. Greater precision in this evaluation would necessitate more measurements ofthe actual electron concentrations of the flames being examined, a process which remainsan active topic of study, such as the methods in chapter 4. At present, it suffices to saythat consideration of Ohmic heating effects helps to reconcile the discrepancies betweenexperimental results and numerical predictions.

47

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.20.85

0.9

0.95

1

1.05

1.1

E0

A = 1× 10-3

A = 0.01

B = 100

ε = 100 A = 0.1

µ(0)µ(E )

1/2

))

(a) Experimental

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.20.85

0.9

0.95

1

1.05

1.1

E0

A = 0.1

B = 100

ε = 100

Q = 5

Q = 7

Q = 3µ(0)µ(E )

1/2

))

(b) Numerical

Figure 3.13: Computed effect of mixture energy and ionization on flame speed. Adaptedfrom Sanchez-Sanz et al.[26] and Murphy et al. [39]

3.5 Conclusion

In narrow channels, the application of external electric fields strongly affects both the quench-ing distance and propagation speed. Further, this interaction is highly sensitive to the ori-entation of the field, with a field perpendicular to the direction of propagation moderatelyimpeding the reaction while a field across the flame sheet vastly accelerates the reaction.This behavior was observed across a range of equivalence ratios, although it was somewhatweaker for mixtures far from stoichiometric.

The precise control of flame geometry relative to the applied electric field and the channelwall achieved in these experiments serves to isolate potential physical phenomena responsiblefor the observed combustion phenomena. Kinetic enhancement and energy coupling theoriesare inconsistent with the orientation sensitivity shown. As well, hydrodynamic blowingof flames could not occur significantly across the channel or into the closed end of thechamber, due to the very confined geometry involved. Forced transport of electrons or ionsout of the flame sheet would account for the decreased reactivity in the perpendicular fieldconfiguration. Similarly, transport of charged species across the flames sheet would increaseheat transfer and ionization ahead of the flame, causing the higher flames speeds observedin the parallel field configuration.

Future study of millimeter-scale combustion under electric fields should include bothexperimental and modeling efforts. New experimental apparatuses must be developed tosupport the dramatic enhancement effects observed with parallel fields. The addition of

48

Langmuir probes to measure ion currents to and from flames, and microwave interferometryto measure electron density would provide significant insight into the transport and genera-tion of charged species in the flames. Detailed modeling, in the style of Hu et al.[40] or Belhiet al.[41], would greatly assist in isolating the precise cause of the electric-flame interactionsfound here.

49

3.6 References

[1] RG Fowler and SJB Corrigan. “Burning-Wave Speed Enhancement by Electric Fields”.In: Physics of Fluids (1958-1988) 9.10 (1966), pp. 2073–2074.

[2] HC Jaggers and A Von Engel. “The effect of electric fields on the burning velocity ofvarious flames”. In: Combustion and Flame 16.3 (1971), pp. 275–285.

[3] ER Place and FJ Weinberg. “Electrical control of flame carbon”. In: Proceedings ofthe Royal Society of London. Series A. Mathematical and Physical Sciences 289.1417(1966), pp. 192–205.

[4] PJ Mayo and FJ Weinberg. “On the size, charge and number-rate of formation ofcarbon particles in flames subjected to electric fields”. In: Proceedings of the RoyalSociety of London. A. Mathematical and Physical Sciences 319.1538 (1970), pp. 351–371.

[5] Benjamin Wolk et al. “Enhancement of flame development by microwave-assisted sparkignition in constant volume combustion chamber”. In: Combustion and Flame 160.7(2013), pp. 1225–1234.

[6] AH Epstein and SD Senturia. “Macro power from micro machinery”. In: SCIENCE-NEW YORK THEN WASHINGTON- (1997), pp. 1211–1211.

[7] Kelvin Fu et al. “Microscale combustion research for applications to MEMS rotary ICengine”. In: Proceedings of NHTC, 2001 national heat transfer conference, Anaheim,CA. 2001, pp. 10–12.

[8] S Bennett Sprague et al. “Development and characterisation of small-scale rotary en-gines”. In: International Journal of Alternative Propulsion 1.2 (2007), pp. 275–293.

[9] A Carlos Fernandez-Pello. “Micropower generation using combustion: issues and ap-proaches”. In: Proceedings of the Combustion Institute 29.1 (2002), pp. 883–899.

[10] Paul D Ronney. “Analysis of non-adiabatic heat-recirculating combustors”. In: Com-bustion and Flame 135.4 (2003), pp. 421–439.

[11] VV Zamashchikov. “Experimental investigation of gas combustion regimes in narrowtubes”. In: Combustion and Flame 108.3 (1997), pp. 357–359.

[12] GA Fateev, OS Rabinovich, and MA Silenkov. “Oscillatory combustion of a gas mixtureblown through a porous medium or a narrow tube”. In: Symposium (International) onCombustion. Vol. 27. 2. Elsevier. 1998, pp. 3147–3153.


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[15] DL Wisman, SD Marcum, and BN Ganguly. “Electrical control of the thermodiffusiveinstability in premixed propane–air flames”. In: Combustion and Flame 151.4 (2007),pp. 639–648.

[16] SH Won et al. “Effect of electric fields on the propagation speed of tribrachial flamesin coflow jets”. In: Combustion and Flame 152.4 (2008), pp. 496–506.

[17] SD Marcum and BN Ganguly. “Electric-field-induced flame speed modification”. In:Combustion and Flame 143.1 (2005), pp. 27–36.

[18] Sara McAllister, Jyh-Yuan Chen, and A Carlos Fernandez-Pello. Fundamentals of com-bustion processes. Springer Science & Business Media, 2011.


[20] Irvin Glassman, Richard A Yetter, and Nick G Glumac. Combustion. Academic press,2014.

[21] Gary S Settles. Schlieren and shadowgraph techniques. 2001.

[22] MATLAB. version 7.10.0 (R2010a). Natick, Massachusetts: The MathWorks Inc.,2010.

[23] Robert D Richtmyer and Keith W Morton. “Difference methods for initial-value prob-lems”. In: Malabar, Fla.: Krieger Publishing Co.,— c1994, 2nd ed. 1 (1994).

[24] J Westerweel. “Fundamentals of digital particle image velocimetry”. In: Measurementscience and technology 8.12 (1997), p. 1379.

[25] Mary C Seiler and Fritz A Seiler. “Numerical recipes in C: the art of scientific com-puting”. In: Risk Analysis 9.3 (1989), pp. 415–416.


[27] JW Dold et al. “From one-step to chain-branching premixed flame asymptotics”. In:Proceedings of the Combustion Institute 29.2 (2002), pp. 1519–1526.

[28] G Joulin and G Sivashinsky. “Influence of momentum and heat losses on the large-scalestability of quasi-2d premixed flames”. In: Combustion science and technology 98.1-3(1994), pp. 11–23.

[29] Jingyi Zhu. “A numerical study of chemical front propagation in a Hele-Shaw flowunder buoyancy effects”. In: Physics of Fluids (1994-present) 10.4 (1998), pp. 775–788.

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[30] Philip Geoffrey Saffman and Geoffrey Taylor. “The penetration of a fluid into a porousmedium or Hele-Shaw cell containing a more viscous liquid”. In: Proceedings of theRoyal Society of London. Series A. Mathematical and Physical Sciences 245.1242(1958), pp. 312–329.

[31] P Aghalayam, PA Bui, and DG Vlachos. “The role of radical wall quenching in flamestability and wall heat flux: hydrogen-air mixtures”. In: Combustion Theory and Mod-elling 2.4 (1998), pp. 515–530.

[32] Craig M Miesse et al. “Submillimeter-scale combustion”. In: AIChE Journal 50.12(2004), pp. 3206–3214.

[33] Ahmet N Eraslan and Robert C Brown. “Chemiionization and ion-molecule reactionsin fuel-rich acetylene flames”. In: Combustion and flame 74.1 (1988), pp. 19–37.

[34] Timothy Wayne Pedersen. “Ionic structure of methane flames”. In: (1991).

[35] GJM Hagelaar and LC Pitchford. “Solving the Boltzmann equation to obtain electrontransport coefficients and rate coefficients for fluid models”. In: Plasma Sources Scienceand Technology 14.4 (2005), p. 722.

[36] AV Phelps and LC Pitchford. “Anisotropic scattering of electrons by N 2 and its effecton electron transport”. In: Physical Review A 31.5 (1985), p. 2932.

[37] JP Boeuf, LC Pitchord, and WL Morgan. Siglo cross sections database.

[38] S Pancheshnyi et al. “The LXCat project: Electron scattering cross sections and swarmparameters for low temperature plasma modeling”. In: Chemical Physics 398 (2012),pp. 148–153.

[39] DC Murphy, M Sanchez-Sanz, and C Fernandez-Pello. “An experimental and numer-ical study of flames in narrow channels with electric fields”. In: Journal of Physics:Conference Series. Vol. 557. 1. IOP Publishing. 2014, p. 012076.

[40] Jing Hu, Boris Rivin, and Eran Sher. “The effect of an electric field on the shape ofco-flowing and candle-type methane–air flames”. In: Experimental Thermal and FluidScience 21.1 (2000), pp. 124–133.

[41] Memdouh Belhi, Pascale Domingo, and Pierre Vervisch. “Direct numerical simulationof the effect of an electric field on flame stability”. In: Combustion and flame 157.12(2010), pp. 2286–2297.

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CHAPTER 4

MICROWAVE INTERFEROMETRY

4.1 Introduction

The electrical aspects of flames have attracted much interest over the years, but there re-mains considerable uncertainty regarding the processes responsible for the observed phenom-ena. Detailed physical models are crucial in developing a clear understanding of these effects,but they must be supported by experimental measurements. Specifically, the rates of thechemical kinetics processes responsible for flame chemi-ionization must be determined accu-rately. Free electrons, the primary negative charge carrier in flames, have proven difficult tomeasure accurately and unobtrusively.

Outside of combustion, in the general study of plasmas, numerous techniques have beendeveloped to characterize the electrical properties of ionized media. A classic tool is theLangmuir probe, which measures the current vs. voltage properties of a conductive probeimmersed in plasma. It has the potential to measure multiple properties, including elec-tron number density and temperature. This method does, however, necessitate the use ofan intrusive probe which may physically and electrically disrupt the plasma, introducingsystematic measurement error. Further, the operation of Langmuir probes at high speedspertinent to combustion chemistry has proven to be technically challenging and remains anactive topic of research [1, 2]. A great many of these techniques, such as Stark and Zeemanspectroscopy, or Faraday rotation [3, 4], would seem to be effective but require intense electricor magnetic fields to function. In theory, Thomson scattering may directly measure electronsbut its low sensitivity generally requires both high electron densities and high powered lasersto function.

In the study of plasmas, interferometric techniques have been been applied to shocktube studies of thermal ionization in inert gases [5–9]. These efforts generally involvedexceptionally high ionization rates relative to combustion conditions and could make useof visible wavelengths with photographic recording. Work by Kelly and Harwell at the

53

California Institute of Technology made use of microwave interferometry for comparativelycool plasmas. Even that work was beyond the scales of combustion chemi-ionization, but issimilar enough to inform the testing and validation of combustion diagnostics.

As a result of these challenges, techniques relying on radio and microwave frequencies,which can be both sensitive and non-intrusive, have been applied to examine the electricalproperties of plasmas and to clarify the underlying chemical processes. Notable efforts haveparticularly centered on the use of absorption radio waves to measure free electrons[10, 11]and somewhat more conventional techniques to detect heavy ions [12, 13]. While microwaveabsorption can be effective in some cases, the low electron densities found in combustionnecessitate the use of quite long wavelengths, thus compromising spatial resolution. Onegroup sought to improve upon absorption techniques through an interferometric method[14], which offered a significantly higher sensitivity at a given wavelength. They developed amicrowave frequency (19 GHz) interferometer for use in a shock tube and used it to evaluateionization in a number of different chemical reactions. However, their method relied upon atransmission wire pair passing radially through the shock tube, the effect of which has notbeen evaluated.

Motivated by a desire for highly sensitive measurements of chemi-ionization kinetics andinspired a by diagnostic tool developed for the study of ion-thrusters [15], we have devel-oped a fully non-intrusive microwave interferometer (MWI) for use in shock tube studiesof combustion kinetics. Bellow, we introduce the theory and operating parameters of thisinterferometer. This includes validation and calibration of the system by measurement ofthermal ionization of argon. Primarily, we present experimental measurements of free elec-trons in shock tube oxidation of hydrocarbon fuels and compare those measurements topredictions from a modified ARAMCO 1.3 [16, 17] mechanism with chemi-ionization reac-tions and demonstrate that this mechanism represents an significant improvement over aearlier efforts.

4.1.1 Plasma Dynamics

Flames and other combustion reactions involving the oxidation of hydrocarbon fuels containa small, but non-negligible, quantity of ions due to chemi-ionization and, to a much lesserextent, thermal dissociation. Such systems may be regarded as a plasma or a partially ionizedgas [4]. In practice, combustion is characterized by temperatures and degrees of ionization(θi the fraction of particles which are ions) that are relatively low when compared to themajority of work in the field of plasma physics. As such, we will generally find ourselveson the fringes of standard models and methods that have been developed. There are still agood number of tools developed as diagnostics for conventional plasmas that may be usedto characterize the chemi-ionization process found in combustion.

The presence of unbound electrons changes the complex conductivity (σ) of a gas orplasma. Even very low electron concentrations, such as those encountered in flames, can

54

cause changes in the conductivity large enough to alter the propagation of an electromagneticwave significantly. Specifically, electrons will alter both the phase and amplitude of any wavepassing through them.

Conceptually, the observed interactions between plasma and an incident electromagneticwave are the result of oscillatory motion of unbound electrons and collision of those electronswith heavy species (ions and neutrals). Oscillation of the electrons produces a secondaryelectric field with the same frequency as the incident field, but with a phase lag of 3π/4 orπ/4. The combined effect of the two fields produces a wave of apparently advanced phase.Collisions between electrons and heavy particles dissipates the energy imparted to them bythe incident electric field and causes a net reduction in the total field strength.This process can be more formally described by the equations of plasma dynamics. To arriveat an expression for wavenumber (k), we begin with Maxwell-Ampere equation:

∇2E−∇(∇ · E) = µ0∂J

∂t+ εµ0

∂2E

∂t2(4.1.1)

which is combined with an approximate form of Ohms law in a homogeneous plasma withnegligible magnetic field:

1

ν

∂J

∂t≈ σE− J (4.1.2)

From these two equations, along with the assumption of plane wave solutions have fre-quency ω and wavenumber k, we can arrive at the dispersion relation:

c2k2

ω=

[η2 − (ωp/ω)2

1 + (ν/ω)2

]+ i

[(ωp/ω)2(ν/ω)

1 + (ν/ω)2

](4.1.3)

having introduced η, the index of refraction due to neutral species, and ωp, the plasmafrequency (defined below).Conceptually, the plasma frequency describes the characteristicrate at which the plasma responds to electrical disturbance, which the the electron collisionrate, ν, describes the rate at which energy is exchanged between electrons and the bulkplasma. When an electromagnetic wave interacts with a plasma, its tendency to undergophase shifts and attenuation is determined by ratios of its frequency ω to ωp and ν. Thesetwo effects are described by the complex dielectric constant K = KR + iKI .

ωp =

(nee

2

εme

) 12

KR = η2 − (ωp/ω)2

1 + (ν/ω)2KI =

(ωp/ω)2(ν/ω)

1 + (ν/ω)2(4.1.4)

Variations in phase and amplitude resulting from passage through a plasma can be de-scribed in terms of the the attenuation and phase constants α and β, such that the resulting

55

1000 2000 3000 4000 5000 6000 7000 8000

Temperature (K)

10-14

10-13

Reduced Collision

Rate

Ar

C2H2

CH4

CO

CO2

H

H2

H2O

He

Kr

N2 N2O

Ne

NH3

NO

O

O2 Xe

Figure 4.1: Pre-computed reduced collision rates for a variety of atomic and molecularspecies.

wave is, E electric field varies as:

E = E0 e−αxei(βx−ωt) (4.1.5)

The appropriate reference condition for measurement of all these properties is that of avacuum. For which ωp = ν = (η − 1) = (K − 1) = α = 0. and β = β0 = ω/c. For a knownpath length l (in our case 142mm) the signal loss in decibels dB and the phase shift relativeto vacuum are:

dB = 0.868αl ∆φ = βl − β0l (4.1.6)

We have not yet discussed the electron collision rate ν in great detail. Its computation isnot entirely trivial due to the fact that the collision cross section Q depends upon the neutral(or ion) species and the velocity of the electron. At equilibrium, we assume a Maxwelliandistribution of electron energies and may compute the reduced collision rate ν/ntotal by:

ν

ntotal=

k∑i=1

ni

∫ +∞

0

vQi(v)f(v)dv (4.1.7)

for k species. For convenience, normalized collision rates, ν/nk may be precomputed as afunction of electron temperature. Such results computed using cross sections from [18] areplotted for a number of species in Figure 4.1.While the formulation presented above is valid for a wide range of plasma conditions, it is

more detailed than is strictly required for most conditions encountered in our experiments.

56

For modest electron densities and collision rates, such that ωp << ω and ν << ω, theexpression for phase shift may be simplified as:

∆φ =e2 l ne

2me c ω(4.1.8)

This relationship clarifies that phase shift increases with electron concentration and de-creases with frequency of the incident wave. It is the second point here which drives the useof microwaves, rather than infrared or optical wavelengths, to measure the low concentra-tions of electrons found in combustion.

4.1.2 Thermal Ionization

While the primary goal of the present work the is characterization of combustion, it foundto be useful to apply the microwave interferometer (MWI) to the measurement of plasmasformed by the thermal ionization of inert gas. This process is well understood at a theoret-ical level and has received extensive attention in both experimental [5, 9, 19] and numericalstudies [20, 21]. It provides a convenient and consistent means of producing ionized gaswhich may be compared against published data. Further, the process of inducing thermalionization in a shock tube is comparatively simple and may be easily performed over a widerange of conditions. The ability to produce extremely high ionization serves the secondarypurpose, beyond validation of the MWI, of producing phase shifts greater than 2π, the mea-surement of which can be used to characterize the intrinsic, non-ideal calibration parametersof the MWI components. In our discussion, we use argon as an example, because it was usedin our experiments. The process would be structurally identical for other noble gases andonly somewhat more complicated in the case of polyatomic gases.

Statistically, a macroscopic quantity of gas at any finite temperature is expected to havesome fraction its population in an ionized state. However, the energy required to ionize anatom is generally quite large and it is only at very high temperatures (> 5000K) that theseions are numerous enough to be detected easily. The computation of this fraction comesfrom Maxwell-Boltzmann statistics. This begins with a simple expression for ratio of thenumber of particles in any two given states (ni,nj):

ninj

= exp

(−(εi − εj)kBT

)(4.1.9)

This form is not directly useful for the determination of the total number of ionized atoms(and thus the number of free electrons) because it is limited to only two states. Both theneutral atoms and the ions themselves possess a large number of possible electronic states

57

and it is necessary to resolve the statistics of all states in the neutral atoms, ions and thefree electrons [22]. For the neutral atoms and the ions, in which these states are discrete andfinite in number, we sum over all possible states to arrive at the partition function (Z):

Z =∑j

giexp

−εikBT

(4.1.10)

where εi and gi denote the energy (relative to the ground) and degeneracy of each microstate.The accounting is slightly more complicated for free electrons, which may exist in a contin-uum of energy states. Their partition function is an integral over momentum (p) space,recognizing that εi = p2/(2me).

Ze(T ) =

∫ ∞0

ge(pe)exp

−p2

2mekBT

d3p

=2

ne~(2πmekBT )3/2 (4.1.11)

Given that the masses of neutral atoms and their ions are nearly identical (to within themass of a electron or two), we can safely neglect the differences in their partition functions.We will also only consider the first ionization state, such that ne = n+, due to the largeenergy gap (generally 10 eV) between the first and second ionization states. Under theseassumptions the ionization fraction, αI , can be computed:

fI =

[1 +

P h3

(2π me)3/2(kBT )5/2

ZINTN

ZINT+

exp

(εikBT

)]−1/2

(4.1.12)

where P, h,me, kB, T, ZINTN , ZINT

+ and ei are pressure, the Planck constant, the electronmass, the Boltzmann constant, temperature, electronic partition functions of the neutraland ion, and the ionization energy, respectively. For these modest temperatures, the ratio ofpartition functions for Ar and Ar+ is found in [23] to be relatively constant with temperatureat ∼ 0.189. Coupled with the ideal gas law, we arrive at an expression for the equilibriumnumber density of electrons ne in terms of pressure and temperature:

ne =P αI(T, P )

kB T(4.1.13)

[Note:] Care must be taken with comparison to conditions specified by Glass et al.[5].Their shock was not reflected, their conditions will not be those expected from the T5/T1 cal-culations used below. Also, the Saha equation corresponds to the equilibrium post-ionizationtemperature. The temperature immediately behind shock will be somewhat higher until ion-ization occurs and equilibrium is reached. For highly ionized gases (ne/n ∼ 0.1), such as

58

those used by Glass et al. this effect causes very large changes in temperature. For thepresent work, with ne/n ∼ 10−7, this effect could be included in the final analysis but canbe neglected in selecting experimental conditions. The equilibrium electron concentrationprovides a reasonable characteristic value for the post-shock conditions and we have used itto select our experimental parameters.

The process of ionization by a shock wave is a transient problem described by the re-distribution of kinetic energy among the electronic states involved. The initial compressionleaves the argon in a non-equilibrium state. Initially, the neutral-neutral ionization processAr +Ar ⇐⇒ Ar +Ar+ + e− is relatively slow. Thus, there is a brief induction time duringwhich the gas remains effectively neutral. Once a modest number of electrons have beenproduced, the chain-branching reaction Ar+ e− ⇐⇒ Ar+ + e−+ e− takes hold. This resultsin the so-called electron avalanche, which rapidly brings the electron concentration to near-equilibrium conditions. The subsequent decay of electrons is a relatively slow and will resultin a stable and quasi-homogeneous electron cloud for the MWI to measure.

4.2 Numerical Methodology

The great advantage of shock tube experiments from the perspective of chemical kinetics isthe simplicity with which they can be modeled. The conditions behind the reflected shockare homogeneous in the radial and azimuthal directions. Variation in the axial direction isgenerally small enough to be inconsequential (except for very fast reactions, as discussedin subsection 4.3.2). Thus, the problem becomes effectively zero-dimensional in space. Thefunction of the shock is to suddenly raise both the temperature and pressure of the gases as itpasses. It does so, traveling at 1000 m/s and being 1 µm thick, in a matter of nanoseconds:faster than any pertinent chemical reactions. The result is that the reactants are instantlybrought from a ”frozen” (effectively non-reacting) state to a highly energetic state wherereactions proceed rapidly. Thus, we may pose the initial conditions of the chemical reactionas possessing pure, unburnt reactants

It has been known for some time that hydrocarbon flames exhibit a degree of ionizationsignificantly higher than equilibrium values. This behavior had been identified to be theresult of intermediate reactions that decompose neutral species into a variety of ionizedcompounds, including unbound electrons. Early efforts in this field focused on catalogingpotential reactions leading to chemi-ionization and attempted to identify the most significantreactants involved in this process [24, 25]. The resulting consensus is that the reaction:

CH + O CHO+ + e− (4.2.1)

Is the major source of ionization while the subsequent reactions:

CHO+ + H2O CO + H3O+ (4.2.2)

59

H3O+ + e− Neutrals (4.2.3)

are primary paths to the consumption of free electrons. There are, in fact, a host of ion-related reactions that may be included to characterize the electrical properties of flames.

We have used the AramcoMech 1.3 reaction mechanism [16] as the primary basis for ourmodeling efforts herein. It provides a detailed chemical mechanism for the combustion of C1-C4 hydrocarbons, including 1542 reactions among 253 species, and has been validated againstexperiments using shock tubes, jet stirred reactors, flow reactors and burner-stabilized flames.Although the mechanism is generally very robust in many applications, its performance inthe prediction of CH was not particularly accurate. Given the importance of CH in thechemi-ionization, the existing neutral chemistry of the mechanism was modified to includethe formation of OH∗. Also, the rates for the reactions CH3 + OH → CH2(S) + H2O andCH2 +H → CH +H2 were modified. These additions improved the prediction of CH2 andCH to achieve better agreement with literature values[26].

This modified mechanism was then supplemented with an ion chemistry model based onthe structure proposed by Prager [27], but with reaction rates updated according to data inthe UMIST database [28]. The details of this mechanism are described by Han et al. [29],with the exception that improvements based on average dipole orientation (ADO) theoryhad not been incorporated at the time of these simulations.

The vagaries of shock tube experiments preclude the precise production of desired testconditions. Rather, one must target an intended T5 and P5 and then rely on measurementsafter the fact to determine exactly what conditions were reached. From these measurements,each individual shock tube experiment was modeled to produce electron number densityhistories equivalent to those measured with the MWI. These simulations were performed inCantera 2.1 [30] using a 0-D, homogeneous, perfectly-stirred reactor with constant internalenergy and volume (UV). Initial mixture compositions were those specified in Table 4.1.Pressures and temperatures in the post-reflected-shock region (P5, T5) are also given.

Some initial and supplementary simulations were performed using a similar mechanismbased on the neutral chemistry mechanism of GRI 3.0 [31]. While this mechanism is wellestablished in the combustion community, it is not intended for hydrocarbons larger thanC2. It has also received little validation for the combustion of acetylene, which is known tobe an excellent source of chemi-ions.

60

4.3 Experimental Methodology

4.3.1 Shock Tube

The experiments were conducted using the Low Pressure Shock Tube (LPST) facility at theKing Abdullah University of Science and Technology (KAUST)[32]. This electro-polishedstainless steel shock tube has an internal diameter of 14.2 cm and a driven length of 9 m. Thedriver section is composed of multiple sections to control test time. In these experiments,a driver length of 4 m was used. In all experiments helium was used as the driver gasand shocks were generated by the bursting of a thin plastic diaphragm. Bursting of thediaphragm was controlled by selection of its thickness and positioning of a cutting blade onthe driven side. This method, while reliable, introduced some variability into the speed of theshocks generated, making it difficult to produce exact replicates of any given experimentalcondition.

All experimental conditions (T5 and P5 at which reactions occur) given below are thosebehind the shock after reflecting off of the end wall of the shock tube. Pre-shock pressureswere measured using a Baratron pressure gauge. Incident shock velocities were measuredusing five PCB 113B26 piezoelectric pressure transducers read by four 350 MHz Agilent53220A frequency counters. Determination of the initial conditions was made by applyingthe ideal equations for incident and reflected shocks given in by Ben-Dor et al. [33], for theknown conditions T0, P0, and incident shock Mach number (Ms),

P5

P0

=

[2γ1M

2s − (γ1 − 1)

γ + 1

] [−2(γ1 − 1) +M2

s (3γ1 − 1)

2 +M2s (γ1 − 1

)

](4.3.1)

T5

T0

=[2(γ1 − 1)M2

s + 3− γ1][(3γ1 − 1)M2s − 2(γ1 − 1)]

(γ1 + 1)2M2s

(4.3.2)

Uncertainty in the following test conditions is primarily driven by error in the shockMach number, initial pressure, and initial temperature, for which the uncertainties were ±0.28, ± 0.665 Pa, and ± 1 K, respectively. Generally, in combustion research, it is sufficientto account for the Ms error only, as the other two quantities contribute very little the finalresult. For thermal ionization, however, very high values of Ms are required and the errorsin T0 are amplified to the point of being quite significant. For combustion conditions used(T5 ∼ 2000 K, P5 ∼ 31,000 Pa), the final uncertainties were ± 15 K and ± 300 Pa, while,for thermal ionization (T5 ∼ 7500 K, P5 ∼ 36,000 Pa), they were ± 60 K and ± 300 Pa. Toillustrate the significance of these errors, they may be applied to simplified models of thechemical reactions to be studied. For combustion, we use the rate constant k from a shocktube global reaction rate proposed by Westbrook and Dryer [34]. For argon ionization, asimple Arrhenius expression using the first ionization state as an activation energy will sufficeat this level of accuracy. Distributions of expected rate constants were computed by uniform,

61

0.985 0.990 0.995 1.000 1.005 1.010 1.015

Normalized Reaction Rate (k/kave)

0.1

0.2

Frequency

Combustion Conditions

UncertainVariable

MS

MS ,P0

MS ,P0 ,T0

0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25

Normalized Reaction Rate (k/kave)

0.05

0.1

Frequency

Thermal Ionization Conditions

UncertainVariable

MS

MS ,P0

MS ,P0 ,T0

Figure 4.2: Approximate variation in reaction rates due to uncertainty in experimental con-ditions for combustion conditions and thermal ionization of argon. Each dataset computedfrom 24,000 random samples within the listed uncertain input values.

random sampling of the uncertain values (24,000 points per trial) and applying the resultingT5 and P5 values. The results from these computations are shown in Figure 4.2. From thesecomputations, we expect that deviations from nominal conditions will produce an error of± 1.5 % in reaction rates for combustion, which is quite manageable and ± 21 % in thethermal ionization cases, which is more problematic.

4.3.2 Test Conditions

The intent behind the selection of gas mixtures and test conditions, temperature and pres-sure, was to maximize the detectable electron concentrations and to probe the key chemi-ionization reactions responsible for the production of electrons. Doing so primarily involvedstriking a balance between increasing reactivity and producing controlled and measurablereactions. The process was significantly guided by estimates provided by chemical kineticssimulations of shock experiments using the model described in section 4.2. Figure 4.3 showsrepresentative data from one of the parametric studies conducted. These simulations consis-tently indicated that it was possible to increase peak electron number density by increasing:the concentrations of the reactants, temperature, pressure and, to a lesser extent, the reac-tivity of the mixture (as controlled by fuel-air ratio Φ = 1). However, these changes carrythe inconvenient side-effect of increasing the rate of reaction.

Fast reactions are problematic because of the inherent relationship between time and

62

1µs 10µs 100µs0

5 ·1017

1 ·1018

Electron Number Density

(m−

3) 0D Reactor

C2H2 +N2O 1.2 bar

0.5 1 1.5

Φ

2800

2300

1800

T(K

)

Figure 4.3: A parametric study of the effect of temperature and stoichiometry on the timeevolution of electron number density in the reaction of acetylene with nitrous oxide. Com-putations were performed using Cantera[30] using a 0-dimensonal, constant volume/temper-ature reactor. The chemical mechanism was GRI 3.0 [31], augmented with ion-chemistry

space in a shock tube. Although the arrival of a shock at a point causes an almost instan-taneous change in the pressure and temperature, the transit of the shock is certainly notinstantaneous. Points farther from the end wall are heated slightly later than those nearerand will undergo reaction slightly later, producing a degree of spatial inhomogeneity in themixture. This is not generally an issue for laser diagnostics, which use very narrow beamsacross which the shock passes very quickly. The 1 cm beam width of the MWI poses moreof a problem. For the shock speeds used here (1000 to 2600m/s), reaction events less than10µm would not be uniformly distributed across the beam and could not be correctly re-solved. Even if the beam width concern had not been present, there would have also beenlimitations due to the potential for detonation to occur in the rather energy-dense mixturesdesired.

The practical balance reached is evident in the test conditions reported in Table 4.1. Themost reactive mixtures (e.g. stoichiometric acetylene) were tested at very low temperatures(∼1300 K) while less reactive mixtures (e.g lean methane) could be subject to fairly hightemperatures (up to 2027 K). The reported conditions are for those tests which producedignition delay times in the range 100 µs-2.5 ms.

At the request of our colleagues developing chemical mechanisms, fuel-oxygen mixtureswere chosen to provide variation in the relative quantities of the intermediate chemical species

63

responsible for chemi-ionization. Use of a single fuel or stoichiometry would have limitedthe potential for improving distinct reaction rates, making the mechanism robust across awide range of conditions. The basic tool to produce meaningful differences was control ofthe ratio of hydrogen to carbon present in the fuels. Light hydrocarbon fuels from methane(H/C = 4) to acetylene (H/C = 1) provided excellent coverage of the ratios expected inmost practical fuels. As indicated in Equation 4.2.1, CH radicals play a critical role in theinitial formation of free electrons, such that both carbon and hydrogen must be present forsignificant ionization to occur. Separately, we note that the subsequent consumption of thoseelectrons through Equation 4.2.2 and Equation 4.2.3 depends on the presence of H2O, whichcan be produced in the absence of carbon.

As the hydrogen/carbon ration increases, amount of H2O produced naturally increases.Additionally, greater quantities ofH2O become available early in the reaction process and cancontribute to the electron capture process. Although the full chemical process is significantlymore complicated, peak concentration of electrons is expected to increase as H/C decreasestoward unity and that the strength of this dependence will be sensitive to the rates of thechemi-ionization reactions.

64

Mixture # 1 2 3 4 5 6 7 8 9 10Component CH4 O2 CH4 O2 CH4 O2 C2H4 O2 C2H2 O2 C3H8 O2 CH4 O2 CH4 O2 CH4 O2 ArNominal % 2 8 2 4 2 2 1 3 1 2.5 0.66 3.33 0.75 3 2 5.33 2 2.66 100Actual % 2 8.03 2 4 2 3.25 1 3.04 1 2.5 0.66 3.33 0.751 3.01 2 5.33 2 2.68 100

T(K) P(bar) T(K) P(bar) T(K) P(bar) T(K) P(bar) T(K) P(bar) T(K) P(bar) T(K) P(bar) T(K) P(bar) T(K) P(bar) 5278 1.031498 1.21 1898 1.24 1697 1.07 1416 1.14 1513 1.03 1346 1.18 2699 0.95 1619 1.23 1896 1.5 5026 1.171494 1.14 1694 1.2 2027 0.98 1464 1.33 1173 0.82 1335 1.11 1885 1.16 1618 1.13 1621 1.2 4971 1.311610 1.16 1605 1.24 1843 1 1411 1.32 1312 1.08 1356 1.1 2241 0.99 1731 1.095 1802 1.26 5156 1.081725 1.15 1759 1.49 1552 1.07 1375 1.45 1247 1.14 1265 1.17 2415 1.35 1915 1.1 1886 1.206 5232 0.9891816 1.13 1576 1.28 1597 1.06 1241 1.33 1120 1.25 1408 1.08 2750 0.96 1576 1.23 1587 0.9 5901 0.911949 1.13 1929 2.8 1276 1.54 1445 1.06 1697 1.53 2016 1.27 5901 0.911539 1.46 1763 2.96 1266 1.51 1328 0.85 2023 1.19

1630 3.14 1350 1.4 1493 1.031883 1.21785 1.411562 1.26

Table 4.1: Gas mixtures used in the shock tube chemi-ionization experiments. Temperatures and pressures are thosecomputed to be behind the reflected shock based on initial pressure (P0) and shock Mach number Ms

65

The full accounting of the gas mixtures used is given in Table 4.1. Acetylene, ethylene,propane and methane were all reacted with oxygen at stoichiometric conditions. The con-centration of fuel was adjusted for each fuel such that number of moles of carbon atoms permole of the mixture remained constant at 0.02, in order to maintain comparable potential forthe production of electrons. Additional experiments were conducted with methane at twolean fuel-air rations (Φ = 0.5, 0.66) and two rich fuel-air ratios (Φ = 1.25, 1.5). A mixtureof more dilute methane was tested in the hopes of comparing to work by [17], although theresults of these tests suffered from severe noise.

In all cases, oxygen was used as the oxidizer and argon was used as a bath gas. Eachtest mixture was initially formed in a 24 L magnetically stirred mixing vessel. Gases wereadmitted to achieve the desired composition by the method of partial pressures and werestirred for a minimum of 1 hour prior to use.

4.3.3 Microwave Interferometer

The Microwave Interferometer (MWI) is fundamentally equivalent to an optical Mach-Zehnder, with differences chiefly arising from the practicalities of generating, directing andmeasuring microwaves. All components of the MWI were manufactured and tested by Mil-litech Inc. The system is diagrammed in Figure 4.5, which identifies the primary componentsof the system, all of which were connected by electroformed WR-10 waveguides. Function-ally, the MWI begins at the Gunn diode oscillator (O), which generates the 94 GHz signalused throughout. This signal is then amplified (A) and then split at a short slot hybridcoupler (C), producing two signals; a test wave, to be transmitted through the plasma, anda reference wave, to be routed around the plasma. The test wave is transmitted through theplasma and received on the opposite side by a pair of identical 20° beam-width scalar feedhorn antennae separated by 157 mm. The received test wave then passes through an irisbandpass filter (94 GHz center frequency 1.06 GHz bandwidth) before entering the mixer(M). The reference wave is routed, via waveguides, around the test region (the shock tube)to reach the mixer. The power of the reference wave is controlled by passing through afixed amplifier (A) followed by a variable attenuator (At). The test and reference waves arerecombined at the mixer to measure their relative phase and amplitude.

Due to differences in path length and intervening materials, there will always be somephase difference between the test and reference waves. This initial shift is simply taken as anew zero condition, such that the electric fields (E) test and reference waves may be writtenas:

~Eref = 1/2Einiteiωt (4.3.3a)

~Etest = 1/2Einiteiωt+∆φ (4.3.3b)

In the mixer, the test signal and reference waves are each split to create two outputsignals. The first is the result of the test wave being modulated by the reference wave. The

66

(a)

(b)

Figure 4.4: a) Photograph of the MWI in operation on the shock tube. b) Rendering of theMWI mounting structure with a detail view of the bushing used to secure the horns to theshock tube.

second is identical except that the reference wave is first phase shifted by π/2. These aretermed I and Q and correspond to a measure of real and imaginary components of Etest·ERef ,such that:

I = Re(~Etest · ERef

)= | ~Etest|| ~Eref |cos(∆φ) (4.3.4a)

Q = Im(~Etest · ERef

)= | ~Etest|| ~Eref |sin(∆φ) (4.3.4b)

In practice, the measured I and Q channels exhibit a number of non-ideal behaviors:differing sensitivities (AI ,AQ), non-zero voltage offsets (I0,Q0) and a phase mismatch (δφ).

By introducing these terms, including | ~Eref | and a normalization constant for | ~Etest| in the

67

A C

A

At

MIs I

QO:A:C:Is:

M:At:

I:Q:

MixerAttenuatorIn-PhaseOut-of-Phase

OscillatorAmplifilerCouplerIsolator

ShockWTube

O

WReferenceWWave

Figure 4.5: Simplified Diagram of the microwave interferometer, demonstrating basic struc-ture and operating principles.

sensitivity factors AI and AQ it is possible to form a more accurate model for I and Q:

I = | ~Etest|AIcos(∆φ) + I0 (4.3.5a)

Q = | ~Etest|AQsin(∆φ+ δφ) +Q0 (4.3.5b)

which is plotted in Figure 4.6 to clarify the meaning of these parameters. Figure 4.6 alsoprovides a vector representation of two I/Q signals and the phase difference between them.

Theoretically, it should be possible to recover our primary quantity of interest usingonly one of these, if attenuation of the test beam is small (which was the case in all of thecombustion experiments). In practice, asymptotes in the derivatives of inverse trigonometricfunctions introduce unacceptable sensitivity to noise and require that both equations beused. With some manipulation, Equation 4.3.5a and Equation 4.3.5b may be combined andinverted to determine both phase shift and amplitude.

∆φ = tan−1

[ Q−Q0

AQ− I−I0

AI− sin(δφ)

I−I0AI

cos(δφ)

](4.3.6a)

| ~Etest| =

√(I − I0

AI

)2

+

(Q−Q0

AQcos(δφ)− I − I0

AItan(δφ)

)2

(4.3.6b)

As these new added parameters are the result of deviation from ideal and nominal values,they were not known in advance and needed to be determined experimentally in order to cal-ibrate the MWI. Conveniently, these equations describe elliptical patterns in I/Q space and

68

Q

I

∆φ

Q0

I0

δφ

AQ

AI

Figure 4.6: Phasor representation of I/Q signals, showing the improved mathematical modelMWI response and demonstrating phase shift from an arbitrary initial condition (teal) toanother (gold)

their values are bounded by the case of an unattenuated test beam. As such, if experimentalconditions can be produced which produce significant ∆φ with little to no attenuation, theresulting signals will lie entirely along the perimeter of the largest possible ellipse describedby the equations. Reed et al. [15] briefly discuss the process of determining parameters byfitting sinusoids to the I/Q signals produced by inducing phase shifts. In their work, theyhad the advantage of calibrated, adjustable phase shifters with which they could producesmall, well controlled shifts. They have also neglected to incorporate the phase mismatchterm, δϕ, in their equations, which inherently introduces a non-linear periodic error in thedetermination of ∆φ.

We have used data from our experiments on the shock induced-thermal ionization of argon(see subsection 4.4.1) as the source of data for the calibration for our present work. The earlystages of the ionization process produced an easily resolvable, gradual increase in the freeelectron population. Owing to the comparatively low collision cross section between argonand electrons, attenuation effects were minimal for the early stages of the ionization eventand the I/Q channels could be observed to trace multiple cycles around the ellipse beforebeginning to spiral inward toward I0, Q0. The free electrons began to produce sufficientlyhigh real conductivity in the plasma, that the test signal was reflected back at the side wallof the shock tube, leading to complete attenuation of the beam.

69

The fitting procedure was performed in Python using a Nelder-Mead simplex method [35]included in scipy.optimize[36]. To do this, a fitness function f(x) was formulated to measureaccuracy with which a given set of parameters fit the experimental data and conformed tothe expectation that all data points should lie within or on the boundary the ellipse. Thespecific form required for the algorithm is that f(x) should return a strictly non-negativescalar that increases as the quality of the fitting function decreases. The problem is to findthe values of AI ,AQ,I0,Q0,δφ that minimize f(AI , AQ, I0, Q0, δφ; I,Q), where I,Q are sets

of experimental data (n.b. for clarity, geometric vectors are given the notation ~X whileabstract vectors, such as data, are written X and the ith component of either is Xi). Wecan then make use of Equation 4.3.6b to evaluate the distance of experimental points fromI0,Q0. Those distances are assigned a fitness score by and arbitrarily defined function, find.Overall fitness is simply the sum of the scores.

f(AI , AQ, I0, Q0, δφ; I,Q) =∑i

[find

(| ~Etest|(AI , AQ, I0, Q0, δφ; Ii, Qi)

)](4.3.7)

where the individual fitness function, find, selected was

find(x) ≡

(2− 2e−100(x−1)2)(1− 2x2 + e(x−1)4) (x ≤ 1.15)

(2− 2e−100(1.15−1)2)(1− 2 · 1.152 + e(1.15−1)4) (x > 1.15)(4.3.8)

the form of which is shown in Figure 4.7a. In Figure 4.7b, we give a representation of thefitting procedure, in which a curve defined by initial estimates of the parameters (gold) isshifted to reflect new parameters (red) to better reflect data (teal) from the thermal ionizationof argon.

An additional practical matter to consider is that the MWI is quite sensitive to motion.Changes in the effective path length as small as 100 µm would produce spurious phaseshifts comparable to the true phase shifts induced by the presence of electrons generatedby combustion. Production and reflection of shocks is an energetic process that exertssignificant loads on the shock tube itself. To minimize stress on mechanical components, thetube is seated on rollers which permit small amounts of axial motion. Conventional opticaldiagnostics are normally mounted to an isolated optical below the test section. Any lasersor detectors are directed through windows in the sides of the tube, and there is no directmechanical connection between the optics and the tube itself.

Initial attempts to use the MWI as one would use an optical diagnostic were unsuccessfuldue motion of the shock tube during experiments. This produced tremendous noise in the Iand Q and made detection of chemi-ionization phenomena impossible. In order to minimizerelative motion this motion, the MWI assembly was mounted directly to the shock tube.Also, the horn antennas were cantilevered from the MWI support and were designed to floatfreely within ports in the shock tube side wall, providing no vibration damping. Vibrationisolation was added by replacing conventional side wall windows in the shock tube with boredplugs of Rexolite, a microwave compatible polystyrene. The plugs functioned as windows

70

0 1 2x

2

4

6

find

(a) Plot of the fitness function used for MWIcharacterization.

−0.2 −0.1 Q 0.1 0.2

−0.2

−0.1

I

0.1

0.2

(b) Example of a parameter ellipse fit to argonshock data.

Figure 4.7: Illustration of the data fitting process used to determine the intrinsic parametersof the MWI.

to seal the shock tube and as a bushing between the horns and the walls of the shocktube to guarantee a tight fit. This interface is illustrated in Figure 4.4b. This modificationsignificantly reduced the noise, making the MWI a viable diagnostic tool. That having beensaid, the I/Q signals detected in combustion reactions are still subject to significant noise.It remains unclear whether this is due to minute vibration still present, inhomogeneity inthe reaction front or any as yet unidentified sources of error.

4.4 Results and Discussion

4.4.1 Thermal Ionization

At the first level of analysis, one simply wishes to verify that the MWI is fundamentallyoperating as a means of detecting electrons in the shock tube. The thermal ionization of anoble gas, such as Argon, is a very consistent and reliable method of producing electronsfor this purpose. More importantly, the phenomenon has been studied previously and thereexist datasets and analysis against which MWI measurements may be validated. Primarily,the prior art has focused on very intense ionization, such that optical interferometry couldbe used [5, 6, 20, 37]. The work by Harwell [9] and Kelly [8] extends down to the relativelymodest temperature range of 5500 - 10000 K, providing conditions comparable to thoseproduced in the present. Figure 4.8 shows direct ne history from the MWI with somewhat

71

50 100 200

Time µs

1016

1017

1018

1019


(m−

3)

5278 K ,1.03 bar

5026 K, 1.17 bar

4971 K, 1.31 bar

5156 K, 1.08 bar

5232 K, 0.99 bar

5901 K, 0.91 bar

5375 K, 1.13 bar

5344 K, 1.12 bar

Kelly (1965) 8400 K, 0.6 bar

Harwell (1963) 8000 K, P1 = 0.07 bar

Figure 4.8: Comparison of ne measurements with the MWI to similar experiments fromliterature.

similar shock-induced ionization events reported in the literature.

A simple comparison of ne indicates reasonable behavior of the system, but is not partic-ularly meaningful, due to the effects of temperature and pressure on ionization rates. Evenwithin the MWI dataset, it is difficult to discern consistent behavior (e.g. the ne histories for5901 K and 5375 are nearly identical, despite the large difference in their initial conditions).Fortunately, Harwell formulated a basic model for the initial ionization rate of shock-heatedargon[9]. This analysis is partially based in theory, but also partially empirically derivedfrom his measurements. It is also limited in scope to relatively low levels of ionization, suchthat electron production is dominated by a single atom-atom collision reaction.

The purely theoretical component of the model comes from the kinetic theory of gases,and indicated that the rate of atom-atom collisions will scale with the average speed of theatoms v and the square of number density of atoms NA. Strictly, from here one should beable to construct an Arrhenius expression for the reaction rate, based on the atom-atomcross sections and the ionization energy [38]. In practice, Harwell found that an empiricallydetermined cross section Q, still having an Arrhenius from, but with arbitrarily determinedparameters, matched his results more effectively. The resulting expression for the rate of

72

ionization became:

ne = Q(T ) v(T )N2A = Q0exp

(−7.2eV

kBT

)[kBT

m

] [P

kBT

]2

(4.4.1)

For the purposes of comparing datasets, only the functional dependencies on temperatureand pressure are strictly important. One may group many of the constant terms (Q0,m,etc.) as well as unit conversions and arrive at a more compact expression that reduces therates to a nominal constant value, C.

exp

(−7.2eV

kBT

)P 2

n T 3/2= C (4.4.2)

Applying this transformation across datasets provides a level ground, so to speak, acrossa variety of temperatures and pressures. This method greatly improved the consistency ofthe MWI data but did not entirely remove the temperature dependence, as it did with therates given by Harwell. Values from Kelly, Harwell and the MWI are shown in Figure 4.9,a plot of the value C with respect to the inverse of temperature. Values from the MWIare consistently lower than the others, are certainly of the appropriate scale. Further, thereappears to be a temperature dependent trend to the MWI data, which crosses through therange of values given by Harwell. This may indicate that his proposed fit of Q(T ), does notextend well to the low temperature regime. It is also possible that the MWI experimentsmay have suffered minute amounts of contamination related to the combustion research forwhich the LPST is normally used. Harwell observed significant changes in ionization ratesfrom contamination as little as 10 parts per million. In as much as is possible, given the weakagreement among literature values and the potential of error unrelated to the MWI, thesethermal ionization experiments confirmed the function of the MWI as a tool for measuringfree electrons in shock tube and allowed the study of combustion to proceed.

4.4.2 Combustion

Figure 4.10 shows typical results from the shock-induced reaction of stoichiometric methaneand oxygen. At t=0, the incident shock arrives and produces a sudden rise in pressure andtemperature (to ∼ 0.3 bar and ∼ 970 K). This brief intermediate state does not increasereactivity sufficiently to cause any significant changes in the mixture (they remain effectively”frozen”). Very shortly thereafter shock passes the test location, having reflected off of theendwall of the shock tube, bringing the fuel-air mixture to the desired conditions. This pointis arguably the beginning of the experiment.

After the combustible mixture has reached sufficient conditions for chemical reactionsto proceed in earnest, there is some delay before ignition is observed (ignition delay time,τig). This period is characterized primarily by chain-initiation reactions (e.g. CH4 + O2 →CH3+HO∗2) decomposing relatively stable molecules into more reactive radical species. Once

73

1.25 1.5 1.75 2.0

104 /T(K)

1 ·10−17

2 ·10−17

3 ·10−17

C

MWI Data

Harwell (1963)

Kelly (1965)

Fit to MWI Data

Figure 4.9: Argon ionization rates adjusted according toQ,v,and nAr, as proposed by Harwell[9]

a sufficient pool of radicals develops, chain-branching reactions, in which radical-moleculecollision results in two or more radicals, become dominant and overall reactivity increasesexponentially and ignition occurs[39, 40]. This process is further accelerated as temperatureincreases due to the exothermicity of the combustion process. Ignition appears in Figure 4.10as a sudden rise in pressure and a spike in CH∗ chemiluminescence. The pressure rise is aresult of the energy release and remains after the combustion event has completed. CH∗,and electronically excited form of the more common CH radical decays after its peak asit and other intermediate species are consumed and the final products (primarily CO2 andH2O) are formed.

The rise and fall of the free electron number density, ne, is almost exactly coincidentwith the detected CH∗ chemiluminescence although very slightly delayed. This result washighly consistent across all experiments and was not, itself, coincidence. As discussed insection 4.2, the reaction of CH + O → CHO+ + e− is believed to be the most significantsource of electrons in hydrocarbon combustion. It is only natural that forms of CH be foundimmediately prior to the detection of electrons. The subsequent consumption of electronsrequires some intermediate reactions to occur and should be expected to occur at a somewhatmore languid pace than that of CH∗ and it does so. This sort of evaluation is not quantitativein the way that our subsequent comparison to simulations with detailed chemical mechanismscan be, but does provide a ready means of verifying that the results are consistent with thetheory.

Reported estimates of electron concentrations in combustion are in the range of 10−8-10−7 [41–43] for flames burning in air at atmospheric pressure. Measurements via the MWI

74

0 1 2Time (ms)

0

5e17

1e18

ne (m−

3)

T0 =1760K P0 =1.49 bar 2% CH4 4% O2

CH ∗(A.U.)

0

0.5

1

1.5

Pressure

(bar)

Figure 4.10: Experimental results from a single methane-oxygen combustion shock experi-ment. Combustion occurs at ∼ 0.7 ms, as evidenced by pressure rise, CH∗ luminescence.

are mostly consistent with this, with concentrations peaking around 10−7 for most of themixtures studied. These values are actually somewhat higher than expected, given that thefuel and oxygen used here are between four and eight times more dilute than they wouldbe in air. This concern is only slightly mitigated by the fact that the shock experimentsprimarily occurred above atmospheric pressure. A possible factor is that the experimentalmethods used had low spatial resolution or had a disruptive effect on the flame structure,although this is not supported by successful numerical replication of those results [44]. Ourown computations using the detailed mechanism for ion-chemistry, which is substantiallybased on measurements from flames, agree with the MWI results.

A more consistent explanation is that there is a fundamental difference in the devel-opment of electron concentrations in flames and shock tube experiments. While a shockexperiment is very nearly spatially homogeneous, flames vary across length scales deter-mined by the diffusion of heat and chemical species (usually on the order of millimeters).In the shock experiments, species only enter or leave the test region by chemical action.In flames, those same species are free to diffuse from regions of high concentration to low.Consider, for example the shock tube measurements by Chang et al. [45] which report peakCH3 concentrations around 3 · 10−4 while Kee et al. [46] report 2 · 10−3. After accountingfor dilution by a factor of 105 used by Chang et al.[45] one finds that peak concentrationsof the intermediate species to be roughly ten times higher in the shock tube than in flames,which is similar to what has been observed with electrons via the MWI. It is possible thatelectrons would be even more resistant to forming high concentrations in flames due to theirvery high mobility [47], although their electrical interaction with less mobile positive ionswould limit that tendency somewhat, as was studied by Sanchez-Sanz et al. [48].

While there is value in the above comparison as a high-level validation of the MWI results,

75

Figure 4.11: Electron number density from the stoichiometric combustion of methanefrom experiments (solid) and simulations using the enhanced AramcoMech 1.3 mechanism(dashed). Locations of peak electron density are marked and connected for each pair.

the primary purpose of these experiments is as a means to improve kinetics mechanisms.The essence of this process is to compare measurements from the MWI to predicted electronhistories from the simulations described in section 4.2. The complete set of experimentalelectron number density histories, along with their numerical equivalents simulated withthe enhanced AramcoMech 1.3, are given in Appendix D. In the interest of clarity, threerepresentative experiment/simulation pairs are reproduced in Figure 4.11 to show the typicallevel of agreement found.

4.4.2.1 Peak Electron Density

Direct, point by point comparison of the datasets is not particularly useful due significantnoise in the MWI measurements and rather large variation between the broader aspects ofkinetics simulations. A particular challenge is that the time at which combustion occursis rarely exceptionally well matched between models and experiments. This discrepancyoriginates in the existing neutral chemistry and is not affected by the addition of ion species,which generally make upO(100 ppb) of the total mixture. Reported experimental measures ofτig exhibit noticeable variability [32, 45, 49, 50]. Theoretical reaction mechanisms attemptto reproduce such experiments as faithfully as possible but are limited by the quality ofexperimental data, the breadth of conditions to be satisfied and the level of sophisticationused to define the mechanism [16, 31].

As indicated above, production of electrons was always observed to occur at ignitionand very closely followed a peak in CH∗ chemiluminescence, which is commonly used tomeasure τig. Therefore, the time at which electron concentrations occur provides an equiva-

76

100µs 1msExperiments

EnhancedAramco

1.3

2

3456

CH4 Φ =1

CH4 Φ =0.75

CH4 Φ =0.5

CH4 Φ =1 (Dilute)

C2H4 Φ =1

100µs 1msGRI 3.0

EnhancedAramco

1.3

2

3456

CH4 Φ =1

CH4 Φ =0.75

CH4 Φ =0.5

CH4 Φ =1 (Dilute)

C2H4 Φ =1

Figure 4.12: A comparison of ignition delay times, as measured by occurrence peak electronconcentration among experiments, enhanced 1.3 predictions and enhanced GRI 3.0 predic-tions.

lent measure of τig. Using this measure, values of τig were extracted from the experiments,enhanced AramcoMech 1.3 simulations and from additional set of simulations performedusing GRI 3.0 and the most current ion-chemistry model. The results in Figure 4.12 showsthat both models tended to under-predict τig. The greatest error is found for the dilutedmethane experiments, which were aimed at reproducing the conditions used by Karasevichand Yu[17]. Experiments for that mixture were conducted at significantly higher temper-ature than the others and were found to be quite variable, with many attempted shocksresulting in detonation rather than controlled combustion.

In the interest of decoupling the present analysis from details of the neutral chemistrymechanism we have focused on more tractable measures of the accuracy of the ion chemistrythat are independent of τig. The peak number density of electrons was one such quantityof interest. In addition to being readily identified in both experiments and simulations, thisvalue was expected to be naturally sensitive to the rate of Equation 4.2.1.

In the mechanism used, reactions 4.2.1 and 4.2.3 were the primary paths of electronproduction and consumption, respectively. Their rates of progress were defined by a modifiedArrhenius equation of the form

k = A0Tb exp

(−EaRT

)(4.4.3)

77

5 ·1016 1 ·1017 5 ·1017

Experiments

1 ·1017

5 ·1017

Enhanced

AramcoMech

1.3

Peak Electron Number Density

CH4 φ=0.5

CH4 φ=0.66

CH4 φ=1.0

C2H4 φ=1.0

C2H2 φ=1.0

C3H8 φ=1.0

CH4 φ=1.0 (Dilute)

Figure 4.13: Evaluation of peak electron number density as measured using the MWI andpredicted by the enhanced 1.3 mechanism. Square markers denote heavier fuels for which themechanism may not be completely accurate. Added reference lines, y = 1/2x, y = x, y =2x,

where A0, R and Ea are the frequency (or pre-exponential) factor, gas constant and activationenergy, which come from collision theory for particles having a Maxwell-Boltzmann distri-bution[51]. The additional term, T b, is an empirical artifact used to better fit experimentaldata. Simulations of methane combustion (the first test of mixture 2), were recomputedwith A0 increased and decreased by an order of magnitude for both reactions (separately).Peak ne values increased with A0 for reaction 4.2.1, varying by a factor of ∼

√3 for each

decade of A0. The result was less sensitive to reaction 4.2.3, with ne decreasing by 20 % andincreasing by 3 % as A0 was increased and decreased. Given that ne is much more sensitiveto production rates than consumption rates, it is primarily to tool for the estimation ofparameters in reaction 4.2.1.

The collected values of ne for the majority of the test conditions are presented in Fig-ure 4.14. The mechanism used is still under development at the time of this writing and ourcolleagues caution that its applicability for larger hydrocarbons and rich combustion con-ditions could be limited due the absence of the C3H

+3 ion which appear in large quantities

in rich flames [12, 52]. As such, we have omitted the two rich methane mixtures from this

78

1 2 3 4

GRI 3.0 /Experiments

1

2

3

4AramcoMech

1.3/Experiments

Peak Electron Concentration

CH4 φ=0.5

CH4 φ=0.66

CH4 φ=1.0

C2H4 φ=1.0

Figure 4.14: Comparison peak electron density between the two chemical mechanisms. Re-sults normalized by experimental values.

figure and have differentiated between the lighter and heavier fuels with circular and squaremarkers, respectively.

For ethylene and methane (stoichiometric and both lean conditions), experimental andnumerical values of ne agree to within a factor of 2. Also, the mechanism does not appear tohave any bias towards over-predicting or under-predicting, which would suggest the presenceof systematic error. The same cannot be said for either propane (C3H3) or acetylene (C2H2)for which the model consistently over-predicted and under-predicted ne, respectively. Inaddition to the aforementioned absence of C3H

+3 , the deviation here may be tied strongly

to the neutral chemistry. In the supplementary material released with the mechanism [16],comparison to methane and ethylene experiments receives approximately three times as muchdetail (each) as acetylene experiments do, while no comparison to propane experiments ispresented.

It is possible to further investigate the merits of the ion chemistry model by performing

79

equivalent analyses with different neutral chemistry models applied. This is shown in Fig-ure 4.14, where numerical results computed using 1.3 and GRI 3.0 are normalized by theirequivalent MWI measurements. Due to the limits of GRI 3.0, only results for methane andethylene are included. Unsurprisingly, the underlying neutral chemistry process stronglyinfluences the prediction of ne, with results from GRI 3.0 being consistently biased towardsover-prediction by a factor of 2 or more.

While measurements from the MWI are quite noisy, they provide useful data to select thebetter of two possible mechanisms for the simulation of chemi-ionization. In the future, asfurther sophistications are added to extend the applicability of the mechanism to acetylene,ethylene and rich conditions, the same analysis may be used to validate those improvementsand make full use of the present experiments.

4.4.2.2 Electron Consumption

In subsubsection 4.4.2.1, we discussed the sensitivity of ne to reactions and found that itwas relatively insensitive to the rate of electron combination. As a result, ne is a poor toolfor the evaluating the quality of electron capture reactions. Instead we have attempted toobserve the electron-capture process directly.

There are many paths by which the free electrons measured by the MWI can be lost [12,24, 52]. Even the most significant path, reaction 4.2.3, has been left with the intentionallyvague product: ”neutrals” to reflect that a variety of neutral species (H2O + H,OH + 2H)may result. There are also similar electron-cation reactions (e.g. C2H3O+e− → CO+CH3)as well as three body reactions, in which a third species is required to absorb the energyreleased upon electron capture (e.g. O2 + e− + O → O−2 + O). A full accounting of thesepaths involves far too many degrees of freedom to study in detail with a single diagnostic.Instead, we have utilized a simplified model of the recombination process. To do so, we limitthe scope of our analysis to a two-body recombination reaction having the form[53]:

ne = −nenCA0Tb exp

(−EaRT

)(4.4.4)

to give the rate of electron loss ne, using the rate constant from Equation 4.4.3 and thenumber density of some cation species nC .

Under ideal conditions, it would be reasonable to perform simple differencing of themeasured ne(t) values to recover ne and determine any unknown parameters. The severenoise and relatively low values of ne make that method, which generally exacerbates theeffect of noise, impractical in this case. Instead, we sought to describe ne(t) in a compactfunctional form informed by the underlying physics.

Without simultaneous diagnostics for their detection, we were unable to prescribe valuesfor nC . Within the effectively homogeneous reacting volume, charge must be conserved

80

50µs 100µs 150µs

0

2.5 ·1017

ne (m−

3)

Reacton : ne =−A0n2e

Fitting : ne (τ) =[A0 τ+ne (τ=0)

]−1

Figure 4.15: Late stage decay of electron populations in the combustion of acetylene. Ex-perimental measurements (solid) and fitted forms using Equation 4.4.6

such that, in the absence of multiply-charged ions, anions and cations must exist in equalquantities. If the status of electrons as the dominant (majority) anion species is extendedto the assumption that all anions in the mixture are electrons, the approximation nC ≈ neemerges. This was not meant to be a highly accurate model, only as a tool to constructa functional form that would be more physically meaningful than a naıve polynomial orexponential decay.

The process of electron-ion recombination is extremely favorable from a thermodynamicperspective, with ionization energies for most molecules being around 1 MJ/mole. There iseffectively no activation energy associated process required for the reaction to occur uponcollision. The exponential term in Equation 4.4.4 reduces to unity. The term T b is a reaction-specific empirical term with little meaning in the global sense and will be neglected. Underthese assumptions, we reduce Equation 4.4.4 to the much more tractable form:

ne = −n2eA0 (4.4.5)

At the later stages of the combustion process, the primary sources of electron productionare absent (note the rapid decay of CH∗ chemiluminescence in Figure 4.10) and one mayassume that pure recombination governs the decay of the electron population. So, from anypoint in this phase, tR, Equation 4.4.5 can be easily integrated to solve for ne. The resultmay be given in terms of τ = t− tR

ne(τ) =1

A0τ + ne(τ = 0)(4.4.6)

81

0.5 .81000/T(K)

1017

1018

Rate

Constant

(cm

3/mol−s)

Guo and Goodings

Prager

UMIST

Calcote′s Upper Limit

Calcote′s Lower Limit

CH4 φ=0.5

CH4 φ=0.66

CH4 φ=1.0

C2H4 φ=1.0

C2H2 φ=1.0

Figure 4.16: Electron decay rate constants. MWI observations (points) and literature values(lines).

The fitting procedure used subsection 4.3.3 to calibrate the MWI signals was similarlyapplied here to recover A0 and ne(τ = 0) from MWI data taken after the CH∗ disappeared.A constant offset, n0, was introduced to accommodate drift. Leaving ne(τ = 0) as a freeparameter, rather than using the first data point, is important to prevent an unjustified over-emphasis of the weight of that one measurement. The resulting values of the rate constant,A0, shown in Figure 4.16 with equivalent measures from multiple literature sources shownfor reference. Agreement across the MWI results was not particularly strong, with a factorof ∼4.5 between the highest and lowest values. No doubt, this was heavily related to noisein the MWI signals themselves. To estimate the inaccuracy introduced by the simplifiedrecombination method, the same fitting procedure was applied to the simulated datasets.The resulting estimates of A0 differed by as much factor of 1.67. As imprecise as thesemeasurements are, they fall well within the range of values anticipated from the literature,which vary by more than an order of magnitude. As such, these results serve to confirm thegeneral validity of the MWI, but may be of limited value in refining chemi-ionization models.

4.5 Conclusions

The microwave interferometer has been demonstrated as a non-intrusive method for measur-ing the free electrons in shock tube experiments. Thermal ionization of inert gases have beenused as reference subject for two purposes. First as a means of calibrating the MWI in place

82

on the shock tube, compensating for internal and external sources of error simultaneously.Second to provide experiments which could be compared against known results from existingliterature. These efforts were successful on both fronts, the MWI reliably produced electronmeasurements consistent with the prior art.

The driving purpose of the work was provide electron histories against which chemicalkinetic mechanisms can be validated. Mechanisms developed to date are reasonable in theirpredictions and, in fact, served as an effective tool to guide the selection of experimentalparameters for maximum sensitivity. Shock-induced combustion of multiple fuels and fuel-oxidizer mixtures have produced a substantial dataset to support modeling efforts. At thea basic level, electron production histories agreed very well with simultaneous detection ofCH∗, a key ion, strongly supporting successful detection chemi-ionization reactions, furthervalidating the method. More importantly, the electron histories were compared to comple-mentary chemical kinetics simulations using recent chemi-ionization models. Peak electronconcentrations between experiments and the most recent mechanism agree to within a factorof two for the fuels and mixtures appropriate for the model. An earlier model was shown toconsistently over-predict electron formation. Electron recombination rates compared reason-ably with those reported in the literature, although range of possible values is quite wide.

83

4.6 References

[1] SH Lam. “Unified theory for the Langmuir probe in a collisionless plasma”. In: Physicsof Fluids (1958-1988) 8.1 (1965), pp. 73–87.

[2] Robert B Lobbia and Alec D Gallimore. “High-speed dual Langmuir probe”. In: Reviewof Scientific Instruments 81.7 (2010), p. 073503.

[3] Ferdinand Hofmann and Gilbert Tonetti. “Tokamak equilibrium reconstruction usingFaraday rotation measurements”. In: Nuclear Fusion 28.10 (1988), p. 1871.

[4] Morton Mitchner and Charles H Kruger. Partially ionized gases. Vol. 8. Wiley NewYork, 1973.

[5] II Glass and WS Liu. “Effects of hydrogen impurities on shock structure and stabilityin ionizing monatomic gases. Part 1. Argon”. In: Journal of Fluid Mechanics 84.01(1978), pp. 55–77.

[6] II Glass, WS Liu, and FC Tang. “Effects of hydrogen impurities on shock structure andstability in ionizing monatomic gases: 2. Krypton”. In: Canadian Journal of Physics55.14 (1977), pp. 1269–1279.

[7] Paul H Wojciechowski and Helmut D Weymann. “Multistep initial ionization be-hind strong shock waves in argon”. In: The Journal of Chemical Physics 61.4 (1974),pp. 1369–1379.

[8] Arnold J Kelly. “Atom Atom Ionization Cross Sections of the Noble GasesArgon,Krypton, and Xenon”. In: The Journal of Chemical Physics 45.5 (1966), pp. 1723–1732.

[9] Kenneth Edwin Harwell. “Initial ionization rates in shock-heated argon, krypton, andxenon”. PhD thesis. California Institute of Technology, 1963.

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CHAPTER 5

CONCLUSIONS

As intended, this work has presented a set of experiments to support the ongoing studyof electrical effects in combustion. As appropriate, the experimental data have been sup-plemented with numerical modeling to aid in their analysis. The first two experimentalendeavors are of the macroscopic type described in chapter 1. They provide informationabout the overall response of flames to electric fields. While the analysis derived from themdoes not probe deeper than the heat and mass transfer of flames, they will serve as a ref-erence for future modeling efforts. They also highlight practical uses of electrical effectsin the control and miniaturization of combustion power systems. The last experiment, themicrowave interferometer, is well removed from real-world application but is immediatelyuseful to the detailed modeling chemi-ionization kinetics.

5.1 Flame Stability

Chapter 2 demonstrated the use of non-uniform electric fields as a means of controlling theinstability of a combustion system, without the need for mechanical baffles or actuators.Brief periods of electrical enhancement served to disturb flames, promoting wrinkling andinducing turbulent combustion in a geometry that would otherwise remain laminar. Uponapplication of sustained electrical enhancement, flames were forced to adopt an asymmetric,but laminar, propagation mode. In that mode, overall burning rate was increased relative toan axisymmetric flame but the transition to turbulence was completely suppressed. Thereis value in both scenarios. Enhancing turbulence in a combustor promotes faster mixing ofreactants and can increase its total heat release rate (power). On the other hand, practicalcombustors often exhibit undesirable instabilities that are difficult to suppress efficiently[1].Electric fields could be used to produce rapid, non-intrusive actuation of the flames to achievestable operation by passive or active control.

The first step in harnessing these observed effects is to understand how enhancementeffects interact with existing combustion processes. The numerical model constructed takes

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the classic combustion problem of an premixed flame prone to thermo-diffusive instability andintroduces external reaction rate modifications comparable to those in the experiments. Itwas found that, depending on their spatial disposition, these enhancements can serve to eitherquell or enhance the existing instability. As in the experiments, the formation of winkles inthe flame sheet was inhibited when they would result in a less efficient use to the externalenhancement. In addition to providing an interpretation of the present experiments, thismodel suggests that the broader study of inhomogeneous combustion enhancement effectsmay identify novel means of control.

5.2 Quenching

Flame quenching and propagation were the focus of chapter 3. The experiments have shownthat the application of external electric fields strongly affects both the quenching distanceand propagation speed of premixed flames in narrow channels. Further, this interactionwas found to be highly sensitive to the orientation of the field. Fields perpendicular tothe direction of propagation moderately impeded the reaction, while fields parallel to thedirection of propagation could vastly accelerate the reaction, for certain polarities. On arelative basis, more energetic flames were more sensitive to enhancement in this fashion.The only significant limitation of these results was that the enhancement produced was sogreat that, in some cases, quenching did not occur and quenching distance could not bemeasured directly.

Comparison to a theoretical model based on these experiments showed partial agreement.Both show that positive, parallel electric fields increase flame propagation speed (decreasingquenching distance) and indicate that more energetic fuel-air mixtures should have a greaterresponse to electrical effects. The two diverge in that the experiments found no upper limit onthe degree of enhancement possible, as was predicted by the model. Nor did the experimentsfind that negative fields reduced propagation appreciably. These two points of disagreementmay be partially explained by Ohmic of the flame sheet, which is non-negligible for the fieldstrengths and electron concentrations involved in the experiments.

5.3 Microwave Interferometry

In chapter 4, the microwave interferometer was demonstrated as a non-intrusive method formeasuring the free electrons in shock tube experiments. Thermal ionization of inert gases wasused as reference subject for two purposes. The first was to provide a means of calibratingthe MWI in place on the shock tube, compensating for internal and external sources of errorsimultaneously. The second to provide experiments which could be compared against knownresults from the literature. These efforts were successful on both fronts, the MWI reliablyproduced electron measurements consistent with the prior art.

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The driving purpose of the MWI was to provide electron histories against which chemicalkinetic mechanisms can be validated. Mechanisms developed to date are reasonable in theirpredictions and, in fact, served as an effective tool to guide the selection of experimentalparameters for maximum sensitivity. Shock-induced combustion of multiple fuels and fuel-oxidizer mixtures have produced a substantial dataset to support modeling efforts. At thea basic level, electron production histories agreed very well with simultaneous detection ofCH∗, a key ion precursor, strongly supporting successful detection chemi-ionization reac-tions and further validating the device. More critically, the electron histories were comparedagainst complementary chemical kinetics simulations using recent chemi-ionization models.Peak electron concentrations found in experiments and via the most recent mechanism agreeto within a factor of two for the fuels and mixtures appropriate for use with that mechanism.An earlier model was shown to consistently over-predict electron formation. Electron recom-bination rates compared reasonably with those reported in the literature, although range ofboth detected and reported values is quite wide.

5.4 Future Work

All three of these topics are ripe for further investigation in with experiments and modeling.Flame instability modification could be tested in more detail with active, closed-loop controlmethods to minimize power requirements and maximize stability. Modeling of that processcould be extended to a wider class of problems and should include more concrete definitionsof the reaction enhancement effects involved.

It has already been noted that the V-channel quenching apparatus was unable to observequenching at the very small scales believed to be accessible with electric field. A newexperiment, designed to test quenching below the millimeter scale, could more throughly anddirectly explore the limits of electrically-assisted microcombustors. It may also be possibleto apply electric fields directly to existing microcombustor designs. Modeling of this processwould, ideally, be extended to include heat loss terms across the flame, to resolve quenchingbehavior. The importance of Ohmic heating should not be overlooked in future efforts,provided a manageable approximation of the electron collision terms can be developed.

Some existing data from the MWI are simply waiting for models capable of simulatingtheir conditions (e.g. for rich mixtures). It would also be quite reasonable to conduct new setsof shock tube experiments. New fuels or mixtures of those already used could certainly be ofinterest, as would the addition of simultaneous laser diagnostics, to measure other chemicalspecies. Future experiments could also attempt to identify mixtures able to produce higherelectron densities, to improve the signal-to-noise ratio of the data. As with the initial designof the shock tube experiments, doing so will require a careful balance between signal qualityand the physical limitations of the equipment involved. In any case, additional data willonly serve to strengthen the foundation of future modeling efforts.

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5.5 References

[1] Michel Cazalens et al. “Combustion instability problems analysis for high-pressure jetengine cores”. In: Journal of Propulsion and Power 24.4 (2008), pp. 770–778.

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Appendix A

Flame stability code

This is an example input file of FreeFEM++ code used to study the effect of spatially non-uniform reaction enhancement effect on the development of flame front wrinkling due to athermodiffusive instability.

bool read = 1 , write = 1 , bl=1, PerInp=1;string rdname = ”D 1 33PerGrFn” , wrname = ”D 1 33PG0 0 tight ” ;

real dti=30, u = 0.948385 , sig = 0 , alpha=0, d=1,tm=0;real Li = 1.66 , beta=10.0 , btsq=beta∗beta , gamma=.8 , theta = 0 . 5 ;i n t n=3000 , nls=3;real [ i n t ] loc (n ) ,t (n ) ;

//Mesh Remappingreal Ly=35.5;real a= 1.29224038 , ai= −0.0437077 , c= 34 .353974 ;real a2= 0.99654768 , ai2= 37.65137063 ;func map = a∗tan (2∗ ( x− .5) ∗( pi/2− .02) )+ai ∗2∗(x− .5)+c∗pow (2∗ ( x− .5) , 3 ) ;func map2 = a2∗tan (2∗ ( x− .5) ∗( pi/2− .02) )+ai2∗pow (2∗ ( x− .5) , 3 ) ;

i n t nx=100 ,ny=100;mesh Th = square (nx , ny , [ map2 , Ly ∗(y− .5) ] ) ;mesh TUh = square (nx , ny , [ map , Ly ∗(y− .5) ] ) ;

mesh Rh = square (nx , ny ) ;

fespace Vh (Th , P2 , periodic=[ [1 , x ] , [ 3 , x ] ] ) ;Vh T , To , TT , C , Co , CC , K , Ko , Vect ;fespace Sh (Rh , P2 , periodic=[ [1 , x ] , [ 3 , x ] ] ) ;Sh Pa , Pb ;

func k = 0.5∗ Li∗btsq∗exp ( beta ∗(T−1)/(1+gamma ∗(T−1) ) ) ;func ko = 0.5∗ Li∗btsq∗exp ( beta ∗(To−1)/(1+gamma ∗(To−1) ) ) ;

func textent = 1 .0∗ ( x>0)+0;i f ( ! read )

// func t i n i t= 0 . 5∗ ( tanh (x )+1) ;

92

func tinit= 0 .5∗ ( tanh (x+0.000∗sin (y ∗ . 3 ) )+1) ;T=tinit ;To=tinit ;C=1−tinit ;Co=1−tinit ;K=k ;

e l s e ifstream hh ( rdname+” . txt ” ) ;i f ( PerInp==1)

fespace UUh ( TUh , P2 , periodic=[ [1 , x ] , [ 3 , x ] ] ) ;UUh Ti , Ci ;hh >> Ti [ ] ;hh >> Ci [ ] ;T=textent ;C=1−textent ;T=Ti ;C=Ci ;

e l s e hh >> T [ ] ;hh >> C [ ] ;

To=T ;Co=C ;

// Introduce pe r tu rba t i on s by convect ing the e x i s t i n g s o l u t i o ni f ( bl ) func prt = 0.02∗ sin ( .179008∗ y ) ;

Vect=prt ;Co=convect ( [ Vect , 0 ] , 1 , Co ) ;To=convect ( [ Vect , 0 ] , 1 , To ) ;C=convect ( [ Vect , 0 ] , 1 , C ) ;T=convect ( [ Vect , 0 ] , 1 , T ) ;

Pa [ ]= T [ ] ; plot (Pa , cmm=” t = 0” ) ;

problem Temp (T , TT )=// B i l i n e a rint2d ( Th ) (dti∗T∗TT

+theta ∗(dx (T ) ∗dx ( TT )+d∗dy (T ) ∗dy ( TT )+u∗dx (T ) ∗TT ))// Linear+int2d ( Th ) (−dti∗To∗TT

−theta ∗(1.0−sig ) ∗C∗K∗TT

+(1−theta ) ∗(dx ( To ) ∗dx ( TT )+d∗dy ( To ) ∗dy ( TT )−(1.0−sig ) ∗Co∗Ko∗TT+u∗dx ( To ) ∗TT )

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)+on (4 , T=0)+on (2 , T=1);

problem Con (C , CC )=int2d ( Th ) (dti∗C∗CC

+theta ∗(Li∗dx (C ) ∗dx ( CC )+Li∗d∗dy (C ) ∗dy ( CC )+C∗K∗CC+u∗dx (C ) ∗CC ))

+int2d ( Th ) (−dti∗Co∗CC

+(1−theta ) ∗(dx ( Co ) ∗dx ( CC )+d∗dy ( Co ) ∗dy ( CC )+Co∗Ko∗CC+u∗dx ( Co ) ∗CC ))+on (4 , C=1)+on (2 , C=0);

//Th=square (100 , 100 , [map , Ly∗(y− .5) ] ) ;

real loc2 ;i n t plt=100;real [ i n t ] yy ( plt ) , frnt ( plt ) ;ofstream ff ( wrname+”Data”+” . txt ” ) ;

f o r ( i n t jj=0; jj<n ; jj++)Ko=ko ;f o r ( i n t kk=0; kk<nls ; kk++)

K=k ;Con ;Temp ;

loc [ jj ]=int2d ( Th ) (x∗T∗C ) /int2d ( Th ) (T∗C ) ;tm+=1.0/dti ;t [ jj ]=tm ;To=T ;Co=C ;i f ( ( jj+1)%100==0)Pa [ ]= T [ ] ; plot (Pa , cmm=” t=”+jj/dti ) ;i f ( jj%2)u−=.5∗dti ∗( loc [ jj ]−loc [ jj−1]) ;i f ( jj%100==0)

ff<<tm<<endl ;Ko=C∗K ;loc2=int2d ( Th ) (x∗Ko ) /int2d ( Th ) ( Ko ) ;f o r ( i n t rr=0; rr<plt ; rr++)

x=loc2 ;y=Ly ∗ (1 . 0∗ rr /(1 . 0∗ plt ) −.5) ;yy [ rr ]=y ;frnt [ rr ]=Ko ;ff<<frnt [ rr]<<endl ;

94

cout<<” j j = ”<<jj<<endl ;

i f ( write )

ofstream gg ( wrname+” . txt ” ) ;gg << T []<<endl ;gg << C [ ] ;gg << To [ ] ;gg << Co [ ] ;savemesh (Th , wrname+” .msh” ) ;

cout<<”Speed = ”<<u<<endl ;

Pa [ ]= T [ ] ;plot (Pa , fill=true ) ;

\endverbatim

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Appendix B

Image Analysis Code

B.1 Flame Tracking with Background Subtraction

The flame tracking method used in chapter 2. Under the assumption that the flames areluminous and transient, the code begins by generating background image corresponding tothe minimum intensity observed at each location throughought the video. This image issubrtracted from each frame, resulting images of the flames only.

The flame is detected by intensity thresholding to form a binary image, and enmeratingall contiguous objects and selecting the largest by area.

c l e a rsplee=d i r ( 'T∗ ' ) ;doc=struct ( ' drct ' ,1 , 'name ' ,1 , 'Cx ' ,1 , 'Cy ' ,1 , 'Bx ' ,1 , 'By '

,1 , 'Bdx ' ,1 , 'Bdy ' ,1 , ' frame ' ,1 ) ;f o r n=3: l ength ( splee )

cd ( splee (n ) . name )splaa=d i r ( ' ∗ . TIF ' ) ;%sample f o l d e r f o r background imagesf o r j=1:4

A=imread ( splaa (3+(j−1)∗30) . name ) ;i f j==1

Filter=A ;e l s e

Filter=min( Filter , A ) ;end

end%generate background f i l t e rimwrite ( Filter , ' S t a tF i l t e r .bmp ' )f o r k=3: l ength ( splaa )

F=(imread ( splaa (k ) . name )−Filter ) ∗3 ;imwrite (F , strcat ( splaa (k ) . name ( 1 : 6 ) , ' . jpg ' ) , 'JPEG ' )B=im2bw (F , . 1 0 ) ;[ L , N ]=bwlabel (B ) ;R=regionprops (L , 'Area ' , ' Centroid ' , 'BoundingBox ' ) ;i f N˜=1

[ x , y ]=max ( [ R ( : ) . Area ] ) ;

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R=R (y ) ;enddocc=struct ( ' drct ' ,pwd , 'name ' , strcat (

splaa (k ) . name ( 1 : 6 ) , ' . jpg ' ) , 'Cx ' ,R . Centroid (1 ) , 'Cy ' ,R . Centroid (2 ) , 'Bx ' ,R . BoundingBox (1 ) , 'By ' ,R . BoundingBox (2 ) , 'Bdx ' ,R . BoundingBox (3 ) , 'Bdy ' ,R . BoundingBox (4 ) , ' frame ' ,k−2) ;

doc=[doc ; docc ] ;endcd . .

endsave ( 'PicDoc .mat ' , ' doc ' )

B.2 Video Enhancement

Code to process raw image files from the high speed camera into video files. Performs rotationand adaptive hisogram equalization to improve viewability.

Directory = 'G:\Flame Behavior\ ' ;X = d i r ( Directory ) ;tests=length (X )−3;velocity=ze ro s ( tests , 2 ) ;

% path i s the f o l d e r l o c a t i o n where the image f i l e s are s to r ed .f o r jj=1:tests

jjpath = [ Directory getfield (X ( jj+3) , 'name ' ) ' \ ' ] ;Test = getfield (X ( jj+3) , 'name ' ) ;velocity (jj , 1 ) = str2num ( Test ( 5 : end ) ) ;

% This s e c t i o n w i l l import the images and s t o r e them as the four% dimens iona l matrix Images .files = d i r ( [ path ] ) ;files = files ( 3 : end ) ;files = so r t ( files . name ) ;Pos = cell2mat ( files (1 ) ) ;Pos = str2num ( Pos ( 2 : 6 ) ) ;%dea l with negat ive timestamp va luesi f sum( ismember ( files , 'F00000 . TIF ' ) )==1

z = f ind ( ismember ( files , 'F00000 . TIF ' )==1) ;files ( 1 : z−1) = files (z−1:−1:1) ;

e l s e i f Pos<0z = length ( files )+1;files ( 1 : z−1) = files (z−1:−1:1) ;

endsizes = s i z e ( imread ( [ path files 1 ] ) ) ;Images = zero s ( [ sizes (2 ) , sizes (1 ) , sizes (3 ) , l ength ( files ) ] , ' uint8 ' ) ;

f o r i = 1 : l ength ( files )Images ( : , : , : , i ) = imrotate ( imread ( [ path filesi ] ) , 90) ;

end

f o r ii = 1 : l ength ( files )J = adapthisteq ( Images ( : , : , 2 , ii ) ) ;

97

Images ( : , : , 2 , ii ) = J ;end

name=strcat ( 'Flame ' , num2str ( jj ) , ' . av i ' )

Object = avifile ( name )f o r ii = 1 : l ength ( files )−10Object = addframe ( Object , Images ( : , : , : , ii ) ) ;endObject = c l o s e ( Object )

end

B.3 V-channel Flame Tracking

% Direc tory i s the base f o l d e r l o c a t i o n where the image f i l e s are s to r edDirectory = 'C:\ Users \Angel\Documents\ gnoe l \Experiments\Flame Quenching←

\121119\121119\ ' ;X = d i r ( Directory ) ;

% Pre−a l l o c a t e matr i ce s f o r use l a t e r in the codeQuench_Location = zero s ( l ength (X ) −3 ,1) ;Quench_Diameter = zero s ( l ength (X ) −3 ,1) ;Location = zero s ( l ength (X ) −3 ,1) ;Number = zero s ( l ength (X ) −3 ,1) ;PPI = zero s ( l ength (X ) −3 ,1) ;x_crit = zero s ( l ength (X ) −3 ,1) ;B = zero s (121 ,151 , l ength (X )−3) ;A2 = zero s (512 ,301 , l ength (X )−3) ;

% path i s the s p e c i f i c f o l d e r l o c a t i o n where the image f i l e s are s to r ed f o r % ←each i nd i v i dua l t e s t .

f o r jj = 1 : l ength (X )−3path = [ Directory getfield (X ( jj+2) , 'name ' ) ' \ ' ] ;Test = getfield (X ( jj+2) , 'name ' ) ;Number ( jj ) = str2num ( Test ( 5 : end ) ) ;

% This s e c t i o n w i l l import the images and s t o r e them as the four% dimens iona l matrix Images .files = d i r ( [ path ] ) ;files = files ( 3 : end ) ;files = so r t ( files . name ) ;Pos = cell2mat ( files (1 ) ) ;Pos = str2num ( Pos ( 2 : 6 ) ) ;i f sum( ismember ( files , 'F00000 . TIF ' ) )==1



endsizes = s i z e ( imread ( [ path files 1 ] ) ) ;i f l ength ( sizes ) == 2

sizes (3 ) = 1 ; %Pretend we have a th i rd dimension i f the re i sn ' t one .endImages = zero s ( [ sizes , l ength ( files ) ] , ' uint8 ' ) ;

98

f o r i = 1 : l ength ( files )Images ( : , : , : , i ) = imread ( [ path filesi ] ) ;

end

% This s e c t i o n enhances the con t ra s t in the images us ing a contras t−l im i t ed % ←adapt ive histogram equa l i z a t i o n bu i l t−in .

f o r ii = 1 : l ength ( files )J = adapthisteq ( Images ( : , : , 2 , ii ) ) ;Images ( : , : , 2 , ii ) = J ;J2 = adapthisteq ( Images ( : , : , 3 , ii ) ) ;Images ( : , : , 3 , ii ) = J2 ;

endc l e a r J ; c l e a r J2 ;

% M i s a vec to r used to f i nd the c e n t e r l i n e o f the channel by the s i gna tu r e % o f ←the images in a g iven t e s t .

M = mean( double ( Images ( 5 5 : 8 5 , 6 0 : 1 40 , 2 , 1 ) ) . ˆ 2 , 2 ) ;CL = round (54+(1: l ength (M ) ) ∗M/sum(M ) ) ;

% PPI i s the number o f p i x e l s per inch o f phy s i c a l space .PPI ( jj ) = 96 ;

% x c r i t ( j j ) i s the l o c a t i o n at which s l o t s in the p l a s t i c begin in number o f % ←p i x e l s from the l e f t s i d e o f the image , which are used to l o c a t e the V % ←por t i on o f the channel .

xcrit ( jj ) = 48 ;

% Where the s l ope s t a r t s and ends in p i x e l number ( 0 . 1 inche s from the top% and bottom of the s l o t )SlopeStart = x_crit ( jj ) + 4 .1∗ PPI ( jj ) ;SlopeEnd = x_crit ( jj ) + 0 .1∗ PPI ( jj ) ;% The width o f the channel in mmWidth_SS = 5 ;Width_end = 2 ;

% Show the f lame f r on t as a func t i on o f time with in the channel .A = squeeze ( Images (CL , : , 2 , : ) ) ;

% F i l t e r the f lame f r on t to ex t r a c t only the in fo rmat ion that i s r e l e van t to % the←quenching and v e l o c i t y c a l c u l a t i o n s .

M = median (A , 2 ) ;f o r mm = 1 : s i z e ( Images , 4 )

A ( : , mm ) = A ( : , mm )−M ;endA (A<0.4∗max(A ( : ) ) ) = 0 ;A = medfilt2 (A , [ 3 , 3 ] ) ;

% Ca lcu la t e the quenching l o c a t i o n in terms o f p i x e l number from the l e f t % ←s i d e o f the image , the time step o f the quench , the quenching diameter % (←based on the l o c a t i o n with in the experiment , and the l o c a t i o n in inche s % ←from mouth o f the experiment .

[ x , y ] = f i nd (A , 100000) ;Quench_Location ( jj ) = min (x ) ;time_Quench ( jj ) = f i nd (A (min (x ) , : ) , 1 ) ;Quench_Diameter ( jj ) = Width_SS−(SlopeStart−Quench_Location ( jj ) ) ∗( Width_SS−←

Width_end ) /( SlopeStart−SlopeEnd ) ;Location ( jj ) = ( SlopeStart−Quench_Location ( jj ) ) /PPI ( jj ) ;

99

% Generate i n t e r r o g a t i o n windows around the quenching l o c a t i o n which can be % ←used to c a l c u l a t e the v e l o c i t y o f the f lame f r on t in the channel .

A2 ( : , : , jj ) = A ( : , end−300: end ) ;B ( : , : , jj ) = A ( (min (x ) ) : ( min (x )+120) , ( time_Quench ( jj )−170) : time_Quench ( jj )−20) ;

end

B.4 Correlation Flame Velocimetry

Directory = 'C:\ Users \Dan\Desktop\Summer13Images\130708\TestBed\ ' ;X = d i r ( Directory ) ;Location = zero s ( l ength (X ) −3 ,1) ;Number = zero s ( l ength (X ) −3 ,1) ;PPI = zero s ( l ength (X ) −3 ,1) ;Top = zero s ( l ength (X ) −3 ,1) ;Bottom = zero s ( l ength (X ) −3 ,1) ;x_Top = zero s ( l ength (X ) −3 ,1) ;x_Bottom = zero s ( l ength (X ) −3 ,1) ;x_crit = zero s ( l ength (X ) −3 ,1) ;B = zero s (121 ,151 , l ength (X )−3) ;A2 = zero s (512 ,301 , l ength (X )−3) ;Br=struct ( 'nm ' , , ' Shock ' , ) ;

% path i s the f o l d e r l o c a t i o n where the image f i l e s are s to r ed .jj=2%fo r j j = 1 : l ength (X)−3path = [ Directory getfield (X ( jj+2) , 'name ' ) ' \ ' ] ;Test = getfield (X ( jj+2) , 'name ' ) ;Number ( jj ) = str2num ( Test ( 5 : end ) ) ;

% This s e c t i o n w i l l import the images and s t o r e them as the four% dimens iona l matrix Images .files = d i r ( [ path ] ) ;files = files ( 3 : end ) ;files = so r t ( files . name ) ;Pos = cell2mat ( files (1 ) ) ;Pos = str2num ( Pos ( 2 : 6 ) ) ;i f sum( ismember ( files , 'F00000 . TIF ' ) )==1



end

sizes = s i z e ( imread ( [ path files 1 ] ) ) ;

ymin=50;ymax=65;xmin=90;xmax=350;

i f l ength ( sizes ) == 2sizes (3 ) = 1 ; %Pretend we have a th i rd dimension i f the re i sn ' t one .

end

100

Images = zero s ( [ [ ( ymax−ymin ) +1 ,(xmax−xmin ) +1] , l ength ( files ) ] , ' uint8 ' ) ;

f o r i = 1 : l ength ( files )

Iint=imread ( [ path filesi ] ) ;Images ( : , : , i ) = Iint ( ymin : ymax , xmin : xmax , 2 ) ;

end

pause (1 )f o r i=1:s (3 )p l o t ( medfilt1 (median ( Images ( : , : , i )−Im ( : , : , i ) , 1 ) ) )ax i s ( [ 0 261 −10 10 ] ) ;pause (1/10)end

dt=12;n=s (3 )−dt ;corr=ze ro s (n , 2 ) ;snr=ze ro s (n , 1 ) ;

f o r i=1:nI1=Images ( : , : , i )−Im ( : , : , ( i ) ) ;I2=Images ( : , : , i+dt )−Im ( : , : , ( i+dt ) ) ;cr=xcorr2 (I1 , I2 ) ;mx=max( cr ( : ) ) ;

nz=std ( cr ( : ) ) ;[ corr (i , 2 ) , corr (i , 1 ) ]= f i nd ( cr==mx (1 ) ,1 ) ;snr (i , 1 )=mx/mn ;pause (1/10) ;i

end

subplot ( 3 , 1 , 1 ) ; p l o t ( corr ( : , 2 ) ) ;subp lot ( 3 , 1 , 2 ) ; p l o t ( corr ( : , 1 ) ) ;subp lot ( 3 , 1 , 3 ) ; p l o t ( snr ( : ) ) ;

x=(sum( corr ( : , 1 ) .∗ snr ) /sum( snr )−261)/dt

101

Appendix C

Electron Density Measurements

100µs 1ms

0

1 ·1017

2 ·1017

3 ·1017


(m−

3)

Experiment

Simulation

1490.0

1720.0

1950.0

T(K)

1.46

1.295

1.13P(bar)

Figure C.1: Electron histories mixture 1: 2% CH4, 8% O2

102

100µs 1ms

0

1 ·1017

2 ·1017

3 ·1017


(m−

3)

Experiment

Simulation

1562.0

1745.0

1929.0

T(K)

3.14

2.17

1.2P(bar)

Figure C.2: Electron histories mixture 2: 2% CH4, 4% O2

100µs 1ms

0

1 ·1017

2 ·1017

3 ·1017


(m−

3)

1552.0

1790.0

2027.0

T(K)

1.07

1.025

0.98P(bar)

Figure C.3: Electron histories mixture 3: 2% CH4, 3.25% O2

103

100µs 1ms

0

1 ·1017

2 ·1017

3 ·1017


(m−

3)

Experiment

Simulation

1241.0

1353.0

1464.0

T(K)

1.54

1.43

1.32P(bar)

Figure C.4: Electron histories mixture 4: 1% C2H4, 3% O2

100µs 1ms

0

1 ·1017

2 ·1017

3 ·1017


(m−

3)

Experiment

Simulation

1120.0

1317.0

1513.0

T(K)

1.246

1.033

0.82P(bar)

Figure C.5: Electron histories mixture 5: 1% C2H2, 2.5% O2

104

100µs 1ms

0

1 ·1017

2 ·1017

3 ·1017


(m−

3)

1265.0

1379.0

1493.0

T(K)

1.18

1.0135

0.847P(bar)

Figure C.6: Electron histories mixture 6: 0.66% C3H8, 3.33% O2

100µs 1ms

0

1 ·1017

2 ·1017

3 ·1017


(m−

3)

Experiment

Simulation

1885.0

2318.0

2750.0

T(K)

1.35

1.15

0.95P(bar)

Figure C.7: Electron histories mixture 7: 0.75% CH4, 1.5% O2

105

100µs 1ms

0

1 ·1017

2 ·1017

3 ·1017


(m−

3)

Experiment

Simulation

1576.0

1800.0

2023.0

T(K)

1.53

1.3125

1.095P(bar)


100µs 1ms

0

1 ·1017

2 ·1017

3 ·1017


(m−

3)

1587.0

1802.0

2016.0

T(K)

1.5

1.2

0.9P(bar)


106

Appendix D

MWI-Related Codes

D.1 Kinetics Simulations

%pylab inlineimport sysimport cantera as ctLnClr = '#53626F 'Clr3 = '#003262 'Clr2 = '#FDB515 'Clr1 = '#00B2A5 'Clr4 = '#ED4E33 'AlmostBlack = '#262626 '

de f ShockTubeUV ( Mixture , T5 , P5 , t=3.6 ,m=360 ,Mech= 'GrIon . xml ' ) :' ' 'UV reac t o rPass mixture as s t r i n g ( e . g . 'CH4: 9 . 1 , O2 : 1 8 . 2 , AR: 7 2 . 8 , H2O: 0 . 0 0 1 , N2 ' )T5 (K) , P5 ( bar ) , time (ms) , number o f po in t sr e tu rn s t ,T: temperature h i s t o r y' ' 'import cantera as ctgas1 = ct . Solution ( Mech )gas1 . TPX = T5 , P5∗ct . one_atm , Mixture

#gas2 = ct . So lu t i on ( ' argon . xml ' )#gas2 .TPX = 300 .0 , P5∗ ct . one atm , 'AR:1 '

r1 = ct . IdealGasReactor ( gas1 )r1 . volume = 0.1

net = ct . ReactorNet ( [ r1 ] )

Results = zeros ( ( m , 6 ) )

f o r n in xrange (m ) :time = (n+1) ∗1 .0 e−3∗t/mnet . advance ( time )tim = time ∗ 1000xCH = gas1 . X [ gas1 . species_index ( 'CH ' ) ]xO = gas1 . X [ gas1 . species_index ( 'O ' ) ]

107

xE = gas1 . X [ gas1 . species_index ( 'E− ' ) ]xH3O = gas1 . X [ gas1 . species_index ( 'H3O+ ' ) ]Results [ n , : ] = [ tim , r1 . T , xCH , xO , xE , xH3O ]#t2 . append ( r1 .T)#v1 . append ( r1 . volume )#v2 . append ( r1 . volume − 100 .0 )#v . append ( r1 . volume + r1 . volume − 100 .0 )#xch4 . append ( r1 . thermo [ 'CH4 ' ] .X[ 0 ] )#xh2 . append ( r1 . volume / . 1 )

re turn Results

de f ShockTubeUVTig ( Mixture , T5 , P5 , t=3.6 ,m=360 ,Mech= 'GrIon . xml ' ) :' ' 'UV reac t o r f o r i g n i t i o n de lay t imesPass mixture as s t r i n g ( e . g . 'CH4: 9 . 1 , O2 : 1 8 . 2 , AR: 7 2 . 8 , H2O: 0 . 0 0 1 , N2 ' )T5 (K) , P5 ( bar ) , time (ms) , number o f po in t s' ' 'import cantera as ctgas1 = ct . Solution ( Mech )gas1 . TPX = T5 , P5∗ct . one_atm , Mixture

gas2 = ct . Solution ( ' argon . xml ' )gas2 . TPX = 300 .0 , P5∗ct . one_atm , 'AR:1 '

r1 = ct . IdealGasReactor ( gas1 )r1 . volume = 0.1

net = ct . ReactorNet ( [ r1 ] )

tim = [ ]t1 = [ ]

f o r n in xrange (m ) :time = (n+1)∗1e−3∗t/mnet . advance ( time )tim . append ( time ∗ 1000)i f ( r1 . T>1.05∗T5 ) :

tm=timebreak

return tm

de f ShockTubeDefineP ( Mixture , T5 , tp , Pp , n=360 ,Mech= 'GrIon . xml ' ) :' ' 'UV reac t o rPass mixture as s t r i n g ( e . g . 'CH4: 9 . 1 , O2 : 1 8 . 2 , AR: 7 2 . 8 ' )T5 (K) , t imes o f p r e s c r i b ed pressure , p r e s c r i b ed pressure , time (ms) , number ←

o f po in t smechanism

A con t r o l l e d p r e s su r e r e a c t o r system .

The bu i l t−in r e a c t o r systems in cantera do not in c lude the f a c i l i t y top r e s c r i b e p r e s su r e h i s t o r i e s . However they are e x c ep t i o na l l y e f f i c i e n tr e l a t i v e to us ing matlab ode s o l v e r s .

Here we ' l l a ch i eve the de s i r ed pr e s su r e c on t r o l by way o f a two r ea c t o rsystem . The f i r s t r e a c t o r w i l l conta in the mixture o f i n t e r e s t andw i l l comunicate with the second r ea c t o r by a wal l to maintain constantp r e s su r e between them . The second r ea c t o r conta in s a very l a r g e

108

quatn i ty o f i n e r t i s o the rma l argon connected to a second wal l with ade f ined v e l o c i t y p r o f i l e . Through con t r o l o f the v e l o c i t y p r o f i l e wegain con t r o l o f the p r e s su r e in the second r ea c t o r . Due to the l a r g er a t i o o f volumes (1000) between the two reac to r s , we assume that theo v e r a l l p r e s su r e changes due to the f i r s t r e a c t o r are n e g l i g i b l e .

141029 − being lazy now , i gno r i ng t h i s f o r now .' ' 'import cantera as ctfrom scipy . optimize import fmin as Fm

gas1 = ct . Solution ( 'GrIon . xml ' )gas1 . TPX = T5 , 1 .8∗ ct . one_atm , Mixturegas2 = ct . Solution ( ' argon . xml ' )gas2 . TPX = 300 .0 , 1 .8∗ ct . one_atm , 'AR:1 'r1 = ct . IdealGasReactor ( gas1 )r1 . volume = 0.1

#Dummy Reactor : Huge b a l l o f i s o the rma l argonr2 = ct . IdealGasReactor ( gas2 , energy= ' o f f ' )r2 . volume = 100

#Placeho lder environmentenv = ct . Reservoir ( gas2 )

#Moving wal l − moves r ap id l y due to p r e s su r e d i f f e r e n t i a l between r1 , r2w = ct . Wall (r1 , r2 , K=1.0e4 )

#Se t t i ng up a v e l o c i t y p r o f i l e#Current ly a guass ian , to g ive us a smoothed step changePCurve = lambda P , t : P [ 0 ] + P [ 1 ] ∗ ( 1 + exp ( −(t−P [ 2 ] ) /P [ 3 ] ) )PFitness = lambda P , t , y : ( ( PCurve (P , t )−y ) ∗∗2) . sumP0 = [ Pp [ 0 ] , ( Pp [−1]−Pp [ 0 ] ) , ( tp [−1]−tp [ 0 ] ) /2 , ( tp [−1]−tp [ 0 ] ) / 1 0 . 0 ]

P1 = Fm ( PFitness , P0 , args=(tp , Pp ) )

' ' 'vI = lambda A, t : A[ 0 ] ∗ exp ( −(t−A[ 1 ] ) ∗∗2/A[ 2 ] )A = [ 0 , 1 . 5 e−3, 1e−4]v0 = lambda t : vI (A, t )' ' 'w2 = ct . Wall (r2 , env , velocity=0)

net = ct . ReactorNet ( [ r1 , r2 ] )

tim = [ ]t1 = [ ]t2 = [ ]v1 = [ ]v2 = [ ]v = [ ]xch4 = [ ]xh2 = [ ]

f o r n in xrange (3000) :time = (n+1)∗0.000001net . advance ( time )tim . append ( time ∗ 1000)t1 . append ( r1 . T )

109

t2 . append ( r2 . T )v1 . append ( r1 . volume )v2 . append ( r2 . volume − 100 .0 )v . append ( r1 . volume + r2 . volume − 100 .0 )xch4 . append ( r1 . thermo [ 'CH4 ' ] . X [ 0 ] )xh2 . append ( r1 . volume / . 1 )

#Perform Cantera Runs and Save timt−to−peak−e l e c t r o n sTarget = 2G = genfromtxt ( ' L i s t '+s t r ( Target )+ 'Both . csv ' , delimiter=” , ” )G2 = G . copy ( )Count = shape (G ) [0]−1#pr in t G[ : , : 8 ]f o r ii in range ( Count ) :

p r i n t ii#pr in t G[ i i , 2 ] ,G[ i i , 3 ]R = ShockTubeUV ( 'CH4: 2 , O2 : 4 , AR:96 , HE: 0 . 0 0 1 , H2O: 0 . 0 0 1 , N2 : 0 . 0 0 1 ' ,G [ ii+1 ,2] ,←

G [ ii+1 ,3] , t=3.6 ,m=360∗8 ,Mech= ' aramco−i on . xml ' )mx = R [ : , 5 ] . max( )loc = array ( find (R [ : ,5 ]== mx ) ) [ 0 ]G2 [ ii+1 ,4] = R [ loc , 0 ]#ax . semi logx (R[ 1 0 : , 0 ] ,R[ 1 0 : , 5 ] )

#ax . ax i s ( [ 3 e−2,6,−1e−6,2e−6])savetxt ( ' L i s t '+s t r ( Target )+ 'Both2 . csv ' ,G2 , delimiter=” , ” )

D.2 Intrinsic Parameter Calibration

Due to non-ideal behavior, the MWI does not produce perfect I/Q. We can describe thesignals that we measure by the following two equations.

I = α AI cos(∆Φ) + I0

Q = α AQ sin(∆Φ + δφ) +Q0

Where the signal amplitude α and phase shift ∆Φ are the true measures of the recievedmicrowave signals and the remaining parameters AI , AQ, I0, Q0 and δφ are the unknownproperties of the system. By fitting these functional forms for I/Q data from an argonionization shock we are able to determine these unknown properties and calibrate the MWI.

If we assume that no gases or plasmas introduced into the system will decrease thesignal attenuations then α will never be less than its initial value, when the shock tube isnear vacuum. Under that assumption, all I/Q pairs measured during the shock should liewithin an ellipse described by the above equations. Further, we expect from the equations ofplasma dynamics that, during an argon ionization event of the kind used here, the populationof electrons will initially produce signficant phase shifts in the microwave beam, but little

110

attenuation. Therefore, the initial stage of the ionization event should begin by tracing thedesired ellipse adn then proceed to spiral inward toward the offset voltages I0 and Q0

We define a fitness function by which proposed system parameters may be tested againstshock data in order to evaluate their ability to match the predicted behavior. When properlyselected, these values will produce an ellipse that fits tightly around the data set, such thatpoints from the early portion of ionization event lie on the ellipse and all others lie withinthe ellipse.

#Module imports%pylab inlineimport osfrom scipy . optimize import fminimport scipy . ioimport matplotlib . patches as mpatches

de f movingaverage ( interval , window_size ) :window= ones ( i n t ( window_size ) ) / f l o a t ( window_size )re turn convolve ( interval , window , ' same ' )

Fitness function definitionde f alphaSq (AI , AQ , I0 , Q0 , dphi , xp , yp ) :

' ' ' For a given e l l i p s e , t h i s measures the ' rad iu s ' o f a po int at xp , ypaccounts f o r vo l t age o f f s e t s , d i f f e r e n t channel s e n s i t i v i t yr e tu rn s the square o f the ' rad iu s '' ' 'x = xp − I0y = yp − Q0xnor = x/AIynor = (y/AQ − xnor∗sin ( dphi ) ) /( cos ( dphi ) +.00001∗( cos ( dphi==0)) )a2 = xnor∗∗2 + ynor∗∗2return a2

de f ftnss ( a2 ) :' ' ' Measures the qua l i t y o f f i t f o r a g iven ' rad iu s 'Rewards po in t s that l i e on the e l l i p s emoderately p ena l i z e s those with inSTRONGLY pena l i z e s those out s id er e tu rn s a p o s i t i v e or zero value . Zero i n d i c a t e s a2=1, a p e r f e c t f i t' ' 'ct=1.15fa = ((2 − 2∗exp(−(a2−1.0) ∗∗2∗100) ) ∗ (1 − 2∗a2∗∗2 + exp ( ( a2−1)∗4) ) ) ∗(a2<=ct )fb = ((2 − 2∗exp(−(ct−1.0) ∗∗2∗100) ) ∗ (1 − 2∗ct∗∗2 + exp ( ( ct−1)∗4) )+4∗(a2−ct )←

) ∗(a2>ct )re turn fa+fb

de f Ieq (AI , AQ , I0 , Q0 , dphi , Phi , alpha ) :r e turn alpha∗AI∗cos ( Phi ) + I0

de f Qeq (AI , AQ , I0 , Q0 , dphi , Phi , alpha ) :r e turn alpha∗AQ∗sin ( Phi+dphi ) + Q0

#Import data and smoothGasName= 'Ar 'n=6Count=[n , n , n ]

111

dat = genfromtxt ( GasName+s t r ( Count [ 0 ] )+ ' CH1 Wfm . csv ' , delimiter= ' , ' , skiprows=22)data0=zeros ( ( shape ( dat ) [ 0 ] , 5 ) )data0 [ : , : 2 ] = datdat = genfromtxt ( GasName+s t r ( Count [ 1 ] )+ ' CH2 Wfm . csv ' , delimiter= ' , ' , skiprows=22)data0 [ : , 2 ]= dat [ : , 1 ]dat = genfromtxt ( GasName+s t r ( Count [ 2 ] )+ ' CH3 Wfm . csv ' , delimiter= ' , ' , skiprows=22)data0 [ : , 3 ]= dat [ : , 1 ]dat = genfromtxt ( GasName+s t r ( Count [ 2 ] )+ ' CH4 Wfm . csv ' , delimiter= ' , ' , skiprows=22)data0 [ : , 4 ]= dat [ : , 1 ]p r i n t GasName+s t r ( Count [ 1 ] )+ ' CH2 Wfm . csv '

d10=movingaverage ( data0 [ : , 1 ] , 2 5 )d20=movingaverage ( data0 [ : , 2 ] , 2 5 )

it=Noneextents = [1 e4 , 5 e4 ]DI = d10 [ extents [ 0 ] : extents [ 1 ] : 1 0 ]#concatenate ( ( d103 [ 3 0 : 5 0 0 0 0 ] , d100 [30 : −15000 :10 ] )←

)DQ = d20 [ extents [ 0 ] : extents [ 1 ] : 1 0 ]#concatenate ( ( d203 [ 3 0 : 5 0 0 0 0 ] , d200 [30 : −15000 :10 ] )←

)

Imean = mean ( DI )Qmean = mean ( DQ )Iextent = ( DI .max( )−DI . min ( ) ) /2 .0Qextent = ( DQ .max( )−DQ . min ( ) ) /2 .0

A0=array ( [ 1∗ Iextent , 1∗ Qextent , Imean , Qmean , 0 . 0 ] )

ToFit = lambda A , x , y : ftnss ( alphaSq (A [ 0 ] , A [ 1 ] , A [ 2 ] , A [ 3 ] , A [ 4 ] , x , y ) ) . sum( )

Af = fmin ( ToFit , A0 , args=(DI , DQ ) , maxiter=it , disp=0)#Af=A0

fig , ax = subplots ( )fig . set_size_inches (10 ,10)

AI = Af [ 0 ]AQ = Af [ 1 ]I0 = Af [ 2 ]Q0 = Af [ 3 ]dphi = Af [ 4 ]alpha = 1

PhiRange = linspace (0 , pi ∗2 .0 , 200 )IRange = Ieq ( Af [ 0 ] , Af [ 1 ] , Af [ 2 ] , Af [ 3 ] , Af [ 4 ] , PhiRange , alpha )QRange = Qeq ( Af [ 0 ] , Af [ 1 ] , Af [ 2 ] , Af [ 3 ] , Af [ 4 ] , PhiRange , alpha )

mags = alphaSq (AI , AQ , I0 , Q0 , dphi , DI , DQ )fitz = ftnss ( mags )

p r i n t shape ( mags ) , shape ( fitz )

p r i n t 'AI = ' , AI , ' \n 'pr in t 'AQ = ' , AQ , ' \n 'pr in t ' I0 = ' , I0 , ' \n 'pr in t 'Q0 = ' , Q0 , ' \n 'pr in t ' dPhi = ' , dphi , ' \n '

112

D.3 I/Q data processing

Imports and smooths raw IQ signals

#module imports and ba s i c d e f i n i t i o n simport osfrom scipy . optimize import fminimport scipy . iode f movingaverage ( interval , window_size ) :

window= ones ( i n t ( window_size ) ) / f l o a t ( window_size )re turn convolve ( interval , window , ' same ' )

from scipy . optimize import curve_fit

de f pol5 (x , a5 , b4 , c3 , d2 , e1 , f0 ) :r e turn a5∗x∗∗5 + b4∗x∗∗4 + c3∗x∗∗3 + d2∗x∗∗2 + e1∗x +f0

de f pol4 (x , b4 , c3 , d2 , e1 , f0 ) :r e turn b4∗x∗∗4 + c3∗x∗∗3 + d2∗x∗∗2 + e1∗x +f0

import matplotlib . patches as mpatches

de f Flimp ( GasName ) :Drct=os . listdir ( ' . ' )Count=0f o r s in Drct :

i f GasName in s :nm= ' CH1 Wfm . csv 'i f nm in s :

Count=Count+1f o r ii in ( 1 , 2 , 3 , 4 ) :

Target= 'C:\ Users \Dan\Downloads\Tek CH '+s t r ( ii )+ ' Wfm. csv 'New=GasName+s t r ( Count )+ ' CH '+s t r ( ii )+ ' Wfm. csv 'os . rename ( Target , New )p r i n t ii

re turn Count

ei = 15 .2∗1 . 6 e−19e0=8.85e−12c=3.0e8kB=1.38e−23Na=6.02e23ec=1.6e−19me=9.1e−31R=8.314h=6.62e−34l_path=.142lmb=c/94e9

PhaseConst = (4 . 0∗ pi∗me∗e0∗c∗∗2) /( ec∗∗2∗lmb∗l_path )

de f f_Ion (T , P ) :' ' ' Saha Equation , computes equ i l i b r i um e l e c t r on concent ra t i on f o r argon at ←

giventemperature and Pressure . Should upgrade to dea l with other gase s' ' 'F1 = P∗h∗∗3 / ( (2∗ pi∗me ) ∗∗ ( 1 . 5 ) ∗ ( kB∗T ) ∗∗2 .5 )PartitionRatio = 1/5.3 #from \ c i t e d r e l l i s h a k 2 0 0 4p a r t i t i o n f o r Argon

113

Expo = exp ( ei/kB/T )re turn 1/sqrt(1+F1∗PartitionRatio∗Expo )

de f NeAr (T , P ) :' ' ' Uses Id ea l Gas Law and Saha equat ion to g ive number dens i ty o f e l e c t r o n stakes T,P' ' 're turn f_Ion (T , P ) ∗P/kB/T

VNeAr=vectorize ( NeAr )AlphAr=vectorize ( f_Ion )

de f dPhi (Ne , nu ) :' ' 'Computes phase s h i f t ( rad ) and at tenuat ion ( f r a c t i o n o f amplitude )f o r a number dens i ty o f e l e c t r o n s and c o l l i s i o n f requencyAssumes p r op e r t i e s o f the MWI as mounted on the shock tube .' ' 'Om=2∗pi∗94e9OmP=sqrt ( Ne∗ec ∗∗2/( e0∗me ) )

KR = 1 − ( OmP/Om ) ∗∗2 / (1+ ( nu/Om ) ∗∗2)KI = ( OmP/Om ) ∗∗2∗( nu/Om ) / (1+ ( nu/Om ) ∗∗2)K = KR + (1j ) ∗KIKmag=sqrt ( KR∗∗2 + KI ∗∗2)

alphaPhi = Om/c ∗ sqrt ( ( Kmag − KR ) /2)beta = Om/c ∗ sqrt ( ( Kmag + KR ) /2)beta0 = 1/(c/Om )

d=0.142

dP = ( beta − beta0 ) ∗ddB = 8.68∗ alphaPhi∗d

#trans = sq r t (10∗∗(−dB/10 .0 ) )trans = exp(−alphaPhi∗d )

re turn dP , trans , dB

VdPhi=vectorize ( dPhi )

de f IQ (A , DPhi , I0=0,Q0=0,aI=0.3 , aQ=0.3 , dp=0.0) :I = I0 + aI∗A∗cos ( DPhi )Q = Q0 + aQ∗A∗sin ( DPhi + dp )re turn I , Q

#For ca s e s without PressueGasName= 'Ar 'n=0#n = Flimp (GasName)Count=[n , n , n ]dat = genfromtxt ( GasName+s t r ( Count [ 0 ] )+ ' CH1 Wfm . csv ' , delimiter= ' , ' , skiprows=22)data0=zeros ( ( shape ( dat ) [ 0 ] , 5 ) )data0 [ : , : 2 ] = datdat = genfromtxt ( GasName+s t r ( Count [ 1 ] )+ ' CH2 Wfm . csv ' , delimiter= ' , ' , skiprows=22)data0 [ : , 2 ]= dat [ : , 1 ]dat = genfromtxt ( GasName+s t r ( Count [ 2 ] )+ ' CH3 Wfm . csv ' , delimiter= ' , ' , skiprows=22)data0 [ : , 3 ]= dat [ : , 1 ]dat = genfromtxt ( GasName+s t r ( Count [ 2 ] )+ ' CH4 Wfm . csv ' , delimiter= ' , ' , skiprows=22)

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data0 [ : , 4 ]= dat [ : , 1 ]p r i n t GasName+s t r ( Count [ 1 ] )+ ' CH2 Wfm . csv '

d10=movingaverage ( data0 [ : , 1 ] , 2 5 )d20=movingaverage ( data0 [ : , 2 ] , 2 5 )phi0=arctan ( ( d10− .1) / .166 / ( ( d20 ) / . 23 ) )ne=−phi0∗PhaseConstne=ne−ne [ 1000 : −1000 ] .min ( )

D.4 Electron Decay Fitting

Electron decay rate constant estimation by fitting data with the functional form describedin subsubsection 4.4.2.2.

Recomb = lambda Coefs , t : 1 . 0 / ( Coefs [ 0 ] ∗ t+Coefs [ 1 ] )Fitness = lambda Coefs , t , y : mean ( ( Recomb ( Coefs , t )−y ) ∗∗2)

p r i n t Recomb ( [ 1 e−13 ,3.0/1 e17 ] , 1 e−4)

de f Decay ( mixture , FileList , bl=[12000 ,14000 ] , times=[9e−6,3e−3]) :#Study the Recombination ra t eFl = genfromtxt ( FileList , delimiter=” , ” , skiprows=1)Cntr = genfromtxt ( ' L i s t '+s t r ( mixture )+ 'Maxes . csv ' , delimiter=” , ” )

Fits = zeros ( ( size ( Fl [ : , 1 ] ) , 3 ) )

#go f i g u r efig , ax =subplots ( )fig . set_size_inches (16 ,6 )

x0=Fl [ : , 2 ] . min ( )x1=Fl [ : , 2 ] . max( )y0=Fl [ : , 3 ] . min ( )y1=Fl [ : , 3 ] . max( )

p r i n t x0 , x1 , y0 , y1

f o r ii in range (0 , size ( Fl [ : , 1 ] ) ) :t , n , p , ch = Import ( mixture , Fl [ ii , 1 ] )nm = mean (n [ bl [ 0 ] : bl [ 1 ] : 1 0 ] )C=TwoColor ( Fl [ ii , 2 ] , Fl [ ii , 3 ] , x0 , x1 , y0 , y1 )p r i n t ii , Cn=n−nmym=n .max( )im=find (n==ym )imo= im i f ( size ( im )==1) e l s e im [ 0 ]xm=t [ im ] i f ( size ( im )==1) e l s e t [ im [ 0 ] ]T = t [ imo :−300:25]−t [ imo ]N =n [ imo : −300 :25 ]C0 = [ 1 . 0 e4/N [ 0 ] , 1 . 0 / N [ 0 ] ]C1 = fmin ( Fitness , C0 , args = (T , N ) )#, maxiter=10)Nf0 = Recomb (C0 , T )Nf = Recomb (C1 , T )ax . plot (T , N , color=C , linewidth=3,alpha=.55)#ax . p l o t (T, Nf0 , '−− ' , c o l o r=C, l i n ew id th=1, alpha =.9)

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ax . plot (T , Nf , '−− ' , color=C , linewidth=4,alpha=1)Fits [ ii , : ] = [ C1 [ 0 ] ∗ 6 e23∗1e6 , Fl [ ii , 2 ] , Fl [ ii , 3 ] ]#ax . s c a t t e r (0 ,ym)

#savetxt ( ' L i s t '+ s t r ( mixture )+'Both . csv ' , Cntr , d e l im i t e r =” ,”)

#ax . semi logx ( t r [ : : 2 5 ] , nr [ : : 2 5 ] , c o l o r ='k ' , l i n ew id th=2)#ax i s l andax . axis ( [ 0 , 2 e−4,−.5e17 , 0 . 3 e18 ] )ax . spines [ ' top ' ] . set_visible ( False )ax . spines [ ' r i g h t ' ] . set_visible ( False )ax . set_yticks ( ( 0 , 2 . 5 e17 ) )ax . set_yticklabels ( ( ' $0$ ' , ' $2 . 5 \ cdot 10ˆ17$ ' ) , fontsize=20)ax . set_xticks ( (5 e−5,1e−4 ,1.5e−4) )ax . set_xticklabels ( ( ' $50 \mu s$ ' , ' $100 \mu s$ ' , ' $150 \mu s$ ' ) , fontsize=24)ax . yaxis . set_ticks_position ( ' l e f t ' )ax . xaxis . set_ticks_position ( ' bottom ' )ax . set_ylabel ( ' $n e$ $ (mˆ−3)$ ' , fontsize=24,rotation=90)ax . yaxis . set_label_coords ( −0 .03 , . 5 )#ax . s e t t i t l e ( '$MWI$ $Electron$ $Experiments$ ' , f o n t s i z e =28)ax . text ( 0 . 85 e−4 ,2.3e17 , r ' $Reacton : $ $\dot n e = −A 0 n e ˆ2$ ' , fontsize=20)ax . text ( 0 . 85 e−4 ,1.7e17 , r ' $F i t t i ng : $ $ n e (\ tau ) = \ l e f t [ A 0 \ tau + n e (\ tau=0)←

\ r i g h t ]ˆ−1$ ' , fontsize=20)savefig ( 'DecayFit . pdf ' , bbox_inches= ' t i g h t ' )re turn Fits

D.5 Shock tube uncertainty estimation

Code to estiamte uncertaintly in observed reactions in the shock tube due to uncertainty ininitial temperature, pressure and Mach number (T0,P0,Ms).

#Plasmafig , ax = subplots ( )fig . set_size_inches (10 ,10)

n=24000P0 = zeros (n )T0 = zeros (n )Td = zeros (n )Pd = zeros (n )Md = zeros (n )Prates = zeros ( ( n , 3 ) )

#ax . s c a t t e r (1 , 1 , s=50, c='k ' , l i n ew id th s =0, zorder=7)ax . scatter (1 , 1 , s=100 ,c=Clr1 , linewidths=0,zorder=7)ax . scatter (1 , 1 , s=100 ,c=Clr2 , linewidths=0,zorder=4)

f o r ii in range (n ) :Md [ ii ] = uniform (5 .7∗ (1 −0 .0026) ,5 .7∗ (1+0.0026) )T0 [ ii ] = 296#uniform (295 . 5 , 2 9 6 . 5 )P0 = 1.25/760 .0#(1 + uniform (−0 .0025 ,0 .0025) )Tr , Pr = STRel ( Md [ ii ] , 300 ,101325)Td [ ii ] = T0 [ ii ]∗ TrPd [ ii ] = P0∗Pr

a=0

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Prates [ : , 0 ] = ( Pd/Td ) ∗∗2∗exp (−15.75∗1.6e−19/1.38e−23/Td )#ra t e s [ : , 0 ] = (Pd/Td) ∗∗2∗ exp (30∗4 .2/8 .314/Td)#ax . s c a t t e r (Td,Pd , s=10, c='k ' , l i n ew id th s =0, alpha=1, zorder=6)ax . plot ( [ Td . min ( ) , Td .max( ) ] , [ Pd . min ( ) , Pd .max( ) ] , c= 'k ' , linewidth=3,alpha=1,zorder←

=6)

f o r ii in range (n ) :Md [ ii ] = uniform (5 .7∗ (1 −0 .0026) ,5 .7∗ (1+0.0026) )T0 [ ii ] = 296#uniform (295 . 5 , 2 9 6 . 5 )P0 = 1.25/760 .0∗ (1 + uniform (−0 .0025 ,0 .0025) )Tr , Pr = STRel ( Md [ ii ] , 300 ,101325)Td [ ii ] = T0 [ ii ]∗ TrPd [ ii ] = P0∗Pr

Prates [ : , 1 ] = ( Pd/Td ) ∗∗2∗exp (−15.75∗1.6e−19/1.38e−23/Td )#ra t e s [ : , 1 ] = (Pd/Td) ∗∗2∗ exp (30∗4 .2/8 .314/Td)ax . scatter (Td , Pd , s=100 ,c=Clr1 , linewidths=0,alpha=0.55 , zorder=5)

f o r ii in range (n ) :Md [ ii ] = uniform (5 .7∗ (1 −0 .0026) ,5 .7∗ (1+0.0026) )T0 [ ii ] = uniform (295 ,297)P0 = 1.25/760 .0∗ (1 + uniform (−0 .0025 ,0 .0025) )Tr , Pr = STRel ( Md [ ii ] , 300 ,101325)Td [ ii ] = T0 [ ii ]∗ TrPd [ ii ] = P0∗Pr

Prates [ : , 2 ] = ( Pd/Td ) ∗∗2∗exp (−15.75∗1.6e−19/1.38e−23/Td )#ra t e s [ : , 2 ] = (Pd/Td) ∗∗2∗ exp (30∗4 .2/8 .314/Td)ax . scatter (Td , Pd , s=100 ,c=Clr2 , linewidths=0,alpha=0.55 , zorder=5)ax . set_xticks ( [ 7 340 , 7390 , 7440 ] )ax . set_yticks ( [ . 3 5 8 , . 3 6 1 , . 3 6 4 ] )ax . yaxis . set_ticks_position ( ' l e f t ' )ax . xaxis . set_ticks_position ( ' bottom ' )ax . set_ylabel ( ' $Pressure$ ' , fontsize=18)ax . set_xlabel ( ' $Temperature$ ' , fontsize=18)ax . spines [ ' top ' ] . set_visible ( False )ax . spines [ ' r i g h t ' ] . set_visible ( False )

ax . axis ( [ 7 3 2 0 , 7 4 7 0 , . 3 5 7 , . 3 6 5 ] )#ax . ax i s ( [ 2 1 2 5 , 2 1 6 5 , . 3 1 , . 3 1 7 ] )

Prates = Prates/mean ( Prates . ravel ( ) )

leg = ax . legend ( [ ' $M S$ ' , ' $M S , P 0$ ' , ' $M S , P 0 , T 0$ ' ] , title= ' $Uncertain \ , ←Var iab le$ ' , fontsize=22,loc=2,framealpha=0,scatterpoints=1)

ax . set_title ( ' $Plasma \ , Condit ions$ ' , fontsize=20)

setp ( leg . get_title ( ) , fontsize=20)

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The Measurement and Application of Electric E ects in ... · Appendix C {Electron Density...

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